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Quantum dynamics of spatial decoherence of two atoms in a ring cavity

We study the spatial decoherence dynamics for the relative position of two atoms in a singlemode ring cavity. We find that the spatial decoherence of the two atoms depends strongly on their relative position. Taking into account the spatial degrees of freedom, we investigate the entanglement dynamics of the internal states of the two atoms. It is shown that the entanglement decays to almost zero in a finite time, and the disentanglement time depends on the width of the wave packets describing the atomic spatial distribution.


I. INTRODUCTION
Superposition and its many-particle version-entanglement-are two basic features of quantum physics, distinguishing the quantum world from the classical world. Because of its intriguing properties, quantum entanglement has attracted considerable attention as an important resource for quantum information processing [1,2]. However, quantum coherence can be destroyed due to the physical system interacting with the environment, which has been recognized as a main obstacle to realizing quantum information processing. Hence a better understanding of the mechanisms of quantum decoherence is not only crucial for the understanding of the quantum-classical transition (see, e.g., [3][4][5]), but also essential for the implementation of quantum information processing.
The influence of atomic spatial motion on quantum dynamics has been considered in different contexts (see, e.g., Refs. [18,19]). For a system of two cold atoms placed in a noisy vacuum field, the back-action of emitted photons on the wave packet evolution about the relative position of the two cold atoms was discussed in Ref. [20]. It was shown that the photon recoil resulting from the atomic spontaneous emission can induce the localization of the relative position of the two atoms, through the entanglement between the spatial motion of individual atoms and their emitted photons.
In contrast with previous works, here we consider the case where a pair of atoms simultaneously interact with a single-mode ring cavity. In fact, this kind of system is a typical system in which the cavity field acts as a data bus inducing two qubits to be entangled [21,22]. The system is efficiently used in many schemes for proposing how to realize quantum logic gates and teleportation in cavity QED [23]. In this paper, we will study the spatial decoherence of the atomic relative position and the disentanglement of the internal states of these two atoms induced by the back-action of the photons emitted from these atoms. We find that the spatial decoherence of the two atoms depends strongly on their relative position. Our results show that the entanglement of the internal states decays to almost zero in a finite time, and the disentanglement time is determined by the width of the atomic wave packets. This paper is organized as follows. In Sec. II we present an effective Hamiltonian of the total system and then give the solution for the eigenequation of the effective Hamiltomian. In Sec. III we discuss the time evolution of the density operator of the total system and obtain the reduced density matrix in the relative-coordinate picture. Furthermore, a decoherence factor for arbitrary cavity-field and general atomic spatial states is introduced. In Sec. IV, under the assumption that the two atoms' relative position is initially in a superposition state of two Gaussian wave packets, we demonstrate the spatial decoherence when the cavity field is initially in a coherent state. In Sec. V we study the disentanglement dynamics of the two atoms' internal states. Finally, a concluding summary is given in Sec. VI.

