Macroscopic Quantum Tunneling in a Non-Equilibrium Environment

Macroscopic Quantum Tunneling in a Non-equilibrium Environment

Abstract

In this study, we examine the generic dynamics of a macroscopic quantum system in interaction with a non-equilibrium environment. The system is conveniently described as a particle confined in a symmetric double-well potential and the environment is minimally modeled as two different harmonic reservoirs. We employ the time-dependent second-order perturbation theory to describe the dynamics without any explicit Born and Markov assumptions. We demonstrate that in the case of identical reservoirs the short-time dynamics is affected by the interference between two reservoirs through the system. In the non-equilibrium environment, the quantum coherence of the system essentially has an oscillatory dependence on the difference between two reservoirs. Nonetheless, for a wide range of differences, the non-equilibrium environment enhances the quantum coherence of the system. This effect is weakened by the macroscopic force the system exerts on reservoirs. Keywords: Macroscopic Quantum Systems, Non-equilibrium Environment, Decoherence Theory

I 1. Introduction

Quantum coherence is the distinguishing feature of isolated microscopic systems. In macroscopic systems, however, it is destroyed due to the inevitable interaction between the system and the surrounding environment, a phenomenon called decoherence (1); (2). This interaction brings out some novel physics, such as lasing without inversion or extracting work from a single heat bath (3); (4); (5); (6). In the theory of open quantum systems, it is usually assumed that the environment is in thermal equilibrium (7); (8). Such an assumption not only faithfully projects most environments but also greatly simplifies theoretical analysis. Nonetheless, there are physical and especially biological situations where the environment is not in thermal equilibrium. In these cases, the environment has the opportunity to influence the quantum evolution in a manner that is more rich and complex than simply acting to randomize relative phases and dissipate energy. Light-induced ultra-fast coherent electronic processes in chemical and biological systems, superconducting circuits and quantum dots are examples where non-equilibrium effects are important (9); (10); (11); (12); (13); (14).
The dynamics of quantum systems in non-equilibrium environments has been studied in a number of contexts. Myatt and co-workers studied the decoherence of trapped ions coupled to engineered reservoirs, where the internal state and the coupling strength can be controlled (15). Schriefl and co-workers studied the dephasing of a two-level system, coupled to non-stationary noise modeling interacting defects (16) and to non-stationary classical intermittent noise (17). Kohler and Sols reported the emergence of recoherence in the dissipative dynamics of a harmonic oscillator, coupled linearly through its position and momentum to two independent heat baths at the same temperatures (18). Gordon and co-workers discussed the control of quantum coherence and the inhibition of dephasing using stochastic control fields (19). Clausen and co-workers demonstrated a bath-optimized minimal energy control scheme to use arbitrary time-dependent perturbations to slow decoherence of quantum systems interacting with non-Markovian but stationary environments (20). The well-known increase of the decoherence rate with the temperature, for a quantum system coupled to a linear thermal bath, no longer holds for a different bath dynamics. This is shown by means of a simple classical nonlinear bath (21). Emary considered two examples of nonlinear baths weakly coupled to a quantum system and showed that the decoherence rate is a monotonic decreasing function of temperature (22). Beer and Lutz discussed decoherence in a general non-equilibrium environment consisting of several equilibrium baths at different temperatures, described as a single effective bath with a time-dependent temperature (23). Martens studied the non-equilibrium response of the environment by a non-stationary random function which offers the possibility of the control of quantum decoherence by the detailed properties of the environment (24). Li and co-workers concluded that the amount of the steady quantum coherence increases with the temperature difference of the two heat baths coupled to a three-state system (25). If two heat baths have the same temperature, all quantum coherence vanishes and the dynamics returns to the equilibrium case. Moreno and co-workers showed that nested environments can improve coherence of a central system as the coupling between near and far environment increases (26).
Non-equilibrium quantum dynamics is also attractive from fundamental point of view. Ludwig and co-workers showed that there is an optimal dissipation strength for which the entanglement between two coupled oscillators interacting with a non-equilibrium environment is maximized (27). Castillo and co-workers demonstrated that under non-equilibrium thermal conditions, in a certain range of temperature gradients, Leggett-Garg inequality violation can be enhanced (28). In all such studies, in principle, the possibility of controlling decoherence is of great importance.
Among all the researches mentioned, lack of a macroscopic quantum system as the case study is deeply felt. Such a system is of great importance not only in fundamental physics but also in modern quantum technology (29); (30). To be macroscopic, a quantum system should have macroscopically distinguishable, entangled states (31). The effective dynamics of the macroscopic system, especially the phenomenon of macroscopic quantum tunneling, can be studied in the typical double-well potential. At sufficiently low temperatures, the system’s Hamiltonian can be expanded by two first states of energy. When the two-level system couples linearly to an oscillator environment, the result is the renowned Spin-Boson model (32); (7). Due to decoherence, then, a mixture of localized states of the system emerges.
The standard approach to open quantum systems is based on Born and Markov approximations. Both are essentially justified in most equilibrium environments. This is not the case for a non-equilibrium environment. It is not stationary, thus Born approximation is not justified. The dynamical feature of such environment also implies the importance of non-Markovian effects (33). The limitations of Markovian master equation approach in unfolding the non-equilibrium dynamics are discussed in a number of contexts. Wichterich and co-workers pointed out that the standard microscopic derivation of the weak-coupling Born-Markov-secular master equation (which is in Lindblad form) is inadequate for the description of non-equilibrium properties (34). Also, Ludwig and co-workers showed that commonly employed Lindblad approach fails to give even a qualitatively correct picture of the effect of a non-equilibrium environment on the entanglement in the system (27).
Here, we examine the effective dynamics of a macroscopic quantum system, in interaction with a non-equilibrium environment. The system is conveniently modeled by the motion of a particle in a symmetric double-well potential and the non-equilibrium environment is minimally described by two independent reservoirs with different spectral densities. The experiments devoted to the studies of macroscopic quantum coherence - most notably, those involving superconductivity, superfluity, and single-domain magnet - requires temperatures close to absolute zero in order to operate (for more detail see (35), Ch. 3). Hence, we assume that the reservoirs are initially prepared in the ground state. To address non-equilibrium effects, we employ the time-dependent second-order perturbation theory without any explicit Born and Markov approximation. Note that since the non-equilibrium environment is not stationary, we can not define a temperature for it.
The paper is organized as follows. In the next section, we describe the physical model of the whole system, consisting of the macroscopic quantum system and surrounding reservoirs. We examine the kinematics and dynamics of the system in sections 3 and 4, respectively. The parameters of the model are estimated in section 5. The results are discussed in section 6. Our concluding remarks are presented in the last section.

