We propose to probe the quantum ground state of a spin-1 Bose-Einstein condensate with the transmission spectra of an optical cavity. By choosing a circularly polarized cavity mode with an appropriate frequency, we can realize coupling between the cavity mode and the magnetization of the condensate. The cavity transmission spectra then contain information of the magnetization statistics of the condensate and thus can be used to distinguish the ferromagnetic and antiferromagnetic quantum ground states. This technique may also be useful for continuous observation of the spin dynamics of a spinor Bose-Einstein condensate.
We examine the multistability associated with miscibility-immiscibility conditions for a two-component Bose-Einstein condensate coupled to the light field in an optical cavity. For a strongly immiscible condition, the system exhibits a variety of density structures, including separated state, stripe state, and their coexistence. The multistability arises from these spatial structures of the two-component condensate, which significantly alter the hysteresis curve with respect to the intensity of cavity pumping. We present a variational approach to confirm our numerical results.
The dispersive interaction of a Bose-Einstein condensate with a single mode of a high-finesse optical cavity realizes the radiation pressure coupling Hamiltonian. In this system the role of the mechanical oscillator is played by a single condensate excitation mode that is selected by the cavity mode function. We study the effect of atomic s-wave collisions and show that it merely renormalizes parameters of the usual optomechanical interaction. Moreover, we show that even in the case of strong harmonic confinement---which invalidates the use of Bloch states---a single excitation mode of the Bos…
We study cavity quantum electrodynamics of Bose-condensed atoms that are subjected to continuous monitoring of the light leaking out of the cavity. Due to a given detection record of each stochastic realization, individual runs spontaneously break the symmetry of the spatial profile of the atom cloud and this symmetry can be restored by considering ensemble averages over many realizations. We show that the cavity optomechanical excitations of the condensate can be engineered to target specific collective modes. This is achieved by exploiting the spatial structure and symmetries of the collecti…
We study the normal mode splitting in a system consisting of a Bose Einstein condensates (BECs) trapped inside a Fabry Perot cavity driven by a single mode laser field. We analyze the variations in frequency and damping rate of the collective density excitation of a BEC imparted by the optical field. We study the occurrence of normal mode splitting which appears as consequences of the hybridization of the fluctuations of the intracavity field and the condensate mode. It is shown that normal mode splitting vanishes for weak coupling between the condensate mode and the intracavity field. Moreove…
We provide an exact solution for the interplay between Bose-Einstein condensation and the Dicke-Hepp-Lieb self-organization transition of an ideal Bose gas trapped inside a single-mode optical cavity and subject to a transverse laser drive. Based on an effective action approach, we determine the full phase diagram at arbitrary temperature, which features a bi-critical point where the transitions cross. We calculate the dynamically generated band structure of the atoms and the associated supression of the critical temperature for Bose-Einstein condensation in the phase with a spontaneous period…
We determine the quantum ground-state properties of ultracold bosonic atoms interacting with the mode of a high-finesse resonator. The atoms are confined by an external optical lattice, whose period is incommensurate with the cavity mode wave length, and are driven by a transverse laser, which is resonant with the cavity mode. While for pointlike atoms photon scattering into the cavity is suppressed, for sufficiently strong lasers quantum fluctuations can support the build-up of an intracavity field, which in turn amplifies quantum fluctuations. The dynamics is described by a Bose-Hubbard mode…
We investigate the dynamical phase transition of two-component Bose-Einstein condensate with nonlinear tunneling, which is trapped inside a double-well and dispersively coupled to a single mode of a high-finesse optical cavity with one moving end mirror driven by a single mode standing field. The nonlinear tunneling interaction leads to an increase of stability points and riches the phase diagram of the system. It is shown that the appearance of the moving end mirror speeds up the tunneling of Bose-Einstein condensates, which makes population difference between two wells and regulates the numb…
