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Cavity quantum electrodynamics - Laboratoire Kastler Brossel

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Accueil du site > Atomes, cavités et photons > Principes de l’electrodynamique quantique en cavité > Spin et oscillateur : états habillés

Spin et oscillateur : états habillés

The coupled atom-field system is conveniently described in terms of the dressed states. We use them to investigate the quantum Rabi oscillation at resonance and the dispersive shifts in the non-resonant case.

The Jaynes-Cummings hamiltonian

The complete atom-cavity hamiltonian writes :

$H=H_a+H'_c+H_{ac}\ ,$

where $H_a$ and $H'_c$ are the atom and cavity hamiltonians. The coupling term, $H_{ac}$ , is $-D\cdot E_c$ , where $E_c$ the electric field operator at the atomic location. The hamiltonian $H$ has been introduced by Jaynes and Cummings as an approximation to matter-field coupling in free space. It is only in a cavity that it provides a precise description of the atomic dynamics, since the interaction with a single mode dominates the evolution.

We assume first that the atom is sitting at cavity centre [ $|f(r)|=1$ ]. The coupling hamiltonian is then :

$H_{ac}=-d\left[\epsilon_a \sigma_-+\epsilon_a^*\sigma_+ \right]\cdot iE_0\left[\epsilon_c a-\epsilon_c^* a^\dagger \right]\ ,$

where $\epsilon_c$ is the unit vector describing the field polarization at cavity centre. Its expansion involves four terms. Two are proportional to $\sigma_-a$ and $\sigma_+a^\dagger$ . The first corresponds to a transition from $e$ to $g$ , together with the annihilation of a photon. The second describes the emission of a photon by an atom in a transition from $g$ to $e$ . When the cavity and atomic transition frequencies, $\omega_c$ and $\omega_{eg}$ , are comparable, these terms correspond to non-resonant processes. They can be neglected (Rotating Wave Approximation). The two others, proportional to $\sigma_+ a$ and $\sigma_-a^\dagger$ , correspond to photon absorption or emission and the atom-cavity coupling reduces to :

$H_{ac}=-i\hbar\frac{\Omega_0}{2}\left[ a\sigma_+-a^\dagger\sigma_- \right]\ ,$

where we introduce the `single photon Rabi frequency’, $\Omega_0$ :

$\Omega_0=2\frac{dE_0\epsilon_a^*\cdot\epsilon_c}{\hbar}\ .$

We assumed that $\epsilon_a^*\cdot\epsilon_c$ is real and positive, hence $\Omega_0$ . The frequency $\Omega_0$ measures the strength of the atom-field coupling. It is proportional to the interaction energy of the atomic dipole with a classical field corresponding to a single photon stored in $C$ . In our experiments $\Omega_0=2\pi\times50$ kHz, a rather large value. It is much larger than the cavity relaxation rate, achieving the `strong coupling regime’ of cavity QED.

The `uncoupled states’, eigenstates of $H_a+H'_c$ , are the tensor products $|{e,n}\rangle$ and $|{g,n}\rangle$ of atomic energy eigenstates and cavity Fock states, with the energies $\hbar(\omega_{eg}/2+n\omega_c)$ and $\hbar(-\omega_{eg}/2+n\omega_c)$ . The ground state of $H_a+H'_c$ is the non-degenerate $|{g,0}\rangle$ state. When the atom-cavity detuning, $\Delta_c=\omeg-\omega_c$ , is much smaller than the atomic frequency, the uncoupled states $|{e,n}\rangle$ and $|{g,n+1}\rangle$ are degenerate ( $\Delta_c=0$ ) or nearly degenerate. They form a ladder of equally-spaced two-level manifolds, which are not coupled by $H_{ac}$ . The diagonalization of the full hamiltonian amounts to solving separate two-level problems.

