Coherent effect of vacuum fluctuations on driven atoms

We study the coherent effect of the Casimir-Polder interaction on the oscillations of two-photon driven atoms. We find that, for oscillations between two degenerate states in $\mathrm{\Lambda }$ configuration, shifts on the Rabi frequency may be induced by nonadditive level shifts. For oscillations between two Rydberg states in ladder configuration, shifts on the Rabi frequency may be induced by the effective renormalization of the laser interaction.

Figure 1

(a) Schematic representation of a $\mathrm{\Lambda }$ system, where two Raman lasers of frequencies ${\omega }_p$ and ${\omega }_s$ couple the states $|g\rangle$ and $|e\rangle$ to a third intermediate state $|i\rangle$ with coupling strengths ${\mathrm{\Omega }}_{gi}^p$ and ${\mathrm{\Omega }}_{ei}^s$ respectively. (b) Schematic representation of a ladder system. In this case the intermediate state $|i\rangle$ lays between $|g\rangle$ and $|e\rangle$.

Figure 2

(a) Schematic representation of the effective vertices of the Rabi Hamiltonian, $H_R^{\mathrm{int}}{|}_{eg}\left(t\right)$ and $H_R^{\mathrm{int}}{|}_{ge}\left(t\right)$. (b) Diagrammatic representation of the light shifts, $\delta {\omega }_g$ and $\delta {\omega }_e$.

Figure 3

(a) Atom (in red) placed at $\mathbf{R}$ in front of a plane surface perpendicular to the $\widehat{\mathbf{Z}}$ axis, with relative permittivity ${\epsilon }_r$ and permeability ${\mu }_r$. (b) Feynman diagram of a multiple reflection process which contributes to ${}_g{U}_g^W$ in the nondegenerate case, ${\omega }_{eg}T\gg 1$. The diagram equals three one-particle-irreducible additive phase shift factors, ${\left(\delta E_{gg}^g\right)}^3$—dissipative factors are omitted here for brevity. (c) Feynman diagram of a single reflection process which contributes to the nonadditive shift factor $\delta E_{ge}^g$. Left-handed (L) and right-handed (R) photons are created or annihilated at the time the atom transits from $|g\rangle$ to $|i\rangle$ and from $|i\rangle$ to $|e\rangle$ respectively. (d) Feynman diagram of a multiple reflection process which contributes to ${}_e{U}_g^W$ in the degenerate case, ${\omega }_{eg}T\ll 1$. The diagram equals three one-particle-irreducible nonadditive phase shift factors, ${\left(\delta E_{ge}^{\mathcal{E}}\right)}^2\delta E_{eg}^{\mathcal{E}}$—dissipative factors are omitted here for brevity.

Figure 4

(a) Diagrammatic representation of a multiple reflection process in which additive CP terms induce a shift on the Rabi frequency of the kind ${\delta }_1{\mathrm{\Omega }}_R$ of Eq. (22). One-loop shifts in the levels $E_g,E_e$, and $E_i$ are included. (b) Diagrammatic representation of a multiple reflection process in which nonadditive CP terms induce a shift on the Rabi frequency of the kind ${\delta }_2{\mathrm{\Omega }}_R$ of Eq. (28) for ${\omega }_{eg}\ll \left|\mathrm{\Omega }\right|$. (c) Diagrammatic representation of the one-loop vertex renormalization processes which generates $\delta H_{ig}^p$ and $\delta H_{ig}^s$ respectively. They give rise to a shift on the Rabi frequency of the kind ${\delta }_3{\mathrm{\Omega }}_R$ of Eq. (33). The transient states $\tilde{i},\tilde{g}$, and $\tilde{e}$ belong to the same energy levels as the states $i,g$, and $e$ respectively.

