Following the recent paper (Teittinen et al 2019 New J. Phys.21 123041), one can see that in general there is no simple relation between non-Markovianity and quantum speed limit. Here, we investigate the connection between quantum speed limit time and non-Markovianity of an atom in structured environments (reservoirs) whose dynamics is governed by an exact pseudomode master equation (Garraway 1997 Phys. Rev. A 55 2290). In particular, we find an inverse relation between them, which means that the non-Markovian feature of the quantum process leads to speedup of evolution. Thus, there is a link between quantum speedup and memory effects for specific cases of dynamical evolution. Our results might shed light on the relationship between the speedup of quantum evolution and the backflow of information from the environment to the system.
Maryam Hadipour, Soroush Haseli, Saeed Haddadi. Quantum speedup and non-Markovianity of an atom in structured reservoirs: pseudomodes as a good description of environmental memory[J]. Communications in Theoretical Physics, 2024, 76(8): 085105. DOI: 10.1088/1572-9494/ad54ae
1. Introduction
How to accelerate the evolution of quantum systems is one of the most important questions in quantum mechanics theory. Based on this theory, quantum speed limit (QSL) quantifies the minimum evolution time for a quantum system to pass through a predetermined distance [1–8]. It refers to the shortest possible time required for a quantum system to evolve from an arbitrary initial state to a target final state [8–15]. QSL time and its applications have been extensively studied experimentally and theoretically [16–20]. Moreover, QSL time plays a particularly important role in the development of quantum information protection and quantum optimal theory as well [21–24]. In recent years, there has been a particular focus on utilizing the advantages of QSL time in quantum batteries [25–28]. A deeper knowledge of QSL has led to the expansion of the speed limit to classical systems [29–31]. There is a strong correlation between the vanishing QSL times and the emerging classical behavior due to the reduced uncertainty in quantum observables [32]. In [33], the application of QSL time to quantum resource theories has been investigated. In the beginning, only isolated systems were studied in accordance with the concept of QSL time [34–36].
Initially, Mandelstam and Tamm (MT) introduced the minimal time for evolution from an initial to an orthogonal state for a closed quantum system as $\tau \geqslant \tfrac{\pi {\hslash }}{2{\rm{\Delta }}E}$, where ${\left({\rm{\Delta }}E\right)}^{2}=\left\langle \psi \left|{H}^{2}\right|\psi \right\rangle -{\left(\langle \psi | H| \psi \rangle \right)}^{2}$ is the energy variance of the system [34]. This bound is known as MT bound. Another bound related to the mean energy E = 〈ψ∣H∣ψ〉 was then derived by Margolus and Levitin (ML) as $\tau \geqslant \tfrac{\pi {\hslash }}{2E}$, where it is assumed that the ground state energy E0 is equal to zero [35]. This bound is known as ML bound. It is possible to implement a unified QSL bound for the unitary evolution of closed quantum systems by combining both ML and MT bounds as [36]
The ML and MT bounds have been expanded and improved using a wide range of metrics, including trace distance, Bures angle, relative purity, and many other metrics [37–40]. The distance among generalized Bloch vectors for both unitary and non-unitary processes can also be used to obtain tight QSL bounds [41–46].
It should be noted that in the real world, quantum systems are open and their evolution is non-unitary [47]. A variety of approaches can be used to analyze the behavior of open systems, including the master equation formalism and the quantum trajectory method. The master equation is a central tool for studying the dynamics of open systems. It provides a mathematical description of how the system's density matrix changes over time. The nature of an interaction between a quantum system and its environment can influence its dynamics and evolution. The system-environment interaction is most often analyzed from both the non-Markovian and Markovian perspectives [47]. In a Markovian regime, the evolution of the system is purely determined by its current state. In this case, the system-environment interactions occur over a much shorter timescale than the significant system evolution. However, in a non-Markovian regime, system-environment interactions are longer in time than the intrinsic evolution of the system [48–55]. An important feature of non-Markovian regimes is that the interaction behavior of the system with its environment in the past determines the evolution of the system in the future. Due to the unavoidable interaction between a quantum system and its environment, the decoherence effect caused by this coupling can have a significant effect on quantum statistics.
