Input-output theory of cavities in the ultrastrong coupling regime: The case of time-independent cavity parameters

We present a full quantum theory for the dissipative dynamics of an optical cavity in the ultrastrong light-matter coupling regime, in which the vacuum Rabi frequency is a significant fraction of the active electronic transition frequency and the antiresonant terms of the light-matter coupling play an important role. In particular, our model can be applied to the case of intersubband transitions in doped semiconductor quantum wells embedded in a microcavity. The coupling of the intracavity photonic mode and of the electronic polarization to the external, frequency-dependent, dissipation baths is taken into account by means of quantum Langevin equations in the input-output formalism. In the case of a time-independent vacuum Rabi frequency, exact analytical expressions for the operators are obtained, which allows us to characterize the quantum dissipative response of the cavity to an arbitrary initial condition (vacuum, coherent field, thermal excitation). For a vacuum input in both the photonic and electronic polarization modes, the ground state of the cavity system is a two-mode squeezed vacuum state with a finite population in both photonic and electronic modes. These excitations are, however, virtual and cannot escape from the cavity: for a vacuum input, a vacuum output is found, without any trace of the intracavity squeezing. For a coherent photonic input the linear optical response spectra (reflectivity, absorption, transmission) have been studied, and signatures of the ultrastrong coupling have been identified in the asymmetric and peculiar anticrossing of the polaritonic eigenmodes. Finally, we have calculated the electroluminescence spectra in the case of an incoherent electronic input: the emission intensity in the ultrastrong coupling regime results in being significantly enhanced as compared to the case of an isolated quantum well without a surrounding cavity.

Figure 1

(Color online) A sketch of the investigated model with the corresponding operators and frequencies. The system is represented by a planar cavity mode embedding a doped multiple-quantum-well structure. The cavity photon field is coupled to the quantum well electronic polarization associated with a resonant intersubband transition in the doped quantum wells. The Hopfield Hamiltonian for the cavity system includes the antiresonant light-matter interaction terms, which are significant in the ultrastrong coupling regime (vacuum Rabi frequency ${\Omega }_{R,k}$ comparable to the transition frequency ${\omega }_{12}$). The cavity mode is coupled to the extracavity electromagnetic field. The electronic polarization is coupled to a bath of electronic excitations.

Figure 2

(Color online) A recapitulative sketch of the input-output scheme. The initial condition on the extracavity photon operators and the electronic bath operators at $t=-\infty$ represents the input. The intracavity dynamics for the cavity mode and electronic polarization field is described by quantum Langevin equations including the antiresonant terms of the light-matter interaction. The radiative and nonradiative baths produce a complex frequency-dependent damping as well as frequency-dependent Langevin forces. The bath operators at $t=+\infty$ represent the radiative and nonradiative output.

Figure 3

Quantum properties of the intracavity ground state for input fields in the vacuum state. In this case, the output is also in the vacuum state, which implies that the cavity photons present in the cavity are purely virtual ones and cannot escape the cavity. Left panel: cavity photon number $⟨a_{\mathbf{k}}^{†}{a}_{\mathbf{k}}⟩$ (solid line) and the real part of the anomalous expectation value $⟨a_{\mathbf{k}}{a}_{-\mathbf{k}}⟩$ (dashed line) as a function of the normalized vacuum Rabi frequency ${\Omega }_R∕{\omega }_{12}$. Right panel: corresponding cavity photon correlation $C_{\mathbf{k},-\mathbf{k}}$ (solid line) and the normalized difference noise $F_{\mathrm{diff}}$ (dashed line). ${\omega }_{\mathrm{cav},k}={\omega }_{12}$, and the damping rates are taken as frequency-independent ones. The calculations have been performed for a symmetric double-sided cavity with $\mathrm{Re}\left\{{\tilde{\Gamma }}_{\mathrm{cav},\mathbf{k}}\right(\omega >0\left)\right\}={\overline{\Gamma }}_{\mathrm{cav},k}=\mathrm{Re}\left\{{\tilde{\Gamma }}_{\mathrm{cav},\mathbf{k}}^{\prime }\right(\omega >0\left)\right\}={\overline{\Gamma }}_{\mathrm{cav},k}^{\prime }=0.04{\omega }_{12}$, and $\mathrm{Re}\left\{{\tilde{\Gamma }}_{12,k}\right(\omega >0\left)\right\}={\overline{\Gamma }}_{12,k}=0.08{\omega }_{12}$.