Figure 5

Schematic representation of the action of two Raman lasers which drive a ${}^{87}\mathrm{Rb}$ atom between two Zeeman sublevels of the ground state, $|g\rangle =|5{}^2{S}_{1/2},F=1,m_F=-1\rangle$ and $|e\rangle =|5{}^2{S}_{1/2},F=1,m_F=+1\rangle$. The quantization axis is taken along the $\widehat{\mathbf{X}}$ direction. The lasers have opposite polarization, ${\sigma }^{+}$ and ${\sigma }^{-}$, equal frequency, $\lambda \simeq 795$ nm such that ${\mathrm{\Delta }}_0=2\pi \times 10.0$ GHz and equal intensity such that ${\mathrm{\Omega }}_0=2\pi \times 8.0$ MHz. They couple both states to a pair of common intermediate states of the level $|5{}^2{P}_{1/2}\rangle ,|i_1\rangle =|5{}^2{P}_{1/2},F=1,m_F=0\rangle$ and $|i_2\rangle =|5{}^2{P}_{1/2},F=2,m_F=0\rangle$, with transition frequencies ${\omega }_{5S5P_{1/2}}^1$ and ${\omega }_{5S5P_{1/2}}^1+\delta {\omega }_1$ respectively, where ${\omega }_{5S5P_{1/2}}^1=2\pi \times 377.107$ THz and the hyperfine interval is $\delta {\omega }_1=2\pi \times 0.817$ GHz. Grey lines represent secondary couplings of the three Zeeman sublevels of the ground state, $|5{}^2{S}_{1/2},F=1\rangle$, to other states in $|5{}^2{P}_{1/2}\rangle$. They are relevant for the computation of light shifts. In addition to the $D1$ transition line, relevant to the calculation of ${\delta }_2{\mathrm{\Omega }}_R$ are the $D2$ line transitions from $|g\rangle$ and $|e\rangle$ to the Zeeman sublevels of $5P_{3/2}$ with $m_F=0$. For those transitions, ${\omega }_{5S5P_{3/2}}^1=2\pi \times 384.230$ THz, $\delta {\omega }_{2a}=-2\pi \times 0.072$ GHz, $\delta {\omega }_{2b}=2\pi \times 0.157$ GHz. Also relevant to ${\delta }_1{\mathrm{\Omega }}_R$ are the transitions $5P_{1/2}\leftrightarrow 6S_{1/2}$ and $5P_{1/2}\leftrightarrow 4D_{3/2}$.

Figure 6

Graphical representation of the Rabi frequency shifts as a function of the distance to the reflecting surface for the ${}^{87}\mathrm{Rb}$ atom driven in $\mathrm{\Lambda }$ configuration.

Figure 7

Schematic representation of the action of a $\pi$-polarized microwave source which drives the oscillations of a ${}^{87}\mathrm{Rb}$ atom between two Rydberg levels, $|g\rangle =|48{}^2{S}_{1/2},m_j=-1/2\rangle$ and $|e\rangle =|49{}^2{S}_{1/2},m_j=-1/2\rangle$. The quantization axis is taken along the $\widehat{\mathbf{X}}$ direction. The intermediate states are $|i_1\rangle =|48{}^2{P}_{1/2},m_j=-1/2\rangle$ and $|i_2\rangle =|48{}^2{P}_{3/2},m_j=-1/2\rangle$. Exact a priori resonant condition is assumed, ${\omega }_0={\omega }_{eg}/2=2\pi \times 35.2387$ GHz. Hence, ${\mathrm{\Delta }}_{g{i}_{1,2}}={\mathrm{\Delta }}_{e{i}_{1,2}}$.

Figure 8

Schematic representation of three processes which contribute to (a) $\delta H_{g{i}_2}^s$, (b) $\delta H_{e{i}_2}^s$, and (c) $\delta H_{g{i}_1}^p$, respectively. Wavy lines depict virtual transitions driven by vacuum photons. Dashed lines depict actual transitions driven by the microwave source. In the nonretarded regime (a) is negligible and, after considering all kinds of transitions, only processes of the kind of (c) survive.

Figure 9

Graphical representation of the Rabi frequency shifts as a function of the distance to the reflecting surface for the ${}^{87}\mathrm{Rb}$ atom driven ladder configuration.