Cavity-enhanced metrology in an atomic spin-1 Bose−Einstein condensate

Abstract

Atom interferometer has been proven to be a powerful tool for precision metrology. Here we propose a cavity-aided nonlinear atom interferometer, based on the quasi-periodic spin mixing dynamics of an atomic spin-1 Bose−Einstein condensate trapped in an optical cavity. We unravel that the phase sensitivity can be greatly enhanced with the cavity-mediated nonlinear interaction. The influence of encoding phase, splitting time and recombining time on phase sensitivity are carefully studied. In addition, we demonstrate a dynamical phase transition in the system. Around the criticality, a small cavity light field variation can arouse a strong response of the atomic condensate, which can serve as a new resource for enhanced sensing. This work provides a robust protocol for cavity-enhanced metrology.

Graphical abstract

Keywords

nonlinear atom interferometer / spin-1 Bose−Einstein condensate / spin-mixing dynamics / quantum Fisher information / parameter estimation

1 Introduction

Atom interferometry has become an indispensable tool for both fundamental physics research [1–7], and practical applications [8] such as gravimeters [9, 10], gradiometers [11, 12], and gyroscopes [13–15]. The unique quantum properties of ultracold atomic system provide an underlying platform to improve the sensitivity of atom interferometers beyond standard quantum limit (SQL) [16–21]. Spinor Bose−Einstein condensates (BECs) have been proven to be an ideal candidate for enhancing the sensitivity of atom interferometers beyond SQL owing to the generation of entanglements exploiting spin-exchange collisions [22, 23]. The realization of entanglement enhancement in spinor BECs through parametric amplifiers [24], spin nematic squeezing [25], and quantum phase transitions [26–28] have been demonstrated. Recently, nonlinear interferometry based on time reversal protocols was proposed in which the improvement of the signal-to-noise ratio results from amplifying the signal instead of reducing the quantum noise to circumvent low-noise detection [29–31]. Spin-exchange dynamics of spinor BECs can also be used as a nonlinear mechanism for realizing time reversal readout by controlling phase imprinting [32]. However, the difficulty of inverting the sign of Hamiltonian in an interacting many-body system with large particle numbers poses a challenge to realize time-reversed operation.

To circumvent the difficulty, a closed-cyclic nonlinear interferometer based on the quasi-periodic spin mixing dynamics in a three-mode ⁸⁷Rb atomic spinor condensate was proposed [33] and implemented [34]. The mechanism employs cyclic dynamics which automatically drives the system back to the vicinity of the initial state, thus bypassing time reversal operation. A nonlinear interferometer based on quasi-periodic spin mixing dynamics consists of the following processes shown in Fig.1(a): (i) Initialization with all atoms in the |m_F=0⟩ component. (ii) Splitting through creating paired atoms in |m_F=\pm 1⟩ components based on spin mixing dynamics with a duration time t_1. (iii) Encoding with a phase shift \varphi. (iv) Recombing with a duration time t_2. (v) Readout. When the encoded phase is \varphi =0, the system will return back to the initial state after recombining owing to the periodic dynamics, while it evolves to another state if \varphi \ne 0. Experimentally, the nonlinear interferometer hiring cyclic dynamics has achieved a phase sensitivity of 5 decibels beyond the SQL for a large number of atoms [34]. However, the ubiquitous decoherence [35–37] will hinder the achievable phase sensitivity especially in the long evolution time required for cyclic dynamics.

In previous works [38, 39], we have studied the nonlinear dynamics by coupling the spinor condensate to an optical cavity. The nonlinear interactions between atoms and the intracavity light field provide an extra control knob for the spin mixing dynamics, giving rise to novel many body nonlinear phenomena such as strong matter wave bistability and spin domain formation. Motivated by that, here we explore the new possibility of cavity-aided atom interferometer. It has been proved that the coupling of atoms to cavity can give rise to entangled states useful in quantum metrology [40–42]. Optical cavities can be used to effectively create spin-squeezing, squeezed-state atomic clocks and highly non-classical states through interaction with atoms [43–45]. In this work, we theoretically propose a cavity-aided spin-1 nonlinear atom interferometer based on quasi-periodic spin mixing dynamics. We study the spin mixing dynamics of a spin-1 ⁸⁷Rb atomic condensate in a cavity and investigate the entanglement of the probe state characterized by quantum Fisher information (QFI). The results indicate that the phase sensitivity can be enhanced with cavity aiding. The influence of encoding phase, splitting time and recombining time on phase sensitivity are discussed in detail. Furthermore, we investigate that dynamical phase transition (DPT) in the cavity-condensate coupling system, which is usually defined in terms of nonanalytic behavior of a time-averaged order parameter at a critical point [46–49], can be characterized by an abrupt increase of the QFI, indicating an underlying connection to the bistable phase transition [38]. Criticality has been proved to be a useful resource for enhanced quantum sensing in a spinor condensate [50–53]. We explore the parameter estimation near the critical point with DPT, which provides a new platform for an interferometric protocol that can enable DPT for enhanced sensing.

