# Quantum Optomechanics with Single Atom

###### Abstract

The recently increasing explorations for cavity optomechanical coupling assisted by a single atom or an atomic ensemble have opened an experimentally accessible fashion to interface quantum optics and nano (micro) -mechanical systems. In this paper, we study in details such composite quantum dynamics of photon, phonon and atoms, specified by the triple coupling, which only exists in this triple hybrid system: The cavity QED system with a movable end mirror. We exactly diagonalize the Hamiltonian of the triple hybrid system under the parametric resonance condition. We find that, with the rotating-wave approximation, the hybrid system is modeled by a generalized spin-orbit coupling where the orbital angular momentum operator is defined through a Jordan-Schwinger realization with two bosonic modes, corresponding to the mirror oscillation and the single mode photon of the cavity. In the quasi-classical limit of very large angular momentum, this system will behave like a standard cavity-QED system described by the Jaynes-Cummings model as the angular momentum operators are transformed to bosonic operators of a single mode. We test this observation with an experimentally accessible system with the atom in the cavity with a moving mirror.

###### pacs:

42.50.Tx, 03.67.Bg, 32.80.Qk, 85.85.+j## I Introduction

Nowadays, there are an increasing number of researches on cavity optomechanical systems assisted by atoms or atomic ensembles Tombesi ; meystre ; Ian . Such hybrid systems show a convergence between quantum optics and nano (micro) -mechanical systems. In this context, a Fabry-Perot cavity is applied to form an approximate standing light wave while one of the mirrors at the end of the cavity is allowed to oscillate. Furthermore, the novel quantum natures are discovered in this composite system by placing an atomic ensemble confined in a gas chamber inside the cavity. With the helps of the atomic ensemble, theoretical explorations have been made to show the possibilities to create not only the entanglement of the cavity field and a macroscopical object Vitali ; mancini ; penrose ; knight2 , i.e., a mirror, but also the entanglement of atom-light-mirror Tombesi ; Ian . In this paper, we will consider this atom assisted optomechanical (AAOPM) system with the strong coupling of a single atom to the photon field inside the cavity, which is modified by the moving end mirror of the cavity (see Fig. 1). We find that a three body coupling term of photon, phonon and atom will play the crucial role in the quantum dynamics of the triple system.

The triple system we will refer to contains a two-level system interacting with a single mode electromagnetic field inside the cavity with an oscillating mirror. Due to the vibration of the cavity length, which is conventionally thought to induce the light pressure term mancini2 ; mancini ; ian2 ; meystre2 , we illustrate that the vibrating length can also result in the triple coupling term of photon-atom-mirror. Under a certain condition concerning the frequencies and the coupling strength in the so-called parametric resonance, we can exactly diagonalize the Hamiltonian to obtain the eigenstates and the eigenvalues. Here, the eigenstate is the direct product of the mirror’s state and the state of a two body hybrid system, which consists of the photon and the atom, and can be described by the Jaynes-Cummings (J-C) model. We consider how the mirror’s oscillation affects the cavity QED subsystem, by studying the quantum decoherence zhang of this subsystem. To this end, we calculate the time evolution of the total system with the mirror initially prepared in a coherent state. We use the decoherence factor (or the Loschmidt echo (LE), the absolute square of the decoherence factor quan ), to characterize the influence of the mirror’s motion, which is the overlap of two wavefunctions of the mirror driven respectively by two conditional Hamiltonians derived from two dressed states. We find that, though the decoherence factor depends on the initial state of the mirror, the LE has nothing to do with this initial coherent state. This means the mirror’s initial position only affect the decoherence factor in the form of a phase factor.

The above condition for exact solution seems too special to be realized, thus we consider a more general case with the rotating-wave approximation, which only requires a set of matching frequencies rather than the coupling strength, among the atom, the cavity field and the mirror. In this case, our model is reduced to the generalized spin-orbit coupling model where the spin is referred to as the internal energy level of the atom, while the orbit is depicted by the quasi-orbital angular momentum defined by two bosonic modes (the mirror oscillation and the single-mode photon of the cavity) through the Jordan-Schwinger representation.

