Quantum mixed phases of a two-dimensional polarized degenerate Fermi gas in an optical cavity

The coupling of ultracold fermions to a high-finesse optical cavity can result in novel many-body phenomena, and has attracted significant interests at present. Here we consider a realization of the Fermi-Dicke model with controllable parameters, based on a two-dimensional polarized degenerate Fermi gas coupled to an optical cavity. We analytically investigate the ground-state properties of such system under the mean-field approximation. We find the system can exhibit a rich phase diagram depending on the fermion-photon coupling strength and the atomic resonant frequency. Contrasting to the bosonic counterpart, a first-order quantum phase transition between the superradiant phase and the normal phase featuring two Fermi surfaces can occur for the weak atomic resonant frequency, and there is a unique mixed phase where this normal phase and the superradiant phase coexist. The experimental detection of our results is also discussed.

in the yz plane along with a tightly-radial confinement in the x direction. The fermions are coupled to a high-finesse optical cavity, where the cavity mode is driven by a linearly polarized laser (with frequency ω l ) propagating along the x direction. Two transverse pumping lasers (with frequencies ω A and ω B ), which are leftand right-handed circular polarized in the yz plane, propagate along the x direction. As a result, two Raman processes are induced. In order to obtain a time-independent Hamiltonian, the condition ω ω ω = + ( ) /2 l A B is required; see the detailed derivation in the main text. A magnetic field B is applied along the positive z direction, producing a Zeeman shift between two hyperfine ground states. (b) The atomic energy levels and their transitions. Each fermionic atom has two ground states (|↑〉 and |↓〉) and two excited states (|1〉 and |2〉). The |↓〉 ↔|1〉 and |↑〉 ↔|2〉 transitions (red solid lines) are caused by the quantized cavity field with fermion-photon coupling strengths g 1 and g 2 . The |↑〉 ↔|1〉 and |↓〉 ↔|2〉 transitions (green dashed lines) are governed by the transverse pumping lasers with Rabi frequencies Ω 1 and Ω 2 . ω ω ω = − ↑↓ ↑ ↓ is the resonant frequency between the ground states |↑〉 and |↓〉 with eigenfrequencies ω ↑ and ω ↓ . ∆ 1 and ∆ 2 are the detuning from the excited states |1〉 and |2〉.
ScIentIfIc REPORtS | 7: 10568 | DOI: 10.1038/s41598-017-10686-3 The considered system can be effectively described by the following 2D Hamiltonian Here, ψ † r ( ) i and ψ r ( ) i ( = ↓ ↑ i 1, 2, , ) are the field operators for the fermionic atoms with mass M, μ is the chemical potential, and ω i denotes the frequency of the ith eigenstate. The quantized cavity field and the driving laser are described by the Hamiltonian where ˆ † a (â) creates (annihilates) the quantized cavity field with frequency ω c , and ε (ω l ) labels the magnitude (frequency) of the driving laser. The Hamiltonian Ĥ AR represents the interaction between fermions and the transverse pumping lasers, which under the rotating-wave approximation can be written as with Ω 1 and Ω 2 (ω A and ω B ) being the Rabi frequencies (frequencies) of the two lasers. Finally, Hamiltonian Ĥ AP describes the interaction between fermions and the quantized cavity fields, i.e., with g 1 and g 2 labeling the fermion-photon coupling strengths associated with the two pumping lasers, respectively. It is more transparent to recast the time-dependent Hamiltonian (1) into a time-independent form. Introducing a unitary transformation Here, ω ω ω = −  c l is the effective cavity frequency, ) denotes the detuning from the excited state |1〉 (|2〉), and ω = describes the effective frequency associated with the internal state |↑〉 (|↓〉).
To proceed, we recall that in the experiments 3-5 , a weak driving (ε → 0) and large detuning  Here, the factor N 1/ , with N being the total atom number, is introduced so as to ensure a finite free energy per fermion in the thermodynamic limit 9,10 . Furthermore, ) /2 is the effective resonant frequency between the ground states |↑〉 and |↓〉, which in Eq. (10) acts as an effective Zeeman field. Without loss of generality, below we shall take ω > 0 0 . In addition, the parameter η = Ω ∆ ⁎ N g /(2 ) 2 is the effective fermion-photon coupling strength, and ω ζ ω = +  N labels the atom-number dependent cavity frequency with We remark that all the parameter choice here are motivated by the experimental considerations [3][4][5] .
