Dynamics of a quantum phase transition in the random Ising model: Logarithmic dependence of the defect density on the transition rate

Dziarmaga, Jacek

doi:10.1103/physrevb.74.064416

Cited by 101 publications

(124 citation statements)

References 37 publications

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“…In the case of the LMG model the gain is stronger, from a cubic to a linear dependence, outlining the fact that the limit of the optimization is set by a constant value of the action s * = T * ∆. As a last remark, from the previous discussion it can be argued that optimized evolutions achieve a substantial speed up only when the minimum gap closes polynomially with the size; in the case of an exponential closure, even an optimized process leads to a total evolution time exponentially diverging with N [41][42][43][44][45].…”

Section: B Dynamical Regime Classificationmentioning

confidence: 99%

Speeding up critical system dynamics through optimized evolution

Caneva¹,

Calarco²,

Fazio³

et al. 2011

Phys. Rev. A

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The number of defects which are generated on crossing a quantum phase transition can be minimized by choosing properly designed time-dependent pulses. In this work we determine what are the ultimate limits of this optimization. We discuss under which conditions the production of defects across the phase transition is vanishing small. Furthermore we show that the minimum time required to enter this regime is T ∼ π/∆, where ∆ is the minimum spectral gap, unveiling an intimate connection between an optimized unitary dynamics and the intrinsic measure of the Hilbert space for pure states. Surprisingly, the dynamics is non-adiabatic, this result can be understood by assuming a simple two-level dynamics for the many-body system. Finally we classify the possible dynamical regimes in terms of the action s = T ∆. PACS numbers:The rapid progress in the experimental realization and manipulation of quantum systems [1] is opening the rich and intriguing perspective of the exploitation of quantum physics to realize quantum technologies like quantum simulators [2] and quantum computers [3,4]. These achievements pave the way to the simulation of condensed matter systems giving the possibility of studying different states of matter in controlled experiments [5]. Despite the impressive results obtained so far, this is a formidable technological and theoretical challenge due to the complexity of the systems in analysis and the experimental requirements. Indeed, the level of control needed on the quantum system is unprecedented: one should be able to prepare a system in a desired initial state, perform the desired evolution and finally measure the state in a very precise way. Moreover, the whole experiment should be performed faster than the system decoherence time that eventually will destroy any quantum information capability. Quantum optimal control (OC) theory, the study of optimization strategies to improve the outcome of a quantum process, can be an extremely powerful tool to cope with these issues [6-10]. It allows not only to optimize the desired experiment outcome but also to speed up the process itself. Traditionally employed in atomic and molecular physics [11,12], OC has been recently applied with success to the optimization of the dynamics of many-body systems [13][14][15], allowing to achieve the ultimate bound imposed by quantum mechanics, the so called quantum speed limit (QSL) [16]. Indeed as intuitively suggested by the time-energy uncertainty principle, the time required by a state to reach another distinguishable state has to be longer than the inverse of its energy fluctuations [17]. This implies that a quantum system cannot evolve at an arbitrary speed in its Hilbert space, but a minimum time is required to perform a transformation between orthogonal states [18][19][20][21][22]. For time-independent Hamiltonians this bound has been exactly determined [16]; the QSL has been formally generalized also to time-dependent Hamiltonians, but so far has been computed only in a few simple cases [13,[23][24][25][26]. A s...

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Section: B Dynamical Regime Classificationmentioning

confidence: 99%

Speeding up critical system dynamics through optimized evolution

Caneva¹,

Calarco²,

Fazio³

et al. 2011

Phys. Rev. A

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“…By now, the original KZS has been confirmed for a variety of models involving a regular isolated QCP [14][15][16][17][18] , and extensions have been introduced for more general adiabatic dynamics, including repeated 19 , non-linear 20 , and optimal 21 quench processes. In parallel, departures from the KZS predictions have emerged for more complex adiabatic scenarios, involving for instance quenches across either an isolated multicritical point (MCP) 20,[22][23][24][25] or non-isolated QCPs (that is, critical regions) [26][27][28][29] , as well as QPTs in infinitely-coordinated 30 , disordered 31 , and/or spatially inhomogeneous systems 12,32 . A main message that has emerged from the above studies is that, unlike in the standard KZS of Eq.…”

Section: Introductionmentioning

confidence: 99%

Dynamical critical scaling and effective thermalization in quantum quenches: Role of the initial state