It is well known that the quantum system with a large angular momentum can be regarded as a classical rotor when the angular momentum approaches infinity in the classical limit sun3 . This is because the component variations , and become vanishingly small in comparison with the large angular momentum . When the angular momentum is large enough, but not infinite, the ladder operators of any large angular momentum can behave as the creation and annihilation operators of a single-mode boson jin ; liu . This point can be seen from the Holstein-Primakoff transformation straightforwardly. This case is named the quasi-classical limit and its significance in many-body physics can be understood as the low-energy excitation above the ordered ground states. This quasi-classical reduction of large angular momentum has been extensively studied and applied in quantum storage sun ; sun2 ; he . Here, we study this quasi-classical reduction by referring to the experimentally accessible parameters in triple hybrid system. In this triple system, when the frequency of the electromagnetic field almost matches the eigenfrequency of the energy level spacing of the atom plus the mirror oscillation, the photon and the mirror oscillation are coupled to form a composite object, which exactly behaves as an angular momentum. Then our triple system is modeled by the generalized spin-orbit coupling similar to the Hamiltonian for the Paschen-Back effect without the radial-dependence landau . This observation means that the triple coupling system will be reduced to a two-part coupling system described by the J-C model in cavity QED. At last, we show this quasi-classical reduction indeed works well when the hybrid excitation of the mirror plus photon is large enough.

The paper is organized as follows. In Sec. II, we model the AAOPM coupling and reduce it to the generalized spin-orbit coupling under the frequency matching condition. In Sec. III, we exactly solve the AAOPM model under the parametric resonance condition and show the decoherence in Sec. IV. In Sec. V, we compare the generalized spin-orbit coupling model to the J-C model with their eigenstates and eigenvalues. Furthermore, in Sec. VI, we study the dynamics of the generalized spin-orbit coupling model to demonstrate the similarities and differences to that of the J-C model. In Sec. VII, we summarize our results.

## Ii Modeling the triple coupling of atom-photon-mirror

In this section, we study an experimentally accessible AAOPM system, as illustrated in Fig. 1. This system consists of three parts: an atom, the photons inside a cavity, and a movable end mirror. We show that such a hybrid system can be modeled by a spin-orbit coupling system, where the orbital angular momenta are realized by the phonon of the mirror dressed by the photon of the light field inside the cavity.

The single mode electric field inside the cavity along the -axis is quantized as

(1) |

where is the position of the atomic center of mass, and . The frequency of the cavity field is dependent of the cavity length, ; is the dielectric constant in vacuum; and are the length and the volume of the cavity, respectively. Here, we omit the polarization of the field, but this will not affect our final conclusion for a practical system. When the length of the cavity slightly changes from to due to the mirror’s displacement (see Fig. 2), the electric field becomes

(2) |

where

Then, the total Hamiltonian reads

(3) | |||||

where is the creation operator of the oscillating mode of the cavity field (mirror); , and are the spin operators, which represent the transitions among the atomic inner states. is the atomic-position-dependent coupling strength between the cavity field and the atom, where is the electric-dipole transition matrix element. , where is the frequency of the mirror’s oscillation and is the mass of the mirror, describes the light pressure, while denotes the coupling strength of “three body” due to the vibration of the mirror. With some experimentally feasible parameters, there exists the situation where is on the same order of magnitude of , for which the strong coupling region meystre3 is reached (e.g., and , then ). In this case, the triple coupling term

(4) |

should not be neglected. In the following discussion, we will focus on this strong coupling case.

It follows from the Hamiltonian in Eq. (3) that, when , i.e., , the J-C type interaction vanishes, but the three-body interaction remains. Near the photon-phonon resonance case where the frequencies satisfy , the rotating-wave approximation reduces the Hamiltonian to

(5) |

In the following discussions, we invoke the Jordan-Schwinger representation of the SO(3) group sakurai

(6) |

where the commutation relations of the angular momenta

(7) |

are satisfied due to the generic commutation relations between the and bosons. Then, the Hamiltonian can be rewritten as

(8) |

where , , and .

