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Influence of periodically modulated cavity field on the generation of atomic-squeezed states

We investigate the influence of periodically time-modulated cavity frequency on the generation of atomic squeezed states for a collection of N two-level atoms confined in a non-stationary cavity with a moving mirror. We show that the two-photon character of the field generated from the vacuum state of field plays a significant role in producing the atomic or spin squeezed states. We further show that the maximum amount of persistent atomic squeezing is obtained for the initial cavity field prepared in the vacuum state.

Squeezed states of electromagnetic field have also attracted much attention for decades [19][20][21]. In particular, squeezing of light can be generated by exploiting the optomechanical coupling between light and movable mirror. The radiation pressure creates correlations between the field quadratures: amplitude fluctuations move one mirror through radiation pressure force, which leads to phase fluctuations. In this way one can obtain radiation with fluctuations in one quadrature smaller than the Standard Quantum Limit (SQL), which is basically termed as ponderomotive squeezing [22][23][24]. The extraction of pondermotively squeezed light generated by radiation-pressure induced rigidity in an interferometer has also been experimentally proposed [24]. Squeezed optical states have numerous applications which includes the improvement of the signal-to-noise ratio in optical interferometers [25,26] and absorption spectroscopy [27].
The spin squeezing in atomic ensembles can be generated via light-atom interactions. It involves the very natural idea of transferring of squeezing from light to atoms [3,12,13,[28][29][30]. There have been many proposals which focusses on transferring squeezing to the atoms via electromagnetically induced transparency [31][32][33]. Moreover, the effect of the phase of the external light fields on the dynamics of spin squeezing in a coupled two-component BoseEinstein condensates has been studied in [34]. The generation of spin-squeezed states in a two-componenet BEC via nonlinear interaction between them has also been investigated [6,35,36]. Experimental realizations of spin-squeezed states in a BEC were also reported [35][36][37][38]. The spin-squeezed states in BEC have been used to detect weak forces [39] and to perform sub-shot-noise measurements [40,41].
Motivated by these interesting developments in this field, we propose an optomechanical system consisting of an elongated cigar-shaped gas of two-level BEC atoms interacting with a single mode of a lossless optical cavity with one movable mirror. We study this model to generate the correlated-particle states or atomic-squeezed states in both the x-and y-directions in the small-angle approximation by considering the cavity field to be initially prepared in the squeezed vacuum state. Because of the limited validity of small-angle approximation, we also investigate the generation of squeezed-spin states by numerically solving the Schrodinger's equation for the three special cases of initial states of cavity field, namely, a coherent state, a squeezed vacuum state and a squeezed state. We further discuss how the optomechanical coupling acts as a new additional handle in significantly varying the squeezing in spins along both the x-and y-directions. We also study the additional effect of nonlinear atom-atom interaction on spin-squeezing. Such squeezed-spin states are more convenient for storing quantum information since squeezing of photons is not easy to preserve. Many efforts have been done to use atomic ensembles in order to realize the quantum memories [42][43][44], and using photons as the information carriers.

II. MODEL HAMILTONIAN
In this section, we introduce the basic model and describe the Hamiltonian for our system. The schematic representation of the system is depicted in fig.(1). The optomechanical system considered here consists of a lossless Fabry-Perot optical cavity having perfectly reflecting cavity mirrors with one heavy fixed mirror and another light end mirror which is allowed to move freely. We have in addition an elongated cigar-shaped gas of N BEC atoms of 87 Rb with two different hyperfine levels |F = 2, m f = −1 > and |F = 2, m f = 1 > having transition frequency ω a and mass m [45]. Each atomic mode is associated with an annihilation operator c j (j = 1, 2) in the two-mode approximation. The cloud of BEC interacts strongly with a single quantized cavity mode having frequency ω c of an optical cavity of length L. A single-mode quantized cavity field is equivalent to a single quantum-mechanical harmonic oscillator of unit mass. The system is driven by an external field through the fixed cavity mirror which acts as a constant source of photons for the cavity. These photons are scattered by the BEC atoms along the axis of the cavity. Multiple reflections from the cavity mirrors result in the formation of one-dimensional optical lattice between the two mirrors. We consider the movable mirror as a single mechanical oscillator of frequency ω m and mass m ′ . The consideration of single vibrational mode of the movable mirror can be realized experimentally by using a bandpass filter in the detection loop such that the other mechanical degrees of freedom arising from the radiation pressure can be neglected [46].
