Mechanical backreaction effect of the dynamical Casimir emission

We consider an optical cavity enclosed by a freely moving mirror attached to a spring and we study the quantum friction effect exerted by the dynamical Casimir emission on the mechanical motion of the mirror. Observable signatures of this simplest example of backreaction effect are studied in both the ring-down oscillations of the mirror motion and in its steady-state motion under a monochromatic force. Analytical expressions are found in simple yet relevant cases and compared to complete numerical solution of the master equation. In order to overcome the experimental difficulties posed by the weakness of the backreaction effect in current setups, a promising circuit-QED device allowing for the observation of an analog backreaction effect with state-of-the-art technology is proposed and theoretically characterized.

Figure 1

Illustrative representation of a generic optomechanical system. One of the cavity mirrors is allowed to harmonically oscillate around the position of equilibrium and is optomechanically coupled to the cavity mode via the radiation pressure.

Figure 2

Free evolution of the interacting cavity field-mirror system. At the initial time $t=0$, the cavity field is in its vacuum state, while the mechanical oscillator is in a coherent state of amplitude $b_0=2$. Solutions are given in terms of the number of photons in the cavity $\langle {\widehat{a}}^{†}\widehat{a}\rangle$ and of mechanical oscillator quanta $\langle {\widehat{b}}^{†}\widehat{b}\rangle$. The different panels (a)–(d) are for growing values of the ratio ${\omega }_c/{\gamma }_0=0.1,0.5,5,10$. In each panel, the solution (L) of the linearized equations is plotted as a blue fine dashed line, the solution (NL) of the nonlinear mean-field equations is shown as a dashed red line, and the full numerical solution (ME) of the master equation Eq. (7) is shown as a continuous (black) line. In the panels for $\langle {\widehat{b}}^{†}\widehat{b}\rangle$, the dotted green lines [not distinguishable in (a)] show the evolution of the mirror in the absence of optomechanical coupling to the cavity field, that is for ${\omega }_c=0$.

Figure 3

Stationary state in the presence of a monochromatic mechanical drive acting on the mirror, in terms of the number of photons in the cavity as a function of the drive frequency $\omega$. All curves are for the resonant case ${\omega }_b=2{\omega }_a$, while different values ${\lambda }^{\prime }=F_0/{\left(F_0^{\text{th}}\right)}^{\prime }=1,3,5$ for the drive amplitude are used. The presence of multiple solutions for the same drive frequency and amplitude indicate multistable behaviors, the dynamically unstable branches being highlighted by symbols.

Figure 4

Stationary state of the driven-dissipative evolution in the presence of a monochromatic drive acting on the mirror in a fully resonant condition $\omega ={\omega }_b=2{\omega }_a$. (a) The amplitude of the mirror oscillations as a function of the drive amplitude. The solid line is the mean-field solution of (10) and (11), squares show the analytical nonlinear expression (40), and the circles indicate the full numerical prediction of the master equation. Different colors refer to different values of the optomechanical coupling ${\omega }_c/{\gamma }_0=0.5$ (red) and 2 (black). (b) The relative deviation between the analytical nonlinear solutions and the full numerical solution. (c) and (d) The numerical solutions for the normalized correlations between the field and the mirror, as defined in Eq. (44). Vertical dashed lines indicate the points of maximum deviation $\mathrm{\Delta }b/〈b〉$.

Figure 5

Steady-state amplitude of the mirror oscillation as a function of the frequency $\omega$ of the monochromatic drive. We consider the mechanical oscillator resonant with the cavity field: ${\omega }_b=2{\omega }_a$, and use different values for the optomechanical coupling and of the drive amplitude. (a)–(c) ${\omega }_c/{\gamma }_0=0.5$ and $F_0/{\left(F_0^{\text{th}}\right)}^{\prime }=0.5$ (a), $F_0/{\left(F_0^{\text{th}}\right)}^{\prime }=2$ (b), and $F_0/{\left(F_0^{\text{th}}\right)}^{\prime }=5$ (c). (d) ${\omega }_c/{\gamma }_0=2$ and $F_0/{\left(F_0^{\text{th}}\right)}^{\prime }=3$. The black symbols indicate the numerical solution of the master equation (7). The prediction of the linearized theory is shown as a solid green line. The prediction of the nonlinear mean-field model is shown as a dashed red line. The response of the bare mirror for a vanishing optomechanical coupling is shown as a dotted blue line.

Figure 6

(a) Pictorial representation of the LC resonator magnetically coupled to the coplanar waveguide. (b) Sketch of the equivalent circuit. (c) Effective cavity with moving mirror in correspondence of the SQUID.

Figure 7

Discrete model of the CPW magnetically coupled with the LC circuit.