Tunable Bistability in Hybrid Bose-Einstein Condensate Optomechanics

Cavity-optomechanics, a rapidly developing area of research, has made a remarkable progress. A stunning manifestation of optomechanical phenomena is in exploiting the mechanical effects of light to couple the optical degree of freedom with mechanical degree of freedom. In this report, we investigate the controlled bistable dynamics of such hybrid optomechanical system composed of cigar-shaped Bose-Einstein condensate (BEC) trapped inside high-finesse optical cavity with one moving-end mirror and is driven by a single mode optical field. The numerical results provide evidence for controlled optical bistability in optomechanics using transverse optical field which directly interacts with atoms causing the coupling of transverse field with momentum side modes, exited by intra-cavity field. This technique of transverse field coupling is also used to control bistable dynamics of both moving-end mirror and BEC. The report provides an understanding of temporal dynamics of moving-end mirror and BEC with respect to transverse field. Moreover, dependence of effective potential of the system on transverse field has also been discussed. To observe this phenomena in laboratory, we have suggested a certain set of experimental parameters. These findings provide a platform to investigate the tunable behavior of novel phenomenon like electromagnetically induced transparency and entanglement in hybrid systems.

In this report, we investigate tunable bistable behavior of optomechanical system which consists of one fixed mirror, one moving-end mirror with maximum amplitude q 0 1 and Bose-Einstein condensate (BEC) trapped inside the cavity, driven by a single mode optical field 38,39 . The idea of present work is motivated by the findings of Ref. 39. They demonstrated controlled optical switching of field inside an optical cavity containing ultra-cold atoms. They used transverse optical field to control bistable behavior of intra-cavity photons induced by external pump field. We use this approach of transverse optical field to control the bistable behavior of hybrid optomechanical system and show that the optical bistability or photons bistability inside the cavity is increased by increasing the strength of transverse field. The demonstration of results shows that the transverse optical field can also be used to control the bistable dynamics of moving-end mirror as well as BEC. We also investigate how this bistable behavior can affect the effective potential of the system. The set of experimental parameters which make these results experimentally feasible in laboratory has also been provided.

BEC Optomechanics
Optomechanical system which is a Fabry-Pérot cavity of length L consists of a fixed mirror and a moving-end mirror driven by a single mode parallel pump field with frequency ω p , as shown in Fig. 1. Cigar-Shaped BEC, with N-two level atoms, is trapped inside optical cavity 2,40,41 . Moreover, optomechanical system is also driven by transverse optical field which interacts perpendicularly with BEC with intensity η^ and frequency ω^3 8,39,42 . Counter propagating field inside the cavity forms a one-dimensional optical lattice. The moving-end mirror possesses harmonic vibrations under radiation pressure with frequency ω m , with maximum amplitude q 0 and exhibits Brownian motion in the absence of intra-cavity field. To make this study of tunable bistability in hybrid BEC-Optomechanics experimentally feasible, we suggest a set of particular parameters from present available experimental setups 1,2,41,43,44 . We consider N = 2.3 × 4 87 Rb atoms trapped inside Fabry-Pérot cavity with length L = 1.25 × 10 −4 , driven by single mode external field with power P in = 0.0164mW, frequency ω p = 3.8 × 2π × 10 14 Hz and wavelength λ p = 780nm. The frequency of intra-cavity field is ω c = 15.3 × 2π × 10 14 Hz, with decay rate κ = 1.3 × 2πkHz. Intra-cavity field produces recoil of ω r = 3.8 × 2πkHz in atoms trapped inside cavity. Intra-cavity atomic mode performs harmonic motion with damping rate γ a = 0.21 × 2πkHz. The coupling of intra-cavity field with BEC is ξ sm = 4.4 MHz. Detuning of the system is taken as Δ = Δ c + U 0 N/ 2 = 0.52 × 2πMHz, where vacuum Rabi frequency of the system is U 0 = 3.1 × 2πMHz. The moving-end mirror of cavity should be a perfect reflector that oscillates with a frequency ω m = 15.2 × 2πMHz and damping γ m = 1.1 × 2πkHz. strength of its coupling with intra-cavity field is considered as ξ = 3.8 MHz.
