Optical bistability in a $\Lambda$-type atomic system including near dipole-dipole interaction

TL;DR

This paper investigates how near dipole-dipole interactions influence optical bistability in a three-level atomic system with electromagnetically induced transparency, revealing enhanced control and multistability effects.

Contribution

It introduces the role of near dipole-dipole interactions in controlling optical bistability and multistability in a three-level EIT atomic system, extending previous two-level models.

Findings

Near dipole-dipole interaction modifies absorption, dispersion, and nonlinearity.

Optical bistability can be controlled and transitioned to multistability.

Enhanced controllability of OB through atomic coherence and NDD effects.

Abstract

The advantage of optical bistability (OB) using three-level electromagnetically induced transparency (EIT) atomic system over the two-level system is its controllability, as absorption, dispersion, and optical nonlinearity in one of the atomic transitions can be modified considerably by the field interacting with nearby atomic transitions. This is due to induced atomic coherences generated in such EIT system. The inclusion of near dipole-dipole (NDD) interaction among atoms further modifies absorption, dispersion, and optical nonlinearity of three-level EIT system and the OB can also be controlled by this interaction, producing OB to multistability.

Figures

Equations

$${\dot{\rho }}_{22}-{\dot{\rho }}_{11}=$$

$$+i{\mu }_{23}({\epsilon }_L^C{)}^{\ast }{\rho }_{23}-i{\mu }_{23}({\epsilon }_L^C){\rho }_{32}-{\gamma }_{31}({\rho }_{33}-{\rho }_{11})-{\gamma }_{21}^D|{\rho }_{21}{|}^2,$$

$${\dot{\rho }}_{22}-{\dot{\rho }}_{33}=$$

$$+2i{\mu }_{23}({\epsilon }_L^C{)}^{\ast }{\rho }_{23}-2i{\mu }_{23}({\epsilon }_L^C){\rho }_{32}-{\gamma }_{31}({\rho }_{11}-{\rho }_{33})-{\gamma }_{23}^D|{\rho }_{23}{|}^2,$$

$${\dot{\rho }}_{23}=$$

$$+i{\mu }_{23}({\epsilon }_L^C)({\rho }_{22}-{\rho }_{33})-i{\mu }_{12}({\epsilon }_L^P){\rho }_{13},$$

$${\dot{\rho }}_{21}=$$

$$+i{\mu }_{12}({\epsilon }_L^P)({\rho }_{22}-{\rho }_{11})-i{\mu }_{23}({\epsilon }_L^C){\rho }_{31},$$

$${\dot{\rho }}_{31}=$$

$$-i{\mu }_{23}({\epsilon }_L^C{)}^{\ast }{\rho }_{21}+i{\mu }_{12}({\epsilon }_L^P){\rho }_{32},$$

$$\vec{\mathbf{E}}=({\vec{\mathbf{E}}}_P{e}^{-i{\omega }_{P}t}+{\vec{\mathbf{E}}}_C{e}^{-i{\omega }_{C}t}+c.c.).$$

$$E_P^T$$

$$E_P(0,t)$$

$$\frac{\partial E_P}{\partial t}+c\frac{\partial E_P}{\partial z}=2\pi i{\omega }_P{\mu }_{12}P({\omega }_P),$$

$$E_P^T$$

$$E_P(0)$$

Full text

1 Introduction

Atomic optical bistability (AOB) is a phenomenon that exploits both the cooperative nature of the interaction between a group of atoms with the field and its strong nonlinearity. The essential element in AOB is the nonlinear relation between the applied electromagnetic field and the electromagnetic field that is radiated by the atomic dipoles, when the actions of all the other dipoles are taken into account. Cooperative radiation therefore plays a key role in bistability [1].

The optical bistability (OB) in an ensemble of two-level atomic systems contained inside an optical cavity has been extensively studied in the decade of eighties and early years of nineties. The studies were focused mainly on to utilize the phenomenon of OB in optical memories, optical transistor-like systems and all-optical switches [2, 3]. The observed phenomenon of OB is classified as absorptive OB if the atomic transition is saturated. On the other hand the intensity-dependent refractive index of the media is the mechanism for the generating dispersive OB [2, 3, 4, 5, 6, 7].