2 Model

We consider a model of a tightly trapped spin-1 ⁸⁷Rb atomic condensate trapped in a unidirectional ring cavity, as shown in Fig.1(b). Due to the tight trapping, we can apply the single-mode approximation (SMA), under which all spin components have the same spatial wavefunction \phi \left(\mathit{r}\right). The cavity is driven by a coherent laser field with amplitude {\epsilon }_p and frequency {\omega }_p. The cavity mode is described by an annihilation operator \hat{d}, which is \pi-polarized and characterized by a frequency {\omega }_c and a decay rate \kappa. The Hamiltonian under the SMA can be written as [38]

(1)\hat{H}={\hat{H}}_0+\left[U_0\left({\hat{a}}_1^{†}{\hat{a}}_1+{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}\right)-{\delta }_c\right]{\hat{d}}^{†}\hat{d}+\mathrm{i}{\epsilon }_p\left({\hat{d}}^{†}-\hat{d}\right),

with {\hat{H}}_0 describing the dynamics of a spin-1 condensate [54]

(2)\begin{array}{rl}{\hat{H}}_0=& {\lambda }_a({\hat{a}}_1^{†}{\hat{a}}_1^{†}{\hat{a}}_1{\hat{a}}_1+{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}{\hat{a}}_{-1}+2{\hat{a}}_0^{†}{\hat{a}}_0{\hat{a}}_1^{†}{\hat{a}}_1\\ & +2{\hat{a}}_0^{†}{\hat{a}}_0{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}-2{\hat{a}}_1^{†}{\hat{a}}_1{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}+2{\hat{a}}_0^{†}{\hat{a}}_0^{†}{\hat{a}}_1{\hat{a}}_{-1}\\ & +2{\hat{a}}_1^{†}{\hat{a}}_{-1}^{†}{\hat{a}}_0^{}{\hat{a}}_0)+q\left({\hat{a}}_1^{†}{\hat{a}}_1+{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}\right),\end{array}

where {\hat{a}}_{\alpha } (\alpha = \pm 1, 0) is the annihilation operator associated with the condensate mode. Here 2{\lambda }_a=c_2\int \mathrm{d}\mathit{r}{\left|\phi \left(\mathit{r}\right)\right|}^4 is the spatially integrated interaction strength with the spin-exchange interaction coefficient c_2 defined in terms of the s-wave scattering lengths [the explicit form of c_2 is given as c_2= 4\pi {\hslash }^2\left(a_0-a_2\right)/(3m_a), where m_a is the mass of each particle and a_0 (a_2) is the s wave scattering length for spin-1 atoms colliding in symmetric channels of total spin J=0 (J=2)]. q is the quadratic Zeeman shift. Here the magnetization {\hat{a}}_1^{†} {\hat{a}}_1- {\hat{a}}_{-1}^{†}{\hat{a}}_{-1} is conserved such that the linear Zeeman shift can be eliminated via a unitary transform. {\delta }_c= {\omega }_p-{\omega }_c is the cavity-pump detuning and U_0= g^2/\left({\omega }_p-{\omega }_a\right) is the strength of the atom−photon coupling, with g being the dipole coupling constant and {\omega }_a the atomic transition frequency. Furthermore the photon frequency is assumed to be detuned away from the atomic transition frequency such that the atomic upper energy level can be eliminated adiabatically and the photon−atom interaction is essentially of dispersive nature. The transition selection rule allows transitions between |F_g=1, m_g=\pm 1 ⟩ and corresponding states in the excited manifold with the same magnetic quantum numbers |F_e= 1,m_e= \pm 1⟩ while dipole transitions between |F_g=1, m_g=0⟩ and any excited states are forbidden. On the other hand, the atomic population can be redistributed in the ground state manifold via spin exchange collisions, which is utilized to implement nonlinear splitting and recombining in the interferometer. By considering the fact that the cavity decay rate is much larger than the spin oscillation frequency, the cavity field always follows adiabatically the atomic dynamics:

(3)\hat{d}={\displaystyle \frac{{\epsilon }_p}{\kappa -\mathrm{i}\left[{\delta }_c-U_0\left({\hat{a}}_1^{†}{\hat{a}}_1+{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}\right)\right]}}.

