# Exponential stabilization of quantum systems under continuous non-demolition measurements

## TL;DR

This paper introduces a feedback control method using Brownian motion to exponentially stabilize a specific eigenstate in quantum systems under continuous non-demolition measurements, ensuring high-fidelity state preparation.

## Contribution

It proposes a novel feedback strategy that employs Brownian noise to achieve exponential stabilization of a quantum eigenstate, advancing quantum control techniques.

## Key findings

- Proves exponential convergence to the target eigenstate.
- Demonstrates the effectiveness of Brownian-driven control in quantum stabilization.
- Suggests simplified filters for practical implementation.

## Abstract

We present a novel continuous-time control strategy to exponentially stabilize an eigenstate of a Quantum Non-Demolition (QND) measurement operator. In open-loop, the system converges to a random eigenstate of the measurement operator. The role of the feedback is to prepare a prescribed QND eigenstate with unit probability. To achieve this we introduce the use of Brownian motion to drive the unitary control actions; the feedback loop just adapts the amplitude of this Brownian noise input as a function of the system state. Essentially, it "shakes" the system away from undesired eigenstates by applying strong noise there, while relying on the open-loop dynamics to progressively reach the target. We prove exponential convergence towards the target eigenstate using standard stochastic Lyapunov methods. The feedback scheme and its stability analysis suggest the use of an approximate filter which only tracks the populations of the eigenstates of the measurement operator. Such reduced filters should play an increasing role towards advanced quantum technologies.