The effect of atomic collisions on the quantum phase transition of a Bose-Einstein condensate inside an optical cavity
TThe effect of atomic collisions on the quantum phase transition of a Bose-Einsteincondensate inside an optical cavity
A. Dalafi , ∗ M. H. Naderi , , and M. Soltanolkotabi , Department of Physics, Faculty of Science, University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran Quantum Optics Group, Department of Physics, Faculty of Science,University of Isfahan, Hezar Jerib, 81746-73441, Isfahan, Iran (Dated: November 8, 2018)In this paper, we investigate the effect of atomic collisions on the phase transition form the normalto the superradiant phase in a one-dimensional Bose-Einstein condensate (BEC) trapped inside anoptical cavity. Specifically, we show that driving the atoms from the side of the cavity leads to theexcitation of modes in the edges of the first Brillouin zone of every energy band, which results inthe two-mode approximation of the BEC matter field in the limit of weak coupling regime. Thenonlinear effect of atom-atom interaction shifts the threshold of the quantum phase transition of theBEC and also affect the power low behavior of quantum fluctuations in the total particle number.Besides, we show the possibility of controlling the quantum phase transition of the system throughthe s -wave scattering frequency when the the strength of the transverse pumping has been fixed. PACS numbers: 03.75.Hh, 37.30.+i, 05.30.Rt, 34.10.+xKeywords: Bose-Einstein condensate, atoms in cavities, quantum phase transition, atomic collisions
I. INTRODUCTION
Ultracold atoms trapped inside high-Q cavities aresuitable systems for studying the interaction of light withmatter in the regime where not only their mutual ef-fects are simultaneously observable but also their quan-tum mechanical properties are manifested in the samelevel [1, 2]. Such an interaction is inherently nonlinearwhich is due to the mutual matter-light coupling [3]. Onthe other hand, if the density of the atomic ensemble ishigh enough then another kind of nonlinearity manifestsitself which is due to the atom-atom interaction. Such asituation is usually manifested in a BEC [4].Furthermore, ultracold atoms trapped inside opticalcavities exhibit phenomena typical of solid-state physicslike the formation of energy bands [5] and Bloch oscil-lations [6]. One of the other features of these so-calledhybrid systems is their similarities with the optomechan-ical systems (optical cavities with moving mirror) [7, 8].In the hybrid system the excitation of a collective modeof the cold gas plays the role of the vibrational mode ofthe moving mirror of the optomechanical cavity [9–13].Besides, the nonlinear effects of atom-atom interactionin systems consisting of BEC affect the optical bistabil-ity [14] and the squeezing of the vibrational modes of thesystem [15].Hybrid systems consisting of BEC have also attractedconsiderable attention in connection with quantum phasetransition phenomena. One celebrated example is thetransition from a superfluid to a Mott insulator phase inthe Bose-Hubbard model [16–19] that has been observedexperimentally in a gas of ultracold atoms trapped insidean optical lattice [20]. Another kind of quantum phase ∗ adalafi@yahoo.co.uk transition from the homogeneous into a periodically pat-terned distribution can be observed in hybrid systemsconsisting of a BEC whose atoms are coherently pumpedfrom the side of the cavity [Fig.1]. It has been shown[21] that this kind of phase transition is very similar tothat of the Dicke model from the normal to the superra-diant phase. When the intensity of the transverse pumplaser is below a critical value, the system is in the normalphase where all the atoms populate the ground state ofthe optical lattice (distributed homogeneously) and theradiation field inside the cavity is in the vacuum state.However, above the critical point some atoms are excitedto higher motional modes and the mean value of the opti-cal field gets nonzero (superradiant phase). In this phase,the spatial distribution of atoms gets a periodic pattern[22].Near the critical point, the second-order correlationfunctions