We remark that the three body interaction in the AAOPM system can be approximately modeled with the x-y coupling

(9) |

This is a kind of “spin-orbit coupling” referred to as Paschen-Back effect landau . The “orbital” angular momentum defined by and essentially results from the joint excitation of photon and phonon. Physically, this excitation can be understood as a kind of effective mechanical oscillation of the mirror, which is dressed by the single mode photon. Such kinds of dressed bosons just satisfy the angular momentum algebra.

## Iii Exact solutions for parametric resonance

From Eq. (3), obviously the particle number operator is conserved, i.e., . Then, in the subspace , where denotes that the photon is prepared in a Fock state while the atom in the ground (excited) state. In the following we consider the situation in which the eigen-equation of the Hamiltonian in Eq. (3) can be solved exactly. To this end, we first show that the Hamiltonian is formally expanded as follows:

(10) |

where

(11) |

and

(12) |

In Eq. (12), is the atom-photon detuning. The above argument shows that in the subspace the total system can be reduced to a spin-boson model defined by Eqs. (10-12).

Now let us temporarily leave the above concrete system to consider a more general spin-boson model with the Hamiltonian

(13) |

where is a function that only depends on the boson model and is spin-dependent. There is a seemingly trivial proposition for this spin-boson system: If the coupling can be factorized as

(14) |

with () depending on boson (spin) only, then can be exactly diagonalized through the diagonalizations of the two pure boson systems with branch Hamiltonians

(15) |

where and are the eigenvalues of the C-number coefficient matrix in the basis . The proof of this proposition is rather straightforward and we only need to diagonalize first.

Next we return to the concrete example.

We explore when can be factorized to , where is a function of and , and is a -number matrix. Actually when the detuning and the three coupling coefficients have the relation

(16) |

which we call the parametric resonance, the Hamiltonian indeed becomes the form of Eq. (14):

(17) |

where

(18) |

and

(19) |

Thus, to diagonalize the Hamiltonian , all we need to do is to diagonalize the matrix and the left quadratic part formed by and . Then, the eigenvalues of are obtained as

(20) | |||||

for , where

(21) |

and

(22) |

Here, represents the quantum number of the photons (phonons). Correspondingly, the eigenstates of the AAOPM system are , where the photon dressed state

(23) |

and

(24) |

are defined by the mixing angle :

(25) |

while the mirror’s states are

(26) |

with the displacement operator

If , the atom-assisted optomechanical coupling model goes back to the J-C model, and the eigenvalues, together with the eigenstates, degenerate to that of the J-C model. However, due to the vibration of the mirror, there emerge fruitful results in our model due to the complex three-body coupling. First, we examine the realization of the parametric resonance condition, (16), in experiments. Substituting the experimental feasible parameters into the parametric resonance condition, we know that it is easily satisfied if the detuning is adjusted properly to adapt to different positions of the atom. In experiments, and can reach Hz, while is on the order of Hz, thus can be on any order that lies on the atomic position. In the special case when , i.e., , is sufficient to meet the condition in Eq. (16).

It is observed from Eq. (20) that the terms in the second line obviously differ from the eigenvalues of the J-C model. The first term is the mirror’s eigenvalue, and the second one (without the sign) is expanded as

(27) |

where the first term describes the energy of the light pressure term knight2 ; knight , but the left terms are induced by the atom-assisted optomechanical coupling.

## Iv Conditional dynamics for decoherence

In this section we demonstrate the parametric resonance will lead to a conditional dynamics with respect to two superpositions of atomic inner states and , which is described by a non-demolition Hamiltonian.

From the above argument about the exact solvability of the AAOPM system, we can find that the operator-valued Hamiltonian matrix

(28) |

is diagonalized with respect to the basis for the spin part. Obviously this is a non-demolition Hamiltonian with respect to the basis vectors and and thus results in the corresponding decoherence.

Driven by this non-demolition Hamiltonian, the factorized initial state for the cavity QED system

(29) |

will evolve into an entanglement state

(30) |

where

and the extent of decoherence due to this quantum entanglement is characterized by the so-called decoherence factor

(31) | |||||

and its norm square is the so-called Loschmidt echo quan .

