Dynamical tunneling of a nanomechanical oscillator

The study of the quantum to classical transition is of fundamental as well as technological importance, and focuses on mesoscopic devices, with a size for which either classical physics or quantum physics can be brought to dominate. A particularly diverse selection of such devices is available in cavity quantum optomechanics. We show that these can be leveraged for the study of dynamical tunneling in a quantum chaotic system. This effect probes the quantum to classical transition deeply, since tunneling rates sensitively depend on the ability of the quantum system to resolve the underlying classical phase space. We show that the effective Planck's constant, which determines this phase space resolution, can be varied over orders of magnitude as a function of tunable parameters in an optomechanical experiment. Specifically, we consider a membrane-in-the-middle configuration of a mechanical oscillator within an optical cavity, where the intracavity field is modulated periodically by the external laser source. We demonstrate that a mixed regular and chaotic phase space can be engineered in one spatial dimension, through a significant quartic optomechanical interaction. For that case, we explore the expected dynamical tunneling rates using Floquet theory and map out values of the effective Planck's constant that should be within practical reach.

Figure 1

(a) Membrane-in-the-middle within a cavity as a quantum harmonic oscillator with tunable, light-driven anharmonicity of the potential $V(x,t)$. The typical modulation range of the oscillator potential is sketched below. A fairly large cavity decay-rate ${\gamma }_c$ makes sure cavity dynamics follows the external drive modulation. (b) Stroboscopic Poincaré section of this system with contours of the Husimi function of a Floquet state (magenta, cyan, orange) involved in dynamical tunneling for decreasing effective Planck's constant $h_{\text{eff}}$ as shown.

Figure 2

Varying phase space of the classical version of Hamiltonian (8) with the potential strength $\kappa$ and modulation amplitude $\epsilon$ chosen as (a) $\kappa =0.2, \epsilon =0.7$, (b) $\kappa =0.2, \epsilon =0.9$, (c) $\kappa =1.2, \epsilon =0.7$, and (d) $\kappa =1.2, \epsilon =0.9$. Labels ${\mathrm{\Delta }}_{\pm }$ mark period one islands of stability. All panels show a stroboscopic Poincaré section as discussed in the text.

Figure 3

Examples of the evolution of Floquet states (a) $n=2$ and (b) $n=8$ of the Hamiltonian (8) over one period $T=2\pi$ for $\kappa =1.2, \epsilon =0.9$, and ${\hslash }_{\text{eff}}=0.5$. The indices $n$ order Floquet states by increasing quasienergy. The color map represents position-space density ${\rho }_n(x,t)={\left|\langle x|{\mathrm{\Phi }}_n\left(t\right)\rangle \right|}^2$.

Figure 4

The support regions of the Husimi distribution of selected Floquet states (b)–(f) for $\kappa =1.2, \epsilon =0.9, {\hslash }_{\text{eff}}=0.5$ align themselves with features of the classical phase space shown in (a) [identical to Fig. 2]. (b)–(f) $\mathcal{Q}(x,p)$ from (17) as color shade. These Floquet states are associated with regular (b)–(d) and chaotic (e) and (f) regions of phase space. (d) A tunneling state defined in the next section.

Figure 5

The probability density in the tunneling state $|{\mathrm{\Phi }}_{+}\rangle$ at $t=0$ for $\kappa =1.2, \epsilon =0.9$, and (a) ${\hslash }_{\text{eff}}=0.5$, (c) ${\hslash }_{\text{eff}}=0.05$. (b) and (d) Demonstration of dynamical tunneling by considering the stroboscopic evolution of the initial states in (a) and (c) in position space. We show the probability density $\rho (x,sT)={\left|\langle x|{\mathrm{\Phi }}_{+}\left(sT\right)\rangle \right|}^2$ after an integer $s$ of modulation periods $T$, for the parameters as in the corresponding left panel.

Figure 6

Variations in dimensionless quasienergy splitting $\mathrm{\Delta }E=E_u-E_v$ with changed effective Planck's constant ${\hslash }_{\text{eff}}$ (blue dotted) for $\kappa =1.2$ and $\epsilon =0.9$ (left axis, blue). The red axis shows the minimal oscillator $Q$ factor required for the oscillator lifetime to exceed the dynamical tunneling period ${\tau }_m>T_{\text{tun}}$, see text.

Figure 7

(a) The Husimi distribution (17) of the initial tunneling state (blue) and two simpler approximations (red, magenta) for ${\hslash }_{\text{eff}}=0.5$, overlayed as FWHM contour on the classical phase space for the same parameters as in Fig. 2. The approximations correspond to the ground state in a modified initial harmonic potential as discussed in the text, with ${\kappa }_{\text{ini}}=1.2$ (red dashed) and ${\kappa }_{\text{ini}}=150.0$ (magenta dotted). The stroboscopic evolution from the approximate initial tunneling states is shown in (b) for ${\kappa }_{\text{ini}}=1.2$ and (c) for ${\kappa }_{\text{ini}}=150.0$.

Figure 8

The Husimi distribution of the odd tunneling state $|{\mathrm{\Phi }}_u\rangle$ at $t=0$ for $\kappa =1.2, \epsilon =0.9$, and (a) ${\hslash }_{\text{eff}}=0.5$, (b) ${\hslash }_{\text{eff}}=0.05$.

Figure 9

Tunability of the effective Planck's constant ${\hslash }_{\text{eff}}$. (a) As a function of the input laser power $P_0$ and oscillator frequency ${\omega }_m$, while cavity decay rate ${\gamma }_c$ and modulation frequency $\mathrm{\Omega }$ are kept at a fixed ratio with ${\omega }_m$, see (C3). Other parameters are held constant at $m=1\text{pg}, g^{\left(4\right)}/2\pi =1\text{kHz}{\text{nm}}^{-4}$, laser wavelength = 1064 nm. ${\hslash }_{\text{eff}}\in [0.1,0.001]$ on the blue line. (b) ${\hslash }_{\text{eff}}$ as a function of the laser field modulation frequency $\mathrm{\Omega }$ and oscillator frequency ${\omega }_m$, while ${\gamma }_c$ is kept at a fixed ratio with ${\omega }_m$, see (C1). We use $P_0=0.5\text{mW}$ and other fixed parameters as in (a).

Figure 10

Tunability of the effective Planck's constant ${\hslash }_{\text{eff}}$ as a function of (a) the input laser power $P_0$ and cavity decay-rate ${\gamma }_c$, and (b) the oscillator mass $m$ and frequency ${\omega }_m$. Constant parameters in (a) $m=1$ pg, ${\omega }_m/2\pi =10\text{kHz}, g^{\left(4\right)}/2\pi =1\text{kHz}{\text{nm}}^{-4}$, laser wavelength = 1064 nm, see (C2). Constant parameters in (b) $P_0=0.5\text{mW}, g^{\left(4\right)}/2\pi =1\text{kHz}{\text{nm}}^{-4}$, laser wavelength = 1064 nm, while modulation frequency $\mathrm{\Omega }$ and ${\gamma }_c$ are kept at a fix ratio with ${\omega }_m$, see (C3).