OB is a phenomenon that exploits both the cooperation among atoms and the strong nonlinearity produced by the interacting field. The two-level OB systems show first-order phase transition like behavior because in Focker–Plank equation the diffusion coefficient is intensity dependent [8, 9]. Several interesting theoretical and experimental works reported for the two-level OB systems in past are well documented in the literature. However, experiments with two-level OB systems have limitations due to the lack of control in generating desirable OB curves with the help of only one field. The electromagnetically induced transparency (EIT) phenomena and related quantities, e.g., dispersion, nonlinearity etc. in multi-level systems have given a significant impact on the control of OB and optical multistability in such systems [10, 11, 12, 13, 14, 15, 16, 17] . In multi-level systems, the two or more than two interacting fields are generating the induced atomic coherence, which modifies the linear absorption, dispersion and enhances the third-order Kerr nonlinear index of refraction.

Some studies on the behavior of nonlinearity with intensities and frequency detunings of both probe and coupling beams were reported [18] in the $\Lambda$-type atomic system to demonstrate the controllability of OB during experiments in such system. Controlling of OB hysteresis cycle is useful in all-optical switches, optical memories, optical transistors, and logic circuits. This will avoid optical–electronic–optical conversion of information in the form of signals. The atomic coherence in EIT systems enhances Kerr nonlinearity, and this brings down the switching thresholds of such devices. Thus the control becomes easier and more efficient, at very low intensity levels of light.

The cavity output field of atom-field composite system for some parametric conditions becomes unstable, giving rise to dynamic instability and chaos as the optical pumping in the coupling transition competes with the saturation in the probe transition [19, 20]. This is achieved by adjusting the coupling beam Intensity and/or frequency detuning. The phenomenon of stochastic resonance [21] in optical bistability is also reported in three-level systems. The noise induced switching phenomenon [22, 23] in three-level bistable systems is also observed, not available with the two-level systems conveniently.

Propagation of electromagnetic field can generate near dipole-dipole (NDD) interaction in a collection of atoms. The NDD effects can introduce the inversion-dependent-chirping of the atomic resonance frequency in a two-level atomic sample. The microscopic field-atom coupling can be modified due to NDD effect. The microscopic field results from the composition of macroscopic field with the induced polarization [24]. The NDD interaction is realized in a dense medium from entities confined in a very small volume of the order of a cubic wavelength. The modified Maxwell-Bloch equation in which NDD effects are included gives some extraordinary results such as invariant pulse propagation that departs from the hyperbolic secant pulse shape of self-induced transparency (SIT) [25] and self-phase modulation in SIT [26]. The interaction of external driving field with a atomic sample generates reaction field that works against driving field causing reduction of total field. The effect of strong external driving field in comparison to generated reaction field due to the dipole-dipole interaction, results in a first-order phase transition far away from the equilibrium due to the suppression of reaction field [27, 28]. The Bose Einstein condensates can be realized at high temperatures [29] with the help dipole-dipole interaction. In systems used for quantum information processing the entanglement of qubits [30] can be achieved through dipole-dipole interaction. The dipole-dipole interaction is also important during the photosynthesis process [31, 32]. The desired value of dipole–dipole or NDD interaction among entities can be achieved by change of the number density. The dipole–dipole interaction strength and its direction depends on environmental electromagnetic field modes, which can be obtained by directional stop-gaps or manipulating degeneration of the field modes [33, 34]. By using specific optical confinement structures, the desired dipole–dipole interaction can be realized [35, 36].

Recently, the dynamical evolution of a three-level $\Lambda$-type atomic system including near-dipole–dipole interaction was investigated. The linear and third order non-linear susceptibilities for the probe transition was studied and details of absorption and dispersion in the probe transition and dynamic evolution of the system were obtained [37]. The aim of this work is to study the phenomenon of OB in a three-level $\Lambda$-type system including NDD interaction among atoms and thus demonstrate the controllability of OB via NDD interaction.

The remaining paper is organized as follows. In Section 2 the model for this work for OB is described. Numerical results for OB are discussed in Section 3 by the solution of the exact density matrix equation. In Section 4 some concluding remarks are presented.

