A csoport tudományos kutatásainak fókuszában a rezonátoros kvantumelektrodinamika és az ultrahideg atomok állnak. A fény-anyag kölcsönhatást vizsgáljuk abban a végső határesetben, amikor mindkét komponens dinamikailag változó kvantumrendszer. Az anyagi összetevő lehet egy atom, kis számú atom, vagy atomok Bose-Einstein kondenzátuma. Az ultrahideg atomok, amelyek hőmérséklete az abszolút zérus felett mindössze néhány nanokelvin, fizikája elvezet kritikus jelenségekhez, fázisátalakulásokhoz és a kvantummechanikai soktestprobléma módszereihez. A csoport 2016-ban elkezdte egy laboratórium felépítését Rb atomokkal végzett optikai kísérletekhez.

Domokos Péter

Clark Thomas

Dombi András

Jakab Dávid

Kónya Gábor

Nagy Dávid

Szirmai Gergely

Vidák Rózsa

Vukics András

(* ***2016**)

**Cavity Quantum Electrodynamics, quantum critical phenomena. ** — Non-equilibrium phase transitions exist in damped-driven open quantum systems, when the continuous tuning of an external parameter leads to a transition between two robust steady states.

In second-order transitions, this change is abrupt at a critical point, whereas in first-order transitions, the two phases can co-exist in a critical hysteresis domain. In collaboration with the experimental group of the ETH Zürich, we found a first-order dissipative quantum phase transition in a driven-circuit quantum electrodynamics (QED) system. It takes place when the photon blockade of the driven cavity-atom system is broken by increasing the drive power. The observed experimental signature is a bimodal phase space distribution with varying weights controlled by the drive strength. The measurements showed an improved stabilization of the classical attractors up to the millisecond range when the size of the quantum system is increased from one to three artificial atoms. The theoretical work included the fitting of the experimental data as well as it contributed to prove the phase-transition character of the effect. Furthermore it was possible to prove theoretically that the photon-blockade-breakdown effect relies on a given range of the parameters of a three-level atomic system. We showed that the parameters of the actual experimental setup happen to correspond to this range [Phys. Rev. X 7, 011012 (2017) ].

**Figure 1.** (a) Simulated histogram of the output intensity as a function of the coupling constant g_{2} between the two excited atomic states |eñ and |fñ at a given driving amplitude h with red representing high probability and blue indicating zero probablility. This plot shows that only a certain range of the ratio g_{2}/g_{1} gives rise to bistability.

*(b) Measured vacuum Rabi spectra for various input powers with all three atoms in resonance with the cavity. For better visibility the shown spectra are offset by 1.6 nW from each other. The sharp transmission peak shown in the inset appears stochastically. In this particular measurement (orange line at 4.4 fW input power), we observe only two frequency points with small but finite switching probability and we sample over multiple switching events resulting in a certain mean detected power. At lower drive power, no transmission is observed (no switching). At higher drive powers, the transmission peak approaches the cavity linewidth and scales linearly with input power (no switching again).*

*(c) Measured histogram of the detected power as a function of the cavity input power for a single transmon (density plot). The most likely photon numbers (line plots) are extracted from this measurement (red) and two similar measurements taken with 2 (orange) and 3 qubits (green) in resonance with the cavity mode. Simulation results for the single qubit case are shown with connected black symbols for comparison. The dashed line is for reference and represents the response of the empty cavity.*

**Ultracold gases, Bose-Einstein condensates.** — Modeling the coupling between a trapped Bose-Einstein condensate and a current carrying nanowire, we studied, in collaboration with an experimental group at Tübingen, the magneto-mechanical interaction by means of classical radio-frequency sources. We performed the spectral analysis and the local measurement of intensity correlations of microwave fields using ultracold quantum gases. The fluctuations of the electromagnetic field induce spin flips in a magnetically trapped quantum gas and generate a multimode atom laser. The output of the atom laser was measured with high temporal resolution on the single-atom level, from which the spectrum and intensity correlations of the generating microwave field have been reconstructed in accordance with our recently proposed scheme. We gave the theoretical description of the atom-laser output and its correlations in response to resonant microwave fields and verified the model with measurements on an atom chip. The measurement technique is applicable for the local analysis of classical and quantum noise of electromagnetic fields, for example, on chips, in the vicinity of quantum electronic circuits[Phys. Rev. A 95, 043603 (2017)].