II. MODEL
Our system consists of two identical two-level atoms that interact with a single-mode cavity field (Fig. 1). The two atoms are denoted here as atom 1 and atom 2. The mass of each atom is denoted by m 0 and their atomic transition frequency is ω 0 . Assume that the two atoms are spatially separated and located at the positionsx 1 andx 2 respectively, and the corresponding momenta for atom 1 and atom 2 arep 1 andp 2 . Then the momentum for the center of mass (c.m.) of the two atoms isP =p 1 +p 2 , and the relative momentum of the two atoms isp = (p 1 −p 2 )/2. The c.m. position and the relative position areX = (x 1 +x 2 ) /2 andx =x 1 −x 2 , respectively.
Under the rotating-wave approximation, the Hamiltonian of this system reads aŝ are the atomic operators for the i-th atom with respect to the excited state |e and the ground state |g of the atoms,â † andâ are the creation and annihilation operators for the cavity field, ω and k are the frequency and wave number of the cavity field, and g is the atom-field coupling constant. There may be a counter-propagating running-wave mode with a wave vector −k in such a ring cavity, which may affect the atom-field dynamics and contribute to the interaction part. However, as discussed in Ref. [18,24], we can consider that an atom traverses only an arm of the optical ring cavity and only one cavity mode is excited by an external laser. Thus, it is reasonable to consider a single propagating (e.g., a clockwise-running) wave mode in a ring cavity.
We first factorize the evolution operatorÛ (t) = exp(−iĤt/ ) into a product aŝ whereŴ (x i ) (i = 1, 2) is a unitary transformation defined bŷ which concerns the coupling of the internal levels with the spatial degrees of the atom i. The operatorÛ e (t) = exp −iĤ e t/ is easily proved to be determined by the effective HamiltonianĤ e =Ĥ 0 +Ĥ 1 of the system witĥ Here, Ω 1 = ω 0 + p 1 k/m 0 and Ω 2 = ω 0 + p 2 k/m 0 . In the following discussion, we consider that the atomic transition is resonant with the cavity mode, i.e., ω 0 = ω. In this case, we have ω 0 = ck (c is the light velocity in vacuum). Note that p i k/m 0 = V i k (i = 1, 2), where V i is the velocity of the atom i. Thus, when the velocity of the atom i is far smaller than the light velocity in vacuum (i.e., V i << c), we have p i k/m 0 << ω 0 . On the other hand, the condition p i k/m 0 << g needs to be met, such that the influence of the momentum-dependent energy shifts of the atomic internal levels on the system dynamics is negligibly small. To see the availability of this condition, let us consider a 87 Rb atom with two circular Rydberg levels |g and |e (corresponding to principal quantum numbers 50 and 51). The transition frequency between |g and |e is ∼ 51.1 GHz [25]. Thus, we have k ∼ 10 3 m −1 for the case when the transition between |g and |e is resonant with the cavity mode. The coupling constant g is on the order of 10 5 Hz [25][26][27]. For the laser-cooled and optically trapped 87 Rb atom, we can assume that the atom has a measured temperature of T ∼ 180 µK [28]. A simple calculation shows that the atomic velocity is V ∼ 23 c.m./s. Hence, we have V k ∼ 230 Hz << 10 5 Hz, which demonstrates that the approximation of the condition |p i k/m 0 | << g could be satisfied in practice. The analysis given here shows that the term p i k/m 0 in Ω i can be neglected, leading to We note that the same approximation was made in Ref. [20]. The total excitation numberâ †â + |e 1 e 1 |+ |e 2 e 2 | is conserved during the interaction. By resolving the eigenequation ofĤ 1 , in the subspace spanned by states with the total excitation number (n + 2), the following eigenstates of H 1 are obtained: with the eigenvalues where A n = 2(2n + 3)g, f 1n = (n + 1) / [2 (2n + 3)], f 2n = (n + 2) / [2 (2n + 3)] and n is an arbitrary nonnegative integer.

III. DECOHERENCE FACTOR
In this section, we discuss the time evolution of the system, and investigate the spatial decoherence factor for the atom-atom relative position.
Assume that the initial density operator of the whole system is given bŷ whereρ s (0) = |ψ (0) ψ (0)| is the initial density operator for the spatial motion of the two atoms,ρ i (0) = |φ (0) φ (0)| is the initial density operator for the internal state of the two atoms, andρ f (0) is the initial density operator for the cavity field.
Let us now assume that the state |ψ (0) is expanded (in the momentum representation) as where |p 1 is the momentum eigenstate of atom 1, |p 2 is the momentum eigenstate of atom 2, and C p1p2 is the distribution function satisfying the normalization condition ∞ −∞ dp 1 dp 2 |C p1p2 | 2 = 1. The state |φ (0) is assumed to be a pure separable state |e 1 , e 2 . In terms of Fock states |n and |n ′ , the density operatorρ f (0) is, in general, written asρ The time evolution ofρ (0) can be obtained byρ (t) =Û (t)ρ (0)Û † (t). Following the above eigenstates (6)-(9), eigenvalues (10) ofĤ 1 , and based on Eqs. (2)(3) and (11)(12)(13), we find that the density operatorρ (t) can be written aŝ where and We assume that the distribution function C p1p2 in Eq. (12) satisfies C p1p2 = C P C p ; i.e., the initial state |ψ (0) for the spatial motion of the two atoms can be written as |ψ (0) = |µ (0) ⊗ |ϕ (0) , with |µ (0) = ∞ −∞ dP C P |P describing the initial c.m. state of the atoms with a momentum distribution function C P corresponding to the c.m. momentum eigenstate |P , and |ϕ (0) = ∞ −∞ dp C p |p describing the initial relative position state of the atoms with a distribution function C p corresponding to the relative momentum eigenstate |p . Both C P and C p satisfy the normalization condition, i.e., ∞ −∞ |C P | 2 dP = 1 and ∞ −∞ |C p | 2 dp = 1. By tracing over the cavity field, the c.m. motion, and the internal states of the atoms, we obtain from Eq. (14) the following elements of the reduced density matrix (in a relative coordinate picture) where ϕ (x, t) = ∞ −∞ dp C p exp −ip 2 t/ (m 0 ) + ipx/ is the free-evolution state of |ϕ (0) expressed in a relative coordinate picture, and F (x, x ′ , t) is the decoherence factor, which is given by The decoherence factor F (x, x ′ , t) given here will be used below in our analysis of the spatial decoherence of two atomic wave packets.