Deng¹,

Ortíz²,

Viola³

2011

Phys. Rev. B

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We explore the robustness of universal dynamical scaling behavior in a quantum system near criticality with respect to initialization in a large class of states with finite energy. By focusing on a homogeneous XY quantum spin chain in a transverse field, we characterize the non-equilibrium response under adiabatic and sudden quench processes originating from a pure as well as a mixed excited initial state, and involving either a regular quantum critical or a multicritical point. We find that the critical exponents of the ground-state quantum phase transition can be encoded in the dynamical scaling exponents despite the finite energy of the initial state. In particular, we identify conditions on the initial distribution of quasi-particle excitation which ensure Kibble-Zurek scaling to persist. The emergence of effective thermal equilibrium behavior following a sudden quench towards criticality is also investigated, with focus on the long-time expectation value of the quasi-particle number operator. Despite the integrability of the XY model, this observable is found to behave thermally in quenches to a regular quantum critical point, provided that the system is initially prepared at sufficiently high temperature. However, a similar thermalization behavior fails to occur in quenches towards a multi-critical point. We argue that the observed lack of thermalization originates in this case in the asymmetry of the impulse region that is also responsible for anomalous multicritical dynamical scaling.

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“…According to the Kibble-Zurek argument, if a parameter of the Hamiltonian is varied as t/τ , the density of defects (n) in the final state is expected to scale as n ∼ τ −νd/(νz+1) , where d is the spatial dimensionality of the system 6,7,8,9 .The above scaling form has been verified for quantum spin systems quenched through critical points 10,11,12,13 and also generalized to the cases of non-linear quenching 14 when a parameter is quenched as h(t) ∼ |t/τ | |a| sgn(t), for gapless systems 15 and also for quantum systems with disorder 16 and systems coupled to external environment 17 . Recently, a generalized form of the Kibble-Zurek scaling has been introduced which includes a situation where the system is quenched through the multicritical point 18 which shows that the general expression for kink density can be given as n ∼ τ −d/2z2 , where z 2 determines the scaling of the off-diagonal term of the equivalent Landau-Zener problem close to the critical point.…”

Section: Introductionmentioning

confidence: 94%

Effects of interference in the dynamics of a spin- 1/2 transverseXYchain driven periodically through quantum critical points

Mukherjee

Dutta

2009

J. Stat. Mech.

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We study the effects of interference on the quenching dynamics of a one-dimensional spin 1/2 XY model in the presence of a transverse field (h(t)) which varies sinusoidally with time as h = h0 cos ωt, with |t| ≤ t f = π/ω. We have explicitly shown that the finite values of t f make the dynamics inherently dependent on the phases of probability amplitudes, which had been hitherto unseen in all cases of linear quenching with large initial and final times. In contrast, we also consider the situation where the magnetic field consists of an oscillatory as well as a linearly varying component, i.e., h(t) = h0 cos ωt + t/τ , where the interference effects lose importance in the limit of large τ . Our purpose is to estimate the defect density and the local entropy density in the final state if the system is initially prepared in its ground state. For a single crossing through the quantum critical point with h = h0 cos ωt, the density of defects in the final state is calculated by mapping the dynamics to an equivalent Landau Zener problem by linearizing near the crossing point, and is found to vary as √ ω in the limit of small ω. On the other hand, the local entropy density is found to attain a maximum as a function of ω near a characteristic scale ω0. Extending to the situation of multiple crossings, we show that the role of finite initial and final times of quenching are manifested non-trivially in the interference effects of certain resonance modes which solely contribute to the production of defects. Kink density as well as the diagonal entropy density show oscillatory dependence on the number of full cycles of oscillation. Finally, the inclusion of a linear term in the transverse field on top of the oscillatory compo nent, results to a kink density which decreases continuously with τ while increases monotonically with ω. The entropy density also shows monotonous change with the parameters, increasing with τ and decreasing with ω, in sharp contrast to the situations studied earlier. We do also propose appropriate scaling relations for the defect density in above situations and compare the results with the numerical results obtained by integrating the Schrödinger equations.

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Dynamics of a quantum phase transition in the random Ising model: Logarithmic dependence of the defect density on the transition rate

Cited by 101 publications

References 37 publications

Speeding up critical system dynamics through optimized evolution

Speeding up critical system dynamics through optimized evolution

Dynamical critical scaling and effective thermalization in quantum quenches: Role of the initial state

Effects of interference in the dynamics of a spin- 1/2 transverseXYchain driven periodically through quantum critical points

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