Optical-bistability-enabled control of resonant light transmission for an atom-cavity system

The control of light transmission through a standing-wave Fabry-Pérot cavity containing atoms is theoretically and numerically investigated, when the cavity mode beam and an intersecting control beam are both close to specific atomic resonances. A four-level atomic system is considered and its interaction with the cavity mode is studied by solving for the cavity field and atomic state populations. The conditions for optical bistability of the atom-cavity system are obtained. The response of the intracavity intensity to an intersecting beam on atomic resonance is understood in the presence of stationary atoms (closed system) and nonstatic atoms (open system) in the cavity. The nonstatic system of atoms is modelled by adjusting the atomic state populations to represent the exchange of atoms in the cavity mode, which corresponds to a thermal environment where atoms are moving in and out of the cavity mode volume. The control behavior with three- and two-level atomic systems is also studied, and the rich physics arising out of these systems for closed and open atomic systems is discussed. The solutions to the models are used to interpret the steady-state and transient behavior observed by Sharma et al. [Phys. Rev. A 91, 043824 (2015)].

Figure 1

(a) Schematic diagram of the system. The atoms are enclosed in a cell, located within the Fabry-Pérot cavity. The filled circles represent atoms overlapped with the mode and the empty circles represent the background atom vapor. The incident probe beam on cavity resonance is transmitted through. A transverse control beam (dashed line) intersects the cavity mode volume. On atomic resonance, cavity mode light intensity is affected by the presence of the control light beam, which affects the light transmitted through the cavity. The inset (b) shows the atomic level scheme. The probe laser beam is near resonant with transition $\left|1\right.〉\leftrightarrow \left|3\right.〉$ with frequency ${\omega }_p$ and detuning ${\mathrm{\Delta }}_p$, and the control laser beam is near resonant with $\left|2\right.〉\leftrightarrow \left|4\right.〉$ with frequency ${\omega }_c$ and detuning ${\mathrm{\Delta }}_c$.

Figure 2

Transmitted power of cavity vs input power after solving Eq. (4) numerically and converting to SI units. The different curves correspond to different detunings of the probe laser from the atomic resonance. When there are no atoms, the response is linear. As ${\mathrm{\Delta }}_p$ changes, the nature of the hysteresis and therefore the region for bistable behavior changes. All curves are for control power corresponding to $\mathrm{\Omega }/2\pi =-50$ GHz and detuning ${\mathrm{\Delta }}_c=0$.

Figure 3

Bistable curves for two different control beam powers with ${\mathrm{\Delta }}_p=2\mathrm{\Gamma }$.

Figure 4

Hysteresis for the two different control beam powers in Fig. 3, with probe beam detuning ${\mathrm{\Delta }}_p=2\mathrm{\Gamma }$. The hysteresis shown here is a result of following the change in output as the input is increased and decreased adiabatically.

Figure 5

Bistability in case of a three-level system for different values of detuning. Here $\left|\mathrm{\Omega }\right|/2\pi =5.15$ MHz. Unlike the four-level system, the bistable curves have two distinct regions of negative slope in some cases. The four curves illustrate the different cases of probe and control detuning, as specified in the legend.

Figure 6

Double hysteresis for the case of ${\mathrm{\Delta }}_p=0,{\mathrm{\Delta }}_c=2\mathrm{\Gamma }$ and $\left|\mathrm{\Omega }\right|/2\pi =5.15$ MHz in Fig. 5. The increasing and decreasing legend represents the sense of the change in the input probe power.

Figure 7

Bistability when atoms are flowing in and out of cavity. The two curves correspond to the condition when the control light is off and on as seen in the legend.

Figure 8

Time evolution of cavity output power when the cavity input laser is switched by the control beam. Here the control beam off is $\left|\mathrm{\Omega }\right|/2\pi =0$ and the control beam on is $\left|\mathrm{\Omega }\right|/2\pi =1.76$ MHz. The input power of $430 \mu \mathrm{W}$ corresponds to the bistability parameter region in Fig. 7. The submillisecond slow response of the output power is clearly seen. The inset shows the submicrosecond response of the transmitted power, where the left ordinate axis is the power in the control off case and right axis is the power for the control on. The time axis is in milliseconds. Clearly the short-time behavior is complex.

Figure 9

Bistability for atoms with exchange when control beam is in resonance with levels $\left|1\right.〉$ and $\left|3\right.〉$, making it a two-level system with loss. Here the fraction of atoms addressed by control beam is 0.2, and all other parameters are the same as the values used for Fig. 7.

Figure 10

Schematic diagram for closed system of atoms. The black spot can represent an ensemble of laser cooled and trapped atoms or ions, or atoms in a dipole trap.

Figure 11

Fast switching times for closed four-level system. Here the control beam off is $\left|\mathrm{\Omega }\right|/2\pi =0$ and the control beam on is $\left|\mathrm{\Omega }\right|/2\pi =31$ MHz. The input power of $1 \mu \mathrm{W}$ corresponds to the lower part of the second bistability curve in Fig. 3 when control is on and the no-atoms curve in Fig. 2 when control is off.

Figure 12

Rise and decay of transmitted intensity when cavity laser is switched on and off respectively. The input power of $300 \mu \mathrm{W}$ corresponds to the lower branch of hysteresis curve in Fig. 7 with $\mathrm{\Omega }=0$.