C. S. Hamilton, R. Kruse, L. Sansoni, S. Barkhofen, Ch. Silberhorn, I. Jex.
Gaussian Boson Sampling.
PHYSICAL REVIEW LETTERS, 119, 170501, October 2017
[ bib ]
Boson sampling has emerged as a tool to explore the advantages of quantum over classical computers as it does not require universal control over the quantum system, which favors current photonic experimental platforms. Here, we introduce Gaussian Boson sampling, a classically hardtosolve problem that uses squeezed states as a nonclassical resource. We relate the probability to measure specific photon patterns from a general Gaussian state in the Fock basis to a matrix function called the Hafnian, which answers the last remaining question of sampling from Gaussian states. Based on this result, we design Gaussian Boson sampling, a #P hard problem, using squeezed states. This demonstrates that Boson sampling from Gaussian states is possible, with significant advantages in the photon generation probability, compared to existing protocols. Keywords: quantum walks, driven walks, sampling 
S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. Gabris, I. Jex, C. Silberhorn.
Measuring topological invariants in disordered discretetime quantum walks.
PHYSICAL REVIEW A, 96, 033846, September 2017.
[ bib ]
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks, we implement the scattering scheme proposed by Tarasinski et al. [Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to readout the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted, we also find localized states at the edges between two bulks belonging to different topological phases. In order to analyze the impact of disorder, we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample. Keywords: quantum walks, invariants 
S. Barkhofen, T. J. Bartley, L. Sansoni, R. Kruse, Hamilton,I. Jex, and Ch. Silberhorn.
Driven Boson Sampling.
PHYSICAL REVIEW LETTERS, 118, 020502, January 2017.
[ bib ]
Sampling the distribution of bosons that have undergone a random unitary evolution is strongly believed to be a computationally hard problem. Key to outperforming classical simulations of this task is to increase both the number of input photons and the size of the network. We propose driven boson sampling, in which photons are input within the network itself, as a means to approach this goal. We show that the mean number of photons entering a boson sampling experiment can exceed one photon per input mode, while maintaining the required complexity, potentially leading to less stringent requirements on the input states for such experiments. When using heralded singlephoton sources based on parametric downconversion, this approach offers an similar to efold enhancement in the input state generation rate over scattershot boson sampling, reaching the scaling limit for such sources. This approach also offers a dramatic increase in the signaltonoise ratio with respect to higherorder photon generation from such probabilistic sources, which removes the need for photon number resolution during the heralding process as the size of the system increases. Keywords: quantum walks, driven walks, sampling 
C. S. Hamilton, S. Barkhofen, L. Sansoni, I. Jex, and Ch. Silberhorn.
Driven Discrete Time Quantum Walks.
NEW JOURNAL OF PHYSICS, 18, 073008, July 2016.
[ bib ]
Sampling the distribution of bosons that have undergone a random unitary evolution is strongly believed to be a computationally hard problem. Key to outperforming classical simulations of this task is to increase both the number of input photons and the size of the network. We propose driven boson sampling, in which photons are input within the network itself, as a means to approach this goal. When using heralded singlephoton sources based on parametric downconversion, this approach offers an ∼efold enhancement in the input state generation rate over scattershot boson sampling, reaching the scaling limit for such sources. More significantly, this approach offers a dramatic increase in the signaltonoise ratio with respect to higherorder photon generation from such probabilistic sources, which removes the need for photon number resolution during the heralding process as the size of the system increases. Keywords: quantum walks, driven walks, sampling 
T. Nitsche, F. Elster, J. Novotný, A. Gábris, I. Jex, S. Barkhofen, Ch. Silberhorn.
Quantum Walks with Dynamical Control: Graph Engineering, Initial State Preparation and State Transfer.
NEW JOURNAL OF PHYSICS, 18, 063017, June 2016.
[ bib ]
Quantum walks are a wellestablished model for the study of coherent transport phenomena and provide a universal platform in quantum information theory. Dynamically influencing the walker's evolution gives a high degree of flexibility for studying various applications. Here, we present timemultiplexed finite quantum walks of variable size, the preparation of nonlocalized input states and their dynamical evolution. As a further application, we implement a state transfer scheme for an arbitrary input state to two different output modes. The presented experiments rely on the full dynamical control of a timemultiplexed quantum walk, which includes adjustable coin operation as well as the possibility to flexibly configure the underlying graph structures. Keywords: quantum walks, dynamical control 
M. Stefanak, and I. Jex.
Persistence of unvisited sites in quantum walks on a line.
PHYSICAL REVIEW A, 93, 032321, MARCH 2016.
[ bib ]
We analyze the asymptotic scaling of persistence of unvisited sites for quantum walks on a line. In contrast to the classical random walk, there is no connection between the behavior of persistence and the scaling of variance. In particular, we find that for a twostate quantum walk persistence follows an inverse power law where the exponent is determined solely by the coin parameter. Moreover, for a oneparameter family of threestate quantum walks containing the Grover walk, the scaling of persistence is given by two contributions. The first is the inverse power law. The second contribution to the asymptotic behavior of persistence is an exponential decay coming from the trapping nature of the studied family of quantum walks. In contrast to the twostate walks, both the exponent of the inverse powerlaw and the decay constant of the exponential decay depend also on the initial coin state and its coherence. Hence, one can achieve various regimes of persistence by altering the initial condition, ranging from purely exponential decay to purely inverse powerlaw behavior. Keywords: quantum walks, persistence 
M. Stefanak, J. Novotny, and I. Jex.
Percolation assisted excitation transport in discretetime quantum
walks.
NEW JOURNAL OF PHYSICS, 18, 023040, FEB 2016.
[ bib ]
Coherent transport of excitations along chains of coupled quantum systems represents an interesting problem with a number of applications ranging from quantum optics to solar cell technology. A convenient tool for studying such processes are quantum walks. They allow us to determine all the process features in a quantitative way.Westudy the survival probability and the transport efficiency on a simple, highly symmetric graph represented by a ring. The propagation of excitation is modeled by a discretetime (coined) quantum walk. For a twostate quantum walk, where the excitation (walker) has to leave its actual position to the neighboring sites, the survival probability decays exponentially and the transport efficiency is unity. The decay rate of the survival probability can be estimated using the leading eigenvalue of the evolution operator. However, if the excitation is allowed to stay at its present position, i.e. the propagation is modeled by a lazy quantum walk, then part of the wavepacket can be trapped in the vicinity of the origin and never reaches the sink. In such a case, the survival probability does not vanish and the excitation transport is not efficient. The dependency of the transport efficiency on the initial state is determined. Nevertheless, we show that for some lazy quantum walks dynamical, percolations of the ring eliminate the trapping effect and efficient excitation transport can be achieved. Keywords: percolation; quantum walks, transport efficiency 
Andras Gilyen, Tamas Kiss, Igor Jex.
Exponential Sensitivity and its Cost in Quantum Physics.
SCIENTIFIC REPORTS,
6, 20076, FEB 2016.
[ bib ]
State selective protocols, like entanglement purification, lead to an essentially nonlinear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such nonlinear dynamics, is a promising perspective for applications. Here we demonstrate that chaotic behaviour is a rather generic feature in state selective protocols: exponential sensitivity can exist for all initial states in an experimentally realisable optical scheme. Moreover, any complex rational polynomial map, including the example of the Mandelbrot set, can be directly realised. In state selective protocols, one needs an ensemble of initial states, the size of which decreases with each iteration. We prove that exponential sensitivity to initial states in any quantum system has to be related to downsizing the initial ensemble also exponentially. Our results show that magnifying initial differences of quantum states (a Schrödinger microscope) is possible; however, there is a strict bound on the number of copies needed.Keywords: quantum chaos; iterative maps 
Jaroslav Novotny, Gernot Alber, Igor Jex. Universality in random quantum networks.
PHYSICAL REVIEW A,
92, 062335, DEC 2015.
[ bib ]
Networks constitute efficient tools for assessing universal features of complex systems. In physical contexts, classical as well as quantum networks are used to describe a wide range of phenomena, such as phase transitions, intricate aspects of manybody quantum systems, or even characteristic features of a future quantum internet. Random quantum networks and their associated directed graphs are employed for capturing statistically dominant features of complex quantum systems. Here, we develop an efficient iterative method capable of evaluating the probability of a graph being strongly connected. It is proven that random directed graphs with constant edgeestablishing probability are typically strongly connected, i.e., any ordered pair of vertices is connected by a directed path. This typical topological property of directed random graphs is exploited to demonstrate universal features of the asymptotic evolution of large random qubit networks. These results are independent of our knowledge of the details of the network topology. These findings suggest that other highly complex networks, such as a future quantum internet, may also exhibit similar universal properties.Keywords: quantum networks; asymptotics, universality 
Regina Kruse, Linda Sansoni, Sebastian Brauner, Raimund Ricken, Craig S. Hamilton, Igor Jex, Christine Silberhorn.
Dualpath source engineering in integrated quantum optics.
PHYSICAL REVIEW A,
92, 053841, NOV 2015.
[ bib ]
Quantum optics in combination with integrated optical devices shows great promise for efficient manipulation of single photons. New physical concepts, however, can only be found when these fields truly merge and reciprocally enhance each other. Here we work at the merging point and investigate the physical concept behind a twocoupledwaveguide system with an integrated parametric downconversion process. We use the eigenmode description of the linear system and the resulting modification in momentum conservation to derive the state generation protocol for this type of device. With this new concept of state engineering, we are able to effectively implement a twoinone waveguide source that produces the useful twophoton NOON state without extra overhead such as phase stabilization or narrowband filtering. Experimentally, we benchmark our device by measuring a twophoton NOON state fidelity of F=(84.2±2.6)% and observe the characteristic interferometric pattern directly given by the doubled phase dependence with a visibility of VNOON=(93.3±3.7)%.Keywords: parametric down conversion; NOON states 
Fabian Elster, Sonja Barkhofen,Thomas Nitsche, Jaroslav Novotny, Aurel Gabris, Igor Jex,Christine Silberhorn.
Quantum walk coherences on a dynamical percolated graph.
SCIENTIFIC REPORTS,
5, 13495, AUG 2015.
[ bib ]
Coherent evolution governs the behaviour of all quantum systems, but in nature it is often subjected to influence of a classical environment. For analysing quantum transport phenomena quantum walks emerge as suitable model systems. In particular, quantum walks on percolation structures constitute an attractive platform for studying open system dynamics of random media. Here, we present an implementation of quantum walks differing from the previous experiments by achieving dynamical control of the underlying graph structure. We demonstrate the evolution of an optical timemultiplexed quantum walk over six double steps, revealing the intricate interplay between the internal and external degrees of freedom. The observation of clear nonMarkovian signatures in the coin space testifies the high coherence of the implementation and the extraordinary degree of control of all system parameters. Our work is the proofofprinciple experiment of a quantum walk on a dynamical percolation graph, paving the way towards complex simulation of quantum transport in random media.Keywords: quantum walk; percolation 
Iva Bezdekova, Martin Stefanak and Igor Jex.
Suitable bases for quantum walks with Wigner coins.
PHYSICAL REVIEW A,
92(2):022347, AUG 2015.
[ bib ]
The analysis of a physical problem simplifies considerably when one uses a suitable coordinate system. We apply this approach to the discretetime quantum walks with coins given by 2j + 1dimensional Wigner rotation matrices (Wigner walks), a model which was introduced in Miyazaki et al. [Phys. Rev. A 76, 012332 (2007)]. First, we show that from the three parameters of the coin operator only one is physically relevant for the limit density of the Wigner walk. Next, we construct a suitable basis of the coin space in which the limit density of the Wigner walk acquires a much simpler form. This allows us to identify various dynamical regimes which are otherwise hidden in the standard basis description. As an example, we show that it is possible to find an initial state which reduces the number of peaks in the probability distribution from generic 2j + 1 to a single one. Moreover, the models with integer j lead to the trapping effect. The derived formula for the trapping probability reveals that it can be highly asymmetric and it deviates from purely exponential decay. Explicit results are given up to dimension 5. Keywords: quantum walk; localization 
Pavel Bazant, Holger Frydrych, Gernot Alber and Igor Jex.
Suppressing systematic control errors to high orders.
PHYSICAL REVIEW A,
92(2):022325 , AUG 2015.
[ bib ]
Dynamical decoupling is a powerful method for protecting quantum information against unwanted interactions with the help of openloop control pulses. Realistic control pulses are not ideal and may introduce additional systematic errors. We introduce a class of selfstabilizing pulse sequences capable of suppressing such systematic control errors efficiently in qubit systems. Embedding already known decoupling sequences into these selfstabilizing sequences offers powerful means to achieve robustness against unwanted external perturbations and systematic control errors. As these selfstabilizing sequences are based on singlequbit operations, they offer interesting perspectives for future applications in quantum information processing.Keywords: dynamical decoupling; error correction 
Jiri Maryska, Jaroslav Novotny and Igor Jex.
Dominant couplings in qubit networks with controlled interactions.
JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL,
48(21):215301 , MAY 2015.
[ bib ]
Systems evolving under the influence of competing twoand threebody interactions are of particular interest in exploring the stability of the equilibrium states of a strongly interacting manybody system. We present a solvable model based on qubit networks, which allows us to investigate the intricate influence of these couplings on the possible asymptotic equilibrium states. We study the asymptotic evolution of finite qubit networks under two and threequbit interactions. As representatives of threequbit interactions we choose controlled unitary interactions (cuinteractions) with one and two control qubits. It is shown that networks with purely threequbit interactions exhibit different asymptotic dynamics depending on whether we deal with interactions controlled by one or two qubits. However, when we allow threequbit interactions next to twoqubit interactions, the asymptotics is dictated by twoqubit interactions only. Finally, we prove that the simultaneous presence of two types of threequbit interactions results in the asymptotic dynamics characteristic for twoqubit cuinteractions.Keywords: quantum networks; threeparticle interactions 
Balint Kollar, Tamas Kiss and Igor Jex.
Strongly trapped twodimensional quantum walks
PHYSICAL REVIEW A,
91(2):022308, JAN 2015.
[ bib ]
Discrete time quantum walks (DTQWs) are nontrivial generalizations of random walks with a broad scope of applications. In particular, they can be used as computational primitives, and they are suitable tools for simulating other quantum systems. DTQWs usually spread ballistically due to their quantumness. In some cases, however, they can remain localized at their initial state (trapping). The trapping and other fundamental properties of DTQWs are determined by the choice of the coin operator. We introduce and analyze an up to now uncharted type of walks driven by a coin class leading to strong trapping, complementing the known list of walks. This class of walks exhibit a number of exciting properties with the possible applications ranging from light pulse trapping in a medium to topological effects and quantum search. Keywords: quantum walk; localization 
Holger Frydrych, Antonin Hoskovec, Igor Jex and Gernot Alber.
Selective dynamicsl decoupling for quantum state transfer.
JOURNAL OF PHYSICS B,
48(2):025501, JAN 2015.
[ bib ]
State transfer across discrete quantum networks is one of the elementary tasks of quantum information processing. Its aim is the faithful placement of information into a specific position in the network. However, all physical systems suffer from imperfections, which can severely limit the transfer fidelity. We present selective dynamical decoupling schemes which are capable of stabilizing imperfect quantum state transfer protocols on the model of a bent linear qubit chain. The efficiency of the schemes is tested and verified in numerical simulations on a number of realistic cases. The simulations demonstrate that these selective dynamical decoupling schemes are capable of suppressing unwanted errors in quantum state transfer protocols efficiently. Keywords: quantum state transfer; errors 
Martin Stefanak, Iva Bezdekova, Igor Jex, and Stephen M. Barnett.
Stability of point spectrum for threestate quantum walks on a line.
QUANTUM INFORMATION & COMPUTATION,
14(1314):12131226, OCT 2014.
[ bib ]
Evolution operators of certain quantum walks possess, apart from the continuous part, also a point spectrum. The existence of eigenvalues and the corresponding stationary states lead to partial trapping of the walker in the vicinity of the origin. We analyze the stability of this feature for threestate quantum walks on a line subject to homogenous coin deformations. We find two classes of coin operators that preserve the point spectrum. These new classes of coins are generalizations of coins found previously by different methods and shed light on the rich spectrum of coins that can drive discretetime quantum walks. Keywords: quantum walk; localization 
Craig S. Hamilton, Regina Kruse, Linda Sansoni, Christine Silberhorn, and Igor
Jex.
Driven Quantum Walks.
PHYSICAL REVIEW LETTERS, 113(8), AUG 21 2014.
[ bib ]
We introduce the concept of a driven quantum walk. This work is motivated by recent theoretical and experimental progress that combines quantum walks and parametric downconversion, leading to fundamentally different phenomena. We compare these striking differences by relating the driven quantum walks to the original quantum walk. Next, we illustrate typical dynamics of such systems and show that these walks can be controlled by various pump configurations and phase matchings. Finally, we end by proposing an application of this process based on a quantum search algorithm that performs faster than a classical search.

