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

**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.