of the system show a power law behavior. Thisbehavior has been investigated for the stationary state ofthe driven and damped hybrid system in the thermody-namic limit [23] and also for finite-size systems consistingof low-density Bose-Einstein condensates where the effectof atomic collisions is dispensable [24].In this paper we are going to study the nonlinear effectof atom-atom interaction on the threshold of quantumphase transition of a BEC trapped inside an optical cav-ity when the atoms are driven coherently from the sideof the cavity. Besides, we show that the atomic collisionscan affect the scaling law behavior of quantum fluctua-tions in the total number of atoms. In addition to theeffect of atomic collisions, here we also consider the effectof the damping of the matter field of the BEC due to theleakage of atoms to other modes [25]. For this purpose weconsider the discrete mode approximation for the BEClike that was studied in our previous paper [26] with thedifference that here we no longer consider the Bogoliubovapproximation. a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov FIG. 1. (Color online) The quantum phase transition ofa BEC inside an optical cavity from the normal phase (left),when the transverse pumping rate η t is below the critical value η c , to the superradiant phase (right) when η t > η c . In Refs.[26, 27] it has been shown how the atoms arescattered to the modes with momenta 2 n (cid:126) k when thecavity is pumped from one of its mirrors ( k is the wavenumber of the optical mode and n is the band index).Here, using the theoretical description of the band struc-ture of a one-dimensional BEC inside an optical latticedeveloped in the two above-mentioned references, we willspecifically show that driving the atoms from the side ofthe cavity leads to the excitation of modes in the edges ofthe first Brillouin zone of every energy band. It is shownthat the linear combination of modes in the two oppositeedges in the Brillouin zones of two adjacent energy bandsforms the modes with momenta n (cid:126) k which are coupled tothe central modes (with zero quasi mementum) of eachenergy band.The paper is structured as follows. In Sec.II we de-scribe the Hamiltonian of the system and show how thetrapped atoms are scattered to the modes with momenta n (cid:126) k due to the transverse pumping. In Sec. III we usethe two-mode approximation for describing the BEC andsolve for the mean-field values of the system through thenumerical self-consistent method considering atom-atominteraction. Then, we investigate the effect of atomic col-lisions on the threshold of quantum phase transition andon the power law behavior of quantum fluctuations. Fur-thermore, we investigate the quantum phase transitionof the system in terms of the s -wave scatterin frequencywhen the strength of the transverse pumping has beenkept on a fixed value. Finally, our conclusions are sum-marized in Sec. IV. II. DESCRIPTION OF THE SYSTEM
We consider a system consisting of a BEC of N two-level atoms inside an optical cavity with length L wherethe atoms are coherently driven from the side by a laserwith frequency ω p , and wave number k = ω p /c , directedperpendicularly to the cavity axis (Fig.1). We assumethe BEC to be confined in a cylindrically symmetric trapwith a transverse trapping frequency ω ⊥ and negligiblelongitudinal confinement along the x direction [4]. In this way we can describe the dynamics within an effec-tive one-dimensional model by quantizing the atomic mo-tional degree of freedom along the x axis only.In the dispersive regime, where the laser pump is de-tuned far below the atomic resonance (the absolute valueof ∆ a = ω p − ω a < γ byorders of magnitude), the excited electronic state of theatoms can be adiabatically eliminated and spontaneousemission can be neglected [28]. In the frame rotating atthe pump frequency, the many-body Hamiltonian reads H = − (cid:126) ∆ c a † a + (cid:90) L − L Ψ † ( x ) (cid:104) − (cid:126) m a d dx + (cid:126) U cos ( kx ) a † a + (cid:126) η t cos( kx )( a † + a ) + 12 U s Ψ † ( x )Ψ( x ) (cid:105) Ψ( x ) dx. (1)Here, a is the annihilation operator of the optical field,∆ c = ω p − ω c is the cavity-pump detuning, U = g / ∆ a is the optical lattice barrier height per photon which rep-resents the atomic back action on the field, g is the vac-uum Rabi frequency, U s = π (cid:126) a s m a w , a s is the two-body s -wave scattering length [28, 29], and w is the waist ofthe optical potential.The first term in the second lineof Eq.(1) describes the effect of the transverse pumpfield which drives the atoms with the constant amplitude η t = Ω R g/ ∆ a where Ω R is the Rabi frequency. A. The matter field of the BEC
