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Abstract
We study the effects of dissipation induced blockade on the dynamics of an open quantum system having two qubits in nonequilibrium independent baths. The qubits are driven by a classical field with a temporally modulated detuning. The introduction of blockade induced via two effective baths together with the effect of the driving field enable us to observe maximal entanglement oscillation of about unity that decays with a quasi-steady entanglement state oscillating about the 1/2 limit with adjustable decay rate. When the temperature difference between two baths is not large, maximal entanglement oscillation can still be observed in the model. In addition, the adjustment of the nonequilibrium thermal baths by modulating the dissipation and the application of time-dependent detuning give rise to rich entanglement dynamics. We further demonstrate numerically the practical implementation of the proposed scheme with a universal cavity QED setting.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quantum reservoir engineering has been used to control quantum systems [1–3]. Generally, decoherence restricts quantum coherence to the low temperature limit whenever there is interaction with hot environment [4–7]. Such limitation might be overcome by the engineering of quantum reservoir in special condition. For example, the use of nonequilibrium state for two coupled, parametrically driven, dissipative harmonic oscillators [4] showed that at high temperatures steady state entanglement can be achieved. Thermal entanglement in nonequilibrium [8–11] quantum systems has been extensively studied with a focus on steady state entanglement. In experiment, flexibility needs to be improved in the manipulation and control of the quantum system parameters [3,12]. Interestingly, a coupled qubit system interacting with two reservoirs is the simplest model for the investigation of entanglement in nonequilibrium steady states [13,14]. Hot reservoirs and entropy production, pushed the theoretical limit to $C$ = 1/2 ($C$ is the concurrence), beating the limits set by classic thermal operations on an equivalent system [15].
Rydberg blockade effects have been investigated widely since it is an important physical phenomenon and has essential applications in control of quantum state and performing quantum information processing [16–22]. In principle, the Rydberg blockade is a consequence of the powerful Rydberg–Rydberg interaction which induces level shift and therefore suppresses the simultaneous excitation of two nearby atoms into Rydberg states [23,24]. The Rydberg blockade effects have been observed in different experimental systems [25–27]. Moreover, in the presence of Rydberg-Rydberg interactions, some interesting dynamics phenomena can occur, especially the blockade enhancement and antiblockades. For example, a recent work has reported that by driving an array of Rydberg atoms with a temporally modulated atom-field detuning [28], both the Rydberg blockade and antiblockade can be engineered in different regimes based on the modulation parameters. On the other hand, dissipation processes including dephasing and decay of individual Rydberg atoms in most Rydberg systems typically compete with the Rydberg interactions, which enables rich driven-dissipative many-body dynamics, such as dissipation-induced dipole blockade and antiblockade [29], dephasing controlled excitation statistics [30], dephasing-enhanced blockade [31], and others [32]. In particular, in Ref. [31] the authors proposed to enhance the Rydberg blockade via strong two-body dephasing that is induced in a regime where dipole-dipoles interaction couple nearly degenerate Rydberg pair states. In comparison with the proposals that require direct interactions (van der Waals and dipole-dipole interactions) between atoms, we mainly focus on a basic bipartite system containing two independent qubits, based on which a similar dynamics phenomenon to the Rydberg blockade can be induced using incoherent resources.
In this paper, we study the effects of dissipation induced blockade on the dynamics of an open quantum system consisting of two qubits in nonequilibrium independent effective baths. The qubits are driven by a classical field with a periodic modulated detuning. Our model is inspired directly by the recent work with respect to entanglement generation in a driven-dissipative system [15], while significant differences can be identified. Specifically, for the common resonant case without qubit-field detuning, the combined effects of the dissipation induced blockade and the driving field acting on qubits allow us to observe remarkable entanglement oscillation with a maximal value close to unity, with quasi-steady state entanglement oscillating about the 1/2 limit. Interestingly, when introducing the time-dependent qubit-field detuning, rich entanglement dynamics can be observed based on the modulation parameters determining two different regimes. In addition, we discuss briefly the realization of our model with a cavity QED setting consisting of two independent and incoherently pumped qubits coupled to a leaky cavity mode, and show that the system could be implemented in practice.
