Nonexponential Quantum Decay under Environmental Decoherence

Abstract

A system prepared in an unstable quantum state generally decays following an exponential law, as environmental decoherence is expected to prevent the decay products from recombining to reconstruct the initial state. Here we show the existence of deviations from exponential decay in open quantum systems under very general conditions. Our results are illustrated with the exact dynamics under quantum Brownian motion and suggest an explanation of recent experimental observations.The exponential decay law of unstable systems is ubiquitous in Nature and has widespread applications [1][2][3]. Yet, in isolated quantum systems deviations occur at both short and long times of evolution [4][5][6]. Short time deviations underlie the quantum Zeno effect [7, 8], ubiquitously used to engineer decoherence free-subspaces and preserve quantum information. Long-time deviations are expected in any nonrelativistic systems with a ground state; they slow down the decay and generally manifest as a power-law in time [9]. Both short and long-time deviations are present as well in manyparticle systems [11][12][13][14][15][16]. Indeed, the latter signal the advent of thermalization in isolated many-body systems [17,18]. In quantum cosmology, power-law deviations constrain the likelihood of scenarios with eternal inflation [10]. They also rule the scrambling of information as measured by the decay of the form factor [19][20][21][22] in blackhole physics and strongly coupled quantum systems described by AdS/CFT, that are believed to be maximally chaotic [23].Given a unstable quantum state |Ψ 0 prepared at time t = 0, it is customary to describe the closed-system decay dynamics via the survival probability, which is the fidelity between the initial state and its time evolution S(t) := |A(t)| 2 = | Ψ 0 |Ψ(t) | 2 .Explicitly, the survival amplitude reads A(t)is the time evolution operator generated by the Hamiltonian of the systemĤ. Short time deviations are associated with the quadratic decayand are generally suppressed by the coupling to an environment that induces the appearance of a term linear in t, see, e.g. [2, 3,24,25]. The origin of the long-time deviations can be appreciated using the Ersak equation for the survival amplitude [4,26,27]that follows from the unitarity of time evolution in isolated quantum systems. The memory term readsHere, we denote the projector onto the space spanned by the initial state byP ≡ |Ψ 0 Ψ 0 | and its orthogonal complement byQ ≡ 1 −P. As a result, the memory term m(t, t ) represents the formation of decay products at an intermediate time t and their subsequent recombination to reconstruct the initial state |Ψ 0 . The suppression of this term leads to the exponential decay law for A(t) and S(t), as an ansatz of the form A(t) = e −γt is a solution of Eq. (3) with m(t, t ) = 0, i.e, A(t) = A(t − t )A(t ). [4,27]. In addition, using the definition of the survival probability and Eq. (3), it has been demonstrated that the long-time non-exponential behavior of S(t) is dominated by |m(t, t )| 2 . The onset of lon...

Criterion for quantum Zeno and anti-Zeno effects

“…Thus one can assume that the total system collapses to its initial state after each measurement. Moreover, the decay pattern of an openquantum system is often nonexponential [54]. One can always use the effective decay rate to describe the system evolution in a moderate time scale under the weakcoupling approximation.…”

Decay of a thermofield-double state in chaotic quantum systems

“…with C, γ > 0 and q < 1. In physical terms, the origin of this slower decay can be traced back to the possibility that the time-evolving state reconstructs the initial state [38,49,50]. The specific form of the long-time decay depends on the behavior of the LDOS near any edges.…”

Quantum Speed Limits across the Quantum-to-Classical Transition

“…While QSL provide fundamental constraints to the pace at which quantum systems can change, a plethora of applications have been found that well extend beyond the realm of quantum dynamics. Indeed, QSL provide limits to the computational capability of physical devices [5], the performance of quantum thermal machines in finite-time thermodynamics [6,7], parameter estimation in quantum metrology [8,9], quantum control [10][11][12][13][14], the decay of unstable quantum systems [15][16][17][18] and information scrambling [19], among other examples [3,4,20].Specifically, QSL are derived as upper bounds to the rate of change of the fidelity F(τ) = | ψ 0 |ψ τ | 2 ∈ [0, 1] between an initial quantum state |ψ 0 and the corresponding time-evolving state |ψ τ =Û(τ, 0)|ψ 0 , whereÛ(τ, 0) is the time-evolution operator. More generally quantum states need not be pure, and given two density matrices ρ 0 and ρ τ =Û(τ, 0)ρ 0Û (τ, 0) † the fidelity readsThe fidelity is useful to define a metric between quantum states in Hilbert space, known as the Bures angle, [24,25] This gives a geometric interpretation of speed limit as the minimum time required to sweep out the angle L (ρ 0 , ρ τ ) under a given dynamics [26].…”