Quantum Dynamics of Cavity–Bose–Einstein Condensates in a Gravitational Field

Abstract

We theoretically studied the quantum dynamics of a cavity–Bose–Einstein condensate (BEC) system in a gravitational field, which is composed of a Fabry–Pérot cavity and a BEC. We also show how to deterministically generate the transient macroscopic quantum superposition states (MQSSs) of the cavity by the use of optomechanical coupling between the cavity field and the BEC. The quantum dynamics of the cavity–BEC system specifically include phase space trajectory dynamics, system excitation number dynamics, quantum entanglement dynamics, and quantum coherence dynamics. We found that the system performs increasingly complex trajectories for larger values of the Newtonian gravity parameter. Moreover, the number of phonon excitations of the system can be increased by coupling the cavity–BEC system to Newtonian gravity, which is analogous to an external direct current drive. The scattering of atoms inside the BEC affects the periodicity of the quantum dynamics of the system. We demonstrate a curious complementarity relation between the quantum entanglement and quantum coherence of cavity–BEC systems and found that the complementarity property can be sustained to some extent, despite being in the presence of the cavity decay. This phenomenon also goes some way to show that quantum entanglement and quantum coherence can be referred to together as quantum resources.

1. Introduction

The research on the interaction of light and matter is a key research area in quantum physics [1,2,3]. Bose–Einstein condensates (BECs) trapped inside a high-finesse optical cavity interact with the dispersion of the optical field inside the cavity to form an effective optomechanical coupling, in which the fluctuations of the BEC atomic field (Bogoliubov modes) act as vibrational modes for the moving mirrors inside the optomechanical cavity [4,5]. Compared with the traditional optomechanical system, the optomechanical coupling between the cavity field and BEC Bogoliubov mode can be increased to strong coupling by increasing the number of atoms [6,7]. Also, there are different kinds of nonlinearities in cavity–BEC systems with optomechanical coupling. The nonlinear properties like the optical Kerr effect [8,9], optical–atomic cross-Kerr effect [10,11], and nonlinear atom–atom interaction [12,13] emerge naturally due to the direct interaction between the atoms and the optical field of the cavity. Thus, the cavity–BEC system with optomechanical coupling is an ideal platform for studying light–matter interactions.

In previous studies, we noted that cavity–BEC systems with optomechanical coupling, utilizing the interaction between the optical and Bogoliubov mode of the BEC, have greatly extended the fundamental theoretical research and practical applications of coherent light–matter interactions, such as optical bistability [14,15], the anti-bunching effect [16], bipartite entanglement [17], quadrature squeezing [18], and phonon cooling [19].

Recent studies have shown that the system response of traditional optomechanical systems can be further modified with an additional mechanical pump. In the presence of a mechanical pump, the group delay of the probe field can be controlled [20]. The mechanical phase, amplitude, and frequency can be controlled independently by electrically driving the mechanical mode in a piezoelectric optomechanical crystal [21]. Moreover, it is possible to replace optical pumping with mechanical pumping for nonreciprocal mode conversion and the breaking of the time-reversal symmetry in an optomechanically active racetrack resonator [22]. However, in previous studies, the mechanical pump has mostly been used in conventional optomechanical systems. We propose that the mechanical pump can also be exploited to manipulate the quantum dynamics of a cavity–BEC system with optomechanical coupling. Here, it is Newtonian gravity that plays the role of the mechanical pump.

Motivated by the above achievements, we theoretically investigated the quantum dynamics of the cavity–BEC system with optomechanical coupling. Different from previous works, we mainly researched the dynamic response of the system to the Newtonian gravity drive and the nonlinear atom–atom interactions within the BEC. We show that the phase space trajectory and system excitation number dynamics can be effectively tuned by the Newtonian gravity drive and atomic scattering inside the BEC. In this article, quantum resources include quantum entanglement and quantum coherence. The concomitant modification of the scattering strength of the atoms inside the BEC leads to a controllable dynamical evolution period of the quantum resource. The tunable range of atomic scattering intensity inside the BEC is large via the Feshbach resonance. Thus, it is shown that the tunable photon–phonon correlation can be easily realized by modulating the atomic parameters.

The remainder of the paper is organized as follows. In Section 2, we introduce the theoretical model of the cavity–BEC system in a gravitational field and obtain the analytical expression of the state vector when the initial state of the system is in the direct product of two coherent states. In Section 3, we show the time trajectories in the phase space and the excitation number dynamics of the system tuned by different parameters. Then, we demonstrate that the multi-component cat states of the cavity field can be prepared deterministically in our system. In Section 4, we first study how to tune the photon–phonon correlation in the cavity–BEC system with the phonon pump and, then, show that the complementarity property of the quantum resource can be sustained to some extent, despite being in the presence of the cavity decay. We finally summarize our work in Section 5.

2. Physical Model and Solution

The cavity–Bose–Einstein condensate (BEC) quantum system is composed of a Fabry–Pérot optical cavity and a cigar-shaped BEC with radiation pressure coupling. As schematically shown in Figure 1, the radiation pressure interaction between the cavity field and the atoms is only in the horizontal direction, which is independent of external gravity. By rotating the whole system around the y-axis, the Newtonian gravity will modify the radiation pressure coupling, as schematically shown in Figure 2. Next, we will introduce the effective Hamiltonian of the cavity–BEC system in the gravitational field and provide the corresponding eigenstates and eigenvalues.

2.1. System Hamiltonian

The cavity–BEC quantum system is composed of a cigar-shaped BEC inside a Fabry–Pérot optical cavity with length L. The cigar-shaped BEC consists of N identical two-level atoms. The transition frequency and mass of the atoms are ${\omega }_b$ and $m_b$, respectively. Thus, the Hamiltonian of the cavity–BEC system in the gravitational field reads

$$H=H_{CB}+H_G.$$

(1)

The BECs have a transverse trapping frequency ${\omega }_{\perp }$, and the longitudinal confinement along the x direction is negligible [23] when it is confined in a cylindrically symmetric trap. Then, the dynamics of the system can be expressed by quantizing the atomic motional degree of freedom along just the x-axis within an effective one-dimensional model. The cavity is driven by a laser through one of the fixed mirrors. The rate of driving is $\eta =\sqrt{2P{\gamma }_a/{\omega }_p}$, where ${\omega }_p$, P, and ${\gamma }_a$ are the laser frequency, power, and decay rate of the cavity, respectively. In our situation, the excited state of the atoms can be adiabatically eliminated. At the same time, the atomic spontaneous emission also can be neglected when the atomic linewidth is orders of magnitude smaller than the detuning ${\Delta }_b={\omega }_p-{\omega }_b$ [24]. We assumed $\hslash =1$ throughout the paper. In the frame rotating at the driving frequency ${\omega }_p$, the Hamiltonian of the cavity–BEC quantum system with radiation pressure coupling [24,25] can be expressed as

$$H_{CB}={\Delta }_a{a}^{†}a+{\int }_{-L/2}^{L/2}d\mathbf{x}{\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)\left[-\frac{1}{2m}{\nabla }^2+V\left(\mathbf{x}\right)+\frac{U_s}{2}{\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)\mathsf{\Psi }\left(\mathbf{x}\right)\right]\mathsf{\Psi }\left(\mathbf{x}\right)+\eta (a+a^{†}),$$