2 The Model

We consider a cavity illustrated in Fig. 1, in which there is a nonlinear medium of length $L$ in one arm, driven by a field $E_{P}^{I}$ that travels the medium in one direction only [23, 38]. We assume that the input and output mirrors, $M_{1}$ and $M_{2}$, have identical reflectance $\sqrt{R}$ and transmittance $\sqrt{T}$ amplitudes (with $T=1-R$), whereas the other two mirrors are fully reflecting ($R=1$). We also assume that the mirror separation $\ell$ is adjusted so that the cavity is tuned to near resonance with the applied fields. The nonlinear medium consists of a three-level atomic system with energy levels $E_{i}$ ($i=1,2,3$) and in a $\Lambda$-type configuration ($E_{2}>E_{3}>E_{1}$), interacting with probe and coupling lasers of frequencies $\omega_{P}$ and $\omega_{C}$, with amplitudes $E_{P}$ and $E_{C}$, respectively (see Fig. 2). The probe laser acts on the dipole-allowed transition $\lvert 1\rangle\rightarrow\lvert 2\rangle$, with energy difference $\omega_{21}$ and frequency detuning $\Delta_{P}=\omega_{P}-\omega_{21}$. The coupling laser couples to the other dipole-allowed transition $\lvert 2\rangle\rightarrow\lvert 3\rangle$, with energy difference $\omega_{23}$ and frequency detuning $\Delta_{C}=\omega_{C}-\omega_{23}$.

In the semiclassical approximation, where the system interacts with the classical electromagnetic fields of the two lasers, the Liouville equations of the density-matrix elements in the dipole and rotating wave approximation, including NDD interaction as described in [37], have the form

[TABLE]

where $\varepsilon_{L}^{P}$ and $\varepsilon_{L}^{C}$ are complex, microscopic, slowly-varying electric field envelopes of the probe and coupling fields, respectively; $\gamma_{21}$ and $\gamma_{23}$ are the radiative decay rates from level $\lvert 2\rangle$ to levels $\lvert 1\rangle$ and $\lvert 3\rangle$, respectively; population decays from level $\lvert 3\rangle$ back to level $\lvert 1\rangle$ at a rate $\gamma_{31}$ in non-radiative manner, and we define $\gamma=\frac{1}{2}(\gamma_{21}+\gamma_{32}+\gamma_{31})$; $\Omega_{P}=2\mu_{12}\,\varepsilon_{L}^{P}$ is the Rabi frequency of the probe field and ($\Omega_{C}=2\mu_{32}\,\varepsilon_{L}^{C}$) the Rabi frequency of the coupling field; $\mu_{12}$ is the dipole matrix element for transition $\lvert 1\rangle\rightarrow\lvert 2\rangle$ and $\mu_{23}$ for transition ($\lvert 2\rangle\rightarrow\lvert 3\rangle$). Equations (1) describe the atomic dynamics of the system. $\gamma_{21}^{D}$ is the dipole-dipole induced decay rate due to NDD interaction to level $\lvert 1\rangle$ and $\gamma_{23}^{D}$ to level $\lvert 3\rangle$ [37].

The combined electric field acting on each atom inside the cavity is the sum of the probe and coupling fields, so that

[TABLE]

The probe field $E_{P}$ at frequency $\omega_{P}$ (interacting with the atomic transition $\lvert 1\rangle\rightarrow\lvert 2\rangle$) circulates inside the cavity acting as the cavity field. The coupling field $E_{C}$ at frequency $\omega_{C}$ (coupling transition $\lvert 3\rangle\rightarrow\lvert 2\rangle$) does not circulate inside the optical ring cavity as shown in Fig. 1. However, the two fields propagate through the nonlinear medium in the same direction, so the first-order Doppler effect in this three-level, EIT system can be eliminated [16]. In fact, the coupling field behaves like a controlling field. The field $E_{P}$ enters the cavity from the partially transparent mirror $M_{1}$ and drives the nonlinear medium. To determine the field inside the unidirectional ring cavity, we write the relations obtained by boundary conditions [38]

[TABLE]

where the length of the atomic sample is specified by the parameter $L$, $\ell$ is the side arm length between $M_{2}$ and $M_{3}$ (and also $M_{4}$ and $M_{1}$), and the time taken by light to travel from mirror $M_{2}$ to $M_{1}$ via mirrors $M_{3}$ and $M_{4}$ is given by $\Delta t=(2\ell+L)/c$. The cavity frequency detuning is defined as $\delta_{0}=(\omega_{\text{cav}}-\omega_{P})L_{T}/c$, where $\omega_{\text{cav}}$ is the frequency of the nearest cavity mode to frequency $\omega_{P}$, and $L_{T}\approx 2(\ell+L)$ represents the round-trip length of the ring cavity.