The complete Hamiltonian of the system consists of three parts, here, H AF describes behavior of the atomic mode (BEC) and its coupling with cavity-optomechanics, H MF is related to the intra-cavity field and its association with the moving-end mirror while, H T accounts for noises and damping associated with the system. Cigar-shaped Bose-Einstein condensate is trapped inside a Fabry-Pérot cavity of length L with a fixed mirror and a moving-end mirror having maximum amplitude of q 0 and frequency ω m , driven by a single mode optical field with frequency ω p . Intra-cavity field interacts with BEC and due to photon recoil, the momentum side modes are generated in the matter wave. We use another transverse field, with intensity η^ and frequency ω^, to control the bistable dynamics of the system. We use strong detuning regime to adiabatically eliminate internal excited levels of atomic mode inside the cavity potential. We further apply rotating frame at external field frequency to derive Hamiltonian H AF having quantized motion of atoms along the cavity axis. To avoid atom-atom interactions in BEC, we assume BEC is dilute enough so that many-body interaction effects can easily be ignored 38,39 .
ψ ψˆ † is bosonic annihilation (creation) operator and U g a 0 0 2 = /∆ , g 0 is the vacuum Rabi frequency and Δ a is far-off detuning between field frequency and atomic transition frequency ω 0 . The mass of an atom is represented by m a and k is the wave number for intra-cavity field. η^ = g 0 Ω p /Δ a is the coupling of BEC with transverse field and represents maximum scattering and Ω p is the Rabi frequency of the transverse pump field. Due to field interaction with BEC, photon recoil takes place that generates symmetric momentum ± 2lħk side modes, where, l is an integer. In the presence of weak field approximation, we consider low photon coupling. Therefore, only lowest order perturbation of the wave function will survive and higher order perturbation will be ignored. So, ψ is defined depending upon these 0 th , 1 st and 2 nd modes 15 as, ere, b 0 , b 1 and b 2 are annihilation operators for 0 th , 1 st and 2 nd modes respectively. By using x ψ( ) defined above in Hamiltonian H a , we write the Hamiltonian governing the field-condensate interaction as, The sum of number of particles in all momentum side modes is , where, N is the total number of bosonic particles. As population in 0 th mode is much larger than the population in 1 st and 2 nd order side mode, therefore, we can comparatively ignore the population in 1 st and 2 nd order side mode and can write b b 0 0 †  N or b 0 and b N 0 → † . This is possible when side modes are weak enough to be ignored. Under these assumptions, we write simplified form of BEC-field Hamiltonian, First term accommodates the effects of condensate on intra-cavity field and Δ a is the atom-field detuning. Second term describes the motion of condensate inside the cavity. =ˆ and Ω = 4ω r = 2ħk 2 /m a , is recoil frequency of an atom. Third term in eq. (5) describes coupled energy of field and condensate with coupling strength ξ sm = (ω c )/(L)(ħ/m bec 4ω r ) 1/2 , where, is the side mode mass of condensate. The last term accounts for the coupling of BEC with transverse field and η eff = (n) 1/2 η^ is transverse coupling strength.
The Hamiltonian for the part of moving-end mirror H MF can be given as 45 , first term is associated with energies of the intra-cavity field where, Δ c = ω c − ω p is external pump field and intra-cavity field detuning, ω c is intra-cavity field frequency and ĉ † (ĉ) are creation (annihilation) operators for intra-cavity field with commutation relation [ĉ,ĉ † ] = 1. Second term describes the motion of moving-end mirror where, q and p are dimensionless position and momentum operators for moving-end mirror respectively, having commutation relation q p =ˆ, which reveals the value of the scaled Planck's constant, ħ = 1. Third term represents coupling of moving-end mirror with field generated by radiation pressure applied by intra-cavity field on moving-end mirror. Here ξ = (2) 1/2 (ω c /L)x 0 is the coupling strength and x 0 = (ħ/2mω m ) 1/2 , is zero point motion of mechanical mirror having mass m. Last term is associated with the coupling of intra-cavity field with external pump field |η| = (Pκ/ħω p ) 1/2 , where, κ is cavity decay rate related with outgoing modes and P is the external pump field power. The Hamiltonian H T describes the effects of dissipation and noises associated with optomechanical system by using standard quantum noise operators 46 . The total Hamiltonian H leads to develop coupled quantum Langevin equations for optical, mechanical (moving-end mirror) and atomic (BEC) degrees of freedom.
