The group deals with elementary quantum systems, bigger networks constructed from these, their quantum informatical applications, and the description of the concrete physical configurations. One of their much examined models is the quantum walk, which, beside the possible quantum search algorithms, is a good model for such rudimentary quantummechanical phenomena like the formation of topologically protected phases. Among their research topics are investigation of vibrational properties of simple molecules, taking part in the experiments for creating non-classical light, and creating quantum states of light, and quantum measurement theories.

On 1. January, 2021, the Complex Systems research group joined this group, forming the Quatum informatics and complex systems research group. This part of the group deals with the cooperative behavior, ordering, equilibrium and far from equilibrium dynamics of interacting systems with many degrees of freedom. Our main research topics are the following.

**1. Disordered quantum magnets**

- application and development of the strong disorder renormalization group method
- critical behavior of long-range interacting systems
- ground state entanglement in inhomogeneous systems

**2. Ground state of interacting classical and quantum systems **

- periodic ordering at finite temperatures
- consequences of Galilei invariance on the ground state superfluidity and ordering

**3. Nonequilibrium relaxation of quantum systems **

- quench dynamics in closed systems
- dynamics of open systems in a fluctuating field

**4.** **Quantum Hall effect **

- study of anomalous quantum Hall effect and topologically protected edge states

**5. Dynamics in nonequilibrium models and networks **

- dynamics and phase transitions of stochastic many-particle systems (lattice gases, models of population dynamics)
- dynamical processes on complex networks

Kiss Tamás

Ádám Péter

Asbóth János Károly

Bodor András

Diósi Lajos

Gábris Aurél Gábor

Gombkötő Ákos

Hanyecz Ottó

Homa Gábor

Iglói Ferenc

Juhász Róbert

Kálmán Orsolya

Kis Zsolt

Koniorczyk Mátyás

Mechler Mátyás Illés

Naszvadi Péter

Ortega Rosales Hugo Adrian

Pitrik József

Portik Attila

Roósz Gergő

Rozgonyi Áron

Tóth Géza

Trényi Róbert

Varró Sándor

**Topological delocalization in the completely disordered two-dimensional quantum walk.** — In low dimensional quantum systems, spatial disorder generically leads to Anderson localization, inhibiting spatial spread of wavefunctions. This also occurs in two-dimensional quantum walks, versatile toy models for periodically driven quantum systems. We have found that increasing spatial disorder to the maximum possible value (Haar random coin operators), leads to a delocalization: a diffusive spread of an initially localized wavepacket instead of Anderson localizaiton. This delocalization happens because maximal disorder places the quantum walk to a critical point between different anomalous Floquet-Anderson insulating topological phases. We have previously observed topological delocalization in a simpler quantum walk, using time-evolution of the wavefunctions and level spacing statistics. We have now calculated the topological invariants of disordered quantum walks using scattering theory, substantiating the topological interpretation of the delocalization, and finding its signatures in the finite-size scaling of transmission. We have shown criticality of the Haar random quantum walk by calculating the critical exponent η in three different ways and found η ≈ 0.52 as in the integer quantum Hall effect. Our results showcase how theoretical ideas and numerical tools from solid-state physics can help us understand spatially random quantum walks.

**Figure 1.** Position distribution of the Haar random quantum walk after 10000 timesteps

**Ex ante versus ex post equilibria in classical Bayesian games with a nonlocal resource.** —The relation of nonclassical correlations such those of Einstein-Podolsky-Rosen type to the classical theory of Bayesian games is getting an increasing research attention. As a contribution to this line of research we have analyzed the difference between the so-called ex ante and ex post equilibria in classical Bayesian games played with the assistance of a nonlocal (quantum or no-signaling) resource. In physics, the playing of these games is known as performing bipartite Bell-type experiments. By analyzing the Clauser-Horn-Shimony-Holt game, we have found a constructive procedure to find two-person Bayesian games with a nonlocal (ie, no-signaling, and, in many cases, quantum) advantage. Most games of this kind known from the literature can be constructed along this principle. Our construction can be applied to find further scenarios and protocols in which quantum correlations can be useful if realized. In addition we have found that the already known scenarios share the property that their relevant ex ante equilibria are ex post equilibria as well. We have found, however, a different type of game, based on the Bell theorem by Vértesi and Bene, which does not have the latter property: The ex ante and ex post equilibria differ.

