The dynamical formation of a complete Bose-Einstein condensate and the asymptotic purification of trapped bosons coupled to a quantized radiation field are rigorously established. By tracking the macroscopic one-particle density matrix
κT through a nonlinear resonance cascade, it is demonstrated that radiative relaxation irreversibly depletes all excited trap modes, even when the initial macroscopic state
κ0 is genuinely mixed. Here
κT is an average, with respect to a de Finetti measure
μ0, over mean-field particle-field orbits, so that the mixedness of
κ0 reflects a statistical ensemble of orbits. Under the assumptions of a non-vanishing initial ground state component and strictly positive downward Fermi-Golden-Rule rates, strong trace-norm convergence to a pure condensate is proven,
limT→∞∥κT−∣χ0⟩⟨χ0∣∥S1(h)=0. Furthermore, the von Neumann entropy of the macroscopic one-particle density matrix is shown to vanish asymptotically,
limT→∞S[κT]=0, without any claim of monotonicity in
T. Thus, any initially mixed, positive-entropy state (
0<S[κ0]<∞) is purified to the ground state
∣χ0⟩⟨χ0∣. For the proof, we derive the Hartree hierarchy of the coupled system of bosons and photons from the finite-
N BBGKY hierarchy of the microscopic quantum many-body system, establishing trace-norm convergence in the mean-field limit as
N→∞. The limits are taken successively; first
N→∞, then the weak-coupling limit, and finally
T→∞. Although the global particle-field evolution remains unitary, the macroscopic one-particle state converges to a pure state as energy and correlations are continuously radiated into the dispersive field.
Bose-Einstein condensation (BEC) is the transition to a macroscopic occupation of the ground state of a system of bosonic particles. In 1995 the first experimental realization of BEC was achieved by condensing a dilute atomic gas [1,2]. It was long thought only massive particles could undergo BEC. However, in 2010 it was shown that photon BEC was achieved in an optical dye-filled microcavity [3]. In 2015, we became the second group to successfully repeat these experiments.\n\nBEC is a phase transition and thus it occurs at thermal equilibrium. To achieve a thermal photon gas the cavity is filled with a fluorescent dye solution. The dye molecules thermalize the photons by means of repeated absorption-emission cycles. We optically pump the cavity. With the pump we determine the number of photons absorbed by the dye, consequently defining a nonvanishing chemical potential. This ensures a conserved photon number. The photons are trapped by the harmonic potential induced by the concave cavity mirrors. Due to the cutoff energy determined by the cavity length, the trapped photons have a dispersion similar to that of a massive particle in a quantum harmonic oscillator. The cavity length is tuned such that the free spectral range is larger than the bandwidth of the dye. Therefore, only one longitudinal mode fits inside the cavity, making the photon gas effectively two-dimensional. We end up with a thermalized two-dimensional photon gas that has the eigenenergies of a quantum harmonic oscillator. The Bose-Einstein distribution function predicts a critical photon number. Pumping the thermal photon gas beyond this critical number will lead to photon BEC.\n\nWe measure the spectral- and spatial distributions of our two-dimensional photon gas. In both distributions we observe photon BEC. Results show the characteristics of a phase transition: the appearance of a sharp peak on top of the thermal cloud that coincides with a converging chemical potential to zero. Both the spectral- and spatial distribution are in agreement with the theoretical distribution of photons with an effective mass in a quantum harmonic oscillator. Expansion of the condensate has been measured. This could indicate nonlinear optical effects such as thermal lensing or repulsive photon-photon interactions.
Abstract Bose-Einstein Condensate (BEC) is a paradigmatic representative of macroscopic quantum states. 39 K atoms serve as an ideal system for realizing a nearly ideal BEC owing to their ultra-small background scattering length. This study further investigates the nonlinear dynamics of 39 K BEC confined in a double-well potential under attractive interactions. In this work, a Gaussian barrier is integrated with a double-well potential, and we innovatively adopt a transient dynamics approach to construct a highly non-equilibrium initial state where all atoms are confined to the left well. We track the full evolutionary process of the system from non-equilibrium excitation to stable self-trapping, elucidating wave function contraction and energy conversion. Under high barrier height conditions, tunneling is substantially suppressed: atoms localize in a single well, the longitudinal position expectation value remains constant, and the phase difference is locked, leading to the formation of a macroscopic quantum self-trapping state. Radial oscillations exhibit reduced amplitude and a slower evolution rate owing to density homogenization. A smooth, fluctuation-free momentum density profile corresponds to the static localized regime of self-trapping. Coherence transitions from dynamic oscillation to static locking, and weak attractive interactions in 39 K atoms further prolong the coherence time. This study elucidates key microscopic mechanisms, including the nonlinear coupling mechanism of attractive interactions, the dynamic-static transition rules of coherence, and momentum density acting as a microscopic probe for tunneling and soliton dynamics, thereby providing a quantitative basis for the preparation of tunable quantum states.
