Coherent control of atomic tunneling

TL;DR

This paper investigates how coupling a two-level atom to a cavity mode influences its tunneling in a double well, revealing quasiperiodic behavior and collapse-revival phenomena, enabling coherent control of atomic position.

Contribution

It introduces a novel analysis of atom tunneling under cavity coupling, showing how internal states affect external motion and enabling coherent manipulation of atomic localization.

Findings

Tunneling becomes quasiperiodic due to cavity coupling.

Collapse and revival of tunneling observed under certain conditions.

Coherent control of atom position via laser access to internal states.

Abstract

We study the tunneling of a two-level atom in a double well potential while the atom is coupled to a single electromagnetic field mode of a cavity. The coupling between internal and external degrees of freedom, due to the mechanical effect on the atom from photon emission into the cavity mode, can dramatically change the tunneling behavior. We predict that in general the tunneling process becomes quasiperiodic. In a certain regime of parameters a collapse and revival of the tunneling occurs. Accessing the internal degrees of freedom of the atom with a laser allows to coherently manipulate the atom position, and in particular to prepare the atom in one of the two wells. The effects described should be observable with atoms in an optical double well trap.

Figures

Equations

$$H=H_A+H_F+H_{AF},$$

$$H_A^{\mathrm{ex}}=\frac{p_x^2}{2m}+V(x),$$

$$H_A^{\mathrm{in}}=\frac{\hslash {\omega }_0}{2}{\sigma }_z^{\mathrm{in}},$$

$$H_F=\hslash \omega a^{†}a,$$

$$H_{AF}=-d.E,$$

$$\mathbf{E}=E_{\omega }\mathbf{\epsilon }\left(a+a^{†}\right)\sin (k(x-x_0))$$

$$H_A^{\mathrm{ex}}=\frac{\hslash \Delta }{2}{\sigma }_z^{\mathrm{ex}}$$

$$|L⟩=(|+⟩-|-⟩)/\sqrt{2},$$

$$|R⟩=(|+⟩+|-⟩)/\sqrt{2}.$$

$$H_{AF}=-\hbar g(a+a^{\dagger})\Big{[}\sin\chi\cos\kappa\;\sigma_{x}^{\mathrm{in}}-\cos\chi\sin\kappa\;\sigma_{x}^{\mathrm{ex}}\sigma_{x}^{\mathrm{in}}\Big{]}$$

$$\chi =k{x}_0,\kappa =kb/2.$$

$$H$$

$${\lambda }_{\rho \mu }=(N-1/2)\hslash \omega +\rho \frac{\hslash {\Omega }_{\mu }}{2},$$

$${\Omega }_{\pm }$$

$${\Omega }^2$$

$$\displaystyle U_{11}=-\frac{i}{2\Lambda}\sum_{\mu=\pm}\Big{[}$$

$$\displaystyle-i\mu\Omega_{+}\Omega_{-}C_{\mu}(\delta\Delta-\mu\Omega^{2})\Big{]},$$

$$\displaystyle U_{12}=\frac{-i\sqrt{N}g\cos\kappa}{\sqrt{2}\Lambda}\sum_{\mu=\pm}\Big{[}$$

$$\displaystyle+\mu\Omega^{2})+i\mu\Omega_{+}\Omega_{-}\Delta C_{\mu}\Big{]},$$

$$U_{13}=$$

$$U_{23}=$$

$$\xi =\Delta ({\delta }^2+2N{g}^2{\cos }^2\kappa -\delta \Delta ),$$

$$\Lambda ={\Omega }_{+}{\Omega }_{-}{\Omega }^2,$$

$$C_{\pm }=\cos ({\Omega }_{\pm }t/2),S_{\pm }=\sin ({\Omega }_{\pm }t/2).$$

$$⟨x⟩=\frac{b}{2}{\mathrm{Tr}}_{\mathrm{ex}}({\rho }^{\mathrm{ex}}{\sigma }_x^{\mathrm{ex}})=\frac{b}{2}(1-2{\rho }_{LL})$$

$${\rho }_{LL}=\frac{{\Delta }^2}{{\Delta }^2+N{g}^2}{\sin }^2\left(\frac{{\Omega }_{\mathrm{tun}}t}{2}\right)$$