M. Stefanak, I. Bezdekova, and I. Jex.
Limit distributions of threestate quantum walks: The role of coin
eigenstates.
PHYSICAL REVIEW A, 90(1), JUL 31 2014.
[ bib ]
We analyze two families of threestate quantum walks which show the localization effect. We focus on the role of the initial coin state and its coherence in controlling the properties of the quantum walk. In particular, we show that the description of the walk simplifies considerably when the initial coin state is decomposed in the basis formed by the eigenvectors of the coin operator. This allows us to express the limit distributions in a much more convenient form. Consequently, striking features which are hidden in the standard basis description are easily identified. Moreover, the dependence of moments of the position distribution on the initial coin state can be analyzed in full detail. In particular, we find that in the eigenvector basis the even moments and the localization probability at the origin depend only on incoherent combination of probabilities. In contrast, odd moments and localization outside the origin are affected by the coherence of the initial coin state.

Balint Kollar, Jaroslav Novotny, Tamas Kiss, and Igor Jex.
Discrete time quantum walks on percolation graphs.
EUROPEAN PHYSICAL JOURNAL PLUS, 129(5), MAY 30 2014.
[ bib ]
Randomly breaking connections in a graph alters its transport properties, a model used to describe percolation. In the case of quantum walks, dynamic percolation graphs represent a special type of imperfections, where the connections appear and disappear randomly in each step during the time evolution. The resulting open system dynamics is hard to treat numerically in general. We shortly review the literature on this problem. We then present our method to solve the evolution on finite percolation graphs in the long time limit, applying the asymptotic methods concerning random unitary maps. We work out the case of onedimensional chains in detail and provide a concrete, stepbystep numerical example in order to give more insight into the possible asymptotic behavior. The results about the case of the twodimensional integer lattice are summarized, focusing on the Grovertype coin operator.

B. Kollar, J. Novotny, T. Kiss, and I. Jex.
Percolation induced effects in twodimensional coined quantum walks:
analytic asymptotic solutions.
NEW JOURNAL OF PHYSICS, 16, 023002, FEB 2014.
[ bib ]
Quantum walks on graphs can model physical processes and serve as efficient tools in quantum information theory. Once we admit random variations in the connectivity of the underlying graph, we arrive at the problem of percolation, where the longtime behaviour appears untreatable with direct numerical methods. We develop novel analytic methods based on the theory of random unitary operations which help us to determine explicitly the asymptotic dynamics of quantum walks on twodimensional finite integer lattices with percolation. Based on this theory, we find new unexpected features of percolated walks like asymptotic position inhomogeneity or special directional symmetry breaking.

R. Kruse, F. Katzschmann, A. Christ, A. Schreiber, S. Wilhelm, K. Laiho,
A. Gabris, C. S. Hamilton, I. Jex, and C. Silberhorn.
Spatiospectral characteristics of parametric downconversion in
waveguide arrays.
NEW JOURNAL OF PHYSICS, 15, 083046, AUG 2013.
[ bib ]
High dimensional quantum states are of fundamental interest for quantum information processing. They give access to large Hilbert spaces and, in turn, enable the encoding of quantum information on multiple modes. One method to create such quantum states is parametric downconversion (PDC) in waveguide arrays (WGAs) which allows for the creation of highly entangled photon pairs in controlled, easily accessible spatial modes, with unique spectral properties. In this paper we examine both theoretically and experimentally the PDC process in a lithium niobate WGA. We measure the spatial and spectral properties of the emitted photon pairs, revealing correlations between spectral and spatial degrees of freedom of the created photons. Our measurements show that, in contrast to prior theoretical approaches, spectrally dependent coupling effects have to be taken into account in the theory of PDC in WGAs. To interpret the results, we developed a theoretical model specifically taking into account spectrally dependent coupling effects, which further enables us to explore the capabilities and limitations for engineering the spatial correlations of the generated quantum states.

Peter P. Rohde, Andreas Schreiber, Martin Stefanak, Igor Jex, Alexei Gilchrist,
and Christine Silberhorn.
Increasing the Dimensionality of Quantum Walks Using Multiple
Walkers.
JOURNAL OF COMPUTATIONAL AND THEORETICAL NANOSCIENCE,
10(7, SI):16441652, JUL 2013.
[ bib ]
We show that with the addition of multiple walkers, quantum walks on a line can be transformed into lattice graphs of higher dimension. Thus, multiwalker walks can simulate singlewalker walks on higher dimensional graphs and vice versa. This exponential complexity opens up new applications for presentday quantum walk experiments. We discuss the applications of such higherdimensional structures and how they relate to linear optics quantum computing. In particular we show that multiwalker quantum walks are equivalent to the BOSONSAMPLING model for linear optics quantum computation proposed by Aaronson and Arkhipov. With the addition of control over phasedefects in the lattice, which can be simulated with entangling gates, asymmetric lattice structures can be constructed which are universal for quantum computation. Keywords: Quantum; Walk; Complexity 
J. Novotny, G. Alber, and I. Jex.
Asymptotic properties of quantum Markov chains.
JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL,
45(48), DEC 7 2012.
[ bib ]
The asymptotic dynamics of discrete quantum Markov chains generated by the most general physically relevant quantum operations is investigated. It is shown that it is confined to an attractor space in which the resulting quantum Markov chain is diagonalizable. A construction procedure of a basis of this attractor space and its associated dual basis of 1forms is presented. It is applicable whenever a strictly positive quantum state exists which is contracted or left invariant by the generating quantum operation. Moreover, algebraic relations between the attractor space and Kraus operators involved in the definition of a quantum Markov chain are derived. This construction is not only expected to offer significant computational advantages in cases in which the dimension of the Hilbert space is large and the dimension of the attractor space is small, but it also sheds new light onto the relation between the asymptotic dynamics of discrete quantum Markov chains and fixed points of their generating quantum operations. Finally, we show that without any restriction our construction applies to all initial states whose support belongs to the socalled recurrent subspace.

G. M. Nikolopoulos, A. Hoskovec, and I. Jex.
Analysis and minimization of bending losses in discrete quantum
networks.
PHYSICAL REVIEW A, 85(6), JUN 21 2012.
[ bib ]
We study theoretically the transfer of quantum information along bends in twodimensional discrete lattices. Our analysis shows that the fidelity of the transfer decreases considerably as a result of interactions in the neighborhood of the bend. It is also demonstrated that such losses can be controlled efficiently by the inclusion of a defect. The present results are of relevance to various physical implementations of quantum networks, where geometric imperfections with finite spatial extent may arise as a result of bending, residual stress, etc.

B. Kollar, T. Kiss, J. Novotny, and I. Jex.
Asymptotic Dynamics of Coined Quantum Walks on Percolation Graphs.
PHYSICAL REVIEW LETTERS, 108(23), JUN 5 2012.
[ bib ]
Quantum walks obey unitary dynamics: they form closed quantum systems. The system becomes open if the walk suffers from imperfections represented as missing links on the underlying basic graph structure, described by dynamical percolation. Openness of the system's dynamics creates decoherence, leading to strong mixing. We present a method to analytically solve the asymptotic dynamics of coined, percolated quantum walks for a general graph structure. For the case of a circle and a linear graph we derive the explicit form of the asymptotic states. We find that a rich variety of asymptotic evolutions occur: not only the fully mixed state, but other stationary states; stable periodic and quasiperiodic oscillations can emerge, depending on the coin operator, the initial state, and the topology of the underlying graph.

M. Stefanak, I. Bezdekova, and I. Jex.
Continuous deformations of the Grover walk preserving localization.
EUROPEAN PHYSICAL JOURNAL D, 66(5), MAY 2012.
[ bib ]
The threestate Grover walk on a line exhibits the localization effect characterized by a nonvanishing probability of the particle to stay at the origin. We present two continuous deformations of the Grover walk which preserve its localization nature. The resulting quantum walks differ in the rate at which they spread through the lattice. The velocities of the left and righttraveling probability peaks are given by the maximum of the group velocity. We find the explicit form of peak velocities in dependence on the coin parameter. Our results show that localization of the quantum walk is not a singular property of an isolated coin operator but can be found for entire families of coins.

Andreas Schreiber, Aurel Gabris, Peter P. Rohde, Kaisa Laiho, Martin Stefanak,
Vaclav Potocek, Craig Hamilton, Igor Jex, and Christine Silberhorn.
A 2D Quantum Walk Simulation of TwoParticle Dynamics.
SCIENCE, 336(6077):5558, APR 6 2012.
[ bib ]
Multidimensional quantum walks can exhibit highly nontrivial topological structure, providing a powerful tool for simulating quantum information and transport systems. We present a flexible implementation of a twodimensional (2D) optical quantum walk on a lattice, demonstrating a scalable quantum walk on a nontrivial graph structure. We realized a coherent quantum walk over 12 steps and 169 positions by using an optical fiber network. With our broad spectrum of quantum coins, we were able to simulate the creation of entanglement in bipartite systems with conditioned interactions. Introducing dynamic control allowed for the investigation of effects such as strong nonlinearities or twoparticle scattering. Our results illustrate the potential of quantum walks as a route for simulating and understanding complex quantum systems.