The ratio between the QSL time and the actual time quantifies the possibility of speeding up the quantum evolution. QSL bounds are saturated with no potential for speedup when the ratio equals one. However, the QSL bound is unsaturated if the ratio is less than one, and there exists potential for speeding up the evolution of the dynamical system. In [41], it has been shown that the non-Markovian effect can accelerate quantum evolution [41]. Both theoretically and experimentally, non-Markovianity regulation has been proposed as a mechanism for regulating dynamical speedup. Moreover, it has been shown that non-Markovian effects induce unsaturated QSL bounds in open dynamics [56–64]. It is important to note that dynamical speedup in this way depends very much on the feedback from the environment. Therefore, it will be essential to investigate other sources that can generate unsaturated QSL bounds besides non-Markovianity as a source of unsaturated QSL bounds [65–69].
Motivated by this, we investigate the possibility of quantum speedup using a convenient cavity-based engineered environment. We show how the Markovian environments consisting of two cavities can accelerate the dynamics of an artificial atom interacting with a pseudomode. In the considered model, the coupling between the pseudomodes, atom-pseudomode coupling, and detuning between atom and pseudomode can have significant impacts on QSL time and non-Markovianity.
The work is organized as follows. First, we present a physical model and its solution in section 2. Then, we discuss the QSL time and non-Markovianity for the considered model in section 3. Next, we turn to the results and discussion in section 4. Finally, we provide a brief conclusion of the most important results in section 5.
2. Model and pseudomode method
The considered model consists of a two-level atom interacting with a Lorentzian structured environment. Figure 1 shows the schematic representation of the scheme. For this model, the Hamiltonian can be expressed as follows [70, 71]
where ${\hat{\sigma }}_{z}=| e\rangle \langle e| -| g\rangle \langle g| $ is the ordinary Pauli operator in z-direction of atom, ω0 is the transition frequency of the atom, σ+(σ−) is raising (lowering) operator of the atom, and ω1 and ω2 are the frequencies of the modes inside the Markovian environments shown by cavity 1 and cavity 2, respectively. Let us take ω1 = ω2 = ω and set ω0 = ω + δ, where δ is atom-pseudomode detuning. Besides ${\hat{a}}_{1}$(${\hat{a}}_{1}^{\dagger }$) and ${\hat{a}}_{2}$(${\hat{a}}_{2}^{\dagger }$) are the annihilation (creation) operators.
Figure 1. Schematic representation of the model. Two-level atom interacts with pseudomode 1 (coupling strength k) which leaks into cavity 1 via dissipation rate Γ1, and pseudomode 1 is coupled to pseudomode 2 with coupling strength J which the latter leaks into cavity 2 via dissipation rate Γ2.
In what follows, we study the dynamics using the pseudomode theory [72–74]. This approach relies on the relationship between the atom dynamics and the shape of the spectral distribution of the environment. The significance of precise approaches is of current interest due to the experimental realization of quantum systems. As shown in figure 1, the environment can be depicted by two nondegenerate pseudomodes that leak into the Markovian environments (cavities 1 and 2) with dissipation rates Γ1 and Γ2. Here, the two-level atom only interacts with the first pseudomode Pm1 (the strength of the coupling k) which is in turn coupled to the second pseudomode Pm2 (the strength of the coupling J).
The dynamics of the whole system for a Lorentzian spectral distribution can be obtained by the following exact pseudomode master equation
The above master equation represents the coherent interaction between the two-level atom and two pseudomodes in the presence of the pseudomodes decay caused by the interaction with Markovian environments.
Here, it is assumed that the atom is initially in an excited state ∣ea〉 while both pseudomodes Pm1 and Pm2 are in the ground state $| {g}_{{\mathrm{Pm}}_{1}}{g}_{{\mathrm{Pm}}_{2}}\rangle $, that is, the initial state of the overall system is assumed to be $\rho (0)=| {e}_{a}{g}_{{\mathrm{Pm}}_{1}}{g}_{{\mathrm{Pm}}_{2}}\rangle \langle {e}_{a}{g}_{{\mathrm{Pm}}_{1}}{g}_{{\mathrm{Pm}}_{2}}| $. Considering that there is only one excitation in the whole system including a two-level atom and two cavities, the state of the system at time t can be written as
where $\widetilde{h}(t)=\sqrt{1-\eta (t)}h(t)$ is the probability amplitude that the two-level atom is in an excited state while ${\tilde{c}}_{n}(t)\equiv \sqrt{1-\eta (t)}{c}_{n}(t)$ (with n = 1, 2) is the probability amplitude that the pseudomodes being in their excited states. Using the above unnormalized state, the density matrix of the total system can be rewritten as
The time-dependent amplitudes of $\tilde{h}(t)$, ${\tilde{c}}_{1}(t)$, and ${\tilde{c}}_{2}(t)$ can be obtained by solving the following coupled differential equations
In the above set of coupled differential equations, the coefficients $\widetilde{h}(t)$, ${\tilde{c}}_{1}(t)$, and ${\tilde{c}}_{2}(t)$ can be obtained using the standard Laplace transformations. By knowing these, the time-evolved density matrix of the two-level atom can be obtained by taking the partial trace over two pseudomodes, which we will provide in the next section.
