Enhancing photon generation in cavity through antiresonant terms of the vacuum Rabi coupling

Abstract

The Rabi model describes the simplest interaction between a two-level system and a bosonic mode beyond the rotating wave approximation. The antiresonant terms that result from this coherent interaction play an important role. In this paper, we go beyond the rotating wave approximation even for the interaction with vacuum. This leads to the ’incoherent’ antiresonant terms. Using the master equation which includes both coherent and incoherent antiresonant terms, we numerically compute the mean photon number and show that these incoherent antiresonant terms enhance the generation of mean photon number. Moreover we study numerically the effect of the detuning and show that it also enhances the generation of photons. Finally, we generalize our result to two two-level and two-mode systems.

Appendix A. Equation of motion using modified input-output theory

The antiresonant terms that result from the interaction of a system with vacuum are usually ignored in the derivation of master equation [30]. In this Appendix we show how to generalize this master equation by taking into account these antiresonant terms in the quantum Ito stochastic differential equation.

Our model consists of a system interacting with a bath. The Hamiltonian of the system-bath is typically written as the sum of three Hamiltonians

$$\begin{aligned} H = H_{sys} + H_B + H_{int}, \end{aligned}$$

(A.1)

where $H_{sys}$ is the Hamiltonian of the system which will specify it later. The Hamiltonian of the bath is given by

$$\begin{aligned} H_B = \hbar \int _{0}^{\infty } d\omega \, \omega b^\dagger (\omega ) b(\omega ), \end{aligned}$$

(A.2)

while, $H_{int}$ is the interaction Hamiltonian consisting of the sum of two Hamiltonians, the Hamiltonian $H_{int}^{res}$ which corresponds to the resonant terms and $H_{int}^{anti}$ corresponds to the anti-resonant terms

$$\begin{aligned} H_{int}= & {} H_{int}^{res}+{H_{int}^{anti}}, \end{aligned}$$

(A.3)

where

$$\begin{aligned} H_{int}^{res}= & {} i\hbar \int _{0}^{\infty } d\omega \,\kappa (\omega ) \left( b^\dagger (\omega )\,c -c^\dagger b(\omega )\right) , \end{aligned}$$

(A.4)

$$\begin{aligned} H_{int}^{anti}= & {} i\hbar \int _{0}^{\infty } d\omega \,\kappa (\omega ) \left( cb(\omega )-b^\dagger (\omega )c^\dagger \right) . \end{aligned}$$

(A.5)

In the standard input-output theory [30], the Hamiltonian $H_{int}^{anti}$ is completely ignored. This is the well know the RWA, which is valid if the coupling strength is weak. Here we go beyond the RWA and keep all the antiresonant terms in the Hamiltonian $H_{int}^{anti}$. The equations of motion for a and b are given by

$$\begin{aligned} {\dot{b}}= & {} -i \omega b +\kappa (\omega )c-{\kappa (\omega )c^\dagger }, \end{aligned}$$

(A.6)

$$\begin{aligned} {\dot{a}}= & {} \frac{i}{\hbar }[H_{sys},a] +\int _{0}^\infty d\omega \kappa (\omega ) \left\{ b^\dagger (\omega )[a,c]-[a,c^\dagger ]b(\omega )\right\} \nonumber \\&+ { \int _{0}^\infty d\omega \kappa (\omega ) \left\{ [a,c]b(\omega ) -b^\dagger (\omega )[a,c^\dagger ] \right\} }, \end{aligned}$$

(A.7)

where a is an arbitrary system operator. The solution of Eq. (A.6) is

$$\begin{aligned} b(\omega )= & {} \mathrm{e}^{-i\omega (t-t_0)}b_0(\omega )+\kappa (\omega ) \int _{t_0}^t \mathrm{e}^{-i\omega (t-t')}c(t') dt'\nonumber \\&-\kappa (\omega ) \int _{t_0}^t \mathrm{e}^{-i\omega (t-t')}c^\dagger (t') dt'. \end{aligned}$$

(A.8)

The last term in Eq. (A.8) is due to the anti-resonant term. So, in weak interaction regime, one can ignore this term and write

$$\begin{aligned} b(\omega ) \approx \mathrm{e}^{-i\omega (t-t_0)}b_0(\omega )+\kappa (\omega ) \int _{t_0}^t \mathrm{e}^{-i\omega (t-t')}c(t') dt'. \end{aligned}$$

(A.9)

