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Superradiant Phase Transition with Cavity Assisted Dynamical Spin-Orbit Coupling

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Superradiant Phase Transition with Cavity Assisted Dynamical Spin-Orbit Coupling Superradiant phase transition with cavity assisted dynamical spin-orbit coupling Ying Lei1 and Shaoliang Zhang1, 2, ∗ 1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan, 030006,China (Dated: December 21, 2020) Superradiant phase transition represents an important quantum phenomenon that shows the col- lective excitations based on the coupling between atoms and cavity modes. The spin-orbit coupling is another quantum effect which induced from the interaction of the atom internal degrees of free- dom and momentum of center-of-mass. In this work, we consider the cavity assisted dynamical spin-orbit coupling which comes from the combination of these two effects. It can induce a series of interesting quantum phenomena, such as the flat spectrum and the singularity of the excitation energy spectrum around the critical point of quantum phase transition. We further discuss the influence of atom decay and nonlinear coupling to the phase diagram. The atom decay suppresses the singularity of the phase diragram and the nonlinear coupling can break the symmetric properties of the phase transition. Our work provide the theoretical methods to research the rich quantum phenomena in this dynamic many-body systems. I. INTRODUCTION such as the stripe phase [23] and algebra quantum liquids [24], etc. Recently, the cavity-assisted dynamical spin- The cavity quantum electrodynamics (CQED) has a orbit coupling which induced by a superradiant phase wide range of applications in quantum information and transition has been realized in experiment by replacing quantum simulations, because the interaction between one of the Raman laser with an optical cavity [25{27]. hyperfine spin states of atom and quantum photon field In this system, the cavity modes not only couple with has a great enhancement by the cavity. A famous the- the hyperfine spin states of atoms, but also interact with oretical model which used to describe this interaction is the atomic COM momentum. The interplay of cavity called Dicke model [1]. It is well known to undergo a modes, atomic internal and external degrees of freedom quantum phase transition from normal phase to superra- are expected to lead to novel quantum phenomena [28] diant phase called "superradiant phase transition" [2,3]. and pave a way for the research of quantum simulation Over the last few years, based on the development of ex- in topological matters and many-body physics. perimentally techniques, there are a series of works focus Our propose in this work is to understand the interplay on researching the properties of this system [4{8] and of these degrees of freedom. In the system with cavity- superradiant phase transition has also be observed ex- assisted dynamical SOC, the SOC comes from the col- perimentally first by the group at ETH [9{11] then many lective excitation of cavity photons and all atoms | the other groups [12{14]. Many interesting quantum phe- superradiance, which is very different from the coupling nomena has been observed such as supersolid-like phase, between Raman laser and single atom in static SOC. Al- and research about their applications has also been de- though the translation invariance is still conserved, the veloped rapidly [15]. energy dispersion is very different and some exotic quan- The standard Dicke model only includes the coupling tum phenomena emerges, such as the dynamical SOC between cavity modes and atomic internal degrees of free- induced flat spectrum. In this system, the superradi- dom. But in ultracold atomic system, the coupling be- ant phase transition has some interesting properties be- tween the internal degrees of freedom and the atomic cause of the interplay among cavity modes, hyperfine spin center-of-mass(COM) motion, which is called the spin- states and the COM momentum of atoms, which is very orbit coupling (SOC) effect, has attracted a lot of inter- different from the standard Dicke model. There is a first ests in past two decades[16{18]. With the SOC effect, order phase transition and the energy spectrum of polari- a lot of interesting quantum phenomena, especially the tons has singularity at the critical point. In this work, phenomena related to the topological physics, such as we also discuss the influence of atom decay and nonlinear arXiv:2012.10098v1 [cond-mat.quant-gas] 18 Dec 2020 the synthetic gauge fields [19, 20], chiral edge currents coupling to the phase diagram. We find that the atom [21, 22], has been observed experimentally. In ultracold decay suppresses the singularity of the phase diagram atom systems, the SOC effect can be realized by