**Complete classification of trapping coins for quantum walks on the two-dimensional square lattice. **— One of the unique features of discrete-time quantum walks is called trapping, meaning the inability of the quantum walker to completely escape from its initial position, although the system is translationally invariant. The effect is dependent on the dimension and the explicit form of the local coin. A four-state discrete-time quantum walk on a square lattice is defined by its unitary coin operator, acting on the four-dimensional coin Hilbert space. The well-known example of the Grover coin leads to a partial trapping, i.e., there exists some escaping initial state for which the probability of staying at the initial position vanishes. On the other hand, some other coins are known to exhibit strong trapping, where such an escaping state does not exist. We present a systematic study of coins leading to trapping, explicitly construct all such coins for discrete-time quantum walks on the two-dimensional square lattice, and classify them according to the structure of the operator and the manifestation of the trapping effect. We distinguish three types of trapping coins exhibiting distinct dynamical properties, as exemplified by the existence or nonexistence of the escaping state and the area covered by the spreading wave packet.

**Optimization of multiplexed single-photon sources operated with photon-number-resolving detectors. **— Detectors inherently capable of resolving photon numbers have undergone a significant development recently, and this is expected to affect multiplexed periodic single-photon sources where such detectors can find their applications. We analyze various spatially and time-multiplexed periodic single-photon source arrangements with photon-number-resolving detectors, partly to identify the cases when they outperform those with threshold detectors. We develop a full statistical description of these arrangements in order to optimize such systems with respect to maximal single-photon probability, taking into account all relevant loss mechanisms. The model is suitable for the description of all spatial and time multiplexing schemes. Our detailed analysis of symmetric spatial multiplexing identifies a particular range of loss parameters in which the use of the new type of detectors leads to an improvement. Photon number resolution opens an additional possibility for optimizing the system in that the heralding strategy can be defined in terms of actual detected photon numbers. Our results show that this kind of optimization opens an additional parameter range of improved efficiency. Moreover, this higher efficiency can be achieved by using less multiplexed units, i.e., smaller system size as compared to threshold-detector schemes. We also extend our investigation to certain time-multiplexed schemes of actual experimental relevance. We find that the highest single-photon probability is 0.907 that can be achieved by binary bulk time multiplexers using photon-number-resolving detectors.

**Mesurement-induced nonlinear transformations.** — By involving two or more identical copies of qubits in the same state, it is possible to induce nonlinear quantum state transformations by applying an entangling unitary operation pairwise, combined with a postselection scheme conditioned on the measurement result obtained on one of the qubits of the pair. Such maps play a central role in distillation protocols used for quantum key distribution. We determined that such protocols may exhibit sensitive, quasi-chaotic evolution not only for pure initial states but also for mixed states, i.e., the complex dynamical behavior is not destroyed by small initial uncertainty. We showed that the appearance of sensitive, complex dynamics associated with a fractal structure in the parameter space of the system has the character of a phase transition. The purity of the initial state plays the role of the control parameter, and the dimension of the fractal structure is independent of the purity value after passing the phase transition point (Fig.1). The critical purity coincides with the purity of a repelling fixed point of the dynamics, and we show that all the pre-images of states from the close neighborhood of pure chaotic initial states have purity larger than this. Initial states from this set can be considered as quasi-chaotic .

**Figure 1.** The fractal dimension D_{bc} of the border between the different convergence regions as a function of the initial purity. The error bars indicate the range of numerically calculated values D_{bc}. The black squares correspond to the mean value of the 37 data points for each initial purity. For initial purities above 0.78, D_{bc} is fluctuating around the average value 1.561 (solid line). For P < 0.76, where the structure is no longer a fractal, D_{bc} equals 1 within the statistical error of the numerical method.