The remainder of this paper is organized as follows. In Sec. 2, we first describe the model and give the system Hamiltonian. Then, we illustrate the basic mechanism for achieving improved steady-state entanglement as well as entanglement oscillations. In Sec. 3, we examine the dynamics of the system under both resonant case and dispersive case with time-dependent detuning. In Sec. 4, we discuss the practical realization of the system. Finally, we present a summary of our work in Sec. 5.
2. System and Hamiltonian
We consider a general bipartite system, as illustrated in Fig. 1, which contains two identical qubits. The energy level structure of the qubits is shown in Fig. 1(b), with ground state $\vert g \rangle _{k}$ and excited state $\vert e \rangle _{k}$ $(k = 1, 2)$. The transition between states $|e\rangle$ and $|g\rangle$ is driven by a classical field with frequency $\omega$ and Rabi frequency $\Omega$. To ensure coupling to the antisymmetric Bell state (see below), we assume a $\pi$ phase difference in the driving field between the two qubits. We note that though such a setting has been exploited in several theoretical schemes [33–37], in real experimental systems it requires good control over the relative phase of the driving field [38–40]. The Hamiltonian of the system reads ($\hbar = k_{B}= 1$, hereinafter)
In the Hamiltonian above, $\omega _e(t)=\omega _e+\delta (t)$, where $\omega _e$ is the energy of the excited state $|e\rangle$, and $\delta (t)=\delta \cos (\omega _{0}t)$ is a designed modulated term with amplitude $\delta >0$ and frequency $\omega _{0}$. This modulation could be implemented by subjection to additional radio frequency or microwave fields, creating sidebands in the atomic levels [28]. Assuming $\omega _g=0$ and working in the rotating frame with respect to $H_0=\omega \sum _{k} |e\rangle _k\langle e|$, the system Hamiltonian (1) can be transformed to
Fig. 1. (a) The schematic representing a pair of qubits coupled to thermal reservoir modes pumped by an external drive in nonequilibrium. (b) Energy level transitions for the system.
We assume that there are two independent effective baths coupling the collective degree of freedom of the two qubits, as depicted in Fig. 1(a). The two baths have inverse temperatures $\beta _S$ and $\beta _A$, respectively, which could be taken as negative values [15]. When these baths are considered, the dynamics of the open system could be described by the master equation as follows
withTo understand how to produce steady state entanglement by virtue of the dissipation induced blockade for the proposed model, we plot in Fig. 2 the internal level structure of the composite system exhibiting the two degenerate entangled states $\vert S \rangle$ and $\vert A \rangle$. The red-arrowed lines denote the dissipation channels with rates either $\Gamma ^{-}_{S}$ or $\Gamma ^{-}_{A}$ depending on the entangled states, whereas the green-arrowed lines represent the pumping processes with rates either $\Gamma ^{+}_{S}$ or $\Gamma ^{+}_{A}$. The two blue-arrowed lines correspond to the transitions induced by the external driving field $\Omega$. In the absence of the driving field, nonseparable steady state with the population of either $\vert S \rangle$ or $\vert A \rangle$ is achievable for unbalanced baths. For example, when $\Gamma ^{-}_{S}$ is much lager than $\left \lbrace \Gamma ^{-}_{A}, \Gamma ^{+}_{A}, \Gamma ^{+}_{S} \right \rbrace$, the dissipation channel with $\Gamma ^{-}_{S}$ dominates over others and therefore there hardly exists populations on $\vert ee \rangle$ and $\vert S \rangle$ during the evolution. Then by choosing a negative effective temperature of the bath (i.e., $\beta _A<0$), the system could be pumped into the state $\vert A \rangle$ with 1/2 stationary probability, while the rest is in the ground state $\vert gg\rangle$, in particular with zero probability in $\vert S \rangle$ [15]. When the driving field $\Omega$ comes into play, however, the system shows up different characteristics where maximal value of entanglement oscillation higher than 0.98 during the evolution is observed, with quasi-steady state entanglement reaching the limiting value of about 1/2. We show and discuss these features via detailed numerical simulations in the next section.