The simplest model of such system is provided by the following Hamiltonian in the dipole approximation ( we use = 1 throughout the paper) [47,48]: (1) Figure 1: (color online) Schematic representation of the optomechanical system we are investigating here. It involves the Bose-Einstein Condensate inside a lossless optical cavity driven by an external field. One of the cavity mirrors is movable.
Here, the ensemble of N atoms is described using the picture of a collective spin. The collective spin is basically the sum of effective spins 1/2 that simply describe the internal degrees of freedom of each atom. Employing the second-quantization formalism, the creation (c † 1 , c † 2 ) and annihilation operators (c 1 , c 2 ) for the two modes can be related to three different hermitian spin components (known as Schwinger bosons [49]) as: The cavity photon annihilation and creation operators are represented by a and a † respectively satisfying the commutation relation [a, a † ] = 1. The annihilation and creation operators for the mechanical mode of movable mirror are represented by b and b † respectively such that [b, b † ] = 1. The parameters γ and g 0 , respectively, denote the effective nonlinear atomic interaction strength and atom-photon coupling strength via dipole interaction. Here we assume no direct coupling between the atoms and the cantilever, which can be achieved by assuming that only a few lattice sites are appreciably populated near the centre of the cavity. The nonlinear dispersive coupling between the intensity of the light field and the position quadrature of an oscillating mirror is represented by ǫ, which arises as a result of radiation pressure exerted by the intra-cavity photons on the mirror.
Optomechanical systems do not only measure the mechanical motion of the mirror, but can also manipulate the mirror motion via radiation pressure. In particular, squeezed light can be created by exploiting the optomechanical or radiation pressure coupling between the incident light field and the low-mass moving mirror. The radiation pressure shot noise (fluctuating optical force from quantum laser amplitude noise) induces mechanical motion of the movable mirror well above that of thermally driven motion. This motion imprints a Kerr-like phase shift on the light field, leading to correlations between the amplitude and phase quadratures [51]. At the expense of increasing fluctuations in an orthogonal quadrature, these correlations can suppress the quantum fluctuations on the output optical field below the shot noise level [22,23]. This radiation-pressure induced squeezing of light as a result of the coupling between the optical field and the mechanical motion of the mirror has been termed pondermotive [52] squeezing. Such squeezing implies that our optomechanical system is dominantly driven by quantum fluctuations in radiation pressure (radiation pressure shot noise) [53]. The pondormotively squeezed light has been demonstrated very recently using a mechanical mode of an ultracold atomic gas inside an optical cavity [53] and via optomechanical interaction between coherent light and membrane mechanical resonator embedded in an optical cavity [51]. The quadratures of the light transmitted through the cavity can be experimentally explored with balanced heterodyne detection [53]. The use of a balanced heterodyne configuration reduce the contamination of our measurements by optical technical noise. Pondermotive squeezing is observable when the probe light is detuned from the cavity resonance. In order to maintain a constant probe detuning from the shifted cavity resonance and average intracavity photon occupation, a single-photon counting module (SPCM) can be used. SPCM detects only ten percent of the light exiting the cavity and is fed back to control the frequency of an acoustic-optic modulator (AO). The rest of the cavity output is sent to the heterodyne detection set-up, where the probe and local oscillator are mode-matched through a single-mode fibre. The fibre output is sent to the detector which is balanced by controlling the beam polarization with a liquid-crystal rotator. Now, we rewrite the above Hamiltonian given by eqn. (1) in the interaction picture as: The derivation of this Hamiltonian is described in Appendix A. It is always convenient to work in the interaction picture. From now on, we treat the mirror mode operator b semiclassically such that it can be replaced by its scalar quantity β which is assumed to be real. This is certainly satisfied when the mechanical amplitude of the mirror mode | β |>> 1. Thus, the Hamiltonian given by eqn. (2) can now be rewritten as: Now, the modified mirror-photon coupling is defined as g(t) = 2ǫβcos(ω m t) which is sinusoidally modulated at frequency ω m , with 2ǫβ as the modulation amplitude. This modified mirror-photon coupling is basically related to the time modulated cavity frequency that is arising due to the motion of cavity boundary (mirror motion) [54].