In the context of quantum chaos, the Loschmidt echo characterizes the sensitivity of evolution of quantum system in comparison with the butterfly effect in classical chaos: Starting from the same initial state, the quantum system is separately driven by two slight different Hamiltonians. Quantum chaos is implied by the much larger differences in the two corresponding final states; namely, their overlap (Loschmidt echo) vanishes to illustrate the dynamical sensitivity of the quantum chaos system.

Next we return to the concrete system.

Consider the time evolution of the system when the initial state is as follows:

(32) |

where is a coherent state of phonon that satisfies

(33) |

, , is the dressed state mentioned in last section, and is the weight of each dressed state. Therefore, at time the wave function of the total system is

(34) |

where

(35) | |||||

The mirror’s motion will result in the collapse of the decoherence, with the Loschmidt echo (LE) being

(36) | |||||

where

(37) |

We notice that does not play any role in the LE. Note that the mirror’s initial coherent state evolves to another coherent state, and only determine the initial position of the center of the wavepacket . Thus the overlap of the two wavepackets and is independent of . Physically, this fact shows that the decoherence of the cavity-QED system is irrelevant to the phonon excitations of the mirror if its wavepacket is Gaussian.

In Fig. 3, for different photon-phonon excitations, we plot the time evolution of the Loschmidt echo with the parameters set to Hz, , m, kg, Hz. From Fig. 3, we see that all the curves have the same period, i.e., , and the larger the difference of the photon number is, the larger the amplitudes of the curve are. We remark that the large photon number means a classical electromagnetic field, and thus the quantum decoherence of the atomic inner states reflects the classical transition of optical field from the quantum regime.

## V Generalized spin-orbit coupling in comparison with Jaynes-Cummings model

In Sec. II, we have derived the generalized spin-orbit coupling model under the rotating-wave approximation, which can be rewritten as

(38) |

Here, and characterize the coupling of the angular momentum and the spin to the external field, respectively; is the coupling strength of the spin-orbit.

This model can be studied by exactly diagonalizing the model Hamiltonian in Eq. (38) within its invariant subspace spanned by

(39) |

and

(40) |

Here, is the standard angular momentum basis, while and denote the spin-up and spin-down vectors, respectively. In this basis, the spin-orbit coupling Hamiltonian in Eq. (38) is reduced to a quasi-diagonal matrix with an block

(41) |

where and

(42) |

Then it can be further diagonalized to obtain the eigenvectors

(43) |

and

(44) |

with the corresponding eigenvalues

(45) |

where and

From the spectrum structure of the AAOPM system described above, we demonstrate that the triple coupling system can realize the entanglement between an orbital angular momentum and a spin. The -component of the total angular momentum is conserved. Thus, while the orbital angular momentum is flipped from down (up) to up (down), the spin will make a reverse flip.

Now we can consider the quasi-classical limit of the above spin-orbit coupling model for large enough with low excitation. Obviously, the above basis vector in this limit becomes a Fock state, i.e, where ; while . Correspondingly, the eigenstates become the dressed states in the usual J-C model with the eigenvalues

(46) |

where and

We remark that the system we considered in the limit above can also be described by the J-C model

(47) |

The correspondence between the Fock state and the standard angular momentum basis is

(48) |

Actually, the above equivalence of spin-orbit coupling model and J-C interaction can be found directly by considering the Holstein-Primakoff mapping

(49) |

in the large limit.

Next we come back to the practical physics of the AAOPM system. Then the joint states is re-expressed in terms of the two Fock states and as

(50) |

Here and represent the states with photons and phonons of the mirror’s vibration mode respectively.

In Fig. 4 and Fig. 5, with different values of and , we plot as the functions of . Here, we take the physical parameters as Hz, , m, kg, Hz.