A. Schreiber, A. Gabris, P. P. Rohde, K. Laiho, M. Stefanak, V. Potocek,
C. Hamilton, I. Jex, and C. Silberhorn.
Quantum simulations with a twodimensional Quantum Walk.
In 2012 CONFERENCE ON LASERS AND ELECTROOPTICS (CLEO), 345
E 47TH ST, NEW YORK, NY 10017 USA, 2012. IEEE.
Conference on Lasers and ElectroOptics (CLEO), San Jose, CA, MAY
0611, 2012.
[ bib ]
We present an experimental implementation of a quantum walk in two dimensions, employing an optical fiber network. We simulated entangling operations and nonlinear multiparticle interactions revealing phenomena such as bound states. (C) 2011 Optical Society of America

T. Brougham, G. M. Nikolopoulos, and I. Jex. Perfect transfer of multiple excitations in quantum networks (vol 83, artnno 022323, 2011). PHYSICAL REVIEW A, 84(3), SEP 16 2011. [ bib ] 
H. Lavicka, V. Potocek, T. Kiss, E. Lutz, and I. Jex.
Quantum walk with jumps.
EUROPEAN PHYSICAL JOURNAL D, 64(1):119129, SEP
2011.
[ bib ]
We analyze a special class of 1D quantum walks (QWs) realized using optical multiports. We assume nonperfect multiports showing errors in the connectivity, i.e. with a small probability the multiports can connect not to their nearest neighbor but to another multiport at a fixed distance  we call this a jump. We study two cases of QW with jumps where multiple displacements can emerge at one timestep. The first case assumes timecorrelated jumps (static disorder). In the second case, we choose the positions of jumps randomly in time (dynamic disorder). The probability distributions of position of the QW walker in both instances differ significantly: dynamic disorder leads to a Gaussianlike distribution, while for static disorder we find two distinct behaviors depending on the parity of jump size. In the case of evensized jumps, the distribution exhibits a threepeak profile around the position of the initial excitation, whereas the probability distribution in the odd case follows a Laplacelike discrete distribution modulated by additional (exponential) peaks for long times. Finally, our numerical results indicate that by an appropriate mapping a universal functional behavior of the variance of the longtime probability distribution can be revealed with respect to the scaled average of jump size.

T. Kiss, S. Vymetal, L. D. Toth, A. Gabris, I. Jex, and G. Alber.
MeasurementInduced Chaos with Entangled States.
PHYSICAL REVIEW LETTERS, 107(10), AUG 31 2011.
[ bib ]
The dynamics of an ensemble of identically prepared twoqubit systems is investigated which is subjected to the iteratively applied measurements and conditional selection of a typical entanglement purification protocol. The resulting dynamics exhibits strong sensitivity to initial conditions. For one class of initial states two types of islands characterize the asymptotic limit. They correspond to a separable and a fully entangled twoqubit state, respectively, and their boundaries form fractallike structures. In the presence of incoherent noise an additional stable asymptotic cycle appears.

Jaroslav Novotny, Gernot Alber, and Igor Jex.
Entanglement and Decoherence: Fragile and Robust Entanglement.
PHYSICAL REVIEW LETTERS, 107(9), AUG 23 2011.
[ bib ]
The destruction of entanglement of open quantum systems by decoherence is investigated in the asymptotic longtime limit. For this purpose a general and analytically solvable decoherence model is presented which does not involve any weakcoupling or Markovian assumption. It is shown that two fundamentally different classes of entangled states can be distinguished and that they can be influenced significantly by two important environmental properties, namely, its initially prepared state and its size. Quantum states of the first class are fragile against decoherence so that they can be disentangled asymptotically even if coherences between pointer states are still present. Quantum states of the second type are robust against decoherence. Asymptotically they can be disentangled only if also decoherence is perfect. A simple criterion for identifying these two classes on the basis of twoqubit entanglement is presented.

J. Novotny, G. Alber, and I. Jex.
Asymptotic dynamics of qubit networks under randomly applied
controlled unitary transformations.
NEW JOURNAL OF PHYSICS, 13, MAY 26 2011.
[ bib ]
The asymptotic dynamics of manyqubit quantum systems is investigated under iteratively and randomly applied unitary transformations. For a oneparameter family of unitary transformations, which entangle pairs of qubits, two main theorems are proved. They characterize completely the dependence of the resulting asymptotic dynamics on the topology of the interaction graph that encodes all possible qubit couplings. These theorems exhibit clearly which aspects of an interaction graph are relevant and which ones are irrelevant to the asymptotic dynamics. On the basis of these theorems, the local entropy transport between an open quantum system and its environment are explored for strong nonMarkovian couplings and for different sizes of the environment and different interaction topologies. It is shown that although the randomly applied unitary entanglement operations cannot decrease the overall entropy of such a qubit network, a local entropy decrease or `cooling' of subsystems is possible for special classes of interaction topologies.

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, I. Jex, and Ch.
Silberhorn.
Decoherence and Disorder in QuantumWalks: From Ballistic Spread to
Localization.
PHYSICAL REVIEW LETTERS, 106(18), MAY 6 2011.
[ bib ]
We investigate the impact of decoherence and static disorder on the dynamics of quantum particles moving in a periodic lattice. Our experiment relies on the photonic implementation of a onedimensional quantum walk. The pure quantum evolution is characterized by a ballistic spread of a photon's wave packet along 28 steps. By applying controlled timedependent operations we simulate three different environmental influences on the system, resulting in a fast ballistic spread, a diffusive classical walk, and the first Anderson localization in a discrete quantum walk architecture.

M. Stefanak, S. M. Barnett, B. Kollar, T. Kiss, and I. Jex.
Directional correlations in quantum walks with two particles.
NEW JOURNAL OF PHYSICS, 13, MAR 21 2011.
[ bib ]
Quantum walks on a line with a single particle possess a classical analogue. Involving more walkers opens up the possibility of studying collective quantum effects, such as manyparticle correlations. In this context, entangled initial states and the indistinguishability of the particles play a role. We consider the directional correlations between two particles performing a quantum walk on a line. For noninteracting particles, we find analytic asymptotic expressions and give the limits of directional correlations. We show that by introducing deltainteraction between the particles, one can exceed the limits for noninteracting particles.

T. Brougham, G. M. Nikolopoulos, and I. Jex.
Perfect transfer of multiple excitations in quantum networks.
PHYSICAL REVIEW A, 83(2), FEB 24 2011.
[ bib ]
We present a general formalism to the problem of perfect state transfer (PST), where the state involves multiple excitations of the quantum network. A key feature of our formalism is that it allows for inclusion of nontrivial interactions between the excitations. Hence, it is perfectly suited to addressing the problem of PST in the context of various types of physical realizations. The general formalism is also flexible enough to account for situations where multiple excitations are “focused” onto the same site.

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, I. Jex, and
Ch Silberhorn.
Photonic quantum walks in a fiber based recursion loop.
In Ralph, T and Lam, PK, editor, QUANTUM COMMUNICATION,
MEASUREMENT AND COMPUTING (QCMC): THE TENTH INTERNATIONAL CONFERENCE,
volume 1363 of AIP Conference Proceedings, 2 HUNTINGTON
QUADRANGLE, STE 1NO1, MELVILLE, NY 117474501 USA, 2011. Australian Res
Council Ctr Excellence Quantum Comp Technol; Tamagawa Univ; Australian Res
Council Ctr Excellence QuantumAtom Opt; Res Lab Elect Massachusetts Inst
Technol; NewSpec Pty Ltd; Oxford Instruments Pty Ltd; Lastek Australia,
Toptica Photon, AMER INST PHYSICS.
10th International Conference on Quantum Communication, Measurement
and Computing (QCMC), Brisbane, AUSTRALIA, JUL 1923, 2010.
[ bib ]
We present a flexible and robust system for implementing onedimensional coined quantum walks. A recursion loop in the optical network together with a translation of the spatial into the time domain ensures the possible increment of the step number without need of additional optical elements. An intrinsic phase stability assures a high degree of coherence and hence guarantees a good scalability of the system. We performed a quantum walk over 27 steps and analyzed the 54 output modes. Furthermore, we estimated that up to 100 steps can be realized with only minor changes in the used components. Keywords: Quantum walk; Quantum simulations 
Craig S. Hamilton, Aurel Gabris, Igor Jex, and Stephen M. Barnett.
Quantum walk with a fourdimensional coin.
NEW JOURNAL OF PHYSICS, 13, JAN 2011.
[ bib ]
We examine the physical implementation of a discrete time quantum walk with a fourdimensional coin. Our quantum walker is a photon moving repeatedly through a time delay loop, with time being our position space. The quantum coin is implemented using the internal states of the photon: the polarization and two of the orbital angular momentum states. We demonstrate how to implement this physically and what components would be needed. We then illustrate some of the results that could be obtained by performing the experiment.

T. Brougham, V. Kost'ak, I. Jex, E. Andersson, and T. Kiss.
Entanglement preparation using symmetric multiports.
EUROPEAN PHYSICAL JOURNAL D, 61(1):231236, JAN
2011.
[ bib ]
We investigate the entanglement produced by a multipath interferometer that is composed of two symmetric multiports, with phase shifts applied to the output of the first multiport. Particular attention is paid to the case when we have a single photon entering the interferometer. For this situation we derive a simple condition that characterizes the types of entanglement that one can generate. We then show how one can use the results from the singlephoton case to determine what kinds of multiphoton entangled states one can prepare using the interferometer.

Peter P. Rohde, Andreas Schreiber, Martin Stefanak, Igor Jex, and Christine
Silberhorn.
Multiwalker discrete time quantum walks on arbitrary graphs, their
properties and their photonic implementation.
NEW JOURNAL OF PHYSICS, 13, JAN 2011.
[ bib ]
Quantum walks have emerged as an interesting alternative to the usual circuit model for quantum computing. While still universal for quantum computing, the quantum walk model has very different physical requirements, which lends itself more naturally to some physical implementations, such as linear optics. Numerous authors have considered walks with one or two walkers, on onedimensional graphs, and several experimental demonstrations have been performed. In this paper, we discuss generalizing the model of discrete time quantum walks to the case of an arbitrary number of walkers acting on arbitrary graph structures. We present a formalism that allows for the analysis of such situations, and several example scenarios for how our techniques can be applied. We consider the most important features of quantum walksmeasurement, distinguishability, characterization and the distinction between classical and quantum interference. We also discuss the potential for physical implementation in the context of linear optics, which is of relevance to presentday experiments.

L. Kecskes, T. Kiss, M. Stefanak, and I. Jex.
The role of measurement in the recurrence property of discrete timed
quantum walks.
In Prochazka, I and Sobolewski, R and Fiurasek, J, editor,
PHOTON COUNTING APPLICATIONS, QUANTUM OPTICS, AND QUANTUM INFORMATION
TRANSFER AND PROCESSING III, volume 8072 of Proceedings of SPIE,
1000 20TH ST, PO BOX 10, BELLINGHAM, WA 982270010 USA, 2011. SPIE,
SPIEINT SOC OPTICAL ENGINEERING.
Conference on the Photon Counting Applications, Quantum Optics, and
Quantum Information Transfer and Processing III, Prague, CZECH REPUBLIC, APR
1820, 2011.
[ bib ]
The quantum mechanical generalisation of random walks (called Quantum Walks) present us with a broader spectrum of possibilities compared to their classical counterparts. The aim of the presented study is to explore a new portion of this area by incorporating a new step in the process of the Quantum Walk unique to quantum mechanics: the measurement. Our focus lies in the characterising number of the recurrence behaviour of the walk (Polyanumber). We observe the effect of the standard projective measurement, a yesno measurement on the origin and the effect of different measurement schemes (periodic and random) on the definition and the numeric value of the Polyanumber.

Jaroslav Novotny, Gernot Alber, and Igor Jex.
Asymptotic evolution of random unitary operations.
CENTRAL EUROPEAN JOURNAL OF PHYSICS, 8(6):10011014,
DEC 2010.
[ bib ]
We analyze the asymptotic dynamics of quantum systems resulting from large numbers of iterations of random unitary operations. Although, in general, these quantum operations cannot be diagonalized it is shown that their resulting asymptotic dynamics is described by a diagonalizable superoperator. We prove that this asymptotic dynamics takes place in a typically low dimensional attractor space which is independent of the probability distribution of the unitary operations applied. This vector space is spanned by all eigenvectors of the unitary operations involved which are associated with eigenvalues of unit modulus. Implications for possible asymptotic dynamics of iterated random unitary operations are presented and exemplified in an example involving random controllednot operations acting on two qubits. Keywords: random unitary map; asymptotic evolution; iterations; attractor; open dynamics 
P. Adam, T. Kiss, Z. Darazs, and I. Jex.
Conditional generation of optical Schrodinger cat states.
PHYSICA SCRIPTA, T140, SEP 2010.
16th Central European Workshop on Quantum Optics, Turku, FINLAND,
MAY 2327, 2009.
[ bib ]
Given a source of two coherent state superpositions with small separation in a travelling wave optical setting, we show that by interference and balanced homodyne measurement it is possible to conditionally prepare a symmetrically placed superposition of coherent states around the origin of the phase space. The separation of coherent states in the superposition will be amplified during the process.

M. Stefanak, B. Kollar, T. Kiss, and I. Jex.
Full revivals in 2D quantum walks.
PHYSICA SCRIPTA, T140, SEP 2010.
16th Central European Workshop on Quantum Optics, Turku, FINLAND,
MAY 2327, 2009.
[ bib ]
Recurrence of a random walk is described by the Polya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full revival of its quantum state. Localization for twodimensional quantum walks is known to exist in the sense of nonvanishing probability distribution in the asymptotic limit. We show, on the example of the 2D Grover walk, that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for a fourstate quantum walk. Stationary states and revivals result from interference, which has no counterpart in classical random walks.

