Detecting phonon blockade with photons

Measuring the quantum dynamics of a mechanical system, when few phonons are involved, remains a challenge. We show that a superconducting microwave resonator linearly coupled to the mechanical mode constitutes a very powerful probe for this scope. This new coupling can be much stronger than the usual radiation pressure interaction by adjusting a gate voltage. We focus on the detection of phonon blockade, showing that it can be observed by measuring the statistics of the light in the cavity. The underlying reason is the formation of an entangled state between the two resonators. Our scheme realizes a phonotonic Josephson junction, giving rise to coherent oscillations between phonons and photons as well as a self-trapping regime for a coupling smaller than a critical value. The transition from the self-trapping to the oscillating regime is also induced dynamically by dissipation.

Figure 1

(a) Scheme of the system. A mechanical resonator is capacitively coupled to an artificial atom to induce a nonlinearity and to a superconducting microwave resonator to detect phonon blockade. (b) Schematics of the energy spectrum of the system for $\eta \gg g$. The first excited states are the maximally entangled Bell states $|{\Psi }^{\pm }\rangle$ and are off resonance with the states with more than one excitation. When the system is excited at the frequency ${\omega }_d={\omega }_0\pm g$, phonon blockade is observed through the photon statistics.

Figure 2

Second-order correlation functions against the quality factors. The phonotonic Josephson junction works for quality factors down to $Q_{r,c}\sim {10}^3$, where the damping rates are comparable to the coupling strength.

Figure 3

Spectroscopy of the energy spectrum. Steady-state phonon number (solid line) and photon number (dashed line) as a function of the driving frequency. The different peaks correspond to the excitation of the states with one phonon $|1,n_c\rangle$. (Inset) Second-order correlation functions at coinciding times for the phonons $g_a^{\left(2\right)}\left(0\right)$ and for the photons $g_b^{\left(2\right)}\left(0\right)$. Antibunching occurs at ${\omega }_d={\omega }_0\pm g$ where phonon and photon blockade take place. Photon bunching occurs when the states with many photons are excited.

Figure 4

Steady-state phonon and photon numbers (a) and second-order correlation functions (b) as a function of the Kerr nonlinearity. The Bell state $|{\Psi }^{-}\rangle$ ensures a perfect match between the phonon and photon numbers. Many excitations are generated when the nonlinearity is comparable with the damping rates of the resonators, where the excitations are not antibunched anymore. The correlators are plotted for different values of the coupling, from top to bottom, $g/2\pi =1,2,5,20\mathrm{MHz}$, respectively. The correlation of the NMR is essentially coupling independent while for large $\eta$, the correlation of the SMR saturates to a constant value ${(2\epsilon /g)}^2$. The corresponding solutions [Eq. (9)] in the limit $\eta \gg g$ are plotted in dotted lines.

Figure 5

(a) Population distribution of the mechanical oscillator after being driven to a state with ten phonons (points). The statistics is compared to a coherent state with the same mean value (line). (b) Average value of the imbalance $z$ as a function of the parameter $\Lambda$ from the classical dynamics without dissipation, starting from the initial state $z\left(0\right)=1$. The symmetric junction corresponds to interacting photons with a strength $\eta$. (c) Time evolution of the population imbalance in the self-trapping regime ($\eta /2\pi =10\mathrm{MHz}$ and $g/2\pi =1\mathrm{MHz}$), the oscillating regime ($\eta /2\pi =1\mathrm{MHz}$ and $g/2\pi =10\mathrm{MHz}$). The dynamical transition is observed with a symmetric junction, where the photons also interact with a strength $\eta /2\pi =10\mathrm{MHz}$. The coupling is equal to $g/2\pi =10\mathrm{MHz}$ and the quality factors are $Q_{r,c}={10}^4$.