We propose a novel type of composite light-matter magnetometer based on a transversely driven multi-component Bose-Einstein condensate coupled to two distinct electromagnetic modes of a linear cavity. Above the critical pump strength, the change of the population imbalance of the condensate caused by an external magnetic field entails the change of relative photon number of the two cavity modes. Monitoring the cavity output fields thus allows for nondestructive measurement of the magnetic field in real time. We show that the sensitivity of the proposed magnetometer exhibits Heisenberg-like sca…
We present a new generalized Dicke model, an impurity-doped Dicke model (IDDM), by the use of an impurity-doped cavity-Bose-Einstein condensate. It is shown that the impurity atom can induce Dicke quantum phase transition (QPT) from the normal phase to superradiant phase at a critic value of the impurity population. It is found that the IDDM exhibits continuous Dicke QPT with an infinite number of critical points, which is significantly different from that observed in the standard Dicke model with only one critical point. It is revealed that the impurity-induced Dicke QPT can happen in an arbi…
We investigate the effects of dynamical Casimir effect in superradiant light scattering by Bose-Einstein condensate in an optomechanical cavity. The system is studied using both classical and quantized mirror motions. The cavity frequency is harmonically modulated in time for both the cases. The main quantity of interest is the number of intracavity scattered photons. The system has been investigated under the weak and strong modulation. It has been observed that the amplitude of the scattered photons is more for the classical mirror motion than the quantized mirror motion. Also, initially, th…
We show that photon number measurement can be used to detect superfluidity for a two-band Bose-Hubbard model coupled to a cavity field. The atom-photon coupling induces transitions between the two internal atomic levels and results in entangled polaritonic states. In the presence of a cavity field, we find different photon numbers in the Mott-insulating versus superfluid phases, providing a method of distinguishing the atomic phases by photon counting. Furthermore, we examine the dynamics of the photon field after a rapid quench to zero atomic hopping by increasing the well depth. We find a ro…
In this paper we explore the rich structure of macroscopic many-particle quantum states for Bose- Einstein condensate in an optical cavity with the tunable nonlinear atom-photon interaction [Nature (London) 464, 1301 (2010)]. Population inversion, bistable normal phases and the coexistence of normal{superradiant phases are revealed by adjusting of the experimentally realizable interaction strength and pump-laser frequency. For the negative (effective) cavity-frequency we observe remark- ably an inverted quantum phase transition (QPT) from the superradiant to normal phases with the increase of …
We present a novel cavity QED system in which a Bose-Einstein condensate (BEC) is trapped within a high-finesse optical cavity whose length may be adjusted to access both single-mode and multimode configurations. We demonstrate the coupling of an atomic ensemble to the cavity in both configurations. The atoms are confined either within an intracavity far-off-resonance optical dipole trap (FORT) or a crossed optical dipole trap via transversely oriented lasers. Multimode cavity QED provides fully emergent and dynamical optical lattices for intracavity BECs. Such systems will enable explorations…
We investigate the quantum measurement noise effects on the dynamics of an atomic Bose lattice gas inside an optical resonator. We describe the dynamics 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. We assume that the atoms are prepared initially in the ground state of the lattice Hamiltonian and then start to interact with the cavity mode. We show that the cavity field fluctuations originating from the dissipative outcoupling of photons from the resonator lead to vastly different effects …
We explore the many-body phases of a two-dimensional Bose-Einstein condensate with cavity-mediated dynamic spin-orbit coupling. By the help of two transverse non-interfering, counterpropagating pump lasers and a single standing-wave cavity mode, two degenerate Zeeman sub-levels of the quantum gas are Raman coupled in a double-$Λ$-configuration. Beyond a critical pump strength the cavity mode is populated via coherent superradiant Raman scattering from the two pump lasers, leading to the appearance of a dynamical spin-orbit coupling for the atoms. We identify three quantum phases with distinct …