B. Kollar, M. Stefanak, T. Kiss, and I. Jex.
Recurrences in threestate quantum walks on a plane.
PHYSICAL REVIEW A, 82(1), JUL 6 2010.
[ bib ]
We analyze the role of dimensionality in the time evolution of discretetime quantum walks through the example of the threestate walk on a twodimensional triangular lattice. We show that the threestate Grover walk does not lead to trapping (localization) or recurrence to the origin, in sharp contrast to the Grover walk on the twodimensional square lattice. We determine the powerlaw scaling of the probability at the origin with the method of stationary phase. We prove that only a special subclass of coin operators can lead to recurrence, and there are no coins that lead to localization. The propagation for the recurrent subclass of coins is quasione dimensional.

A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. J. Mosley,
E. Andersson, I. Jex, and Ch. Silberhorn.
Photons Walking the Line: A Quantum Walk with Adjustable Coin
Operations.
PHYSICAL REVIEW LETTERS, 104(5), FEB 5 2010.
[ bib ]
We present the first robust implementation of a coined quantum walk over five steps using only passive optical elements. By employing a fiber network loop we keep the amount of required resources constant as the walker's position Hilbert space is increased. We observed a nonGaussian distribution of the walker's final position, thus characterizing a faster spread of the photon wave packet in comparison to the classical random walk. The walk is realized for many different coin settings and initial states, opening the way for the implementation of a quantumwalkbased search algorithm.

Thomas Brougham, Stephen M. Barnett, and Igor Jex.
Interference of composite bosons.
JOURNAL OF MODERN OPTICS, 57(7):587594, 2010.
[ bib ]
We investigate multiboson interference. A Hamiltonian is presented that treats pairs of bosons as a single composite boson. This Hamiltonian allows two pairs of bosons to interact as if they were two single composite bosons. We show that this leads to the composite bosons exhibiting novel interference effects, such as HongOuMandel interference. We then investigate generalisations of the formalism to the case of interference between two general composite bosons. Finally, we show how one can realise interference between composite bosons in the twoatom Dicke model. Keywords: interference of bosons; quantum optics; HongOuMandel effect 
T. Brougham, G. M. Nikolopoulos, and I. Jex.
Communication in quantum networks of logical bus topology.
PHYSICAL REVIEW A, 80(5), NOV 2009.
[ bib ]
Perfect state transfer (PST) is discussed in the context of passive quantum networks with logical bus topology, where many logical nodes communicate using the same shared media without any external control. The conditions under which a number of pointtopoint PST links may serve as building blocks for the design of such multinode networks are investigated. The implications of our results are discussed in the context of various Hamiltonians that act on the entire network and are capable of providing PST between the logical nodes of a prescribed set in a deterministic manner. Keywords: information theory; quantum communication; topology 
I. Marzoli, P. Tombesi, G. Ciaramicoli, G. Werth, P. Bushev, S. Stahl,
F. SchmidtKaler, M. Hellwig, C. Henkel, G. Marx, I. Jex, E. Stachowska,
G. Szawiola, and A. Walaszyk.
Experimental and theoretical challenges for the trapped electron
quantum computer.
JOURNAL OF PHYSICS BATOMIC MOLECULAR AND OPTICAL PHYSICS,
42(15), AUG 14 2009.
[ bib ]
We discuss quantum information processing with trapped electrons. After recalling the operation principle of planar Penning traps, we sketch the experimental conditions to load, cool and detect single electrons. Here we present a detailed investigation of a scalable scheme including feasibility studies and the analysis of all important elements, relevant for the experimental stage. On the theoretical side, we discuss different methods to couple electron qubits. We estimate the relevant qubit coherence times and draw implications for the experimental setting. A critical assessment of quantum information processing with trapped electrons concludes the paper.

J. Novotny, G. Alber, and I. Jex.
Random unitary dynamics of quantum networks.
JOURNAL OF PHYSICS AMATHEMATICAL AND THEORETICAL,
42(28), JUL 17 2009.
[ bib ]
We investigate the asymptotic dynamics of quantum networks under repeated applications of random unitary operations. It is shown that in the asymptotic limit of large numbers of iterations this dynamics is generally governed by a typically low dimensional attractor space. This space is determined completely by the unitary operations involved and it is independent of the probabilities with which these unitary operations are applied. Based on this general feature analytical results are presented for the asymptotic dynamics of arbitrarily large cyclic qubit networks whose nodes are coupled by randomly applied controlledNOT operations.

T. Kiss, L. Kecskes, M. Stefanak, and I. Jex.
Recurrence in coined quantum walks.
PHYSICA SCRIPTA, T135, JUL 2009.
15th Central European Workshop on Quantum Optics, Belgrade, SERBIA,
MAY 29JUN 03, 2008.
[ bib ]
Recurrence of quantum walks on lattices can be characterized by the generalized Polya number. Its value reflects the difference between a classical and a quantum system. The dimension of the lattice is not a unique parameter in the quantum case; both the coin operator and the initial quantum state of the coin influence the recurrence in a nontrivial way. In addition, the definition of the Polya number involves measurement of the system. Depending on how measurement is included in the definition, the recurrence properties vary. We show that in the limiting case of frequent, strong measurements, one can approach the classical dynamics. Comparing various cases, we have found numerical indication that our previous definition of the Polya number provides an upper limit.

M. Stefanak, T. Kiss, and I. Jex.
Recurrence of biased quantum walks on a line.
NEW JOURNAL OF PHYSICS, 11, APR 2009.
[ bib ]
The Polya number of a classical random walk on a regular lattice is known to depend solely on the dimension of the lattice. For one and two dimensions it equals one, meaning unit probability of returning to the origin. This result is extremely sensitive to the directional symmetry, and any deviation from the equal probability of travelling in each direction results in a change of the character of the walk from recurrent to transient. Applying our definition of the Polya number to quantum walks on a line we show that the recurrence character of quantum walks is more stable against bias. We determine the range of parameters for which biased quantum walks remain recurrent. We find that there exist genuine biased quantum walks that are recurrent.

Craig S. Hamilton, Hynek Lavicka, Erika Andersson, John Jeffers, and Igor Jex.
Quantum public key distribution with imperfect device components.
PHYSICAL REVIEW A, 79(2), FEB 2009.
[ bib ]
We examine the operation of a device for a public quantum key distribution network. The recipients attempt to determine whether or not their individual key copies, which are a sequence of coherent states, are identical. To quantify the success of the protocol we use a fidelitybased figure of merit and describe a method for increasing this in the presence of noise and imperfect detectors. We show that the fidelity may be written as the product of two factors: one that depends on the properties of the device setup and another that depends on the detectors used. We then demonstrate the effect various parameters have on the overall effective operation of the device. Keywords: cryptographic protocols; quantum cryptography 
Thomas Brougham, Goce Chadzitaskos, and Igor Jex.
Transformation design and nonlinear Hamiltonians.
JOURNAL OF MODERN OPTICS, 56(14):15881597, 2009.
[ bib ]
We study a class of nonlinear Hamiltonians, with applications in quantum optics. The interaction terms of these Hamiltonians are generated by taking a linear combination of powers of a simple `beam splitter' Hamiltonian. The entanglement properties of the eigenstates are studied. Finally, we show how to use this class of Hamiltonians to perform special tasks such as conditional state swapping, which can be used to generate optical cat states and to sort photons. Keywords: quantum optics; nonlinear optics; quantum information 
V. Potocek, A. Gabris, T. Kiss, and I. Jex.
Optimized quantum randomwalk search algorithms on the hypercubel.
PHYSICAL REVIEW A, 79(1), JAN 2009.
[ bib ]
Shenvi, Kempe, and Whaley's quantum randomwalk search (SKW) algorithm [Phys. Rev. A 67, 052307 (2003)] is known to require O(root N) number of oracle queries to find the marked element, where N is the size of the search space. The overall time complexity of the SKW algorithm differs from the best achievable on a quantum computer only by a constant factor. We present improvements to the SKW algorithm which yield a significant increase in success probability, and an improvement on query complexity such that the theoretical limit of a search algorithm succeeding with probability close to one is reached. We point out which improvement can be applied if there is more than one marked element to find.

M. Stefanak, T. Kiss, and I. Jex.
Recurrence properties of unbiased coined quantum walks on infinite
ddimensional lattices.
PHYSICAL REVIEW A, 78(3), SEP 2008.
[ bib ]
The Polya number characterizes the recurrence of a random walk. We apply the generalization of this concept to quantum walks [M. Stefanak et al., Phys. Rev. Lett. 100, 020501 (2008)] which is based on a specific measurement scheme. The Polya number of a quantum walk depends, in general, on the choice of the coin and the initial coin state, in contrast to classical random walks where the lattice dimension uniquely determines it. We analyze several examples to depict the variety of possible recurrence properties. First, we show that for the class of quantum walks driven by Hadamard tensorproduct coins, the Polya number is independent of the initial conditions and the actual coin operators, thus resembling the property of the classical walks. We provide an estimation of the Polya number for this class of quantum walks. Second, we examine the twodimensional Grover walk, which exhibits localization and thus is recurrent, except for a particular initial state for which the walk is transient. We generalize the Grover walk to show that one can construct in arbitrary dimensions a quantum walk which is recurrent. This is in great contrast with classical walks which are recurrent only for the dimensions d=1,2. Finally, we analyze the recurrence of the 2D Fourier walk. This quantum walk is recurrent except for a twodimensional subspace of the initial states. We provide an estimation of the Polya number in its dependence on the initial state.

T. Kiss, I. Jex, G. Alber, and E. Kollar.
Properties of complex chaos in conditional qubit dynamics.
INTERNATIONAL JOURNAL OF QUANTUM INFORMATION,
6(1):695700, JUL 2008.
International Meeting on Noise, Information and Complexity AT
Quantum Scale, Erice, ITALY, NOV 0210, 2007.
[ bib ]
Complex chaos is specified by an iterated mapping on complex numbers. It has recently been found in the dynamics of qubits where each time step is conditioned on a measurement result on part of the system. We analyse the simplest case of one qubit dynamics with one complex parameter in some detail. We point out that two attractive cycles can exist and provide examples how the fractal like Julia set divides the areas of corresponding initial states. We show how to determine the set of parameters for which one, two or no stable fixed cycles exists and provide the numerically calculated images of the sets. The results can be relevant for the quantum state purification protocol based on the similar dynamics of two or more qubits and in general for any protocol based on conditioned nonlinear dynamics where truly chaotic behavior may occur. Keywords: quantum chaos; complex chaos; purification 
D. Geberth, O. Kern, G. Alber, and I. Jex.
Stabilization of quantum information by combined dynamical
decoupling and detectedjump error correction.
EUROPEAN PHYSICAL JOURNAL D, 46(2):381394, FEB
2008.
[ bib ]
Two possible applications of random decoupling are discussed. Whereas so far decoupling methods have been considered merely for quantum memories, here it is demonstrated that random decoupling is also a convenient tool for stabilizing quantum algorithms. Furthermore, a decoupling scheme is presented which involves a random decoupling method compatible with detectedjump error correcting quantum codes. With this combined error correcting strategy it is possible to stabilize quantum information against both spontaneous decay and static imperfections of a qubitbased quantum information processor in an efficient way.

M. Stefanak, I. Jex, and T. Kiss.
Recurrence and polya number of quantum walks.
PHYSICAL REVIEW LETTERS, 100(2), JAN 18 2008.
[ bib ]
We analyze the recurrence probability (Polya number) for ddimensional unbiased quantum walks. A sufficient condition for a quantum walk to be recurrent is derived. As a byproduct we find a simple criterion for localization of quantum walks. In contrast with classical walks, where the Polya number is characteristic for the given dimension, the recurrence probability of a quantum walk depends in general on the topology of the walk, choice of the coin and the initial state. This allows us to change the character of the quantum walk from recurrent to transient by altering the initial state.

S. Vymetal, T. Kiss, I. Jex, G. Alber, A. Gabris, and T. Langrova.
Chaos in the conditional dynamics of two qubits purification
protocol.
In Robnik, M and Romanovski, VG, editor, LET'S FACE CHAOS
THROUGH NONLINEAR DYNAMICS, volume 1076 of AIP Conference
Proceedings, pages 255261, 2 HUNTINGTON QUADRANGLE, STE 1NO1,
MELVILLE, NY 117474501 USA, 2008. Slovenian Res Agcy; GEN Energija;
Telekom Slovenije; Nova KBM, AMER INST PHYSICS.
7th International Summer School and Conference on Lets Face Chaos
through Nonlinear Dynamics, Univ Maribor, Maribor, SLOVENIA, JUN 29JUL 13,
2008.
[ bib ]
The presence of complex chaos in iterative apphcations of selective dynamics on quantum systems is a novel form of quantum chaos with true sensitivity to initial conditions. Techniques for the study of pure states are extended to the twoqubit case(1). Keywords: complex chaos; quantum chaos; purification protocol 
A. Gabris, T. Kiss, and I. Jex.
Scattering quantum randomwalk search with errors.
PHYSICAL REVIEW A, 76(6), DEC 2007.
[ bib ]
We analyze the realization of a quantumwalk search algorithm in a passive, linear optical network. The specific model enables us to consider the effect of realistic sources of noise and losses on the search efficiency. Photon loss uniform in all directions is shown to lead to the rescaling of search time. Deviation from directional uniformity leads to the enhancement of the search efficiency compared to uniform loss with the same average. In certain cases even increasing loss in some of the directions can improve search efficiency. We show that while we approach the classical limit of the general search algorithm by introducing random phase fluctuations, its utility for searching is lost. Using numerical methods, we found that for static phase errors the averaged search efficiency displays a damped oscillatory behavior that asymptotically tends to a nonzero value.