$${\Omega }_{\mathrm{tun}}=\frac{1}{2}\left({\Omega }_{+}+{\Omega }_{-}\right),$$

$${\rho }_{ee}=\frac{1}{2}+\frac{{\sum }_{\mu =\pm }\left({\Omega }_{\mu }^2-{\Delta }^2\right)\cos ({\Omega }_{\mu }t)+4{\Delta }^2\cos \left(\frac{{\Omega }_{+}-{\Omega }_{-}}{2}t\right)}{8(N{g}^2+{\Delta }^2)}.$$

$$t_c\sim \frac{1}{g}\left(1+\frac{(\Delta /g{)}^2+3/4}{2⟨n⟩}\right)+\mathcal{O}(⟨n{⟩}^{-2})$$

$$t_r\simeq \frac{4\pi \sqrt{⟨n⟩}}{g}\left(1+\frac{(\Delta /g{)}^2+1/2}{2⟨n⟩}+\mathcal{O}(⟨n{⟩}^{-2})\right)$$

$$U_{13}=$$

$$U_{33}=$$

$$\bar{\Omega }=\frac{\sqrt{{\delta }^2{\Delta }^2+N^2{g}^4{\sin }^2(2\kappa )}}{\delta },$$

$${\rho }_{LL}=\frac{1}{2}-\frac{N\delta \Delta \sin (2\kappa )}{2{\bar{\Omega }}^2(\delta /g{)}^2}\left[1-\cos \left(\bar{\Omega }t\right)\right],$$

$$\rho_{LL}=\frac{1}{2}\mp\frac{1}{4}\left[1-\cos\big{(}\sqrt{2}\Delta t\big{)}\right].$$

Full text

I Introduction

The tunneling effect is considered one of the hallmarks of quantum mechanical behavior. Historically, tunneling was first examined for single particles (e.g. $\alpha$ particles Gamow (1928), electrons in field emission Guth and Mullin (1942) and later in mesoscopic circuits Devoret et al. (1990)), for Cooper pairs Josephson (1974), and for molecular groups Hueller (1980); Würger (1989); Braun and Weiss (1994). Recently the tunneling of atoms has attracted substantial attention Louis and Sethna (1995); Meier and Zwerger (2001); Luxat and Griffin (2002); Albiez et al. (2005). Dynamical (chaos assisted) tunneling of ultracold atoms between different islands of stability in phase space was analyzed in Grossmann et al. (1991); Averbukh et al. (2002) and has been observed experimentally Steck et al. (2001); Hensinger et al. (2001). Resonantly enhanced tunneling of atoms between wells of a tilted optical lattice has also been observed very recently Sias et al. (2007). In all of these examples, the atoms have been considered internally as inert, and only the center of mass coordinate of the atom was of interest. In Haycock et al. (2000) it was shown that by taking into account the internal degrees of freedom of atoms, an atom/optical double well potential could be created in which tunneling atoms see their internal and external states correlated (such an effect is also known from other contexts Salzburger and Ritsch (2004)). Mechanical effects of light in optical resonators were also investigated in Domokos and Ritsch (2003), but no tunneling was considered.

Here we show that the tunneling effect can be drastically modified if an internal transition of the atom is coupled to a single electromagnetic mode in a cavity, such that photon emission is a reversible and coherent process. The resulting Rabi oscillations between states with the excitation in the atom and states with a photon in the cavity modulate the periodic tunneling motion. Depending on the frequencies involved, a rich quasi-periodic behavior can result. If the cavity is fed with a coherent state, collapse and revival of the tunneling effect can occur. Moreover, we show that one may profit from access to the internal degrees of freedom of the atom (e.g. with a laser) to control the atomic motion in the external potential.