We investigate the energy structures and the dynamics of a Bose-Einstein condensates (BEC) in a triple-well potential coupled a high finesse optical cavity within a mean field approach. Due to the intrinsic atom-cavity field nonlinearity, several interesting phenomena arise which are the focuses of this work. For the energy structure, the bistability appears in the energy levels due to this atoms-cavity field nonlinearity, and the same phenomena can be found in the intra-cavity photons number. With an increase of the pump-cavity detunings, the higher and lower energy levels show a loop structu…
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 a…
In this paper, we have investigated theoretically the influence of atomic collisions on the behaviour of a one-dimensional Bose-Einstein condensate inside a driven optical cavity. We develop the discrete-mode approximation for the condensate taking into account the interband transitions due to the s-wave scattering interaction. We show that in the Bogoliubov approximation the atom-atom interaction shifts the energies of the excited modes and also plays the role of an optical parametric amplifier for the Bogoliubov side mode which can affect its normal-mode splitting behaviour. On the other han…
Using parametric conversion induced by a Shapiro-type resonance, we produce and characterize a two-mode squeezed vacuum state in a sodium spin 1 Bose-Einstein condensate. Spin-changing collisions generate correlated pairs of atoms in the $m=\pm 1$ Zeeman states out of a condensate with initially all atoms in $m=0$. A novel fluorescence imaging technique with sensitivity $ΔN \sim 1.6$ atom enables us to demonstrate the role of quantum fluctuations in the initial dynamics and to characterize the full distribution of the final state. Assuming that all atoms share the same spatial wave function, w…
Dicke model predicts a quantum phase transition from normal to superradiant phases for a two-level atomic ensemble coupled with an optical cavity at zero temperature. In a recent pioneer experiment [Nature 464, 1301 (2010)], such a phase transition has been observed using a Bose-Einstein condensate (BEC) in an optical cavity. Compared with the original Dicke model, the experimental system features finite temperature and strong atom-photon nonlinear interaction in BEC. In this Letter, we develop a finite temperature theory for the Dicke phase transition of a BEC in an optical cavity, taking int…
We study 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, are identified. All the first-order correlation functions are presented by means of the Keldysh Green's function technique. Beyond confirming the existence of the resonant enhancement of Beliaev damping, we find a very structured spectrum of fluctuations. There is a spectral hole burning at half of the recoil frequency r…
We probe the interface between a phase-separated Bose-Fermi mixture consisting of a small BEC of $^{41}$K residing in a large Fermi sea of $^6$Li. We quantify the residual spatial overlap between the two components by measuring three-body recombination losses for variable strength of the interspecies repulsion. A comparison with a numerical mean-field model highlights the importance of the kinetic energy term for the condensed bosons in maintaining the thin interface far into the phase-separated regime. Our results demonstrate a corresponding smoothing of the phase transition in a system of fi…
We investigate a system consisting of a two-component Bose-Einstein condensate interacting dispersively with a Fabry-Perot optical cavity where the two components of the condensate are resonantly coupled to each other by another classical field. The key feature of this system is that the atomic motional degrees of freedom and the internal pseudo-spin degrees of freedom are coupled to the cavity field simultaneously, hence an effective spin-orbital coupling within the condensate is induced by the cavity. The interplay among the atomic center- of-mass motion, the atomic collective spin and the c…
We study spatial spin and density self-ordering of a two-component Bose-Einstein condensate via collective Raman scattering into a linear cavity mode. The onset of the Dicke superradiance phase transition is marked by a simultaneous appearance of a crystalline density order and a spin-wave order. The latter spontaneously breaks the discrete $\mathbf{Z}_2$ symmetry between even and odd sites of the cavity optical potential. Moreover, in the superradiant state the continuous $U(1)$ symmetry of the relative phase of the two condensate wavefunctions is explicitly broken by the cavity-induced posit…
We present a detailed study to analyse the Dicke quantum phase transition within the thermodynamic limit for an optomechanically driven Bose-Einstein condensates in a cavity. The photodetection-based quantum optical measurements have been performed to study the dynamics and excitations of this optomechanical Dicke system. For this, we discuss the eigenvalue analysis, fluorescence spectrum and the homodyne spectrum of the system. It has been shown that the normal phase is negligibly affected by the mechanical mode of the mirror while it has a significant effect in the superradiant phase. We hav…