IV. SPATIAL DECOHERENCE OF TWO ATOMIC WAVE PACKETS
We consider that the initial state |ϕ (0) of the two atoms' relative position is a superposition of two Gaussian wave packets centered at a and −a, respectively; i.e., we have (in the x representation) where G ± (x) = √ 2πd −1/2 exp −(x ± a) 2 / 4d 2 and δ = 1 + exp −a 2 / 2d 2 . Here δ is a normalization constant and, for simplicity, we have assumed that the two Gaussian distributions have the same spread d, which is limited in Here, d min is much smaller than the wavelength λ of the cavity field, but a zero spread is not permitted to avoid the atoms indistinguishable by quantum dispersion [29]. Assume that the time for the atoms staying in the cavity is so short that we can assume the spatial distribution remains unchanged with respect to the initial spatial state for the atomic free evolution. Then Eq. (19) can be written as a product of the initial state and the decoherence factor where ρ (x, x ′ , 0) = ϕ (x, 0) ϕ * (x ′ , 0) describes the initial state of the atomic relative position. The initial state is illustrated in Fig. 2. Here, the two peaks, along the x ′ = x direction, correspond to the diagonal terms of ρ (x, x ′ , 0), while the other two peaks along the x ′ = −x direction correspond to the off-diagonal terms of ρ (x, x ′ , 0) , which represent the coherence between the two wave packets. Following Eq. (20), it can be proved that for the case of x ′ = x, F (x, x, t) = 1, thus we have ρ (x, x, t) = ρ (x, x, 0), i.e., the diagonal terms of the density matrix ρ (x, x ′ , t) remain unchanged during the time evolution. However, for the off-diagonal terms with x ′ = −x = mλ (m is an arbitrary nonzero integer), it can be seen that |F (x ′ = −x, t)| ≤ 1. Thus we will mainly analyze the evolution of the off-diagonal terms of the reduced density matrix below.
For the cavity field initially in a coherent state |α , we have c n,n = exp −|α| 2 α 2n /n!, where α is a complex number. Thus, Eq. (20) becomes Along the x ′ = −x direction, the decoherence factor takes the following form which is shown in Fig. 3. It can be seen that:  Fig. 3(c)].
Above, we presented an analysis on the decoherence factor, for the case of the initial state ϕ(x, 0) described by Eq. (21). According to the above analysis, it can be concluded that: (i) for the case a = mλ, the two peaks along the x ′ = −x direction remain unchanged during the time evolution. So the relative motion of the two atoms decouples with the cavity field, and thus there is no decoherence induced by the photon recoil.
(ii) for the case a = mλ, the two peaks along the x ′ = −x direction partially decay in a finite time and there is no revival, and the decay of the two peaks depends on a in a cosine law, with a maximum decay at a = (m + 1/2) λ.
The evolution of ρ(x ′ = −x, t) for the initial state ϕ(x, 0) with a = mλ is shown in Fig. 4, which demonstrates that the initial state ϕ(x, 0) undergoes a partial decoherence after a finite time and a revival does not occur. Thus, the coherence of the atomic relative position is strongly destroyed. The wigner functions for the initial density matrix ρ(x, x ′ , 0) shown in Fig. 2  out that the oscillations are partially damped by decoherence [ Fig. 5(b)]. This may be induced by the entanglement between the two-atom spatial motion and the cavity field resulting from the photonic back-action. This entangling process inevitably destroys the coherence of the spatial motion of the two atoms. Since the single-mode cavity field is just an environment with a few degrees of freedom, the spatial coherence is not completely destroyed.