Fig. 2. The schematic of energy level transitions for the system showing the symmetric $\vert S \rangle$ and the antisymmetric $\vert A \rangle$ states.
3. Dynamical evolution of the system
3.1 Resonant case
In this subsection, we consider resonant case with $\Delta (t)=0$ and examine the effects of various parameter adjustments, including the classical field, dissipation and pumping rates, that are amenable to the improvement of entanglement oscillation and quasi-steady state entanglement.
3.1.1 Variation of the classical field
We introduce the concurrence as an entanglement measure in this work [41], which is defined for a two-qubit system as $C(\rho )=\max \left \{0, \lambda _{1}-\lambda _{2}-\lambda _{3}-\lambda _{4}\right \}$, where $\lambda _z^2$ ($z=1,2,3,4$) are the eigenvalues of the Hermitian matrix $\rho \tilde {\rho }$ in decreasing order, and $\tilde {\rho }=\left (\sigma _{y} \otimes \sigma _{y}\right ) \rho ^{*}\left (\sigma _{y} \otimes \sigma _{y}\right )$ [15]. For a maximal entangled pure state, the concurrence has the maximum value of 1, and for an unentangled pure state, its concurrence is equal to zero. Figure 3 displays the concurrence as a function of evolution time $t$ for various values of classical-field Rabi frequency $\Omega$. The brown curve shows quasi-steady state concurrence of about 0.3297 without the influence of the external field ($\Omega =0$), which is similar to the observation made in [15]. When the external field with $\Omega =\Omega _{0}$ is introduced, as displayed by the red curve, an obvious entanglement oscillation occurs with a maximal value of about 0.9853, which decays into a quasi-steady state with $C$ of about 0.4967. When further increasing the driving field to $\Omega =2\Omega _{0}$ and $\Omega =3\Omega _{0}$, the values of maximal entanglement oscillation change a little, while the time to steady state is reduced greatly with increasing $\Omega$. Specifically, Fig. 3(b) exemplifies the trend observed by increasing the external field on the maximal entanglement oscillation. Clearly, we see that when $\Omega =\Omega _{0}$ we obtain the highest value of about 0.9853 and subsequent increase in $\Omega$ yields a lower value 0.9794 with faster oscillations. This is reasonable because more population leaks into the excited state $|ee\rangle$ for larger $\Omega$, while the entanglement oscillation between states $|A\rangle$ and $|gg\rangle$ can be accelerated in case of a stronger drive. In order to scrutinize the steady state, we increase the resolution and observe that the classical field has negligible effect on the dynamics as seen in Fig. 3(c).
Fig. 3. (a-c) Concurrence against evolution time for various values of $\Omega$. (b) and (c) display the magnified views of the main figure for the maximal entanglement oscillation and steady state entanglement, respectively. The parameters are $\Delta _{0}=\delta =\omega _{0}=0$, $\Gamma ^{+}= \Gamma ^{-}_{A}=0.001\Omega _{0}$, and $\Gamma ^{-}_{S}=100\Omega _{0}$.
Based on the steady-state solution of the system, the corresponding steady-state two-body correlation can be calculated via $C_{ss}=\langle \sigma _1^{ee}\sigma _2^{ee}\rangle /\langle \sigma _1^{ee}\rangle \langle \sigma _2^{ee}\rangle$ [31], where $\sigma _k^{ee}=|e\rangle _k\langle e|$. The value of $C_{ss}$ can be used to evaluate the effect of blockade: $C_{ss}<1$ ($C_{ss}>1$) corresponds to the anti-bunching (bunching) effect, where double qubit excitation is suppressed (enhanced). Taking the case with $\Omega =\Omega _0$ as an example, the steady-state correlation is calculated as $5.26\times 10^{-4}$, which indicates a strong blockade effect induced in the system.