We introduce a detector that selectively probes the phononic excitations of a cold Bose gas. The detector is composed of a single impurity atom confined by a double-well potential, where the two lowest eigenstates of the impurity form an effective probe qubit that is coupled to the phonons via density-density interactions with the bosons. The system is analogous to a two-level atom coupled to photons of the radiation field. We demonstrate that tracking the evolution of the qubit populations allows probing both thermal and coherent excitations in targeted phonon modes. The targeted modes are se…
We explore the asymmetric sequential Landau-Zener (LZ) dynamics in an ensemble of interacting Bose condensed two-level atoms coupled with a cavity field. Assuming the couplings between all atoms and the cavity field are identical, the interplay between atom-atom interaction and detuning may lead to a series of LZ transitions. Unlike the conventional sequential LZ transitions, which are symmetric to the zero detuning, the LZ transitions of Bose condensed atoms in a cavity field are asymmetric and sensitively depend on the photon number distribution of the cavity. In LZ processes involving singl…
The influence of mirror motion on the spatial self organization of a Bose-Einstein condensate (BEC) in an optical cavity is studied. We show that the mirror dynamics tends to destroy the process of self organization. An additional external phonon pump is shown to modify the critical photon pump needed to observe the onset of self organization.
We present a general theory for calculating the damping rate of elementary density wave excitations in a Bose-Einstein condensate strongly coupled to a single radiation field mode of an optical cavity. Thereby we give a detailed derivation of the huge resonant enhancement in the Beliaev damping of a density wave mode, predicted recently by Kónya et al., Phys.~Rev.~A 89, 051601(R) (2014). The given density-wave mode constitutes the polariton-like soft mode of the self-organization phase transition. The resonant enhancement takes place, both in the normal and ordered phases, outside the critical…
The coupled nonlinear dynamics of ultracold quantum matter and electromagnetic field modes in an optical resonator exhibits a wealth of intriguing collective phenomena. Here we study a $Λ$-type, three-component Bose-Einstein condensate coupled to four dynamical running-wave modes of a ring cavity, where only two of the modes are externally pumped. However, the unpumped modes play a crucial role in the dynamics of the system due to coherent back-scattering of photons. On a mean- field level we identify three fundamentally different steady-state phases with distinct characteristics in the densit…
We study the spin orbit coupled ultra cold Bose-Einstein condensate placed in a single mode Fabry-Pérot cavity. The cavity introduces a quantum optical lattice potential which dynamically couples with the atomic degrees of freedom and realizes a generalized extended Bose Hubbard model whose zero temperature phase diagram can be controlled by tuning the cavity parameters. In the non-interacting limit, where the atom-atom interaction is set to zero, the resulting atomic dispersion shows interesting features such as bosonic analogue of Dirac points, cavity controlled Hofstadter spectrum which bea…
We investigate the superfluid behavior of a Bose-Einstein condensate of $^6$Li molecules. In the experiment by Weimer et al., Phys. Rev. Lett. 114, 095301 (2015) a condensate is stirred by a weak, red-detuned laser beam along a circular path around the trap center. The rate of induced heating increases steeply above a velocity $v_c$, which we define as the critical velocity. Below this velocity, the moving beam creates almost no heating. In this paper, we demonstrate a quantitative understanding of the critical velocity. Using both numerical and analytical methods, we identify the non-zero tem…
Phase transition, phase separation and mode softening of a two-component Bose-Einstein condensate in an optical cavity
College of Physics and Energy, Fujian Normal University（福建师范大学物理与能源学院）；Fujian Provincial Key Laboratory of Quantum Manipulation and New Energy Materials（福建省量子调控与新能源材料重点实验室）；Fujian Provincial Engineering Technology Research Center of Solar Energy Conversion and Energy Storage（福建省太阳能转换与储能工程技术研究中心）
We propose a novel scheme to generate the spin-orbit coupling for a condensate placed inside an optical cavity by using a standing wave and a traveling wave. It is shown that the interplay of the laser lights and the cavity gives rise to rich quantum phases. Our scheme also generates an nontrivial lattice of the magnetic flux, which may facilitate the study of the exotic quantum phases.