Ii 2. Model

The Hamiltonian of the whole system composed of the macroscopic quantum system and the reservoirs and is conveniently defined as

 H=HS+∑RHR+∑RHSR (1)

where . The system is modeled as a quasi-particle with effective mass in a symmetric double-well potential. To quantify the extent to which the system exhibits quantum coherence, we incorporate the dimensionless form of the model. The potential has the characteristic energy (height of the barrier) and the characteristic length (distance between two minima) which we adopt as the units of energy and length, respectively. The corresponding characteristic time can be defined as which we consider as the unit of time. Likewise, the unit of momentum is taken as . We then define the dynamical variables, and , as and , respectively. The corresponding commutation relation is defined as , where the Planck constant is redefined as , which we call “reduced Planck constant”.
In the limit (with as Boltzmann constant, and as temperature), the states of the system are confined in the two-dimensional Hilbert space spanned by the lowest localized eigenstate of each well. The effective Hamiltonian of the system in the localized basis would then be

 HS=−Δσx (2)

where is the tunneling strength between two localized states and is the component of the spin Pauli operator. For an isolated system, the probability of the tunneling from the left state to the right one is given by

 PL→R=sin2(Δt2) (3)

A frequently employed model for a reservoir is a collection of harmonic oscillators. The -th harmonic oscillator in the reservoir is characterized by its natural frequency, , and position and momentum operators, and, , respectively, according to the Hamiltonian

 HR=∑α12(p2α,R+ω2α,Rx2α,R−hωα,R) (4)

The last term, which merely displaces the origin of energy, is introduced for later convenience. For the reservoir , we define as the vacuum eigenstate and as the excited eigenstate with energy .
The interaction between the macroscopic quantum system and the reservoir has the form (35)

 HSR=−∑α(ω2α,Rfα,R(σz)xα,R+12ω2α,Rf2α,R(σz)) (5)

according to which the macroscopic system displaces the origin of the oscillator of reservoir with the spring constant by , as it can be recognized from

 HR+HSR=∑α12[p2α,R+ω2α,R(x2α,R−fα,R(σz))2−hωα,R] (6)

For simplicity, we assume that the interaction model is separable (, where is the coupling strength) and bilinear ().

Iii 3. Kinematics

The shift in the system’s energy due to the perturbation up to the second order is obtained as

 δEn,R ≃R⟨0|⟨n|HSR|n⟩|0⟩R+∑m≠nα≠0∣∣R⟨α|⟨m|HMR|n⟩|0⟩R∣∣2En−(Em+Eα,R) =12∑r|σrn|2Ωrn∑αγ2α,Rω2α,RωαR+Ωrn (7)

where , . The state with the energy shifted to due to the perturbation is not actually stationary and decays with a finite lifetime given by the Fermi’s golden rule as

 Γn,R ≃2πh∑m≠nα≠0∣∣R⟨0|⟨n|HSR|n⟩|0⟩R∣∣2δ(En−(Em+Eα,R)) =πh∑r|σrn|2Ωrn∑αγ2α,Rω2α,Rδ(Ωrn−ωα,R) (8)

Iv 4. Dynamics

We assume that the initial state of the whole system is

 |Ψ(0)⟩=|ψ⟩|0⟩A|0⟩B (9)

where is an arbitrary state of the macroscopic system. The state of the whole system at time is obtained by

 |Ψ(t)⟩=e−iH0t/hUI(t)|Ψ(0)⟩ (10)

where we defined and is the time evolution operator in the interaction picture. We expand up to the second order with respect to the interaction Hamiltonian to find

 UI(t)≃1−ih∫t0dt1Hint(t1)−1h2∫t0dt2∫t20dt1Hint(t2)Hint(t1) (11)

We first calculate

 UI(t)|0⟩A|0⟩B≃U0A,0B(t)|0⟩A|0⟩B+UαA,αB(t)|α⟩A|α⟩B+U0A,αB(t)|0⟩A|α⟩B+UαA,0B(t)|α⟩A|0⟩B (12)

where

 U0A,0B(t)= 1−i2h∑R,α∫t0dt1f2α,R(t1)−12h∑R,αωα,R∫t0dt2∫t20dt1e−i(t2−t1)ωα,Rfα,R(t2)fα,R(t1) −14h2∑α∫t0dt2∫t20dt1{f2α,A(t2),f2α,B(t1)} UαA,αB(t)= −12h∑αω1/2α,Aω1/2α,B∫t0dt2∫t20dt1{eiωα,At2fα,A(t2),eiωα,Bt1fα,B(t1)} U0A,αB(t)= i√2h∑αω1/2α,B∫t0dt1eiωα,Bt1fα,B(t2)+1√8h3∑αω1/2α,B∫t0dt2∫t20dt1{f2α,A(t2),eiωα,Bt1fα,B(t1)} (13)

to find

 |Ψ(t)⟩=2∑n=1e−iEnt/h {|0⟩A|0⟩B⟨n|U0A,0B(t)|ψ⟩+∑α|0⟩A|α⟩B⟨n|U0A,αB(t)|ψ⟩ +∑α|α⟩A|0⟩B⟨n|UαA,0B(t)|ψ⟩+∑α|α⟩A|α⟩B⟨n|UαA,αB(t)|ψ⟩} (14)

where we defined , and . Note that can be obtained by interconverting within in (IV).
The problem is now reduced to the evaluation of matrix elements of interaction Hamiltonian in (IV). The potential being an even function, the energy eigenstates have definite parity. Because and are even functions and and are odd functions, the following selection rules are identified

 ⟨m|U0A,0B|n⟩=⟨m|UαA,αB|n⟩=0 ⟨n|U0A,αB|n⟩=⟨n|UαA,0B|n⟩=0 (15)

The diagonal matrix elements of are evaluated as

 ⟨n|U0A,0B(t)|n⟩=1 −ith∑R{δE(1)n,R−1π∑mσ2mn∫∞0dωRJ(ωR)ωR+Ωmn} (16) −1πh∑mσ2mn∫∞0dωRJ(ωR)1−eıt(ωR+Ωmn)(ωR+Ωmn)2−t2h2δE(1)n,AδE(1)n,B

where is the spectral density of the reservoir , corresponding to a continuous spectrum of environmental frequencies, , defined as

 J(ωR)=π2∑αγ2α,Rω3α,Rδ(ωR−ωα,R)≡JRωsRRe−ωR/ΛR (17)

where , and are type, coupling strength and cut-off strength of reservoir . The different types of reservoirs are characterized by the value of parameter as sub-ohmic (), ohmic () and super-ohmic (). We consider the ohmic case which is the most common choice.
The quantity embraced by the bracket on the right-hand side of (16) coincides with in (III). Thus, the following expression is valid up to the second order