The dynamical evolution of the probe field inside the optical cavity is described by the Maxwell’s equation (in the slowly-varying envelope approximation) [38]

[TABLE]

where $P(\omega_{P})=N\mu_{21}\rho_{12}$ is the induced atomic polarization responsible for AOB, and $N$ is the number density of atoms. In order to obtain the polarization $P(\omega_{P})$, one needs to first solve the set of density-matrix equations (1) in the steady-state limit, then integrate (5) over the length of the atomic sample using (3) and (4). The steady-state boundary conditions can be written as [38]

[TABLE]

3 Results and Discussion

We solved equations (1) numerically and plotted the cavity output field ($E_{P}^{T}$) versus the cavity input field ($E_{P}^{I}$) for various parametric values of the coupling field and NDD interaction parameter. The output-input features for a system showing OB are shown in Figs. 3–6.

Figure 3(a) shows the curves for a three-level system in Raman configuration of its levels with no dipole-dipole interaction for different $\Omega_{C}$ values of the coupling field. The atomic sample is kept inside the cavity (see Fig. 1). The parameters used were $\gamma^{D}_{21}=\gamma^{D}_{23}=0$, $\Delta_{P}=\Delta_{C}=0$, $\epsilon_{P}=\epsilon_{C}=0$, and $C=150$ as the cooperative parameter for OB  [38]. We changed $\Omega_{C}$ to $0.05$, $0.1$, $0.2$, $0.5$, and $1.0$. For a small $\Omega_{C}$, the OB threshold was large (solid line curve). When we increased $\Omega_{C}$, the threshold reduced, as seen in the figure. For a very large $\Omega_{C}$, the OB feature disappeared. The effect of a finite detuning of the coupling field is shown in Fig. 3(b). Here, the only parameter modified was $\Delta_{C}=1.0$ leaving the other parameters unchanged. The non-zero coupling field detuning enhanced the OB threshold, as it can be observed by comparing the solid lines in Figs. 3(a) and 3(b). This threshold decreased for bigger values of the coupling field. The different curves correspond to values of $\Omega_{C}=1.0$, $3.0$, $5.0$, $7.0$, and $10$ (see [38]).

Figures 4(a,b) showed the impact of non-zero, real and imaginary parts of NDD interaction. We used parameters $\gamma^{D}_{21}=\gamma^{D}_{23}=0$, $\Delta_{P}=\Delta_{C}=0$, $\Omega_{C}=2.0$, and $C=150$. The effect of the imaginary part of NDD interaction is depicted in the curves plotted (Fig. 4(a)) for $\epsilon_{P}=\epsilon_{C}=0.1$, $0.5$, $1.0$, $1.5$, and $2.0$. For smaller values of $\epsilon$ ($0.1$, $0.5$), the curves showed OB. For a further increase in the value of $\epsilon$ ($1.0$, $1.5$, $2.0$), we observed the interesting phenomenon of optical multistabilty in the curves. The threshold of all these OB and multistability curves can be altered by modifying the value of the coupling field because this changes the induced coherence and hence the threshold. One such interesting study is shown in Fig. 4(b), where we changed $\Omega_{C}$ to $3.0$ but kept all other parameters the same. By doing so, the threshold of OB and multistability curves came down. Again, this was due to the altered induced coherence.

In Fig. 5(a,b), we showed the controllability of OB by keeping the values of an NDD interaction parameter fixed. Parameters used for Fig. 5(a) were $\gamma^{D}_{21}=\gamma^{D}_{23}=0$, $\Delta_{P}=\Delta_{C}=0$, $C=150$, and $\epsilon_{C}=\epsilon_{P}=1.0$. We plotted curves for $\Omega_{C}=1.0$, $3.0$, $5.0$, $7.0$ and $10.0$. At lower values of $\Omega_{C}$ (1.0 and 3.0), we found multistability that gradually changed to OB as $\Omega_{C}$ further increased to 5.0, 7.0 and 10.0. Thus for a fixed value of the NDD parameter $\epsilon$ (imaginary part), the transition from OB to multistability can be controlled by changing the coupling field strength. However, when we introduced the real part of a NDD parameter, such as $\gamma^{D}_{21}=\gamma^{D}_{23}=1.5$ in Fig. 5(b), the shape of all curves remained similar to those in Fig. 5(a) but the OB threshold of all curves increased. This effect is because of the associated increase in the system’s dampening that reduces the induced coherence.