The Hamiltonian (10) represents the paradigmatic Fermi-Dicke model 31 describing the fermion-photon interaction. We emphasize that our setup allows flexible controllability of all Hamiltonian parameters. For example, both ω 0 and ω can be tuned by modifying the frequencies of the driving laser and the transverse pumping lasers, while η can be controlled via the Rabi frequencies of the transverse pumping lasers.
Ground-state properties. Our goal is to investigate the ground state of the Hamiltonian (10). To this end, it is more convenient to transform to the momentum space representation. Writing where σ C k, annihilates a fermion in the internal state σ with momentum k and S is the system size (hereafter S = 1 is set for convenience), we obtain is the kinetic energy, and π π being the Fermi energy and K F being the Fermi momentum. In solving the ground state of the Hamiltonian (12), we will rely on the mean-field approximation, i.e., by replacing â with its steady-state value. Specifically, we write down the Heisenberg-Langevin equation for the cavity field operator â 32, 33 , taking into account of the cavity decay with rate κ, i.e., () (13) in For the quantum noise operator γˆt ( ) in , we have γ Since quantum noise usually occurs on a much shorter time scale than 1/κ 34 , its average effect can be generically ignored on the time scale relevant for the steady state. When 1/κ is much shorter than the time scales of system dynamics, a steady-state solution 3,35 to Eq. (13) can be found, i.e., Motivated by the experiments 3, 4 which shows that the mean-photon number α = † a a 2 determines the SR properties, we henceforth refer to it as the SR order parameter.
By approximating ≈â a in Eq. (12) using Eq. (14), we obtain a quadratic Hamiltonian is the standard Nambu spinor, and M k is the photon-number dependent BdG matrix, i.e., with η η α α = + ⁎ n ( ) / . Now, the Hamiltonian (15) can be easily diagonalized as k   k  k  k  k  k  k  ,  ,  ,  ,  ,  ,  2 where α ± k, describe the fermionic Bogoliubov quasiparticles, whose energy is given by . Note that ± E k, can be either positive or negative depending on parameter choices. Regrouping terms with positive and negative energies using the Heaviside step function Θ x ( ), we recast Eq. (17) as k, where E G is the ground-state energy expressed by Despite the formal expression of the ground-state energy, Eq. (20) contains two variables, i.e., μ and |α|, which are to be determined from the particle number equation and χ η ω = + 2 0 2 . In deriving Eqs (21) and (22), we have used the identity being the delta function. Equations (20)(21)(22) allow complete specification of the ground state: The ground-state energy can be obtained in a self-consistent manner by solving Eqs (21) and (22) for a fixed atom density n. It is important to bear in mind that solutions of above equations must be supplemented with a careful stability analysis, which we describe below.
Phase diagram. We now detail our analysis on the ground-state properties. We shall be interested in four quantities: the ground-state energy per fermion = E E n / G G , the chemical potential μ, the scaled mean-photon number α 2 , and the scaled polarization 36 measuring the response to the effective Zeeman field defined by As we shall see, their ground-state values depend crucially on the strength of the effective Zeeman field ω 0 and the effective fermion-photon coupling strength η, in particular, new features compared to their boson counterpart are found in the regime ω < E F 0 . Note that all energies will be measured in units of E F hereupon. In order to gain some intuition, let us first consider the simplest case η = 0 corresponding to a free Fermi gas. Transforming Eq. (20) into an integral over momenta as usual, we calculate the ground-state energy per fermion as which excludes existence of real fermions 37, 38 , we shall limit our subsequent discussions in the regime µ ω + > 0 0 . There, the Fermi gas can exhibit different ground states depending on whether ω Based on Eqs (21), (23) and (25), we obtain Thus the Fermi gas is in a normal phase characterized by a full polarization and one Fermi surface with µ = ↑ E 2 F , and we shall refer to it as the N-I phase.
In contrast, when ω µ < Based on Eqs (21), (23) and (25), we have Hence the Fermi gas is only partially polarized and exhibits two Fermi surfaces, i.e., µ We shall call it as the N-II phase. It is clear from Eqs (26) and (28) that, by tuning the parameter ω 0 across the critical point ω = E c F 0 37 , the Fermi gas undergoes a first-order transition between the N-I to N-II phases. Now, suppose there exists a weak effective coupling between the photon and fermions, i.e., η is small. Still, the noninteracting terms in the Hamiltonian (12) dominate the system dynamics, such that |α| 2 = 0 is expected to persist for the ground state and the Fermi gas remains in the N-I or N-II phase (depending on values of ω 0 ).
