Method for observing robust and tunable phonon blockade in a nanomechanical resonator coupled to a charge qubit

Phonon blockade is a purely quantum phenomenon, analogous to Coulomb and photon blockades, in which a single phonon in an anharmonic mechanical resonator can impede the excitation of a second phonon. We propose an experimental method to realize phonon blockade in a driven harmonic nanomechanical resonator coupled to a qubit, where the coupling is proportional to the second-order nonlinear susceptibility ${\chi }^{\left(2\right)}$. This is in contrast to the standard realizations of phonon and photon blockade effects in Kerr-type ${\chi }^{\left(3\right)}$ nonlinear systems. The nonlinear coupling strength can be adjusted conveniently by changing the coherent drive field. As an example, we apply this model to predict and describe phonon blockade in a nanomechanical resonator coupled to a Cooper-pair box (i.e., a charge qubit) with a linear longitudinal coupling. By obtaining the solutions of the steady state for this composite system, we give the conditions for observing strong antibunching and sub-Poissonian phonon-number statistics in this induced second-order nonlinear system. Besides using the qubit to produce phonon blockade states, the qubit itself can also be employed to detect blockade effects by measuring its states. Numerical simulations indicate that the robustness of the phonon blockade, and the sensitivity of detecting it, will benefit from this strong induced nonlinear coupling.

Figure 1

Schematic diagram of the coupled system of a NAMR and a Cooper-pair box working as a charge qubit. The two Josephson junctions (red rectangles) with tunneling energy $E_J$ and capacitance $C_J$ form a SQUID loop. The perpendicular static magnetic field $B_0$, together with the current $I\left(t\right)$ through the NAMR, produces the Lorentz force to drive the NAMR. Here a gate voltage $V_g$ and a static voltage $V_0$ are applied to the capacitances $C_g$ and $C_0$, respectively. The microwave line is located right above the charge qubit. The time-dependent current $I_{\text{p}}\left(t\right)$ and static current $I_{\text{s}}$ in the microwave line induce a magnetic flux ${\mathrm{\Phi }}_x$ through the SQUID loop.

Figure 2

Schematic energy-level diagrams [in (a), (b), and (c)] explaining the occurrence of phonon blockade in our approach. The red and black arrows represent the drivings for the NAMR and qubit with strengths $\epsilon$ and ${\mathrm{\Omega }}_{\text{p}}$, respectively. (a) The energy shift ${\mathrm{\Delta }}_s=\tilde{\mathrm{\Delta }}-\mathrm{\Delta }$ (blue two-direction arrows) of the energy gap for the qubit is due to the side-band driving. (b) The induced nonlinear strength $\lambda$ leads to the transfer between states $|e,0\rangle \leftrightarrow |g,2\rangle$ ($|e,1\rangle \leftrightarrow |g,3\rangle$) with rate $\sqrt{2}\lambda$ ($\sqrt{6}\lambda$). (c) Due to the energy level splitting between the dressed states $\left(\right|e,n\rangle \pm |g,n+2\rangle )/\sqrt{2}$, there is no corresponding transition level for a multiphonon being excited. (d) The time evolution of the probabilities $P_{2g}$ (black curve) and $P_{0e}$ (red curve) demonstrate the Rabi oscillations between the states $|g,2\rangle$ and $|e,0\rangle$ without considering any decay channels.

Figure 3

(a) Probabilities $P_n$ of measuring $n$ phonons and the average phonon number $\langle n\rangle$ as a function of the evolution time. (b) Phonon antibunching is revealed by the second-order correlation function versus the rescaled delay time $\kappa \tau .$ Other parameters are: $\mathrm{\Gamma }/\left(2\pi \right)=1$ MHz, ${\mathrm{\Gamma }}_f=0, \epsilon /\left(2\pi \right)=0.2$ MHz, ${\mathrm{\Delta }}_d=0$, and $n_{\text{th}}=0.$

Figure 4

The fidelity $F$, given by Eq. (34), of state truncation vs (a) the drive strength $\epsilon$ and (b) the effective Rabi frequency ${\mathrm{\Omega }}_{\text{p}}$ of the drive field. The other parameters used here are the same as those in Fig. 3.

Figure 5

(a) Mean phonon number $\langle n\rangle$ and (b) zero-delay time second-order correlation function $g_2\left(0\right)$ as functions of the frequency detuning ${\mathrm{\Delta }}_d/\left(2\pi \right)$ of the driving force.

Figure 6

(a) Dependence of the sub-Poissonian dip $g_2\left(0\right)$ on the decay rate of the qubit, for different values of the driving strength ${\mathrm{\Omega }}_{\text{p}}/\left(2\pi \right)=50$ MHz (red dashed curve), ${\mathrm{\Omega }}_{\text{p}}/\left(2\pi \right)=100$ MHz (black dot curve), and ${\mathrm{\Omega }}_{\text{p}}/\left(2\pi \right)=200$ MHz (blue solid curve)$.$ (b) $g_2\left(0\right)$ vs thermal phonon number $n_{\text{th}}$ for different values of the quality factor $Q=5\times {10}^3$ (blue dashed curve) and $Q=5\times {10}^4$ (black solid curve)$.$ The other parameters used here are the same as those in Fig. 3.

Figure 7

Steady-state second-order correlation function $g_2$ of the mechanical mode vs the rescaled delay time $\kappa \tau ,$ assuming various values of the pure-dephasing rate ${\mathrm{\Gamma }}_f=0$ (black solid curve), ${\mathrm{\Gamma }}_f=0.2\mathrm{\Gamma }$ (red dot curve), and ${\mathrm{\Gamma }}_f=2\mathrm{\Gamma }$(blue dashed curve), where $\mathrm{\Gamma }$ is the qubit decay rate and $\kappa$ is the decay rate of the NAMR. Here the other parameters are the same as those in Fig. 3.

Figure 8

(a) Probabilities $P_e\simeq {\left|C_{0e}\right|}^2$ and $P_2\simeq {\left|C_{2g}\right|}^2$, as defined in Eq. (21), as a function the zero-delay-time second-order correlation function $g_2\left(0\right)$ under the driving strength ${\mathrm{\Omega }}_{\text{p}}/\left(2\pi \right)=100$ MHz. (b) Ratio $R=P_e/P_2$ vs $P_2.$ Here we set ${\mathrm{\Omega }}_{\text{p}}/\left(2\pi \right)$ equal to 100 MHz (blue solid curve) and 200 MHz (red dashed curve). The other parameters used here are the same as those in Fig. 3.