Deterministic quantum controlled-PHASE gates based on non-Markovian environments

We study the realization of the quantum controlled-PHASE gate in an atom-cavity system beyond the Markovian approximation. The general description of the dynamics for the atom-cavity system without any approximation is presented. When the spectral density of the reservoir has the Lorentz form, by making use of the memory backflow from the reservoir, we can always construct the deterministic quantum controlled-PHASE gate between a photon and an atom, no matter the atom-cavity coupling strength is weak or strong. While, the phase shift in the output pulse hinders the implementation of quantum controlled-PHASE gates in the sub-Ohmic, Ohmic or super-Ohmic reservoirs.

In the study of quantum cavity dynamics involving the dissipation, the Markovian approximation is introduced into the method to solve the evolution of such atom-cavity systems [20,21]. Given that the secondorder perturbation of the system-reservoir coupling is applied, the feasibility of the Markovian approximation demands that the couplings between systems and the surrounding reservoirs be weak enough. Meanwhile, the characteristic time of the reservoir is much shorter than that of systems, which leads to the negligible memory backflow from the reservoir to the system. However, many studies have been demonstrated that the method with the Markovian approximation cannot provide the correct descriptions of the dynamics of the open quantum systems in cases that Markovian conditions do not hold [22][23][24][25][26]. Hence, the precise structures and memory effects of the reservoirs should be considered. Typical examples include a single trapped ion interacting with an engineered reservoir [22,23], spontaneous emission atomic dynamics in photonic crystals [24], and microcavities coupling with a coupled resonator optical waveguide or photonic crystals [25,26], and so on. Atom-cavity system is very promising in realizing the quantum controlled-PHASE gate [9]. Here in this article we investigate the quantum controlled-PHASE gate in non-Markovian regime. We shall count in and more importantly, make use of the memory backflow from the reservoir to realize a deterministic quantum controlled-PHASE gate even though the atom-cavity coupling strength is weak. It has been shown that, depending on the atomic states, the input single photon can be directly reflected unchanged or transmitted through the cavity with π phase-shift [9]. Such study of quantum controlled-PHASE gates extends to the area of non-destructively detection of the photon [5,6], the heralded storage of the photonic polarization states [27]. Moreover, following the spirit of designing quantum controlled-PHASE gates between the atom and the Figure 1. Configuration of the system. A Λ-type atom couples resonantly to the single cavity mode with coupling strength g. The atom also interacts with the reservoir 1, while the single cavity mode is coupled to the reservoir 2. The light is reflected perfectly when reaching the left side of the cavity, and the light can transmit or reflect through the right side of cavity.
in the interaction picture can be written as ( ) is the transition frequency between state kñ | and lñ | of the atom, k l kl s = ñá | |is the atomic operator. The single cavity mode is denoted by the creation and annihilation operators a † and a, 0 w is its frequency. The parameter g represents the coupling strength between the cavity mode and an atom. The two reservoirs are modeled by a collection of harmonic oscillators. b 1 w ( )and b 1 w ( ) † are the annihilation and creation operators of reservoir 1. Similarly, the external field (reservoir 2) is denoted by b 2 w ( ) and b 2 w ( ) † . The atomic transitions ab s and ac s are coupled to the reservoir 1 with coupling strength g 1 w ( ) and g 1 w( ), respectively.
2 k w ( ) is the coupling strength between the external field and single cavity mode.
We assume that the input optical field has only one photon initially and the atom is not excited. The intracavity field and the reservoir 1 are vacuum at first. In our article, we only concern the case that the interaction forms in the total system satisfy the rotating-wave approximation. In this case the population of the total system is conserved. The total system therefore has only one excitation in the evolution. The total state can be generally expressed as The time evolution of total system is governed by the equation Hence, we substitute t y ñ | ( ) (equation (2)) into this equation, and obtain the equations of motion c t c t g i e , 3 t 300 1 i ab 200 0 By integrating the equations (3)-(6), respectively, we get For high frequency optical systems, 0 w is so large. One can shift the integration c 0 e d t 0 )is the response function between the intracavity field and external field. The normalization of input optical fields a t t d 1 , which makes the initial states of total systems normalized. Similarly, by using equations (13a), (13b) and adopting the same procedure as before, we obtain Here, a t iñ ( )and a t out ( ) characterize the envelope shapes of input and output fields, when an atom is in state cñ | . Substituting equation (11a) into equations (7), (10) and (12) into equation (8), and equation (13a) into equation (9), we finally get It is noted that all of the derivations above do not introduce any approximation, including the Markovian approximation, so the obtained results above can be applied to the general discussion of the atom-cavity system. In our study, we want to investigate the feasible construction of quantum gates for the high-speed quantum communication. The characteristic time of the system is comparable with the correlation time of the reservoir, and the non-Markovian information backflow needs to be counted. In the equations above (equations (19)-(21)), the integral kernels involve the structured reservoirs. The non-Markovian memory effects are therefore taken into account by these correlation functions The equations derived above describe interactions between atom-cavity system and general reservoirs. By adopting these integrodifferential equations, one can investigate the dynamic process of the total system beyond Markovian approximation. Following the calculation above, we will discuss the realization of the controlled-PHASE gates with different structured reservoirs.