3. QSL time and non-Markovianity
3.1. QSL time
We first deal with QSL time using the method introduced in [42] for open quantum systems. A comprehensive bound of the QSL time for any arbitrary initial state can be defined as
where ${\rm{\Theta }}({\rho }_{0},{\rho }_{t})=\arccos \left(\sqrt{\tfrac{\mathrm{tr}[{\rho }_{0}{\rho }_{t}]}{\mathrm{tr}\left[{\rho }_{0}^{2}\right]}}\right)$ is a function of relative purity between initial state ρ0 and the state at arbitrary time ρt that determined by the time-dependent non-unitary master equation ${\dot{\rho }}_{t}={{ \mathcal L }}_{t}({\rho }_{t})$ and
where $\parallel {{ \mathcal L }}_{t}({\rho }_{t}){\parallel }_{\mathrm{op}}={\lambda }_{1}$, $\parallel {{ \mathcal L }}_{t}({\rho }_{t}){\parallel }_{\mathrm{tr}}={\sum }_{i}{\lambda }_{i}$ and $\parallel {{ \mathcal L }}_{t}({\rho }_{t}){\parallel }_{\mathrm{hs}}\,=\sqrt{{\sum }_{i}{\lambda }_{i}^{2}}$ are operator norm, trace norm and Hilbert-Schmidth norm, respectively. Also, λi's are singular values of ${{ \mathcal L }}_{t}({\rho }_{t})$ and λ1 is the largest singular value of ${{ \mathcal L }}_{t}({\rho }_{t})$. When the denominator of the fraction is ${{\rm{\Lambda }}}_{\tau }^{\mathrm{op}}$ and ${{\rm{\Lambda }}}_{\tau }^{\mathrm{tr}}$, we have a generalized ML type of QSL bound for open quantum systems, whereas when it is ${{\rm{\Lambda }}}_{\tau }^{\mathrm{hs}}$, we have MT type bound on the QSL time. According to [42], the ML type bound based on the operator norm provides the tightest QSL bound.
Using the recently achieved bound for QSL time and a given actual driving time τ, one can estimate the intrinsic speed of a dynamical evolution. In the case of τQSL = τ, the further speedup is not possible and the speedup is not observed. In contrast, when τQSL < τ, this indicates the possibility of quantum dynamics speeding up. Based on the assumption that the two-level atom is initially in the excited state as mentioned in the previous section, the time-evolved density matrix of the atom takes the following form
During evolution, the backflow of information from the environment to the system is responsible for the non-Markovian feature of the process. For quantification, the degree of non-Markovianity can be defined as [50]
where $D\left[{\rho }_{1}(t),{\rho }_{2}(t)\right]=\tfrac{1}{2}\mathrm{tr}\left|{\rho }_{1}(t)-{\rho }_{2}(t)\right|$ is the trace distance of a pair of states with the trace norm of an operator as $| \hat{A}| =\sqrt{\hat{A}{\hat{A}}^{\dagger }}$.
Note that when $\sigma \left[t,{\rho }_{1}(0),{\rho }_{2}(0)\right]\gt 0$, we have the backflow of information from the environment to the system and so, the evolution is non-Markovian ${ \mathcal N }\ne 0$. However, if $\sigma \left[t,{\rho }_{1}(0),{\rho }_{2}(0)\right]\lt 0$ then ${ \mathcal N }=0$, meaning that the evolution is Markovian, information irreversibly flows from system to environment, and backflow does not occur. As a result, non-Markovianity indicates the total backflow of information from the environment to the system.