As shown in the figures 4 and 5, when is fixed, the spectrum diagram of the AAOPM system looks quite like that of the J-C model’s orszag . However, in general, its number of energy levels are much more than that in the J-C model’s within the same energy range. Furthermore, we can see from the several lowest levels that there exist small differences under different values of , such as vs. , vs. and so on. Accordingly, the larger the value of (the orbital angular momentum) is, the closer the spectrum is to the corresponding ones in the J-C model. For evidence, we have plotted the curves with a much more larger range of the variable that is not valid in our rotating wave approximation that requires .

## Vi Quasi-classical dynamics via Jaynes-Cummings Model

In the above sections, we have shown the similarity between the triple hybrid system and the J-C model in their energy spectra. Now we continue to consider this similarity in quantum dynamics, which is referred to as the so-called quasi-classical one as we make the analysis for very large angular momentum.

We consider generally the system described by the Hamiltonian Eq. (38) is initially prepared in the state represented by the density matrix

(51) |

where the joint states denote the initial factorized structure of the triple system. According to the similarity between the generalized L-S coupling system and the triple coupling system we mentioned in the last section, the density matrix for the time evolution reads

(52) |

where the branch wavefunctions

(53) | |||||

for , are determined by the time dependent parameters defined by

(54) |

(55) |

(56) |

(57) |

It is noticed that the above Eqs. (53-57) have forms similar to that for the J-C model in the limit that . We remark that this limit means a low excitation of the joint system consisting of the photon and phonon, i.e., there are only a few photons stimulated so that the joint system is almost prepared in the lowest weight state . The above equations give us rich information on the complex system, from which all the physically relevant quantities to the cavity field, the atom and the mirror can be obtained. In Eq. (53), () is proportional to the probability that at time , there are photons, quanta of the mirror’s vibration mode and one atom in the excited (ground) state. Therefore the probability that photons are measured is

(58) | |||||

where , and

Another important quantity is the population inversion depending on the probability amplitudes and as

(59) | |||||

Note that if the initial state of the mirror is the vacuum state, i.e., , then it follows from Eqs. (58) and (59) that and both of which are time independent no matter which state the light field is initially in. This result can be explained as follows: with the rotating-wave approximation, we only hold the slowly varying terms in the original Hamiltonian to obtain effective Hamiltonian (5). These terms make a transition from the atom’s upper level state to the ground state , together with a decrement of the quanta of the mirror’s vibration mode and an increment of the photon number, or vice versa. Thus when initially the mirror is in vacuum and the atom is in the excited state, the total state can not evolve to the “dressed state”, but only stays in the initial state companied by a dynamical phase factor.

The above obtained results are very similar to that of the J-C model: and also contain many Rabi oscillations with various frequencies, and in different initial states, and behave differently.

Next we consider that the mirror is initially in a thermal state, while the photon field is in one of several different states: a thermal state, a Fock state and a coherent state . We obtain different initial parameters as follows:

(60) |

(61) |

(62) |

where , is the temperature.

In Fig. 6, 7 and 8, we plot the evolution of in the three initial states mentioned above with the parameters K, kg, m, Hz, , Hz, , . It can be seen from the figures that in each case, as time increases, collapses and revivals appear cyclically, but the time duration in which each collapse and each revival take place differs from each other because of different that represents the weight of the Rabi oscillation with fixed frequency. This behavior of collapse and revival of inversion is repeated with increasing time, with the amplitude of Rabi oscillations decreased and the time duration in which the revival takes place increased and ultimately overlapping with the earlier revival.

Note that the temperature is so low that , and thus the case described in Fig. 6 reflects the phenomenon that the atomic transition between the upper and the lower levels can happen even when the light field is initially prepared in the vacuum state. This is obviously a purely quantum effect to prove the role of vacuum. In Fig. 7 we observe that even if the cavity field in a Fock state, the collapse and revival appear explicitly. This case differs from the J-C model based collapse and revival phenomenon, in which the evolution of inversion is just a cosine curve when the field in a Fock state.