 ⟨n|U0A,0B(t)|n⟩=∑Re−itδEn,R/h{ 1−1πh∑mσ2mn∫∞0dωRJ(ωR) ×(2sin2{(ωR+Ωmn)t/2}(ωR+Ωmn)2−isin{(ωR+Ωmn)t}(ωR+Ωmn)2−t2h2δE(1)n,AδE(1)n,B)}−1 (18)

We assume that the reservoir’s cut-off strength is much higher than the system’s characteristic strength , so that at times much higher than the first term of the integral in (IV) can be approximated by a delta function . Obviously, the result of the corresponding integral would be , which is zero for . The elements of are then reduced to

 ⟨1|U0A,0B(t)|1⟩ =∑Re−itδE1,R/h{1+iπh∫∞0dωRJ(ωR)sin{(ωR+Δ)t}(ωR+Δ)2−t2h2δE(1)1AδE(1)1B}−1 ⟨2|U0A,0B(t)|2⟩ =∑Re−itδE2,R/h{1−Γ2,Rt2+iπh∫∞0dωRJ(ωR)sin{(ωR−Δ)t}(ωR−Δ)2−t2h2δE(1)2AδE(1)2B}−1 (19)

If the coupling between the system and each reservoir is weak, we have . At the temporal domain , we finally obtain

 ⟨1|U0A,0B(t)|1⟩ ≈∑Rexp{−ih(tδE1,R−1π∫∞0dωRJ(ωR)sin{(ωR+Δ)t}(ωR+Δ)2)−t2h2δE(1)1AδE(1)1B}−1 ⟨2|U0A,0B(t)|2⟩ ≈∑Rexp{−Γ2,Rt2−ih(tδE2,R−1π∫∞0dωRJ(ωR)sin{(ωR−Δ)t}(ωR−Δ)2)−t2h2δE(1)1AδE(1)1B}−1 (20)

The elements of are obtained as

The elements of are evaluated as

 ⟨m|U0A,αB(t)|n⟩=2√πh(i+thδE(1)n,A)∫∞0dωBJ(ωB)sin{(ωB+(−1)mΔ)t/2}ωB+(−1)mΔei(ωB+(−1)mΔ)t/2 (22)

The elements of are calculated by interconverting within in (22).
We suppose that the initial state of the system is the left-handed state . The evolved state of the whole system in the localized basis of the macroscopic system is written as

 Missing or unrecognized delimiter for \Big (23)

where we defined

 |χn(t)⟩=e−iEnt/h( |0⟩A|0⟩B⟨n|U0A,0B(t)|L⟩+|0⟩A|α⟩B⟨n|U0A,αB(t)|L⟩ +|α⟩A|0⟩B⟨n|UαA,0B(t)|L⟩+|α⟩A|α⟩B⟨n|UαA,αB(t)|L⟩) (24)

for . We are interested in the probability of finding the macroscopic system in the right-handed state, i.e.,

 PR(t)=|⟨R|Ψ(t)⟩|2=12(⟨χ1(t)|χ1(t)⟩+⟨χ2(t)|χ2(t)⟩+Re[⟨χ1(t)|χ2(t)⟩]) (25)

V 4. Estimation of Parameters

To examine the dynamics of open macroscopic quantum system, we first estimate the parameters relevant to our analysis. We start with the parameters of the system. In dimensionless form, the macroscopic symmetric double-well potential can be represented by a forth-order polynomial as , where () is the curvature at the minima. This special function turns out to approximate the potential fairly well for many macroscopic systems if experiments are appropriately designed. We are interested only in the lowest eigenstate of each well, thus we can expand the potential around to find with the corresponding localized ground states . The first two states of energy are symmetric and anti-symmetric maximal superpositions of localized states . In first-order perturbation theory, the tunneling strength is obtained as . Approximating the integrand by a simple Gaussian, we finally obtain . Note that up to this point our relations are quite general and not yet explicitly restricted to any particular system. The parameter quantifies the macroscopicity of the system in question. The system in which is called the quasi-classical system. The typical range of for a macroscopic quantum system is ((35), Ch.2). For most discussed macroscopic systems, e.g, superconducting devices, is estimated as about ((35), Ch. 3). Of course, quantum phenomena is impossible to detect if is too small. Accordingly, we choose .
The properties of the reservoir are projected in the parameters of the corresponding spectral density: coupling strength and cut-off strength . Since we have employed the perturbation theory, the system should be weakly coupled to the reservoirs, i.e. . Also, in the previous section we pointed out that the reservoirs are able to resolve the systems’s states, i.e. .