As is shown in Ref.[26] the matter field of the BEC canbe expanded in terms of plane waves in the following wayΨ( x ) = 1 √ L (cid:88) n ∈ Z + l/ (cid:88) m = − l/ b n,m e i ( n + m/l )2 kx , (2)where l = 2 L/λ is the number of periods of the mat-ter field inside the cavity which is assumed to be even,and b n,m is the annihilation operator for the atomic fieldthat annihilates a particle in a state determined with theBloch band index n and quasimomentum q m = 2 mk/l .In the limit of weak photon-atom coupling, where U (cid:104) a † a (cid:105) ≤ ω R ( ω R = (cid:125) k m a is the recoil frequency of thecondensate atoms), the above expansion can be restrictedto the lowest band numbers n = 0 , ±
1. Besides, if thesystem starts from a homogeneous BEC, only the cosineparts of the exponential functions are excited because ofthe parity conservation[30]. Therefore, the matter fieldcan be written in the following formΨ( x )= (cid:114) L C + (cid:114) L C cos(2 kx )+ (cid:114) L l/ (cid:88) m =1 (cid:104) C m cos (cid:16) ml kx (cid:17) + C m cos (cid:16) ml (cid:17) kx + C , − m cos (cid:16) − ml (cid:17) kx (cid:105) , (3)where the new modes C nm have been defined in terms ofthe following Bogoliubov transformations C = b ,C m = 1 √ b m + b , − m ) = C , − m ,C m = 1 √ b m + b − , − m ) ,C , − m = 1 √ b , − m + b − ,m ) . (4)Confining the matter field to some discrete modes likeEq.(3), is called the discrete-mode approximation (DMA)[3]. In the following we will derive the Hamiltonian (1)in terms of the discrete modes. B. System Hamiltonian in the discrete-modeapproximation
In order to see which modes of the BEC are exciteddue to the transverse pumping of the atoms let us firstinvestigate that part of Eq.(1) which is related to thetransverse pumping i.e., H tp = (cid:126) η t ( a † + a ) (cid:90) L/ − L/ Ψ † ( x ) cos( kx )Ψ( x ) dx. (5)Substituting Eq.(3) into Eq.(5) we will have H tp = (cid:126) η t ( a + a † ) × (cid:104) √
22 ( C † ,l/ + C † , − l/ ) C + 12 ( C † ,l/ + C † , − l/ ) C + √ C † ( C ,l/ + C , − l/ ) + 12 C † ( C ,l/ + C , − l/ ) (cid:105) . (6)As is seen from this equation, driving the atoms fromthe side of the cavity makes the modes C , C ,l/ and C , − l/ be excited while leaving the other modes unex-cited. Besides, it leads to a coupling between the linearcombination C ,l/ + C , − l/ and the two modes C and C (the central modes of the bands n = 0 and n = 1).Based on Eq.(3) both C ,l/ (the mode in the tail of theBrillouin zone of the band n = 0) and C , − l/ (the modein the head of the Brillouin zone of the band n = 1)have the same mode function, i.e, cos( kx ). Therefore, ifwe consider c ≡ C ,l/ + C , − l/ as a new mode withquasimomentum (cid:126) k and redefine the two central modeswith momenta 0 and 2 (cid:126) k by c ≡ C and c ≡ C respectively, then Eq.(6) reads H tp = 12 (cid:126) η t ( a + a † ) (cid:104) √ c † c + c † c )+( c † c + c † c ) (cid:105) . (7)Considering higher bands of energy, it can be easily seenthat other modes on the edges of the Brillouin zone de-fined as c n +1 ≡ C n,l/ + C n +1 , − l/ are coupled to thecentral modes c n ≡ C n, for ( n = 1 , , ... ) which leads to the appearnce of terms as c † n c n +1 + c † n +1 c n inside thebrackets of Eq.(7). Therefore, by pumping the atomstransversely the modes in the edges of the first Brillouinzones of the energy bands are excited so that the linearcombination of modes in the two opposite edges in theBrillouin zones of two adjacent energy bands form themodes with momenta n (cid:126) k which are coupled to the cen-tral modes (with zero quasi mementum) of each energyband.In the same way one can calculate that part of theHamiltonian (1) corresponding to the interaction be-tween the atoms and intracavity photons, i.e,. H at − ph = (cid:126) U a † a (cid:90) L/ − L/ Ψ † ( x ) cos ( kx )Ψ( x ) dx. (8)Again, substituting Eq.