## Vii Conclusion and Remarks

We have shown a AAOPM coupling in the triple hybrid system composing of atoms, a cavity field and a movable mirror and discovered that under parametric resonance condition, this complicated model can be solved exactly. Furthermore, we have demonstrated that this triple hybrid system can be modeled by generalized L-S coupling under the rotating-wave approximation. It is shown that the composite object formed by the cavity-field-dressed mirror acts like an orbital angular momentum. Then we studied the physically intrinsic relation between the generalized L-S coupling system and the J-C model, i.e., when the orbital angular momentum is large enough, the former is quite like the latter. Same to the generalized L-S coupling system, in quasi-classical limit, the ladder operators behave as the bosonic operators and thus the large angular momentum can be regarded as “excitons” in the low excitation limit. We also investigated some characteristic properties of the J-C model in our triple hybrid system and discovered their similarities and differences.

When preparing this paper we find a paper x x yi , where the triple interaction was introduced.

###### Acknowledgements.

C. P. Sun acknowledges supports by the NSFC with Grants No. 10474104, No. 60433050, and No. 10704023, NFRPCNo. 2006CB921205 and 2005CB724508.## References

- (1) C. Genes, D. Vitali, and P. Tombesi, Phys. Rev. A 77, 050307 (2008).
- (2) D. Meiser and P. Meystre, Phys. Rev. A 73, 033417 (2006).
- (3) H. Ian, Z. R. Gong, Yu-xi Liu, C. P. Sun and Franco Nori, Phys. Rev. A 78, 013824 (2008).
- (4) D. Vitali, S. Gigan, A. Ferreira, H. R. Böhm, P. Tombesi, A. Guerreiro, V. Vedral, A. Zeilinger, and M. Aspelmeyer, Phys. Rev. Lett. 98, 030405 (2007).
- (5) W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Phys. Rev. Lett. 91, 130401 (2003).
- (6) S. Bose, K. Jacobs, and P. L. Knight, Phys. Rev. A 59, 3204 - 3210 (1999).
- (7) S. Mancini and P. Tombesi, Phys. Rev. A 49, 4055 (1994).
- (8) S. Mancini1, V. Giovannetti, D. Vitali, and P. Tombesi, Phys. Rev. Lett. 88, 120401 (2002).
- (9) Z. R. Gong, H. Ian, Yu-xi Liu, C. P. Sun, Franco Nori, arXiv:0805.4102 (2008).
- (10) M. Bhattacharya and P. Meystre, Phys. Rev. Lett. 99, 073601 (2007).
- (11) P. Zhang, X. F. Liu, and C. P. Sun , Phys. Rev. A 66, 042104 (2002).
- (12) C. P. Sun, Phys. Rev. A 48, 898 (1993).
- (13) Y. X. Liu, N. Imoto, Ş. K. özdemir, G. R. Jin, and C. P. Sun, Phys. Rev. A 65, 023805 (2002).
- (14) G. R. Jin, P. Zhang, Yu-xi Liu, and C. P. Sun, Phys. Rev. B 68, 134301 (2003).
- (15) C. P. Sun, Y. Li and X. F. Liu, Phys. Rev. Lett. 91, 147903 (2003).
- (16) Y. Li and C. P. Sun, Phys. Rev. A 69, 051802 (2004).
- (17) L. He, Y. X. Liu, S. Yi, C. P. Sun, and F. Nori, Phys. Rev. A 75, 063818 (2007).
- (18) J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley, Reading, Ma, 1994).
- (19) L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-relativistic Theory) (Butterworth-Heinemann, Oxford, 1977) Third Edition.
- (20) D. Meiser and P. Meystre, Phys. Rev. A 74, 065801 (2006)
- (21) S. Bose, K. Jacobs, and P. L. Knight, Phys. Rev. A 56, 4175 - 4186 (1997).
- (22) H. T. Quan, Z. Song, X. F. Liu, P. Zanardi, and C. P. Sun, Phys. Rev. Lett. 96, 140604 (2006) .
- (23) M. Orszag, Quantum Optics: Including Noise Reduction, TrappedIons, Quantum Trajectories and Decoherence (Springer-Verlag, Berlin Heidelberg, 2000).
- (24) X. X. Yi, H. Y. Sun, and L. C. Wang, arXiv:0807.2703 (2008).