II Model

II.1 Derivation of the Hamiltonian

Consider a trapped two-level atom (with levels $|g\rangle$, $|e\rangle$ of energy $\mp\hbar\omega_{0}/2$ respectively) interacting with a standing wave (with wave number $k$ and frequency $\omega$) inside a cavity as illustrated in Fig. 1. The atom is assumed to be bound in the $y-z$ plane at the equilibrium position $y=z=0$ and to experience a symmetric double well potential $V(x)$ along the $x$ direction. We denote by $\Delta$ the tunnel splitting, i.e. the energy spacing between the two lowest energy states (the symmetric $|-\rangle$ and antisymmetric $|+\rangle$ states) of this double well potential. Below we also allow the trapped atom to interact resonantly with an external laser. The Hamiltonian of this system is given by

[TABLE]

where $H_{A}=H_{A}^{\mathrm{ex}}+H_{A}^{\mathrm{in}}$ is the Hamiltonian of the trapped atom, $H_{F}$ is the Hamiltonian of the free field and $H_{AF}$ is the interaction Hamiltonian describing the atom-field interaction. We have

[TABLE]

where $\mathbf{d}$ denotes the atomic dipole,

[TABLE]

is the electric field operator, with $E_{\omega}=\sqrt{\frac{\hbar\omega}{\epsilon_{0}V}}$, where $\epsilon_{0}$ is the permittivity of free space, $V$ the electromagnetic mode volume, $x_{0}$ the abscissa at the left cavity mirror ($x_{0}<0$), and $\bm{\varepsilon}$ the electric field polarization vector. We have introduced the operators $\sigma_{i}^{\mathrm{in}}$ (resp. $\sigma_{i}^{\mathrm{ex}}$) for $i=x,y,z$ as the Pauli spin operators in the basis $\{|e\rangle,|g\rangle\}$ (resp. $\{|+\rangle,|-\rangle\}$). The operator $x$ stands for the center-of-mass position of the atom, $p_{x}$ is the conjugate momentum along the $x$ axis, $m$ denotes the atomic mass, and $a$ ($a^{\dagger}$) the annihilation (creation) operator of the cavity radiation field.

We adopt the two-level approximation which consists of taking into account only the two lowest motional energy states. This requires the Rabi frequency $\sqrt{4g^{2}+\delta^{2}}$ (with $\delta=\omega-\omega_{0}$ the detuning between the cavity field and the atomic transition frequencies) to be much smaller than the frequency gap $\tilde{\Delta}$ between the upper motional states and the ground state doublet (see Fig. 1). Within this approximation, Hamiltonian $H_{A}^{\mathrm{ex}}$ becomes

[TABLE]

and the position operator takes the form $x=\frac{b}{2}\sigma_{x}^{\mathrm{ex}}$ with $b/2=\langle+|x|-\rangle$. We can form states that are mainly concentrated in the left/right wells,

[TABLE]

The average position of a particle localized in the right well is then given by $b/2$ (see Fig. 1) and $\sigma_{x}^{\mathrm{ex}}=|R\rangle\langle R|-|L\rangle\langle L|$. The interaction Hamiltonian $H_{AF}$ can then be written

[TABLE]

with the atom-field coupling strength $g=-\langle e|\mathbf{d}|g\rangle\mathbf{.}\bm{\varepsilon}E_{\omega}/\hbar$, and

[TABLE]

For long wavelengths ($\kappa\ll 1$), or $\kappa=n\pi$ with integer $n$, the left and right sites of the double well are indistinguishable to the cavity photon and $H_{AF}$ reduces to Jaynes-Cummings Hamiltonian without rotating wave approximation (with a sine varying coupling constant), $-\hbar g\sin\chi\;(a+a^{\dagger})\sigma_{x}^{\mathrm{in}}$. Note that $\kappa\ll 1$ would normally be identified with the Lamb-Dicke regime. Here the situation is more subtle as the level spacings between the tunneling split ground state doublet and the next excited states can be very different such that the recoil energy $\hbar\omega_{\rm recoil}$ satisfies $\Delta\ll\omega_{\rm recoil}\ll\tilde{\Delta}$. One may thus be in the Lamb-Dicke regime concerning transitions to higher vibrational states but have a significant mechanical effect on the atomic tunneling. Furthermore, since there is only one photon mode, the recoil energy cannot vary continuously and exciting higher vibrational levels requires $\omega_{\rm recoil}$ close to a level spacing. Our numerical calculations show that even for $\kappa\sim 1$ the two-level approximation can still work very well (see Fig. 4).