(3) into Eq.(8) and after somesimplification we will have H at − ph = 12 (cid:126) U a † a × (cid:104) c † c + 32 c † c + c † c + 1 √ c † c + c † c ) + B (cid:105) , (9)in which the operator B has been defined as B = l/ − (cid:88) m =1 (cid:104) C † m C m + C † m C m + C † , − m C , − m + 12 ( C † m C m + C † m C , − m + C † ,m C m + C † , − m C m ) (cid:105) . (10)Finally, the kinetic energy part of the Hamiltonian (thefirst term inside the integral in Eq.(1)) takes the followingform H kin = (cid:126) ω R ( c † c + 4 c † c + 4 F ) , (11)where F = l/ − (cid:88) m =1 (cid:104)(cid:16) ml (cid:17) C † m C m + (cid:16) ml (cid:17) C † m C m + (cid:16) − ml (cid:17) C † , − m C , − m (cid:105) . (12)Based on Eqs.(7),(9), and (11) the total Hamiltonianof the system, i.e., Eq.(1), can written as H = − (cid:126) ∆ c a † a + (cid:126) ω R ( c † c + 4 c † c )+ 12 (cid:126) U a † a (cid:104) c † c + 32 c † c + c † c + 1 √ c † c + c † c ) (cid:105) + 12 (cid:126) η t ( a + a † ) (cid:104) √ c † c + c † c ) + ( c † c + c † c ) (cid:105) +4 (cid:126) ω R F + 12 (cid:126) U a † aB + H aa , (13)where H aa is the atom-atom interaction part of theHamiltonian which is given by the last term of Eq.(1).The Hamiltonian obtained in Eq.(13), except for the lastthree terms, is just like Eq.(3.3) of Ref.[22] and alsoEq.(8) of Ref.[31] . C. The Heisenberg equations of motion
Since the photon-field operator a and the matter-fieldoperators c , c and c commute with the operators F and B , the Heisenberg equations of the matter fields donot change with or without the presence of the termsconsisting of these operators in the total Hamiltonian(13). The only modification due to these additional termsis the term − U aB which appears in the Heisenbergequation of the photon-field operator˙ a = i (cid:104) δ c + iκ − U ( √ c † c + c † c ) + c † c + 2 B ) (cid:105) a − iη t (cid:104) √ c † c + c † c ) + ( c † c + c † c ) (cid:105) + ξ. (14)Here, δ c = ∆ c − N U and κ is the dissipation rate ofthe cavity field. The cavity-field quantum vacuum fluctu-ation ξ ( t ) satisfies the Markovian correlation functions,i.e., (cid:104) ξ ( t ) ξ † ( t ) (cid:105) = 2 κ ( n ph + 1) δ ( t − t (cid:48) ), (cid:104) ξ † ( t (cid:48) ) ξ ( t )) (cid:105) =2 κn ph δ ( t − t (cid:48) ) with the average thermal photon num-ber n ph which is nearly zero at optical frequencies [32].As is seen From Eq.(14), the optical mode is coupledto the modes C ,m and C ,m through the operator B .On the other hand, the Heisenberg equations for C ,m , C ,m without considering the last term of the Hamilto-nian (13), i.e, H aa , reads˙ C ,m = − i ω R (cid:16) ml (cid:17) C ,m − iU a † a (cid:16) C ,m + 12 C ,m + 12 C , − m (cid:17) , (15a)˙ C ,m = − i ω R (cid:16) ml (cid:17) C ,m − iU a † a (cid:16) C ,m + 12 C ,m (cid:17) . (15b)Based on these equations, there is no direct coupling be-tween the modes C nm and c n in the absence of atomiccollisions. In fact, the only way that these modes can becoupled to each other in the absence of atom-atom inter-action is through the optical mode, i.e., through the term (cid:126) U a † aB in the Hamiltonian (13), which is negligible inthe limit of weak photon-atom coupling. However, dueto the atomic interactions the modes c n can be coupledto the modes C n,m in a very complicated way. In fact,the modes C n,m as well as the harmonic trapping poten-tial of the BEC act as a kind of reservoir which lead todamping and broadening of these modes [25, 26]. In therest of this paper we will set aside the modes C n,m andconsider a two-mode approximation for the matter fieldof the BEC. The effects of the extra modes C n.m as wellas that of the harmonic trapping potential are simulatedas a damping process which leads to the fluctuation anddissipation of the two modes of the matter-field. III. THE TWO-MODE MODEL OF THE BEC