Agnieszka Tereszkiewicz, Anatol Odzijewicz, Maciej Horowski, Igor Jex, and Goce
Chadzitaskos.
Explicitly solvable models of a twomode coupler in Kerr media.
PHYSICAL REVIEW A, 75(6), JUN 2007.
[ bib ]
We present exact solutions for two nonlinear models each of which describes parametric down conversion of photons as well as the Kerr effect. The Hamiltonians for these models are related to the dual Hahn finite orthogonal polynomials. Explicit expressions are obtained for the spectra and for the eigenvectors of the Hamiltonians. A discussion of the physical characteristics of the systems is presented.

Stefan T. ProbstSchendzielorz, Alexander Wolf, Matthias Freyberger, Igor Jex,
Bing He, and Janos A. Bergou.
Unambiguous discriminator for unknown quantum states: An
implementation.
PHYSICAL REVIEW A, 75(5), MAY 2007.
[ bib ]
We give a scheme for a physical implementation of the programmable state discriminator that unambiguously discriminates between two unknown qubit states with optimal probability of success. One copy of each of the unknown states is provided as input, or program, for the two program registers, and the data state, which is guaranteed to be prepared in one of the program states, is fed into the data register of the device. This device will then tell us, in an optimal way, which of the templates stored in the program registers the data state matches. The implementation is based on Neumark's theorem. We introduce a single qubit as ancilla and a unitary operator that entangles the system with the ancilla in such a way that projective measurements performed in the computational basis of the system plus ancilla transform the initial system states according to the optimal positiveoperatorvalued measure.

V. Kostak, G. M. Nikolopoulos, and I. Jex.
Perfect state transfer in networks of arbitrary topology and
coupling configuration.
PHYSICAL REVIEW A, 75(4), APR 2007.
[ bib ]
A general formalism of the problem of perfect state transfer is presented. We show that there are infinitely many Hamiltonians that may provide a solution to this problem. In a first attempt to give a classification of them we investigate their possible forms and the related dynamics during the transfer. Finally, we show how the present formalism can be used for the engineering of perfect quantum wires of various topologies and coupling configurations.

J. Novotny, G. Alber, and I. Jex.
Network implementation of covariant twoqubit quantum operations.
PHYSICAL REVIEW A, 75(4), APR 2007.
[ bib ]
A sixqubit quantum network consisting of conditional unitary gates is presented which is capable of implementing a large class of covariant twoqubit quantum operations. Optimal covariant NOT operations for one and twoqubit systems are special cases contained in this class. The design of this quantum network exploits basic algebraic properties which also shed light onto these covariant quantum processes.

M. Stefanak, T. Kiss, I. Jex, and B. Mohring.
The meeting problem in the quantum walk.
JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL,
39(48):1496514983, DEC 1 2006.
[ bib ]
We study the motion of two noninteracting quantum particles performing a random walk on a line and analyse the probability that the two particles are detected at a particular position after a certain number of steps (meeting problem). The results are compared to the corresponding classical problem and differences are pointed out. Analytic formulae for the meeting probability and its asymptotic behaviour are derived. The decay of the meeting probability for distinguishable particles is faster than in the classical case, but not quadratically. Entangled initial states and the bosonic or fermionic nature of the walkers are considered.

T. Kiss, I. Jex, G. Alber, and S. Vymetal.
Complex chaos in the conditional dynamics of qubits.
PHYSICAL REVIEW A, 74(4), OCT 2006.
[ bib ]
We analyze the consequences of iterative measurementinduced nonlinearity on the dynamical behavior of qubits. We present a onequbit scheme where the equation governing the time evolution is a complexvalued nonlinear map with one complex parameter. In contrast to the usual notion of quantum chaos, exponential sensitivity to the initial state occurs here. We calculate analytically the Lyapunov exponent based on the overlap of quantum states, and find that it is positive. We present a few illustrative examples of the emerging dynamics.

Erika Andersson, Marcos Curty, and Igor Jex.
Experimentally realizable quantum comparison of coherent states and
its applications.
PHYSICAL REVIEW A, 74(2), AUG 2006.
[ bib ]
When comparing quantum states to each other, it is possible to obtain an unambiguous answer, indicating that the states are definitely different, already after a single measurement. In this paper we investigate comparison of coherent states, which is the simplest example of quantum state comparison for continuous variables. The method we present has a high success probability, and is experimentally feasible to realize as the only required components are beam splitters and photon detectors. An easily realizable method for quantum state comparison could be important for real applications. As examples of such applications we present a “lock and key” scheme and a simple scheme for quantum public key distribution.

J. Novotny, G. Alber, and I. Jex.
Completely positive covariant twoqubit quantum processes and
optimal quantum NOT operations for entangled qubit pairs.
PHYSICAL REVIEW A, 73(6), JUN 2006.
[ bib ]
The structure is investigated of all completely positive quantum operations that transform pure twoqubit input states of a given degree of entanglement in a covariant way. Special cases thereof are quantum NOT operations which transform entangled pure twoqubit input states of a given degree of entanglement into orthogonal states in an optimal way. Based on our general analysis all covariant optimal twoqubit quantum NOT operations are determined. In particular, it is demonstrated that only in the case of maximally entangled input states can these quantum NOT operations be performed perfectly.

J Novotny, M Stefanak, T Kiss, and I Jex.
Control of entanglement in Isingtype networks with one and two
excitations.
JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL,
38(41):90879103, OCT 14 2005.
[ bib ]
We analyse the dynamics of single and twoparticle states in Isingtype networks. The mutual entanglement is quantified using the concept of concurrence. We derive explicit expressions for the concurrence for single and twoparticle initial states in arbitrary passive networks and specify the result for Isingtype networks. We show how to design a network to prepare a prescribed pattern of entanglement for one excitation and study the maximum attainable entanglement for passive optical networks in general. The effect of network randomization on the average entanglement is also studied.

E Andersson, J Bergou, and I Jex.
Comparison of unitary transforms using Franson interferometry.
JOURNAL OF MODERN OPTICS, 52(10):14851494, JUL 10
2005.
[ bib ]
Unknown unitary transforms may be compared to each other in a way which makes it possible to obtain an unambiguous answer, indicating that the transforms are different, already after a single application of each transform. Quantum comparison strategies may be useful for example if we want to test the performance of individual gates in a quantum information or quantum computing network. It is then possible to check for errors by comparing the elements to a master copy of the gate, instead of performing a complete tomography of the gate. In this paper we propose a versatile linear optical implementation based on the Franson interferometer with short and long arms. A click in the wrong output port unambiguously determines that the tested gate is faulty. This setup can also be used for a variety of other tasks, such as confirming that the two transforms do not commute or do not anticommute.

J Novotny, G Alber, and I Jex.
Optimal copying of entangled twoqubit states.
PHYSICAL REVIEW A, 71(4), APR 2005.
[ bib ]
We investigate the problem of copying pure twoqubit states of a given degree of entanglement in an optimal way. Completely positive covariant quantum operations are constructed which maximize the fidelity of the output states with respect to two separable copies. These optimal copying processes hint at the intricate relationship between fundamental laws of quantum theory and entanglement.

A Chefles, E Andersson, and I Jex.
Unambiguous comparison of the states of multiple quantum systems.
JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL,
37(29):73157340, JUL 23 2004.
[ bib ]
We consider N quantum systems initially prepared in pure states and address the problem of unambiguously comparing them. One may ask whether or not all N systems are in the same state. Alternatively, one may ask whether or not the states of all N systems are different. We investigate the possibility of unambiguously obtaining this kind of information. It is found that some unambiguous comparison tasks are possible only when certain linear independence conditions are satisfied. We also obtain measurement strategies for certain comparison tasks which are optimal under a broad range of circumstances, in particular when the states are completely unknown. Such strategies, which we call universal comparison strategies, are found to have intriguing connections with the problem of quantifying the distinguishability of a set of quantum states and also with unresolved conjectures in linear algebra. We finally investigate a potential generalization of unambiguous state comparison, which we term unambiguous overlap filtering.

I Jex, E Andersson, and A Chefles.
Comparing the states of many quantum systems.
JOURNAL OF MODERN OPTICS, 51(4):505523, MAR 10
2004.
[ bib ]
We investigate how to determine whether the states of a set of quantum systems are identical or not. This paper treats both errorfree comparison, and comparison where errors in the result are allowed. Errorfree comparison means that we aim to obtain definite answers, which are known to be correct, as often as possible. In general, we will also have to accept inconclusive results, giving no information. To obtain a definite answer that the states of the systems are not identical is always possible, whereas in the situation considered here, a definite answer that they are identical will not be possible. The optimal universal errorfree comparison strategy is a projection onto the totally symmetric and the different nonsymmetric subspaces, invariant under permutations and unitary transformations. We also show how to construct optimal comparison strategies when allowing for some errors in the result, minimizing either the error probability, or the average cost of making an error. We point out that it is possible to realize universal errorfree comparison strategies using only linear elements and particle detectors, albeit with less than ideal efficiency. Also minimumerror and minimumcost strategies may sometimes be realized in this way. This is of great significance for practical applications of quantum comparison.

E Andersson, I Jex, and SM Barnett.
Comparison of unitary transforms.
JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL,
36(9):23252338, MAR 7 2003.
[ bib ]
We analyse the problem of comparing unitary transformations. The task is to decide, with minimal resources and maximal reliability, whether two given unitary transformations are identical or different. It is possible to make such. comparisons without obtaining any information about the individual transformations. Different comparison strategies are presented and compared with respect to their efficiency. With an interferometric setup, it is possible to compare two unitary transforms using only one test particle. Another strategy makes use of a twoparticle singlet state. This strategy is more efficient than using a nonentangled twoparticle test state, thus demonstrating the benefit of entanglement. Generalizations to higher.dimensional transforms and to more than two transformations are made.

SM Barnett, A Chefles, and I Jex.
Comparison of two unknown pure quantum states.
PHYSICS LETTERS A, 307(4):189195, FEB 3 2003.
[ bib ]
Can we establish whether or not two quantum systems have been prepared in the same state? We investigate the possibility of universal unambiguous state comparison. We show that it is impossible to conclusively identify two pure unknown states as being identical, and construct the optimal measurement for conclusively identifying them as being different. We then derive optimal strategies for state comparison when the state of each system is one of two known states. (C) 2002 Elsevier Science B.V. All rights reserved. Keywords: quantum mechanics; quantum information; quantum measurement 
I Jex, G Alber, SM Barnett, and A Delgado.
Antisymmetric multipartite quantum states and their applications.
FORTSCHRITTE DER PHYSIKPROGRESS OF PHYSICS,
51(23):172178, 2003.
9th Central European Workshop on Quantum Optics, SZEGED, HUNGARY,
MAY 0306, 2002.
[ bib ]
Entanglement is a powerful resource for processing quantum information. In this context pure, maximally entangled states have received considerable attention. In the case of bipartite qubitsystems the four orthonormal Bellstates are of this type. One of these Bell states, the singlet Bellstate, has the additional property of being antisymmetric with respect to particle exchange. In this contribution we discuss possible generalizations of this antisymmetric Bellstate to cases with more than two particles and with singleparticle Hilbert spaces involving more than two dimensions. We review basic properties of these totally antisymmetric states. Among possible applications of this class of states we analyze a new quantum key sharing protocol and methods for comparing quantum states.

S Meneghini, I Jex, WP Schleich, and VP Yakovlev.
Reshaping of atomic waves with twodimensional optical crystals.
JOURNAL OF OPTICS BQUANTUM AND SEMICLASSICAL OPTICS,
4(3):165171, JUN 2002.
[ bib ]
We study the dynamics of atomic waves in a twodimensional light crystal formed by two crossed standing laser fields. The longitudinal modulation of the crystal with the Doppler frequency significantly influences the transversal spatial modulation of the atomic wave. Near the doppleron resonance the atomic density shows a fractional space period. In this case a normally incident wave gives rise to an almost perfect conversion into the first momentum components and the light crystal acts as a highly efficient beamsplitter. The crossing angle, determining the Doppler frequency, is the easytocontrol parameter of the system. Keywords: atom optics; Bragg diffraction; population conversion 
P Torma, I Jex, and WP Schleich.
Localization and diffusion in Isingtype quantum networks.
PHYSICAL REVIEW A, 65(5, A), MAY 2002.
[ bib ]
We investigate the effect of phase randomness in Isingtype quantum networks. These networks model a large class of physical systems. They describe micro and nanostructures or arrays of optical elements such as beam splitters (interferometers) or parameteric amplifiers. Most of these stuctures are promising candidates for quantum information processing networks. We demonstrate that such systems exhibit two very distinct types of behavior. For certain network configurations (parameters), they show quantum localization similar to Anderson localization whereas classical stochastic behavior is observed in other cases. We relate these findings to the standard theory of quantum localization.