Combining Eq. (3) and the corresponding Heisenberg equations of motion for the condensate mode operators, the effective Hamiltonian {\hat{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}} reads

(4){\hat{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}= {\hat{H}}_0- {\displaystyle \frac{{\epsilon }_p^2}{\kappa }}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left[{\displaystyle \frac{{\delta }_c-U_0\left({\hat{a}}_1^{†}{\hat{a}}_1+{\hat{a}}_{-1}^{†}{\hat{a}}_{-1}\right)}{\kappa }}\right].

We anticipate that the proposed scheme can be readily implemented experimentally with recent advances in coupling a ring cavity with cold atoms [55] and BECs [56]. A mean-field treatment is adopted by replacing the operators \hat{d} and {\hat{a}}_{\alpha } with the corresponding c-numbers D=⟨ \hat{d}⟩ and {\chi }_{\alpha }=\sqrt{N_{\alpha }}\exp \left(\mathrm{i}{\theta }_{\alpha }\right) respectively, where N_{\alpha } and {\theta }_{\alpha } represent the number and the phase of internal mode \alpha. Under this replacing, one can obtain the mean-field effective Hamiltonian as [57, 58]

(5)\begin{array}{rl}\frac{H}{N\kappa }=& \overline{q}\left(1-{\rho }_0\right)+c{\rho }_0\left[1-{\rho }_0+\sqrt{{\left(1-{\rho }_0\right)}^2-m^2}\cos \theta \right]\\ & +U\left({\rho }_0\right),\end{array}

with the normalized spin-0 population {\rho }_0=N_0/N and the spinor phase \theta ={\theta }_1+ {\theta }_{-1}- 2{\theta }_0. N=N_1+N_0+ N_{-1} is the total atomic number and m=M/N is the atomic polarization with M=N_1- N_{-1} being the magnetization. N and M are two conserved quantities. The effective cavity-atom counterpart U\left({\rho }_0\right) can be derived by the equations of motion \mathrm{d}{\rho }_0/\mathrm{d}\tau =-2\mathrm{\partial }H/\mathrm{\partial }\theta and \mathrm{d}\theta /\mathrm{d}\tau =2\mathrm{\partial } H/ \mathrm{\partial }{\rho }_0 [58] of two conjugate variables {\rho }_0 and \theta as

(6)U\left({\rho }_0\right)={\eta }^2\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left[{\overline{U}}_0\left(1-{\rho }_0\right)-{\overline{\delta }}_c\right]/N,

where we have introduced dimensionless quantities

c={\displaystyle \frac{2N{\lambda }_a}{\kappa }}, \overline{q}={\displaystyle \frac{q}{\kappa }}, {\overline{U}}_0={\displaystyle \frac{N{U}_0}{\kappa }},\eta ={\displaystyle \frac{{\epsilon }_p}{\kappa }},{\overline{\delta }}_c={\displaystyle \frac{{\delta }_c}{\kappa }}.

3 Spin mixing dynamics

In this section, we illustrate the spin-mixing dynamics of a spin-1 ⁸⁷Rb atomic condensate inside a cavity in a semiclassical phase space [59], to understand the physical origin for achieving high phase sensitivity of our nonlinear interferometer. We consider a ⁸⁷Rb condensate of N=20000 atoms confined in a cavity and a bias magnetic field fixed at 0.23 G corresponding to a quadratic Zeeman shift of q = 2\pi \times 3.84 Hz [34]. The atoms are prepared initially in the |F_g=1, m_g=0⟩ hyperfine ground state (polar state). We adjust the spin-exchange interaction coefficient c_2 and the cavity decay rate \kappa to satisfy \overline{q}=\left|c\right|, at which the initial polar state undergoes a quasi-periodic oscillation [60]. Here we use the semiclassical truncated Wigner approximation (TWA) [61] to simulate the quantum dynamics (see Appendix B).

