Time Dependence of Correlation Functions Following a Quantum Quench

Abstract

We show that the time-dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some hamiltonian and then evolves without dissipation according to some other hamiltonian, may be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system. PACS numbers: 73.43. Nq, 11.25.Hf. 64.60.Ht Suppose that an extended quantum system in d dimensions (for example a quantum spin system), is prepared at time t = 0 in a pure state |ψ 0 which is the ground state of some hamiltonian H 0 (or, more generally, in a thermal state at a temperature less than the gap m 0 to the first excited state.) For times t > 0 the system evolves unitarily according to the dynamics given by a different hamiltonian H, which may be related to H 0 by varying a parameter such as an external field. This variation, or quench, is supposed to be carried out over a time scale much less than m −1 0 . How do the correlation functions, expectation values of products of local observables, then evolve? The answer to this question would appear to depend in detail on the system under consideration. It was first addressed in the context of the quantum Ising-XY model in Refs.[1] (see also [2]). Until recently it was, however, largely an academic question, because the time scales over which most condensed matter systems can evolve coherently without coupling to the local environment are far too short, and the effects of dissipation and noise are inescapable. However, with the development of experimental tools for studying the behavior of optical lattices of ultracold atoms, and quantum phase transitions in these systems [3], there has been renewed interest in this theoretical problem (see, for example, [4,5].)In this Letter we study such problems in general and argue that, if H is at or close to a quantum critical point (while H 0 is not), there is a large degree of universality in the behavior at sufficiently large distances and late times, despite the fact that correlations typically fall off exponentially, rather than the power laws characteristic of the ground state near a quantum critical point. Our arguments are based on the path integral approach and the well-known mapping of the quantum problem to a classical one in d+1 dimensions. The initial state plays the role of a boundary condition, and we are able to then use the renormalisation group (RG) theory of boundary critical behavior (see, e.g., [6]). From this point of view, particularly powerful analytic results are available for d = 1 and when the quantum critical point has dynamic exponent z = 1 (or, equivalently, a linear quasiparticle dispersion relation ω = v|...

Dynamic correlations, fluctuation-dissipation relations, and effective temperatures after a quantum quench of the transverse field Ising chain

“…The grey vertical arrows indicate the time evolution, which takes the form of Eq. (36) in the basis of the quasi-particles {γ Γ k ,γ Γ † −k }. In order to solve the dynamics of the model, one first expresses these quasi-particles…”

“…On the other hand, the dynamics after the quench takes a particular simple form [see Eq. (36)] if the operators one is interested in are expressed in terms of the quasi-particles of the final HamiltonianĤ(Γ). Figure 1 summarizes schematically the relations between the various operators: black arrows indicate the transformations R which connect them, given explicitly in Eq.…”

“…In Sec. II we review several definitions of effective temperatures proposed in the context of quantum quenches [17,18,20,21,30,31,36] and classical and quantum dissipative glassy dynamics [12-14, 43, 51]. Section III summarizes those features of the static and dynamic behavior of a quantum Ising chain that are relevant to our study.…”

“…Instead, local quantities such as the transverse magnetization do not show thermal behavior with the notable exception of quenches to the critical point. In fact, compared to non-critical quenches, the critical point shows some remarkable properties [31,36,37] that can be attributed to the gapless spectrum and the linearity of the dispersion relation at low momenta. This is clearly seen in the scaling limit of one-dimensional models with these properties, for which the results of Conformal Field Theory (CFT) [36] suggest the emergence of a unique scale which plays the role of a temperature, at least for certain (non-generic) initial states [23,38].…”

Species-selective confinement of atoms dressed with multiple radiofrequencies

“…With each species in an adjustable well, as per the two rf scheme, the system can be used to examine quantum quenches [58], the self-trapping of impurities in the condensate [59], or the superfluid drag force exerted on impurities [60]. The inherent smoothness of dressed potentials generated from macroscopic coils is ideally suited for mechanical rotation experiments, and for species where the mass differs by only a few percent the centrifugal separation of the mixture is minimal.…”