However, the ground-state properties of the Fermi gas can change drastically from both normal phases for a strong photon-fermion coupling when η is increased above a threshold. As we show now, when the interaction part of the Hamiltonian (12) dominates, the system can acquire a macroscopic collective excitation with |α| 2 ≠ 0, i.e., a quantum phase transition into the SR phase occurs when η is above some critical values. Integrating Eq. (20) over the momenta, we obtain  and Eqs (21-23) take the following form We can obtain two set of solutions, namely, Obviously, while solutions (34) are associated with the normal phase, solutions (35) exhibit SR property. Furthermore, as mentioned earlier, the ground-state solutions must satisfy the stability condition as defined by . Consequently, we find that the stability condition for the normal solution [see Eq.

(a)] and thereby
, the ground state of the Fermi gas is in the N-I phase for η η < < 0 c (1) , but transits into the SR phase for η η > c ; see summarized results below with corresponding relevant quantities given by The ground-state properties described by Eqs (38)-(41) exhibit two features. First, the quantum phase transition between the N-I and SR phases is second order in nature: while the first-order derivative of E G is continuous with respect to η, its second order derivative develops discontinuity at the critical point η c (1) , as in the case with ultracold Bose atoms 9,10 . Second, in contrast to the N-I phase, the SR phase features macroscopic collective excitations for both fermions and photons, i.e., we have both α ≠ 0 and ≠ m 0. Also note that while η ∼ m 2 , we see µ η ∼ −2 . These analytical results are confirmed by our numerical calculations as illustrated by Fig. 3, where we plot E G , μ, α 2 , and m as functions of η, respectively.
, and µ < − 0 for η η > c (2) and µ > − 0 for η η < < 0 c (2) . Imposing the stability condition , and keeping in mind µ ≥ − 0, we find that in the regimes ω < E F 0 and η η < < 0 c (2) , the relevant ground-state solutions are Fig. 2(b)], where the ground-state solutions are of the form of Eq. (35). Thus in contrast to the case ω ≥ E F 0 , the ground state of the Fermi gas with ω < E F 0 exhibits the N-II phase for η η < < 0 c (2) and the SR phase for η η > c (2) . We summarize results below We see that a quantum phase transition from the N-II phase to the SR phase occurs at the critical point η c (2) , which is first order in nature and is accompanied by a sudden change in α 2 [see also red-dashed curves in Fig. 3(c)].
Interestingly, at the critical regime η η = c (2) , the scaled ground-state energies of the N-II and SR phases become equal, meaning both phases coexist. We shall therefore call it the N-II-SR mixed phase. In order to characterize this mixed phase, let x 0 stand for the fraction of the N-II part in the mixed phase, which can take arbitrary value in the regime ≤ ≤ x 0 1 0 . We find (see Methods Section for detailed derivation) Since both the SR and N-II phases exhibit ≠ m 0 [see Eqs (35) and (28)], the former due to the the macroscopic collective excitation of fermions while the later induced by the effective Zeeman field, it is naturally expected that both mechanisms contribute to the nonzero polarization of the N-II-SR mixed phase [see Eq. (48)]. In addition, we find two first-order quantum phase transitions, from the N-II-SR phase to the N-II phase and to the SR phase, respectively. This is different from the boson counterpart, as well as the previously discussed N-I phase, where the corresponding transitions are second order 9, 10 .
Collecting all above results, we plot in Fig. 4 the phase diagram in the entire parameter regimes of ω E / F 0 and η E / F . As predicted previously, while the quantum phase transition from the N-I phase to the SR phase is second order, a first-order transition occurs between the N-II and SR phases due to coexistence of both phases at the critical line. In addition, the phase diagram exhibits a tri-critical point (see the red dotted curves), where the quantum phase transition changes its character from the first to the second order.
Considering the optical properties of the 40 K D1-line, we take the cavity length as 178 μm and the wavelengths of the transverse pumping lasers as 770 nm. This ensures that both g 1 and g 2 characterizing the fermion-photon coupling strength are at the order of MHz, such that the rotating-wave approximation leading to Eqs (4) and (5) is valid. For a cavity with a waist radius 27 μm and a finesse ~10 5 , we estimate its decay rate κ is at the order of MHz. For the effective fermion-photon coupling strength η = Ω ∆ . Finally, the effective resonant frequency ) /2 and the atom-number dependent cavity frequency ω ζ ω = +  N can be easily controlled by tuning the frequencies of the driving and transverse pumping lasers. In the experiments 3 , both ω 0 and ω can be tuned from − GHz to GHz and even beyond.