For ω a = ω c , the resulting simplified Hamiltonian is plus the high-frequency terms which we neglect (rotating wave approximation). Last term in the above Hamiltonian is basically the Tavis-Cummings interaction [55] which is of considerable interest in quantum optics. It is valid for small values of atom-photon coupling [56].
In the next section, we will study the production of spin squeezing when an ensemble of two-level BEC atoms (spins) held within the optomechanical cavity interacts with the harmonic oscillator (single-mode quantized cavity field) through the interaction H 1 . The squeezed-spin states prepared this way could then be used in Ramsey spectroscopy.

III. APPROXIMATE SOLUTION
In this section, we study how the squeezing can be transferred to the spins from the harmonic oscillator (single-mode quantized cavity field) through the interaction H 1 . Without making the problem mathematically complicated, we consider the following special case to get a feeling for how this system works. The spin is assumed to be initially prepared in the Dicke state |J, is the square root of the variance of any operator A. Also, the harmonic oscillator is assumed to be initially prepared in a squeezed vacuum state characterized by < x(0) >=< p(0) >= 0 and ∆x(0) < ∆x(coherent state)= x 0 , where x 0 is its zero-point amplitude. The squeezing of harmonic oscillator can be characterized by the parameters χ x (t) = ∆x(t)/x 0 or χ p (t) = ∆p(t)/x 0 . The condition χ x < 1 represents amplitude squeezing and χ p < 1 represents momentum squeezing (or velocity squeezing). The Heisenberg equations of motion obtained by using the Hamiltonian H 1 are given as follows: Similarly, we can find their adjoint expressions. The averages < J x >, < J y >, < x > and < p > are assumed to be zero at all times. Now, we make the small-angle approximation that J z (t) = −JI, where I is an identity operator. This approximation is valid under the assumption that the number of atoms N is large enough such that the value of < J z (t) > does not change appreciably during the time H 1 is applied. With this approximation and by rewriting the cavity field operators and the angular momentum operators in terms of their quadratures as , the Heisenberg equations of motion becomes:ẋ Depending on the context in which squeezing is considered, different definitions of spin squeezing can be used. The squeezing parameter given by Wineland [2,3] This parameter is intimately connected to the improvement of the sensitivity of angular-momentum states to rotations. The atomic-squeezing condition in terms of this parameter is given by the condition ξ R < 1. On the other hand, another squeezing parameter ξ S given by Kitagawa and Ueda [1] is defined as ξ S = 2 J ∆J ⊥ where the subscript ⊥ refers to an axis perpendicular to the mean angular momentum < J > where the minimum value of ∆J is obtained. The parameter ξ S is used to indicate the degree of quantum correlations among the elementary spins. Here also, ξ S < 1 signifies atomic squeezing. A squeezed-spin state with ξ S < 1 can be rotated so that < J >=< J z >ẑ and ∆J ⊥ = ∆J y and can then be used in Ramsey spectroscopy with ξ R = (J/ |< J z >|)ξ S . In the context of these definitions, the squeezing parameters considered are defined as follows: Further note that ξ R,x = ξ S,x = ξ x and ξ R,y = ξ S,y = ξ y under the small-angle approximation J z (t) = −JI. Now using the eqns.(7)-(10) and the definitions of squeezing parameter for spins and harmonic oscillator with J x → J x √ N and J y → J y √ N , we obtain the following equations of motion: We now solve the coupled eqns. of motion (15)-(24) numerically using MATHEMATICA 9.0 for different values of mirror-photon coupling by assuming that the oscillator is initially in an amplitude-squeezed vacuum state where χ x (0) = 0.632. Experimentally, the frequency of mechanical oscillator may vary from 2π × 10kHz [57], 2π × 155.5kHz [53] to 2π×73.5MHz [58]. In an optomechanical system, the mirror-photon coupling rate can take the value 2π×54kHz [53]. The intracavity field interacting with the cloud of BEC held within the optomechanical cavity may have a coherent coupling strength of 2π×5.86kHz [59] (2π×14.4MHz [60]). In fig.2, we have shown the plots of squeezing parameter for harmonic oscillator χ x (t) versus scaled time (ω m t) for ǫ = 0.2ω m (thin dashed line) and the spin squeezing parameters ξ x (t), ξ y (t) as a function of scaled time (ω m t) for three different values of mirror-photon coupling with ǫ = 0.2ω m (solid line), ǫ = 0.4ω m (thick dashed line) and ǫ = 0.6ω m (dot dashed line). Squeezed-spin states can be created by transferring the squeezing from the harmonic oscillator (χ x (0) < 1) to the spins through H 1 , which is clearly depicted in fig.2(b). This figure shows the back and forth transferring of squeezing between the spins and the harmonic oscillator as the modified mirror-photon coupling g(t) changes with time. The radiation pressure, or pondermotive, force acting on the end mirror as a result of coupling between the optical field and the mirror motion helps in generating further squeezing in harmonic oscillator, known as pondermotive squeezing [22][23][24], which is then transferred to the spins. The transfer of squeezing to the spins in this small-angle approximation is like wave-function exchange between coupled harmonic oscillators [61]. After the particles are put in squeezed-spin states, the correlations exist between them even though they do not interact. Moreover, figs. 2(a) and 2(b) show that when the value of ξ x increases with time, then the corresponding value of ξ y decreases and vice-versa. This is due to the fact that J x and J y follow the uncertainty relation given by ∆J x ∆J y ≥|< J z > /2 | from the commutation relations for angular momentum. Further notice from fig.2(a) that the increase in mirror-photon coupling ǫ enhances the spin squeezing along x-direction such that ξ x attains a minimum value for ǫ = 0.6ω m . However, squeezing parameter ξ y reaches a minimum value for ǫ = 0.2ω m (see fig.2(b)). This implies that the squeezing of spins can be coherently controlled by the mirror-photon coupling. Hence, mirror-photon coupling acts as a new additional handle in significantly varying the atomic-squeezing in both the x-and y-directions.
In the next section, we discuss the generation of squeezed-spin states by numerically solving the Schrodinger's equation because of the limited validity of small-angle approximation for H 1 .

IV. NUMERICAL SOLUTION
Here, we study the generation of atomic-squeezed states for some special cases by numerically solving the Schrodinger's equation for the interaction H 1 . The wave function for the combined spin ensemble held within the optomechanical cavity and the cavity mode(treated as harmonic oscillator) can be written as: where we use the notation |M > (= |J, M >) since J is a constant of motion for H 1 . Here, |n > represent the cavity mode eigen states such that a|n >= √ n|n − 1 > and a † |n >= √ n + 1|n + 1 >. We further assume that the cavity mode and BEC atoms are uncorrelated for the initial wave function such that the wave function can be written as a direct product where ψ(0) = |J = N/2, M = −N/2 > and C n are the initial cavity mode wave function coefficients which are discussed in Appendix B. The basis of cavity mode states are also to be truncated of course by determining an adequate number of states until the result remains unchanged. The equation of motion for Hamiltonian H 1 which is evaluated using Schrodinger's equation is given as: We are considering here J = 10. H 1 couples only those states where n+M is always a constant. The time-dependent wave function for the combined cavity mode and spin ensemble confined within the optomechanical cavity can be written as a sum over all the possible eigen states. In order to obtain the time evolution of the wave function, the time dependent wave-function coefficients are evaluated using eqn. (27) with the help of MATHEMATICA 9.0. Using this wave function, various squeezing parameters defined in eqns.(11)- (14) can be evaluated with time for the Hamitonian H 1 . Now, we show the time evolution of squeezing parameter ξ S in order to study the generation of squeezed-spin states by coupling a single-mode quantized cavity field to the condensate atoms confined within the optomechanical cavity via interaction H 1 .
