New Class of Eigenstates in Generic Hamiltonian Systems

Abstract

In mixed systems, besides regular and chaotic states, there are states supported by the chaotic region mainly living in the vicinity of the hierarchy of regular islands. We show that the fraction of these hierarchical states scales ash −α and relate the exponent α = 1 − 1/γ to the decay of the classical staying probability P (t) ∼ t −γ . This is numerically confirmed for the kicked rotor by studying the influence of hierarchical states on eigenfunction and level statistics.Typical Hamiltonian systems are neither integrable nor ergodic [1] but have a mixed phase space, where regular and chaotic regions coexist. The regular regions are organized in a hierarchical way [2] (see, e.g., Fig. 2a) and chaotic dynamics is clearly distinct from the dynamics of fully chaotic systems. In particular, chaotic trajectories are trapped in the vicinity of the hierarchy of regular islands. The most prominent quantity reflecting this, is the probability P (t) to be trapped longer than a time t, which decays as [3]in contrast to the typically exponential decay in fully chaotic systems. While the power-law decay is universal, the exponent γ is system and parameter dependent. The origin of the algebraic decay are partial transport barriers [4], e.g., Cantori, leading to a hierarchical structure of the chaotic region [5]. Quantum mechanically the classical algebraic decay of P (t) is mimicked at most until Heisenberg time [6].Even after two decades of studying quantum chaos, the search for quantum signatures of this universal power-law trapping is still in its infancy: In fact, only conductance fluctuations of open systems have been investigated so far. It was semiclassically derived that these fluctuations should have a fractal dimension D = 2 − γ/2 [7], which was confirmed in gold nanowires [8], semiconductor nanostructures [9], and numerics [10]. Quite recently, a second type of conductance fluctuations in mixed systems has been discovered numerically [11,12], namely isolated resonances. There the classical exponent γ seems to appear in the scaling of the variance of conductance increments, surprisingly, on scales below the mean level spacing, what is not understood so far. Thus, even for this subject there is a lack of basic understanding.

Flooding of Chaotic Eigenstates into Regular Phase Space Islands

“…Our analysis applies as well to hierarchical states [22], which are confined by partial transport barriers with turnstile areas smaller than h. We predict the additional condition γ < 1/τ H for their existence, where γ describes the decay through these partial barriers. For regular states on island chains within that hierarchical region, condition (1) applies, with τ H given by the mean level spacing of the surrounding hierarchical states.…”

Spectral backbone of excitation transport in ultracold Rydberg gases

“…Rydberg gases therefore constitute an ideal testbed to study the physics of coherent (or incoherent) energy transfer which, due to the combined influence of disorder and coherence, gives rise to an intriguing variety of phenomena * Please address correspondence to: torsten.scholak@googlemail.com ranging from diffusive (where the excitation, if initially localized at a single atom, eventually spreads over the whole cloud) to localized transport (where, even at long times, the excitation remains localized in a certain sub-region of the cloud) [20][21][22][23]. While disorder and interaction-induced transport phenomena represent a long-standing and central theme of condensed matter theory [20,21,24,25], mesoscopic physics [26][27][28][29][30][31], light matter interaction [32][33][34][35][36][37][38][39] and, more recently, quantum simulation [40][41][42], one can argue that cold Rydberg gases offer the specific advantage to address rather subtle issues of quantum transport theory in disordered systems, which hitherto could not be addressed. In our present contribution, we will focus on the tunability of the spectral structure of these experimental objects.…”

“…The first term in Eq. (27) generates not only all irreducible three-center diagrams, but also some that do not contain any three-center loops and are thus reducible. These are already generated by F 1 and are subsequently subtracted in order to prevent them from being counted twice.…”

Decay Rates and Survival Probabilities in Open Quantum Systems

“…With these premises, our initial assertion on the crucial importance of microscopic dynamics for macroscopic transport leads to the guiding question of our present contribution [9,10]: How do classically mixed regular-dynamics affect these global and local spectral properties, and, consequently, the time dependence of P surv ? Whilst, recently, a considerable corpus of literature has addressed this problem within the context of simple models of mesoscopic transport [7,10,11], we shall here underpin our rather general answer by a numerically exact treatment of highly excited three dimensional Rydberg states under strong microwave driving [12]. These are paradigmatic objects to (theoretically and experimentally) study [6,9,12,13] quantum probability decay in the presence of tunneling as well as semiclassical and dynamical localization, and allow a natural and clean definition of P surv and the associated projection P Ω , without any approximations.…”

“…As already anticipated in the introduction, such observation does not come as a surprise, since periodically driven Rydberg states exhibit a mixed regular chaotic phase space. Indeed, it is known from experimental as well as exact theoretical/numerical studies [9,12,15] of strongly perturbed atomic Rydberg systems, and, more recently, from mesoscopic systems with decay [10], that the eigenstates of such quantum systems can be classified according to their localization [9,10,16] (i) on regular and/or elliptic regions, (ii) along remnants of regular motion immerged in the chaotic sea ("separatrix/hierarchical states" living on "can-tori"), and (iii) in the chaotic domain of phase space. Furthermore, elliptic regions as well as remnants of invariant tori are rather robust under changes of some control parameter such as F in our present case [9].…”