Regulation of a continuously monitored quantum harmonic oscillator with inefficient detectors

Abstract

We study the control problem of regulating the purity of a quantum harmonic oscillator in a Gaussian state via weak measurements. Specifically, we assume time-invariant Hamiltonian dynamics and that control is exerted via the back-action induced from monitoring the oscillator's position and momentum observables; the manipulation of the detector measurement strengths regulates the purity of the target Gaussian quantum state. After briefly drawing connections between Gaussian quantum dynamics and stochastic control, we focus on the effect of inefficient detectors and derive closed-form expressions for the transient and steady-state dynamics of the state covariance. We highlight the degradation of attainable purity that is due to inefficient detectors, as compared to that dictated by the Robertson-Schrödinger uncertainty relation. Our results suggest that quantum correlations can enhance the purity at steady-state. The quantum harmonic oscillator represents a basic system where analytic formulae may provide insights into the role of inefficient measurements in quantum control; the gained insights are pertinent to measurement-based quantum engines and cooling experiments.

Figures (5)

• Figure 1: Schematic of the Wigner distribution of a quantum harmonic oscillator Gaussian state subject to the simultaneous continuous monitoring of its position and momentum observables $\hat{X}$ and $\hat{P}$, respectively. The system is coupled to two detectors monitoring the system with strength $k$ and efficiency $\eta$. The Gaussian packet is confined in a harmonic potential and rotates about the trajectory of the unitary dynamics. The roughness of the paths is a direct manifestation of the stochastic back-action caused by continually probing the system via an apparatus that provides measurement readouts $R_t^x$ and $R_t^p$. Such measurement schemes have been successfully implemented on qubits hacohen2016quantum.
• Figure 2: The purity $p$ at steady-state versus the product and ratio $q=k_xk_p$ and $s=k_x/k_p$, respectively. We take $m=\omega =1$, $\hbar = 2$, $\eta_x = 0.1$, and $\eta_p$ = 0.9.
• Figure 3: The purity $p$ at steady-state versus the product $q=k_xk_p$ for fixed values of the ratio $s=k_x/k_p$. We take $m=\omega =1$, $\hbar = 2$, $\eta_x = 0.1$, and $\eta_p$ = 0.9.
• Figure 4: The purity $p$ at steady-state versus the ratio $s=k_x/k_p$ for fixed values of the product $q=k_xk_p$. We take $m=\omega =1$, $\hbar = 2$, $\eta_x = 0.1$, and $\eta_p$ = 0.9.
• Figure 5: Purification/squeezing of a zero-correlation initial Gaussian state (shown in black) by slowly varying $s$ from $3$ to $0.0001$ with $q$ fixed at $1$. Starting from the zero correlation purity $\sqrt{\sqrt{\eta_x}\sqrt{\eta_p}}\approx 0.55$ where $s = 3$, the maximal value $\sqrt{\eta_p}=\sqrt{0.9}\approx 0.95$ of $p_{\infty}$ is approached as $s$ decrease, while the correlation goes from $0$ to $-1.04$.

Theorems & Definitions (8)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof