**2018**

**Experiment on Cavity Quantum Electrodynamics with cold Rb atoms** ― We set up a new laboratory for cavity QED experiments with ultracold Rb atoms. We realized the UHV system which is operated in the pressure range below 1e-10 mbar, and includes the magneto-optical trap and the in-vacuo high-finesse single-mode optical resonator. We completed the optical system which is based on three laser sources that are referenced to Rb resonance line by Doppler-free non-linear spectroscopy. By means of acousto-optical modulators we provide the narrow linewidth phase-locked laser sources at five different frequencies for atomic manipulation.

**FIG. 1.** Photo of the actual status of the cavity QED laboratory.

**Cavity Quantum Electrodynamics, quantum critical phenomena. **― The quantum measurement back action noise effects on the dynamics of an atomic Bose lattice gas inside an optical resonator has been described by means of a hybrid model consisting of a Bose–Hubbard Hamiltonian for the atoms and a Heisenberg–Langevin equation for the lossy cavity field mode. Considering atoms initially being prepared in the ground state of the lattice Hamiltonian, we calculated the transient dynamics due to the interaction with the cavity mode. We showed that the cavity field fluctuations originating from the dissipative outcoupling of photons from the resonator lead to vastly different effects in the different possible ground state phases, i.e., the superfluid, the supersolid, the Mott- and the charge-density-wave phases. In the former two phases with the presence of a superfluid wavefunction, the quantum measurement noise appears as a driving term leading to depletion of the ground state. The time scale for the system to leave the ground state was presented in a simple analytical form. For the latter two incompressible phases, the quantum noise results in the fluctuation of the chemical potential. We derived an analytical expression for the corresponding broadening of the quasiparticle resonances. [Phys. Rev. A 97, 063602 (2018) ]

**FIG. 2**. Illustration of the coupled cavity Bose-Hubbard model setup. An atomic cloud is loaded into a square optical lattice, which is inside a single-mode high-Q Fabry-Pérot resonator. The period of the cavity mode is approximately equal to that of the optical lattice. The cavity is pumped from the side by light scattering off the atoms. The system is open, together with the external drive, the photons leak out from the cavity, resulting in heating and decoherence, or, in other terms, quantum measurement back action since the out coupled photons can be measured by classical detectors.

**Ultracold gases, Bose-Einstein condensates.** ― We studied quasiparticle scattering effects on the dynamics of a homogeneous Bose-Einstein condensate of ultracold atoms coupled to a single mode of an optical cavity. The relevant excitations, which are polariton-like mixed excitations of photonic and atomic density-wave modes, have been identified. All the first-order correlation functions were presented by means of the Keldysh Green’s function technique. Beyond confirming the existence of the resonant enhancement of Beliaev damping, we found a very structured spectrum of fluctuations. There is a spectral hole burning at half of the recoil frequency reflecting the singularity of the Beliaev scattering process. The effects of the photon-loss dissipation channel and that of the Beliaev damping due to atom-atom collisions could have been well separated. We showed that the Beliaev process does not influence the properties of the self-organization criticality. [Phys. Rev. A 98, 063608 (2018)]

**FIG. 3** (a) The schematic picture of an atomic ensemble inside a Fabry-Pérot cavity is pumped from the side by a laser close to resonance with the cavity. The atomic gas undergoes self-organization for strong enough laser drive: the atoms scatter photons from the laser to the cavity and may create a classical field serving as an optical lattice trapping them even further in the optimal scattering positions. (b) Spectrum of the quasiparticle excitation. Large detuning ∆C = 100 implies little mixing of the condensate quasiparticle with the photon mode. The enhanced Beliaev scattering process is manifested by the large peak, which exhibits a hole burning at 0.5. (All angular frequencies are expressed in units of the recoil frequency.) The correlation function is plotted for different coupling strengths y approaching the critical value yc: y = 0 (solid red), y/yc = 0.5 (short-dashed green), 0.7 (dotted blue), 0.8 (dashed-dotted orange), 0.9 (dashed-double-dotted brown), 0.95 (long-dashed magenta).

We studied the spin-1 bilinear–biquadratic model on the complete graph of N sites, i.e. when each spin is interacting with every other spin with the same strength. Because of its complete permutation invariance, this Hamiltonian can be rewritten as the linear combination of the quadratic Casimir operators of SU(3) and SU(2). Using group representation theory, we explicitly diagonalized the Hamiltonian and mapped out the ground-state phase diagram of the model. Furthermore, the complete energy spectrum, with degeneracies, was obtained analytically for any number of sites. [J. Phys. A: Math. Theor. 51, 105201 (2018) ]

**2017**

**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.

**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.

**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.

**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).