Using the approximations mentioned in the previoussection one can set aside the terms containing F and B in Eq.(13) and also limit the matter field of the BEC justto the modes c and c so that it can be written asΨ( x ) = 1 √ L c + (cid:114) L c cos( kx ) . (16)In this way, we will obtain the following Hamiltonian fora two-mode BEC H = − (cid:126) ∆ c a † a + (cid:126) ω R c † c + 12 (cid:126) U a † a (cid:16) c † c + 32 c † c (cid:17) + 1 √ (cid:126) η t ( a + a † )( c † c + c † c ) + (cid:126) ω sw N (cid:16) c † c + c † c + c † c + 4 c † c c † c + 32 c † c (cid:17) . (17)Here, the terms proportional to the s-wave scattering fre-quency, i.e., ω sw = 8 π (cid:126) a s N/m a Lw , have come throughthe atom-atom interaction part, i.e., the last term ofEq.(1). The Heisenberg equations of motion in the grandcanonical formalism are written in terms of the grandcanonical Hamiltonian, i.e., H − µ ˆ N , where µ is the chem-ical potential andˆ N = (cid:90) Ψ † ( x )Ψ( x ) dx = c † c + c † c (18)is the particle number operator whose expectation valueis the mean value of total perticles, i.e., (cid:104) ˆ N (cid:105) = N . There-fore, the time evolution of the optical and the matter-fieldmodes are obtained as follows˙ a = i ∆ c a − iU a (cid:16) c † c + 32 c † c (cid:17) − i √ η t ( c † c + c † c ) − κa + ξ ( t ) , (19a)˙ c = (cid:104) i (cid:16) µ (cid:126) − U a † a (cid:17) − γ c (cid:105) c − i √ η t ( a † + a ) c − iω sw N (cid:16) c † c + c † c + 2 c c † c (cid:17) + f ( t ) , (19b)˙ c = (cid:104) i (cid:16) µ (cid:126) − ω R − U a † a (cid:17) − γ c (cid:105) c − i √ η t ( a † + a ) c − iω sw N (cid:16) c † c + 2 c † c c + 32 c † c (cid:17) + f ( t ) , (19c)where, γ c is the dissipation of the collective density ex-citations of the matter field and f ( t ) and f ( t ) are thethermal noise inputs for the two modes of BEC which sat-isfy the same Markovian correlation functions as thoseof the optical noise, i.e., (cid:104) f j ( t ) f † j ( t ) (cid:105) = 2 γ c δ ( t − t (cid:48) ), (cid:104) f † j ( t (cid:48) ) f j ( t )) (cid:105) = 0 for ( j = 0 , a = √ N α + δa, c = √ N β + δc and c = √ N β + δc into Eqs.(19) we will obtain a set ofnonlinear algebraic equations for the steady-state meanvalues (cid:104) i ( δ c − uβ ) − κ (cid:105) α = iyβ β , (20a) M ( α, β ) β = µ (cid:126) β, (20b)where the parameters u = N U , y = √ N η t , and β =( β , β ) T is a column vector. M ( α, β ) is a matrix whichreads M = (cid:18) u | α | + w yα R yα R ω R + 3 u | α | + w (cid:19) , (21)where w = ω sw (1 + 2 β ), w = ω sw (1 − β ), and α R is the real part of the complex number α . On the other hand, the linearized quantum Langevineequations (QLEs) for the quadrature fluctuations of theoptical field, i.e., δX a = √ ( δa + δa † ) , δP a = √ i ( δa − δa † ), and the quadrature fluctuations of the matter field,i.e., δX j = √ ( δc j + δc † j ) , δP j = √ i ( δc j − δc † j ) with j = (0 , δ ˙ Z ( t ) = AδZ ( t ) + n ( t ) , (22)where δZ = [ δX a , δP a , δX , δP , δX , δP ] T is the vectorof continuous variable fluctuation operators and n ( t ) = [ ξ x ( t ) , ξ p ( t ) , f x ( t ) , f p ( t ) , f x ( t ) , f p ( t )] T , (23)is the corresponding vector of noises. Besides, the 6 × A is the drift matrix given by A = − κ − ˜ δ c uβ α I uβ α I δ c − κ − ( yβ + 4 uβ α R ) 0 − ( yβ + 6 uβ α R ) 00 0 − γ c Ω + ω sw yα R + β β ω sw − ( yβ + 4 uβ α R ) − uβ α I − (Ω + ω sw ) − γ c − ( yα R + 3 β β ω sw ) 00 0 0 yα R + β β ω sw − γ c Ω − ( yβ + 6 uβ α R ) − uβ α I − ( yα R + 3 β β ω sw ) 0 − Ω − γ c , (24)where˜ δ c = δ c − uβ , (25a)Ω = 2 u | α | − µ (cid:126) , (25b)Ω = ω R − µ (cid:126) + 3 u | α | + 12 ω sw (cid:16) β + β + 3 (cid:17) . (25c) A. Mean field solutions
