Atom-field dressed states in slow-light waveguide QED

We discuss the properties of atom-photon bound states in waveguide QED systems consisting of single or multiple atoms coupled strongly to a finite-bandwidth photonic channel. Such bound states are formed by an atom and a localized photonic excitation and represent the continuum analog of the familiar dressed states in single-mode cavity QED. Here we present a detailed analysis of the linear and nonlinear spectral features associated with single- and multiphoton dressed states and show how the formation of bound states affects the waveguide-mediated dipole-dipole interactions between separated atoms. Our results provide both a qualitative and quantitative description of the essential strong-coupling processes in waveguide QED systems, which are currently being developed in the optical and microwave regimes.

Figure 1

(a) Sketch of a strongly coupled waveguide QED setup with bound atom-photon dressed states around the atomic locations. The slow-light waveguide can be modeled as a large array of coupled optical resonators with nearest-neighbor coupling $J$. (b) Band structure of the waveguide without atoms. (c) Single-photon (i.e., single-excitation) spectrum as a function of the atom-photon coupling $g$ in the case of a single atom (with ${\omega }_a={\omega }_c$) coupled to a cavity array according to Hamiltonian (1).

Figure 2

(a) Correlated decay rates ${\mathrm{\Gamma }}_{ij}$ against the (discrete) interatomic distance $|x_i-x_j|$ and (b) coherent dipole-dipole interactions $U_{ij}$ versus $|x_i-x_j|$ for different detunings $\delta ={\omega }_a-{\omega }_c$. The solid lines are a guide to the eye obtained from a continuous interpolation of Eq. (6). In each case, the photon loss rate has been set to ${\gamma }_c/\left(2J\right)=0.14$.

Figure 3

(a) Atomic population $p_a^{+}={cos}^2\left({\theta }_{+}\right)$ in the upper bound state as a function of the coupling constant $g$ and the atom-field detuning $\delta$. (b) The width of the photonic wave packet in the upper bound state ${\lambda }_{+}$ is plotted as a function of $g$ and for three different detunings $\delta$.

Figure 4

Atomic excitation spectrum $S_a\left(\omega \right)$ (in logarithmic scale) as a function of $g$ and for an atom-cavity detuning (a) $\delta =0$ and (b) $\delta =2J$. The dotted lines show the bound-state energies $E_{\pm }$ in the absence of loss, while the dashed lines correspond to the waveguide band edges. In both cases, we have set ${\gamma }_a/\left(2J\right)=0.1$ and ${\gamma }_c/\left(2J\right)=0.2$.

Figure 5

Dependence of the atomic excitation spectrum $S_a\left(\omega \right)$ near the band edge and for (a) $\delta =0$ and (b) $\delta =2J$. In (a) the values $g/\left(2J\right)=0.3$ and ${\gamma }_a/\left(4J\right)=0.02$ and in (b) the values $g/\left(2J\right)=0.2$ and ${\gamma }_a/\left(4J\right)=0.05$ have been assumed, and in both cases the spectrum is plotted for different cavity decay rates ${\gamma }_c$.

Figure 6

Sketch of the single- and two-excitation spectrum in a finite-bandwidth waveguide coupled to an atom for $\delta =0$. See main text for more details.

Figure 7

The $N_e$-photon bound-state energies $E_{-}^{\left(N_e\right)}$ obtained from a variational approach are plotted for $N_e=1,\cdots ,8$ in descending order and for (a) $\delta =0$ and (b) $\delta =-2J$. The dashed lines in the insets show the exact numerical results for $N_e=2$ and $N_e=3$.

Figure 8

Sketch of the first three photonic wave functions that appear in the variational ansatz, Eq. (41), for the multiphoton bound states. Here we have set $g/\left(2J\right)=0.6$ and (a) $\delta =0$ and (b) $\delta =-2J-0^{+}$. (c) and (d) The exponential decay length ${\overline{\lambda }}_{N_e}$ as a function of $g$ for $N_e=1,2$, and 3 photons and for $\delta =0$ and $\delta =-2J-0^{+}$, respectively. The dashed line shows the result for $\overline{\lambda }$ obtained numerically for the case $N_e=2$. (e) and (f) The atomic population $p_a={cos}^2\left[\theta \left(E_{-}^{N_e}\right)\right]$ plotted against $g$ for $\delta =0$ and $\delta =-2J-0^{+}$, respectively.

Figure 9

The nonlinearity parameter ${\mathrm{\Delta }}_{\mathrm{nl}}\left(N_e\right)$ as defined in Eq. (44) is plotted for $N_e=2$ and different atom-photon detunings $\delta$.

Figure 10

The bound-state energy levels $E_{-,s}$ (left column) and the corresponding atomic populations $p_a={cos}^2\left[\theta \left(E_{-,s}\right)\right]$ (right column) are plotted as a function of the interatomic distance for the case of two atoms and for three representative values of $g/\left(2J\right)$. For all plots $\delta =0$ is assumed. For comparison, in each panel the dashed line indicates the corresponding bound-state energy or atomic population for a single atom.

Figure 11

Spatial profile of the photonic wave function $u_s\left(x\right)=\langle x|{\mathrm{\Phi }}_s\rangle$ corresponding to the even (red solid line) and odd (green dashed line) lower band bound states in the case of two atoms for different coupling strengths and interatomic distances. For all plots $\delta =0$ is assumed.

Figure 12

Single-excitation energy spectrum in the case of $N_a=40$ equally spaced atoms as a function of the atomic nearest-neighbor distance $\mathrm{\Delta }x$. Note the appearance of upper and lower metabands of bound states. For this plot $\delta /\left(2J\right)=0.6$ and $g/\left(2J\right)=1$ have been assumed.