Observing controlled state collapse in a single mechanical oscillator via a direct probe of energy variance

Due to their central role in our classical intuition of the physical world and their potential for interacting with the gravitational field, mechanical degrees of freedom are of special interest in testing the nonclassical predictions of quantum theory at ever larger scales. The projection postulate of quantum theory predicts that, for certain types of measurements, continuously measuring a system induces a stochastic collapse of the state of the system toward a random eigenstate. Here we propose an optomechanical scheme to observe this fundamental effect in a vibrational mode of a mechanical membrane. The observation in the scheme is direct (it is not inferred via an a priori assumption of the projection postulate for the mechanical mode) and is made possible through an in situ probe of the mechanical energy variance. In the scheme, quantum theory predicts that a steady state is reached as the measurement-induced collapse is counteracted by dissipation to the unmonitored environment. Numerical simulations show this to result in a monotonic decrease in the time-averaged energy variance as the ratio of continuous measurement strength to dissipation is increased. The measurement strength in the proposed scheme is tunable in situ, and the behavior predicted by the simulations therefore implies a way to verifiably control the time-averaged variance of a mechanical wave function over the course of a single quantum trajectory. The scheme's ability to directly probe the energy variance of the mechanical mode may also enable further investigations of the effects on the mechanical state of coupling the mechanical mode to other quantum systems.

Figure 1

A depiction of the membrane-in-the-middle system wherein the mechanical membrane (black) is suspended in a cavity (gray) orthogonal to its axis. The $a_{1,L\left(R\right)}$ [red, solid (dashed)] and $a_{2,L\left(R\right)}$ [blue, solid (dashed)] optical modes shown with respective frequencies ${\omega }_1$ and ${\omega }_2$ arise on each side of the membrane if it is perfectly opaque and still. Due to the finite transparency, however, the normal modes $a_{1,\pm }=(1/\sqrt{2})(a_{1,L}\pm a_{1,R})$ and $a_{2,\pm }=(1/\sqrt{2})(a_{2,L}\pm a_{2,R})$ become the physically relevant modes. The finite transparency of the membrane lifts the degeneracy between $a_{j,+}$ and $a_{j,-}$, and the optomechanical interaction introduces a modulation of the spectra that is proportional to $x^2$ for $a_{1,\pm }$ and $x^4$ for $a_{2,\pm }$. ${\omega }_1\ne {\omega }_2$ such that all normal modes may be driven and monitored independently.

Figure 2

Simulated collapse of the quantum state in the energy eigenbasis with increasing $n_b$ measurement strength (${\Gamma }_1$) and constant $n_b^2+A{n}_b$ measurement strength (${\Gamma }_2$). For each value of ${\Gamma }_1$, the system is first allowed to reach steady state; then $\overline{{\langle n_b\rangle }_c}$, $\overline{{\langle n_b\rangle }_c^2}$, and $\overline{{\langle n_b^2\rangle }_c}$ are obtained by computing ${\langle n_b\rangle }_c$ and ${\langle n_b^2\rangle }_c$ (the subscript $c$ denotes that the quantity is conditioned upon the measurement record, as required by the projection postulate) at each subsequent time step and averaging for a sufficiently long time. The steady-state conditional variance decreases monotonically as a function of ${\Gamma }_1$ and becomes much less than 1 when ${\Gamma }_1\gg 4{\gamma }_{\mathrm{th}}{\overline{n}}_{\mathrm{th}}$, the regime in which quantum jumps are predicted to arise in ${\langle n_b\rangle }_c\left(t\right)$ [25]. Simulation parameters are ${\Gamma }_2={10}^{-7}\left(4{\gamma }_{\mathrm{th}}{\overline{n}}_{\mathrm{th}}\right)$, ${\overline{n}}_{\mathrm{th}}=5\times {10}^3$, ${\gamma }_{\mathrm{cool}}={\gamma }_{\mathrm{th}}{\overline{n}}_{\mathrm{th}}$, and $A=1$.