In fig.3, we plot ξ S (t) versus scaled time (ω m t) in the absence of cavity-mirror coupling ǫ = 0 (solid line) and for two different values of mirror-photon coupling ǫ, ǫ = 0.2ω m (dashed line) and ǫ = 0.4ω m (dot dashed line). Here, we assume ψ(0) = |J, −J > and the cavity mode to be initially prepared in a coherent state |α > with an average number of photon |α| = 1 such that < x(0) > = 0, < p(0) >= 0 and χ x (0) = χ p (0) = 1. Variation of ξ S,x (t) and ξ S,y (t) with time in the presence of effective nonlinear atom-atom interaction are shown in figs.3(a) and 3(b) respectively. However, figs.3(c) and 3(d) represent the time evolution of ξ S,x (t) and ξ S,y (t) in the absence of effective nonlinear atomic interaction. Fig.3 shows a considerable amount of spin-squeezing even in the absence of mirror-photon coupling. In the presence of optomechanical coupling, the most important observation apparent from fig.3(a) is that even though we start with a coherent state, squeezing along the x-direction increases significantly with the increase in light-mirror coupling. This is due to the fact that the radiation pressure (pondermotive) force acting on the end mirror generates squeezing in the optical cavity field, which is transferred to the condensate atoms through the interaction H 1 . One could argue that the nonlinear atom-atom interaction could also be responsible for squeezing. To confirm that pondermotively squeezed state is also involved in generating atomic-squeezed states, figs.3(c) demonstrates spin squeezing in the absence of nonlinear atom-atom interaction. This validates our argument that the spin-squeezed states can be generated inside the optomechanical cavity using the input coherent state via pondermotive force. However, spin-squeezing along the y-direction deteriorates with the increase in mirror-photon coupling and shows the maximum amount of squeezing for ǫ = 0 (see figs.3(b) and 3(d)). A comparison of figs.3(a) and 3(c) shows that by switching off the atom-atom interaction (γ = 0), squeezing in the x-direction is enhanced for higher light-mirror coupling (ǫ = 0.4ω m ). Further, we observe that for γ = 0 and ǫ = 0.4ω m , spins in the x-direction remain squeezed for large times (persistent spin squeezing). Thus, for an initially coherent state, maximum squeezing along the x-direction (ξ S,x ) can be achieved for γ = 0 and high ǫ, whereas, maximum squeezing along y-direction can be obtained for γ = 0 and low ǫ. Hence squeezing in spins along both the x-and y-directions can be significantly altered by choosing the appropriate values of cavity-mirror coupling and nonlinear atomic interaction. Fig.4 illustrates the time evolution of spin-squeezing parameter ξ S (t) in the absence of mirror-photon coupling (solid line) and for two different values of light-mirror coupling ǫ, ǫ = 0.2ω m (dashed line) and ǫ = 0.4ω m (dot dashed line) by assuming ψ(0) = |J, −J > and the oscillator to be initially prepared in an amplitude squeezed vacuum state where < x(0) >=< p(0) >= 0 and χ x (0) < 1. Figs.4(a) and 4(b) represent the plot of ξ S,x (t) and ξ S,y (t) versus scaled time (ω m t) respectively in the presence of nonlinear atomic interaction. We have also shown the time evolution of ξ S,x (t) and ξ S,y (t) in the absence of atom-atom interaction in figs.4(c) and 4(d) respectively. As before, we observe that higher value of mirror-photon coupling favors spin squeezing in the x-direction while lower value of ǫ favors spin squeezing along the y-direction. However, no squeezing along x-direction is observed in the absence of cavity-mirror coupling. Further note that the persistent squeezing along x-direction is observed for high ǫ and γ = 0, whereas, persistent and high squeezing is attained in the y-direction for ǫ = 0. We further show the variation of ξ S (t) with time in fig.5 by considering ψ(0) = |J, −J > and the cavity mode to be initially prepared in an amplitude squeezed state with < x(0) = 0 >, < p(0) >= 0 and χ x (0) < 1. Here, figs.5(a) and 5(b) depict the respective variation in ξ S,x (t) and ξ S,y (t) with time in the presence of effective nonlinear atom-atom interaction, whereas, figs.5(c) and 5(d) illustrate the respective time evolution of ξ S,x (t) and ξ S,y (t) in the absence of effective nonlinear atom-atom interaction. These figures illustrate a similar kind of behavior as in the previous cases but more important is the fact that, in the presence of optomechanical coupling, initially squeezed state is able to produce a much higher degree of persistent spin squeezing compared to the previous cases. This is due to the radiation pressure force acting on the movable mirror (cavity boundary) that helps in producing further squeezing in cavity mode. This squeezing is drained away from the harmonic oscillator and is transferred to the condensate atoms. Hence, mirror-photon coupling plays a vital role in coherently controlling the atomic spin-squeezing. This is the main result that we have demonstrated in this paper. Hence, we can generate the atomic-squeezed states in an optomechanical cavity due to the transfer of squeezing from cavity mode to spins by considering the cavity mode to be initially prepared either in a coherent state, squeezed vacuum state or a squeezed state. Maximum spin-squeezing in the x-direction is obtained for higher light-mirror coupling in the case of initial cavity field prepared in a squeezed state, whereas, maximum squeezing along the y-direction is attained in the absence of optomechanical coupling for the initial coherent state. In a realistic system, decay of the cavity mode and thermal noise will deteriorate the atomic spin squeezing but substantial squeezing can still be achieved during the initial dynamics.

