Defect Production in Nonlinear Quench across a Quantum Critical Point

Sen, Diptiman; Sengupta, K.; Mondal, Sayan

doi:10.1103/physrevlett.101.016806

Cited by 198 publications

(209 citation statements)

References 34 publications

(52 reference statements)

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“…For the linear ramp r = 1 this result reduces to Eq. (1) and for the exponential ramp r → ∞ we obtain linear scaling as generally expected [19,20] in 1D systems with the linear spectrum.…”

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confidence: 99%

Adiabatic Nonlinear Probes of One-Dimensional Bose Gases

2008

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We discuss two complimentary problems: adiabatic loading of one-dimensional bosons into an optical lattice and merging two one-dimensional Bose systems. Both problems can be mapped to the sine-Gordon model. This mapping allows us to find power-law scalings for the number of excitations with the ramping rate in the regime where the conventional linear response approach fails. We show that the exponent of this power law is sensitive to the interaction strength. In particular, the response is larger, or less adiabatic, for strongly (weakly) interacting bosons for the loading (merging) problem. Our results illustrate that in general the nonlinear response to slow relevant perturbations can be a powerful tool for characterizing properties of interacting systems.Dynamics of one-dimensional (1D) cold gases [1,2,3] is characterized by a strong response to external perturbations [4,5] compared to higher-dimensional systems, due to the large density of low-energy states. For example, for slow processes the excitations are strongly enhanced and can qualitatively affect the dynamics and lead to the breakdown of adiabatic approximation [6]. Understanding slow dynamics has two-fold implications. First, such dynamics can be used as a probe of interacting systems in the regimes where the linear response (Kubo-type) analysis fails. Second, this understanding can help to devise optimal ways for minimizing heating in the system during slow processes. This minimization can be important for various applications including adiabatic preparation of strongly correlated systems and adiabatic quantum computing. In this Letter we will demonstrate both such implications using two related problems of loading 1D interacting bosons into an optical lattice and merging (or splitting) two 1D condensates.The physical parameter that defines the properties of 1D bosonic systems is the ratio of the interaction and kinetic energies γ = mc/( 2 ρ), where m is the mass of a particle, c is the interaction strength [7], and ρ is the density of particles. In the spatially uniform system, the energy spectrum is gapless in the whole range of the parameter γ, which at low energies is reflected in the Luttinger liquid description [8]. In the extreme limit of strong interactions, γ ≫ 1, Tonks-Girardeau (TG) gas [9], the particles behave as hard-core spheres, with the repulsion playing a role similar to the Pauli exclusion principle. This regime has been recently realized in cold atoms [10,11].External perturbations can significantly modify the phase structure of bosonic gases.A bosonic system undergoes the transition from superfluid to Mottinsulator [12,13], driven by the competition of the kinetic and interaction energy: in deep lattices the insulating phase is realized when the interaction energy becomes larger than the tunneling (kinetic) energy. This mean-field picture of the transition is modified in one dimension. If the repulsion between particles is strong enough, an arbitrarily small periodic potential com- mensurate with the boson density stabilizes th...

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confidence: 99%

Adiabatic Nonlinear Probes of One-Dimensional Bose Gases

2008

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“…Suppressing excitations is also of interest to variety of operations in the laboratory, like entangling strings of atoms [11]. This has motivated studies including the use of the energy gap arising from the finite size of the system [12], optimal non-linear passage across a QCP [13,14], inhomogeneous quenches [15][16][17], and optimal quantum control strategies [18]. All those approaches can be regarded as strategies to exploit or engineer a spectral gap.…”

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confidence: 99%

Assisted Finite-Rate Adiabatic Passage Across a Quantum Critical Point: Exact Solution for the Quantum Ising Model

Campo¹,

Rams²,

Zurek³

2012

Phys. Rev. Lett.

291

350

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The dynamics of a quantum phase transition is inextricably woven with the formation of excitations, as a result of the critical slowing down in the neighborhood of the critical point. We design a transitionless quantum driving through a quantum critical point that allows one to access the ground state of the broken-symmetry phase by a finite-rate quench of the control parameter. The method is illustrated in the one-dimensional quantum Ising model in a transverse field. Driving through the critical point is assisted by an auxiliary Hamiltonian, for which the interplay between the range of the interaction and the modes where excitations are suppressed is elucidated.PACS numbers: 03.75. Kk, The complexity involved in describing a generic manybody quantum system prompted Feynman to suggest the use of a highly controllable quantum system as a simulator of another, generally complicated, quantum system of interest [1]. From this perspective, interesting quantum systems are those with a large amount of entanglement and hardly tractable in classical computers [2]. Quantum simulation has become an exciting field of research, which is being developed experimentally by exploring a variety of platforms including ultracold atoms, trapped ions, photonic quantum systems and superconducting circuits, among others. Simulation of manybody interacting systems is particularly advanced in implementations with trapped ions [3] where the building blocks of a digital quantum simulator for both closed [4] and open [5] quantum systems have been demonstrated. Moreover, while early experimental efforts have been limited to somewhat low number of qubits, the simulation of few-hundreds of spins with variable-range spin-spin Ising-type interactions has recently been reported [6].In a continuous quantum phase transition, divergence of length and time scales across a quantum critical point (QCP) leads inevitably to non-adiabatic dynamics. When a parameter λ of the Hamiltonian is changed across its critical value λ c , the energy gap between the ground and the first excited state vanishes, and adiabaticity breaks down. The KibbleZurek mechanism (KZM), originally developed for classical and continuous phase transitions [7,8], predicts that the resulting density of excitations obeys a power-law scaling with the quench rate. The power-law exponent is expressed using the critical exponents at equilibrium and the dimensionality of the system [9, 10]. As a result, quantum quenches are useful to characterize universal features of a system, and to shed some light on its dynamics out of equilibrium.The inevitable formation of excitation is however undesirable for a wide range of applications, such as the preparation of novel quantum phases in quantum simulation, and adiabatic quantum computation. Suppressing excitations is also of interest to variety of operations in the laboratory, like entangling strings of atoms [11]. This has motivated studies including the use of the energy gap arising from the finite size of the system [12], optimal non-linear pass...

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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: 98%

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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Defect Production in Nonlinear Quench across a Quantum Critical Point

Cited by 198 publications

References 34 publications

Adiabatic Nonlinear Probes of One-Dimensional Bose Gases

Adiabatic Nonlinear Probes of One-Dimensional Bose Gases

Assisted Finite-Rate Adiabatic Passage Across a Quantum Critical Point: Exact Solution for the Quantum Ising Model

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

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