Gernot Alber, Aldo Delgado, and Igor Jex.
OPTIMAL TWOPARTICLE ENTANGLEMENT BY UNIVERSAL QUANTUM PROCESSES.
QUANTUM INFORMATION & COMPUTATION, 1(3):3351, NOV
2001.
[ bib ]
Within the class of all possible universal (covariant) twoparticle quantum processes in arbitrary dimensional Hilbert spaces those universal quantum processes are determined whose output states optimize the recently proposed entanglement measure of Vidal and Werner. It is demonstrated that these optimal entanglement processes belong to a oneparameter family of universal entanglement processes whose output states do not contain any separable components. It is shown that these optimal universal entanglement processes generate antisymmetric output states and, with the single exception of qubit systems, they preserve information about the initial input state. Keywords: Entanglement; universal quantum processes; quantum cloning 
G Alber, A Delgado, N Gisin, and I Jex.
Efficient bipartite quantum state purification in arbitrary
dimensional Hilbert spaces.
JOURNAL OF PHYSICS AMATHEMATICAL AND GENERAL,
34(42):88218833, OCT 26 2001.
[ bib ]
A new purification scheme is proposed which applies to arbitrary dimensional bipartite quantum systems. It is based on the repeated application of a special class of nonlinear quantum maps and a single, local unitary operation. This special class of nonlinear quantum maps is generated in a natural way by a Hermitian generalized XORgate. The proposed purification scheme offers two major advantages, namely it does not require local depolarization operations at each step of the purification procedure and it purifies more efficiently than other known purification schemes.

I Jex. Foreword. FORTSCHRITTE DER PHYSIKPROGRESS OF PHYSICS, 49(1011):899, 2001. [ bib ] 
A Bandilla, G Drobny, and I Jex.
Parametric downconversion and maximal pump depletion.
JOURNAL OF OPTICS BQUANTUM AND SEMICLASSICAL OPTICS,
2(3):265270, JUN 2000.
[ bib ]
Using the truncated Wigner function approach and the Bloembergen solutions for nondegenerate downconversion we calculate the conversion efficiency of spontaneous parametric downconversion. In addition we determine higher moments of the pump and signal photon number. We find the upper bound for the efficiency of energy transfer and give a physically intuitive explanation for its existence. The depletion of the pump mode is immediately characterized by a strong destruction of its coherence. Keywords: nonlinear optics; parametric downconversion; process efficiency 
I Jex and G Alber.
Rotational wavepacket dynamics of the C(60) molecule.
JOURNAL OF PHYSICS BATOMIC MOLECULAR AND OPTICAL PHYSICS,
33(8):16631674, APR 28 2000.
[ bib ]
The wavepacket dynamics of (nonstationary) rotational quantum states of the C(60) molecule are investigated. It is demonstrated that the icosahedral symmetry of this molecule gives rise to a variety of peculiar coherence phenomena which are characteristic for this specific discrete symmetry group. On the one hand, the dynamics of these wavepackets reflects the underlying classical dynamics on the associated manifold of group space. On the other hand, these dynamics also exhibit characteristic quantum features such as the appearance of fractional revivals, of quantum tunnelling between different classically accessible regions of the associated manifold of group space and of catlike quantum states which are generated in the course of the time evolution.

P Torma and I Jex.
Hamiltonians for Isingtype quantum networks.
JOURNAL OF MODERN OPTICS, 47(1):110, JAN 15 2000.
[ bib ]
We derive Hamiltonians for quantum networks of Isingtype structure. These networks can be composed of simple optical elements such as beam splitters or parameteric amplifiers, similar structures appear also in a variety of systems with many intersecting energy levels. We give exact forms for the Hamiltonians and discuss their implications in certain limits.

S Meneghini, I Jex, KAH van Leeuwen, MR Kasimov, WP Schleich, and VP Yakovlev.
Atomic motion in longitudinally modulated light crystals.
LASER PHYSICS, 10(1):116122, JANFEB 2000.
[ bib ]
We study the dynamics of atomic waves in longitudinally modulated light crystals using an effective complex potential with time (space) dependent detuning. In the weak coupling regime we predict a sharp asymmetry of the scattering probability near the Bragg angle. In the strong coupling case we introduce the concept of a light crystal with “complex kick.” We have found that the initially blue detuned crystal is much more transparent for atomic waves. Moreover, the probability distribution of the outgoing wave shows carpetlike interference structure.

A Bandilla, G Drobny, and I Jex. Energy transfer in parametric downconversion. In Kumar, P and DAriano, GM and Hirota, O, editor, QUANTUM COMMUNICATION, COMPUTING, AND MEASUREMENT 2, pages 521525, 233 SPRING ST, NEW YORK, NY 10013 USA, 2000. Tamagawa Univ; NW Univ; Natl Secur Agcy; USN Off Res, KLUWER ACADEMIC/PLENUM PUBL. 4th International Conference on Quantum Communication, Measurement, and Computing, NORTHWESTERN UNIV, EVANSTON, IL, AUG 2227, 1998. [ bib ] 
M Horne, I Jex, and A Zeilinger.
Schrodinger wave functions in strong periodic potentials with
applications to atom optics.
PHYSICAL REVIEW A, 59(3):21902202, MAR 1999.
[ bib ]
When an atom diffracts in intense standing light, the periodic potential can be too strong for known solutions of the Schrodinger equation. We present general solutions of Schrodinger's equation in strong sinusoidal media, thus generalizing dynamical diffraction theory. The solutions exhibit rich generalizations of the pendellosung phenomena. [S10502947(99)012032].

P Torma and I Jex.
Properties of Isingtype linear networks.
JOURNAL OF OPTICS BQUANTUM AND SEMICLASSICAL OPTICS,
1(1):813, FEB 1999.
[ bib ]
We study some of the mathematical properties of quantum networks with Isingtype nearestneighbour structure. Based on the formal properties the physical behaviour of the network, such as destructive and constructive interference, is discussed. Keywords: Ising model; linear quantum networks 
M Konopka and I Jex.
Exact solutions for nonlinear Hamiltonians.
CZECHOSLOVAK JOURNAL OF PHYSICS, 48(12):15371553,
DEC 1998.
[ bib ]
We find the eigenvalues and eigenvectors of two nonlinear Hamiltonians describing the interaction between a twolevel system and a quantized linear harmonic oscillator. In the first case we obtain exact isolated solutions for the Hamiltonian used as a model of an ion in a harmonic trap and interacting with a laser field, not restricted to the LambDicke limit. After projecting these eigenstates onto one of the levels of the twolevel system the oscillator state is described by a finite superposition of Fock states. In the second case me consider a Hamiltonian, with a squeeze operator in the interaction part. We give perturbation results in the weakcoupling limit and results obtained by numerical diagonalization for the strong coupling limit. Nonclassical results are pointed out also in this case.

A Bandilla, G Drobny, and I Jex.
The classical parametric approximation for threewave interactions.
OPTICS COMMUNICATIONS, 156(13):112122, NOV 1
1998.
[ bib ]
We study sufficient conditions for the applicability of the classical parametric approximation in threewave interactions when the pump intensity is very large compared to signal and idler intensity. To derive such conditions we express the exact classical solutions given by Jacobian elliptic functions in terms of hyperbolic functions. Thereby the first minimum of the pump intensity is correctly described but the periodicity is lost. We derive new approximations for the initial conditions using pump coordinate scaling and find the interval that defines complete pump depletion. We show that the classical parametric approximation with a fixed and sharp pump amplitude and phase can be used for an increasing fraction of this interval if the pump intensity is made to grow. By choosing higher and higher pump intensities the nonlinearity is shifted to the end of that interval. As an instructive example for the application of these findings the generation of twomode squeezing is briefly considered. (C) 1998 Elsevier Science B.V. All rights reserved.

A Bandilla, G Drobny, and I Jex.
Quantum description of nonlinearly coupled oscillators via classical
trajectories.
In Kasperkovitz, P and Grau, D, editor, 5TH WIGNER SYMPOSIUM,
PROCEEDINGS, pages 389391, PO BOX 128 FARRER RD, SINGAPORE 9128,
SINGAPORE, 1998. WORLD SCIENTIFIC PUBL CO PTE LTD.
5th Wigner Symposium, VIENNA, AUSTRIA, AUG 2529, 1997.
[ bib ]
We discuss the usefulness of the Wigner function description via classical trajectories in the example of the threewave interaction with a strong coherent pump.

I Tittonen, S Stenholm, and I Jex.
Stepwise phase modulation of atoms coupled to a quasicontinuum of
states in a cavity.
JOURNAL OF MODERN OPTICS, 45(1):2333, JAN 1998.
[ bib ]
We study the interaction of a small number of resonant twolevel atoms in a cavity coupled to a quasicontinuum of cavity modes. During the interaction we induce externally a phase step to one of the atoms. As a result an enhancement in the photon emission to the field is observed. The phase step is obtained by a pulse which causes the atom to be offresonance for a short lime interval. The effect is strongest in cases where the number of atoms is small. The enhancement is not restricted to a small parameter range but should be observable in many systems even though the pi phase shift gives the best enhancement.

H Paul, P Torma, T Kiss, and I Jex.
Twomode state reconstruction using photon chopping.
JOURNAL OF MODERN OPTICS, 44(1112):23952404,
NOVDEC 1997.
[ bib ]
The reconstruction of the complete quantum state of a purestate twomode field is shown to be possible by the photonchopping method, that is, by spreading the signal with balanced multiports over an array of detectors and by measuring coincidences. In particular, correlations between the two modes can be detected with high efficiency.

H Paul, P Torma, T Kiss, and I Jex.
Multiple coincidences and the quantum state reconstruction problem.
PHYSICAL REVIEW A, 56(5):40764085, NOV 1997.
[ bib ]
We analyze the recently proposed method [H. Paul et al., Phys. Rev. Lett. 76, 2464 (1996)] for reconstructing the quantum state of a light field from multiple coincidences measured at the outputs of a passive multiport. We show that applying a large multiport the reconstruction of a pure state becomes possible using avalanche photodiodetype detectors. The presented simulations show that the photon chopping scheme is appropriate for the indirect measurement of the photon statistics of a weak nonclassical signal. [S10502947(97)075070].

P Torma and I Jex.
Twomode entanglement in passive networks.
JOURNAL OF MODERN OPTICS, 44(5):875882, MAY 1997.
[ bib ]
We discuss the degree of entanglement for symmetric passive optical networks. As a measure of the entanglement we use the information entropy and compare our results for a general number N of inputs with the known results for a twomode beam splitter. Special cases of interest are pointed out.

G Drobny, A Bandilla, and I Jex.
Quantum description of nonlinearly interacting oscillators via
classical trajectories.
PHYSICAL REVIEW A, 55(1):7893, JAN 1997.
[ bib ]
We investigate systems of few harmonic oscillators with mutual nonlinear coupling. Using classical trajectoriesthe solutions of Hamiltonian equations of motion for a given nonlinear systemwe construct the approximate quasiprobability distribution function in phase space that enables a quantum description. The nonclassical effects (quantum noise reduction) and their scaling laws can be so studied for high excitation numbers. In particular, the harmonic oscillators represent modes of the electromagnetic field and the Hamiltonians under consideration describe representative nonlinear optical processes (multiwave mixings). The range of the validity of the approximation for Wigner and Husimi functions evolved within the classical Liouville equation is discussed for a diverse class of initial conditions, including those without classical counterparts, e.g., Fock states.

P Torma and I Jex.
Plate beam splitters and symmetric multiports.
JOURNAL OF MODERN OPTICS, 43(11):24032408, NOV
1996.
[ bib ]
We show that for experimental arrangements with few nonvacuum inputs the fully symmetric multiport can be replaced by a simpler partially symmetric multiport. This device is equivalent to a beam splitter arrangement where the number of necessary beam splitters increases linearly with increasing number of inputs, that is it is considerably simpler to realize owning to the smaller number of components needed.

A Bandilla, G Drobny, and I Jex.
Phasespace motion in parametric threewave interaction.
OPTICS COMMUNICATIONS, 128(46):353362, JUL 15
1996.
[ bib ]
We study the interaction of three modes in phi((2)) media in the classical as well as in the quantum picture. We show that it is possible to distinguish two basic forms of phasespace motion depending on the value of the integral of motion Gamma in the classical formulation or the (initial) mean value of the interaction Hamiltonian in the quantum formulation. The quantum and classical pictures are compared using a simulation of the Husimi Q function via classical trajectories for an ensemble of phasespace points. The very good correspondence between the two pictures is shown to last for a significant interaction time and can be investigated for arbitrary initial intensities.

H Paul, P Torma, T Kiss, and I Jex.
Photon chopping: New way to measure the quantum state of light.
PHYSICAL REVIEW LETTERS, 76(14):24642467, APR 1
1996.
[ bib ]
We propose the use of a balanced 2Nport as a technique to measure the pure quantum state of a singlemode light field. In our scheme the coincidence signals of simple, realistic photodetectors are recorded at the output of the 2Nport. We show that applying different arrangements both the modulus and the phase of the coefficients in a finite superposition state can be measured. In particular, the photon statistics can be so measured with currently available devices.

I Tittonen, S Stenholm, and I Jex.
Effect of a phase step on twolevel atoms in a cavity.
OPTICS COMMUNICATIONS, 124(34):271276, MAR 1
1996.
[ bib ]
We analyze the dynamics of an excited twolevel atom in the presence of other deexcited atoms in a cavity. We show, that due to an instantaneous phase shift experienced by one of the atoms, the probability for emitting a photon into the cavity can be increased. In the special case of only two atoms in the cavity we show that the system with certainty can release a photon that is otherwise partially trapped.

P Torma, I Jex, and S Stenholm.
Beam splitter realizations of totally symmetric mode couplers.
JOURNAL OF MODERN OPTICS, 43(2):245251, FEB 1996.
[ bib ]
We discuss the symmetric multiport and show in a constructive way how it can be decomposed into a set of beam splitters. Based on the decomposition we estimate the number of beam splitters needed. It is shown that for more than three input ports the number of beam splitters needed is less than the number indicated by the standard decomposition scheme.