Dressed states

We consider here the restriction $H_n$ of the Jaynes and Cummings hamiltonian to the manifold spanned by $|{e,n}\rangle$ and $|{g,n+1}\rangle$ . Taking the energy reference at the mid-point between these levels, $H_n$ writes in matrix form :

$H_n=\frac{\hbar}{2} \left(\begin{array}{cc} \Delta_c&-i\Omega_n\\i\Omega_n&-\Delta_c\end{array}\right)\ ,$

where

$\Omega_n=\Omega_0\sqrt{n+1}\ .$

The Rabi frequency $\Omega_n$ is proportional to $\sqrt{n+1}$ , relative field amplitude in the $n$ -photons Fock state.

The eigenvalues of $H_n$ are :

$E_n^\pm=\pm\frac{\hbar}{2}\sqrt{\Delta_c^2+\Omega_n^2}\ ,$

and the eigenvectors, the `dressed states’ :

$|{\pm,n}\rangle=\cos\theta_n^\pm|{e,n}\rangle+i\sin\theta_n^\pm|{g,n+1}\rangle\ ,$

are generally entangled atom-cavity states. The mixing angles $\theta_n^\pm$ are given by

$\tan\theta_n^\pm=\pm\frac{\sqrt{\Delta_c^2+\Omega_n^2}\mp\Delta_c}{\Omega_n}\ .$

The positions of the dressed energies are represented as a function of $\Delta_c$ on the figure above. For large detunings, the dressed energies almost coincide with the uncoupled ones $\pm\hbar\Delta_c/2$ . At zero detuning, the uncoupled levels cross. The atom-cavity coupling transforms this crossing into an avoided crossing, the minimum distance between the dressed states being $\hbar\Omega_n$ . We examine now two limiting cases : the resonant case ( $\Delta_c=0$ ) and the detuned case ( $\Delta_c\gg\Omega_0$ ).

Resonant coupling

In the resonant case, the mixing angles are $\theta_n^\pm=\pm\pi/4$ and :

$|{\pm,n}\rangle=\frac{1}{\sqrt{2}}\left[|{e,n}\rangle\pm i|{g,n+1}\rangle \right]\ .$

The separation of the dressed states at resonance, $\hbar\Omega_n$ , corresponds to the frequency of the atom-field reversible energy exchange. We consider an atom in state $e$ at time $t=0$ inside the cavity containing $n$ photons. The initial state, $|{\Psi_e(0)}\rangle=|{e,n}\rangle$ , expands on the dressed states basis as :

$|{\Psi_e(0)}\rangle=\frac{1}{\sqrt 2}\left[|{+,n}\rangle+|{-,n}\rangle \right]\ ,$

and becomes at time $t$ :

$|{\Psi_e(t)}\rangle=\frac{1}{\sqrt 2}\left[ |{+,n}\rangle e^{-i\Omega_n t/2}+|{-,n}\rangle e^{i\Omega_n t/2} \right]\ .$

The probabilities for finding the atom in $e$ or $g$ are obtained by returning to the uncoupled basis :

$|{\Psi_e(t)}\rangle=\cos\frac{\Omega_n t}{2}|{e,n}\rangle+\sin\frac{\Omega_n t}{2}|{g,n+1}\rangle\ .$

The coupling results in a reversible exchange between $|{e,n}\rangle$ and $|{g,n+1}\rangle$ at frequency $\Omega_n$ . This is the Rabi oscillation phenomenon. It can be understood as a `quantum beat’ between the two dressed states. For the initial state $|{e,0}\rangle$ (or $|{g,1}\rangle$ ), this oscillation occurs at the frequency $\Omega_z$ .