3.1.2 Variation of dissipation $\Gamma ^{-}_{S} > \Gamma ^{+}_{A}$
The values of steady-state and maximal entanglement are closely related to the parameter relationship between dissipation and pumping rates. In principle, the larger the difference between $\Gamma ^{-}_{S}$ and $\Gamma ^{+}_{A}$, the higher the concurrence. To verify this, we plot the time evolution of concurrence for various values of $\Gamma ^{-}_{S}$, as shown in Fig. 4. Increase in $\Gamma ^{-}_{S}$ results in asymptotic increase in the maximal entanglement oscillation [see Fig. 4(b)]. When the dissipation rate is sufficiently large, for example when $\Gamma ^{-}_{S}=100\Omega _{0}$ and $200\Omega _{0}$, the increase in maximal entanglement oscillations becomes negligible, which suggests that there is an upper limit for a given set of system parameters. Moreover, it is shown that entanglement oscillations persist for longer times at higher dissipation rates before attaining the quasi-steady state. Figure 4(c) shows that the quasi-steady state asymptotically increases with increase in the dissipation rate to a maximal value of about 0.5. It is worthwhile to note that the maximal entanglement oscillation is large (about 0.9) even when the dissipation rate is relatively small, e.g., 10$\Omega _0$.
Fig. 4. (a-c) Concurrence against evolution time for various values of $\Gamma ^{-}_{S}$. (b) and (c) display the magnified views of the main figure for the maximal entanglement oscillation and steady state entanglement, respectively. The parameter is $\Omega =\Omega _{0}$ and other parameters are the same as in Fig. 3.
3.1.3 Variation of dissipation $\Gamma ^{-}_{S} < \Gamma ^{+}_{A}$
In the two subsections above, we assume $\Gamma ^{-}_{S}>\Gamma ^{+}_{A}$ which gives maximally entangled state $\vert A\rangle$ with approximately 0.5 stationary probability. In what follows, we discuss the case with an inverse qualification, i.e., $\Gamma ^{-}_{S}<\Gamma ^{+}_{A}$. In Fig. 5, we plot the concurrence versus evolution time for various values of pumping rates $\Gamma ^{+}_{A}$. When the pumping rate is relatively small, $\Gamma ^{+}_{A}=10\Omega _{0}$, the quasi-steady state entanglement has a value of about 0.32. Increase in the pumping rate results in a slight increase in the quasi-steady state entanglement as exhibited by the curves with $\Gamma ^{+}_{S}=50\Omega _{0}, 100\Omega _{0}$ and 200$\Omega _{0}$. The steady state entanglement asymptotically approaches a maximal value of about 1/3 and we observe no entanglement oscillation even when the driving field is present. This is a consequence of the counteraction between the driving $\Omega$ and a large pumping rate $\Gamma ^{+}_{A}$. In this case the population of the Bell states is unbalanced with 1/3 in the state $\vert S\rangle$ while the rest is in the excited state $\vert ee\rangle$. Note that an improved steady state entanglement is also achievable for this particular case with $\Gamma ^{-}_{S}<\Gamma ^{+}_{A}$, which can be done by ensuring the driving part in the Hamiltonian (2) being the form of $(\Omega /2) \sum _{k=1}^{2} \sigma ^{x}_{k}$, that is, removing the $\pi$ phase difference of the driving field between the two qubits. In doing so, we observe an improved steady state entanglement to about 0.5, accompanied by the entanglement oscillations with a maximal value over 0.98, as depicted by the gray curve in Fig. 5. This result manifests again that the introduction of classical driving field is able to increase the critical value of entanglement.
Fig. 5. Concurrence against evolution time for various values of $\Gamma ^{+}_{A}$. The lower four curves without entanglement oscillations are obtained for the case where a $\pi$ phase difference in the drive between the two qubits is assumed, whereas the grey oscillating curve corresponds to the case without relative phase in the drive between the two qubits. The inset displays a magnified view of the main figure for quasi-steady state entanglement. The parameters are $\Omega =\Omega _{0}$, $\Gamma ^{-}_{A}=\Gamma ^{+}_{S}=\Gamma ^{-}_{S}=0.001\Omega _{0}$, and other parameters are the same as in Fig. 3.