For $\delta$, $\Delta\ll\omega$, $\omega_{0}$, a rotating wave approximation is justified, which consists in eliminating the energy non-conserving terms $a\sigma_{\pm}^{\mathrm{ex}}\sigma_{-}^{\mathrm{in}}$ and $a^{\dagger}\sigma_{\pm}^{\mathrm{ex}}\sigma_{+}^{\mathrm{in}}$ with $\sigma_{+}^{\mathrm{in}}=|e\rangle\langle g|$, $\sigma_{-}^{\mathrm{in}}=\sigma_{+}^{\mathrm{in}\,\dagger}$ and $\sigma_{+}^{\mathrm{ex}}=|+\rangle\langle-|$, $\sigma_{-}^{\mathrm{ex}}=\sigma_{+}^{\mathrm{ex}\,\dagger}$. Within this approximation, the total Hamiltonian reads

[TABLE]

Thus, depending on the parameters $\chi$ and $\kappa$, the cavity photon may induce internal transitions in the atom only ($\cos\chi\sin\kappa=0$), or induce transitions between internal and external states at the same time ($\cos\chi\sin\kappa\neq 0$) even for a vanishing detuning ($\delta=\omega-\omega_{0}=0$). This is in contrast to conventional sideband transitions of harmonically bound atoms or ions in the Lamb-Dicke regime which require an appropriate value of the detuning. For a fixed potential center (and thus fixed $\chi$), $\kappa$ can be changed through a modulation of the well-to-well separation $b$. We will neglect in the following the effects of decoherence, which means that not only $g$ but also $\Delta$ should be much larger than the rate of spontaneous emission $\Gamma$, and the cavity decay rate $\kappa_{\rm cav}$.

We denote the global state of the atom-field system by $|n,i,j\rangle\equiv|n\rangle\otimes|i\rangle\otimes|j\rangle$ where $|n\rangle$ stands for the cavity field eigenstates, $|i\rangle\in\{|-\rangle,|+\rangle\}$ for the external motional states, and $|j\rangle\in\{|g\rangle,|e\rangle\}$ for the internal states. The total excitation number $N$ is given by $a^{\dagger}a+\sigma_{+}^{\mathrm{in}}\sigma_{-}^{\mathrm{in}}$.

II.2 Energy levels

The states $|0,\pm,g\rangle$ are eigenstates of $H$ with eigenvalue $(-\hbar\omega_{0}\pm\hbar\Delta)/2$, i.e. these states remain uncoupled and represent the two lowest energy states in the regime $\delta$, $\Delta\ll\omega$, $\omega_{0}$. It is straightforward to verify that the Hamiltonian (II.1) only induces transitions between states with the same number of excitations $N$, $\{|N-1,+,e\rangle,|N,+,g\rangle,|N-1,-,e\rangle,|N,-,g\rangle\}\equiv\{|1\rangle,|2\rangle,|3\rangle,|4\rangle\}$. It is therefore sufficient to solve the dynamics in this subspace. In doing so, we obtain the eigenvalues of $H$,

[TABLE]

for $\rho,\mu\in\{\pm\}$, $N=1,2,\ldots$, and with

[TABLE]

For a vanishing tunnel splitting ($\Delta=0$), $\Omega_{\pm}$ reduces to the maximum (minimum) of the two Rabi frequencies of the Jaynes-Cummings models in the right and left wells. For $\cos\kappa=1$, the decoupling of external and internal degrees of freedom manifests itself also in the eigenvalues with $\Omega_{\pm}=|\sqrt{4Ng^{2}\sin^{2}\chi+\delta^{2}}\pm\Delta|$.

II.3 Evolution operator

The whole dynamics of the system can be described by means of the evolution operator $U(t)=e^{-iHt/\hbar}$ with components $U_{ij}=\langle i|U(t)|j\rangle=U_{ji}$, which can be calculated exactly. In order to simplify the expressions, we restrict ourselves in the following to $\chi=-\pi/4-2n\pi$ (integer $n$). We find, up to a an overall phase $e^{-i(N-1/2)\omega t}$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

with

[TABLE]

and where all time dependence is in the coefficients

[TABLE]

The remaining components can be deduced from the relations $U_{22}(\delta,\Delta)=U_{33}(-\delta,-\Delta)=U_{44}(\delta,-\Delta)=U_{11}(-\delta,\Delta)$, $U_{24}(\delta,\Delta)=U_{13}(-\delta,\Delta)$, $U_{14}(\delta,\Delta)=U_{23}(\delta,-\Delta)=U_{23}(-\delta,\Delta)$, and $U_{34}(\delta,\Delta)=U_{12}(\delta,-\Delta)$, valid for any $\chi$, where we have made explicit the dependence of the $U_{ij}$ on $\delta$ and $\Delta$.

