A csoport elemi kvantumos rendszerekkel, ezekből felépített nagyobb hálózatokkal, ezek kvantuminformatikai alkalmazásával és kapcsolódó konkrét fizikai elrendezések leírásával foglalkozik. Egyik sokat vizsgált modellünk a kvantumos bolyongás, amely a lehetséges kvantumos keresési algoritmusokon túl jól modellez olyan alapvető kvantummechanikai jelenségeket, mint a topológikusan védett fázisok kialakulása. Kutatási témáink között megjelenik az egyszerű molekulák vibrációs tulajdonságainak vizsgálata, a nemklasszikus fény létrehozását célzó kísérletekben való részvétel, illetve a fény kvantumállapotainak előállítása és a kvantummérés-elmélet.

Tamás Kiss

Péter Ádám

János Asbóth

Péter Boross

Aurél Gábris

András Gilyén

Orsolya Kálmán

Zsolt Kis

Miklós Kornyik

Pavlo Pyshkin

Viktor Szalay

Géza Tóth

(* ***2016**)

**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.