We conclude this section by briefly discussing how to probe the predicted quantum phases and phase diagrams. As has been elaborated, the physics of the ground-state phases is determined by the scaled mean-photon number α 2 and the scaled polarization m. Building on the development of the state of the art experimental techniques, m can be measured by observing the different density distributions between the two-component Fermi gas 39,40 , while α can be detected using the calibrated single-photon counting modules which allows for the in situ monitor of the intra-cavity light intensity 3 . We thus expect that our results to be feasible within the experimental capabilities.

Discussion
In realistic experiments, there typically exists a shallow harmonic external confining potential, which can be modeled by ω = ⊥ V r r ( ) /2 2 2 , where ω ⊥ is the harmonic trap frequency and r is the harmonic trap radius. In order to estimate its effect, we first use the local density approximation 41 to obtain an effective chemical potential , where μ 0 is the chemical potential at the center of the harmonic trap and μ(r) determines the total density = + ↑ ↓ n n n . Then, the density distribution n(r) and the magnetization distribution m(r) can be solved from 42 In Fig. 5, we plot the density distributions n(r)/n 0 in the radial direction for (a) ω ≥  is the Fermi energy in the trapped systems. As shown in Fig. 5(a), for ω ≥  E F 0 (fully polarized fermions), the density profile does not depend on the effective fermion-photon coupling strength η, which implies that the critical line η η = c (1) for the second-order quantum phase transition from the N-I phase to the SR phase is unaffected by the trapping potential. However, for ω <  E F 0 (partially polarized fermions), the density profile depends strongly on the effective fermion-photon coupling strength η, leading to the modification of the critical line for the first-order quantum phase transition from the N-II phase to the SR phase, as shown in Fig. 5(b). Moreover, the predicted N-II-SR mixed phase only exists near the center of the trap and the corresponding density profile has thus a jump discontinuity, which provides an experimentally observable signature. We also note that the presence of the harmonic trap renders the system to become of finite size, i.e., the density profile vanishes for size > r R r R / / T c T , where r c denotes the size of the trap, as shown in both Fig. 5(a) and 5(b).
In addition, when the decay rate κ, which has the order of MHz in experiments, is much larger than the external atomic degrees of freedom, the relatively slow atoms at times κ  t 1/ feel an average affect of steady-state photons, i.e., Eq. (14). Furthermore, when the system enters the SR phase with a large mean-photon number, the commutator =ˆ † a a [ , ] 1 can be ignored at the leading order results. Thus, it is reasonable to neglect the non-commutability between â and ˆ † a and just set them to their steady-state averaged value 3, 4, 12-25 . Based on the above two arguments, the employed mean-field approximation, which assumes to approximate the photon annihilation operator by its steady-state averaged value, provides a good description of the system within our context. along the radial direction r R / T . The atom-number dependent cavity frequency ω and the cavity decay rate κ are the same as those in Fig. 3. The scaled polarization is taken as = .
In summary, we have analytically investigated the ground-state properties of a 2D polarized degenerate Fermi gas coupled to a high-finesse optical cavity. By solving the photon-number dependent BdG equation, we have found rich quantum phases and phase diagrams, which depend crucially on the fermion-photon coupling strength and the atomic resonant frequency (effective Zeeman field). In particular, for the weak atomic resonant frequency, we have shown that there exists a mixed phase where the N-II and SR phases coexist. In addition, we have revealed a first-order quantum phase transition from the N-II phase to the SR phase, which contrasts to the familiar second-order transition for the case with bosons. Finally, we have presented a parameter estimation and have addressed briefly how to detect these predicted quantum phases and phase diagrams in experiments.

Methods
In this Section, we present a detailed derivation of Eqs (45)-(48). We denote by µ N II SR N II SR the chemical potential in the N-II-SR mixed phase, and let − n N II (n SR ) and µ − N II (μ SR ) denote the atom density and the chemical potential in the N-II (SR) phase, respectively. In terms of the fraction ≤ ≤ , μ SR , α SR , n SR , and m SR are given by  2 corresponding to x 0 = 0, the phase boundary between the N-II-SR mixed phase and the SR phase is also described by η η = c (2) . In addition, for the N-II-SR mixed phase, we can determine µ − − N II SR from the relation Finally, let us prove that the ground state of the N-II-SR mixed phase is stable. Due to the existence of both N-II and SR phases, the scaled ground-state energy in this mixed phase is defined by 37,42 ω η µ µ ω Thus we obtain which are either less than or equal to zero, i.e., the ground state of the N-II-SR mixed phase is stable at η η = c (2) . Substituting Eqs (28), (51), and (56) into Eqs (53) and (57), we derive Eqs (45)-(47). In addition, according to Eq. (50), we obtain Eq. (48).