P Torma, S Stenholm, and I Jex. Quantum theory of optical networks. In Eberly, JH and Mandel, L and Wolf, E, editor, COHERENCE AND QUANTUM OPTICS VII, pages 415416, 233 SPRING ST, NEW YORK, NY 10013, 1996. Amer Phys Soc; Opt Soc Amer; Int Union Pure & Appl Phys; Univ Rochester, PLENUM PRESS DIV PLENUM PUBLISHING CORP. 7th Rochester Conference on Coherence and Quantum Optics, UNIV ROCHESTER, ROCHESTER, NY, JUN 0710, 1995. [ bib ] 
A Bandilla, G Drobny, and I Jex. Threewave mixing with entangled and disentangled states. In Eberly, JH and Mandel, L and Wolf, E, editor, COHERENCE AND QUANTUM OPTICS VII, pages 429430, 233 SPRING ST, NEW YORK, NY 10013, 1996. Amer Phys Soc; Opt Soc Amer; Int Union Pure & Appl Phys; Univ Rochester, PLENUM PRESS DIV PLENUM PUBLISHING CORP. 7th Rochester Conference on Coherence and Quantum Optics, UNIV ROCHESTER, ROCHESTER, NY, JUN 0710, 1995. [ bib ] 
U Leonhardt and I Jex. Quantumstate tomography and quantum communication. In Eberly, JH and Mandel, L and Wolf, E, editor, COHERENCE AND QUANTUM OPTICS VII, pages 675676, 233 SPRING ST, NEW YORK, NY 10013, 1996. Amer Phys Soc; Opt Soc Amer; Int Union Pure & Appl Phys; Univ Rochester, PLENUM PRESS DIV PLENUM PUBLISHING CORP. 7th Rochester Conference on Coherence and Quantum Optics, UNIV ROCHESTER, ROCHESTER, NY, JUN 0710, 1995. [ bib ] 
A Bandilla, G Drobny, and I Jex.
Nondegenerate parametric interactions and nonclassical effects.
PHYSICAL REVIEW A, 53(1):507516, JAN 1996.
[ bib ]
We consider the classical and quantummechanical processes of threewave interactions in different phase regimes and present numerical calculations for the quantum case, where all three modes are sizably excited from the beginning. These excitations are coherent so that various important phase regimes can be adjusted. In addition, one mode can also be prepared in a squeezed or Kerr state. The classical solutions are well known and are briefly summarized, but certain phase regimes are classically unexplored and we show here that they give interesting and surprising results. In the outofphase regime (where the photon numbers do not change in the first order of time) we get, with an initial Kerr state, strongly subPoissonian photon statistics in the signal after a short interaction time. This effect is limited by the classically described phase shifts that are present even in the parametric approximation. This nonclassical phenomenon (due to the Kerr state) helps us to understand similar nonclassical effects generated by entangled states of the pump and signal during sumfrequency generation.

P TORMA, S STENHOLM, and I JEX.
MEASUREMENT AND PREPARATION USING 2 PROBE MODES.
PHYSICAL REVIEW A, 52(6):48124822, DEC 1995.
[ bib ]
We consider the simultaneous measurement of two conjugate variables by coupling the system of interest to two independent probe modes. Our model consists of linearly coupled boson modes that can be realized by quantum optical fields in the rotatingwave approximation. We approach the setup both as a device to extract observable information and to prepare an emerging quantum state. The initial states of the probe modes and the coupling are utilized to optimize the operation in the various regimes. In contrast to the Arthurs and Kelly ideal scheme [Bell. Syst. Tech. J. 44, 725 (1965)], our more realistic coupling does not allow perfect operation but the ideal situations can be approximated closely. We discuss the conditions for maximum information transfer to the probe modes, information extraction with minimum disturbance of the system mode, and optimal state preparation for subsequent measurements. The minimum disturbance operation can be made to approximate a nondemolition measurement, especially when the information is carried in one quadrature component only. In the preparation mode, we find that the recording accuracy of the probe signals plays an essential role. We restrict the discussion to the first and second moments only, but the method can easily be generalized to any situation, Choosing all modes to be in squeezed coherent states originally, we can carry out analytical considerations; other cases can be treated numerically. The results are presented and discussed in detail as the paradigm of a class of realizable measurements.

P TORMA, S STENHOLM, and I JEX.
HAMILTONIAN THEORY OF SYMMETRICAL OPTICAL NETWORK TRANSFORMS.
PHYSICAL REVIEW A, 52(6):48534860, DEC 1995.
[ bib ]
We discuss the theory of extracting an interaction Hamiltonian from a preassigned unitary transformation of quantum states. Such a procedure is of significance in quantum computations and other optical information processing tasks. We particularize the problem to the construction of totally symmetric 2N peas as introduced by Zeilinger and his collaborators [A. Zeilinger, M. Zukowski, M. A. Home, H. J. Bernstein, and D. M. Greenberger, in Fundamental Aspects of Quantum Theory, edited by J. Anandan and J. J. Safko (World Scientific, Singapore, 1994)]. These are realized by the discrete Fourier transform,which simplifies the construction of the Hamiltonian by known methods of Linear algebra. The Hamiltonians found are discussed and alternative realizations of the Zeilinger class transformations are presented. We briefly discuss the applicability of the method to more general devices.

A BANDILLA, G DROBNY, and I JEX.
SUBPOISSONIAN PHOTON STATISTICS IN A 3WAVE INTERACTION STARTING IN
THE OUTOFPHASE REGIME.
PHYSICAL REVIEW LETTERS, 75(22):40194022, NOV 27
1995.
[ bib ]
We present for the first time numerical and analytical calculations for the nonlinear interaction of three quantized waves all sizably excited from the beginning and having different phase relations. With a Kerrstate ansatz for the signal we get strongly subPoissonian photon statistics and conclude on similar effects by initially entangled states.

I JEX, P TORMA, and S STENHOLM.
MULTIMODE COHERENT STATES.
JOURNAL OF MODERN OPTICS, 42(7):13771386, JUL
1995.
[ bib ]
We define multimode (entangled) coherent states as properly chosen linear superpositions of suitable composition states. Our definition is equivalent to the definition of coherent states as eigenvectors of a corresponding generalized annihilation operator. In certain limit cases we discuss the statistical properties of the states defined.

I JEX, S STENHOLM, and A ZEILINGER.
HAMILTONIAN THEORY OF A SYMMETRICAL MULTIPORT.
OPTICS COMMUNICATIONS, 117(12):95101, MAY 15
1995.
[ bib ]
We analyze a simple Hamiltonian model for a passive multiport. We show that such a model can be made symmetric with respect to the outputs for certain (interaction) times up to a certain number of inputoutput ports. The limit of infinitely many inputoutput channels is discussed.

A Bandilla, G Drobny, and I Jex. Threewave mixing with entangled and disentangled states. In Betak, E, editor, ACTA PHYSICA SLOVACA, VOL 45, NO 3, JUNE 1995, pages 335340, DUBRAVSKA CESTA 9, 842 28 BRATISLAVA, SLOVAKIA, 1995. SLOVAK ACADEMY SCIENCES INSTITUTE PHYSICS. 3rd CentralEuropean Workshop on Quantum Optics, BUDMERICE CASTLE, SLOVAKIA, APR 28MAY 01, 1995. [ bib ] 
G Drobny and I Jex. Field state manipulation using a multiatomic system. In Betak, E, editor, ACTA PHYSICA SLOVACA, VOL 45, NO 3, JUNE 1995, pages 347351, DUBRAVSKA CESTA 9, 842 28 BRATISLAVA, SLOVAKIA, 1995. SLOVAK ACADEMY SCIENCES INSTITUTE PHYSICS. 3rd CentralEuropean Workshop on Quantum Optics, BUDMERICE CASTLE, SLOVAKIA, APR 28MAY 01, 1995. [ bib ] 
I JEX and A ORLOWSKI.
WEHRLS ENTROPY DYNAMICS IN A KERRLIKE MEDIUM.
JOURNAL OF MODERN OPTICS, 41(12):23012306, DEC
1994.
[ bib ]
We report on the time evolution of the Wehrl entropy in a Kerrlike medium. We show that the Wehrl entropy gives a clear signature for the formation of finite superpositions of coherent states (catlike states). In addition, the actual value of the Wehrl entropy at the time of a superposition formation gives the number of coherent components taking part in the superposition.

G DROBNY and I JEX.
NONDEGENERATE 2PHOTON DOWNCONVERSION  COHERENT INPUTS AND
NONCLASSICAL EFFECTS.
CZECHOSLOVAK JOURNAL OF PHYSICS, 44(9):827842, SEP
1994.
[ bib ]
We study the nondegenerate twophoton down conversion described by a quantum trilinear Hamiltonian. The idler mode is initially prepared in the vacuum while the pump (laser) and the signal mode are prepared in coherent states which at high intensities resemble classical inputs. Such setup with a coherent signal mode allows us to scan the dynamics from the regime of the down conversion (empty signal) up to the frequency conversion (highly excited signal). The analysis concentrates on the entanglement properties of the modes which are compared with their other statistical properties such as squeezing and antibunching to give a more complete characterization of the modes. We show that the single mode nonclassical effects (squeezing and antibunching) disappear when an initial signal intensity highly exceeds that of the pump. In this regime the numerical results are confirmed by approximate analytical calculations. We point out that initially comparable intensities of the signal and pump mode lead to the effect of the `'spontaneous disentanglement” of the signal mode from others and to the production of its squeezed and subPoissonian state which is pure to a good approximation.

U LEONHARDT and I JEX.
WIGNER FUNCTIONS AND QUADRATURE DISTRIBUTIONS FOR QUANTUMOSCILLATOR
STATES WITH RANDOMPHASE.
PHYSICAL REVIEW A, 49(3):R1555R1557, MAR 1994.
[ bib ]
Applying a result of Vogel and Risken [Phys. Rev. A 40, 2847 (1989)] to quantumoscillator states with random phase, we found surprisingly simple integral relations between the Wiper function and the quadrature distribution. In particular, we have shown that the balance of increasing and decreasing sections of the quadrature distribution decides the sign of the Wiper function.

G DROBNY, T GANTSOG, and I JEX.
PHASE PROPERTIES OF A FIELD MODE INTERACTING WITH N 2LEVEL ATOMS.
PHYSICAL REVIEW A, 49(1):622625, JAN 1994.
[ bib ]
We analyze the phase properties of a strong cavity field interacting with an ensemble of initially excited N twolevel atoms. Using the PeggBarnett phase formalism [Phys. Rev. A 39, 1665 (1989)], we calculate the phase probability distribution as well as the phase variance. The phase probability density exhibits a (N + 1)peak structure at the initial stages of the evolution. The phase variance is used to illustrate the progressive randomization of the phase on the longtime evolution. The difference in the phase dynamics for the N even and the N odd case is pointed out.

I JEX and H PAUL.
LONGTIME BEHAVIOR OF THE FIELD IN A LOSSLESS MICROMASER WITH
INTENSITYDEPENDENT COUPLINGCONSTANT.
JOURNAL OF MODERN OPTICS, 40(12):23992406, DEC
1993.
[ bib ]
Using the asymptotic form of the density matrix the role of the nondiagonal elements of the density matrix is estimated in the lossless micromaser model based on the JaynesCummings model with intensity dependent coupling constant. It is shown that a certain type of a (almost pure) multiphoton state can be generated in this system.

I JEX, M MATSUOKA, and M KOASHI.
PHASE OF THE FIELD IN THE INTERACTION WITH 2 2LEVEL ATOMS.
QUANTUM OPTICS, 5(5):275286, OCT 1993.
[ bib ]
We present phase properties of a field mode interacting with two twolevel atoms using the Pegg and Barnett formalism. We derive analytical and approximate expressions for the phase probability distribution, and calculate the mean value of the phase and their fluctuations. A discussion with respect to the coupling constant of the atoms is given.

G DROBNY and I JEX.
THE SYSTEM OF N 2LEVEL ATOMS INTERACTING WITH A FIELD MODE 
ENTANGLEMENT AND PARAMETRIC APPROXIMATION.
OPTICS COMMUNICATIONS, 102(12):141154, SEP 15
1993.
[ bib ]
We study the entanglement of the cavity field mode to the ensemble of twolevel atoms when their interaction is ruled by the Dicke model. Our investigation is focused mainly on initial states with fully excited atomic system and an intense cavity field mode. We give an analysis of the entanglement on the short as well as the long time scale. We derive limit expressions for the entanglement for the first moments of the evolution where the entanglement reaches constant value. We show that in the early moments of the evolution an almost pure state can be generated under suitable conditions when the intensity of the field is approximately N. On the long time scale we point out the appearance of a clear decrease of the entanglement associated with the collapserevival phenomenon of the mean photon number in this model. The bifurcation of initially a gaussian Qfunction of the field mode into N + 1 peaks of different weight factors is shown. Finally the results known for the JaynesCummings model are compared with the obtained results.

I JEX.
REVIVAL TIMEESTIMATION IN THE PROCESS OF DOWNCONVERSION.
JOURNAL OF MODERN OPTICS, 40(9):17531761, SEP
1993.
[ bib ]
We analyse the collapserevival phenomenon in the process of kphoton down conversion with quantized pump for initial Fock states. We give an explanation for the appearance of the revivals by analysing the decomposition of the initial state into the interaction Hamiltonian eigenvectors. A quantitative estimation of the revival times is given as well.