Analysis of the non-Markovian controlled-PHASE gates in different reservoirs
In the following, we discuss the realization of the quantum controlled-PHASE gates with different structured reservoirs. Numerical simulations are completed with typical spectral densities, e.g. Lorentzian spectral density, Ohmic, sub-Ohmic and super-Ohmic spectral densities. We consider the atom serves as control qubits, and photonic polarizations as target qubits. For an ideal controlled-PHASE gate, when an atom is in state b c ñ ñ | (| ), the input single photon coupled with(decoupled from) the atomic transition, will be immediately reflected (transmitted) by the cavity with a phase-shift zero p ( ). The corresponding initial conditions are a t t d 1

Lorentzian reservoir
We suppose that both the two reservoirs stated before have Lorentzian spectrums where g g (˜) and 1 1 l l (˜) represent the strength and spectral width of the coupling between atomic transition ab ac s s ( )and reservoir 1, respectively. κ denotes coupling strength between intracavity and external fields, and 2 l is the spectral width. By substituting (17), the response function can be expressed as We can also obtain the time correlation functions of the reservoirs by use of equations (25)-(27), It should be noted that we take 2 2 0 ) and shift the integration limit (0, +¥) in equation (22) (equations (23), (24) For high frequency optical systems, 0 w ( , ab ac w w ) is so large that one can extend the lower limit to -¥ as a good approximation [17,21] and use the Fourier transformations to obtain the above equations. Specifically, the parameters , 1 1 l l and 2 l are inversely proportional to the reservoir correlation times. When , 1 1 l l and 2 l converge to infinite, the reservoirs become memoryless. Combining with and assuming , 2 1 1 a g g l l = = = g˜h ereafter, one can describe the Markovian dynamics of the system as follows, These equations are same as those in [11].
In the following, we consider the memory backflow from the reservoirs and discuss the realization of the quantum controlled-PHASE gates with these reservoirs. We suppose that the input single photon pulse has a Gaussian temporal shape a t a t ce , 3 3 in in where t D is its full-width-half-maximum temporal duration. t 0 represents the time the pulse peak enters the cavity.  (16) and (18)-(21), one can easily find that the values of c c , 200 300 and a t out ( ) are always zero, and therefore reservoir 1 is vacuum all the time. Hence, dynamics of the total system can be described only with equations (18), (21) and (28), (29). We just need to discuss the influences brought by the various couplings between intracavity and external fields, because atomic transition ac s is decoupled from the cavity mode and reservoir 1 is vacuum at first. The detailed numerical results are shown in figures 2 and 3. For an ideal quantum controlled-PHASE gate, the output a t out ( ) should be same as the input a t iñ ( )when an atom is in state cñ | . This perfect performance can be achieved in a Markovian environment (equation (32)), as plotted in figure 2. Here, the output field is a little delayed in relation to the input one, which is mainly due to the time spent to propagate in the cavity. We define parameter R 2 2 2 k l = to characterize the coupling between intracavity and external fields. For an input Gaussian pulse with t 1 D = μs, when the coupling strength 2 1MHz k p =´, the output optical pulses under different R 2 are shown in figure 2(a), the corresponding probabilities to find a single photon in cavity P t c t 500 500 2 = ( ) | ( )| are illustrated in figure 2(d). When R 2 goes to zero, the reservoir 2 converges to a memoryless one. In this case, a t out ( ) and P t 500 ( ) obtained in non-Markovian approximation will converge to the results in the Markovian limit. In our chosen parameters, the numerical results are almost same as the corresponding Markovian limit when R 2 equals 0.1. As R 2 gets larger, the Markovian approximation fails and one might expect the memory backflow induced by the non-Markovian reservoirs. The population P t 500 ( ) keeps oscillating and then decays to zero in this situation. The temporal shape of output pulse a t out ( ) is no longer Gaussian and its phase-shift is finite, as shown in figures 2(a) and (d) and also in figures 2(c) and (f), in which the parameters 2 10MHz k p =´, t 0.1 D = μs. We furthermore plot figures 2(b) and (e), the coupling strength κ is same as that in figures 2(c) and (f), but the time duration t D of the optical pulse resembles the one in figures 2(a) and (d). One can find that the numerical results in non-Markovian approximation are almost same as the corresponding Markovian limit, even when R 2 is as large as 10. It is reasonable since the coupling strength κ is larger and therefore a wider spectral width 2 l is necessary to keep the parameters R 2 same. Trade off between κ and 2 l leads to the results in figures 2(b) and (e). For a shorter input optical pulse, as shown in figures 2(c) and (f), the characteristic time scale is shorter and become comparable with the reservoir correlation time as R 2 gets larger. Hence, the non-Markovian effects become more visible in the short pulse case. We concern about the  output field a t out ( ), as it can exhibit whether a quantum controlled-PHASE gate is successful. By setting the same parameters used in plotting figures 2(c) and (f), one can obtain the simulation results in figure 3. When R 2 is smaller than 1.6, the output light a t out ( ) has exactly the same Gaussian temporal shape as the input one a t iñ ( ), and its phase-shift is zero all the time. However, as R 2 keeps increasing and becomes larger than 1.6, the non-Markovian memory effect gets stronger, as previously said. It means that, when an atom is in state cñ | , we can always achieve an ideal output pulse by appropriately manipulating R 2 2 l ( )for a determined input pulse a t iñ ( ) and coupling strength κ.
Now we focus on the other case, in which an atom is in state bñ | initially. By applying the same procedure as before, one can also find that a t out ( ) and c t 500 ( ) are always zero. Dynamics of the total system can be described with equations (16), (19), (20) and (28)- (31). In this situation, both the interactions between the atom and reservoir 1, and intracavity-external fields coupling should be taken into account. Similarly, we define parameter R 2 a 1 1 g l = to characterize the interactions between an atom and reservoir 1. As the atom interacts with reservoir 1 and also with the cavity mode, the corresponding coupling strength are set as 2 6MHz a g p =´and g 2 13MHz p =´, respectively, in figures 4 and 5. We first investigate how the non-Markovian reservoir 1 influences a quantum controlled-PHASE gate. When 2 1MHz k p =´, t 1 D = μs, the output a t out ( ) and populations P t c t P t c t , . The non-Markovian numerical results are almost same as the corresponding Markovian limit, even when R 1 is as large as 10. For a stronger coupling strength κ, a shorter input optical pulse can be considered, as illustrated in figures 4(c) and (d). The non-Markovian numerical results are in good agreement with those in Markovian approximation when R 0.1 1 = . As R 1 increases to 10, the memory effects of the non-Markovian environment appear apparently in this short input pulse case. Because the characteristic time scales become comparable with the correlation time of reservoir 1. The excitations which have been leaked into the reservoir 1 can flow back into the atom-cavity system. By the coupling between the single cavity mode and output fields, this memory effect can finally result in a much larger reflectivity, even when the atom-cavity system is in the weak coupling regime. It is quite good for high-speed quantum computation and communication. As we may acquire a short output pulse with 100% reflectivity and π phase-shift, through manipulating parameter R 1 1 l ( ) appropriately, when the atom-cavity coupling is weak. This typical non-Markovian feature would never appear in a Markovian case. For the mutual information exchange between the atom and reservoir 1, more excitations are provided by the cavity field. Thus, as R 1 increases, the population P t 200 ( ) decreases while P t 300 ( ) increases in figure 4(d). Now we take into account the non-Markovian effects caused by the coupling between intracavity and external fields. The parameters in figure 5 are set as same as figure 4, except that we choose R 10 1 = this time. A larger R 2 leads to a longer memory time of the external field, which enhances the back-flowing effects. Hence, the reflectivity increases and populations P t P t , 200 300 ( ) ( ) decreases, as shown in figure 5.
From the above, it is possible to achieve a deterministic fast controlled-PHASE gate by modulating the reservoirs with the Lorentz type spectral density. When one incidents a short input pulse, the corresponding coupling strength between the cavity and environment should be large. By appropriately increasing R 1 (decreasing 1 l ) and decreasing R 2 (increasing 2 l ), we can implement an ideal deterministic controlled-PHASE gate, even the atom-cavity coupling is weak. As demonstration, for a shorter input pulse with duration time t 0.05 D = μs, the numerical results are illustrated in figure 6, under the parameters 2 20MHz k p =´, The simulation results of the corresponding Markovian limits are also shown here as comparison. Here we take the parameters . Though a π phase shift can be acquired as an atom initially in state bñ | , the phase shift of the output pulse is time-varied with the atom at state cñ | . Such phase shift in the output pulse hinders the implementation of an ideal deterministic controlled-PHASE gate in sub-Ohmic, Ohmic or super-Ohmic environments.

