The quantum Zeno and anti-Zeno effects in the strong coupling regime

Abstract

It is well known that repeated projective measurements can either speed up (the Zeno effect) or slow down (the anti-Zeno effect) quantum evolution. Until now, however, studies of these effects for a two-level system interacting strongly with its environment have focused on repeatedly preparing the excited state of the two-level system via the projective measurements. In this paper, we consider the repeated preparation of an arbitrary state of a two-level system that is interacting strongly with an environment of harmonic oscillators. To handle the strong interaction, we perform a polaron transformation, and thereafter use a perturbative approach to calculate the decay rates for the system. Upon calculating the decay rates, we discover that there is a transition in their qualitative behaviors as the state being repeatedly prepared moves away from the excited state towards a superposition of the ground and excited states. Our results should be useful for the quantum control of a two-level system interacting with its environment. Introduction By subjecting a quantum system to frequent and repeated projective measurements, we can slow down its temporal evolution, an effect referred to as the quantum Zeno effect (QZE). Contrary to this effect is the quantum anti-Zeno effect (QAZE), via which the temporal evolution of the system is accelerated due to repeated projective measurements separated by relatively longer measurement intervals. Both these effects have garnered great interest not only due to their theoretical relevance to quantum foundations but also due to their applications to quantum technologies. For example, the QZE has shown to be a promising resource for quantum computing and quantum error correction. The QAZE, on the other hand, has interestingly been useful in, say, accelerating chemical reactions, suggesting the possibility of quantum control of a chemical reaction. By and large, studies of the QZE and the QAZE have focused on population decay and pure dephasing models. While the former studies measurements performed on a single two-level system, the latter considers the effect of dephasing on the QZE and the QAZE. A few works have gone beyond these regimes. Ref. [46], for instance, presents a general framework to calculate the effective decay rate for an arbitrary system-environment model in the weak coupling regime and finds it to be the overlap of the spectral density of the environment and a filter function that depends on the system-environment model, the measurement interval, and the measurement being performed. This approach, however, fails in the strong coupling regime where perturbation theory cannot be applied in a straightforward manner. For a single two-level system coupled strongly to an environment of harmonic oscillators, Ref. [47] makes the problem tractable by going to the polaron frame and finding that for the excited state, the decay rate very surprisingly decreases with an increase in the system-environment coupling strength. This effect is further investigated in Ref. [48], which studies a two-level system coupled simultaneously to a weakly interacting dissipative-type environment and a strongly interacting dephasing-type one. It is found that even in the presence of both types of interactions, the strongly coupled reservoir can inhibit the influence of the weakly coupled reservoir on the central quantum system. To date, the role of the state that is repeatedly prepared has been left unexplored, especially in the strong coupling regime. For example, it remains unanswered whether increasing the coupling strength of a strongly coupled reservoir would lead to the decay rate decreasing for states other than the excited state. This forms the basis of our investigation in this paper. We work out the decay rates for a two-level system, strongly interacting with a bath of harmonic oscillators, that is repeatedly prepared in an arbitrary quantum state. To make the problem tractable, we first go to the polaron frame, where the system-environment coupling is effectively weakened, and thereafter use time-dependent perturbation theory to evolve the system state and find its decay rate. While we reproduce the results presented for an excited state in Ref. [47], we observe a stark