is the effective detuning of the system and a in is Markovian input noise associated with intra-cavity field. The term γ m describes mechanical energy decay rate of the moving-end mirror and f B is Brownian noise operator associated with the motion of moving-end mirror 47,48 . The term γ a represents damping of BEC due to harmonic trapping potential which affects momentum side modes while, f M 1 and f M 2 are the associated noise operators assumed to be Markovian. As we focused on the the bistable behavior of optomechanics and its classical dynamics, we consider positions and momenta as classical variables. To write the steady-state values of the operator we assume optical field decay at its fastest rate so that, we can set time derivative to zero in equation (7). The steady-state values of operators are given as, ( ) The steady-state photons number of intra-cavity field is described as,

Optical, BEC and moving-end mirror Bistability
The hybrid BEC-optomechanical system shown in Fig. 1 is simultaneously driven by external pump field with frequency ω p . When intra-cavity radiation pressure oscillates with frequency (ω c ) is equal to the frequency of moving-end mirror ω m , it generates the Stokes and anti-Stokes scatterings of light from the intra-cavity field potential. Conventionally, optomechanical systems are operated in resolved-sideband regime κ < < ω m , which is off-resonant with Stokes scattering, therefore, the Stokes scattering is strongly suppressed and only anti-Stokes scattering will survive inside the cavity potential which causes the conversion of external pump photons to the intra-cavity photons. Therefore, by increasing the external pump field strength, the anti-Stokes scattering of intra-cavity photon number will be increased and will cause to move system in bistable regime. Optical bistability has been discussed in optomechanics in Ref 36. by considering double cavity configuration and the effects of s-wave scattering on bistable behavior of intra-cavity photon number have been studied in Ref. 49. In our optomechanical configuration, we consider transverse optical field to control the bistable behavior of the system. The idea is motivated by the findings of Ref. 39, where, they use transverse optical field to control bistable behavior of intra-cavity photons induced by external pump field. This mechanism provides another source of inducing photons in the cavity-optomechanics. It can be easily noticed from mathematical description that how this perpendicular channel of photon influence the optomechanical system and how it is connected to the displacement of atomic mode of the system. Therefore, the transverse photons will scatter inside the cavity depending on the configuration of intra-cavity condensate mode and will behave as nonlinear factor to the intra-cavity photon number.
To obtain the steady-state relation for Intra-cavity photon number, we use expression (17) and substitute the steady state values of position corresponding to moving-end mirror and condensate.  Figure 2 shows the bistable behavior of intra-cavity photon number under the influence of transverse optical field. We obtain optical bistability results by solving equ. (18). In Fig. 2(a), the blue curve represents intra-cavity photon number as function of external pump field strength (η/κ) when the transverse field strength is zero (η eff /κ = 0). We can observe, third order roots exist for photon number corresponding to single value of external pump field, which cause the appearance of bistability in intra-cavity photon number 39 . In Fig. 2(a), red curve shows optical bistability when the value of transverse optical field intensity is η eff /κ = 200 and green curve shows optical bistability when transverse optical field intensity is η eff /κ = 400. By observing the red and the green curves in Fig. 2(a), one can easily note that the strength of optical bistability is changed by changing the intensity of transverse optical field. The transverse optical field directly interacts with the intra-cavity atomic mode or BEC, trapped inside optomechanical system and is the reason for scattering of photons inside the cavity. Therefore, optical bistability curve is modified by increasing the strength of perpendicular nonlinearities. In short, photon bistability of intra-cavity potential is tunable with the help of perpendicular field. 2(b) shows the continuous behavior of intra-cavity photon bistability as a function of external pump field and transverse optical field, which provides better understanding of the control of intra-cavity optical mode with transverse optical field strength.
Further, to understand the influence of transverse field on intra-cavity optical mode, we study the optical behavior of the system with different values of effective detuning. Fig. 2(c) shows the intra-cavity optical bistability as a function of external pump field and effective detuning of the system in the absence of transverse field. We observe the tilted behavior of photonic peaks which represents bistable behavior of photon number. Fig. 2(d) and Fig. 2(e) represent similar behavior for different values of transverse field intensity η eff /κ = 40,50, respectively. These findings more efficiently clarify the intra-cavity optical mode dependence on perpendicularly interacting photons. Initially, intra-cavity optical mode exits only when it is resonant with external pump field. But when we increase the strength of external pump field, a tilt appear in the optical mode due to the influence of atomic mode and by increasing the value of transverse field, the strength of optical mode show enhancement and it is more tilted towards negative detuning of the system. By following these results, we can confidently state that the bistability of optical mode is tunable with the help of transverse optical field.