(2)

where a ($a^{†}$) and $\mathsf{\Psi }\left(\mathbf{x}\right)$$\left({\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)\right)$ are denoted as the annihilation (creation) operations of the cavity and atomic field, respectively. The detuning between the cavity and pump laser is ${\Delta }_a={\omega }_a-{\omega }_p$. The atomic back-action on the field is $V\left(\mathbf{x}\right)=U_0{cos}^2\left(k\mathbf{x}\right)a^{†}a$. The optical lattice barrier height per photon is $U_0=g_0^2/{\Delta }_b$, where $g_0$ indicates the vacuum Rabi frequency. The nonlinear interaction within the BEC is expressed as $U_s=4\pi a_s/m_b{L}_0^2$, where $a_s$ indicates the s-wave scattering length [26,27,28], and $L_0$ indicates the waist radius of the optical potential. The external driving process is described by $\eta (a+a^{†})$.

Under weak driving conditions, the intracavity photon number is very low. The optical lattice barrier height $U_0\langle a^{†}a\rangle \le 10{\omega }_R$. ${\omega }_R=k^2/2m_b$ indicates the recoil frequency of the condensate atoms. We only considered the first two symmetric momentum side modes. The momenta of the side modes are $\pm 2k$. The momentum side modes can be generated by the atom–cavity field interaction [29]. Then, the atomic annihilation (creation) field operators $\mathsf{\Psi }\left(\mathbf{x}\right)$$\left({\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)\right)$ of the BEC can be expanded as the single-mode quantum fields due to the parity conservation and the Bogoliubov approximation. The field operators $\mathsf{\Psi }\left(\mathbf{x}\right)$$\left({\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)\right)$ can be expressed as

$$\begin{array}{ccc}\hfill \mathsf{\Psi }\left(\mathbf{x}\right)& =& \sqrt{\frac{N}{L}}+\sqrt{\frac{2}{L}}cos\left(2k\mathbf{x}\right)b,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill {\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)& =& \sqrt{\frac{N}{L}}+\sqrt{\frac{2}{L}}cos\left(2k\mathbf{x}\right)b^{†},\hfill \end{array}$$

(4)

where the first term $\sqrt{N/L}$ indicates a constant number, which is the condensate mode. The second-term annihilation (creation) operator b$\left(b^{†}\right)$ expresses the Bogoliubov mode of the BEC, which corresponds to the quantum fluctuations of the atomic field around the classical condensate mode [29]. By substituting the atomic field operator $\mathsf{\Psi }\left(\mathbf{x}\right)$ and ${\mathsf{\Psi }}^{†}\left(\mathbf{x}\right)$ into Hamiltonian $H_{CB}$, the Hamiltonian of the cavity–BEC quantum system with radiation pressure coupling can be expressed as

$$H_{CB}^{\prime }={\delta }_a{a}^{†}a+{\mathsf{\Omega }}_b{b}^{†}b+{\lambda }_s(b^2+b^{†2})+{\lambda }_{CB}{a}^{†}a(b+b^{†})+{\lambda }_{ck}{a}^{†}a{b}^{†}b+{\lambda }_{sk}{b}^{†2}{b}^2,$$

(5)

where ${\delta }_a=N{U}_0/2+{\Delta }_a$ denotes the Stark-shifted frequency of the cavity. This frequency shift is generated by the cavity–BEC interactions and is related to the number of atoms in the condensate. ${\mathsf{\Omega }}_b=4({\omega }_R+{\lambda }_s)$ indicates the frequency of the Bogoliubov mode, where the s-wave scattering frequency of the atomic collisions is expressed as ${\lambda }_s=2\pi a_{s}N/L{m}_b{L}_0$. ${\lambda }_{CB}=\sqrt{2N}{U}_0/4$ indicates the radiation pressure interaction between the cavity mode and the Bogoliubov mode of the BEC. ${\lambda }_{ck}=U_0/2$ denotes the nonlinearity cross-Kerr coupling between the cavity modes and the BEC. The intrinsic–Kerr nonlinearity interaction in the BEC is ${\lambda }_{sk}=3\pi a_s/L{m}_b{L}_0^2=3{\lambda }_s/8N$.

One can easily find that the ratio of the cross-Kerr interaction ${\lambda }_{ck}$ to the radiation pressure interaction ${\lambda }_{CB}$ is of the order of $1/\sqrt{N}$. Intrinsic–Kerr coupling ${\lambda }_{sk}$ to atom–atom interactions ${\lambda }_s$ is of the order of $1/N$. Thus, the radiation pressure coupling and atomic collisions are dominant for very large N. The simplified Hamiltonian of the cavity–BEC quantum system with radiation pressure coupling can be expressed as

$$H_{CB}^{″}={\delta }_a{a}^{†}a+{\mathsf{\Omega }}_b{b}^{†}b+{\lambda }_s(b^2+b^{†2})+{\lambda }_{CB}{a}^{†}a(b+b^{†}).$$

(6)

This study focuses on the dynamical evolution of the system. Thus, the cavity is driven by the external laser drives for a limited time until the cavity mode and the Bogoliubov mode of the BEC are prepared in the specified initial states. Then, the external drives will be shut down, and the system will continue to evolve by itself.

As schematically shown in Figure 2, the gravitational potential is added to the Hamiltonian. The gravitational potential is expressed as

$$H_G=xmgcos\theta .$$

(7)

where m denotes the mass of the mechanical oscillator, g denotes the gravitational acceleration, $\theta$ is the angle from the horizontal axis, and x is the position operator [30,31]. The radiation pressure interaction is established between the cavity field and the Bogoliubov mode of the BEC. The coupling constant ${\lambda }_{CB}$ between the cavity and the BEC can be on the order of ${10}^8$ Hz. The gravity coupling strength and the cavity–BEC coupling strength will be approximately equal when the rotation angle of the cavity is close to $\pi /2$ [31]. The Bogoliubov mode of the BEC plays the role of the vibrational mode of a moving mirror or a membrane [4,5,6]. Thus, the position operator $x=(b+b^{†})/\sqrt{2m{\mathsf{\Omega }}_b}$ in our situation [31,32]. With the addition of Newtonian gravity, the Hamiltonian of the system is

$$\begin{array}{ccc}\hfill H_{tot}& =& H_{CB}^{″}+H_G\hfill \\ & =& {\delta }_a{a}^{†}a+{\mathsf{\Omega }}_b{b}^{†}b+{\lambda }_s(b^2+b^{†2})+{\lambda }_{CB}{a}^{†}a(b+b^{†})+g_{0}cos\theta (b+b^{†}),\hfill \end{array}$$

(8)

where $g_0=g\sqrt{\frac{m}{2{\mathsf{\Omega }}_b}}$ is the renormalized gravity coupling parameter.

