Single-atom cavity QED and optomicromechanics

In a recent publication [K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, Phys. Rev. Lett. 103, 063005 (2009)] we have shown the possibility to achieve strong coupling of the quantized motion of a micron-sized mechanical system to the motion of a single trapped atom. In the proposed setup the coherent coupling between a SiN membrane and a single atom is mediated by the field of a high finesse cavity and can be much larger than the relevant decoherence rates. This makes the well-developed tools of cavity quantum electrodynamics with single atoms available in the realm of cavity optomechanics. In this article we elaborate on this scheme and provide detailed derivations and technical comments. Moreover, we give numerical as well as analytical results for a number of possible applications for transfer of squeezed or Fock states from atom to membrane as well as entanglement generation, taking full account of dissipation. In the limit of strong-coupling the preparation and verification of nonclassical states of a mesoscopic mechanical system is within reach.

Figure 1

Dynamic intracavity field provides strong interface between the motion of a single trapped atom and the vibrations of a micron-sized membrane.

Figure 2

Linear atom-membrane coupling mediated by two driven cavity modes. (a) One mode is driven on the red side, the other on the blue side. When the mode frequencies shift due to the membrane vibration (dashed line), the cavity response is reduced for one mode and enhanced for the other. (b) Atom and membrane in equilibrium inside the driven cavity. (c) When the membrane vibrates around its equilibrium, the oppositely changing cavity response for the respective modes shifts the equilibrium of the combined atom potential.

Figure 3

Two lasers with frequencies ${\omega }_1$ and ${\omega }_2$ respectively drive two different internal atomic transitions with detunings $\delta$.

Figure 4

Fidelity for transfer of coherent state $|\beta \rangle$ from atom to membrane. (a and b) Fidelity as function of time for transfer of a state with (a) $\beta =1$ and (b) $\beta =5$ for various values of the dissipation ratio $f$, with fixed $G/{\omega }_{\mathrm{m}}=0.034$. Here $f=0.01$ (black solid line), $f=0.05$ (orange dotted line), and $f=0.10$ (blue dashed line). The little wiggles are due to counter-rotating terms. (c) Snapshot at $t=\pi /2G$: fidelity for transfer of state with $\beta =1$ as a function of the dissipation ratio $f$.

Figure 5

Transfer of squeezed atom state with initial variance $\Delta X_{\mathrm{at}}^2=e^{-2}/2$ to the membrane. (a) Snapshot of Wigner functions (upper row, atom; lower row, membrane) for $f=0.05$ at $t=0$, $t=\pi /2G$, and $t=\pi /G$. (b) Minimum membrane variance as function of time for different dissipation ratios $f=0$ (black solid line), $f=0.05$ (orange dotted line), and $f=0.10$ (blue dashed line). (c) Transferred squeezing (in dB; $S(t)=10\log {}_{10}[s(t)](\mathrm{dB})$) as function of the dissipation ratio $f$ for fixed $G/{\omega }_{\mathrm{m}}=0.034$. The initial atom squeezing is given by $s(0)=e^{-2}$ (purple dashed line) corresponding to $-8.7$ dB, and $s(0)=e^{-1}$ (green solid line) corresponding to $-4.3$ dB, respectively.

Figure 6

Transfer of the Fock state $|n=1\rangle$ from atom to ground-state prepared membrane. Snapshot cuts through the Wigner functions at times $t=0$, $t=\pi /2G$, and $t=\pi /G$, for fixed dissipation ratio $f=0.05$ and $G/{\omega }_{\mathrm{m}}=0.034$.

Figure 7

Analytical result for the relative membrane Wigner function negativity $N_{\mathrm{w}}$ at time $t=t_{\mathrm{s}}$ as a function of the dissipation ratio $f$.

Figure 8

Generation of entanglement between atom and membrane by modulating the coupling strength. (a) Logarithmic negativity as a function of time for different values of the dissipation ratio $f=0.1$ (blue curve), $f=0.05$ (red curve), and $f=0.01$ (black curve). (b) Corresponding increase of the atomic excitation number $n_{\mathrm{at}}=\langle a_{\mathrm{at}}^{†}{a}_{\mathrm{at}}\rangle$. For these plots we chose $G/{\omega }_{\mathrm{m}}=0.034$ and ${\omega }_{\mathrm{at}}/{\omega }_{\mathrm{m}}=1.1$ and assume the membrane to be in the ground state initially.

Figure 9

(a) Spatial dependence $u(x)$ of the AC Stark potential along the cavity axis for a cavity length $L\simeq 53$ $\mu$m. The two driven modes are at ${\lambda }_1\simeq 852$ nm and ${\lambda }_2\simeq 888$ nm. Their separation is $q=5$ FSRs. Both the parameter $\theta ({\mathrm{x̄}}_{\mathrm{at}})$ (b) entering the atom-cavity coupling $g_{\mathrm{at}}$ and the parameter $\xi ({\mathrm{x̄}}_{\mathrm{at}})$ (c) entering the atomic dephasing rate ${\Gamma }_{\mathrm{at}}$ can be kept close to 1 at potential wells around the points where $\delta kx=n\pi$ with $\delta k=k_1-k_2$ and $n⩽q$. In (d) the parameter $\zeta (x)$ is shown, which can be well around 50% while $\theta ,\xi \simeq 1$. The resulting loss in trap frequency can be easily compensated for by an increased intracavity amplitude.

Figure 10

Finite elements simulation of membrane heating due to laser absorption. The steady-state temperature distribution $T(y,z)$ in the membrane is shown for the experimental parameters considered in this article. The laser spot of radius $w_0=10$ $\mu$m is indicated by the black circle. The membrane frame is held at fixed $T_0=2$ K.