Vi 5. Results and Discussion

For an isolated macroscopic quantum system, the tunneling process, according to (3), is manifested by symmetrical oscillations between localized states of the system (FIG. 1-a). Since the system is isolated, such symmetrical oscillations are considered as the quantum signature of the system. A quantum system, especially a macroscopic one, is not actually isolated. An environment destroys the quantum coherence between the preferred states of the system. This decoherence process is manifested in the reduction of the amplitude of oscillations, resulting to an equilibrium steady state at long times (36). In our approach, if we eliminate one of the reservoirs, the expected exponential decay of oscillations is observed (FIG. 1-b).

Now we turn to the case where the system interacts with two identical reservoirs. In order to compare the one-reservoir dynamics with the two-reservoir one, we divide the single reservoir discussed in the previous section into two identical reservoirs. In doing so, we should note that, in two-reservoir dynamics, the effect of one reservoir is multiplied by the effect of the other one (see (25)). Hence, to retain the whole environment intact, we should have , and . At the short-time limit of the dynamics, the interference between two reservoirs is manifested as successive interference patterns (FIG. 2-a). Here, the symmetry of the interference patterns can be considered as the signature of the similarity of two reservoirs. At the long-time limit, as expected, the steady state is finally reached. This is in accordance with the work of Li and co-workers, in which a three-level microscopic system in interaction with two heat baths with the same temperature returns to the equilibrium case (25). Nonetheless, if we compare the one-reservoir decoherence time scale in FIG. 1-b with the two-reservoirs one in FIG. 2-a, we realize that the interference between two reservoirs through the system intensifies the decoherence process.
The system being macroscopic exerts a force on reservoir’s particles, which is realized by the second term of (5). It would be interesting to examine the effect of this back-action on the system’s dynamics. The comparison between the dynamics in the presence of this force (FIG. 2a) and in the absence of it (FIG. 2b) clearly shows that such a back-action weakens the overall interference between two reservoirs.

The non-equilibrium environment is realized by partitioning the whole environment into two non-equivalent reservoirs. This can be done thorough the coupling strength and/or cut-off strength of two reservoirs. The probability of the right-handed state is plotted for different values of coupling strength in FIG. 3a. The figure clearly shows that as we strengthen one of the reservoirs (and weaken the other one accordingly), we reach to the one-reservoir dynamics. Nevertheless, at a certain time, the coherence of the system has an oscillatory dependence on the difference between coupling strengths (FIG. 3b). At first, for a wide range of differences, the coherence increases. This is in agreement with the the work of Li and co-workers, in which the coherence increases with the temperature difference of two heat baths coupled to a three-state system (10). According to our analysis, however, when this difference reaches to a critical value, the coherence is non-monotonically decreased. This is because after the critical point the decoherence feature of the stronger reservoir dominates the process and as a result, the coherence is destroyed.

The dynamics is proximately independent of the corresponding cut-off parameter difference.

Vii 6. Conclusion

We examined the dynamics of a macroscopic quantum system in interaction with a non-equilibrium environment. The system is conveniently modeled by the motion of a particle confined in the macroscopic double-well potential and the environment is minimally described by two independent reservoirs with different spectral densities. A non-equilibrium environment has essentially a non-stationary dynamics. Therefore, in order to analyze the dynamics accurately, i.e. without any Born and Markov approximations, both reservoirs are prepared in the ground state. Such low-temperature reservoirs are conveniently realized in the experiments on macroscopic quantum systems. It is believed that the dynamics of a quantum system in interaction with two independent identical stationary reservoirs is reduced to that of a single reservoir. However, for a macroscopic quantum system, we demonstrated that the interaction between two reservoirs through the system manifests itself as successive interference patterns in the dynamics. In the non-equilibrium environment, it is claimed that the quantum coherence increases with the difference between two reservoirs. However, our analysis shows that this is not true for the large differences. In fact, when the difference reaches a critical value, the decoherence feature of the hot reservoir dominates the process and destroys the coherence. We also examined how the macroscopic feature of the system affects the dynamics. In general, a quantum system exerts a force on environmental particles. This system’s back-action intensifies the decoherence process. The environment is usually considered to be large, so for a microscopic quantum system this force is negligible. For a macroscopic quantum system, however, this force affects the dynamics considerably. In fact, in the non-equilibrium environment, the coherence enhancement is decreased with the macroscopicity of the system.

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