3.2 Time-dependent detuning
We then examine dynamical evolution of the system with variation in the time-dependent detuning $\Delta (t)=\Delta _{0}+\delta \cos (\omega _{0}t)$ under various parameter regimes. The Schrödinger equation of the system is written as $\mathrm {i}\vert \dot {\psi }(t) \rangle = H(t)\vert \psi (t) \rangle$. To gain more insight on the interactions with time-dependent detuning, it is convenient to define a rotating frame
where the frame transformation $U(t)$ is given by with $H'(t)=\Delta (t) \sum ^{2}_{k=1}\sigma ^{+}_{k}\sigma ^{-}_{k}.$ In this frame, the effective Hamiltonian of the system is obtained via [28,42]After using the Jacobi-Anger expansion, $e^{\mathrm {i}z\sin \theta }=\sum ^{\infty }_{m=-\infty }J_{m}(z)e^{\mathrm {i}{m}\theta }$, we arrive at
3.2.1 Case I: $\Delta _{0} \gg \omega _{0}$
From Fig. 6(a) maximal entanglement oscillation of about 0.688 is obtained at about $\delta =17\Omega _{0}$ which gradually decreases as $\delta$ increases. To be more clear, we pick up three values of $\delta$ and plot in Fig. 6(b) the corresponding concurrence versus time curve. For $\delta =0$ no entanglement oscillation is observed and the steady state concurrence is about 0.2266 at $t=100/\Omega _0$. This parameter constraint exemplifies a special case where the strength of effective coupling, i.e., $(\Omega /2)J_{m}\left (\delta /\omega _{0}\right )$, is far less than the effective detuning $m\omega _0 + \Delta _0$ due to an extremely small value of Bessel function for $\delta =0$ [i.e., $J_{m}(0)$]. That is, the interactions induced by driving are significantly off resonant. When $\delta$ increases to $17\Omega _{0}$, we observe a slow "saw-tooth" type of oscillation with superimposed fast oscillations. For larger $\delta$ of $100\Omega _{0}$, the maximal entanglement oscillation reduces to about 0.4728, which is a direct consequence of a relatively smaller $J_{m}\left (\delta /\omega _{0}\right )$. Figure 6(c) exhibits the concurrence variation with $\delta$ at different time slots. For $t=40/\Omega _0$ the curve approximatively depicts the envelope of the $m$-order Bessel function $J_{m}\left (\delta /\omega _{0}\right )$ with changing $\delta$. Figures 6(d)–6(f) display the effect of increase in $\Gamma ^{-}_{S}$ on the entanglement dynamics under similar parameters as those in Figs. 6(a)–6(c). From Figs. 6(d) and 6(e) we see that the region of entanglement oscillation broadens and a maximal value of about 0.6675 is obtained at $\delta =17\Omega _{0}$.
Fig. 6. Concurrence as functions of parameter $\delta$ and evolution time. The parameters are (a-c) $\Gamma ^{-}_{S}=20 \Omega _{0}$, (d-f) $\Gamma ^{-}_{S}=50 \Omega _{0}$. The common parameters are $\Omega =\Omega _{0}, \Gamma ^{-}_{A}= \Gamma ^{+}_{S}=0.001 \Omega _{0}, \Gamma ^{+}_{A}=0.01 \Omega _{0}$, $\Delta _{0}=10 \Omega _{0}, \omega _0=0.1 \Omega _{0}$.
3.2.2 Case II: $\Delta _{0} \ll \omega _{0}$
For $\Delta _{0} \ll \omega _{0}$, the effective detuning $m\omega _{0}+\Delta _{0}$ makes near-resonant interaction when and only when the order Bessel function is $m=0$, while for arbitrary $m\ne 0$ the large detuning results in off resonant interaction. Figure 7(a) shows that the concurrence varies periodically with increasing $\delta$ and there exist three minimal values corresponding to different $\delta$. One may verify that these specific values of $\delta$ (more precisely, $\delta /\omega _0$) accord exactly with the first three zeros of $J_{0}\left (\delta /\omega _{0}\right )$. From Fig. 7(b) we see that a maximal entanglement oscillation of about 0.9065 is obtained for $\delta =0$, whereas for $\delta =24\Omega _0$ the entanglement oscillation is diminished. Further on, for $\delta =71\Omega _0$ the entanglement oscillation resurface. Figure 7(c) plots the concurrence versus $\delta$ at different time slots. For $t=2.2/\Omega _0$ the curve has a similar envelope with that of zero-order Bessel function, which confirms the arguments above. Figures 7(d)–7(f) display the effect of increase in $\Gamma ^{-}_{S}$ under similar parameters as those in Figs. 7(a)–7(c). From Fig. 7(e) we observe a slight increase in the maximal entanglement oscillation.
