Driving quantum systems with repeated conditional measurements

Abstract

We investigate the effect of conditional null measurements on a quantum system and find a rich variety of behaviors. Specifically, quantum dynamics with a time independent H in a finite dimensional Hilbert space are considered with repeated strong null measurements of a specified state. We discuss four generic behaviors that emerge in these monitored systems. The first arises in systems without symmetry, along with their associated degeneracies in the energy spectrum, and hence in the absence of dark states as well. In this case, a unique final state can be found which is determined by the largest eigenvalue of the survival operator, the non-unitary operator encoding both the unitary evolution between measurements and the measurement itself. For a three-level system, this is similar to the well known shelving effect. Secondly, for systems with builtin symmetry and correspondingly a degenerate energy spectrum, the null measurements dynamically select the degenerate energy levels, while the non-degenerate levels are effectively wiped out. Thirdly, in the absence of dark states, and for specific choices of parameters, two or more eigenvalues of the survival operator match in magnitude, and this leads to an oscillatory behavior controlled by the measurement rate and not solely by the energy levels. Finally, when the control parameters are tuned, such that the eigenvalues of the survival operator all coalesce to zero, one has exceptional points that corresponds to situations that violate the null measurement condition, making the conditional measurement process impossible.

Quantum random walk and tight-binding model subject to projective measurements at random times

“…In the above backdrop, we remark that our results reported in this work are complementary to those reported in [13][14][15][16][17][18][19][20][21][22][23][24][25][26] because of the very different dynamical set-ups (regular τ ) considered in these work with respect to ours (stochastic τ ). The only exception to the above set of work that considered stochastic τ in the setting of the TBM with periodic boundary conditions is [27].…”

“…The TBM and related systems when subject to projective measurements with time evolution following scheme 2 of dynamics have in recent years been studied to address the issue of when does a quantum particle evolving under the dynamics arrive at a chosen set of sites [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. A crucial difference is that, barring [27], these contributions almost exclusively implement measurements at regular intervals of times, unlike the scheme considered in this work in which sequence of measurements at random times is implemented.…”

“…While the total detection probability P det in classical random walks is unity, the same for a quantum walker, with detection in some state |d , may for certain initial states |ψ in take a value smaller than unity. In [23,25], the authors derived universal bounds for the quantum total detection probability, under scheme 2 of dynamics with measurements after every fixed [26] studied by considering scheme 2 of dynamics the effect of conditional null measurements, done at equal intervals of time of length τ , on a quantum system and displayed a wide variety of rich behavior, e.g., for systems with built-in symmetry and a degenerate energy spectrum, the null measurements are found to dynamically select the degenerate energy levels, while the non-degenerate levels are effectively wiped out by the measurements.…”

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

“…The TBM (and related systems) when subject to projective measurements has in recent years been extensively studied in the context of detection problem corresponding to a quantum particle evolving under the dynamics to arrive at a chosen set of sites [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59]. In the context of the present work, we briefly summarize the contribution of [59] that also considers projective measurements at random times.…”

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

“…The TBM (and related systems) when subject to projective measurements has in recent years been extensively studied in the context of detection problem corresponding to a quantum particle evolving under the dynamics to arrive at a chosen set of sites [22][23][24][25][26][27][28][29][30][31][32][33][34][35]. However, all these papers focus on the first time the quantum particle arrives at a chosen set of detection sites, unlike our treatment in which repeated detections are taken into account.…”