Generating macroscopic superposition states in nanomechanical graphene resonators

A theoretical study of the quantum dynamics of a symmetric nanomechanical graphene resonator with degenerate flexural modes is presented. Applying voltage pulses to two back gates, flexural vibrations of the membrane can be selectively actuated and manipulated. For graphene, nonlinear response becomes important for amplitudes comparable to the magnitude of zero-point fluctuations. We show, using analytical and numerical methods, that this allows for creation of catlike superpositions of coherent states as well as superpositions of coherent catlike nonproduct states.

Figure 1

Cross section of a square graphene membrane resonator. A fully clamped graphene membrane with side $L$ is suspended a distance $u_0$ above the substrate. Below, covering two adjacent quadrants beneath the membrane, are local backgates with time-dependent voltage biases $V_{1,2}\left(t\right)$. By applying pulses to the local gates, catlike superpositions of flexural mode states, as well as superpositions of coherent catlike nonproduct states, can be generated.

Figure 2

(a) False color plots of snapshots of time evolution of the position probability distribution for one flexural mode. The snapshots are taken at the turning points of the corresponding classical trajectory of the system. The positions (y axis) are scaled to the quantum zero point fluctuations $x_0$. (b) Corresponding time evolution of the envelopes of the quadrature $\langle X\rangle$ (continous line) and its associated quantum fluctuations $\langle \Delta X^2\rangle$; $A\left(t\right)$ (dashed line), $B\left(t\right)$ (dash-dotted line). The appearance of a cat state is signalled by a decrease in $\langle X\rangle$ along with an increased contribtion from quantum fluctuations $\langle \Delta X^2\rangle$ in the noise of $\langle X\rangle$. Vertical dashed lines indicate the agreement with (a).

Figure 3

(a) False color plot of reduced Wigner distribution of the time evolved initial state ${|0,0\rangle }_a$, sampled at ${\tilde{t}}_1$ as function of the dimensionless position ${\xi }_1$ and momentum ${\pi }_1$. The Wigner distribution is obtained by numerically integrating the Hamiltonian Eq. (12). The reduced Wigner distributions are identical for both modes $W({\xi }_1,{\pi }_1)=W({\xi }_2,{\pi }_2)$. A bimodal structure is seen. (b) False color plot of Wigner distribution of ${\widehat{P}}_0{\widehat{\rho }}_1({\xi }_1,{\pi }_1,{\tilde{t}}_1){\widehat{P}}_0$, clearly demonstrating bimodality. (c) False color plot of Wigner distribution of the analytic approximation ${|0,0\rangle }_a+i{|-\sqrt{2}{\eta }_0,-\sqrt{2}{\eta }_0\rangle }_a$. The similarity to the projection in (b) is evident. (d) Time evolution of projections $\langle {\widehat{P}}_0\left(t\right)\rangle$, $\langle {\widehat{P}}_2\left(t\right)\rangle$ and $\langle \widehat{I}-{\widehat{P}}_0\left(t\right)-{\widehat{P}}_2\left(t\right)\rangle$. The most significant contribution is seen to come from $\langle {\widehat{P}}_0\left(t\right)\rangle$.