Fig. 7. Concurrence, C as functions of $\delta$ and evolution time, $t$. The parameters are (a-c) $\Gamma ^{-}_{S}=20 \Omega _{0}$, (d-f) $\Gamma ^{-}_{S}=50 \Omega _{0}$. The common parameters are $\Omega =\Omega _{0}$, $\Gamma ^{+}_{A}=0.01 \Omega _{0}$, $\Gamma ^{+}_{S}=\Gamma ^{-}_{A}=0.001\Omega _{0}$, $\Delta _{0}=0.1 \Omega _{0}$, $\omega _{0}=10 \Omega _{0}$.
4. Cavity QED-based implementation
As for implementing the proposed model in practice, we consider a pair of independent and incoherently pumped qubits that are resonantly coupled to a single mode cavity. Such a system could be described by the following two-qubit Tavis-Cummings Hamiltonian
The master equation describing the open system is then given by $\dot {\rho } = -\mathrm {i}[H_{\mathrm {TC}}(t) , \rho ] + L(\rho )$, with the full Liouvillian
Fig. 8. Concurrence versus evolution time for various cavity decay $\kappa$ and pumping rate $\xi$. The parameters are (a,c) $\xi =5 \times 10^{-4}g$ and (b,d) $\xi =1 \times 10^{-4}g$. (a) and (b) represent the resonant case as in Fig. 3 with $\Delta _{0}=\delta =\omega _{0}=0$, while (c) and (d) represent the nonresonant case when $\Delta _{0}=0.2\Omega _0,\delta =0.3\Omega _0$, and $\omega _{0}=0.002\Omega _0$, as in Fig. 6(b). The common parameters are $g=\Omega _0$, $\Omega =0.02\Omega _{0}$, $\gamma =0.001g$.
As an example of the practical cavity QED systems, we consider a general setup where two alkali-metal atoms couple to the evanescent fields of an ultrahigh-$Q$ whispering gallery mode microcavity. For such cavities small mode volumes offer the prospect of atom-field coupling rates achieving hundreds of MHz [45,46]. Also, the spontaneous emission rate for the $D_1$ line of $^{87}$Rb is $\gamma =2\pi \times 5.7$ MHz [45,47], corresponding to a lifetime of about 0.028 $\mu$s. Then, we choose $g=2\pi \times 600$ MHz, $\Omega =0.08g$, $\kappa =3g$, $\xi =1\times 10^{-4}g$ and $\Delta _{0}=\delta =\omega _{0}=0$ for the resonant case. Based on these parameters, the system could attain the steady state with a entanglement value of 0.4798 within 0.016 $\mu$s, which is shorter than the lifetime of atoms. Accordingly, the maximal entanglement oscillation with $C=0.8469$ could occur at the time of about 0.62 ns.
5. Conclusion
In summary, we have studied the effects of disspation induced blockade on the entanglement dynamics of a bipartite quantum system coupled to two nonequilibrium thermal baths. A classical driving field of qubits is introduced to induce entanglement oscillations with highest value approaching unity and improved quasi-steady entanglement reaching the 1/2 limit. Further, periodic modulation of the qubit-field detuning leads to splendid dynamical behaviors of the system. From the experimental point of view, the potential realization of the proposed model based on a universal cavity QED system is verified. The result shows that the simulated physical (cavity-QED) system exhibits a similar behavior as in the primitively effective model. Our work may find its applications in solid-state cavity QED, quantum communication and the design of devices working in noisy nonequilibrium environments.
Funding
National Natural Science Foundation of China (11675046); Program for Innovation Research of Science in Harbin Institute of Technology (A201412); Postdoctoral Scientific Research Developmental Fund of Heilongjiang Province (LBH-Q15060).
Disclosures
The authors declare no conflicts of interest.
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