We experimentally realized a nonlinear quantum protocol for single-photon qubits with linear optical elements and appropriate measurements. Quantum nonlinearity was induced by postselecting the polarization qubit based on a measurement result obtained for the spatial degree of freedom of the single photon which played the role of a second qubit. Initially, both qubits were prepared in the same quantum state and an appropriate two-qubit unitary transformation entangled them before the measurement of the spatial part. We analyzed the result by quantum state tomography of the polarization degree of freedom. We then demonstrated the usefulness of the protocol for quantum state discrimination by iteratively applying it to either of two slightly different quantum states which rapidly converged to different orthogonal states by the iterative dynamics .

**Nonlinear optical processes.** — We implemented the pseudospectral method for the Finite Difference Frequency Domain method (FDFD) and hence the resulting method is called Pseudospectral Frequency Domain method (PSFD). We compared the PSFD method with the FDFD method in terms of the numerical phase velocity and anisotropy. The main advantage of our method compared to finite differences is the significantly larger accuracy at the same grid resolution. We extended our method for the nonlinear process (NL-PSFD) of Second Harmonic Generation (SHG). We showed how a plane wave source at oblique incidence can be implemented and we discussed its performance for tilted Quasi-Phase Matched (QPM) grating. Finally, we proposed a specific method deduced from NL-PSFD for large volume simulation where only second order nonlinearity is structured spatially, and simulated a two-dimensional nonlinear photonic crystal (Fig.2) .

**Figure 2.** SHG process in a 2D nonlinear photonic crystal. The domain inversion is carried out on a centered rectangular lattice, in a circular pattern. The left part of the figure shows the incoming fundamental wave, the middle represents the nonlinear photonic crystal, while the right part shows the generated second harmonic wave.

**Topological quantum gates.** — Topological properties of quantum systems could provide protection of information against environmental noise, and thereby drastically advance their potential in quantum information processing. Most proposals for topologically protected quantum gates are based on many-body systems, e.g., fractional quantum Hall states, exotic superconductors, or ensembles of interacting spins, bearing an inherent conceptual complexity. We proposed and studied a topologically protected quantum gate, based on a one-dimensional single-particle tight-binding model, known as the Su-Schrieffer-Heeger chain. The proposed Y gate acts in the two-dimensional zero-energy subspace of a Y junction assembled from three chains, and is based on the spatial exchange of the defects supporting the zero-energy modes. With numerical simulations, we demonstrated that the gate is robust against hopping disorder but is corrupted by disorder in the on-site energy. Then we show that this robustness is topological protection, and that it arises as a joint consequence of chiral symmetry, time-reversal symmetry, and the spatial separation of the zero-energy modes bound to the defects. This setup will most likely not lead to a practical quantum computer; nevertheless it does provide valuable insight to aspects of topological quantum computing as an elementary minimal model. Since this model is noninteracting and nonsuperconducting, its dynamics can be studied experimentally, e.g., using coupled optical waveguides .

**Measurement-induced nonlinear transformations. **— We considered the task of deciding whether an unknown qubit state falls in a prescribed neighborhood of a reference state. If several copies of the unknown state are given and we can apply a unitary operation pairwise on them combined with a postselection scheme conditioned on the measurement result obtained on one of the qubits of the pair, the resulting transformation is a deterministic, nonlinear, chaotic map in the Hilbert space. We derived a class of these transformations which are capable of orthogonalizing nonorthogonal qubit states after a few iterations. These nonlinear maps orthogonalize states, which correspond to the two different convergence regions of the nonlinear map. Based on the analysis of the border (the so-called Julia set) between the two regions of convergence, we showed that it is always possible to find a map capable of deciding whether an unknown state is within a neighborhood of fixed radius around a desired quantum state. We analyzed which one- and two-qubit operations would physically realize the scheme. It is possible to find a single two-qubit unitary gate for each map or, alternatively, a universal special two-qubit gate together with single-qubit gates in order to carry out the task. We note that it is enough to have a single physical realization of the required gates due to the iterative nature of the scheme.

**Figure 1. **The decomposition of the relevant map f into the subsequent actions of a contracting two-qubit operation and a single-qubit unitary operation.

