Optomechanics with interacting Fermi gases: a new approach to detecting spin–charge separation in one-dimensional ultracold atom systems

We consider a one-dimensional two-component interacting Fermi gas confined in a cavity. We show that, taking account of the polarization of the cavity field, one can realize an effective cavity optomechanical model with the spin and charge modes playing the role of coupled mechanical resonators, which gives rise to multistability in a steady state. Then, we propose that, by tuning the weak probe laser under a pump field, the signal of spin–charge separation could be probed explicitly via transmission spectra in the sideband resolved regime. Moreover, the spin–charge modes can be addressed separately by designing the probe field configurations, which may be beneficial for future studies of atom-cavity systems.

The cavity mode is driven by a 'pump' laser of frequency ω L , and a weak 'probe' laser of frequency ω p is added to stimulate the system's optical response. 'Out' is the transmission field. Both the fields are polarization dependent, illustrated with red and green arrows in a circle. (b) Internal energy levels of the two-component atoms with detuning a = ω L − ω a . Here, ω a is the resonant frequency between the ground |g and excited |e states. explicit signal of spin-charge separation could be probed definitely in the sideband regime. Moreover, the spin/charge modes can be addressed separately by designing the probe field configurations. Therefore this technique offers a new feasible scheme beyond the preceding methods.
This paper is organized as follows. In section 2, we consider a 1D spin-1/2 interacting Fermi gas coupled with an optical cavity dispersively. By taking account of the polarization degree of the light field, we derive an effective two-mode cavity optomechanical model with the collective spin/charge modes playing the role of mechanical oscillators. Then, in section 3, we analyze the steady state of such a model, which shows nontrivial multistability behavior. Based on the steady state solution, we calculate the intracavity field quadrature and transmission spectrum response to a weak probe field in section 4, which encodes the explicit signals of spin/charge separation. We also discuss the experimental-related parameters there. Finally, we give a summary in section 5.

The model
The system under investigation is illustrated in figure 1(a), where two-component fermionic atoms of mass M are confined in a 1D trap inside an optical cavity along the cavity axis. The cavity mode of frequency ω 0 is driven by a pump laser, and we also add a weak probe field to the cavity to stimulate the fluctuations of the system. For the following study, we take account of the circular polarization degree of the cavity field, which can be separately driven and probed by the pump and probe field correspondingly. Each circular polarization state of the cavity field couples to an atomic internal state (see figure 1(b)) with resonant energy ω a and induces a quantized potential on atoms in the far-off resonance limit. Then, in the dipole and rotating-wave approximations, the atomic part of Hamiltonian can be written as [20] Here,ˆ σ (x), σ =↑, ↓ is the pseudo-spin atomic field operator for two hyperfine fermionic atoms,ĉ σ is the cavity field operator for left/right polarization, and U σ 0 = U 0 = g 2 0 / a is the optical dipole potential strength for a single intracavity photon with K = 2π/λ c (λ c = 1200 nm) the wave vector of the cavity mode. g 1D = 4πha s M is the strength of contact interaction between fermions with opposite spin, and a s is the effective 1D low-energy s-wave scattering length, which can be tuned by Feshbach resonance.
First, following the standard procedure, we transform the atomic field operator into momentum representation byˆ σ (x) = L −1/2 kf k,σ e ikx , wheref k,σ is the fermion annihilation operator for a plane wave with wave vector k. Then, Hamiltonian (1) can be rewritten aŝ where (k) =h 2 k 2 /2M is the single particle kinetic energy and σ = ω 0 − ω L + U 0 N σ /2 is the effective cavity detuning. Here, we concern the spin-balanced case with N σ = N and σ = .
We shall work in the low photon numbers limit and consider only the lowest momentum transfer of 2K induced by photons. For low temperature and small momentum K k F = π N /L, the particle-hole excitations occur around the Fermi surface (Fermi points in 1D). One may then implement the bosonization procedure [1] by introducing the following bosonic operators:â Here,ρ ν σ (k) = qf ν † k+q,σf ν q,σ are density operators for the right and left moving fermions with ν = R, L. By further introducing the charge and spin density bosonic operatorsb ν k,λ = 1 √ 2 (â ν k,↑ ±â ν k,↓ ), = c, s, which physically correspond to the total density and relative density of the two spin components respectively, the Hamiltonian (2) in density operators readŝ By performing the Bogoliubov transformationsd R k,λ = cosh γ λb 2π v F ±g 1d , we derive an effective cavity optomechanical model with coupled spin-charge modes (other q = 2K modes are irrelevant) Here, the first term describes the charge/spin fluctuations of the 1D interacting Fermi gas, which play the role of mechanical resonators with frequency ω q=2K , 2 are the sound velocities of the charge and spin excitations for g 1D /πv F 1. The second term is the linear coupling between the mechanical modes and cavity

The steady state
To describe the dynamics of the above driven optomechanical model, we introduce the quadratures of the mechanical oscillatorsX λ = νX Then, we arrive at the coupled Heisenberg-Langevin equations dX λ dt =ω λP λ , where κ is the cavity decay rate and s σ in =s σ + δs σ denotes the total amplitude of the external fields. Here,s σ ≡ s σ in represents the pump field, and s = s p + s in is a small perturbation, with s σ p the weak probe field and s in the Markovian noise satisfying . θ is the tunable coupling parameter. We also introduce an effective mechanical damping λ , which is related to the finite lifetime of the spin/charge modes.
This effective cavity optomechanics exhibits rich physics compared with previous works. First, we briefly consider the steady-state behavior of the coupled system. The mean-field solutions of equations (5) areP λ = 0,X λ = −2 √ 2 U λnλ /ω λ , and with η σ = √ 2θ κs σ . Figures 2(a) and (b) show the mean-field intracavity photon numbers versus the symmetrical pump rate and detuning. We find that, because of the strong coupling between spin and charge modes, there exists exotic optical multistability, see figure 2(c).