III Internal and external dynamics

The reduced density matrix $\rho^{\mathrm{ex}}$ for the atomic center-of-mass motion alone follows from $\rho=|\psi(t)\rangle\langle\psi(t)|$ by tracing out the field and internal degrees of freedom, where the total wave function at time $t$ reads $|\psi(t)\rangle=\sum_{i,j=1}^{4}U_{ij}\langle j|\psi(0)\rangle\>|i\rangle$. The average position of the atom in the double well potential is then given by

[TABLE]

with $\rho_{LL}=\langle L|\rho^{\mathrm{ex}}|L\rangle$. Similarly, we obtain the reduced density matrix $\rho^{\mathrm{in}}$ for the internal atomic state by tracing out the field and external degrees of freedom, and the probability to find the atom in the excited state as $\rho_{ee}=\langle e|\rho^{\mathrm{in}}|e\rangle$.

In the following, we first focus on resonant atom-field interaction ($\omega=\omega_{0}$) before moving to the non-resonant case ($\omega\neq\omega_{0}$). We distinguish three regimes according to the tunnel splitting compared to the Rabi frequency $g$ : the small tunnel splitting regime (when $\Delta/g\ll 1$), the intermediate regime (when $\Delta/g\sim 1$), and the large tunnel splitting regime (when $\Delta/g\gg 1$).

III.1 Resonant atom-field interaction

For resonant atom-field interaction ($\delta=0$), the expressions for $U_{ij}$ can be greatly simplified. If the system is initially prepared in the state $|N-1,R,e\rangle$ and for $\kappa=\pi/4$, we have

[TABLE]

with the tunnel frequency

[TABLE]

and

[TABLE]

The atom position oscillates with a single frequency $\Omega_{\mathrm{tun}}$ given by Eq. (19), whereas $\rho_{ee}$ evolves with three in general incommensurable frequencies $\Omega_{+}$, $\Omega_{-}$, and $(\Omega_{+}-\Omega_{-})/2$ giving rise to a quasi-periodic signal.

For $\Delta/g\ll 1$, Eq. (18) leads to $\rho_{LL}\simeq 0$ (up to order $(\Delta/g)^{2}$), indicating that tunneling is suppressed. This is already obvious from (II.1), as the term responsible for tunneling, $(\hbar\Delta/2)\sigma_{z}^{\mathrm{ex}}=(\hbar\Delta/2)(|R\rangle\langle L|+|L\rangle\langle R|)$ becomes very small compared to the last term, diagonal in $|R\rangle,|L\rangle$ which leads to internal Rabi flopping. Note, however, that tunneling is suppressed on all time scales, even for $t\gg 1/\Delta$, due to the reduced amplitude in Eq. (18), very much in contrast to tunneling without internal degrees of freedom, where only the period of the tunneling motion, but not the amplitude is affected when $\Delta$ is reduced. For $\kappa$ approaching $\pi$, the situation changes because the term $g\cos\chi\sin\kappa\;\sigma_{x}^{\mathrm{ex}}$ of the interaction Hamiltonian inducing transitions between vibrational states becomes small in comparison with $\Delta$ thereby allowing tunneling again.

Because internal and external degrees of freedom are coupled, the tunneling frequency (Eq. (19)) depends on the number of photons inside the cavity. As an example, let us now consider $\Delta\sim g$ and a cavity field initially in a coherent state $|\alpha\rangle=e^{-\frac{1}{2}|\alpha|^{2}}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle$ with $|\alpha|^{2}$ equal to the mean photon number $\langle n\rangle$. Figure 2 shows that the average position of the atom in the double well as a function of time for a coherent state exhibits collapses and revivals. The oscillation amplitude decreases with increasing mean photon number $\langle n\rangle$ and decreasing tunnel splitting $\Delta$ (see Eqs. (18,9)). Since the probability to find the atom in the excited state oscillates with three frequencies, no collapses and revivals are observed for $\rho_{ee}$.

