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Protecting entanglement by adjusting the velocities of moving qubits inside non-Markovian environments

Efficient entanglement preservation in open quantum systems is a crucial scope towards a reliable exploitation of quantum resources. We address this issue by studying how two-qubit entanglement dynamically behaves when two atom qubits move inside two separated identical cavities. The moving qubits independently interact with their respective cavity. As a main general result, we find that under resonant qubit-cavity interaction the initial entanglement between two moving qubits remains closer to its initial value as time passes compared to the case of stationary qubits. In particular, we show that the initial entanglement can be strongly protected from decay by suitably adjusting the velocities of the qubits according to the non-Markovian features of the cavities. Our results supply a further way of preserving quantum correlations against noise with a natural implementation in cavity-QED scenarios and are straightforwardly extendable to many qubits for scalability.


I. Introduction
Quantum mechanics is a fundamental and accurate theory that can predict many striking effects that are counterintuitive and impossible in classical mechanics. One of the most remarkable effects is the existence of entangled states of two or more distant particles. Entanglement, as non-local quantum coherence of a composite system, describes quantum correlations between two or more quantum subsystems. Generally, a multipartite system is said to be entangled, within a mathematical language, if its quantum state cannot be written as the tensor product of the quantum states of the constituent subsystems and is thus a whole object. Nowadays, entanglement is of great importance due to its fundamental role in quantum information tasks such as quantum teleportation [1], quantum error correction [2,3], quantum cryptography [4] and quantum dense coding [5]. However, the unavoidable interaction between any realistic quantum system and its surrounding environment typically causes decoherence and then usually destroys the entanglement of a quantum state. Such an undesired effect has been considered as a pesky problem to the implementation of various quantum information processing schemes. Therefore, finding a way to overcome this drawback is a valuable step for the realization of practical quantum computers [6,7].
Along this route, investigation of entanglement dynamics in Markovian and non-Markovian environments has recently attracted wide attention . It is well known that in Markovian (memoryless) environments the single qubit coherence decays exponentially. As a consequence, two-qubit entanglement will deteriorate in a short time, also completely disappearing at a finite time [65,66]. Nevertheless, researchers have recently succeeded to preserve two-qubit entanglement in Markovian environments [8][9][10][11][12]. For instance, it has been shown that two-qubit entanglement can be protected by quantum interferences [8]. Moreover, two-qubit entanglement can be both generated and kept for a very long time in the ideal lefthanded materials [9].
However, so far the effect of translational motion of qubits on the two-qubit entanglement dynamics in non-Markovian environment has not been addressed. The knowledge of this effect is important in that it can provide novel insights for preserving entanglement against noise by tailoring motional properties of the qubits. This boosts our motivation to address this problem by studying a simple yet paradigmatic system made of two moving qubits independently interacting with their local environments. It is assumed that the environment of each qubit is structured and modeled by a leaky cavity with Lorentzian spectral density and the qubits are initially prepared in a maximally entangled state.
The paper is organized as follows. In Sec. II the Hamiltonian is introduced and a state evolution in the considered system is discussed. In Sec. III concurrence is presented as measure of entanglement. In Sec. IV, we present the results of our numerical simulations illustrating the excellent performance of qubit motion in protecting two-qubit entanglement.
Finally, Sec. V concludes this paper. ,

II. Model and Solution
is the amplitude in the k th-mode. In order to satisfy the boundary conditions at the mirrors, the mode functions ) (z U k can only be of the following form: Here k  has the values 1, -1 going from each mode to the subsequent one, and k M for a ) are the frequencies of the quasi modes and  is the damping of the ( 0 , l ) cavity. In fact,  quantifies photons leakage through cavity mirrors and indicates the spectral width of the coupling.
The qubit (two-level atom) is taken to interact only with the second cavity ( 0 , l ) and it moves along the z-axis with constant velocity v (see figure 1). This condition can be thought to be fulfilled by Stark shifting (for instance, by turning on a suitable external electric field) the atom frequency far out of resonance from the cavity modes until 0  z and then turning off the Stark shift [71]. During the translational motion, the qubits interact with their respective cavity modes. The Hamiltonian of each subsystem ( ) under the dipole and rotatingwave approximation is given by (we take where e ( g ) and 0  are the previously mentioned excited (ground) state and the transition frequency of the qubit, ) is the creation (annihilation) operator for the k -th cavity mode with frequency k  and denotes the coupling constant between the qubit and the cavity modes. Note that, d is the magnitude of electric-dipole moment of the atom qubit and A is the surface area of the cavity mirrors.
The parameter ) (z f k describes the shape function of qubit motion along the z-axis, and it is given by [69] )], Hz [70,71].
It is well known that the translational motion of an atom can be treated classically if the de Broglie wavelength B  of the atom is much smaller than the wavelength 0  of the resonant transition [69,72]. For the considered qubit, this is equivalent to On the other hand, the relative smallness of photon momentum ( c / 0   ) compared to atomic momentum ( mv ) allows one to neglect the atomic recoil resulting from the interaction with the electric field [73]. To satisfy this condition, the velocity of the qubit is required to be 7 10   v m/s.
Owing to the two subsystems are the same and noninteracting, they experience independent evolutions. Therefore, one can analyze only the dynamics of one subsystem and then use it to obtain the evolution of the total system [14,34,35]. The evolution is considered during the flight time of the atom qubit inside the cavity. We assume each subsystem to be initially in a product state with the qubit in a coherent superposition of its states ( the cavity modes in the vacuum state ( 0 ), Hence, at any later time t , the quantum state of a subsystem can be written as (8) in which the cavity field state k 1 describes the presence of a single photon in mode k , and ) ( , t C k g represents its probability amplitude. Using the time-dependent Schrödinger equation, the differential equations for the probability amplitudes Solving Eq. (10) formally and substituting the solution into Eq. (9), one obtains By redefining the probability amplitude as , we can rewrite Eq. (11) as where the kernel ) , , which is the correlation function including the memory effect, is In the continuous limit the kernel above becomes where ) ( k J  is the spectral density of an electromagnetic field inside the ( 0 , l ) cavity and has the form [74] , is the detuning between 0  and the center frequency of the cavity modes is the decay rate of the qubit in the Markovian limit of flat spectrum with the qubit is at rest in a position in which ) holds in the strong coupling regime where non-Markovian effects become relevant [14,34].
Substituting the resulting kernel into Eq. (8) yields Taking the derivative of Eq. (17) with respect to time, we get ], . The quantity ) (t  plays the role of a time-dependent Lamb shift and ) (t  can be interpreted as a time-dependent decay rate [74].
Once known the dynamics of individual subsystems, we can now determine the dynamics of the total system. In the standard product basis , and assuming that the two individual subsystems are the same, one obtains the elements of the time-dependent reduced density matrix of the total system ) (t  as [34] ].

