Efficient all-optical switch using a $\Lambda$ atom in a cavity QED system

We propose an all-optical switch constructed from a two-mode optical resonator containing a strongly coupled, three-state system. The coupling allows a weak, continuous wave laser drive to incoherently control the transmission of a much stronger, continuous wave signal laser into (and through) the resonator. We demonstrate that in this simple setup the presence of a control drive with 1/10$\text{th}$ the power of the signal drive can induce near complete reflection of the signal, while its absence allows for near complete transmission. The switch can also be operated as a set-reset relay with two control inputs that efficiently drive the switch into either the reflecting or the transmitting state.

Figure 1

(a) The switch consists of a three level $\Lambda$ system coupled to two cavity modes with frequencies ${\omega }_{ca}$ and ${\omega }_{cb}$, respectively. The $a$ mode ($b$ mode) is coupled to the $|H\rangle \leftrightarrow |E\rangle$ ($|G\rangle \leftrightarrow |E\rangle$) transition and spontaneous emission into radiation modes accompanied by a transition from $|E\rangle$ to $|H\rangle$ or from $|E\rangle$ to $|G\rangle$ also occurs. When one control field is used, the state (“on” or “off”) of the $a$ drive ideally determines whether the signal field ($b$ drive), which is always on resonance with the bare cavity, is reflected or transmitted through the cavity, and when two control fields are used (one driving the $a$ mode and the other driving the $b$ mode at a different frequency than the signal field), the switch is ideally driven into the reflecting (transmitting) state if a few photons are present in the control field driving the $a$ mode ($b$ mode). (b) The level structure of the seven energy eigenstates of the switch with at most one excitation in the $\Lambda$ system or in one of the cavity modes when all couplings to the environment (including driving) are neglected and states with more excitations are ignored. (We use the parameters in Table and draw the level structure for ${\omega }_{ca}={\omega }_{cb}$.) The three levels for which no quantum states are given are linear combinations of $|E,0,0\rangle$, $|H,1,0\rangle$, and $|G,0,1\rangle$, where $|A,n,m\rangle \equiv |A\rangle \otimes {|n\rangle }_a\otimes {|m\rangle }_b$ and ${|n/m\rangle }_{a/b}$ is a photon number state of cavity mode $a$/$b$. (c) The same as (b), but in the rotating frame defined in Sec. 2a.

Figure 2

The response of the switch to the signal and control drives, depicted using the lowest-lying energy eigenstates from Fig. 1b. Red and blue solid arrows represent the energies of the control (red) and signal (blue) optical drives. Horizontal red and blue lines are the energy eigenstates containing one $\sim \hslash {\omega }_{cb}$ quantum of excitation that can be excited near-resonantly or off-resonantly by the control and signal drives, respectively. Dashed arrows represent decay processes that can change the state of the switch if the associated excited states are populated by the drives (see text). (a) The response of the system when the $\Lambda$ system is in $|H\rangle$: the signal resonantly excites the cavity and is transmitted; when the control drive is present, it near-resonantly excites two excited states that will efficiently pump the switch into the $|G\rangle$ state. (b) The response of the system when the $\Lambda$ system is in $|G\rangle$: the signal drive cannot efficiently excite the device and is thus reflected; however, occasional, off-resonant excitations will cause the $\Lambda$ system to be pumped into the $|H\rangle$ state. The set-reset relay configuration is not depicted.

Figure 3

The expectation value of the number of photons in the cavity modes in steady state as a function of the detuning between the $a$ drive and the cavity resonance. The units ${\langle a^{†}a\rangle }_0$ and ${\langle b^{†}b\rangle }_0$ are the same expectation values for an empty cavity driven on resonance (assuming the $a$ drive is on). The labels “on” and “off” refer to the $a$ drive being on and off, respectively, and the three vertical dotted lines are the energies of the states in Fig. 1c that are linear combinations of $|E,0,0\rangle$, $|H,1,0\rangle$, and $|G,0,1\rangle$ minus the energy $\delta +{\theta }_a$ of $|H,0,0\rangle$ in the rotating frame. The maximum distance $D$ between the solid and the dashed lines quantifies the ability of the switch to control the transmission of the signal field.