2.2. The Eigenvalues and Eigenstates of the System

Through the displacement and squeezing unitary transformation (i.e., see Appendix A), the Hamiltonian of the system will become a diagonal form as follows:

$$\begin{array}{ccc}\hfill H_{tot}^{″}& =& S\left(\zeta \right)D\left(A\right)H_{tot}{D}^{†}\left(A\right)S^{†}\left(\zeta \right)\hfill \\ & =& {\delta }_a^{\prime }{a}^{†}a+{\mathsf{\Omega }}_b^{\prime }{b}^{†}b-{\lambda }_{CB}^{\prime }{\left(a^{†}a\right)}^2.\hfill \end{array}$$

(9)

The eigenvalues and eigenstates of $H_{tot}^{″}$ are

$$\begin{array}{ccc}\hfill E_{n,m}& =& {\delta }_a^{\prime }n+{\mathsf{\Omega }}_b^{\prime }m-{\lambda }_{CB}^{\prime }{n}^2,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill |n,m\rangle & =& |n\rangle \otimes |m\rangle ,\hfill \end{array}$$

(11)

where $|n\rangle$ and $|m\rangle$, respectively, denote the harmonic-oscillator-number states of the optical mode and the Bogoliubov mode. Then, the eigensystem of the Hamiltonian $H_{tot}$ can be expressed by

$$\begin{array}{ccc}\hfill H_{tot}^{″}|n\rangle |m\rangle & =& E_{n,m}|n\rangle |m\rangle ,\hfill \\ \hfill H_{tot}{D}^{†}\left(A\right)S^{†}\left(\zeta \right)|n\rangle |m\rangle & =& E_{n,m}{D}^{†}\left(A\right)S^{†}\left(\zeta \right)|n\rangle |m\rangle ,\hfill \\ \hfill H_{tot}|n\rangle |\tilde{m}\left(n\right)\rangle & =& E_{n,m}|n\rangle |\tilde{m}\left(n\right)\rangle ,\hfill \end{array}$$

(12)

where the n-photon displacement squeezing number states $|\tilde{m}\left(n\right)\rangle$ are defined by

$$\begin{array}{ccc}\hfill |\tilde{m}\left(n\right)\rangle & =& D^{†}\left(A_n\right)S^{†}\left(\zeta \right)|m\rangle ,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill A_n& =& \frac{{\lambda }_{CB}n+g_{0}cos\theta }{2{\lambda }_s+{\mathsf{\Omega }}_b}.\hfill \end{array}$$

(14)

Thus, a general state of the cavity–BEC system in a gravitational field can be expressed as

$$|\phi \rangle =\sum _{n=0}^{\infty }\sum _{m=0}^{\infty }{C}_{n,m}|n\rangle |\tilde{m}\left(n\right)\rangle .$$

(15)

3. Dynamical Characteristics of the System

3.1. State Vector Dynamics

In this section, we are going to study the quantum dynamics of the quantum system in the Schrödinger picture when the damping processes is neglected, i.e., the decoherence times of both the optical and the atomic fields is larger than the time interval in which we observe the system. In order to study the quantum dynamics of the quantum system in the Schrödinger picture, the time evolution operator $U=exp\left(-i{H}_{tot}t\right)$ and the wave function $\left|\psi \right(t)\rangle$ need to be calculated. According to Equation (9), the Hamiltonian $H_{tot}$ can be expressed as

$$H_{tot}=D^{†}\left(A\right)S^{†}\left(\zeta \right)H_{tot}^{″}S\left(\zeta \right)D\left(A\right).$$

(16)

Thus, the time evolution operator can be expressed as

$$\begin{array}{ccc}\hfill U\left(t\right)& =& exp\left(-it{H}_{tot}\right)\hfill \\ & =& D^{†}\left(A\right)S^{†}\left(\zeta \right)exp\left(-it{H}_{tot}^{″}\right)S\left(\zeta \right)D\left(A\right).\hfill \end{array}$$

(17)

Then, we can investigate the quantum dynamics of the state vector of the system by using the time evolution operator Equation (17). In this case, we assumed that both the optical mode and the Bogoliubov mode of the BEC have been initially prepared to Glauber-coherent states. The initial state vector of the system is $|\psi \left(0\right)\rangle ={|\alpha \rangle }_a\otimes {|\beta \rangle }_b$, where the indices a and b are denoted as the optical field of the cavity and the Bogoliubov mode of the condensate, respectively. We assumed $\alpha$, $\beta$, and $\zeta$ are real numbers. Thus, the quantum dynamics of the state vector of the system is

$$\begin{array}{ccc}\hfill \left|\psi \right(t)\rangle & =& U\left(t\right)\left|\psi \right(0)\rangle \hfill \\ & =& D^{†}\left(A\right)S^{†}\left(\zeta \right)exp\left(-it{H}_{tot}^{″}\right)S\left(\zeta \right)D\left(A\right)|\psi \left(0\right)\rangle .\hfill \end{array}$$

(18)

By expanding the Glauber-coherent state of the optical mode in terms of the number states, the state vector of the system can be rewritten as

$$\begin{array}{ccc}\hfill \left|\psi \right(t)\rangle & =& e^{-\frac{{\alpha }^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}{e}^{-it\left({\delta }_a^{\prime }n-{\lambda }_{CB}^{\prime }{n}^2\right)}|n\rangle \hfill \\ & & \otimes D^{†}\left(A_n\right)S^{†}\left(\zeta \right)|\zeta e^{-2it{\mathsf{\Omega }}_b^{\prime }},{\beta }_n{e}^{-it{\mathsf{\Omega }}_b^{\prime }}\rangle ,\hfill \end{array}$$

(19)

where $A_n$ and ${\beta }_n$ are defined as follows:

$$\begin{array}{ccc}\hfill A_n& =& \frac{{\lambda }_{CB}n+g_{0}cos\theta }{{\mathsf{\Omega }}_b+2{\lambda }_s},\hfill \end{array}$$

