Phonon interferometry for measuring quantum decoherence

Experimental observation of the decoherence of macroscopic objects is of fundamental importance to the study of quantum collapse models and the quantum to classical transition. Optomechanics is a promising field for the study of such models because of its fine control and readout of mechanical motion. Nevertheless, it is challenging to monitor a mechanical superposition state for long enough to investigate this transition. We present a scheme for entangling two mechanical resonators in spatial superposition states such that all quantum information is stored in the mechanical resonators. The scheme is general and applies to any optomechanical system with multiple mechanical modes. By analytic and numeric modeling, we show that the scheme is resilient to experimental imperfections such as incomplete precooling, faulty postselection, and inefficient optomechanical coupling. This proposed procedure overcomes limitations of previously proposed schemes that have so far hindered the study of macroscopic quantum dynamics.

Figure 1

Proposed experimental setup. Two mechanical resonators are optomechanically coupled to an optical cavity. Here we show a membrane and a trampoline resonator with a mirror, but the procedure could be used for any two mechanical resonators coupled via an optical cavity field. A continuous-wave laser is sent to an optical pulse generation setup, which produces pulses of varying frequency, duration, and intensity. The light enters the optomechanical cavity and subsequently the reflected light is filtered to remove the control pulses. The filtered signal contains the single photons used for heralding and readout, which are measured with a superconducting single-photon detector (SSPD).

Figure 2

The four control pulses sent into the optomechanical cavity to execute the experiment. The pulses are (i) cooling to the ground state, (ii) excitation to a coherent state, followed by postselection of the first excited state, (iii) a mechanical-mechanical interaction with $Jt=\pi /2$, and (iv) readout of a resonator. On the bottom, the equivalent optics experiment is shown with the corresponding steps. The grayed out detector is the optional addition of a readout pulse for resonator 2.

Figure 3

Expected results of a decoherence measurement with two entangled resonators in which one interacts with a thermal environment. The red (blue) indicates the readout of resonator 1 (2). Dotted lines are the limits set by the analytical model. Three effects are visible: coherent oscillations due to the frequency difference between the resonators, a decay of that coherence due to environmentally induced decoherence, and thermalization with the environment. The parameters for this plot are ${\omega }_1=2$ GHz, $\mathrm{\Delta }\omega =30$ kHz, $\gamma =2$ kHz, $T_{\mathrm{env}}=0.1\mathrm{K}$, and $\eta =0.01$.

Figure 4

Effects of several experimental imperfections on the resulting interference experiment, using the initial visibility as a metric. (a) The two resonators are only cooled to a phonon occupancy of $n_{th}$ in step (i). (b) The detector has a dark count probability of $\xi t_r$ for different probabilities of excitation, $p$ in step (ii). (c) Step (iii) also induces an optical cooling rate $J_c$ and an optical heating rate $J_h$ in addition to the mechanical-mechanical coupling $J$. The grayed out regions indicate regimes in which the dominant behavior is not the desired entangled state. The purple stars indicate parameters already achieved in experiments in (b) Ref. [35] and (c) Ref. [44]. The unvaried parameters for these plot are $n_{th}=0.01$, $p=0.01$, $\xi t_r={10}^{-6}$, $\eta =0.01$, and $J_c=J_h=0$.

Figure 5

Expected results of a decoherence measurement with imperfections present and an initial visibility of 30%. The parameters for this plot are the same as for Fig. 3 with additional imperfections $n_{th}=0.4$, $p=0.1$, and $J_c=J_h=0$. Despite the limited initial visibility, all three effects are still visible: coherent oscillations due to the frequency difference between the resonators, a decay of that coherence due to environmentally induced decoherence, and thermalization with the environment. We estimate that the experimental limit on the initial visibility is around 10%.

Figure 6

Density matrix representation of decoherence and thermalization. Each matrix is plotted during step (iii) after a delay time $\tau$ of (a) 0 ms, (b) 190 ms, (c) 950 ms, and (d) 3.8 s. The states labeled 1–22 in the figure correspond to the basis states $\{00,01,...,09,010,10,11,...,19,110\}$. The relevant parameters are ${\omega }_2=10$ GHz, $\gamma =1$ Hz, and $T_{\mathrm{env}}=0.2\text{K}$.

Figure 7

Visibility as a function of detuning. Grayed out regions have too high $J_c$ or $J_h$ to run the experiment. The value of $J$ depends on the exact experimental parameters, so it is normalized to the highest value. The parameters are ${\omega }_2/{\omega }_1=2$, ${\omega }_1/\kappa =10$, $n_{th}=0.01$, $p=0.01$, $\xi t_r={10}^{-6}$, and $\eta =0.01$.