III. Concurrence
The entanglement of a 2 × 2 quantum system, described by the density matrix  , can be measured by concurrence which is defined as [75] } , 0 max{ ) ( The corresponding concurrence at time t is given by [34]

IV. Results
In this section, we analyze the dynamics of two-qubit entanglement by numerical solution of   An interesting result is obtained for   1 0  . In this case, the entanglement is strongly protected from being lost. Namely, the entanglement does not change appreciably from its initial value as time passes. According to figure 2 one may conclude that a stronger entanglement protection can be obtained by regularly increasing the velocity of the qubits. To deepen this conjecture, we examine the effect of higher velocities on the entanglement dynamics in figure 3. The atomcavity parameters are the same as in figure 2 ). We find out a contrast between trends of maxima and minima by increasing m : the value of the minima is significantly increased by increasing m , whereas the value of the maxima is slightly decreased. To probe the phyisical origin behind the aforementioned entanglement behavior, we plot in figure 5 the decay rate of single qubit ( ) as a function of t  for different velocities of the qubit. As can be seen from this figure, the decay rate can be greatly inhibited for suitable values of qubit velocity. It is known that the negativeness of the decay rate indicates reabsorbtion of emitted photon by the qubit, leading to information backflow from the environment to the qubit and thus signaling non-Markovianity [13][14][15]30]. Figure 5(a) shows that the decay rate of stationary qubit ( 0 0   ) has a periodic behavior with positive and negative spikes in it. The periodicity is increased for   01 . 0 0  , whereas the magnitude of the spikes is decreased.
This change in the evolution of the decay rate explains why the second revival of entanglement for   01 . 0 0  occurs later than 0 0   (see figure 2). In figure 5(b), we see that the spikes are completely disappeared and the decay rate gets oscillating behavior.
However, the amplitude and periodicity of the decay rate are remarkably decreased by    figure 6 reveals that, for a given velocity, the entanglement can be maintained for a longer time by increasing the non-Markoviniaty (memory effects) of the system dynamics (that is, decreasing the cavity bandwidth  ). Furthermore, figure 6 shows that the motion of the qubits with high enough velocities can remarkably affect the entanglemet even in environments with weaker memory effects (larger values of the cavity bandwidth λ). Another fact is that, when  . It can be clearly seen that in the case of stationary qubits, the entanglement increases regularly by gradual growth of the detuning parameter (Figure 7(a)). A similar result was reported before (see Refs. [41,42]). However, such a regular behavior is not observed for moving qubits (Figure 7  (2)) [77,78]: adjusting the position of the qubit linearly with time, so to have a relation like vt z  , will be then sufficient to realize the model here considered.

Conclusion
We have introduced a system containing two moving atom qubits inside two separated identical cavities. The moving qubits independently interact with their respective cavity modes. In the case of resonant interaction, we have found that compared to stationary qubits, quantum entanglement between moving qubits generally remains closer to its initial value over time. Moreover, we have demonstrated that the initial entanglement can be strongly protected for a long time by appropriately setting the velocities of the qubits according to the non-Markovian characteristics of the cavities (e.g., quality factor, spectral bandwidth and mode frequencies). A strong-coupling (low spectral bandwidth) is in general required for a more effective entanglement protection, with an interplay between memory effects and decoherence inhibition indicated by the value of the qubit decay rate. The role of the translational movement of the atoms in controlling the decay rate stems from the presence of atom (qubit) velocities in the intensity of the coupling to the cavities (see equations (4) and (5)). These considerations suggest a quantitative study, to be done elsewhere, of the degree of non-Markovianity by suitable geometric measures [79] in order to assess to what extent memory effects contribute to preserve entanglement for various velocities of the qubits.
Our results supply new knowledge regarding the response of the entanglement dynamics to the velocity of an atomic qubit interacting with a cavity radiation, which suggests a further way of preserving quantum correlations against noise with a natural implementation in cavity-QED scenarios. Thanks to the analogy between cavity-QED and circuit-QED, one can then design feasible circuit-QED schemes which reproduce our model, by exploiting currently available setups with a sinusoidal position-dependent qubit-cavity coupling [77,78] where the qubit position slowly varies linearly with time.
Our model is straightforwardly extendable to a set of many independent qubits, thus satisfying the requirement of scalability. The present study opens the way to investigate other situations of interest, for instance an array of consecutive cavities where each atom qubit can fly in or a set of oscillating atoms each one within its own cavity, which may lead to new efficient strategies for entanglement protection.