(20)

$$\begin{array}{ccc}\hfill {\beta }_n& =& A_n+\beta \hfill \end{array}$$

(21)

The last expression in this equation $|\zeta e^{-2it{\mathsf{\Omega }}_b^{\prime }},{\beta }_n{e}^{-it{\mathsf{\Omega }}_b^{\prime }}\rangle$ is the free evolution of a coherent squeezed state, which is defined as

$$|\zeta e^{-2it{\mathsf{\Omega }}_b^{\prime }},{\beta }_n{e}^{-it{\mathsf{\Omega }}_b^{\prime }}\rangle =e^{-it{\mathsf{\Omega }}_b^{\prime }{b}^{†}b}S\left(\zeta \right)D\left({\beta }_n\right)|0\rangle .$$

(22)

Then, the reduced density operator ${\rho }_b$ of the Bogoliubov mode can be obtained by tracing over the degrees of freedom of the optical mode, i.e., ${\rho }_b={\mathrm{Tr}}_a\left[|\psi \left(t\right)\rangle \langle \psi \left(t\right)|\right]$. Thus, the reduced density operator ${\rho }_b$ of the Bogoliubov mode can be expressed as

$${\rho }_b\left(t\right)=e^{-{\alpha }^2}\sum _{n=0}^{\infty }\frac{{\alpha }^{2n}}{n!}|{\varphi }_n\left(t\right)\rangle \langle {\varphi }_n\left(t\right)|,$$

(23)

where $|{\varphi }_n\left(t\right)\rangle$ is a vector in the Hilbert space of the Bogoliubov mode. The state vector $|{\varphi }_n\left(t\right)\rangle$ is defined as

$$|{\varphi }_n\left(t\right)\rangle =D^{†}\left(A_n\right)S^{†}\left(\zeta \right)|\zeta e^{-2it{\mathsf{\Omega }}_b^{\prime }},{\beta }_n{e}^{-it{\mathsf{\Omega }}_b^{\prime }}\rangle$$

(24)

Analogously, the reduced-density operator ${\rho }_a$ of the cavity mode can be calculated by tracing over the degrees of freedom of the Bogoliubov mode, i.e., ${\rho }_a={\mathrm{Tr}}_b\left[|\psi \left(t\right)\rangle \langle \psi \left(t\right)|\right]$. Thus, the reduced density operator ${\rho }_a$ of the cavity mode can be expressed as

$$\begin{array}{c}\hfill {\rho }_a\left(t\right)=e^{-{\alpha }^2}\sum _{n,n^{\prime }=0}^{\infty }\frac{{\alpha }^n{\alpha }^{n^{\prime }}}{\sqrt{n!n^{\prime }!}}{e}^{it\left[{\lambda }_{CB}^{\prime }\left(n^2-n^{\prime 2}\right)-{\delta }_a^{\prime }\left(n-n^{\prime }\right)\right]}\langle {\varphi }_{n^{\prime }}\left(t\right)|{\varphi }_n\left(t\right)\rangle |n\rangle \langle n^{\prime }|,\end{array}$$

(25)

where $\langle {\varphi }_{n^{\prime }}\left(t\right)|{\varphi }_n\left(t\right)\rangle$ and $f_{n^{\prime }n}$ are, respectively, defined as follows:

$$\begin{array}{ccc}\hfill \langle {\varphi }_{n^{\prime }}\left(t\right)|{\varphi }_n\left(t\right)\rangle & =& exp\left[-\left({\beta }_{n^{\prime }}^2+{\left|f_{n^{\prime }n}+{\beta }_n\right|}^2-2{\beta }_{n^{\prime }}\left(f_{n^{\prime }n}+{\beta }_n\right)-{\beta }_n\left(f_{n^{\prime }n}-f_{n^{\prime }n}^{*}\right)\right)/2\right],\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill f_{n^{\prime }n}& =& \frac{{\lambda }_{CB}{e}^{-\zeta }\left(n^{\prime }-n\right)\left(e^{it{\mathsf{\Omega }}_b^{\prime }}cosh\zeta +e^{-it{\mathsf{\Omega }}_b^{\prime }}sinh\zeta \right)}{{\mathsf{\Omega }}_b+2{\lambda }_s}\hfill \end{array}$$

(27)

In our paper, the dynamical characteristics of the state vector $|\psi \left(t\right)\rangle$ will be visualized by the trajectories in the phase space, phonon number dynamics, and macroscopic quantum superposition.

3.2. Time Trajectories in Phase Space

Phase space trajectories can provide an intuitive understanding of the statistical properties of quantum states. These trajectories describe the path of motion of a quantum system during its time evolution. By counting a large number of trajectories, one can obtain the average or probability distributions of the macroscopic properties of the system, such as the energy, angular momentum, and dynamical behavior.

The dynamics of the state $\left|\psi \right(t)\rangle$ can be visualized by computing the expectation values of the cavity (Bogoliubov) field quadratures $X_a=\left(a^{†}+a\right)/\sqrt{2}\left(X_b=\left(b^{†}+b\right)/\sqrt{2}\right)$ and $P_a=i\left(a^{†}-a\right)/\sqrt{2}\left(P_b=i\left(a^{†}-a\right)/\sqrt{2}\right)$. Combined with the cavity field density operator ${\rho }_a\left(t\right)$ and Bogoliubov mode density operator ${\rho }_b\left(t\right)$, the expectation values of $\langle a\rangle$ and $\langle b\rangle$ are given by

$$\begin{array}{ccc}\hfill \langle a\rangle & =& \mathrm{Tr}\left[{\rho }_a\left(t\right)a\right]\hfill \\ & =& e^{-{\alpha }^2}\sum _{n=1}^{\infty }\frac{{\alpha }^{2n-1}}{\left(n-1\right)!}{e}^{it\left[{\lambda }_{CB}^{\prime }\left(2n-1\right)-{\delta }_a^{\prime }\right]}{e}^{\frac{{\beta }_n}{2}\left(f_{12}-f_{12}^{*}\right)}\hfill \\ & & \times e^{-\frac{1}{2}\left[{\beta }_{n-1}^2+{\left|f_{12}+{\beta }_n\right|}^2-2{\beta }_{n-1}\left(f_{12}+{\beta }_n\right)\right]},\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill \langle b\rangle & =& \mathrm{Tr}\left[{\rho }_b\left(t\right)b\right]\hfill \\ & =& \left(\beta +\frac{{\lambda }_{CB}{\alpha }^2+g_{0}cos\theta }{2{\lambda }_s+{\mathsf{\Omega }}_b}\right)\left[\frac{f_{21}\left(2{\lambda }_s+{\mathsf{\Omega }}_b\right)}{{\lambda }_{CB}}-1\right]+\beta .\hfill \end{array}$$

(29)

The expected value $\langle X_a\rangle =\sqrt{2}\mathrm{Re}\langle a\rangle$, $\langle X_b\rangle =\sqrt{2}\mathrm{Re}\langle b\rangle$ and $\langle P_a\rangle =\sqrt{2}\mathrm{Im}\langle a\rangle$, $\langle P_b\rangle =\sqrt{2}\mathrm{Im}\langle b\rangle$. For evolution without decoherence, the time trajectories of the subsystem in phase space can be seen for different system parameters in Figure 3.