G DROBNY and I JEX.
2MODE SU(2) COHERENT STATES FOR KAPPAPHOTON DOWNCONVERSION
PROCESS WITH QUANTIZED PUMP.
CZECHOSLOVAK JOURNAL OF PHYSICS, 43(8):797806, AUG
1993.
[ bib ]
We analyze the statistical properties of the twomode SU(2) coherent states associated with the process of kphoton down conversion with quantized pump. We show that the modes exhibit subPoissonian photon statistics, anticorrelation and in some particular cases also squeezing. The influence of various initial number states on this effects is analyzed in detail.

G DROBNY, I JEX, and V BUZEK.
MODE ENTANGLEMENT IN NONDEGENERATE DOWNCONVERSION WITH QUANTIZED
PUMP.
PHYSICAL REVIEW A, 48(1):569579, JUL 1993.
[ bib ]
We study entanglement between field modes in the process of nondegenerate twophoton downconversion with quantized pump. We show that due to the quantum nature of the dynamics, strong entanglement between the pump and the signalidler subsystems can be observed. We find that the higher the initial intensity of the pump mode the stronger the entanglement between the pump and the signalidler subsystem is established during the first instants of the time evolution. We also show that the signal and the idler modes are strongly entangled (correlated). This entanglement is much stronger than the entanglement between the pump and the signalidler subsystem. Correlation between the signal and the idler modes leads to a high degree of twomode squeezing, which can be observed during the first instants of the time evolution when the pump mode is still approximately in a pure state. On the other hand, the back action of the signalidler subsystem on the pump mode leads to a strong singlemode squeezing of the pump mode. At the time interval during which squeezing of the pump mode can be observed the pump mode is far from being in the minimum uncertainty state. We also analyze the longtime behavior of the quantumoptical system under consideration and we show that the interesting collapserevival effect in the time evolution of the mean photon number and of the purity parameters of field modes can be observed. Finally, we show that the degree of entanglement between modes in the nondegenerate quantumoptical downconversion strongly depends on the initial state of the system.

I JEX and V BUZEK.
MULTIPHOTON COHERENT STATES AND THE LINEAR SUPERPOSITION PRINCIPLE.
JOURNAL OF MODERN OPTICS, 40(5):771783, MAY 1993.
[ bib ]
We show that multiphoton coherent states can be expressed as linear quantum superpositions of a finite number of generalized coherent states. Quantum interferences between component states lead to appearance of nonclassical effects such as oscillations in the photon number distribution, quadrature squeezing, SU(1, 1) squeezing and others.

I JEX and G DROBNY.
PHASE PROPERTIES AND ENTANGLEMENT OF THE FIELD MODES IN A 2MODE
COUPLER WITH INTENSITYDEPENDENT COUPLING.
PHYSICAL REVIEW A, 47(4, B):32513258, APR 1993.
[ bib ]
We study the phase properties of two quantized modes within the model of a twomode coupler with intensitydependent coupling. The dynamics of the system is solved numerically as an approximate analytical solution is available only for the initial moments of the time evolution. The phase properties are discussed on the basis of the joint phase distribution. Other quantities of the particular modes (measure of entropy and photonnumber distribution) are presented to support the conclusions reached on the basis of the joint phase distribution. The obtained results are compared with the known results for the closely related model of an ordinary linear coupler.

V BUZEK, G DROBNY, and I JEX. SIGNALPUMP ENTANGLEMENT IN QUANTUMOPTICAL PROCESSES. In Akos, G and Lippenyi, T and Lupkovics, G and Podmaniczky, A, editor, 16TH CONGRESS OF THE INTERNATIONAL COMMISSION FOR OPTICS : OPTICS AS A KEY TO HIGH TECHNOLOGY, PTS 1 AND 2, volume 1983 of PROCEEDINGS OF THE SOCIETY OF PHOTOOPTICAL INSTRUMENTATION ENGINEERS (SPIE), pages 9091, BELLINGHAM, 1993. INT COMMISS OPT; SCI SOC OPT ACOUST & FILMTECH; SOC PHOTO OPT INSTRUMENTAT ENGINEERS; OPT SOC AMER; EUROPEAN OPT SOC; HUNGARIAN NATL COMM TECHNOL DEV; INT UNION PURE & APPL PHYS; HUNGARIAN OPT WORKS; LASRAM LASER TECHNOL, SPIE  INT SOC OPTICAL ENGINEERING. 16TH CONGRESS OF THE INTERNATIONALCOMMISSIONFOROPTICS : OPTICS AS A KEY TO HIGH TECHNOLOGY ( ICO16 ), BUDAPEST, HUNGARY, AUG 0913, 1993. [ bib ] 
I JEX, G DROBNY, and M MATSUOKA.
QUANTUM PHASE PROPERTIES OF THE PROCESS OF KAPPAPHOTON DOWN
CONVERSION WITH QUANTIZED PUMP.
OPTICS COMMUNICATIONS, 94(6):619627, DEC 15 1992.
[ bib ]
We study the phase properties of the field modes in the process of kphoton down conversion with quantized pump using the PeggyBarnett phase formalism. The dynamics of the system is solved numerically via the diagonalization of the interaction hamiltonian. The behavior in the short and long time limit is discussed and compared to previous results related to the treated problem. In order to understand our results an exactly soluble model of the linearized kphoton down conversion is presented.

G DROBNY, I JEX, and M MATSUOKA.
COLLAPSE REVIVAL PHENOMENON IN THE PROCESS OF KPHOTON DOWN
CONVERSION WITH SUPERPOSITIONS OF NUMBER STATES.
JOURNAL OF MODERN OPTICS, 39(11):22292243, NOV
1992.
[ bib ]
We calculate numerically the time evolution of the mean photon number in the process of kphoton down conversion process with quantized pump. The pump mode was supposed to be initially in a superposition of number states and the down converted mode in a number state. We analysed in some detail the influence of the initial field statistics of the pump mode as well as the presence of nonvacuum number states in the down converted mode on the appearance of collapses and revivals.

G DROBNY and I JEX.
PHASE PROPERTIES OF FIELD MODES IN THE PROCESS OF KTH
HARMONICGENERATION.
PHYSICS LETTERS A, 169(4):273280, SEP 28 1992.
[ bib ]
We study the phase properties of the two field modes within the frame of the process of kth harmonic generation with a quantized fundamental mode. Using the PeggBarnett phase formalism the joint phase distribution is calculated numerically. It is shown that in the initial moments of the evolution the degree of the process k does not affect significantly the phase distribution.

G DROBNY and I JEX.
QUANTUMPROPERTIES OF FIELD MODES IN TRILINEAR OPTICAL PROCESSES.
PHYSICAL REVIEW A, 46(1):499506, JUL 1 1992.
[ bib ]
We consider a trilinear Hamiltonian in boson operators describing various physical processes such as frequency conversion, Raman or Brillouin scattering, or the interaction of N twolevel atoms with a singlemode radiation field. Due to the fact that two independent integrals of motion can be found, the solution of the dynamics of the system is reduced to the diagonalization of a finite matrix (as was already shown by Walls and Barakat [Phys. Rev. A 1, 446 (1970)]). Performing a numerical diagonalization, we analyze the statistical properties of the field modes (subPoissonian statistics, anticorrelation, squeezing). We also pay attention to the appearance of collapses and revivals in the mean photon number of the modes. The relation of this model to the model of two coupled modes with an intensitydependent coupling constant is pointed out.

I JEX and G DROBNY.
SU(1,1) AND SU(2) SQUEEZING IN TRILINEAR OPTICAL PROCESSES.
JOURNAL OF MODERN OPTICS, 39(5):10431053, MAY
1992.
[ bib ]
The squeezing properties in terms of SU(1, 1) and SU(2) operators for the case of trilinear processes are studied. The initial state of the system is supposed to be a coherent state in one of the modes and number states in the remaining modes. It is pointed out that in several cases a considerable amount of squeezing can be achieved. Due to the common mathematical structure the case of a twomode coupler with intensity dependent coupling is also analysed.

I JEX.
EMISSIONSPECTRA OF A 2LEVEL ATOM UNDER THE PRESENCE OF ANOTHER
2LEVEL ATOM.
JOURNAL OF MODERN OPTICS, 39(4):835844, APR 1992.
[ bib ]
We investigate the spectrum of light emitted by a twolevel atom interacting with another twolevel atom inside an ideal cavity within the frame of generalized JaynesCummings model. The influence of various ratios of the coupling constants of the atoms to the field on the spectrum of the emitted light is studied in detail for the case when the atoms are supposed to be initially in the excited state and the field in a Fock state as well as their superposition.

G DROBNY and I JEX.
STATISTICS OF FIELD MODES IN THE PROCESS OF KAPPAPHOTON
DOWNCONVERSION WITH A QUANTIZED PUMP.
PHYSICAL REVIEW A, 45(7, B):48974903, APR 1 1992.
[ bib ]
We solve numerically the dynamics of a system related to the problem of kphoton downconversion with a quantized pump. We analyze in detail the statistical properties of the field modes. We show that the fields exhibit subPoissonian statistics and anticorrelation. The influence of different initial Fock states as well as their coherent superpositions on the field statistics and squeezing of the modes is analyzed in detail.

G DROBNY and I JEX.
COLLAPSES AND REVIVALS IN THE ENERGY EXCHANGE IN THE PROCESS OF
KAPPAPHOTON DOWNCONVERSION WITH QUANTIZED PUMP.
PHYSICAL REVIEW A, 45(3):18161821, FEB 1 1992.
[ bib ]
We analyze the collapse and revival phenomenon in the energy exchange of two field modes initially prepared in a Fock state, as well as their coherent superposition in the process of kphoton downconversion with quantized pump. The influence of the presence of a Kerrlike medium is discussed.

V BERNAT and I JEX.
QUANTUM COLLAPSE AND REVIVAL OF A 2LEVEL ATOM WITH
INTENSITYDEPENDENT DETUNING IN A FINITE QCAVITY AT FINITE TEMPERATURE.
QUANTUM OPTICS, 4(1):917, FEB 1992.
[ bib ]
We consider a generalization of the JaynesCummings model when the cavity is supposed to be filled with a Kerrlike medium. This nonlinearity induced by the Kerrlike medium leads to an inhibited decay of the initially excited atom. In the present paper we analyse the influence of the quality of the cavity and the influence of a thermofield on the collapses and revivals of the atomic inversion.

V BUZEK and I JEX.
EMISSIONSPECTRA OF A 2LEVEL ATOM IN A KERRLIKE MEDIUM.
JOURNAL OF MODERN OPTICS, 38(5):987996, MAY 1991.
[ bib ]
We have investigated the spectrum of light emitted by a single atom interacting with a single mode of the radiation field in an ideal cavity filled with a Kerrlike medium. It is shown that owing to the Kerrlike nonlinearity in the system the spectrum of the emitted light exhibits a singlepeaked structure for sufficiently high intensities of the initial coherent field instead of the triplet structure in the case of the standard JaynesCummings model.

V BUZEK, I JEX, and M BRISUDOVA.
JAYNESCUMMINGS MODEL WITH DISPLACED NUMBER STATES.
INTERNATIONAL JOURNAL OF MODERN PHYSICS B,
5(5):797814, MAR 1991.
[ bib ]
We investigate the dynamics of the JaynesCummings model with the cavity field initially prepared in the displaced number state. The time evolution of the atomic population inversion, squeezing of the cavity field and the emission spectra from the twolevel atom are studied.

V BUZEK and I JEX.
MULTIPHOTON STATES AND AMPLITUDE KTH POWER SQUEEZING.
NUOVO CIMENTO DELLA SOCIETA ITALIANA DI FISICA BGENERAL
PHYSICS RELATIVITY ASTRONOMY AND MATHEMATICAL PHYSICS AND METHODS,
106(2):147157, FEB 1991.
[ bib ]
On the basis of the work of d'Ariano and coworkers we introduce a new type of multiphoton states. We analyse amplitude kth power squeezing of the multiphoton states. In particular, we show that even if the multiphoton states do not exhibit ordinary squeezing they can be amplitude kth power squeezed. Keywords: QUANTUM THEORY; QUANTUM MECHANICS; NONCLASSICAL PHOTON STATES (INCLUDING ANTIBUNCHED; SQUEEZED; SUBPOISSONIAN) 
V BUZEK and I JEX. DYNAMICS OF A 2LEVEL ATOM IN A KERRLIKE MEDIUM. OPTICS COMMUNICATIONS, 78(56):425435, SEP 15 1990. [ bib ] 
V BUZEK and I JEX. AMPLITUDE KAPPATHPOWER SQUEEZING OF KAPPAPHOTON COHERENT STATES. PHYSICAL REVIEW A, 41(7):40794082, APR 1 1990. [ bib ] 
V BUZEK, I JEX, and T QUANG. KPHOTON COHERENT STATES. JOURNAL OF MODERN OPTICS, 37(2):159163, FEB 1990. [ bib ] 
V BUZEK and I JEX. COLLAPSE REVIVAL PHENOMENON IN THE JAYNESCUMMINGS MODEL INTERACTING WITH THE MULTIPHOTON HOLSTEINPRIMAKOFF SU(2) COHERENT STATE. JOURNAL OF MODERN OPTICS, 36(11):14271434, NOV 1989. [ bib ] 
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