difference when the initial state is chosen to be a superposition of the excited and ground states. To be precise, the qualitative variation of the decay rate with the system-environment coupling gets inverted. To describe these results, we coin the terms “z-type" and “x-type", identifying the behavior displayed by Ref. [47] as the z-type behavior while the inverted behavior is termed as the x-type behavior. We investigate the transition between these behaviors. These results should be useful in the study of open quantum systems in the strong coupling regime. Results Effective decay rate for strong system-environment coupling We start from the the paradigmatic spin-boson model with the system-environment Hamiltonian written as (we work in dimensionless units with h̄ = 1 throughout) HL = ε 2 σz + ∆ 2 σx +∑ k ωkb † kbk +σz(∑ k gkb † k + gkbk), (1) where HS,L = ε 2 σz+ ∆ 2 σx is the system Hamiltonian, HB =∑k=1 ωkb † kbk is the environment Hamiltonian, and VL =σz(∑k g ∗ kb † k + gkbk) is the system-environment coupling. Note that L denotes the lab frame, ε is the energy splitting of the two-level system, ∆ is the tunneling amplitude, and the ωk are frequencies of the harmonic oscillators in the harmonic oscillator environment interacting with the system. The creation and annihilation operators of these oscillators are represented by the b † k and bk operators respectively. In the strong interacting regime, we cannot treat the interaction perturbatively. Moreover, the initial system-environment correlations are significant and thus cannot be neglected to write the initial state as a simple product state. To make the problem tractable, we perform a polaron transformation, which yields an effective interaction that is weak. More precisely, the transformation to the polaron frame is given by H = UPHLU † P , where UP = e − χ2 σz and χ = ∑k( gk ωk bk − gk ωk b † k). We then get the transformed Hamiltonian H = ε 2 σz +∑ k ωkb † kbk + ∆ 2 (σ+e χ +σ−e−χ). (2) For future convenience, we define H0 = ε 2 σz +∑k ωkb † kbk. Now, if ∆ is taken as being small, the system and environment interact effectively weakly in the polaron frame despite interacting strongly in the lab frame. Let |0〉 represent the excited state of our two-level system, and |1〉 be its ground state. Then, writing an arbitrary initial state of the two-level system as |ψ〉 = ζ1 |0〉+ ζ2 |1〉 with ζ1 = cos(θ/2) and ζ2 = eiφ sin(θ/2), we find the time-evolved density matrix by means of time-dependent perturbation theory. It is important to note that while the initial system-environment state cannot simply be taken as a simple uncorrelated product state in the ‘lab’ frame, we can do so in the polaron frame since the system and its environment are interacting weakly in the polaron frame. We subsequently perform repeated measurements after time intervals of duration τ to see if the system state is still |ψ〉. The survival probability at time τ is s(τ) = TrS,B{Pψρ(τ)}, where ρ(τ) is the combined density matrix of the system and the environment at time t = τ in the polaron frame before the projective measurement while Pψ =UP |ψ〉〈ψ |U P . This survival probability is then s(τ) = TrS,B{Pψe Pψ e−β H0 Z Pψe }, (3) with Z a normalization factor. The subsequent detailed calculation is in the Methods section. For the most general case, this yields a rather extensive expression. In this section, however, we present the expressions for some important cases only. First, let us consider the initial state |0〉, that is |ψ〉 with ζ1 = 1 and ζ2 = 0. It is found that s(τ) = 1− 2Re { ∆ 4 ∫ τ 0 dt1 ∫ t1 0 dt2|ζ1|e1 e2C(t2 − t1) } , (4) where C(t2 − t1) is the environment correlation function and is given by C(t2 − t1) = e−Φ ∗ C(t2−t1), where ΦC(τ) = ΦR(τ)− iΦI(τ) with ΦR = 4 ∫ ∞ 0 dωJ(ω) 1−cosωt ω2 coth ( β ω 2 ) and ΦI = 4 ∫ ∞ 0 dωJ(ω) sinωt ω2 . The environment spectral density J(ω) has been introduced as ∑k |gk|(· · · ) → ∫ ∞ 0 dωJ(ω)(· · · ). Since the system-environment coupling in the polaron frame is weak, we can neglect the accumulation of correlations between the system and the environment and write the survival probability at time t = Nτ , or s(t = Nτ), as [s(t)] , where N denotes the number of measurements performed after time t = 0. Now, we may write s(t = Nτ) ≡ e−Γ(τ)Nτ to define an effective decay rate Γ(τ) for our quantum state. It follows that Γ(τ) =