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Chinese Physics B

Energy mechanism of the first-order superradiant phase transition in cavity-BEC system with double asymmetric pump beams

Wei 威 Qin 覃, Dong-Chen Zheng, Jia-Ying 佳颖 Lin 林, Yuan-Hong Chen, Renyuan Liao

Published May 30, 2025• 1 citation
Publisher

Abstract

Abstract We consider a Bose–Einstein condensate loaded inside an optical cavity and exposed to two crossed coherent pump fields with same imbalance parameter γ . We identify different effects between pure standing wave fields ( γ = 1) and the pump beams combining standing wave and running wave ( γ ≠ 1). In particular, for γ = 1, the system only hosts a normal phase and a superradiant phase. In contrast, for γ ≠ 1, the system features three distinctive phases: the normal phase (NP), superradiant phase 1 (SR 1 ), and superradiant phase 2 (SR 2 ). Importantly, the superradiance is subdivided into different types characterized by the photon phase. Furthermore, we determine perturbatively the phase boundary separating the normal phase and the superradiant phases, and find that there exists a competitive relationship of energy minimum on the overlapping region between SR 1 and SR 2 . Interestingly, the transition between the normal phase to SR 1 or SR 2 is identified to be a second-order phase transition, while the transition between SR 1 and SR 2 is a first-order transition. When the first-order phase transition occurs, the phase of the photons changes abruptly from 0 to π /2.

