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. Among 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 $\lambda = \lambda_c$, the correlation length diverges as $\xi \sim |\lambda - \lambda_c|^{-\nu}$, while the gap between the ground state and first excited state vanishes as $\varDelta E \sim \xi^{-z} \sim |\lambda - \lambda_c|^{z\nu}$. Exactly at the critical point $\lambda = \lambda_c$, the energy of the low-lying excitations vanishes at some wave number k ₀ as ${\boldsymbol \omega} \sim |{\textbf k} - {\textbf k}_0|^z$. 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.