Two-mode single-atom laser as a source of entangled light

A two-mode single-atom laser is considered, with the aim of generating entanglement in macroscopic light. Two transitions in the four-level gain medium atom independently interact with the two cavity modes, while two other transitions are driven by control laser fields. Atomic relaxation as well as cavity losses are taken into account. We show that this system is a source of macroscopic entangled light over a wide range of control parameters and initial states of the cavity field.

Figure 1

(Color online) A single four-level atom is trapped in a doubly resonant cavity and interacts with two cavity modes and two classical laser fields. The inset shows the atomic level scheme. The laser field with frequency ${\omega }_3$ and Rabi frequency ${\Omega }_3$ couples to the $\mid a⟩\leftrightarrow \mid d⟩$ transition, and the cavity mode with frequency ${\nu }_1$ and coupling constant $g_1$ interacts with the $\mid a⟩\leftrightarrow \mid c⟩$ transition. ${\Delta }_a$ is the detuning of the fields ${\Omega }_3$ and $g_1$ from state $\mid a⟩$. The laser field with frequency ${\omega }_4$ and Rabi frequency ${\Omega }_4$ drives the $\mid b⟩\leftrightarrow \mid c⟩$ transition, and the second cavity mode with frequency ${\nu }_2$ and coupling constant $g_2$ interacts with the $\mid b⟩\leftrightarrow \mid d⟩$ transition. ${\Delta }_b$ is the detuning of the fields ${\Omega }_4$ and $g_2$ from state $\mid b⟩$. Spontaneous emission is denoted by dashed arrows, and the parameters ${\Gamma }_i$ are the decay rates of the various transitions.

Figure 2

(a) Time evolution of $⟨{\left(\Delta \widehat{u}\right)}^2+{\left(\Delta \widehat{v}\right)}^2⟩$. The mean value of the total number of photons $⟨\widehat{N}⟩$ is shown in (b) on a logarithmic scale. At $t=0$, the cavity field is assumed to be in the vacuum state. The dashed curves were obtained with the density operator of the parametric oscillator in Eq. (30), and the solid curves correspond to the full density operator in Eq. (20). The parameters are $g_1=g_2=g$, $\mid {\Omega }_3\mid =25g$, $\mid {\Omega }_4\mid =2g$, ${\Gamma }_1={\Gamma }_2={\Gamma }_3={\Gamma }_4=5g$, ${\Delta }_a=0$, ${\Delta }_b=40g$, ${\kappa }_1={\kappa }_2={10}^{-3}g$, and ${\varphi }_3+{\varphi }_4=\pi ∕2$.

Figure 3

(Color online) (a) Time evolution of $⟨{\left(\Delta \widehat{u}\right)}^2+{\left(\Delta \widehat{v}\right)}^2⟩$. The mean value of the total number of photons $⟨\widehat{N}⟩$ is shown in (b) on a logarithmic scale. At $t=0$, the cavity field is assumed to be in the vacuum state, and we set ${\Gamma }_1={\Gamma }_2={\Gamma }_3={\Gamma }_4=5g$, $g_1=g_2=g$, ${\kappa }_1={\kappa }_2={10}^{-3}g$ and ${\varphi }_3+{\varphi }_4=\pi ∕2$. The parameters for the curves labeled I are $\mid {\Omega }_3\mid =25g$, $\mid {\Omega }_4\mid =9.8g$, ${\Delta }_a=0$, ${\Delta }_b=43g$, and for II we have $\mid {\Omega }_3\mid =15g$, $\mid {\Omega }_4\mid =6g$, ${\Delta }_a=0$, ${\Delta }_b=32.5g$.

Figure 4

(Color online) (a) Time evolution of $⟨{\left(\Delta \widehat{u}\right)}^2+{\left(\Delta \widehat{v}\right)}^2⟩$. The mean value of the total number of photons $⟨\widehat{N}⟩$ is shown in (b) on a logarithmic scale. At $t=0$, the cavity field is assumed to be in the coherent state $\mid 100,-100⟩$, and we set ${\Gamma }_1={\Gamma }_2={\Gamma }_3={\Gamma }_4=2g$, $g_1=g_2=g$, ${\kappa }_1={\kappa }_2={10}^{-2}g$, and ${\varphi }_3+{\varphi }_4=\pi ∕2$. The parameters for the curves labeled I are $\mid {\Omega }_3\mid =10g$, $\mid {\Omega }_4\mid =5g$, ${\Delta }_a=0$, ${\Delta }_b=15g$, and for II we have $\mid {\Omega }_3\mid =10g$, $\mid {\Omega }_4\mid =2g$, ${\Delta }_a=0$, ${\Delta }_b=15g$.