In [50], it has been proved that the optimal pair of initial states that maximize the degree of non-Markovianity ${ \mathcal N }$ are ρ1(0) = ∣e〉〈e∣ and ρ2(0) = ∣g〉〈g∣. Using the findings of the previous section and taking into account the mentioned optimal initial states, one can find ${ \mathcal D }\left[{\rho }_{1}(t),{\rho }_{2}(t)\right]=| \widetilde{h}(t){| }^{2}$. Therefore, the degree of non-Markovianity is obtained as
Now everything is ready to study the QSL time and non-Markovianity based on the analytical expressions (12) and (15) obtained in the previous section.
In figure 2, the non-Markovianity (15) has been plotted as functions of atom-pseudomode coupling k/Γ1 and pseudomode–pseudomode coupling J/Γ1. From this figure, it can be seen that in the absence of pseudomode coupling strength (J = 0), the evolution is Markovian in a weak coupling regime k/Γ1 < 0.25 and non-Markovian in a strong coupling regime k/Γ1 > 0.25. In general, the non-Markovian feature of evolution is revealed for strong atom-pseudomode coupling for any pseudomode–pseudomode coupling strength. Therefore, figure 2 clearly shows the regions in which the Markovian and non-Markovian dynamics take place.
Figure 2. The degree of non-Markovianity as functions of atom-pseudomode coupling k/Γ1 and pseudomode–pseudomode coupling J/Γ1 with Γ2/Γ1 = 0.5 and δ/Γ1 = 0.
Interestingly, by employing equations (12) and (15), it is possible to connect QSL time and degree of non-Markovianity as
From equation (16), if the evolution is Markovian (${ \mathcal N }=0$), then the actual evolution time is equal to the QSL time, so we have a saturated bound for the QSL time. On the other hand, in a situation where the evolution is non-Markovian (${ \mathcal N }\ne 0$), the QSL time is shorter than the actual evolution time.
In figure 3, the QSL time and non-Markovianity have been plotted as a function of actual evolution time τ in a weak coupling regime with k/Γ1 = 0.1 for different values of pseudomode–pseudomode coupling J/Γ1. From figure 3(a), one can see that the QSL time decreases with increasing J/Γ1. Besides, figure 3(b) shows that in this weak coupling regime, the non-Markovianity feature of the evolution is revealed by increasing J/Γ1. Thus, we conclude that the QSL time decreases with the emergence of the non-Markovian feature of quantum evolution. Notably, one can also see that in Markovian evolution (red solid line), the QSL time is equal to the actual evolution time (τQSL = τ) because ${ \mathcal N }=0$, whose root is in equation (16). In other words, it can be said that the reduction of QSL time originates from the non-Markovian feature of the quantum evolution.
Figure 3. (a) QSL time and (b) non-Markovianity as a function of Γ1τ in a weak coupling regime (k/Γ1 = 0.1) for different values of J/Γ1 with Γ2/Γ1 = 0.5 and δ/Γ1 = 0.
Figure 4 shows the QSL time and non-Markovianity as a function of actual evolution time τ in a strong coupling regime with k/Γ1 = 0.75 for different values of J/Γ1. It seems that the results are similar to what we observed in the weak coupling regime, however, the non-Markovian feature of evolution can be observed in this case even when J/Γ1 is zero, which is in good agreement with the results of figure 2.
Figure 4. (a) QSL time and (b) non-Markovianity as a function of Γ1τ in a strong coupling regime (k/Γ1 = 0.75) for different values of J/Γ1 with Γ2/Γ1 = 0.5 and δ/Γ1 = 0.
In figure 5(a), the QSL time has been plotted as a function of k/Γ1 for both resonant (δ/Γ1 = 0) and off-resonant (δ/Γ1 = 5) cases. The QSL time decreases with increasing k/Γ1 for both resonant and off-resonant cases. Notice that the QSL time in the off-resonant case is shorter than in the resonant case. In figure 5(b), we plot the non-Markovianity versus k/Γ1. From this plot, we observe that the degree of non-Markovianity increases by the strengthening of the coupling between the atom and the first pseudomode for both resonant and off-resonant cases. By comparing figures 5(a) and (b), it can be concluded that there exists an inverse relation between QSL time and non-Markovianity.
Figure 5. (a) QSL time and (b) non-Markovianity as a function of k/Γ1 for zero and nonzero values of δ/Γ1 with Γ2/Γ1 = 0.5, J/Γ1 = 1, and τ = 5.