Fig.2(a) shows the evolution of {\rho }_0 starting from the polar state with all atoms in the |F_g=1,m_g= 0⟩ state in a spin-1 atomic condensate (solid red line) and an atom-cavity coupling system (dashed purple line). The system would not evolve in the mean-field theory due to that polar state serves as fixed point solution in both the case with and without cavity. Here the system would experience quasi-cyclic dynamics and evolve back to the vicinity of the initial state in the first period of the oscillation for both cases. The distribution in the mean field phase space of {\rho }_0 and \theta at different evolution times are shown in Fig.2(b)−(d) (without cavity) and Fig.2(e)−(g) (with cavity aiding). Initially the distribution in {\rho }_0 is tightly packed at the top of the phase space with random spinor phase [shown in Fig.2(b) and (e)]. As evolution proceeds, the distribution converges towards the separatrix (dashed white curve) [58] which divides the phase space into oscillating phases and winding phases [Fig.2(c) and (f)], and disperses along the separatrix [Fig.2(d) and (g)]. The dynamics along the separatrix is highly sensitive to perturbations, which is responsible for the high sensitivity of the nonlinear interferometer. The probability distributions of {\rho }_0 at intermediate evolution times are shown in Fig.2(h) (without cavity) and (i) (with cavity aiding). One can observe a non-Gaussian distribution for both cases. Notably, the distribution in an atom−cavity coupling system can be much broader than that in a bare atomic system, indicating enhanced entanglement resulting from cavity-mediated nonlinear interaction.

4 Phase sensitivity

The phase encoding operator is U_p={\mathrm{e}}^{\mathrm{i}\varphi N_0/2} with \varphi = {\varphi }_1+{\varphi }_{-1}-2{\varphi }_0. Experimentally U_p is realized by quenching the quadratic Zeeman shift q to a large value with microwave dressing for a small variable time \tau, and the relative phase is \varphi = 2q\tau [34]. A nonzero phase shift will break the cyclic dynamics and make the final state phase-dependent. The phase sensitivity of the interferometer can be assessed by Gaussian error propagation as [22]

(7)\mathrm{\Delta }\varphi = \mathrm{\Delta }{\rho }_0/\left|\text{d}⟨{\rho }_0⟩/\text{d}\varphi \right|,

where ⟨{\rho }_0⟩ and \mathrm{\Delta } {\rho }_0 are the mean value and the standard deviation of {\rho }_0 for the final state depending on splitting time t_1 and recombining time t_2.

The optimal phase sensitivity achievable is given by quantum Cramer−Rao bound \mathrm{\Delta }{\varphi }_{\mathrm{Q}\mathrm{C}\mathrm{R}}= 1/ \sqrt{F_Q}, where F_Q is the quantum Fisher information (QFI) [62, 63]. The quantum Fisher information determined by the interferometer output state \hat{\rho }\left(\varphi \right)=\left|\psi \left(\varphi \right)⟩⟨\psi \left(\varphi \right)\right| can be expressed as F_Q\left[\hat{\rho }\left(\varphi \right)\right]=Tr\left[\hat{\rho }\left(\varphi \right){\hat{L}}_{\varphi }^2\right], where {\hat{L}}_{\varphi } is the symmetric logarithmic derivative defined by {\mathrm{\partial }}_{\varphi }\hat{\rho }\left(\varphi \right)=\left[\hat{\rho }\left(\varphi \right){\hat{L}}_{\varphi }+{\hat{L}}_{\varphi }\hat{\rho }\left(\varphi \right)\right]/2 [62, 64]. For pure states {\hat{\rho }}^2\left(\varphi \right)= \hat{\rho }\left(\varphi \right), we have the relation {\hat{L}}_{\varphi }=2{\mathrm{\partial }}_{\varphi }\hat{\rho }\left(\varphi \right). Combining with the definition of the QFI, we can obtain F_Q\left[\hat{\rho }\left(\varphi \right)\right]= 4[⟨{\mathrm{\partial }}_{\varphi } \psi \left(\varphi \right)|{\mathrm{\partial }}_{\varphi }\psi \left(\varphi \right)⟩-{\left|⟨{\mathrm{\partial }}_{\varphi }\psi \left(\varphi \right)|\psi \left(\varphi \right)⟩\right|}^2]. For the output state |\psi \left(\varphi \right)⟩ =\exp \left(i\varphi N_0/2\right)| {\psi }_p⟩ with a probe state |{\psi }_p⟩, the QFI reads

(8)\begin{array}{rl}{F}_Q\left[|{\psi }_p⟩,N_0/2\right]=& 4\left[⟨{\psi }_p|{\left(\frac{N_0}{2}\right)}^2\left|{\psi }_p⟩-⟨{\psi }_p\right|\frac{N_0}{2}|{\psi }_p{⟩}^2\right]\\ =& {\left(\mathrm{\Delta }{N}_0\right)}^2.\end{array}