The collapse time $t_{c}$ of the tunneling motion can be estimated from the condition Scully and Zubairy (1997) $(\Omega_{\mathrm{tun}}(\langle n\rangle+\sqrt{\langle n\rangle})-\Omega_{\mathrm{tun}}(\langle n\rangle-\sqrt{\langle n\rangle}))\,t_{c}\sim 1$ with $\Omega_{\mathrm{tun}}(m)$ given by Eq. (19) for $N=m+1$, which yields, for $\langle n\rangle\gg 1$,

[TABLE]

The time interval between two following revivals, $t_{r}$, follows from $\left(\Omega_{\mathrm{tun}}(\langle n\rangle)-\Omega_{\mathrm{tun}}(\langle n\rangle-1)\right)\,t_{r}=2\pi$, and is given for $\langle n\rangle\gg 1$ by

[TABLE]

For the parameters of Fig. 2, Eq. (22) yields $gt_{r}\simeq 68.23$ for $\Delta/g=2$ and $gt_{r}\simeq 86.70$ for $\Delta/g=5$. Smaller revival times are possible for smaller values of $\langle n\rangle$, but in general the observation of revivals will be quite challenging, as they require $\Delta\sim g\gg\kappa_{\rm cav}$.

For large tunnel splitting, $\Delta/g\gg 1$, $\Omega_{\mathrm{tun}}=\Delta+Ng^{2}/(2\Delta)+\mathcal{O}((g/\Delta)^{3})$, and Eq. (18) reduces to $\rho_{LL}\simeq\sin^{2}(\Delta t/2)$, which is identical to the tunneling of a particle without internal structure. Equation (20) reduces to a Rabi oscillation $\rho_{ee}\simeq\cos^{2}(\sqrt{N}gt/2)$.

III.2 Non-resonant atom-field interaction

For non-resonant atom-field interaction ($\delta\neq 0$), and intermediate tunnel splitting [see Fig. 3 for $\Delta=\delta=g$], $\langle x(t)\rangle$ involves in general the two non-commensurate frequencies $\Omega_{+}$ and $\Omega_{-}$ and varies therefore quasiperiodically as a function of time. Figure 3 also shows that an atom initially located in one of the two wells remains mostly confined to that well.

For small tunnel splitting, $\Delta/g\ll 1$ and large detuning $|\delta|/g\gg 1$ (with $\Delta|\delta|/g^{2}\sim 1$), the matrix elements of $U$ simplify to

[TABLE]

up to corrections of order $\mathcal{O}(\Delta/g)$ and a phase factor $e^{i[(Ng^{2}/\delta+\delta)-(2N-1)\omega]t/2}$ while the components $U_{12}$ and $U_{23}$ are of order $\mathcal{O}(\Delta/g)$. In this situation, the system oscillates only between the two states $|N-1,+,e\rangle$ and $|N-1,-,e\rangle$ with a single frequency

[TABLE]

just as a three-level atom undergoing a Raman transition in the far detuned regime behaves as a two-level system.

If the system is initially in the state $|N-1,-,e\rangle$, we have from Eqs. (23)

[TABLE]

and $\rho_{ee}=1$. For a detuning $\delta=\pm Ng^{2}\sin(2\kappa)/\Delta$,

[TABLE]

This regime may be suitable for coherently manipulating the atom position through access to its internal degrees of freedom with a laser. Coherent manipulation of the position of neutral atoms has been proposed and demonstrated before, see e.g. Mandel et al. (2003); Mompart et al. (2003); Sebby-Strabley et al. (2006). In these examples, the manipulation is done by modifying the external potential. The mechanism we propose here is very different, as the potential remains totally unchanged, and only internal transitions and the tunneling effect are used to move the atom in a controlled way. As an example, we show how the atom can be prepared in the left well starting from the ground state $|0,-,g\rangle$ for $\delta=-g^{2}/\Delta$.

We first apply a $\pi$-pulse with an external laser resonant with the atomic transition. By using a laser with a wave vector perpendicular to the $Ox$-direction, only the atomic internal degree of freedom is affected, resulting in the transition $|0,-,g\rangle\to-i|0,-,e\rangle$. We assumed that the laser Rabi frequency $\Omega_{R}$ is much larger than the tunnel frequency $\Delta$.