V. CONCLUSION
In conclusion, we have shown the generation of correlated-particle states or squeezed-spin states for a BEC confined in a lossless optomechanical cavity by considering the combined effect of cavity-mirror coupling and the nonlinear atomic interaction strength. In particular, we have studied the squeezing in spins for the three special cases-initial cavity field in coherent state, squeezed vacuum state and squeezed state. We have observed that the spin-squeezing in both the x-and y-directions can be significantly altered by choosing the appropriate values of light-mirror coupling and nonlinear atomic interaction strength. Persistent spin-squeezing which is of more practical use can also be generated by the proper choice of the system parameters. The radiation pressure or pondermotive force acting on the movable mirror as a result of coupling between the optical field and the mechanical motion of the mirror generates squeezing in the optical cavity field. This squeezing gets transferred to the condensate atoms due to a Tavis-Cummings type interaction and results in atomic-squeezed states. We found that, even if the initial state of the cavity field is coherent, still a high degree of squeezing is generated and transferred to the atoms, with maximum squeezing along the y-direction in the absence of optomechanical coupling. Maximum amount of spin-squeezing along the x-direction is observed in the presence of optomechanical coupling for the initial cavity field prepared in the squeezed state. Such squeezed-spin states or correlated-particle states have applications in entanglement detection as well as high precision metrology.

VI. ACKNOWLEDGEMENTS
Neha Aggarwal and A. Bhattacherjee acknowledge financial support from the Department of Science and Technology, New Delhi for financial assistance vide grant SR/S2/LOP-0034/2010.

VII. APPENDIX A
We here describe the derivation of Hamiltonian given by eqn. (2). The Hamiltonian of our system is given by eqn. (1) such that where, H f (= ω c a † a + ω a J z + ω m b † b) is the free Hamiltonian illustrating the energy of light field, atomic field and mechanical field of vibrating mirror separately. H i (= γ N J 2 z + g0 √ N (a+a † )J x +ǫa † a(b+b † )) is the interaction Hamiltonian describing the respective atom-atom interaction, atom-cavity field coupling and cavity-mirror field coupling. The Hamiltonian in the interaction picture is given by [62] H int = e iH f t H i e −iH f t .
Using Baker-Haudsorf lemma [49] for any two operators A and B it can be readily seen that e iωca † at a † ae −iωca † at = a † a, e iωaJzt J 2 z e −iωaJzt = J 2 z .
VIII. APPENDIX B Coherent states are the minimum product states with their variances of position and momentum quadratures equal to those of the vacuum state i.e., var(x) = var(p) = 1/2 which yields the minimum product 1/4. In the case where cavity field is assumed to be initially prepared in the coherent state to obtain the squeezed-spin states, the harmonic oscillator wave function coefficients (or the number state coefficients) C n at t = 0 can be obtained from the following recursion relations [63]: and, in general, C n (0) = αC n−1 (0)/ √ n.
From the normalization requirement ∞ n=0 | C n (0) | 2 = 1, the value of C 0 (0) is determined to be C 0 (0) = e −|α|/2 . Here, | α | 2 =< n > where n represents the photon number. However, a state is considered to be squeezed if its oscillating variances become less than the variances of the vacuum state. In the case of squeezed states, the harmonic oscillator wave function coefficients C n at t = 0 can be evaluated from the following recursion relations [63]: with the general expression as Here, β ′ , µ and ν are complex numbers such that | µ 2 | − | ν 2 |= 1. For a given set of numerical values β ′ , µ and ν, we can find the numerical values of all the coefficients recursively by starting with an arbitrary value of C 0 (0). The value of C 0 (0) is then adjusted such that the normalization condition ∞ n=0 | C n (0) | 2 = 1 is satisfied. For the initial cavity field to be present in the squeezed vacuum state, β ′ is to be kept zero.