Figure 4

Reflectivity ${\mathcal{R}}_{\mathbf{k}}\left(\omega \right)$ (left), absorption ${\mathcal{A}}_{\mathbf{k}}\left(\omega \right)$ (center), and transmission ${\mathcal{T}}_{\mathbf{k}}\left(\omega \right)$ (right) spectra as a function of the normalized photon frequency $\omega ∕{\omega }_{12}$ for different values of the normalized detuning $\delta =({\omega }_{\mathrm{cav},k}-{\omega }_{12})∕{\omega }_{12}$ (from $\delta =-0.5$ to $\delta =+0.5$ in equal steps). The different curves are offset for clarity. The bottom curve corresponds to $\delta =-0.5$, while the top one has been obtained for $\delta =0.5$. Parameters: ${\Omega }_{R,k}∕{\omega }_{12}=0.4$. Broadening parameters as in Fig. 3.

Figure 5

Reflectivity ${\mathcal{R}}_{\mathbf{k}}\left(\omega \right)$ (left), absorption ${\mathcal{A}}_{\mathbf{k}}\left(\omega \right)$ (center), and transmittivity ${\mathcal{T}}_{\mathbf{k}}\left(\omega \right)$ (right) spectra as a function of the normalized photon frequency $\omega ∕{\omega }_{12}$ for different values of the normalized vacuum Rabi frequency ${\Omega }_R∕{\omega }_{12}$ for the resonant case ${\omega }_{\mathrm{cav},k}={\omega }_{12}$. The different curves are offset for clarity. The bottom curve corresponds to ${\Omega }_R=0$, while the top curve corresponds to ${\Omega }_R=0.5{\omega }_{12}$ (${\Omega }_R∕{\omega }_{12}$ is increased in equal steps). Same broadening parameters as in Fig. 4.

Figure 6

A logarithmic-scale plot of the dimensionless quantity $\mid {\overline{\mathcal{U}}}_{12}^{double}(\mathbf{k},\omega ){\mid }^2$ (proportional to the electroluminescence spectrum) as a function of the normalized frequency $\omega ∕{\omega }_{12}$ in a weak coupling regime ${\Omega }_R∕{\omega }_{12}=0.001$ (dashed line) and in a strong coupling one ${\Omega }_R∕{\omega }_{12}=0.3$ (solid line). Other parameters as in Fig. 5. The small increase at very low $\omega$ is a consequence of the specific choice of constant values for $\mathrm{Re}\left\{{\tilde{\Gamma }}_{\mathrm{cav},\mathbf{k}}\right(\omega >0\left)\right\}$ and $\mathrm{Re}\left\{{\tilde{\Gamma }}_{12,\mathbf{k}}\right(\omega >0\left)\right\}$, producing a Kramers-Kronig singularity of the imaginary part (Lamb shift) at $\omega =0$.

Figure 7

(Color online) Solid line: the effective spontaneous emission rate ${\gamma }_k^{\mathrm{eff}}$ (in units of ${\omega }_{12}$) as a function of the normalized vacuum Rabi frequency ${\Omega }_{R,k}∕{\omega }_{12}$ for ${\omega }_{\mathrm{cav},k}={\omega }_{12}$. Dashed line: the Purcell rate in Eq. (71). Horizontal line: the strong coupling limit in Eq. (72). Parameters: ${\overline{\Gamma }}_{\mathrm{cav},\mathbf{k}}={\overline{\Gamma }}_{\mathrm{cav},\mathbf{k}}^{\prime }=0.04{\omega }_{12}$ and ${\overline{\Gamma }}_{12,\mathbf{k}}=0.08{\omega }_{12}$.