Stability of quantum motion in regular systems: A uniform semiclassical approach

Abstract

We study the stability of quantum motion of classically regular systems in presence of small perturbations. On the base of a uniform semiclassical theory we derive the fidelity decay which displays a quite complex behaviour, from Gaussian to power law decay t −α with 1 ≤ α ≤ 2. Semiclassical estimates are given for the time scales separating the different decaying regions and numerical results are presented which confirm our theoretical predictions. Stable manipulation of quantum states is of importance in many research fields such as in quantum information processing and in Bose-Einstein condensation. A measure of the stability of quantum motion is the so-called fidelity or quantum Loschmidt echo [1], which characterizes the stability of quantum dynamics under small perturbations of the Hamiltonian that may derive from static imperfections or from interaction with an external environment. From a more general point of view it is interesting to understand the behavior of fidelity in relation to the dynamical properties of the system. While fidelity decay in classically chaotic systems has been extensively studied [2,3,4,5,6,7,8,9,10,11], the situation in regular systems is much less clear [5,10,11,12,13,14,15,16,17,18,19] and only for the particular case of vanishing time-average perturbations a clear understanding has been achieved [12]. Moreover, for a general perturbation and for a single initial Gaussian wave packet, a Gaussian decay has been predicted in some time interval which is not exactly specified [5]. On the other hand, numerical investigations show a much more rich behavior of the fidelity decay ranging from power law to exponential, up to a Gaussian decay, depending on initial conditions, perturbation strength, and time interval [15]. In addition, a somehow unexpected regime in which the fidelity decay in regular systems is faster than in classically chaotic systems has been found in [5]. All the above calls for a theory which can account for these diverse analytical and numerical findings.We would like to draw the reader's attention to the fact that in the general theory of dynamical systems, integrability is the exception rather than the rule and it is therefore extremely rare. However, the theory we present in this paper applies also to integrable regimes of systems with divided phase space in which both chaotic and integrable components are present and this is indeed the typical situation. Moreover as shown in [20], for a proper operability of a quantum computer it is desirable to remain below the border for transition to quantum chaos, a situation which is likely to correspond to quasi integrable behavior. Finally, some quantum algorithm can have a phase space representation and, in particular, the Grover algorithm can be interpreted as a simple quantum map which turns out to be a regular map [21].The above considerations motivate our interest in the stability of integrable motion. In this letter, we develop a uniform semiclassical approach to the fidelity decay in regular systems and we provid...

Complexity and instability of quantum motion near a quantum phase transition

“…In fact, this is the reason we chose the exponent 2/3 in the fitting in Eq. (39). However, we remark that, to see numerically the dependence m 2 t ∝ t 3 , we would need much smaller values of η than those accessible in our calculations.…”

“…In the latter case, for intermediately strong perturbation [28,32] Γ is given by the half-width of the local spectral density of states and for relatively strong perturbation it is perturbation independent [26,35,38]. In integrable systems with one degree of freedom, the LE has a Gaussian decay, followed after a transient region by a power-law decay [39]. In contrast, in integrable systems with many degrees of freedom the LE has an exponential decay [42].…”

“…For m 2 t ≫ 1, the scaling in Eq. (39) reduces to M L ≃ b m 2 −1/3 t . Therefore, given that the LE M L ∝ 1/t [42], it gives m 2 t ∝ t 3 .…”

“…Extensive investigations have been performed in recent years to understand the decaying behaviors of the LE [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. In chaotic systems, roughly speaking, the LE has a Gaussian decay [23] below a perturbative border and has an exponential decay M L (t) ∝ exp(−Γt) above the border.…”

“…The dramatic change of the wave function at a QPT implies a fast decrease of the fidelity when approaching the critical point. Another metric quantity useful for characterizing the occurrence of a QPT is the quantum Loschmidt echo (LE) [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], which provides a measure to the stability of the quantum motion under * Electronic address: wgwang@ustc.edu.cn two slightly different Hamiltonians. This quantity exhibits a dramatic change in its decaying behavior in the neighborhood of critical points.…”