Non-equilibrium dynamics of quantum systems: order parameter evolution, defect generation, and qubit transfer

TL;DR

The article addresses how quantum systems driven across quantum critical points or surfaces respond out of equilibrium, focusing on defect production and order-parameter dynamics under sudden and slow quenches. It develops universal scaling frameworks, including $n \sim \tau^{-d\nu/(z\nu+1)}$ for linear quenches and refined forms for quenches across critical surfaces or with nonlinear time dependences, with concrete demonstrations in the Kitaev honeycomb model, 1D ultracold atoms, and the infinite-range Ising model. It also shows how non-adiabatic dynamics can optimize quantum information tasks, notably qubit transfer fidelity and speed in spin chains via tailored time-dependent couplings. Together, these results illuminate universal non-equilibrium behavior near quantum criticality and offer practical routes to engineer quantum state transfer using controlled quench protocols.

Abstract

In this review, we study some aspects of the non-equilibrium dynamics of quantum systems. In particular, we consider the effect of varying a parameter in the Hamiltonian of a quantum system which takes it across a quantum critical point or line. We study both sudden and slow quenches in a variety of systems including one-dimensional ultracold atoms in an optical lattice, an infinite range ferromagnetic Ising model, and some exactly solvable spin models in one and two dimensions (such as the Kitaev model). We show that quenching leads to the formation of defects whose density has a power-law dependence on the quenching rate; the power depends on the dimensionalities of the system and of the critical surface and on some of the exponents associated with the critical point which is being crossed. We also study the effect of non-linear quenching; the power law of the defects then depends on the degree of non-linearity. Finally, we study some spin-1/2 models to discuss how a qubit can be transferred across a system.

Figures (14)

• Figure 1: Schematic representation of the parent Mott insulating state with $n_0=2$. Each well represents a local minimum of the optical lattice potential --- we number these as 1-5 from the left. The potential gradient leads to a uniform decrease in the on-site energy of an atom as we move to the right. The grey circles are the $d_i$ bosons of Eq. (\ref{['msshd']}). The vertical direction represents increasing energy: the repulsive interaction energy between the atoms is realized by placing atoms vertically within each well, so that each atom displaces the remaining atoms upwards along the energy axis. We have chosen the diameter of the atoms to equal the potential energy drop between neighboring wells --- this corresponds to the condition $U=E$. Consequently, a resonant transition is one in which the top atom in a well moves horizontally to the top of a nearest neighbor well; motions either upwards or downwards are non-resonant.
• Figure 2: Notation as in Fig. \ref{['mssfig1']}. ( a) A dipole on sites 2 and 3; this state is resonantly coupled by an infinitesimal $t$ to the Mott insulator in ( a) when $E=U$. ( b) Two dipoles between sites 2 and 3 and between 4 and 5; this state is connected via multiple resonant transitions to the Mott insulator for $E=U$.
• Figure 3: Evolution of the Ising order parameter in (\ref{['mssisingO']}) under the Hamiltonian $H_{1D} [E_f]$ for $n_0 = 1$. The initial state is the ground state of $H_{1D} [E_i]$. All the plots in this section have $U=40$, $t=1$, and $E_i=32$, and consequently the equilibrium QCP is at $E_c = 41.85$.
• Figure 4: System size ($N$) dependence of the results of Fig. \ref{['mssfig3']} for $E_f=40$. The curves are labeled by the value of $N$.
• Figure 5: The curve labeled 'dynamic' is the long-time limit $\langle O \rangle_t$ of the Ising order parameter in (\ref{['mssising']}) as a function of $E_f$ (for $N=11$), with other parameters as in Fig. \ref{['mssfig1']}. This long-time limit can be obtained simply by setting $m=n$ in (\ref{['mssising']}). For comparison, in the curve labeled 'adiabatic', we show the expectation value of the Ising order $O$ in the ground state of $H_{1D} [E_f]$; such an order would be observed if the value of $E$ was changed adiabatically. Note that the dynamic curve has its maximal value near (but not exactly at) the equilibrium QCP $E_c = 41.85$, where the system is able to respond most easily to the change in the value of $E$; this dynamic curve is our theory of the 'resonant' response in the experiments of Ref. mssbloch1 discussed in Sect. \ref{['mssintro']}. In contrast, the adiabatic result increases monotonically with $E_f$ into the $E>E_c$ phase where the Ising symmetry is spontaneously broken.