(a) The effect of the decomposition on the complex plane (projection of the Bloch sphere), representing the initial states of the qubit. States within the solid circle are matched.

(b) The same decomposition represented on the Bloch sphere.

**Quantum walks **— Measurements on a quantum particle unavoidably affect its state, since the otherwise unitary evolution of the system is interrupted by a nonunitary projection operation. To probe measurement-induced effects in the state dynamics using a quantum simulator, the challenge is to implement controlled measurements on a small subspace of the system and continue the evolution from the complementary subspace. A powerful platform for versatile quantum evolution is represented by photonic quantum walks because of their high control over all relevant parameters. However, measurement-induced dynamics in such a platform have not yet been realized. We participated (at the University of Paderborn) in the implementation of controlled measurements in a discrete-time quantum walk based on time-multiplexing. This was achieved by adding a deterministic outcoupling of the optical signal to include measurements constrained to specific positions resulting in the projection of the walker’s state on the remaining ones (Fig. 2). With this platform and coherent input light, we experimentally simulated measurement-induced single-particle quantum dynamics. We demonstrated the difference between dynamics with only a single measurement at the final step and those including measurements during the evolution. To this aim, we studied recurrence as a figure of merit, that is, the return probability to the walker’s starting position, which was measured in the two cases. We tracked the development of the return probability over 36 time steps and observed the onset of both recurrent and transient evolution as an effect of the different measurement schemes, a signature that only emerges for quantum systems. Our simulation of the observed one-particle conditional quantum dynamics does not require a genuine quantum particle but was demonstrated with coherent light.

**Figure 2.**Schematic of the experimental setup of the time-multiplexed quantum walk with active in- and outcoupling realized by two EOMs (electro-optical modulators). The active control of the switches allows to implement in the time domain both the continual and reset measurement schemes. HWP, half-wave plate; PBS, polarizing beam splitter; SMF, single-mode fiber; SNSPDs, superconducting nanowire single-photon detectors.

**Rotational-vibrational quantum states in molecules. **— We determined the internal-axis system (IAS) of molecules with a large amplitude internal motion (LAM) by integrating the kinematic equation of the IAS by Lie-group and Lie-algebraic methods. Numerical examples on hydrogen peroxide, nitrous acid, and acetaldehyde demonstrate the methods. By exploiting the special product structure of the solution matrix, we devised simple methods for calculating the transformation to the rho-axis system (RAS) along with the value of the parameter *ρ* characterizing a RAS rotational-LAM kinetic energy operator. The parameter *ρ* so calculated agrees exactly with that one obtained by the Floquet method as shown in the example of acetaldehyde. We gave geometrical interpretation of *ρ*. We numerically demonstated the advantageous property of the RAS over the IAS in retaining simple periodic boundary conditions.

**Nanophotonics. **— Non-linear second-harmonic wave generation (SHG) has been thoroughly examined in one dimension both analytically and numerically. Recently, the application of advanced domain poling techniques enabled the fabrication of two-dimensional (2D) patterns of the sign of the nonlinear coefficient in certain non-linear crystals, such as LiNbO_{3} and LiTaO_{3}. This method can be used to achieve quasi-phase-matching in SHG and hence amplification of the second harmonic fields in 2D. We have worked out a true vectorial numerical method for the simulation of SHG by extending the finite-difference frequency-domain method (FDFD). Our nonlinear method (NL-FDFD) operates directly on the electromagnetic fields, uses two meshes for the simulation (for ω and 2ω fields), and handles the non-linear coupling as an interaction between the two meshes. Final field distributions can be obtained by a small number of iteration steps. NL-FDFD can be applied in arbitrarily structured linear media with an arbitrarily structured χ^(2) component both in the small conversion efficiency and the pump-depleted cases.

**Figure 3. **Magnitude of the E_{z} field components for

(a) fundamental wave

(b) second harmonic wave at the resonant frequency for five arrays of cylinders.

We show here the result of a model calculation: the underlying dielectric structure consists of periodic arrays of nonlinear cylinders, it is infinite in the x-direction and there are five periods in the y-direction. The fundamental wave propagates in the positive y-direction, and it can be tilted from normal incidence. At certain angles and frequencies, the structure exhibits double resonance: the reflectivity is close to unity both for the fundamental and for the second harmonic wave.