For separable states, the variance of N_0 is equal to the sum of the variances for individual particles. Then the maximal variance {\left(\mathrm{\Delta }{N}_0\right)}^2 is given by N/4 since the maximal variance of the single particle takes the value of 1 /4. Therefore, the optimal phase sensitivity without quantum entanglement is given by \mathrm{\Delta }{\varphi }_{\mathrm{S}\mathrm{Q}\mathrm{L}}= 2/ \sqrt{N}, which is the standard quantum limit (SQL) for a nonlinear interferometer. We estimate the phase sensitivity with metrological gain as [34]

(9)G=-20{\log }_{10}\left(\mathrm{\Delta }\varphi /\mathrm{\Delta }{\varphi }_{\mathrm{S}\mathrm{Q}\mathrm{L}}\right).

Eq. (9) indicates that higher sensitivity results in larger value of gain.

We first study the quantum Fisher information of the probe state whose entanglement property is responsible for beyond-SQL phase sensitivity. Fig.3(a) shows the normalized QFI F_Q /N of the probe state as a function of spin-mixing time in the first oscillation period without cavity. We find that the phase sensitivity is lower than the SQL () for a probe state with a small evolution time. At larger times, a high phase sensitivity beyond SQL is realized. Fig.3(b) presents F_Q /N as a function of spin-mixing time and cavity-pump detuning in the first oscillation period with cavity aiding. The quantum Fisher information can obtain a larger value as compared with the bare condensate case under appropriate evolution time and cavity-pump detuning, indicating that phase sensitivity can be enhanced with cavity aiding.

We further explore the dependence of metrological gain of a cavity-aided nonlinear interferometer on relative phase. Fig.4(a) shows the metrological gain as a function of relative phase \varphi without cavity (solid red line) and with cavity aiding (dashed purple line). The total evolution time t_1+ t_2 is fixed at 600 ms (without cavity) and 670 ms (with cavity aiding) based on quasi-cyclic dynamics which drives the system back to the vicinity of the initial state [shown in Fig.2(a)]. For a nonlinear interferometer without cavity, the splitting time t_1 is fixed at around 210 ms. While for a cavity-aided nonlinear interferometer, the splitting time t_1 is fixed at around 230 ms and the cavity-pump detuning is set as {\overline{\delta }}_c= -1. These times are chosen such that the QFIs reach the peak values (shown in Fig.3). One can observe that the nonlinear interferometer can beat SQL for small phases (The metrological gain is zero for SQL). We found that the cavity-aided nonlinear interferometer performs better in a small range of the phase. The maximal gain of cavity-aided nonlinear interferometer can be enhanced 3.4 dB at a relative phase \varphi =8.8\times {10}^{-4} rad.

The dependence of phase sensitivity in the cavity-aided nonlinear interferometer on splitting time t_1 is illustrated in Fig.4(b), in which the metrological gain optimized over relative phase \varphi \in \left[0,0.005\right] rad is presented. The solid red line and dashed purple line denote the results without cavity and with cavity aiding respectively. The total evolution time t_1+t_2 is fixed at 600 ms (without cavity) and 670 ms (with cavity aiding). The optimal metrological gain can reach a higher value in the cavity-aided interferometer when choosing an appropriate splitting time, whose performance behaves more capable to saturate the Cramer−Rao bound indicated by the dotted blue line.

We also investigate the dependence of phase sensitivity of the cavity-aided nonlinear interferometer on both splitting time t_1 and recombining time t_2. The results are shown in Fig.5, in which the metrological gain optimized over relative phase \varphi \in \left[0,0.005\right] rad is presented. Fig.5(a) shows the metrological gain of the nonlinear interferometer without cavity aiding, and Fig.5(b)−(d) show the metrological gain of the cavity-aided nonlinear interferometer at {\overline{\delta }}_c=0, {\overline{\delta }}_c= -1 and {\overline{\delta }}_c= -2. We find that the phase sensitivity is beneath the SQL for both small t_1 and t_2 and the first large metrological gain appears at both the spitting time t_1 and the recombining time t_2 around 200 ms. The maximal value of optimal metrological gain of the nonlinear interferometer is marked by a black circle. t_1 and t_2 corresponding to the maxima gain indicate the moments at which the system develops strong entanglement during splitting and returns to the immediate vicinity of the initial polar state. The maximum metrological gain can be improved to above 3 dB. It is noticeable that due to the existence of atomic loss and technical noises, in experiment only a metrological gain below 10 dB can be achieved [34].