We investigate the effects of a movable mirror (cantilever) of an optical cavity on the superradiant light scattering from a Bose-Einstein condensate (BEC) in an optical lattice. We show that the mirror motion has a dynamic dispersive effect on the cavity-pump detuning. Varying the intensity of the pump beam, one can switch between the pure superradiant regime and the Bragg scattering regime. The mechanical frequency of the mirror strongly influences the time interval between two Bragg peaks. We found that when the system is in the resolved side band regime for mirror cooling, the superradiant…
The system of a transversely pumped Bose-Einstein condensate (BEC) coupled to a lossy ring cavity can favor a supersolid steady state. Here we find the existence of supersolid gap soliton in such a driven-dissipative system. By numerically solving the mean-field atom-cavity field coupling equations, gap solitons of a few different families have been identified. Their dynamical properties, including stability, propagation and soliton collision, are also studied. Due to the feedback atom-intracavity field interaction, these supersolid gap solitons show numerous new features compared with the usu…
We investigate the dynamics of a spinor Bose-Einstein condensate inside an optical cavity, driven transversely by a laser with a controllable polarization angle. We focus on a two-component Dicke model with complex light-matter couplings, in the presence of photon losses. We calculate the steady-state phase diagram and find dynamical instabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes which generate non-reciprocal couplings between the two collective spins. Our predic…
We consider a hybrid atom-optomechanical system consisting of a mechanical membrane inside an optical cavity and an atomic Bose-Einstein condensate outside the cavity. The condensate is confined in an optical lattice potential formed by a traveling laser beam reflected off one cavity mirror. We derive the cavity-mediated effective atom-atom interaction potential, and find that it is non-uniform, site-dependent, and does not decay as the interatomic distance increases. We show that the presence of this effective interaction breaks the Z$_2$ symmetry of the system and gives rise to new quantum p…
We propose to thermalize a low-dimensional photon gas and obtain photon Bose-Einstein condensation by optomechanical interactions in a microscopic optical cavity, with a single longitudinal mode and many transverse modes. The geometry of the short cavity is such that it provides a low-frequency cutoff at a photon energy far above the thermal energy, so that thermal emission of photons is suppressed and the photon number is conserved. While previous experiments on photon Bose-Einstein condensation have used dye molecules for photon gas thermalization, we here investigate thermalization owing to…
We report a spin-orbit coupling induced back-action cooling in an optomechanical system, composed of a spin-orbit coupled Bose-Einstein condensate trapped in an optical cavity with one movable end mirror, by suppressing heating effects of quantum noises. The collective density excitations of the spin-orbit coupling mediated hyperfine states - serving as atomic oscillators equally coupled to the cavity field - trigger strongly driven atomic back-action. We find that the back-action not only revamps low-temperature dynamics of its own but also provides an opportunity to cool the mechanical mirro…
The Dicke model and the superradiance of two-level systems in a radiation field have many applications. Recently, a Dicke quantum phase transition has been realized with a Bose-Einstein condensate in a cavity. We numerically solve the many-body Schrödinger equation and study correlations in the ground state of interacting bosons in a cavity as a function of the strength of a driving laser. Beyond a critical strength, the bosons occupy multiple modes macroscopically while remaining superradiant. This fragmented superradiance can be detected by analyzing the variance of single-shot measurements.
