Diagrammatic treatment of few-photon scattering from a Rydberg-blockaded atomic ensemble in a cavity

In a previous Letter [A. Grankin et al., Phys. Rev. Lett. 117, 253602 (2016)] we studied the giant optical nonlinearities of a Rydberg atomic medium within an optical cavity, in the Schwinger-Keldysh formalism. In particular, we calculated the nonlinear contributions to the spectrum of the light transmitted through the cavity. In this article we spell out the essential details of this calculation and we show how it can be extended to higher input photon numbers and higher-order correlation functions. As a relevant example, we calculate and discuss the three-photon correlation function of the transmitted light and discuss its physical significance in terms of the polariton energy levels of the Rydberg medium within the optical cavity.

Figure 1

Atomic level scheme and the setup.

Figure 2

Representation of contour ordering for the multitime correlation function $\langle \tilde{\mathcal{T}}\left\{{\mathcal{O}}^1\left(t_1\right){\mathcal{O}}^2\left(t_2\right){\mathcal{O}}^3\left(t_3\right)\right\}\mathcal{T}\left\{{\mathcal{O}}^4\left(t_4\right){\mathcal{O}}^5\left(t_5\right)\right\}\rangle$.

Figure 3

Feynman diagrams corresponding to (a) first-order, (b) second-order, and (c) third-order contributions in $H_{dd}$. (d) Schematic representation of the sum of all orders starting with the first. (e) Diagrammatic representation of the perturbative expansion of $T_0$. Here the thick arrows represent the Green's functions $G_{ac}$ and $G_{ca}$, thin arrows stand for the polaritonic propagator $G_{cc}$, and the dashed vertical lines represent the Fourier transform of the interaction potential $U$.

Figure 4

Inelastic component of the cavity transmission spectrum ${\mathcal{S}}_i\equiv 2{\gamma }_c^R\int d\nu \langle a^{†}\left(\omega \right),a\left(\nu \right)\rangle$ in logarithmic scale as a function of ${\mathrm{\Omega }}_{cf}$ and the frequency (in the frame rotating at ${\omega }_p$) for (a) the resonant case ${\mathrm{\Delta }}_c={\mathrm{\Delta }}_e={\mathrm{\Delta }}_r=0$ and (b) the detuned case. The transverse curves give $(\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3)$ as functions of ${\mathrm{\Omega }}_{cf}$ (see the text).

Figure 5

Schematic structure of the cavity-atom coupled system restricted to at most three excitations. The (a) two-photon and (b) three-photon cascades allow one to understand the features observed in the cavity transmission spectrum (see Sec. 6) and the three-photon correlation function (Sec. 7)

Figure 6

Diagrammatic expansion of the three-body correlation function: (a) combination of ladder diagrams into two-body $T$ matrices, (b) effective perturbative expansion with respect to $T$ matrices, and (c) expansion of ${\psi }_{12}^{\left(3\right)}$ to the third order with respect to $T$ matrices.

Figure 7

Experimental scheme for measurement of the three-photon correlation function $|{\langle \delta a_{\mathrm{out}}\left({\omega }_1\right)\delta a_{\mathrm{out}}\left({\omega }_2\right)\delta a_{\mathrm{out}}\left({\omega }_3\right)\rangle }^{\left(3\right)}{|}^2$.

Figure 8

Three-photon correlation function $|{\langle \delta a\left({\omega }_1\right)\delta a\left({\omega }_2\right)\delta a(-{\omega }_1-{\omega }_2)\rangle }^{\left(3\right)}{|}^2$ as a function of ${\omega }_1$ and ${\omega }_2$: for the parameters (a) ${\mathrm{\Delta }}_e=-25{\gamma }_e$ and ${\mathrm{\Omega }}_{cf}={\gamma }_e$, (b) ${\mathrm{\Delta }}_e=0$ and ${\mathrm{\Omega }}_{cf}={\gamma }_e$, (c) ${\mathrm{\Delta }}_e=-25{\gamma }_e$ and ${\mathrm{\Omega }}_{cf}=4{\gamma }_e$, and (d) ${\mathrm{\Delta }}_e=0$ and ${\mathrm{\Omega }}_{cf}=4{\gamma }_e$. Also shown is the polariton eigenstate energy for (e) the detuned regime ${\mathrm{\Delta }}_e=-25{\gamma }_e$ and (f) the resonant case ${\mathrm{\Delta }}_e=0$, with ${\epsilon }_1$, ${\epsilon }_2$, and ${\epsilon }_3$ energies given by the blue (dark gray), orange (light gray), and green (gray) curves, respectively. In the far detuned regime one of the polaritons eigenenergy is equal to ${\epsilon }_3\approx -25{\gamma }_e$ (not shown).

Figure 9

Diagrammatic representation of $T[\vec{k},{\vec{k}}^{\prime }]$ in (a) perturbative form and (b) self-consistent form.