DOI
10.1088/1674-1056/addeb7

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Abstract We consider a Bose–Einstein condensate loaded inside an optical cavity and exposed to two crossed coherent pump fields with same imbalance parameter γ . We identify different effects between pure standing wave fields ( γ = 1) and the pump beams combining standing wave and running wave ( γ ≠ 1). In particular, for γ = 1, the system only hosts a normal phase and a superradiant phase. In contrast, for γ ≠ 1, the system features three distinctive phases: the normal phase (NP), superradiant phase 1 (SR 1 ), and superradiant phase 2 (SR 2 ). Importantly, the superradiance is subdivided into different types characterized by the photon phase. Furthermore, we determine perturbatively the phase boundary separating the normal phase and the superradiant phases, and find that there exists a competitive relationship of energy minimum on the overlapping region between SR 1 and SR 2 . Interestingly, the transition between the normal phase to SR 1 or SR 2 is identified to be a second-order phase transition, while the transition between SR 1 and SR 2 is a first-order transition. When the first-order phase transition occurs, the phase of the photons changes abruptly from 0 to π /2.
We consider a Bose-Einstein condensate inside an optical cavity and two crossed coherent pump fields. We determine perturbatively the phase boundary separating the normal superfluid phase and the superradiance phase. In the regime of negative cavity detuning, we map out the phase diagrams for both an attractive and a repulsive optical lattice. It turns out that the situation is quite different in the two cases. Specifically, in the case of an attractive lattice, if a system is in the superradiant phase with one pump laser, adding another pump does not drive the system out of the superradiance phase, while for the repulsive lattice, increasing another pump potential has suppressive effects on the superradiance. We also find that, in the case of an attractive lattice, equally increasing two pump lattice potentials can induce a transition from the normal phase to the superradiance phase. In stark contrast, for the repulsive lattice, the system will remain in the normal phase as the pump depths are tuned within a wide range, independently of the cavity detuning and the decay rate.
Abstract In this paper, we investigate the effect of atomic collisions on the phase transition form the normal to the superradiant phase in a one-dimensional Bose–Einstein condensate (BEC) trapped inside an optical cavity. Specifically, we show that driving the atoms from the side of the cavity leads to the excitation of modes in the edges of the first Brillouin zone of every energy band, which results in the two-mode approximation of the BEC matter field in the limit of weak coupling regime. The nonlinear effect of atom–atom interaction shifts the threshold of the quantum phase transition of the BEC and also affect the power low behavior of quantum fluctuations in the total particle number. Besides, we show the possibility of controlling the quantum phase transition of the system through the s -wave scattering frequency when the strength of the transverse pumping has been fixed.
Abstract We study superradiant phase transitions in a hybrid system of a two-dimensional Bose–Einstein condensate of atoms and two cavities arranged with a tilt angle. By adjusting the loss rate of cavities, we map out the phase diagram of steady states within a mean field framework. It is found that when the loss rates of the two cavities are different, superradiant transitions may not occur at the same time in the two cavities. A first-order phase transition is observed between the states with only one cavity in superradiance and both in superradiance. In the case that both cavities are superradiant, a net photon current is observed flowing from the cavity with small decay rate to the one with large decay rate. The photon current shows a non-monotonic dependence on the loss rate difference, owing to the competition of photon number difference and cavity field phase difference. Our findings can be realized and detected in experiments.
We investigate the superradiant phase transition in a two-component Bose-Einstein condensate with distinct atomic detunings, confined in an optical cavity and driven by a transverse pump laser. By combining perturbation theory and numerical simulations, we demonstrate that the phase transition is dominated by the red-detuned component, resulting in a phase diagram completely different from that of a single-component case under blue-detuned condition. The system exhibits spontaneous phase separation between the two components, manifested as alternating stripe patterns in the normal phase and distinct Bragg gratings in the superradiant phase. Furthermore, the Bogoliubov excitation spectrum reveals roton-type mode softening, indicating that the phase transition also corresponds to the superfluid-to-lattice supersolid transition. Our findings provide insights into the interplay between atomic detunings and collective quantum many-body phenomena, offering potential applications in quantum simulation and optical switching technologies.
The superfluid to Mott insulator transition and the superradiant transition are textbook examples of quantum phase transition and coherent quantum optics, respectively. Recent experiments in ETH and Hamburg have succeeded in loading degenerate bosonic atomic gases in optical lattices inside a cavity, which enables the first experimental study of the interplay between these two transitions. In this Rapid Communication we present the theoretical phase diagram for the ETH Zurich experimental setup, and determine the phase boundaries and the orders of the phase transitions between the normal superfluid phase, the superfluid with superradiant light, the normal Mott insulator, and the Mott insulator with superradiant light. We find that in contrast to the second-order superradiant transition in a weakly interacting Bose condensate, strong correlations in the superfluid nearby a Mott transition can render the superradiant transition to a first-order one. Our results will stimulate further experimental studies of interactions between cavity light and strongly interacting quantum matter.
Dicke model describes a collective interaction between the two-level atoms and the light cavity and has been predicted to show a peculiar quantum phase transition, which is a second-order phase transition from a normal phase (in a weak-coupling strength) to a superradiant phase (in a strong-coupling strength). The model plays an important role in illustrating the quantum ground-state properties of many-body macroscopic quantum states. In the experiment, Dicke quantum phase transition in an open system could be formed by a Bose-Einstein condensate coupled to a high-finesse optical cavity. This experiment on the Bose-Einstein condensate trapped in the optical cavity have opened new frontiers, which could combine the cold atoms with quantum optics and makes it possible to enter into the strongly coupled regime of cavity quantum electrodynamics. In strong coupled regime, the atoms exchange the photons many times before spontaneous emission and cavity losses set in. It has become a hot research topic in recent years and plays an important role in many fields of modern physics, such as condensed matter physics, nuclear physics, etc. It can be applied to the manipulation of the geometric phase and entanglement in quantum information and computing. Quantum phase transition has been widely studied for the Dicke model as a typical example. Many different research methods about the mean-field approximation have been used to analyze the ground state properties of the Dicke model. In this paper, we study the ground state properties of two-component Bose-Einstein condensate in a single-mode cavity. Meanwhile, the associated quantum phase transition is described by the spin-coherent-state variational method, whose advantage is that the ground state energy and wave function can be obtained without the thermodynamic limit. By taking the average in the boson coherent state, we obtain an equivalent effective pesudospin Hamiltonian, which will be diagonalized by using the spin coherent state. Finally, we can obtain the energy functional, which is the basics of the variation to obtain the numerical solution of photon number and the expression of the atomic number and the ground state energy. This paper presents a rich phase diagram, which can be manipulated by changing the atom-field coupling imbalance between two components and the atom-field frequency detuning. While in the single-mode Dicke model there exist only the normal phase and the superradiation phase. When the frequency of one component atom is zero or the frequency of the two component atoms are equal in optical cavity, the system returns to the standard Dicke model, in which there occurs the second-order phase transition from the normal phase to the superradiant phase by adjusting the atom-field coupling. In conclusion, we discover that the stimulated radiation comes from the collective state of atomic population inversion, which does not exist in the single-mode Dicke model. Meanwhile, the new stimulated-radiation state S and S, which can only be produced by one component of the atom, are observed in the two-component Bose-Einstein condensates in the single-mode optical cavity. By adjusting the atom-field coupling imbalance and the atom-field frequency detuning (the blue or red detuning), the order of the superradiation state and the stimulated-radiation states can be exchanged between the two components of the atom.