Quantum Transition Rates in Arbitrary Physical Processes

Abstract

We introduce a framework for computing time-dependent quantum transition rates (QTRs) that describe the pace of evolution of a quantum state from a given subspace to a target subspace. QTRs are expressed in terms of flux-flux correlators and are shown to obey two complementary quantum speed limits. Our framework readily accommodates the generalization of Hamiltonian dynamics to arbitrary open quantum evolution, including quantum measurements. We illustrate how QTRs can be controlled by counterdiabatic driving.

Figures (6)

• Figure 1: QTRs refine QSLs by using the conditional probability $P(B,t|A,0)$. An initial state $\hat{\rho}_0$ is found in the subspace $\mathcal{H}_A$ by means of a projective measurement of $\hat{\Pi}_A$. Time evolution is described by a quantum channel $\mathcal{E}_t$ and a second projective measurement, of $\hat{\Pi}_B$, determines the probability that the state reaches the target subspace $\mathcal{H}_B$.
• Figure 2: QTRs and flux-flux correlation functions for the TFIM for various driving times exhibiting a sharp change around the critical point.
• Figure 3: (Up) Transition probability in the TFIM under CD, precisely following the theoretical prediction for a broad range of cut-off momenta. ($L=1600$). (Down) QTR with similar agreement wit the analytical predictions.
• Figure 4: Time-evolution of the transition probabilities into a target space of energy range $[E_0,E_1]$. (a) Strong suppression is observed for energy windows below zero energy, $E_1<0$, with a precise matching of the analytical results $(N=2000, W=2, \Delta =1)$. (b) For target subspaces involving also positive energies, the shape and order of magnitude of $P(B,T\vert A)$ changes dramatically, exhibiting a sigmoid-type shape and saturating to values close to unity.
• Figure 5: (a) QTRs for target subspace confined to negative energies exhibiting short-time peak in accordance with the transition probabilities. (b) For target subspace entering the positive energy region, a broader QTR traces out in accordance with the sigmoid shape of the transition probabilities.