The cavity light field affects the sensitivity of interferometer in the sense that the photon number term U_0{\left|D\right|}^2 takes the role of an effective quadratic Zeeman energy shift, which is sensitive to the spin distribution of the condensate as indicated in Eq. (3). The entanglement builds through nonlinear cavity−atom coupling system which can help to improve the sensitivity.

5 An interferometer for an initial coherent spin state (CSS) with ρ₀ ≠ 1

In the previous section, we focused on the case that the condensate is initially prepared in the polar state. Here we discuss the general case with the initial state prepared in an arbitrary coherent spin state (CSS) in which N spin-1 atoms point in the same direction with random spin fluctuations perpendicular to the average spin direction [65]. A CSS |\zeta {⟩}^{\otimes N} can be written as [66]

(10)|\zeta {⟩}^{\otimes N}={\displaystyle \frac{1}{\sqrt{N!}}}{\left({\zeta }_1{\hat{a}}_1^{†}+{\zeta }_0{\hat{a}}_0^{†}+{\zeta }_{-1}{a}_{-1}^{†}\right)}^N|\mathrm{v}\mathrm{a}\mathrm{c}⟩,

where (assuming m=0)

(11)\zeta =\left(\begin{array}{c}{\zeta }_1\\ {\zeta }_0\\ {\zeta }_{-1}\end{array}\right)=\left(\begin{array}{c}\sqrt{{\displaystyle \frac{1-{\rho }_0}{2}}}{\mathrm{e}}^{\mathrm{i}{\theta }_1}\\ \sqrt{{\rho }_0}{\mathrm{e}}^{\mathrm{i}{\theta }_0}\\ \sqrt{{\displaystyle \frac{1-{\rho }_0}{2}}}{\mathrm{e}}^{\mathrm{i}{\theta }_{-1}}\end{array}\right).

Such a CSS can be obtained by performing a unitary transformation on a polar state |\psi {⟩}_{\mathrm{P}\mathrm{S}}={\left(N!\right)}^{-1/2}{a}_0^{†N}|\mathrm{v}\mathrm{a}\mathrm{c}⟩ as

(12)|\zeta {⟩}^{\otimes N}={\displaystyle \frac{1}{\sqrt{N!}}}{\left({\mathrm{e}}^{\mathrm{i}\frac{\sigma }{2}{\hat{Q}}_{yz}}{\hat{a}}_0^{†}{\mathrm{e}}^{-\mathrm{i}\frac{\sigma }{2}{\hat{Q}}_{yz}}\right)}^N|\mathrm{v}\mathrm{a}\mathrm{c}⟩,

with \cos \sigma =\sqrt{{\rho }_0} and \sin \sigma = \sqrt{1-{\rho }_0}. The quadrapole operators are {\hat{Q}}_{ij}={\hat{S}}_i{\hat{S}}_j+{\hat{S}}_j{\hat{S}}_i-4/3{\delta }_{ij} with {\hat{S}}_i being the spin operator and {\delta }_{ij} being the Kronecker delta [67]. In TWA simulation, an initial CSS is sampled by performing a unitary transformation on a polar state (see Appendix A).

We use the CSS with {\rho }_0=0.8, m=0 and a spinor phase \theta = -\pi as the initial state to study spin mixing dynamics of a cavity−condensate coupling system in the phase space. In order to clearly visualize the fluctuations in the phase space, we take the atom number of N=1000. The distribution evolutions of {\rho }_0 starting from the CSS at different evolution times are shown in Fig.6(a)−(c) (without cavity) and Fig.6(d)−(f) (with cavity aiding). The cavity-related parameters are set as {\overline{U}}_0=-10, {\overline{\delta }}_c=-3.8, {\eta }^2=0.3. In Fig.6(a) and (d), the initial CSS is exhibited in the phase space as a distribution with minimal uncertainty. A major difference brought out by cavity aiding lies in that cavity-mediated nonlinearity substantially modifies the topology of the phase diagram, which makes the initial CSS reside on the separatrix and thus the dynamics becomes more sensible to perturbations. Fig.6(g) gives the metrological gain for the same initial CSS as a function of relative phase without cavity (solid red line) and with cavity aiding (dashed purple line). The total evolution time t_1+ t_2 and splitting time t_1 are fixed at 260 ms and 100 ms for both cases. We find that the metrological gain of the nonlinear interferometer without cavity is always beneath the SQL, while the cavity-aided interferometer can achieve a metrological gain up to 11 dB beyond the SQL. Fig.6(h) describes the probability distribution of {\rho }_0 at a evolution time of 100 ms with cavity aiding. One can see that the probe state shows a non-Gaussian distribution indicating the presence of entanglement.