Transverse optical field cause the the scattering of photons inside the cavity and this scattering can be controlled by the strength of transverse optical field. If we increase the strength of transverse field to large values, scattered photons inside the cavity will fill the upper branch of bistability and will cause the suppression of optical bistability (see Ref. 39). Figure 3(a) demonstrates such behavior of intra-cavity photon at higher value of transverse field. Blue, yellow, red and green curve corresponds to the bistable behavior of intra-cavity photon number at transverse field strengths η eff /κ = 400, 800, 2000, 3000, respectively. We can note that by increasing strength of perpendicular field, intra-cavity photons start saturating in upper branch of bistability caused by external driving field and the population of photons in upper branch becomes much larger than the photons in lower branch of bistablity which leads to the suppression of lower stable state. Fig.(b) shows the behavior of upper stable state of bistability with large transverse field strengths. Similarly, blue, yellow, red and green curve contain information about the behavior of upper stable state at transverse field strengths η eff /κ = 3000, 3500, 4000, 4500, respectively and show continuous enhancement in the upper branch population with increase in transverse optical field.
To study the bistable dynamics of atomic mode or condensate mode of the system and mechanical mode or moving-end mirror of cavity-optomechanics, we rewrite the steady-state solution for position of both modes as, To discuss bistability effects on the intra-cavity potential, we derive couple equation of motion for both, moving end mirror and BEC by treating their motion of moving-end mirror and BEC classically and assuming time scale very short so that their mechanical damping can be ignored. We write the couple equations of motion by using the quantum Langevin equations as, It is known that mechanical degree of freedom (moving-end mirror) and atomic degree of freedom (condensate) are coupled with each other through intra-cavity optical mode. The modulation in intra-cavity photon number modify the quantity of radiation pressure exerted on atomic and mechanical mode of the system. We have already discussed the bistable behavior of intra-cavity field. Now we illustrate that the steady-state position of both atomic mode (Q s ) as well as moving-end mirror (q s ) possess same properties of bistability as optical mode of the system by using equation (19) and (20). Figure 4 demonstrates the bistable dynamics of moving-end mirror of optomechanical system and Cigar-Shaped Bose-Einstein condensate (BEC) trapped inside optomechanical system in the presence of transverse optical field. Fig. 4(a) and Fig. 4(b) show bistable behavior of moving-end mirror position q and BEC position Q as a function of external field strength in the absence of transverse optical field (η eff /κ = 0). As equation (19) and (20) are third-ordered equations for both moving-end mirror and BEC, respectively, so by solving these equations, we obtain three roots which lead us to the occurrence of bistable behavior. Fig. 4(c) shows the effective potential V as a function of mechanical mode and atomic mode steady-state position. These results are obtained by solving expression for effective potential (24) and steady-state values for the position of moving-end mirror and BEC. The potential V, shown in Fig. 4(c), is generally illustrated as two-dimensional double-well potential. The two local minima A and B corresponding to the stable points {q A ,Q A } and {q B ,Q B } are shown in Fig. 4(a) and Fig. 4(b). These two points represent two stable regions in effective potential of the optomechanical system in accordance to the particular set of values of moving-end mirror position and BEC position. The point C in effective potential V, shown in Fig. 4(c) is a saddle-point corresponding to values {q c ,Q c } in Fig. 4(a) and Fig. 4(b). Saddle-point represents unstable behavior of the effective potential of system which means at certain particular values of moving-end mirror position and BEC position, system possesses chaotic behavior, as illustrated in many classical systems such as 50,51 .   Fig. 4(e) show bistable behavior of moving-end mirror position q and BEC position Q as a function of external pump field with transverse optical field strength η eff /κ = 0.8 and Fig. 4(f) shows their collective influence on the behavior of effective potential of optomechanical system. We observe the similar behavior, as we discuss earlier in Fig. 4(c), but due to the presence of transverse optical field, the stable and unstable regions are now modified and the values of stable points and saddle-point are changed. The photons interacting perpendicularly with system through transverse field are directly associated with intra-cavity atomic momentum side modes which are bounded with mechanical mode (moving-end mirror) in relation of radiation pressure. Therefore, a minute variation in the strengths of transverse field cause modified bistable behavior of intra-cavity atomic mode Q s and mechanical mode q s . Moreover, the perpendicularly scattering photons inside the cavity drive momentum side modes of BEC and apply radiation pressure on mirror even when there is no external pump field. According to equation (24), effective potential of system completely relies on position of moving-end mirror and BEC. Therefore, a small change in bistable behavior of their positions can bring notable modification in effective potential of the system, as shown in Fig. 4(f), which means the stable points A and B, shown in Fig. 4(f), are not the same as in Fig. 4(c) and saddle-point C is also different from the saddle-point in Fig. 4(c) because of the variations in the bistability of moving-end mirror position and BEC position. Figure 4(g) and Fig. 4(h) illustrate the similar behavior of bistability of moving-end mirror position and BEC position at different transverse optical field strength η eff /κ = 1.6 and Fig. 4(i) shows their mutual effects on the effective potential. These results also assert the effectiveness of previous one that the values of both stable points as well as saddle-point are changed due to the increase in transverse field strength which cause modification in stable and unstable regions in effective potential. By observing these results, one can conclude that the bistable dynamics of moving-end mirror as well as BEC are tunable with the help of transverse optical field.
