Quantum nondemolition measurement of Fock states of mesoscopic mechanical oscillators

We investigate a scheme that makes a quantum nondemolition (QND) measurement of the excitation level of a mesoscopic mechanical oscillator by utilizing the anharmonic coupling between two beam bending modes. The nonlinear coupling between the two modes shifts the resonant frequency of the readout oscillator in proportion to the excitation level of the system oscillator. This frequency shift may be detected as a phase shift of the readout oscillation when driven on resonance. We derive an equation for the reduced density matrix of the system oscillator, and use this to study the conditions under which discrete jumps in the excitation level occur. The appearance of jumps in the actual quantity measured is also studied using the method of quantum trajectories. We consider the feasibility of the scheme for experimentally accessible parameters.

Figure 1

Schematic of a QND measurement using two coupled mechanical oscillators.

Figure 2

Plot of the solution of Eq. (53) without the stochastic component, $k=0$, with the initial state $\mid 1⟩$ (solid line) and $\mid 2⟩$ (dashed line).

Figure 3

A plot of a solution to Eq. (53) with $\nu =0$ with an initial state that is thermal.

Figure 4

Plot of $p_n\left(t\right)$ for a simulation of Eq. (53) with $\nu =0$ for the states $\mid 0⟩,\mid 1⟩,\mid 2⟩,\mid 3⟩$. The initial state is a thermal state with the average occupation number 1.63. The figure is for the same simulation as in Fig. 3.

Figure 5

The evolution of the phonon number $⟨a_0^{†}{a}_0⟩\left(t\right)$ given by Eq. (53) using $N_0=1.62$ and $k∕\nu =250$ (first panel) and $k∕\nu =5$ (second panel).

Figure 6

Histogram of $⟨a_0^{†}{a}_0⟩\left(t\right)$ for a long simulation with $k∕\nu =150$ and $N_0=1.62$ (first panel) and $k∕\nu =15$ (second panel).

Figure 7

The deviation of $⟨a_0^{†}{a}_0⟩\left(t\right)$ from integral values as a function of $k∕\nu$. The deviation is defined as the average of ${\mid ⟨a_0^{†}{a}_0⟩\left(t\right)-\mathrm{Int}⟨a_0^{†}{a}_0⟩\left(t\right)\mid }^2$.

Figure 8

Evolution of $⟨a_0^{†}{a}_0⟩\left(t\right)$ at a temperature corresponding to $N_0=1.62$ (first panel) and $N_0=20$ (second panel), with $\nu N_0∕k=0.0108$ and from an initial state $\mid 2⟩$.

Figure 9

Filtered current using a running average over various window sizes, for parameters $k∕\nu =250$ and $N_0=1.62$. The dotted line is $⟨a_0^{†}{a}_0⟩\left(t\right)$ given by the stochastic density matrix. The current was first averaged over a time interval of $k\Delta t=0.15$. First panel: current observed; second panel: window size $k\Delta t=4.5$; third panel: window size $k\Delta t=7.5$; fourth panel: window size $k\Delta t=10.5$.