Figure 3 describes the effects of gravity parameter $\overline{g}=g_{0}cos\theta$ and Bogoliubov mode internal coupling ${\lambda }_s$ on the system dynamics in the phase space. In Figure 3a,b, we plot the time trajectories of the cavity field for one period in phase space, i.e., ${\lambda }_s=0$. One can find that the cavity field exhibits increasingly complex trajectories for larger values of $\overline{g}$. Moreover, the length of the trajectory in the phase space will grow with the value of $\overline{g}$ over the same time period. Also, the overall extent of the trajectory in the phase space decreases as the value of $\overline{g}$ increases, and this is particularly true for the momentum dimension of the trajectory in the phase space.

Comparing Figure 3a,c, we can see that the periodicity of the trajectory of the cavity field in the phase space is destroyed when the Bogoliubov mode internal coupling ${\lambda }_s$ is nonzero. The other parameters are identical in Figure 3a,c. By depicting the phase space trajectory over a long period of time $\left(t>20\pi \right)$, we found that the trajectory does not return to the initial position when the coupling parameter ${\lambda }_s$ is nonzero. The same phenomenon can be found by comparing Figure 3b,d. We conjecture that it may be Bogoliubov mode internal coupling that causes the subsystem of the cavity field to generate chaos. In subsequent studies, we will gradually try to confirm the conclusion.

In Figure 3e, we plot the time trajectories of the Bogoliubov mode for one period in the phase space when the gravity parameter $\overline{g}$ takes various values. The trajectory of the Bogoliubov mode in the phase space is simple compared to the phase space trajectory of the cavity field. Unlike the cavity field, the trajectory of the Bogoliubov mode in the phase space does not exhibit more complexity as the value of $\overline{g}$ increases. The length and overall extent of the trajectory in the phase space will grow with the value of $\overline{g}$ over the same time period. Comparing Figure 3e,f, the Bogoliubov mode internal coupling will also lengthen the period of the trajectory of the Bogoliubov mode in the phase space. This is attributed to the entanglement properties of the photons and phonons in our system. It is worth noting that the phase space trajectory of the Bogoliubov mode can return to the initial position after a slightly longer dynamical evolution when the system parameters are the same as in Figure 3f. It is just going to take more time.

3.3. Excitation Number Dynamics

The total system excitation number is also important for realizing quantum information processing and quantum communication. For example, in quantum entanglement and quantum state transport, the total system excitation number can be used to characterize the properties and coherence of quantum states.

In our system, the total excitation number of the system is derived from the cavity field and Bogoliubov mode, i.e., $N_{tot}=N_a+N_b=\langle a^{†}a\rangle +\langle b^{†}b\rangle$. However, it is worth noting that the number operator $a^{†}a$ is commutated with Hamiltonian $H_{tot}$, i.e., $\left[a^{†}a,H_{tot}\right]=0$. This indicates that the excitation number of the cavity field is conserved during dynamical evolution without decoherence. $N_a$ is decided by the initial states. Thus, the dynamical evolution of the total excitation number will be characterized by the evolution of the Bogoliubov mode excitation number $N_b$. Using the density operator ${\rho }_b\left(t\right)$, the Bogoliubov mode excitation number $N_b$ is calculated in the following way:

$$\begin{array}{ccc}\hfill N_b& =& \langle b^{†}b\rangle =\mathrm{Tr}\left[{\rho }_b\left(t\right)b^{†}b\right]\hfill \\ & =& e^{-{\alpha }^2}\sum _{n=0}^{\infty }\frac{{\alpha }^{2n}}{n!}{\left|\left(A_n+\beta \right)\frac{f_{21}\left(2{\lambda }_s+{\mathsf{\Omega }}_b\right)}{{\lambda }_{CB}}-A_n\right|}^2.\hfill \end{array}$$

(30)

For decoherence-free evolution, the excitation number dynamics can be seen for different system parameters in Figure 4.

Figure 4 describes the effects of the gravity parameter $\overline{g}=gcos\theta$ and Bogoliubov mode internal coupling ${\lambda }_s$ on the Bogoliubov mode excitation number. The excitation number of the Bogoliubov mode exhibits a certain periodicity for decoherence-free evolution. In Figure 4a, the excitation number of the Bogoliubov mode exhibits different periodicities for different values of ${\lambda }_s$. The period of the excitation number is extended with the increase of ${\lambda }_s$. At the middle of each period, the minimum value of the phonon number decreases with increasing coupling parameter ${\lambda }_s$$\left(\mathrm{i}.\mathrm{e}.,D_1>D_2>D_3\right)$. The Bogoliubov mode internal coupling ${\lambda }_s$ leads to a more-complex dynamical vibrational situation for the phonon number, which is analogous to the phase space trajectory. It is evident that the scattering within the atoms has a significant effect on the energy of the system. Thus, the internal coupling ${\lambda }_s$ of the Bogoliubov mode can affect the period of the dynamical evolution of the system. Periodic entanglement and disentanglement of photons and phonons, which are related to ${\lambda }_s$, determine the periodicity of the excitation number of the Bogoliubov mode during its evolution in our situation. In order to see this more clearly, we calculate the linear entropy for the traced-out density operator of the cavity field or Bogoliubov mode in Section 4.1.

In Figure 4b, one can find that the excitation number at some point during the dynamical evolution increases with the gravity parameter $\overline{g}=gcos\theta$, except for the minimum value. The deep value of the phonon number increases with increasing gravity parameter $\overline{g}=gcos\theta$$\left(\mathrm{i}.\mathrm{e}.,D_6>D_5>D_4\right)$ in each period. According to the Hamiltonian $H_{tot}$, the gravity term is analogous to the phonon drive of the direct current type. The gravity drive provides phonons, and this drive has a certain stability, which may be used to resist the negative effects of the external noise on the system. Therefore, it is favorable to tilt the whole device at an angle to increase the number of phonons in our system.

3.4. Transient Cat States

By replacing the time t with the dimensionless time $\tau ={\mathsf{\Omega }}_b^{\prime }t$, the state vector $|\psi \left(t\right)\rangle$ of the system can be rewritten as

$$\begin{array}{ccc}\hfill \left|\psi \right(\tau )\rangle & =& e^{-\frac{{\alpha }^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}{e}^{-i\tau \left(\frac{{\delta }_a^{\prime }}{{\mathsf{\Omega }}_b^{\prime }}n-\frac{{\lambda }_{CB}^{\prime }}{{\mathsf{\Omega }}_b^{\prime }}{n}^2\right)}|n\rangle \hfill \\ & & \otimes D^{†}\left(A_n\right)S^{†}\left(\zeta \right)|\zeta e^{-2i\tau },{\beta }_n{e}^{-i\tau }\rangle ,\hfill \end{array}$$

(31)

We found that a pure state $\left|\psi \right(\tau )\rangle$ completely decouples the cavity field and Bogoliubov mode at $\tau =2k\pi$$(k\in Z)$ for any other system parameters in decoherence-free evolution. When dimensionless time $\tau$ takes the value of $2\pi$, the system state vector $\left|\psi \right(\tau =2\pi )\rangle$ becomes

$$\begin{array}{ccc}\hfill \left|\psi \right(\tau =2\pi )\rangle & =& e^{-\frac{{\alpha }^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}{e}^{-i2\pi \left(\frac{{\delta }_a^{\prime }}{{\mathsf{\Omega }}_b^{\prime }}n-\frac{{\lambda }_{CB}^{\prime }}{{\mathsf{\Omega }}_b^{\prime }}{n}^2\right)}|n\rangle \hfill \\ & & \otimes D^{†}\left(A_n\right)S^{†}\left(\zeta \right)|\zeta ,{\beta }_n\rangle \hfill \\ & =& e^{-\frac{{\alpha }^2}{2}}\sum _{n=0}^{\infty }\frac{{\alpha }^n}{\sqrt{n!}}{e}^{-i2\pi \left({\delta }_a^{″}n-{\lambda }_{CB}^{″}{n}^2\right)}|n\rangle \otimes |\beta \rangle ,\hfill \end{array}$$

(32)

where ${\delta }_a^{″}={\delta }_a^{\prime }/{\mathsf{\Omega }}_b^{\prime }$ and ${\lambda }_{CB}^{″}={\lambda }_{CB}^{\prime }/{\mathsf{\Omega }}_b^{\prime }$. The cavity field part of the state vector $\left|\psi \right(\tau =2\pi )\rangle$ is the Yurke–Stoler-like state [33,34], which can be recognized as a quantum superposition state by a variety of Glauber-coherent states for different values of ${\lambda }_{CB}^{″}$. We assumed ${\lambda }_{CB}^{″}=1/2P$ ($P=2,3,4,\cdots ,\infty$). Then, the phase parameter in $\left|\psi \right(\tau =2\pi )\rangle$ can be rewritten as a sum of P terms [35]:

$$e^{i\pi n^2/P}=\frac{1}{\sqrt{P}}\sum _{k=1}^{P}exp\left(-i{\xi }_k\right){\left[-exp\left(\frac{-2ik\pi }{P}\right)\right]}^n,$$

(33)

where phase factor ${\xi }_k$ is decided by parameter P. For example: ${\xi }_1=-\pi /4$ and ${\xi }_2=\pi /4$ when $P=2$, ${\xi }_1={\xi }_2=-\pi /6$, and ${\xi }_3=\pi /2$ when $P=3$. The above results can be obtained by solving a system of equations consisting of P equations. Thus, the specific value of ${\xi }_k$ can be calculated when the parameter P is determined by using Equation (33). There are many mathematical processes involved. Based on Equation (33), the cavity field part of the state vector $\left|\psi \right(\tau =2\pi )\rangle$ is superimposed by the coherent states of P components:

$$\begin{array}{ccc}\hfill {|\psi (\tau =2\pi )\rangle }_{1/2P}& =& \frac{1}{\sqrt{P}}\sum _{k=1}^{P}exp\left(-i{\xi }_k\right)\left|-\alpha exp\left(-i2\pi {\delta }_a^{″}\right)exp\left(-i2k\pi /P\right)\right.〉\otimes |\beta \rangle \hfill \\ & =& \frac{1}{\sqrt{P}}\sum _{k=1}^{P}exp\left(-i{\xi }_k\right)\left|-{\alpha }_k\right.〉\otimes |\beta \rangle ,\hfill \end{array}$$

(34)

where k is a positive integer and ${\alpha }_k=\alpha exp\left(-i2\pi {\delta }_a^{″}\right)exp\left(-i2k\pi /P\right)$. $\left|-{\alpha }_k\rangle \right.$ is a coherent state with displacement $-{\alpha }_k$. Obviously, the multi-component cat states of the cavity field can be obtained by adjusting the coupling ratio between the effect of the coupling of the cavity–Bogoliubov mode and the frequency of the Bogoliubov mode.

4. Quantum Resource Dynamics

By studying quantum resource dynamics, we can uncover how resources such as energy, information, and entanglement change and flow in quantum systems over time. This helps us gain a better understanding of the fundamental principles and properties of quantum mechanics. Additionally, research in quantum resource dynamics contributes to the development of quantum information science and quantum technology, providing a theoretical foundation and guidance for applications in quantum computing, quantum communication, and quantum simulation, among other fields. Quantum entanglement and coherence have been recognized as important physical resources. Next, we will, respectively, characterize the quantum entanglement and quantum coherence dynamics of the system.

4.1. Quantum Entanglement Dynamics

According to the dynamical characteristics of the system, we found that the cavity field and Bogoliubov mode periodically entangle and disentangle during its evolution without decoherence. In order to illustrate this phenomenon more clearly, the linear entropy $S\left(t\right)$ for the Bogoliubov mode density operator ${\rho }_b\left(t\right)$ will be calculated in the following. The linear entropy can tell us about the entanglement between the cavity field and Bogoliubov mode. The definition of the linear entropy $S\left(t\right)$ is

$$S\left(t\right)=1-\mathrm{Tr}\left[{\rho }_b^2\left(t\right)\right].$$

(35)

The density operator ${\rho }_b\left(t\right)$ of the Bogoliubov mode has the following form:

$${\rho }_b\left(t\right)=e^{-{\alpha }^2}\sum _{n=0}^{\infty }\frac{{\alpha }^{2n}}{n!}|{\varphi }_n\left(t\right)\rangle \langle {\varphi }_n\left(t\right)|.$$

(36)

Next, the linear entropy $S\left(t\right)$ of the Bogoliubov mode is calculated as follows:

$$\begin{array}{ccc}\hfill S\left(t\right)& =& 1-\mathrm{Tr}\left[{\rho }_b^2\left(t\right)\right]\hfill \\ & =& 1-e^{-2{\alpha }^2}\sum _{n^{\prime },n=0}^{\infty }\frac{{\alpha }^{2n^{\prime }}{\alpha }^{2n}}{n^{\prime }!n!}{\left|\langle {\varphi }_{n^{\prime }}\left(t\right)|{\varphi }_n\left(t\right)\rangle \right|}^2,\hfill \end{array}$$

(37)

where the inner product of the last expression in this equation is shown in Equation (26).