**Measurement-induced non-linear transformations. **— We proposed a cavity quantum electrodynamical scenario for implementing a Schrödinger microscope capable of amplifying differences between non-orthogonal atomic quantum states. The scheme involves an ensemble of identically prepared two-level atoms interacting pairwise with a single mode of the radiation field as described by the Tavis-Cummings model (Fig. 1). We showed that by repeated measurements of the cavity field and of one atom within each pair, a measurement-induced non-linear quantum transformation of the relevant atomic states can be realized. The intricate dynamical properties of this non-linear quantum transformation, which exhibits measurement-induced chaos, allow approximate orthogonalization of atomic states by purification after a few iterations of the protocol and, thus, the application of the scheme for quantum state discrimination.

**Figure 1. **Illustration of the scheme. Two two-level atoms in the same state interact with the cavity field prepared in a coherent state. Before the interaction, a unitary gate is applied to one of the atoms, and after the interaction and the projection of the field onto the initial coherent state, this same atom is projected onto its ground state. Finally, the other atom is left in a non-linearly transformed state.

**Topological phases. **— We discovered topological features of the Hofstadter butterfly spectra of periodically driven systems. The butterfly is the fractal spectrum of energy eigenstates of a quantum lattice system in a magnetic field. It was discovered numerically (1976, predating the word "fractal"), analyzed analytically (contributing to the topological understanding of the quantum Hall effect), and it is about to be observed experimentally on laser-trapped cold atoms and in graphene. We found that in periodically driven systems, where the drive is very far from just a perturbation, the Hofstadter butterfly can "take flight", i.e., can "wind" in quasienergy (Fig. 2). This behaviour is closely related to a recently discovered topological invariant unique to such non-perturbatively driven systems, and gives us a way to numerically evaluate and perhaps experimentally observe this invariant in an efficient way.

*Figure 2.** The spectrum of quasienergies of a periodically driven system (quantum walk) can wind as a function of applied magnetic field. The winding is quantized, and reveals a bulk topological invariant of the system. *

**Ro-vibrational quantum states in molecules.** — Recently, a general expression for Eckart-frame Hamilton operators has been obtained by the gateway Hamiltonian method. The kinetic energy operator in this general Hamiltonian is nearly identical to that of the Eckart-Watson operator even when curvilinear vibrational coordinates are employed. Its different realizations correspond to different methods of calculating Eckart displacements. There are at least two different methods for calculating such displacements: rotation and projection. In our work, the application of Eckart Hamiltonian operators constructed by rotation and projection was numerically demonstrated in calculating vibrational energy levels. The numerical examples confirm that there is no need for rotation to construct an Eckart ro-vibrational Hamiltonian. The application of the gateway method is advantageous even when rotation is used since it obviates the need for differentiation of the matrix rotating into the Eckart frame. Simple geometrical arguments explain that there are infinitely many different methods for calculating Eckart displacements. The geometrical picture also suggests that a unique Eckart displacement vector may be defined as the shortest (mass-weighted) Eckart displacement vector among Eckart displacement vectors corresponding to configurations related by rotation. Its length, as shown analytically and demonstrated by numerical examples, is equal to or less than that of the Eckart displacement vector one can obtain by rotation to the Eckart frame.

**Nanophotonics. **— We have worked out a true vectorial numerical method for the simulation of the non-linear second harmonic generation process by extending the finite difference frequency domain method (FDFD). Our non-linear method (NL-FDFD) operates directly on the electromagnetic fields, uses two meshes for the simulation (for ω and 2ω fields) and handles the non-linear coupling as an interaction between the two meshes. Final field distributions can be obtained by a small number of iteration steps. NL-FDFD can be applied in arbitrarily structured linear media with an arbitrarily structured χ (2) component both in the small-conversion-efficiency and the pump-depleted cases.

**Quantum information processing, quantum walks. **— State-selective protocols, like entanglement purification, lead to an essentially non-linear quantum evolution, unusual in naturally occurring quantum processes. Sensitivity to initial states in quantum systems, stemming from such non-linear 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, see Fig. 1; however, there is a strict bound on the number of copies needed.

