Generation of squeezed states of nanomechanical resonators by reservoir engineering

An experimental demonstration of a nonclassical state of a nanomechanical resonator is still an outstanding task. In this paper we show how the resonator can be cooled and driven into a squeezed state by a bichromatic microwave coupling to a charge qubit. The stationary resonator state exhibits a reduced noise in one of the quadrature components by a factor of 0.5–0.2. These values are obtained for a 100 MHz resonator with a $Q$-value of ${10}^4$ to ${10}^5$ and for support temperatures of $T\approx 25\mathrm{mK}$. We show that the coupling to the charge qubit can also be used to detect the squeezed state via measurements of the excited state population. Furthermore, by extending this measurement procedure a complete quantum state tomography of the resonator state can be performed. This provides a universal tool to detect a large variety of different states and to prove the quantum nature of nanomechanical systems.

Figure 1

Representation of the squeezed state $\mid ϵ=r{e}^{i\theta }⟩$ in phase space spanned by the dimensionless position and momentum coordinates. The error ellipse indicates the reduced/enhanced fluctuations for the quadrature components $X_1∕X_2$.

Figure 2

Coherent and incoherent processes as described by master equation (6). The excitations of the qubit coherently decrease ($g_1$, “red sideband”) and increase ($g_2$, “blue sideband”) the phonon number of the resonator. The energy of the excite state is then dissipated with the decay rate of the TLS, $\Gamma$. These two processes relax the system into the desired state, $\rho =\mid ϵ,g⟩⟨ϵ,g\mid$ while the coupling to the thermal phonon bath, $\gamma$ thermalizes the resonator state.

Figure 3

Setup, the mechanical resonator (thin, vertical bar) is biased by the voltage $V_x$ to achieve a coupling to the Cooper pair box (CPB) via the capacitance $C_x$. The state of the CPB can be controlled independently by the gate voltage, $V_g$ and the Josephson energy, $E_J$.

Figure 4

Coherent and incoherent processes in the squeezed frame as described by master equation (18). In this picture the qubit is only excited on the red sideband $\left(\tilde{g}\right)$. Followed by a spontaneous decay $\left(\Gamma \right)$ these transitions cool the resonator to its ground state. The cooling is compensated by the squeezed heat bath which apart from the enhanced heating rate $\left(\gamma \tilde{N}\right)$ also induces coherences between the states $\mid n⟩$ and $\mid n\pm 2⟩\left(\gamma M\right)$.

Figure 5

Minimum values for the noise reduction, ${\mathcal{R}}_{\min }$ for various parameter values, $g_1,\Gamma ,\gamma$ (all in MHz) and $N_B=5$. (a) shows the dependence of ${\mathcal{R}}_{\min }$ on the driving strength, $g_1$ for a fixed $\Gamma =2$, while (b) shows the dependence on the decay rate of the charge qubit, $\Gamma$ for fixed $g_2=5$. The different values for $\gamma$ in (a) and (b) are 0.001 (solid), 0.005 (dashed), 0.01 (dotted) 0.02 (dashed-dotted). In (c) ${\mathcal{R}}_{\min }$ is plotted as a function of the phonon damping rate, $\gamma$ for $g_1=5$ and $\Gamma =2$ (solid), $\Gamma =5$ (dashed), $\Gamma =8$ (dotted).

Figure 6

Correspondence between the excited state population, $p_e$ (solid line) and the degree of squeezing, $\mathcal{R}$ (dashed line) as a function of the detuning, ${\delta }_x$. The values are plotted for the parameter values $\gamma =0.01\mathrm{MHz},r=0.7$, and $N_B=5$ for (a) $g_1=5\mathrm{MHz},\Gamma =2\mathrm{MHz}$ (strong coupling) and (b) $g_1=1.5\mathrm{MHz},\Gamma =5\mathrm{MHz}$ (weak coupling).

Figure 7

Measurement of the resonator occupation numbers, $Q_n$. A squeezed state is generated as explained in the preceding sections with (all in MHz) $g_1=5,g_2=3.5(r=0.88),\Gamma =2,\gamma =0.01$, and $N_B=5\mathrm{MHz}$. For the detection we assume the same values but $g=g_1=8$ and $g_2=0$. (a) shows the exact time evolution of $⟨{\sigma }_z⟩\left(t\right)$ (solid) which is compared to the expected behavior (dashed) as given in Eq. (28). The difference for larger times is due to the deviations of the exact time dependence of $p_0(n,t)$ from Eq. (29) for $\gamma >0$. In (b) the extracted occupation probabilities (dark gray) are compared with the exact values (light gray). The inset shows the estimated number distribution for a ideal squeezed state (top) and a thermal state (bottom) with the same mean occupation number, $⟨n⟩=1$.

Figure 8

Coefficients, $c_n$ which determine the error of the estimated occupation numbers according to Eq. (39). For a large number of measured data points, $N_D$ the $c_n$ are proportional to $1∕\sqrt{N_D}$. See text for details.

Figure 9

Results of the numerical simulation of the state tomography for a squeezed state as described in Sec. 6. (a) shows the reconstructed Wigner function, $\tilde{W}\left(\alpha \right)$. In (b) a cut of $\tilde{W}\left(\alpha \right)$ (solid line) in the $x$ direction $[\mathrm{Im}\left(\alpha \right)=0]$ is compared with the corresponding values of the real Wigner function, $W\left(\alpha \right)$ (dashed line) and the Wigner function of the resonator ground state (dotted line).