The results can be found in Figure 5 for decoherence-free evolution within two cycles. The linear entropy $S\left(t\right)$ increases until $t=\pi$. At this point, the entanglement between the cavity and the BEC reaches its maximum value. As the system continues to evolve, the pure state completely decouples the cavity field and Bogoliubov mode at lab frame time $t=2\pi$ when the atomic scattering inside the Bogoliubov mode is negligible, i.e., ${\lambda }_s=0$. The dynamical evolution of the linear entropy over a period will be complicated when there is atomic scattering inside the Bogoliubov mode. According to Figure 5a, one can find that the linear entropy $S\left(t\right)$ of the Bogoliubov mode exhibits different periodicities and waveforms for different values of ${\lambda }_s$. The undulation of the waveform in a period is greater when the value of ${\lambda }_s$ increases. The period of the linear entropy $S\left(t\right)$ is extended with the increase of ${\lambda }_s$. Thus, atomic scattering inside the Bogoliubov mode will have an effect on the entanglement dynamics between the Bogoliubov mode and the cavity field. This is attributed to the time evolution operator $U\left(t\right)$. The time evolution operator $U\left(t\right)$ can be rewritten in the following direct product form:

$$\begin{array}{ccc}\hfill U\left(t\right)& =& D^{†}\left(A\right)S^{†}\left(\zeta \right)exp\left(-it{H}_{tot}^{″}\right)S\left(\zeta \right)D\left(A\right)\hfill \\ & =& U\left(a\right)\otimes U\left(b\right)\hfill \\ & =& exp\left[-it\left({\delta }_a^{\prime }{a}^{†}a-{\lambda }_{CB}^{\prime }{\left(a^{†}a\right)}^2\right)\right]\hfill \\ & & \otimes D^{†}\left(A\right)S^{†}\left(\zeta \right)exp\left(-it{\mathsf{\Omega }}_b^{\prime }{b}^{†}b\right)S\left(\zeta \right)D\left(A\right).\hfill \end{array}$$

(38)

The last expression in this equation is denoted as $U\left(b\right)$, which is equal to 1 when interacting with a pure state as long as ${\mathsf{\Omega }}_b^{\prime }t=2k\pi$$(k\in Z)$.

In Figure 5b, the effect of the cavity–Bogoliubov mode coupling strength ${\lambda }_{CB}$ on the cavity–Bogoliubov mode entanglement is plotted. The increase in the absolute value of the coupling strength ${\lambda }_{CB}$ leads to entanglement enhancement for decoherence-free evolution. The negative sign is due to the detuning between the eigenfrequency of the cavity and the external driving light. Moreover, the entanglement period of the cavity field with the Bogoliubov mode is independent of ${\lambda }_{CB}$, and ${\lambda }_{CB}$ mainly plays a key role in the formation of the nonlinear energy levels in the cavity field. According to Section 3.4, the Bogoliubov state will return to its original state when the pure state $|\psi \left(t\right)\rangle$ completely decouples the cavity and Bogoliubov mode at $t=2k\pi /{\mathsf{\Omega }}_b^{\prime }$ for any other system parameters. It is worth noting that the gravity drive has no effect on the linear entropy.

4.2. Quantum Coherence Dynamics

In this subsection, we study the quantum resource complementarity between quantum coherence and entanglement in the cavity–BEC system during the dynamic evolution. To do so, we need to calculate the quantum coherence. The first-order quantum coherence of the cavity–Bogoliubov mode can be characterized by one element of the reduced single-particle density matrix $|{\rho }_{12}|$ with the definition [36,37]:

$$C=\frac{\left|\langle a^{†}b\rangle \right|}{N},$$

(39)

where N is the total excitation number, which includes photons and phonons in our system. The total excitation number N can be expressed as

$$\begin{array}{c}\hfill N=N_a+N_b=\langle a^{†}a\rangle +\langle b^{†}b\rangle ,\end{array}$$

(40)

where $\langle a^{†}a\rangle ={\left|\alpha \right|}^2$ and $\langle b^{†}b\rangle$ can be found in Section 3.3.

From Equations (19) and (39), we can obtain the mean value $\langle a^{†}b\rangle$ at a time t:

$$\begin{array}{ccc}\hfill \langle a^{†}b\rangle & =& e^{-{\alpha }^2}\sum _{n=0}^{\infty }\frac{{\alpha }^{2n+1}}{n!}{e}^{it\left[{\delta }_a^{\prime }-{\lambda }_{CB}^{\prime }\left(2n+1\right)\right]}{e}^{\frac{{\beta }_{n+1}}{2}\left(f_{12}^{*}-f_{12}\right)}{e}^{\left[-\left({\beta }_n^2+{\left|f_{12}+{\beta }_{n+1}\right|}^2-2{\beta }_n\left(f_{12}^{*}+{\beta }_{n+1}\right)\right)/2\right]}\hfill \\ & & \times \left[{\beta }_n\left(e^{-it{\mathsf{\Omega }}_b^{\prime }}{cosh}^{2}r-e^{it{\mathsf{\Omega }}_b^{\prime }}{sinh}^{2}r\right)+i\left(f_{12}^{*}+{\beta }_{n+1}\right)sinh2rsin\left({\mathsf{\Omega }}_b^{\prime }t\right)-A_n\right].\hfill \end{array}$$

(41)

Quantum coherence and entanglement have been recognized as quantum resources [38,39]. Making use of Equations (40) and (41), we can obtain the analytical expression of the first-order quantum coherence function $C\left(t\right)$. Figure 6a illustrates the dynamic behavior of quantum coherence $C\left(t\right)$ and entanglement linear entropy $S\left(t\right)$ when the photon dissipation rate $\kappa =0$. We found that the entanglement between the cavity and Bogoliubov mode will strengthen when the quantum coherence of the system becomes weak and even almost disappears. Then, the quantum coherence of the system will revive along with the time evolution. At the same time, the entanglement between the cavity and Bogoliubov mode will weaken until it almost disappears. This phenomenon is a good description of quantum resource transfer within the entire system. The transfer of the quantum resource indicated that the quantum information transfers between the cavity and Bogoliubov mode.