Intracavity field quadrature response and transmission spectrum
Now, we turn to discuss the system's optical response to the small perturbation around a steady state. Such a response generally comprises the contribution of the probe field and the noise. In this paper, we follow the linear response regime, and the contribution of noise can be separated. Thus, to distill the main physics, but without losing generality, in the following we focus on the probe field response. For a symmetrical pump, we consider the stable branch below the threshold of bistability withn ↑ =n ↓ =n, which gives rise ton c = 2n,X c = −4 √ 2 U cn /ω c andn s = 0,X s = 0. Then, the optical response to the probe field is obtained via a linearization of equations (5) around the steady-state: Here, δX c,s = (δX ↑ ± δX ↓ )/ √ 2, δP c,s = (δP ↑ ± δP ↓ )/ √ 2 represent the cavity-field charge-spin quadratures with δX σ = (δĉ † σ + δĉ σ )/ 2 denotes the corresponding probing field terms with δX σ in = (s σ * p + s σ p )/ We note that, although both the mechanical modes are coupled nonlinearly with the cavity field in equations (5), the fluctuations of spin and charge modes in the above equations (7) can be excited independently, which encodes the explicit signal of spin-charge separation. To see this, we transform equations (7) into frequency space in a rotating frame. Here both the fluctuations of mechanical and cavity field variables oscillate at frequencies ± around the steady-state, with = ω p − ω L being the frequency difference between the probe and pump fields. Then, the intracavity field amplitude can be derived as 5 Here, s λ p = (s ↑ p ± s ↓ p )/2 represents the input charge/spin probe field amplitudes. Before proceeding, we consider the following parameters [16]: L ∼ 100 µm, U 0 2π × 20 kHz, and N 5000 alkali metal atoms. For 6 Li, M ∼ 1 × 10 −26 kg. So that U c,s 2π × (0.39, 0.42) MHz and ω c,s 2 × (7.14, 6.08) MHz for g 1D = 0.5v F . 6 Here we should mention that this theoretical atom number is in fact a little bit harder in current experiments because the cooling of the fermionic gas is not as easy as bosonic gas due to the fermionic Pauli exclusion principle. However, with the foreseeable advancement in the state of the art, we expect that this number can be realized in the foreseeable future. To demonstrate the main physics of our work, here we choose more optimistic parameters for better illustration. Correspondingly, the typical cavity damping κ in experiments is about κ = 2π × 1 MHz. Such parameters satisfy κ ω λ and place the system well in the resolved sideband regime, in which the − part of cavity fluctuations can be neglected [21,22]. In this regime, when the coupling between the cavity mode and the mechanical mode is strong, i.e. U λ √n κ, there exists normal mode splitting [21]. Here, we consider the opposite U λ √n κ [22], where the signal of spin-charge separation could be probed explicitly, see below. Also for cold atom systems, the effective mechanical damping λ (i.e. the lifetime of the collective excitations), which broadens the width of the peaks in the transmission spectrum 7 , is estimated to be λ ∼ (k B T / k F ) 2 ω λ ω λ [8], so we neglect the damping effect in the following discussion. 7 In the sideband resolved regime and weak coupling case ( U λ √n , λ κ), the peaks around ω λ in the transmission spectrum can be described well by a Lorentzian form of ∼ 1 (U λ √n /κ+ λ ) 2 + 4( −ω λ ) 2 . It can be seen that adjusting the probe field configuration to be in-phase (φ = 0, dashed line) with equal amplitudes or out-of-phase (φ = π , dash-dotted line) with equal amplitudes, the charge/spin modes can be addressed separately. In figure 4, We also investigate the impact of detuning on the spectra. It can be seen that although the transmission is generally modified, the peaks of the probe spectra always occur at the charge/spin modes, which are independent of detuning . The frequencies of the charge and spin modes versus the interacting parameter g 1D for g 1D /πv F < 1 are shown in figure 5, which could also be observed in experiments.
Until now, we have mainly focused on the fermionic gas. Experimentally, the above scheme could also be realized in two-component Bose gas by implementing the hydrodynamical theory. The velocities of the charge/spin excitations are u c,s = u 0 √ 1 ± g 12 /g, with g and g 12 the intraspecies and interspecies interactions [11]. We find that the effective frequencies of the collective modes in the Luttinger liquid regime also lie well within the resolved sideband limit. Further studies will consider trapping potentials, where the frequency of collective modes have to be time-averaged in a period because of the position-dependent velocities of the excitations [8].

Conclusions
In summary, we have shown that, by tuning the weak probe laser under a pump field, the intriguing spin-charge separation in 1D quantum liquids can be probed definitely via the optomechanical coupled atom-cavity system. Such experiments allow us to determine the spin and charge modes simultaneously by designing the weak probe field. Furthermore, the twomode optomechanics itself, which exhibits optical multistability, may be of interest for future studies of quantum physics.