Insight into the physical mechanism
We have seen that the deterministic quantum controlled-PHASE gates can only be realized in the Lorentzian reservoirs. The key point in this realization is that the memory effects of the Lorentzian reservoir 1 can be utilized, especially when the atom-cavity coupling is weak, as shown in figures 4(c) and 6(a). To confirm the results above, here we present some insights into the physical mechanism, by making use of the pseudomode theory [42][43][44]. In the pseudomode method, the pseudomodes are defined by the positions and residues of the In our article, the atom is not only coupled with these two pseudomodes, but also interacts with the single cavity mode. After considering the effects of the pseudomodes leakage, one can directly analyze the energy flowing between the atom and reservoir 1, from the compensated rate of change of the total pseudomode population b t t b t d d 2 2 + G | ( )| | ( )| [44]. When R 1 equals 0.1, the compensated rate of change b t t b t d d 2 2 + G | ( )| | ( )| is almost as same as the decay rate of the excited state population of the atom in Markovian limit c t 2 a 300 2 g | ( )| , as illustrated in figure 10(a). A larger R 1 (smaller 1 l ) results in a weaker coupling strength 1 W (or 2 W ) between the atom and pseudomodes, while the atom-cavity coupling is strong. In this condition, more excitations tend to transfer between the atom and single cavity mode, while less excitations are exchanged between the atom and pseudomodes, as shown in figure 10. When R 1 gets much larger, the atom is almost decoupled from the pseudomodes, and almost none excitations transfer to the pseudomodes, as depicted in figure 10(c), which brings about the excellent results in figure 6(a). This phenomenon can be viewed as the net effects due to the backflowing of the environments. Actually, the changes of parameter R 1 only affect the memory time ( 1 l ) of reservoir 1, and the coupling strength γ between the atom and reservoir 1 is unchanged. It should be noted that the compensated rate of change has some negative values, which implies the population of the pseudomodes transfer back to the atom. It is a consequence of the memory effects of environments. As the atom-cavity coupling strength gets much weaker, the interaction between the atom and pseudomodes plays a more important role, the compensated rate of change oscillates in a more intuitive way in this time, as shown in figure 10(b). For a reservoir with a Lorentzian spectrum, one can utilize the above pseudomode theory to interpret the physics. However, this pseudomode method can not be used for sub-Ohmic, Ohmic or super-Ohmic reservoirs, as there exists no such poles in these reservoir spectrums. In our study of sub-Ohmic, Ohmic or super-Ohmic reservoirs, the phase shift of the output pulse is time-varied with the atom at state cñ | .

Discussion
As we stated before, the deterministic quantum controlled-PHASE gate can only be constructed in a Lorentzian reservoir, but not a sub-Ohmic, Ohmic or super-Ohmic reservoir. One can identify the Lorentzian reservoir and Ohmic reservoir by making use of the methods of noise spectroscopy. Many experimental and theoretical studies have been focused on measuring the noise spectrums [45][46][47][48][49]. For example, by utilizing the dynamical decoupling, we can recognize an unknown noise spectrum, after applying appropriate pulse sequences and measuring the decay rates [45,46,48,49]. Given that there exists the connection between the noise spectrums and some observables, one can also identify the noise spectrums by measuring the expectation values of observables [47]. These methods stated above can be used for reference in our system to identify noise. In many experimental architectures, due to inevitable effect from the surrounding environment, the real system cannot be seen as a closed system, and the noise should be taken into consideration. The existence of the systems with Lorentz spectrums has been demonstrated in many situations. An atom placed inside a lossy cavity is a typical example of a system interacting with such engineered structured reservoirs [43,44,50]. The electromagnetic field inside a lossy cavity can be viewed as a reservoir with Lorentzian spectrum. By manipulating the parameters of a cavity, one can control the parameters of the Lorentzian reservoir spectrum. Our study can also be applied to many other systems, e.g., a superconducting artificial atom coupled to a superconducting transmission-line resonator (TLR) [14,51]. TLRs can be viewed as microwave cavities, which can filter the input white noise into the non-Markovian noise with Lorentzian spectrums [51].

Conclusion
In conclusion, we theoretically studied the construction of a quantum controlled-PHASE gate between a single photon and an atom coupled to a single-sided cavity. Here, we derive the dynamics of the atom-cavity system under general conditions, and the memory backflow from the non-Markovian reservoir is counted. Both the coupling between an atom and the reservoir, and the interaction between the cavity mode and the environment have been taken into account. We numerically calculated the output field of the controlled-PHASE gate under different parameters, in various structured environments. We found that, when spectral density of the reservoir is the Lorentz type, whatever the coupling strength between the atom and cavity is strong or weak, it is possible to implement an ideal deterministic controlled-PHASE gate by appropriately manipulating the system parameters, which can not happen in study with the Markovian approximation. This feature induced by the back-flowing phenomenon can benefit the rapid quantum communication and computation. Moreover, we also investigated the performance of a quantum controlled-PHASE gate in environments with sub-Ohmic, Ohmic, and super-Ohmic spectrums. We found that, the phase shift in the output pulse hinders the implementation of quantum controlled-PHASE gates in these reservoirs.