This is our first approximation. Substituting the solution for b, Eq. (A.9), into the equation of motion for a, Eq. (A.7), we obtain

$$\begin{aligned} {\dot{a}}= & {} \frac{i}{\hbar }[H_{sys},a] +\int _0^\infty dw \kappa (\omega )\mathrm{e}^{i\omega (t-t_0)}\, b_0^\dagger (\omega )[a,c] +\int _0^\infty dw \kappa ^2(\omega ) \int _{t_0}^t \mathrm{e}^{i\omega (t-t')}dt' c^\dagger (t')[a,c]\nonumber \\&-\int _0^\infty dw \kappa (\omega )\mathrm{e}^{-i\omega (t-t_0)}\, [a,c^\dagger ]b_0(\omega ) -\int _0^\infty dw \kappa ^2(\omega ) \int _{t_0}^t \mathrm{e}^{-i\omega (t-t')}dt' [a,c^\dagger ]c(t') \nonumber \\&+ \int _0^\infty dw \kappa (\omega )\mathrm{e}^{-i\omega (t-t_0)}\, [a,c]b_0(\omega ) + \int _0^\infty dw \kappa ^2(\omega ) \int _{t_0}^t \mathrm{e}^{-i\omega (t-t')}dt' [a,c]c(t') \nonumber \\&-\int _0^\infty dw \kappa (\omega )\mathrm{e}^{i\omega (t-t_0)}\, b_0^\dagger (\omega )[a,c^\dagger ] -\int _0^\infty dw \kappa ^2(\omega ) \int _{t_0}^t \mathrm{e}^{i\omega (t-t')}dt' c^\dagger (t')[a,c^\dagger ]. \end{aligned}$$

(A.10)

Now we use the Markov approximation. This is our second approximation. In this approximation we assume that the bandwidth of the bath is large enough so that the coupling constant $\kappa (\omega )=\sqrt{\varGamma /\pi }$. So, the Langevin equation can be simplified and takes the form

$$\begin{aligned} {\dot{a}}= & {} \frac{i}{\hbar }[H_{sys},a]+ \left( \frac{\varGamma }{2}+i\varDelta \right) c^\dagger [a,c] -\left( \frac{\varGamma }{2}-i\varDelta \right) [a,c^\dagger ]c \nonumber \\&+ { \left( -\frac{\varGamma }{2}-i\varDelta \right) [a,c]c -\left( -\frac{\varGamma }{2}+i\varDelta \right) c^\dagger [a,c^\dagger ]} \nonumber \\&+\sqrt{\varGamma }b_{in}^\dagger [a,c-{ c^\dagger }] -\sqrt{\varGamma }[a,c^\dagger -{c}]b_{in}, \end{aligned}$$

(A.11)

where $\varDelta$ is a result of the integral of the Cauchy principal value. We also note here that the computation of the integrals involving the counter rotating terms should be done with care. In principle, the parameter $\kappa ^2(\omega )$ is proportional to the frequency $\omega$, and these terms are resonant only in the negative frequencies. So, we need to transform these integrals to the negative frequencies first. This is why the negative sign appears before $\varGamma /2$ in Eq. (A.11), that is,

$$\begin{aligned} \int _0^\infty dw \kappa ^2(\omega ) \int _{t_0}^t \mathrm{e}^{-i\omega (t-t')}dt' [a,c]c(t')\approx & {} \left( -\frac{\varGamma }{2}-i\varDelta \right) [a,c]c,\\ \int _0^\infty dw \kappa ^2(\omega ) \int _{t_0}^t \mathrm{e}^{i\omega (t-t')}dt' c^\dagger (t')[a,c^\dagger ]\approx & {} \left( -\frac{\varGamma }{2}+i\varDelta \right) c^\dagger [a,c^\dagger ]. \end{aligned}$$

Transforming the Langevin equation (A.11) to the quantum stochastic differential equation, we get

$$\begin{aligned} {da}= & {} \frac{i}{\hbar }[H_{sys},a]dt- \frac{\varGamma }{2}\left( c^\dagger c a +a c^\dagger c - 2 c^\dagger a c \right) dt -i\varDelta [c^\dagger c ,a]dt \nonumber \\&- { \frac{\varGamma }{2}\left( [a,c]c-c^\dagger [a,c^\dagger ]\right) dt -i\varDelta \left( c^\dagger [a,c^\dagger ]+[a,c]c\right) dt } \nonumber \\&+\sqrt{\varGamma }dB_{in}^\dagger [a,c] -\sqrt{\varGamma }[a,c^\dagger ]dB_{in} -\sqrt{\varGamma }dB_{in}^\dagger [a,{c^\dagger }] +\sqrt{\varGamma }[a,{c}]dB_{in}, \end{aligned}$$

(A.12)

where we use the notation $dB_{in} = b_{in} dt$. Let us focus now on the ordinary vacuum as an input. In this case, the noise operators that depend on dB and $dB^\dagger$ are ignored. Thus, the Ito quantum stochastic differential equation is then given by the same equation (A.12). From this equation one can find the master equation. It takes the form

$$\begin{aligned} {\dot{\rho }}= & {} -\frac{i}{\hbar }[H_{sys},\rho ]- \frac{\varGamma }{2}\left( \rho c^\dagger c +c^\dagger c \rho - 2 c \rho c^\dagger \right) -i\varDelta [c^\dagger c ,\rho ] \nonumber \\&- \frac{\varGamma }{2}\left( c[c,\rho ]-[c^\dagger ,\rho ]c^\dagger \right) -i\varDelta \left( c[c,\rho ] + [c^\dagger ,\rho ]c^\dagger \right) . \end{aligned}$$

(A.13)

The last two terms of the equation above are due to the incoherent antiresonant terms.