The QSL time and non-Markovianity have been plotted as a function of J/Γ1 for two different values of k/Γ1 in figure 6. We see that in a strong coupling regime (k/Γ1 = 0.75), the QSL time is shorter than that of the weak coupling regime (k/Γ1 = 0.1). On the other hand, the non-Markovianity gets its maximum value in a strong coupling regime. Again, we find that there is an inverse relation between non-Markovianity and QSL time. These results are in good agreement with the previous figures.
Figure 6. (a) QSL time and (b) non-Markovianity as a function of J/Γ1 for two different values of k/Γ1 with Γ2/Γ1 = 0.5, δ/Γ1 = 0, and τ = 5.
Finally, the QSL time and non-Markovianity have been illustrated as a function of the detuning parameter δ/Γ1 for two different values of k/Γ1 in figure 7. It is worth noting that the results are similar to the findings shown in the previous plots. Remarkably, the existence of an inverse relation between non-Markovianity and QSL time is also confirmed in this case.
Figure 7. (a) QSL time and (b) non-Markovianity as a function of δ/Γ1 for two different values of k/Γ1 with Γ2/Γ1 = 0.5, J/Γ1 = 1, and τ = 5.
Note that the ratio between the QSL time, τQSL, and the actual time, τ, quantifies the possibility of speeding up the quantum evolution. QSL bounds are saturated with no potential for speedup when τQSL/τ = 1. However, the QSL bound is unsaturated if τQSL/τ < 1, and there is potential for speeding up the evolution of the dynamical system. Our results show that τQSL/τ ≤ 1, indicating that both of these scenarios occurred in our study.
5. Conclusion and outlook
Using a cavity-based engineered environment, we explored the possibility of manipulating the QSL time. In particular, we have observed that by constructing the Markovian environments consisting of two cavities, one can switch between quantum speed-down and speedup regimes for the dynamics of an artificial atom interacting with a pseudomode. Based on memory effects, the dynamics of the open systems are divided into two categories: memory-less (Markovian) and with-memory (non-Markovian) dynamics. Notably, we observed that there exists an inverse relation between the QSL time and non-Markovianity. In other words, the non-Markovianity feature of quantum evolution can lead to quantum speedup in the dynamics. Therefore, this feature leads to an intrinsic interest in the context of controlling the memory effects that are associated with open systems. The parameters included atom-pseudomode coupling, pseudomode–pseudomode coupling, and detuning between atom and pseudomode were used to manipulate the non-Markovianity. We found that in a strong coupling regime, for both atom-pseudomode and pseudomode–pseudomode couplings, the evolution is non-Markovian and the quantum speedup is revealed. As a result of this study, new insights are gained into the control of QSL time for open systems, which opens the door to further experimental development towards the development of quantum thermodynamic devices and quantum computers.
Acknowledgments
SH was supported by Semnan University under Contract No. 21270.
Disclosures
The authors declare that they have no known competing financial interests.
EscherB M, de Matos FilhoR L, DavidovichL2011 General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology Nat. Phys.7 406
CampaioliF, PollockF A, BinderF C, CeleriL, GooldJ, VinjanampathyS, ModiK2017 Enhancing the charging power of quantum batteries Phys. Rev. Lett.118 150601
EktesabiA, BehzadiN, FaiziE2017 Improved bound for quantum-speed-limit time in open quantum systems by introducing an alternative fidelity Phys. Rev. A95 022115
CimmarustiA D, YanZ, PattersonB D, CorcosL P, OrozcoL A, DeffnerS2015 Environment-assisted speed-up of the field evolution in cavity quantum electrodynamics Phys. Rev. Lett.114 233602
ShahriY, HadipourM, HaddadiS, DolatkhahH, HaseliS2023 Quantum speed limit of Jaynes–Cummings model with detuning for arbitrary initial states Phys. Lett. A470 128783
MazzolaL, ManiscalcoS, PiiloJ, SuominenK A, GarrawayB M2009 Pseudomodes as an effective description of memory: non-Markovian dynamics of two-state systems in structured reservoirs Phys. Rev. A80 012104
PleasanceG, GarrawayB M, PetruccioneF2020 Generalized theory of pseudomodes for exact descriptions of non-Markovian quantum processes Phys. Rev. Res.2 043058