For Fig. 6(a), the values for the NDD parameters were $\epsilon_{C}=\epsilon_{P}=2.0$, $\gamma^{D}_{21}=\gamma^{D}_{23}=0.0$, while for Fig. 6(b), the values were $\epsilon_{C}=\epsilon_{P}=2.0$, and $\gamma^{D}_{21}=\gamma^{D}_{23}=3.0$. Figures 6(a,b) showed reduction in the multistability structure for $\Omega_{C}=1.0$ when compared with Fig. 5(a,b). $\Omega_{C}=3.0$ still showed multistability, but the threshold raised as compared to those in Figs. 5(a,b). For further increase in $\Omega_{C}$, all curves displayed OB but with an augmented threshold value. The physical interpretation of these results is implicit in the manipulation of the physical properties of the three-level, EIT medium considered in this work. We achieved this by varying the coupling field and the NDD interaction parameters in the atomic system that changed its generated atomic coherence [37]. While keeping the coupling field constant, we varied the NDD parameters to control absorption, dispersion, and nonlinearity of the medium and to get changes in its OB.

4 Conclusion

In this work, we studied the optical bistability displayed by a three-level, $\Lambda$-type electromagnetic induced transparency medium including near dipole-dipole interaction. The inclusion of NDD interaction provided another control parameter for changing the absorption, dispersion, and nonlinearity of such EIT system. In fact, by modifying the number density of the atomic medium, we may obtain wanted values of NDD parameters. Consequently, by the combined effect of the coupling field and NDD interaction parameters, the system exhibits interesting changes as it passes from OB to optical multistability. This results could be experimentally realized in the three-level atomic system described in Refs. [23, 38].

Acknowledgments

One of us (A.J.) is thankful to G. Duree (RHIT) for his interest and support for this work.

Figure Captions

Figure 1: Schematic diagram of a unidirectional ring cavity having four mirrors ($M_{1}\text{--}M_{4}$) and an atomic vapor cell of length $L$. Mirrors $M_{3}$ and $M_{4}$ are perfectly reflecting mirrors ($R=1$). The incident and transmitted fields are represented by $E_{P}^{I}$ and $E_{P}^{T}$, respectively, while the coupling field is represented by $E_{C}$.

Figure 2: Schematic diagram of the three-level system in a $\Lambda$-configuration driven by probe and coupling lasers of frequencies $\omega_{P}$ and $\omega_{C}$, respectively.

Figure 3: Optical bistability (in terms of output vs. input cavity field) for a three-level system in Raman configuration of its states. The Rabi frequency $\Omega_{C}$ is variable. (a) $\Delta_{C}=0$, (b) $\Delta_{C}=1.0$.

Figure 4: Output vs. input cavity field of the system as a result of non-zero, near dipole-dipole interaction (NDD). NDD (imaginary part) parameter $\epsilon_{C}$, which is varying. (a) $\Omega_{C}=2.0$, (b) $\Omega_{C}=3.0$.

Figure 5: Output vs. input cavity field of the system for non-zero, near dipole-dipole interaction with $\epsilon_{P}=\epsilon_{C}=2.0$. The Rabi frequency $\Omega_{C}$ is variable. NDD (real part) parameters $\gamma_{21}^{D}$ and $\gamma_{23}^{D}$ are changed as: (a) $\gamma_{21}^{D}=\gamma_{23}^{D}=0$, (b) $\gamma_{21}^{D}=\gamma_{23}^{D}=1.5$.

Figure 6: Same as Fig. 5, with $\epsilon_{P}=\epsilon_{C}=2.0$. (a) $\gamma_{21}^{D}=\gamma_{23}^{D}=0$, (b) $\gamma_{21}^{D}=\gamma_{23}^{D}=3.0$.