Figure 4

Particularly illustrative examples of Monte Carlo simulations of the switch dynamics for the $a$ drive being (a) on and (b) off, respectively. The initial state is $|H,0,0\rangle$ in (a) and $|G,0,0\rangle$ in (b). The crosses, circles, and triangles show the time at which the various types of jumps occur as specified in the legend in part (a) of the figure (for the cavity decays, we show only the jumps corresponding to photons transmitted through the cavity).

Figure 5

Time evolution of $\langle b^{†}b\rangle /{\langle b^{†}b\rangle }_0$ when the $a$ drive is suddenly turned on after being off for a long time (solid line) and when the $a$ drive is suddenly turned off after being on for a long time (dashed line). (Note the different scales.)

Figure 6

Dynamics of the set-reset relay. The system is initiated to $|H,0,0\rangle$ at time ${\gamma }_{b}t=0$, and the $a$ drive is turned on from ${\gamma }_{b}t=0$ to ${\gamma }_{b}t=2000$ to drive the relay to the reflecting state (not shown). From ${\gamma }_{b}t=2000$ to ${\gamma }_{b}t=4000$, the c field is on and the $a$ drive is off, and this affects a transition to the transmitting state. The relay stays in this state when both control fields are subsequently turned off. Turning on the $a$ drive from ${\gamma }_{b}t=6000$ to ${\gamma }_{b}t=8000$ leads to a transition to the reflecting state. When the system is in the reflecting state and both control fields are off, there is a decay toward the transmitting state at the rate $1/T_{\mathrm{off}}$, which is much slower than the transition rate seen for ${\gamma }_{b}t\in [2000,4000]$. Note that a time interval of $2000{\gamma }_b^{-1}$ corresponds to the arrival of 20 control photons ($a$ or c) on average if ${\kappa }_{a,\mathrm{in}}={\kappa }_a/2={\gamma }_b/2$ and ${\kappa }_{b,\mathrm{in}}={\kappa }_b/2={\gamma }_b/2$.

Figure 7

Signaling contrast $D$ (see Fig. 3) as a function of $g_b$ for ${\gamma }_a=0.2{\gamma }_b$ (solid lines) and ${\gamma }_a={\gamma }_b$ (dashed lines) and various values of ${\kappa }_a={\kappa }_b$ as indicated. ${\mathcal{E}}_a=\sqrt{0.01{\kappa }_a{\gamma }_b}$, ${\mathcal{E}}_b=\sqrt{0.1{\kappa }_b{\gamma }_b}$, ${\Theta }_b=0$, and ${\Theta }_a$, $g_a$, $\delta$, and $\Delta$ are optimized numerically to maximize $D$.

Figure 8

The optimal choice of ${\Theta }_a$, $g_a$, $\delta$, and $\Delta$ for the curve in Fig. 7 with ${\gamma }_a=0.2{\gamma }_b$ and ${\kappa }_a={\kappa }_b={\gamma }_b$. The inset shows $D$ as a function of $g_a$ for ${\gamma }_a=0.2{\gamma }_b$, ${\kappa }_a={\kappa }_b={\gamma }_b$, and $g_b=10{\gamma }_b$ when only $\Delta$, $\delta$, and ${\theta }_a$ are optimized.

Figure 9

Switching times $T_{\mathrm{on}}$ (a) and $T_{\mathrm{off}}$ (b) for the same parameters as in Fig. 7. Again, ${\gamma }_a=0.2{\gamma }_b$ for the solid curves and ${\gamma }_a={\gamma }_b$ for the dashed curves. [Note that the vertical axis is logarithmic in (b) but not in (a).]

Figure 10

Enlarged view of Fig. 7. The dotted lines show $T_{\mathrm{off}}/(T_{\mathrm{on}}+T_{\mathrm{off}})$.