It is very important to observe time dependent dynamics of atomic mode and mechanical mode of the system. In this section, we will discuss the temporal behavior of Q and q and see how transverse optical field will influence their temporal dynamics. Figure 5 accounts for the time dependent response of moving-end mirror and Bose-Einstein condensate corresponding to different transverse optical field strengths, in quasi-periodic regime as shown in Fig. 4, initially located at {q = 0,Q = 0}. We solve quantum langevin equation to obtain the spatial dynamics of intra-cavity atomic mode and mechanical mode of the system under the assumption that the position operator for BEC (Q) and moving-end mirror (q) behaves like classical variables which allow us to ignore quantum noise effects associated with the optomechanical system. Figure 5(a) represents behavior of moving-end mirror with respect to time ω m t corresponding to different transverse field strengths. Green line shows the behavior of moving-end mirror in the absence of transverse optical field (η eff /κ = 0). The moving-end mirror follows continuous oscillatory behavior with particular amplitudes corresponding with time as discussed in 15 . The red line represents time dynamics of moving-end mirror when transverse field strength is η eff /κ = 0.8. We can clearly detect that the amplitudes of oscillations of moving-end mirror have been increased by increasing transverse optical field, which cause a shift in moving-end mirror position from stable regime to unstable regime (chaotic regime), as shown in Fig. 4(f). Similarly, blue curve demonstrates time dependent dynamics of moving-end mirror for η eff /κ = 1.8 and shows that the oscillatory behavior of moving-end mirror is different from the other curves because of the increase in transverse field. Transverse optical field directly modulates the intra-cavity field potential which effectively brings nonlinearities to the system and therefore, cause chaotic behavior of moving-end mirror. Figure 5(b) illustrates time dependent behavior of intra-cavity atomic mode (BEC) under the influence of transverse field strength. Green curve is associated with time dependent response of BEC in the absence of transverse field (η eff /κ = 0). We can note that atomic mode possesses similar temporal oscillatory behavior with particular amplitudes like moving-end mirror. The red line represents time dynamics of BEC at perpendicular optical field (η eff /κ = 0.8). We observe that the amplitude of oscillations of intra-cavity atomic mode is attenuated because transverse field photons are associated with the position of BEC and cause the suppression of time dependent oscillations of atomic momentum side modes. Similarly, blue curve represents the effects of transverse field (η eff /κ = 1.8) on the spatial dynamics of BEC. Demonstration of these results enable us to conclude that the time dependent dynamics of BEC and moving-end mirror of optomechanics are also controllable using transverse field.