Non-resonant atom/cavity coupling

We give here a perturbative treatment valid for large atom-cavity detunings. We assume $\Delta_c > 0$ , with $\theta_n \ll 1$ . We develop the eigenstates and eigenenergies in powers of $\Omega_n /\Delc$ . To first order for states and to second order for energies :

$\left| { + ,n} \right\rangle \approx \left| {e,n} \right\rangle + \frac{{\Omega_0 \sqrt {n + 1} }}{{2\Delta_c }}\left| {g,n + 1} \right\rangle$

$\left| { - ,n} \right\rangle \approx - \left| {g,n + 1} \right\rangle + \frac{{\Omega_0 \sqrt {n + 1} }}{{2\Delta_c}}\left| {e,n} \right\rangle$

$\frac{1}{\hbar }E_{ + ,n} = n \omega_c + \frac{{\omega_{eg} }}{2} + \frac{{\Omega_0 ^2 (n + 1)}}{{4\Delta_c }}$

$\frac{1}{\hbar }E_{ - ,n} = (n+1) \omega_c - \frac{{\omega_{eg} }}{2} - \frac{{\Omega_0 ^2 (n + 1)}}{{4\Delta_c }}\ .$

The dressed states $|+,n\rangle$ and $|-,n\rangle$ are very close to the uncoupled states, which are slightly `contaminated’ by the atom-field coupling. The energies of these levels are shifted to second order by an amount proportional to $n+1$ , combination of a light shift, proportional to $n$ , and of a Lamb shift effect.

Assume now that we couple an atom in level $e$ with a coherent field $|\alpha \rangle$ and let the two systems interact for a time $t$ . Expanding the coherent state on a Fock state basis and taking into account that $|e,n \rangle$ is very close to the $| +,n\rangle$ dressed state, we get :

$\left| {\Psi _{e,\alpha } (0)} \right\rangle = \left| e \right\rangle \left| \alpha \right\rangle = \sum\limits_n {c_n \left| {e,n} \right\rangle } \rightarrow$

$\left| {\Psi _{e,\alpha } (t)} \right\rangle \approx \sum\limits_n {c_n e^{ - in \omega_c t} e^{ - i\omega_{eg} t /2} e^{ - i\Omega_0 ^2 (n + 1)t/4\Delta_c } \left| {e,n} \right\rangle } \ ,$

which, in interaction picture yields :

$\left| {\widetilde \Psi _{e,\alpha } (t)} \right\rangle \approx \sum\limits_n {c_n \;e^{ - i\Omega_0^2 (n + 1)t/4\Delta_c } \left| {e,n} \right\rangle } \quad = e^{ - i\Omega_0 ^2 t/4\Delta_c } \left| e \right\rangle \otimes \left| \alpha e^{ - i\Omega_0 ^2 t/4\Delta_c} \right\rangle \ .$

Similarly, for an atom initially in level $\g$ we obtain :

$\left| {\widetilde \Psi _{g,\alpha } (t)} \right\rangle \approx \sum\limits_n {_n e^{ + i\Omega_0^2 nt/4\Delta_c} \left| {g,n} \right\rangle } \quad = \left| g \right\rangle \otimes \left| \alpha e^{ + i\Omega_0 ^2t/4\Delta_c} \right\rangle \ .$

The cavity is in a coherent state, phase shifted by an angle $\pm \Phi_0 = \pm \Omega_0^2t/4\Delta_c$ depending upon the atomic state. This effect is interpreted by attributing to the atom an index $N_i=1\pm\Omega_0^2/4\Delc\omc$ , with the $+$ and $-$ signs corresponding respectively to an atom in $\e$ or $\g$ . With the parameters of our experiment we find, for $\Delc = 3\Omega$ , $|N_i-1|\approx 10^{-7}$ , a huge value for a single atom effect. Note that this index is linear for extremely low fields only and saturates for average photon numbers of the order of $(\delta/\Omz)^2$ , since the dispersive regime condition is no longer fulfilled. Note also the global quantum phase shift of the system’s state when the atom is in $\e$ . This is a cavity Lamb shift effect.

Up to now, we have considered the atom as motionless at cavity center. In the actual experiment, the atom flies across the cavity mode waist $w$ . The atom-field coupling, proportional to the relative mode amplitude $f$ , is thus a time-dependent quantity. In the dispersive regime, the previous results still hold, when the time $t$ is replaced by an effective interaction time $t_i^d$ taking into account the integrated atom-field coupling. For a full crossing of the mode :