**Figure 2.** (a) Cold-atom spectrometer (not to scale) consisting of a magnetically trapped Bose-Einstein condensate and an ionization-based single-atom detector. (b) The microwave couples atoms at resonance surfaces given by equipotential surfaces of the atomic Zeeman potential, i.e., magnetic isofield lines (dashed lines). Due to gravity, the BEC is displaced from the magnetic trap center and the resonance surfaces become nearly plane. Without amplitude modulation, the microwave carrier couples atoms from a single resonance surface (red solid thick line) with a position given via ωc. Amplitude modulation at a single frequency generates sidebands to the carrier and outcoupling from two resonance surfaces (green solid thin lines). (c) Normalized spectral response γ(ω)/γmax of a BEC to a single microwave frequency (black dots) and model function (red line).

**Cavity Quantum Electrodynamics, quantum critical phenomena.** ― Quantum phase transitions in driven-dissipative systems opened up a novel research area in the field of critical phenomena. These transitions lie beyond the standard classification of dynamical or equilibrium phase transitions, and define completely new universality classes. In an open quantum system, the critical behaviour appears in the state formed by the dynamical equilibrium of the external driving and dissipation processes. The abrupt symmetry breaking change of such a steady state takes place when the external control parameters are continuously tuned across the critical point. The correlation functions at the critical point are determined by non-equilibrium noise rather than thermal or ground-state quantum fluctuations. We demonstrated that criticality in a driven-dissipative system is strongly influenced by the spectral properties of the bath. We studied the open-system realization of the Dicke model, where a bosonic cavity mode couples to a fictious large spin formed by two motional modes of an atomic Bose-Einstein condensate. The cavity mode is driven by a high-frequency laser and it decays to a Markovian bath, while the atomic mode interacts with a colored bath. We revealed that the soft mode fails to describe the characteristics of the criticality. We calculated the critical exponent of the superradiant phase transition and identified an inherent relation to the low-frequency spectral density function of the colored bath. We showed that a finite temperature of the colored bath does not modify qualitatively this dependence on the spectral density function.

We investigated the possibility of the Dicke-type superradiant phase transition of an atomic gas. We described the ultrastrong coupling limit of the interaction between light and atoms within the regularized electric dipole gauge, in which we can take into account the short-range depolarizing interactions between atoms that approach each other as close as the atomic size scale. By using a mean field model, we find that a critical point does indeed exist, though the atom-atom contact interaction shifts it to a higher value than what can be obtained from the bare Dicke-model. We pointed out the proximity of the critical density to that of solidification, which leads to the conjecture that the system, at the critical density, goes over to the condensed rather than to the “superradiant” phase.

**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. As a continuation of our previous investigations, 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.

We investigated the magnetic properties of strongly interacting four-component spin-3/2 ultracold fermionic atoms in the Mott insulator limit with one particle per site in an optical lattice with honeycomb symmetry. In this limit, atomic tunneling is virtual, and only the atomic spins can exchange. We found a competition between symmetry-breaking and liquid-like disordered phases. Particularly interesting are the non-singlet valence bond states (where the valence bonds have non-zero magnetization) which are situated between the ferromagnetic and conventional valence bond phases. In the framework of a mean-field theory, we calculated the phase diagram and identified an experimentally relevant parameter region where a homogeneous SU(4) symmetric Affleck-Kennedy-Lieb-Tasaki–like valence bond state is present.

The classical ground states of the SU(4) Heisenberg model on the face-centered-cubic lattice constitute a highly degenerate manifold. We explicitly constructed all the classical ground states of the model. To describe quantum fluctuations above these classical states, we applied linear flavor-wave theory. At zero temperature, the bosonic flavor waves select the simplest of these SU(4) symmetry-breaking states, the four-sublattice-ordered state defined by the cubic unit cell of the fcc lattice. Due to geometrical constraints, flavor waves interact along specific planes only, thus rendering the system effectively two dimensional and forbidding ordering at finite temperatures. We showed that longer-range interactions generated by quantum fluctuations can shift the transition to finite temperatures.