Extending the validity range of quantum optical master equations

This paper derives master equations for an atomic two-level system for a large set of unitarily equivalent Hamiltonians without employing the rotating-wave and certain Markovian approximations. Each Hamiltonian refers to physically different components as representing the “atom” and as representing the “field” and hence results in a different master equation, when assuming a photon absorbing environment. It is shown that the master equations associated with the minimal-coupling and the multipolar Hamiltonians predict enormous stationary state narrow-band photon emission rates, even in the absence of external driving, for current experiments with single quantum dots and color centers in diamond. These seem to confirm that the rotating-wave Hamiltonian identifies the components of the atom-field system most accurately.

Figure 1

Logarithmic plot of the transition rate $A_{-}$ in Eq. (B3) for the minimal-coupling, the multipolar, and the rotating-wave Hamiltonian as a function of the upper cutoff frequency ${\omega }_{\max }$, while ${\omega }_{\min }=0$ and ${\omega }_0\Delta t={10}^4$.

Figure 2

Logarithmic plot of the transition rates $A_{+}$ in Eq. (B3) for the minimal-coupling and the multipolar Hamiltonian as a function of the upper cutoff frequency ${\omega }_{\max }$, while ${\omega }_{\min }=0$ and ${\omega }_0\Delta t={10}^4$. These are in very good agreement with the analytical results in Eqs. (B6, B7, B8).

Figure 3

Logarithmic plot of the stationary state photon emission rate $I_{\mathrm{SS}}$ for the minimal-coupling and the multipolar Hamiltonian as a function of ${\omega }_{\max }$, while ${\omega }_{\min }=0$. The graphs are the result of a numerical solution of Eqs. (58) and (B3).

Figure 4

Logarithmic plot of the function $f_{-}\left({\omega }_k\right)$ in Eq. (B4) for the rotating-wave, the minimal-coupling, and the multipolar Hamiltonian.

Figure 5

Logarithmic plot of the function $f_{+}\left({\omega }_k\right)$ in Eq. (B4) for the minimal-coupling and the multipolar Hamiltonian. For the rotating-wave Hamiltonian we have $f_{+}\equiv 0$.

Figure 6

Logarithmic plot of an upper bound for $A_{+}{A}_{-}/{\left|B\right|}^2$ for the multipolar and the minimal-coupling Hamiltonian obtained from a numerical integration of Eqs. (B3) and (53) after taking into account that $|d_1^2+d_2^2+d_3^2{|}^2\le 1$ and that ${\cos }^2\left({\omega }_0\Delta t\right)\le 1$, while assuming ${\omega }_{\min }=0$. The figure confirms that Eq. (C8) holds for the experimental parameters which we consider here and that the corresponding master equations are in general of Lindblad form.