Quantum quasi-Zeno dynamics: Transitions mediated by frequent projective measurements near the Zeno regime

Frequent observation of a quantum system leads to quantum Zeno physics, where the system evolution is constrained to states commensurate with the measurement outcome. We show that, more generally, the system can evolve between such states through higher-order virtual processes that pass through states outside the measurement subspace. We derive effective Hamiltonians to describe this evolution, and the dependence on the time between measurements. We demonstrate application of this phenomena to prototypical quantum many-body system examples, spin chains and atoms in optical lattices, where it facilitates correlated dynamical effects.

Figure 1

Transitions via virtual processes. The quasi-Zeno Hamiltonians give rise to transitions that occur via states outside the measurement subspace, which can be viewed as virtual processes. These can be compared conceptually with Feynman diagrams, where instead of interactions occurring via virtual particles, we instead represent transitions occurring via occupation of virtual states. Single vertex processes (a) are mediated by the standard Zeno Hamiltonian $H_Z^{\left(1\right)}$, while higher-order processes (b) with $n$ vertices and $n-1$ virtual states are mediated by the quasi-Zeno Hamiltonian $H_Z^{\left(n\right)}$. Transitions to a virtual state and back to the initial state can be represented by loop diagrams (c), akin to the representation of self-interaction with Feynman diagrams.

Figure 2

Correlated processes in spin chains. Multiple configurations of spins (a) belong to the same measurement subspace. Processes that ultimately preserve the measurement can take place, at [b($i$)] first or [b($ii),(iii$)] higher order. Here, the measurement is signified by the total magnetization of the green regions.

Figure 3

Correlated atomic processes. (a) Frequent, consistent measurement of the occupation of a set of sites facilitates correlated tunneling of atoms over the boundaries of the measured region, while (b) measuring occupation number differences gives rise to pair process-like effects. Colored boxes indicate the measured regions (green positive, blue negative), and arrows the same color indicate correlated tunneling events.

Figure 4

Simulation of correlated many-body dynamics. (a) Simulation of an effective Hamiltonian for atoms in an optical lattice, for the scheme of Eqs. (5) and (6) and Fig. 3, showing first- and second-order processes. Color map shows average site occupation. (b) Survival probability for the system to remain in the same Zeno subspace. (c) Exact evolution of the same system. (d) Difference between results for exact and effective evolution; the primary error is in capturing the first-order dynamics, and scales linearly with $\delta t$ (see main text). (e) Evolution under the Zeno Hamiltonian of QZD for the same system fails to capture the correlated processes. Simulations use four sites, two atoms, $\delta tJ={10}^{-2},U=0$, with measurement imposing the constraint $N_2+N_3=1$. Initial state $|1,1,0,0\rangle$.