Non-equilibrium Dynamics of Quantum Systems: Order Parameter Evolution, Defect Generation, and Qubit Transfer

Abstract

The properties of systems near quantum critical points (QCPs) have been studied extensively in recent years [1,2]. A QCP is a point across which the symmetry of the ground state of a quantum system changes in a fundamental way; such a point can be accessed by changing some parameter, say λ , in the Hamiltonian governing the system. The change in the ground state across a QCP is mediated by quantum fluctuations. Unlike conventional thermal critical points, thermal fluctuations do not play a crucial role in such transitions. Similar to its thermal counterparts, the low energy physics near a QCP is associated with a number of critical exponents which characterize the universality class of such a transition. Amongst these exponents, the dynamical critical exponent z provides the signature of the relative scaling of space and time at the transition and has no counterpart in thermal phase transitions. The other exponent which is going to be important for the purpose of this review is the well-known correlation length exponent ν. These exponents are formally defined as follows. As we approach the critical point at λ = λ c , the correlation length diverges as ξ ∼ |λ − λ c | −ν , while the gap between the ground state and first excited state vanishes as ∆ E ∼ ξ −z ∼ |λ − λ c | zν . Exactly at the critical point λ = λ c , the energy of the low-lying excitations vanishes at some wave number k 0 as ω ∼ |k − k 0 | z . The critical exponents are independent of the details of the microscopic Hamiltonian; they depend only on a few parameters such as the dimensionality of the system and the symmetry of the order parameter. These features render the low energy equilibrium physics of a quantum system near a QCP truly universal.In contrast to this well-understood universality of the equilibrium properties of a system near a QCP, relatively few universal features are known in the nonequilibrium behavior of a quantum system. Initial studies in this field aimed at understanding the near-equilibrium finite temperature dynamics near a quantum critical

Weak field collapse in AdS: introducing a charge density

“…Recent developments in cold atoms experiments have initiated a very active research where this perturbation occurs abruptly, see e.g. [3][4][5] for a condensed-mater approach and [6][7][8] for a review of the holographic attempts. The other broad issue is to understand the physics of thermalization for strongly coupled system.…”

“…In such a situation, unless we consider a temperature scale much larger than the scale set by the relevant operator, the black hole formation process may be governed by this infrared geometry instead. 3 To avoid this possible subtlety for relevant operators, we will consider an exactly marginal operator, which does not require a hierarchy between the RG-scale and the temperature-scale. Thus the underlying CFT will remain the same and in the gravitational dual it will suffice for us to specify the asymptotically locally AdS condition with a given radius of curvature as the boundary condition.…”

Kibble-Zurek scaling in holographic quantum quench: backreaction

“…This is particularly so for strongly coupled systems. Nevertheless, Kibble-Zurek scaling has been verified by explicit calculations in many models and is now being seen experimentally as well [3][4][5][6]. 2 In [15] a study of this problem in strongly coupled field theories which have gravity duals via AdS/CFT was initiated and continued in [16] and [17].…”

Defect Production Due to Quenching Through a Multicritical Point and Along a Gapless Line

“…1.6 supports this scaling behavior. For a non-linear quench across a multicritical point [36], when the parameter λ is quenched as λ ∼ (t/τ) α sgn(t), both z 1 and z 2 come into play and the scaling of defect density gets altered to n ∼ τ −dαν/[α(z 2 ν+1)+z 1 ν(1−α)] ; this reduces to the form presented above for α = 1 and z 2 ν = 1. …”

“…Studies have also been made to estimate the defect density for quenching with a non-linear form [21], an oscillatory variation of an applied magnetic field [32] or under a reversal of the magnetic field [33]. In the article in this book by Mondal, Sengupta and Sen [36], the quenching dynamics through a gapless phase and quenching with a power-law form of the change of a parameter as well as the possibility of experimental realizations have been discussed in detail. It should be mentioned that in addition to the density of defects in the final state, the degree of non-adiabaticity can also be quantified by looking at various quantities like residual energy [9,37] and fidelity [4].…”