Now we use the coupling between the internal and external degrees of freedom to create a superposition of the $|0,\pm,e\rangle$ states, and then apply a second resonant $\pi$-pulse to get back to the uncoupled states $|0,\pm,g\rangle$. For $\Delta/g\ll 1$, $\delta=-g^{2}/\Delta$ and $\kappa=\pi/4$, the initial state transforms according to

[TABLE]

up to a physically irrelevant phase. Other coherent superpositions of $|0,+,g\rangle$ and $|0,-,g\rangle$ can be obtained by choosing appropriate interaction times.

In order to verify that the two-level approximation for the external motion used in the derivation of the Hamiltonian is a good approximation, we have numerically solved the time dependent Schrödinger equation with Hamiltonian (1) and rotating wave approximation but with the exact external potential $V(x)$ (i.e. with a large number of vibrational states). Figure 4 shows that provided $\tilde{\Delta}\gg\sqrt{4g^{2}+\delta^{2}}$ as stated before, to take only the two lowest vibrational states into account is indeed a good approximation.

We finally comment on possible experimental realizations of our model. Double well potentials with tunable well-to-well separation have been demonstrated with optical dipole traps e.g. in Shin et al. (2004); Sebby-Strabley et al. (2006), and on atom chips e.g. in Hinds et al. (2001); Hänsel et al. (2001). For our model, the double well potential has to be realized inside the cavity. Optical trapping and even cooling of atoms close to their ground state inside a cavity has been achieved in several groups by now Ye et al. (1999); McKeever et al. (2003); Sauer et al. (2004); Maunz et al. (2004), but up to our knowledge double well potentials have not been realized in a cavity so far. However, some of the cavities developed have a very long lateral opening (up to 222 $\mu$m Fortier et al. (2007)) and should allow more complicated trapping potentials (optical lattices intersecting a cavity have been realized in Chapman’s group Fortier et al. (2007)). We remark that it is not essential for our model that the double well potential be aligned with the cavity axes. Any other orientation is possible, and only leads to modified coefficients $\cos\chi\sin\kappa$ and $\sin\chi\cos\kappa$.

At certain “magical wavelengths”, Cs, Yb, Sr, Mg, and Ca atoms in optical traps experience the same potential for ground and excited internal states coupled by a dipole transition McKeever et al. (2003); Katori et al. (2003); Brusch et al. (2006); Barber et al. (2006). In a symmetric potential $V(z)$ the tunneling frequency $\Delta$ is given in WKB approximation by $\Delta\sim\omega_{\mathrm{osc}}\exp(-1/\hbar\int_{-a}^{a}\sqrt{2m(E_{0}-V(z))}\,dz)$ where $E_{0}$ is the ground state energy, $\omega_{\mathrm{osc}}$ the single well harmonic oscillation frequency, and $z=\pm a$ are the corresponding classical turning points delimiting the range of the barrier. The exponential factor can approach unity for a barrier that is only slightly higher than the ground state energy $E_{0}$, in which case cooling to temperatures $k_{B}T<\hbar\Delta$ should be possible with state of the art techniques McKeever et al. (2003). In McKeever et al. (2003) a trap depth $V_{0}/\hbar=47$ MHz was achieved inside a cavity with $1.2$ mW laser power. In any case, the trap frequency and thus the tunneling splitting are determined by the laser power and the focussing (or the wavelength for optical lattices), and can therefore be controlled independently of $\Gamma$, $\kappa_{\rm cav}$, such that there should be no fundamental problem achieving $\Delta\gg\Gamma,\kappa_{\rm cav}$. The detection of the tunneling motion should be possible by optical imaging, i.e. diffusion of laser light from another transition in the optical regime with smaller wavelength than the well separation. Alternatively, one might monitor the transmission through the cavity in the case that it differs for the two locations of the wells Maunz et al. (2005). Another possibility might be using the atomic spin as a position meter Haycock et al. (2000).

Acknowledgements.

We thank Jacques Vigué for an interesting discussion and CALMIP (Toulouse) for the use of their computers. This work was supported by the Agence National de la Recherche (ANR), project INFOSYSQQ, and the EC IST-FET project EUROSQIP.