We study an ultracold atom-cavity coupling system, which had been implemented in experiment to display weak light nonlinearity [S. Gupta \textit{et al}., Phys. Rev. Lett. \textbf{99}, 213601 (2007)]. The model is described by a non-interacting Bose-Einstein condensate contained in a Fabry-Pérot optical resonator, in which two incommensurate standing-wave modes are excited and thus form a quasiperiodic optical lattice potential for the atoms. Special emphasis are paid to the variation of atomic wavefunction induced by the cavity light field. We show that bistability between the atomic localized…
We consider a two-component Bose-Einstein condensate in a one-dimensional optical cavity. Specifically, the condensate atoms are taken to be in two degenerate modes due to their internal hyperfine spin degrees of freedom and they are coupled to the cavity field and an external transverse laser field in a Raman scheme. A parallel laser is also exciting the cavity mode. When the pump laser is far detuned from its resonance atomic transition frequency, an effective nonlinear optical model of the cavity-condensate system is developed under Discrete Mode Approximation (DMA), while matter-field coup…
The paper considers Bose-Einstein condensation (BEC) of light in a cavity with medium. In the framework of two-level model we show the effect of gaseous medium on the critical temperature of light condensation in the system. Transition of the system to the state with released light condensate is illustrated in consequent stages. Analytical expressions for a typical spatial extent of the condensed cloud of photons, as well for spectral characteristics of the condensate peak are derived. Energy and heat capacity of photons as functions of temperature are obtained. Finally, we demonstrate that th…
We study the fidelity decay and its freeze for an initial coherent state of two-mode Bose-Einstein condensates in the Fock regime considering a Bose-Hubbard model that includes two-particle tunneling terms. By using linear-response theory we find scaling properties of the fidelity as a function of the particle number that prove the existence of two-particle mode-exchange when a non-degeneracy condition is fulfilled. Tuning the energy difference of the two modes serves to distinguish the presence of two-particle mode-exchange terms through the appearance of certain singularities. Numerical resu…
We present a theoretical treatment of coherent light scattering from an interacting 1D Bose gas at finite temperatures. We show how this can provide a nondestructive measurement of the atomic system states. The equilibrium states are determined by the temperature and interaction strength, and are characterized by the spatial density-density correlation function. We show how this correlation function is encoded in the angular distribution of the fluctuations of the scattered light intensity, thus providing a sensitive, quantitative probe of the density-density correlation function and therefore…
We study the problem of high temperature Bose-Einstein condensation (BEC) of atom-light polaritons in a waveguide cavity appearing due to interaction of two-level atoms with (non-resonant) quantized optical radiation, in the strong coupling regime, in the presence of optical collisions (OCs) with buffer gas particles. Specifically, we propose a special biconical waveguide cavity (BWC), permitting localization and trapping of low branch (LB) polaritons imposed by the variation of the waveguide radius in longitudinal direction. We have shown that critical temperature of BEC occurring in the syst…
We analyze the optomechanics of an atomic Bose-Einstein condensate interacting with the optical lattice inside a laser-pumped optical cavity and subject to a uniform bias force such as gravity. An atomic wave packet in a tilted lattice undergoes Bloch oscillations; in a cavity the backaction of the atoms on the light leads to a time-dependent modulation of the intracavity lattice at the Bloch frequency. When the Bloch frequency is on the order of the cavity damping rate we find transport of the atoms either up or down the lattice. The transport dynamics can be interpreted as a manifestation of…
We probe the transition between superfluid and Bose glass phases using quantum quenches of disorder in an ultracold atomic lattice gas that realizes the disordered Bose-Hubbard model. Measurements of excitations generated by the quench exhibit threshold behavior in the disorder strength indicative of a phase transition. Ab-initio quantum Monte Carlo simulations confirm that the appearance of excitations coincides with the equilibrium superfluid--Bose-glass phase boundary at different lattice potential depths. By varying the quench time, we demonstrate the disappearance of an adiabatic timescal…