V. DISENTANGLEMENT DYNAMICS OF TWO ATOMS
To begin with, let us assume that: (i) the initial spatial state |ψ (0) of the two atoms is a separable state Here, |µ 1 and |µ 2 represent two Gaussian wave packets describing the spatial distribution of the two atoms, respec- Here, p is the relative momentum of the two atoms and x is the relative position of the two atoms.
tively. In the coordinate representation, |µ i is expressed as (i = 1, 2): where a i is the center of the Gaussian function µ (x i , 0) and d is the width of the Gaussian function µ (x i , 0). The coefficient C pi is given by For convenience, we assume that a 1 = −a 2 = a/2. The initial density operator for the spatial motion of the two atoms isρ s (0) = |ψ (0) ψ (0)|. (ii) the initial internal state |φ (0) of the two atoms is an entangled state, which is given by The initial density operator for the internal state of the two atoms isρ i (0) = |φ (0) φ (0)| .
(iii) the cavity field is initially in a vacuum state, i.e.,ρ f (0) = |0 0|. The initial density operator of the whole system isρ (0) =ρ s (0)ρ i (0)ρ f (0). At time t, the density operator of the system isρ (t) =Û (t)ρ (0)Û † (t). Here,Û (t) is the unitary operator in Eq. (2). After tracingρ (t) over the electromagnetic field and the spatial degrees of freedom of the atoms, we obtain the following reduced density matrix for the internal state of the two atoms:ρ which is written in a basis formed by |e 1 , e 2 , |e 1 , g 2 , |g 1 , e 2 and |g 1 , g 2 . Here, with ξ(t) = s(t) + d 2 k 2 − ika and s(t) = 2 k 2 t 2 / 4d 2 m 2 0 . The parameter s(t) is obtained after tracingρ (t) over the atomic spatial degrees of freedom, which causes the strong time dependence of the concurrence as shown below.
A popular measure of entanglement forρ i (t) is given by the concurrence [30], which is defined as where the λ i are the eigenvalues of the non-Hermitian matrixρ i ρ i in descending order, and ρ i = σ 1 y ⊗σ 2 y ρ * i σ 1 y ⊗σ 2 y . Here,ρ * i is the complex conjugate ofρ i , andσ i y is the usual Pauli operator for atom i (i = 1, 2). The concurrence C varies from C = 0 for a non-entangled state to C = 1 for a maximally-entangled state. The concurrence of the density matrixρ i (t) in Eq. (29) is [31,32] C (ρ i (t)) = 2 max{0, |z| − √ ad, |w| − √ bc}, which is shown in Fig. 6. Eq. (31) tells us that the concurrence evolves with an non-smoothly exponential decay due to the factor e −s as shown in Fig. 6. Alternatively, the initial internal state for the two atoms could be another type of entangled state given by |φ (0) = cos γ |e 1 , g 2 + sin γ |g 1 , e 2 .
The concurrence for the reduced density matrixρ i (t) in Eq. (33) is given by C (ρ i (t)) = 2 |z| , which is shown in Fig. 7. One can see that there is a common factor e −s(t) with s(t) = 2 k 2 t 2 / 4d 2 m 2 0 , appearing in the expression of z above. Because of this factor e −s(t) , the value of |z| decays to zero with time asymptotically. Here, Fig. 6 and Fig.  7 show that the oscillations of the entanglement are damped exponentially, but the decay of the entanglement occurs faster when the width d of the initial Gaussian function becomes smaller, and the entanglement decays to almost zero in a finite time. So the back-action of the emitted photons may induce an entanglement between the two-atom internal states and the two-atom spatial motion states, and this correlation destroys the coherence of the internal states of the two atoms. The smaller the width of the wave packets, the larger the uncertainty of the atomic spatial momentum, which results in a faster destruction of the atomic internal-state entanglement.

VI. CONCLUSION
We have considered a system of two atoms interacting with a single-mode cavity field. Since the spatial degrees of freedom of the atoms are considered, the Hamiltonian of the whole system becomes complicated. To solve the Schördinger equation, we have introduced two unitary transformations (involving the coupling of the internal levels with the spacial degrees of the atoms), presented an effective Hamiltonian of the system, and given the analytic solutions to the eigenequation of the effective Hamiltonian. Based on these, we have investigated the decoherence dynamics for the relative position of the two atoms, and presented a decoherence factor for a general cavity-field state and an arbitrary atomic spatial state, which might be useful for future related works.
Under the assumption that the atomic relative position is in a superposition of two Gaussian wave packets, we have demonstrated the spatial decoherence of the two atoms' relative position, for a cavity field initially in a coherent state. Our results show that the spatial decoherence of the two atoms depends strongly on the relative position of the two atoms. Interestingly, we found that when the relative position of the two atoms is an integral multiple of the wavelength of the cavity field, the spatial coherence of the relative position of the two atoms is not destroyed by the photon recoil; However, when the relative position of the two atoms is not an integral multiple of the wavelength of the cavity field, spatial decoherence of the relative position of the two atoms happens.
Furthermore, we have studied the entanglement dynamics of the internal states of the two atoms interacting with a single-mode cavity field. Our results show that the entanglement, measured by the concurrence, decays to almost zero in a finite time. Thus, the back-action of the photons emitted from the two atoms may be a fundamental process destroying the entanglement of atoms.
From this work, it can be concluded that there exists a phenomenon that the moving qubits (e.g., atoms) placed in a cavity, may suffer from the spatial decoherence. Therefore, it is important to overcome the influence of the spatial motion of qubits on their entanglement.