We note that in the case of finite magnetization (m\ne 0), the principles of enhanced sensitivity would be the same as the case of m=0 discussed above. The essence lies in that the system is initially prepared in a state located on the separatrix of the corresponding phase diagram, rendering its subsequent evolution sensitive to perturbations. For an arbitrary CSS, the extra knob provided by the cavity enables that one can manipulate the phase diagram topology to make the CSS a good initial testing state. The inclusion of a finite magnetization would modify the phase diagram as well, which in turn affects the interferometer sensitivity.

6 Parameter estimation harnessing dynamical phase transitions (DPT)

Our previous works have demonstrated matter wave bistability in the cavity−condensate coupling system [38, 39]. Here we explore the possibility of harnessing the dynamical phase transition (DPT) associated with bistability for parameter estimation. The mean-field bistability can be derived from the stationary solutions of mean-field dynamical equations:

(13a){\displaystyle \frac{\mathrm{d}{\rho }_0}{\mathrm{d}T}}=2c{\rho }_0\sqrt{{\left(1-{\rho }_0\right)}^2-m^2}\sin \theta ,

(13b)\begin{array}{rl}\frac{\mathrm{d}\theta }{\mathrm{d}T}=& -2\left(\overline{q}+\frac{{\overline{U}}_0{\left|D\right|}^2}{N}\right)\\ & +2c\left[1-2{\rho }_0+\frac{\left(1-{\rho }_0\right)\left(1-2{\rho }_0\right)-m^2}{\sqrt{{\left(1-{\rho }_0\right)}^2-m^2}}\cos \theta \right],\end{array}

where T=\kappa t is the dimensionless time. Here we restrict ourselves to the case with zero magnetization m=0. For a ferromagnetic spinor condensate (), we show the stationary solutions of {\rho }_0 versus cavity-pump detuning {\overline{\delta }}_c with \theta = 0 in Fig.7(a), based on a set of parameters: c=-0.002, {\overline{U}}_0=-5, {\eta }^2=0.8, \overline{q}=\left|c\right| and N=1000. In the region of (marked by dashed lines), the system exhibits bistable behavior and first-order phase transition takes place within this region [38, 39].

We characterize DPT with the QFI, which is defined as the fidelity susceptibility for a tunable parameter {\delta }_c [51, 52, 68]

(14)F_Q\left({\delta }_c,t\right)= -4{\displaystyle \frac{{\mathrm{\partial }}^{2}F\left({\delta }_c,\text{d}{\delta }_c,t\right)}{\mathrm{\partial }{\left(\text{d}{\delta }_c\right)}^2}}{|}_{\mathrm{d}{\delta }_{\mathrm{c}}\to 0},

where the fidelity F\left({\delta }_c,\text{d}{\delta }_c,t\right)\equiv \left|⟨\psi \left({\delta }_c,t\right)|\psi \left({\delta }_c+\text{d}{\delta }_c,t\right)⟩\right| is the overlap between two dynamical states that differ by a perturbation d{\delta }_c to cavity-pump detuning, equivalent to a Loschmidt echo (LE) [69, 70] F\left({\delta }_c,\text{d}{\delta }_c,t\right)=\left|⟨{\psi }_0\right|{\mathrm{e}}^{\mathrm{i}\hat{H}\left({\delta }_c\right)t}{\mathrm{e}}^{-\mathrm{i}\hat{H}\left({\delta }_c+\text{d}{\delta }_c\right)t}\left|{\psi }_0⟩\right|. It measures the revival of a state |{\psi }_0⟩ experiencing time-forward propagation under \hat{H}\left({\delta }_c\right) followed by reverse evolution with \hat{H}\left({\delta }_c+\text{d}{\delta }_c\right). One can expect that when the system becomes critical with {\delta }_c\to {\delta }_c^{\mathrm{c}\mathrm{r}}, the quantum state evolution behaves singularly and exhibits quite distinct results even for a small d{\delta }_c, resulting in prominent decrease of the fidelity and a high F_Q. In order to achieve the correspondence between DPT and bistable phase transition, we take the CSS with {\rho }_0= 0.75, m=0 and \theta = 0 as the initial state, which is also the steady state away from the bistable region. We focus on the QFI after quenching the detuning {\delta }_c to the bistable region to diagnose DPT. To calculate the QFI, we use the exact diagonalization method to compute the time-evolved state with eigenvector expansion (see Appendix B). The dynamical behavior of F_Q/t^2 versus {\delta }_c is presented in Fig.7(b), where the QFI is scaled with t^2 to absorb the expected long-time growth of F_Q\propto t^2 (Appendix C). Around the critical point {\overline{\delta }}_c\approx -3.63, we observe a prominent increase in the QFI. This suggests that the equilibrium phase transition can be mapped out through DPT in terms of the QFI.