No quantum system is totally isolated. The interaction between the environment and the quantum system will occur in the quantum dynamical process. It is worth noting that this interaction will cause decoherence in the quantum system. Here, we took into account the influence of the cavity photon dissipation on the quantum resource dynamics. We assumed the weak environment coupling and adopted the Born–Markov approximation. The cavity interacts with the vacuum reservoir. The dominant dissipation source of the cavity is the photon loss due to cavity decay. When the photon dissipation is considered, the quantum dynamics of the system is described by the master equation [40,41,42]:

$$\begin{array}{c}\hfill \dot{\rho }=-\frac{i}{\hslash }\left[H,\rho \right]+\mathcal{L}\left(\rho \right),\end{array}$$

(42)

where the superoperator $\mathcal{L}\left(\rho \right)$ is given by

$$\begin{array}{c}\hfill \mathcal{L}\left(\rho \right)=\kappa \left(a\rho a^{†}-\frac{1}{2}{a}^{†}a\rho -\frac{1}{2}\rho a^{†}a\right),\end{array}$$

(43)

and $\kappa$ is the decay factor of the optical cavity, while a is the cavity field annihilation operator. The master equation and the density operator $\rho \left(t\right)$ of the system can both be calculated by Qutip [43,44].

From Figure 6b, we believe that the complementarity relation between the quantum entanglement and quantum coherence in the cavity–BEC system is generally preserved in the quantum dynamical process, even though there is dissipation in the cavity. The peaks of the quantum coherence $C\left(t\right)$ exactly correspond to the dips of the linear entropy $S\left(t\right)$, although the dissipation of the cavity suppresses the entanglement creation.

The complementarity relation between the quantum entanglement and quantum coherence is the key result obtained in this paper. Figure 6 indicates not only the curious complementarity relation between the quantum entanglement and quantum coherence, but also reveals the physical mechanism for the generation of the cavity–BEC entanglement. We think that the initial quantum coherence is the seed quantum resource to generate the cavity–BEC entanglement. Without the initial seed quantum coherence, the entanglement between the cavity and BEC cannot be produced in the system. The interaction between the cavity and the BEC is the bridge that transforms quantum coherence into quantum entanglement. Atomic scattering inside the BEC can play a role in modulating the rate of conversion of quantum resources.

5. Discussion

Gravity can be coupled to a quantum system by rotating the quantum system. Then, the information about the gravity parameters will be correlated with the parameters of the quantum system. Sougato Bose et al. used the rotation of a nonlinear optomechanics system to measure the value of gravitational acceleration [31]. It is worth noting that Sougato Bose et al. did not consider the effect of atomic scattering inside the BEC on the system. We found that the atomic scattering destroys the periodicity of quantum systems in the phase space. However, it is not clear to us whether atomic scattering is beneficial to the measurement of the value of gravitational acceleration. We will explore this problem in the process of future research.

In addition, they found that the gravity coupling strength and the cavity–BEC coupling strength will be approximately equal when the rotation angle of the cavity is close to $\pi /2$. Therefore, we believe that the gravity has a strong effect on the cavity–BEC quantum systems when the interaction between the cavity and BEC is radiation pressure coupling. In our opinion, the coupling of gravity to quantum systems provides a completely new way to manipulate quantum systems. Theoretically, a gravitational potential energy term is added to the Hamiltonian of the system when a quantum system interacts with gravity. Gravity will be involved in the dynamical evolution of quantum systems. Then, we can manipulate the quantum system through the parameters of gravity. From an experimental point of view, we don not need to change the structure of the quantum system. We can manipulate the dynamical evolution of a quantum system just by rotating it. This may be a simple and effective way.

A. Dalafi et al. comprehensively studied the cavity–BEC system with radiation pressure coupling [10,13,25,45,46]. They typically utilize optical driving for the flow of information between subsystems in a quantum system. This allows the establishment of correlations between subsystems and the realization of indirect manipulation of quantum systems. The effective attractive interaction will be enhanced due to the Floquet side bands of the phonons when the conventional superconductor is driven by the phonons [47]. The optical signals can be phase-sensitive-amplified and -squeezed in the optomechanical systems with two-phonon driving [48]. In fact, the gravitational coupling term in the Hamiltonian is analogous to a direct current phonon drive. Phonon driving and optical driving have their own advantages. We believe that the gravity driving and optical driving can synergistically manipulate quantum systems.

6. Conclusions

In our paper, we first introduced the cavity–BEC quantum system in a Newtonian gravitational field, which consists of a Fabry–Pérot cavity and cigar-shaped BEC. The effective Hamiltonian of the cavity–BEC quantum system in a Newtonian gravitational field was proposed under certain approximations. On this basis, the eigenvalues and eigenstates of the system can be obtained by diagonalizing the effective Hamiltonian. Secondly, according to the Schrödinger equation, we calculated the wave function of the system when the initial state of the system is the direct product of two coherent states. Based on the wave function, we calculated the dynamical evolution trajectory of the system in the phase space, the dynamics of the system excitation number, the dynamics of quantum entanglement, and the dynamics of quantum coherence.

When the cavity–BEC system is coupled to an external Newtonian gravity field, the external gravity acts as a direct current phonon drive. We found that the gravity drive will complicate the evolutionary trajectory of the system in the phase space. The gravity drive can increase the number of phonon excitations in the system, but does not have an effect on the number of photon excitations in the system. The gravity drive has no effect on the energy of the optical field. Thus, the gravity drive can be considered as a single-mode energy drive, even if there is coupling between the photon and phonon modes in our situation. Furthermore, we found that atomic scattering inside the BEC can affect the dynamical evolution period of the whole system. As the strength of atomic scattering inside the BEC increases, the dynamical evolution period of the system will be longer. This is attributed to the fact that intra-atomic scattering causes the internal energy levels of the atom to become nonlinear. This nonlinear energy level structure will reduce the effective coupling between the light field and the BEC.

We also investigated the dynamics of quantum entanglement and quantum coherence in a cavity–BEC system in a gravity field. We found that the Newtonian gravity drive has no effect on the entanglement and coherence between the cavity and the BEC. The intensity of internal atomic scattering can affect the dynamics period of entanglement. It is worth noting that strong photon–phonon entanglement between the cavity and the BEC Bogoliubov mode can be produced by the use of the initial quantum coherence of the system, which acts as the seed quantum resource to create photon–phonon entanglement. We also gave a curious complementarity relation between the photon–phonon entanglement and the quantum coherence of the system. We found that the quantum coherence of the system can transfer into the photon–phonon entanglement. Lastly, we numerically researched the effect of the cavity decay on the complementarity. It was indicated that the complementarity property can be sustained to some extent, even though there is dissipation in the cavity. Our results provide a route for understanding and controlling quantum resources in photon–phonon quantum systems.