coupling which result in the first order phase transition is hard to two different hyperfine spin states using two-photon Ra- be observed experimentally, and the nonlinear coupling man process which also accompanied by the transfer of can break the symmetric form of the superradiant phase COM momentum. In systems with static SOC, the trans- transition. lation invariance is conserved and the energy dispersion Our paper is organized as follows. In Sec. II we'll give has some anomalous properties because of the SOC effect. a brief introduction about the cavity-assisted dynamical It is the source of many interesting quantum phenomena SOC model. Then we calculate the energy spectrum and 2 find the flat spectrum at critical point of superradiant where !, δ, λ and U are the effective cavity frequency, phase transition. We also draw the phase diagram and energy detuning between hyperfine spin states, coupling discuss the first order phase transition. In Sec. III we amplitude and dispersive cavity shift. In equation (1), discuss the energy spectrum of polaritons and find the pj is the momentum operator of the j-th atoms, Sα = singularity induced by the flat spectrum. In Sec. IV we PN j j=1 σα where α = x; y; z and S± = Sx ± iSy. In most discuss the influence of atom decay and cavity photon part of our work, we'll set the nonlinear atom-photon loss to the phase diagram. We find that in this dissipa- interaction U = 0 to simplify our discussion, the influence tive system, the singularity of the phase diagram will dis- of nonzero U will be discussion in the Sec. V. appear because of the atom decay. In Sec. V we discuss In this work, we only consider the case all atoms are the influence of nonlinear coupling to the phase diagram. condensed with the same COM momentum p. So pj in The Sec. VI is devoted to our conclusions. equation (1) can be simplified as a number instead of an operator. To further simplify our discussion, we can set the spin-orbit coupling along y direction and ignore the II. CAVITY-ASSISTED SPIN-ORBIT influence of other direction, the above Hamiltonian can COUPLING further be simplified as p2 pq H = N + r + δ S + !aya eff 2m m z (2) λ y + p (a + a)(S+ + S−) N Different with the original Dicke model, the Hamil- tonian (2) is also dependent on the COM momentum of atoms. The SOC effect doesn't break the transla- tion invariant symmetry, and the commutation relation [p; H ] = 0 is kept, which means the COM momen- FIG. 1: The schematic diagram of our proposal. (a) Experi- eff mental setup; (b) Level diagram of atom and light configura- tum of atom p is still a good quantum number, and the tion. eigenstates of the atoms are still plane waves. Because of the inversion symmetry, the Hamiltonian (2) is conserved with translation of p ! −p, δ ! −δ and Sz ! −Sz, We consider the atoms with two hyperfine spin states which means the ground state has the same energy for (denoted as j "i and j #i) confined in an optical cav- different sign of δ, so only δ ≥ 0 need to be considered. ity, as shown schematically in Fig. 1. The cavity sup- Define the effective energy detuning between j "i and j #i ports a single-mode quantum field with frequency !, two pqr states as !0 ≡ m + δ and ignores the influence of the counter-propogating external lights (red and blue arrows) kinetic energy term, the Eq.(2) becomes to the standard with frequency q+ = q− = qr produces additional clas- Dicke model as sical lights with different frequencies, which couple dif- ferent hyperfine spin states respectively. Each of these y λ y HDicke = !0Sz + !a a + p (a + a)(S+ + S−) (3) two classical lights and cavity quantum field induce a N two-photon Raman transition between different hyper- fine spin states, and spontaneously transfer COM mo- With the analytical calculation in the work of C. Emary mentum of ±q along the direction perpendicular to the and T. Brandes[29], one can get the eigenenergy and cor- r responding wave functions for different phase, the critical cavity axis in average. In the experiment of R. M. Kroeze p et. al[27], the discretized momentum distribution and the point of superradiant phase transition is at λcr = !!0=2 spatial modulation along cavity axis have been observed. in thermodynamic limit. But in this work, we just focus on the properties perpen- Consider the Hamiltonian (2), for an arbitrary cou- dicular to the cavity axis and the spatial modulation can pling strength λ and energy detuning δ, one can get the be ignored along this direction. After an unitary trans- energy spectrum with different COM momentum p first. formation, the effective Hamiltonian can be written as Because of the relative amplitude of λ and !0 at different [4, 27] p, the corresponding eigenstates should be superradiant states or normal states due to different COM momentum m 4λ2 m 4λ2 p.
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