In order to observe the behavior of effective potential V with respect to the intra-cavity photon number, we derive the expression of effective potential (equ. (24)) by considering the steady-state values for the position of moving-end mirror and BEC (equ. (13) and (15)) as,  Figure 6 demonstrates the relation of effective potential of cavity-optomechanics with the intra-cavity photon number and parallel driving field under the influence of transverse optical field. Fig. 6(a) shows that effective potential of the system increases monotonically with increase in external pump field and intra-cavity photon number in the absence of transverse optical field. Fig. 6(b) represents the similar behavior effective potential but in the presence of transverse pump field strength η eff /κ = 0.8. Effective Scientific RepoRts | 5:10612 | DOi: 10.1038/srep10612 potential shows monotonic behavior with increase in parallel field but because of the presence of transverse field another curvature appears in effective potential. When perpendicular optical field interacts with intra-cavity atomic mode, it causes the scattering of photons inside the cavity depending upon the configuration of atomic mode which lead to the modification in intra-cavity effective optical potential. We can note the existence of photons inside the cavity in the absence of external pump field because of the transverse field scattering from condensates. This scattering also causes the existence of homogeneous atomic side modes. 6(b) clearly shows such modulation in the effective potential of the system. Figure 6(c) represents the response of effective potential inside the cavity at different transverse optical field strength η eff /κ = 1.2. We observe that the curvature appeared in effective potential with intra-cavity photon number is enhanced due to the increase in transverse photon scattering. Figure 6(df) illustrate similar behavior of effective intra-cavity potential and photon number with transverse optical field strength η eff /κ = 1.6, η eff /κ = 2 and η eff /κ = 2.4, respectively and show continuous enhancement in nonlinear curvature appearing in effective potential of the system with transverse photon interaction. These findings lead us to control the collective behavior of intra-cavity optical potential with the help of transverse optical field and enable us to understand the explicit nature of hybrid optomechanical systems.

Discussion and Conclusion
In this report, we discuss the bistable dynamics of hybrid BEC-optomechanics. We consider N = 2.3 × 4 87 Rb atoms trapped inside Fabry-Pérot cavity with length L = 1.25 × 10 −4 , with a moving-end mirror and driven by single mode external field with power P in = 0.0164mW, frequency ω p = 3.8 × 2π × 10 14 Hz and wavelength λ p = 780nm. The moving-end mirror of cavity should be a perfect reflector and performs oscillations, because of intra-cavity field radiation pressure, with frequency ω m = 15.2 × 2πMHz and its coupling with intra-cavity field is considered as ξ = 3.8 MHz. The frequency of intra-cavity field is ω c = 15.3 × 2π × 10 14 Hz, with cavity decay rate κ = 1.3 × 2πkHz. Intra-cavity field produces recoil of ω r = 3.8 × 2πkHz in atoms trapped inside cavity. The coupling of intra-cavity field with BEC is ξ sm = 4.4 MHz. Detuning of the system is taken as Δ = Δ c + (U 0 N)/(2) = 0.52 × 2πMHz, where vacuum Rabi frequency of the system is U 0 = 3.1 × 2πMHz. We use a transverse optical field, with frequency ω ⊥ and coupling η ⊥ , to control the bistable dynamics of hybrid system. Our results demonstrate that the optical bistability of the system is controllable by using transverse optical field because the perpendicularly interacting photons are coupled to atomic momentum side modes, exited by the motion of Bose-Einstein condensate due to the interaction of intra-cavity field. We have also discussed the spatial bistability of moving-end mirror and BEC with different values of transverse field strength and thus, have shown that the bistable dynamics of moving-end mirror and BEC are tunable using transverse optical field intensity and studied their effects on effective potential. We further illustrated the temporal behavior of mechanical mode and atomic mode (BEC) of the system and found that the dynamics of mechanical mode can be pushed to chaotic regime by increasing the strength of vertical field. Besides, we studied the variance in effective potential of the system as a function of cavity photon number and external driving field under the influence of transverse field and found out magnificent nonlinear response of effective potential with perpendicularly scattering photons. This method provides us an excellent control over the nonlinear dynamics of such hybrid systems.
In future, we will explore the nonlinear dynamics of hybrid system in good cavity limits and will also study the controllable dynamics by considering interacting Bose-Einstein condensate. Besides, we intend to extend this method to examine the controllability of novel phenomenon like electromagnetically induced transparency and entanglement. Additional goals include the discussion on the effects of spin-orbit coupling using magnetic field in hybrid BEC-optomechanical systems.  Figure 6. Transverse field effect on effective potential. The continuous behavior of effective potential of optomechanical system as a function of intra-cavity photon number and external pump field strength with different transverse field strengths. (a) demonstrates the behavior of intra-cavity effective potential with respect to photon number and external driving field coupling η/κ in the absence of transverse field η eff /κ. (b) shows the effective potential when the transverse field strength is η eff /κ = 0.8. (c) represents the effects of transverse field strength η eff /κ = 1.2 on the effective potential. Similarly, (d) shows effective potential for transverse field η eff /κ = 1.6. (e) and (f) demonstrate the effective potential of the system for transverse optical field η eff /κ = 2 and η eff /κ = 2.4, respectively. The remaining parameters are same as in above discussion.