**Figure 1. **Iterations of an exponentially mixing map. (a–l) Visualisation of the iteratives of f, the complex function defining the dynamics on the Bloch spere. The domains are coloured according to whether |f^{on}|>1 (black) or ≤1 (white), distinguishing the northern and southern half of the Bloch sphere. After a few iterations, even very close states get mapped to different halves of the Bloch sphere as indicated by the rapid alternation of black and white domains.

We considered recurrence to the initial state after repeated actions of a quantum channel. After each iteration, a projective measurement is applied to check recurrence. The corresponding return time is known to be an integer for the special case of unital channels, including unitary channels. We prove that for a more general class of quantum channels, the expected return time can be given as the inverse of the weight of the initial state in the steady state. This statement is a generalization of the Kac lemma for classical Markov chains.

**Topological phases. **— In a collaboration with an experimental group at Bonn University, we studied the expected effect of decoherence on edge states in topologically non-trivial quantum walks, realized on trapped atoms in optical lattices. This is an important issue when quantum walks are used as simulators for model Hamiltonians from solid state physics since the sources of decoherence in these experiments are quite different from those in solid state. We used models for decoherence previously introduced and tested in one-dimensional quantum walk experiments, and studied their effects on edge states in one- and two-dimensional topologically non-trivial quantum walks. We developed a simple analytical model quantifying the robustness of these edge states against either spin or spatial dephasing, predicting an exponential decay of their population. Moreover, we presented a realistic experimental proposal to realize spatial boundaries between distinct topological phases, relying on a new scheme to implement spin-dependent discrete shift operations. This is part of a preparation for the first experimental demonstration of two-dimensional quantum walks in such setups.

**Ultracold gases, Bose-Einstein condensates. **— Bose-Einstein condensates of ultracold atoms can be used to sense fluctuations of the magnetic field by means of transitions into untrapped hyperfine states. It has been shown recently that counting the outcoupled atoms can yield the power spectrum of the magnetic noise. In our work, we calculated the spectral resolution function which characterizes the condensate as a noise measurement device in this scheme. We used the description of the radio-frequency outcoupling scheme of an atom laser which takes into account the gravitational acceleration. Employing both an intuitive and the exact three-dimensional and fully quantum mechanical approach, we derived the position-dependent spectral resolution function for condensates of different size and shape.

*Figure 2. **Sketch of the system and the outcoupled mode for a monochromatic outcoupling field.*

**Single-photon sources.** — We consider periodic single-photon sources with combined multiplexing in which the outputs of several time-multiplexed sources are spatially multiplexed. We give a full statistical description of such systems in order to optimize them with respect to maximal single-photon probability. We carry out the optimization for a particular scenario which can be realized in bulk optics and its expected performance is extremely good at the present state of the art. We find that combined multiplexing outperforms purely spatially or time-multiplexed sources for certain parameters only, and we characterize these cases. Combined multiplexing can have the advantages of possibly using less non-linear sources, achieving higher repetition rates, and the potential applicability for continuous pumping. We estimate an achievable single-photon probability between 85% and 89%.

**Nanophotonics. **— A detailed analysis of the optical reflectivity of a monolithic, T-shaped surface relief grating structure is carried out. It is shown that by changing the groove depths and widths, the frequency-dependent reflectivity of the diffraction grating can be greatly modified to obtain various specific optical elements. The basic T-shaped grating structure is optimized for three specific applications: a perfect mirror with a wide maximal reflection plateau, a bandpass filter, and a dichroic beam splitter. These specific mirrors could be used to steer the propagation of bichromatic laser fields in situations where multilayer dielectric mirrors cannot be applied due to their worse thermomechanical properties. Colored maps are presented to show the reflection dependency on the variation of several critical structure parameters. To check the accuracy of the numerical results, four independent methods are used: finite-difference time-domain, finite-difference frequency-domain, method of lines, and rigorous coupled-wave analysis. The results of the independent numerical methods agree very well with each other indicating their correctness.