Ruelle resonances in kicked quantum spin chain

Prosen, Tomaž

doi:10.1016/j.physd.2003.09.017

Cited by 21 publications

(16 citation statements)

References 24 publications

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“…12, see also subsect.7.5 later) strongly suggest self-similar behaviour upon multiplying the code c by 4 which is a consequence of the fractal structure of the transfer matrix (see illustration in Ref. [77,18]). …”

Section: Relaxation and Quantum Ruelle Resonancesmentioning

confidence: 83%

Chaos and complexity of quantum motion

Prosen¹

2007

J. Phys. A: Math. Theor.

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Abstract. The problem of characterizing complexity of quantum dynamics -in particular of locally interacting chains of quantum particles -will be reviewed and discussed from several different perspectives: (i) stability of motion against external perturbations and decoherence, (ii) efficiency of quantum simulation in terms of classical computation and entanglement production in operator spaces, (iii) quantum transport, relaxation to equilibrium and quantum mixing, and (iv) computation of quantum dynamical entropies. Discussions of all these criteria will be confronted with the established criteria of integrability or quantum chaos, and sometimes quite surprising conclusions are found. Some conjectures and interesting open problems in ergodic theory of the quantum many problem are suggested.

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Section: Relaxation and Quantum Ruelle Resonancesmentioning

confidence: 83%

Chaos and complexity of quantum motion

Prosen¹

2007

J. Phys. A: Math. Theor.

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“…One can now observe that an exponential decay appearing in an equilibrium correlation function [such as Ft] likely indicates that, in the corresponding linear response nonequilibrium problem [10], the leading correction to the equilibrium density matrix of any subsystem of spins evolves as % 0 xe ÿt ; (9) where x is the set of variables describing the density matrix, and % 0 x is some function. An analogous statement applies to the probability distribution of classical spins.…”

Section: Ftmentioning

confidence: 99%

“…In large isolated quantum systems, these eigenvalues are expected to be analogous to Pollicott-Ruelle (PR) resonances [2,3] in classical chaotic systems. PR resonances and their quantum analogs manifest themselves in the behavior of temporal correlation functions, and as such have been investigated, e.g., in the studies of deterministic diffusion [3], microwave billiards [4], spin dynamics [5][6][7][8][9][10], and classical Loschmidt echoes [11]. If an exponential decay appearing in one correlation function originates from a true PR resonance, then the same decay should appear in all correlation functions having the same symmetry properties.…”

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confidence: 99%

Long-Time Behavior of Spin Echo

Fine

2005

Phys. Rev. Lett.

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It is predicted that (i) spin echoes have two kinds of generic long-time decays: either simple exponential, or a superposition of a monotonic and an oscillatory exponential decays; and (ii) the long-time behavior of spin echo and the long-time behavior of the corresponding homogeneous free induction decay are characterized by the same time constants. This prediction extends to various echo problems both within and beyond nuclear magnetic resonance. Experimental confirmation of this prediction would also support the notion of the eigenvalues of time evolution operators in large quantum systems.Calculations of the spin echo (SE) envelope observed by nuclear magnetic resonance (NMR) in solids are, in general, very difficult and, in practice, mostly limited to simple sudden-jump models (e.g., [1]), which cannot capture complex non-Markovian dynamics induced by spin-spin interactions. The latter difficulty is closely related to presently insufficient understanding of the role of chaos in many-body quantum systems. In this Letter, I show that a promising approach, which can both simplify the quantitative analysis of SEs and elucidate the role of chaos, is to exploit the notion of eigenvalues of the time evolution operator. In large isolated quantum systems, these eigenvalues are expected to be analogous to Pollicott-Ruelle (PR) resonances [2,3] in classical chaotic systems. PR resonances and their quantum analogs manifest themselves in the behavior of temporal correlation functions, and as such have been investigated, e.g., in the studies of deterministic diffusion [3], microwave billiards [4], spin dynamics [5-10], and classical Loschmidt echoes [11]. If an exponential decay appearing in one correlation function originates from a true PR resonance, then the same decay should appear in all correlation functions having the same symmetry properties.The specific goal of this Letter is to make the following two propositions:(1) The generic long-time behavior of SE envelope S2 obtained from the sequence [=2-pulse-time --pulse-time -detection] has the form eitheror S2 e ÿ2 C 1 C 2 cos2! S ;where C, C 1 , C 2 , , !, and S are some constants.(2) The constants and ! describing the long-time SE decay are the same as the long-time constants of the corresponding ''homogeneous'' free induction decay (FID) defined by the pulse sequence [=2-pulse-time tdetection] and exhibiting generic long-time behavior [10] of the form either Ft Ae ÿt ;(3) counterpart of Eq. (1), orFt Ae ÿt cos!t F ;(4) counterpart of Eq. (2). Here A and F are some additional constants. (The meaning of the homogeneous FID will be specified below.) An important consequence of the above propositions is that the long-time constants ( and !) of the SE envelope can be obtained from a much simpler FID calculation. At present, a number of approximations exist (see, e.g., [5,12] and references therein), which can be used to calculate the FID long-time constants with reasonable accuracy.In the following, I shall first consider the case of spinspin interaction only, and then...

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“…One such widely studied system is the Ising model, which exhibits a quantum phase transition when a magnetic field is applied in a direction transverse to the interaction [8]. A variant of the transversefield Ising model, when the field is applied impulsively kicked via a Dirac δ comb, has been discussed in the literature [9][10][11]. Even in the presence of time dependence, the transverse Ising model remains integrable via the Jordan-Wigner transformation [12] as the resulting fermions are free.…”

Section: Introductionmentioning

confidence: 99%

Protocol using kicked Ising dynamics for generating states with maximal multipartite entanglement

Mishra

Lakshminarayan²,

Subrahmanyam³

2015

Phys. Rev. A

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We present a solvable model of iterating cluster state protocols that lead to entanglement production, between contiguous blocks, of 1 ebit per iteration. This continues until the blocks are maximally entangled, at which stage an unravelling begins at the same rate until the blocks are unentangled. The model is a variant of the transverse-field Ising model and can be implemented with controlled-NOT and single-qubit gates. The interqubit entanglement as measured by the concurrence is shown to be zero for periodic chain realizations, while for open boundaries there are very specific instances at which these can develop. Thus we introduce a class of simply produced states with very large multipartite entanglement content of potential use in measurement-based quantum computing.

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Ruelle resonances in kicked quantum spin chain

Cited by 21 publications

References 24 publications

Chaos and complexity of quantum motion

Chaos and complexity of quantum motion

Long-Time Behavior of Spin Echo

Protocol using kicked Ising dynamics for generating states with maximal multipartite entanglement

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