We apply the concepts of nonlinear guided-wave optics to a Bose-Einstein condensate (BEC) trapped in an external potential. As an example, we consider a parabolic double-well potential and derive coupled-mode equations for the complex amplitudes of the BEC macroscopic collective modes. Our equations describe different regimes of the condensate dynamics, including the nonlinear Josephson effect for any separation between the wells. We demonstrate macroscopic self-trapping for both repulsive and attractive interactions, and confirm our results by direct numerical solution of the Gross-Pitaevskii-equation.
We investigate a two-component Bose-Einstein condensate (BEC) strongly coupled to a single optical cavity, effectively described by a mean-field Dicke model supplemented with interatomic nonlinearities. Here, we propose a unified theoretical framework demonstrating that macroscopic quantum self-trapping (MQST) natively emerges between two internal atomic energy levels within a single cavity. By deriving the dimensionless semiclassical Josephson equations (SJE) governing this purely internal-state architecture, we analytically determine the critical nonlinear threshold and intrinsic phase shift mechanism for the phase transition. Based on this framework, we present two approaches for manipulating quantum phase transitions: dynamic in-situ tuning via photon pumping and inducing non-equilibrium dynamical phase transitions (DPT) via real-time parameter quenches. Furthermore, we rigorously prove that the effective charging energy driving this system scales exactly as one-quarter of the effective spin-dependent interaction energy -- the precise parameter governing recent spin-orbit coupled (SOC) BEC experiments. Incorporating realistic
87Rb atomic parameters, we substantiate that these single-cavity MQST and transition dynamics are highly feasible for observation under current state-of-the-art cold-atom technologies.
In this paper, we study the interaction-modulated tunneling dynamics of a Bose-Fermi superfluid mixture, where a Bose-Einstein condensate (BEC) with weak repulsive interaction is confined in a symmetric deep double-well potential and an equally populated two-component Fermi gas in a harmonic potential symmetrically is positioned in the center of the double-well potential. The tunneling between the two wells is modulated by fermions trapped in a harmonic potential. When the temperature is adequately low and the bosonic particle number is adequately large, we can employ the mean-field theory to describe the evolution of the BEC in the double-well potential through the time-dependent Gross-Pitaevskii equation. For the Fermi gas in the harmonic potential trap, we consider the case where the inter-fermion interaction is tuned on the deep Bose-Einstein condensate of the inter-fermion Feshbach resonance, where two fermions of spin-up and spin-down form a two-body bound state. Within the regime, the Fermi gas is well described by a condensate of these fermionic dimers, and hence can be simulated as well by a Gross-Pitaevskii equation of dimers. The inter-species interactions couple the dynamics of the two species, which results in interesting features in the tunneling oscillations. The dynamic equations of the BEC in the double-well potential is described by a two-mode approximation. Coupling it with time-dependent Gross-Pitaevskii equation of the harmonically potential trapped molecular BEC, we numerically investigate the dynamical evolution of the Boson-Fermi hybrid system under different initial conditions. It is found that the interaction among fermions in a harmonic potential leads to strong non-linearity in the oscillations of the bosons in the double-well potential and enriches the tunneling dynamics of the bosons. Especially, it strengthens macroscopic quantum self-trapping. And the macroscopic quantum self-trapping can be expressed in three forms: the phase tends to be negative and monotonically decreases with time, the phase evolves with time, and the phase tends to be positive and increases monotonically with time. This means that it is possible the tunneling dynamics of the BEC in double-well potential is adjustable. Our results can be verified experimentally in a Bose-Fermi superfluid mixture by varying different interaction parameters via Feshbach resonance and confinement-induced resonance.