In the estimation theory, the QFI sets the upper bound on the sensitivity of parameter estimation, i.e., \mathrm{\Delta }{\delta }_c\ge \sqrt{F_Q\left({\delta }_c,t\right)}. One can get access to the estimation {\left(\mathrm{\Delta }{\delta }_c\right)}^{-2} through an observable {\hat{N}}_0 as

(15){\left(\mathrm{\Delta }{\delta }_c\right)}_{{\hat{N}}_0}^{-2}= {\displaystyle \frac{{\left|{\mathrm{\partial }}_{{\delta }_c}⟨{\hat{N}}_0⟩\right|}^2}{{\mathrm{\Delta }}^2{\hat{N}}_0}}\le F_Q,

with {\mathrm{\Delta }}^2{\hat{N}}_0=⟨ {\hat{N}}_0^2⟩-⟨{\hat{N}}_0{⟩}^2 representing the variance with respect to the initial CSS. Similar to the definition of the QFI in Eq. (14), the precision estimation also invokes an echo process as follows [Fig.7(d)]: (i) Preparation of initial CSS |{\psi }_0⟩. (ii) Evolution with the unperturbed Hamiltonian \hat{H}\left({\delta }_c\right) for time t. (iii) Backevolution with the perturbed Hamiltonian \hat{H}\left({\delta }_c+\text{d}{\delta }_c\right) for time t. (iv) Measurement of the observable {\hat{N}}_0. To implement the echo experimentally, one needs to reverse the sign of the Hamiltonian \hat{H} such that the system can experience time-reversing evolution. The sign of the Hamiltonian for a spinor condensate {\hat{H}}_0 can be varied via microwave dressing [71–73] and photon-mediated spin-exchange interaction realized through a cavity light field [74–76]. The sign of cavity−condensate coupling Hamiltonian can be manipulated through changing cavity-pump detuning {\delta }_c and pump-atom detuning {\omega }_p- {\omega }_a. In Fig.7(c), we plot the maximum of {\left(\mathrm{\Delta }{\delta }_c\right)}_{{\hat{N}}_0}^{-2}/t^2 over time as a function of {\overline{\delta }}_c and demonstrate that it reproduces the peak in the transient maximum of F_Q/t^2 near {\overline{\delta }}_c\approx -3.63, which identifies the DPT. It shows that the value of {\left(\mathrm{\Delta }{\delta }_c\right)}_{{\hat{N}}_0}^{-2} can approximately approach F_Q, which also indicates that parameter estimation can be enhanced with the onset of criticality. We demonstrate that a small perturbation in the control parameter \mathrm{d}{\delta }_c can give rise to non-negligible variation in the observable, and this is rooted in the sensitive dependence of quantum-state evolution in the deformation of the separatrix, which is well captured through an echo process near the critical points. Our results suggest that DPTs could be harnessed for enhanced sensing by combining dynamical echoes with measurement of simple observables in nonequilibrium many-body systems.

7 Conclusion

In conclusion, we theoretically proposed a cavity-aided nonlinear atom interferometer implemented with an atomic spin-1 Bose−Einstein condensate in an optical cavity. We studied the spin mixing dynamics in this atom−cavity coupling system and the entanglement of the probe state characterized by quantum Fisher information, indicating that the phase sensitivity can be enhanced by cavity aiding. We discussed the influence of encoding phase, splitting time and recombining time on phase sensitivity and demonstrated that cavity aiding can generate a 3 dB improvement of the maximum metrological gain. Furthermore, we investigated the dynamical phase transition in this cavity−condensate coupling system which can be characterized by the quantum Fisher information via a connection to the bistable phase transition. Finally, we demonstrated an enhanced parameter estimation near the dynamical phase transition critical point, which provides a new perspective for an interferometric protocol for criticality enhanced sensing [35, 77]. It is worth noting that one can also take the counter propagating traveling cavity mode into account. In that case we expect that our main results still hold, which require further exploration. Our results not only advance the exploration of cavity-enhanced metrology but also open up new opportunities in experimental investigations in nonlinear atom interferometry and quantum precision metrology.

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