Equation (20b) is a nonlinear eigenvalue problem inwhich β is the eigenvector of the matrix M ( α, β ) and µ/ (cid:126) is the smallest eigenvalue. There are two kinds ofnonlinearity in this eigenvalue problem: the first one isdue to the indirect dependence of the matrix M on β through the mean optical field α which is itself depen-dent on β through Eq.(20a), and the second one which isdue to the direct dependence of M on β through the pa-rameters w and w , has been originated from the effectof atomic collisions.Eqs.(20a, 20b) can be solved by the self-consistentmethod. For this purpose, we first take arbitrary valuesfor α R , α I , and β and calculate the smallest eigenvalueof the matrix M and obtain its corresponding eigenvec-tor β . Next, we substitute the obtained values of β , β into Eq.(20a) and get the output values for α R , α I . If the difference between the input and the output values of α R is larger than a specified value (cid:15) , we repeat this proce-dure until this difference becomes less than (cid:15) . Wheneverthis condition is satisfied we stop the procedure and takethe last obtained values as the self-consistent solutions.The numerical results of the self-consistent solution tothe nonlinear eigenvalue problem [Eqs.(20a, 20b)] havebeen demonstrated in Fig.2 and Fig.3 in which the valueof (cid:15) has be chosen as small as 10 − .The behavior of quantum phase transition can be seenin Figs.2(a) and 2(b) which show repectively, | α | , themean photon number of the cavity normalized to √ N ,and the mean value fraction of atoms versus the pumpingstrength y/ω R for three different values of the s -wavescattering frequency. As is seen from Fig.2(a), the meanvalue of the optical field is zero below a critical value y c and nonzero above that value. In Fig.2(b) the meanvalue fraction of atoms in the condensate mode c andthe Bogoliubov mode c have been plotted versus thepumping strength. As is seen, below the threshold value y c all the atoms are in the condensate state mode andthere is no excitation, but above the critical point theBogoliubov mode is populated and the quantum phasetransition occurs. The results obtained here are in goodaccordance with those obtained in Ref.[22] in the absenceof atom-atom interaction.The important result is that the value of y c depends (cid:72) a (cid:76) Ω sw (cid:61) Ω sw (cid:61) Ω R Ω sw (cid:61) Ω R
16 18 20 22 24 260.00000.00050.00100.00150.0020 y (cid:144) Ω R (cid:200) Α (cid:72) b (cid:76) Β Β
16 18 20 22 24 260.00.20.40.60.81.0 y (cid:144) Ω R Β j FIG. 2. (Color online) (a) The mean photon number ofthe cavity normalized to √ N and (b) the mean value frac-tion of atoms in the condensate mode c (thick red lines) andthe Bogoliubov mode c (blue thin lines) versus the pumpingstrength y/ω R for three different values of the s -wave scat-tering frequency: ω sw = 0 (solid lines), ω sw = 0 . ω R (dashedlines), and ω sw = 0 . ω R (dotted lines). The parameters are δ c = − ω R , u = − ω R , and κ = 200 ω R . on the strength of atomic interaction, i.e., ω sw . In theabsence of atomic interaction, i.e., for ω sw = 0, y c (cid:39) ω R (solid lines), while this value is increased to morethan 23 ω R for ω sw = 0 . ω R (dashed lines), and 24 ω R for ω sw = 0 . ω R (dotted lines). In this way, the threshold ofthe phase transition of the system can be controlled bythe s -wave scattering frequency.In Fig.3 the chemical potential of the BEC (normalizedto (cid:126) ω R ) has been plotted versus the pumping strength forthree different values of the s -wave scattering frequency.As is seen, below the threshold the chemical potential iszero in the absence of atomic collisions but it gets posi-tive values in the presence of atomic interactions. on theother hand, above the threshold the chemical potentialdecreases by increasing the pumping strength and getsnegative values.Below the critical point the BEC behaves like a uni-form system of bosons in the thermodynamic equilibriumin which all of them populate the single-particle statewith the wave function 1 / √ L . In such a situation, theinteraction energy of a pair of particles is U s / L [the lastterm in Eq.(1) with Ψ = 1 / √ L ]. Therefore, the energyof a state with N bosons all in the same state is given by
16 18 20 22 24 26 (cid:45) (cid:45) (cid:45) y (cid:144) Ω R Μ (cid:144) (cid:209) Ω R FIG. 3. (Color online) The chemical potential of theBEC normalized to (cid:126) ω R versus the pumping strength y/ω R for three different values of the s -wave scattering frequency: ω sw = 0 (solid line), ω sw = 0 . ω R (dashed line), and ω sw =0 . ω R (dotted line). The parameters are the same as those ofFig.2. multiplication of this quantity with the number of possi-ble ways of making pairs of bosons, i.e., N ( N − / E = 12 N ( N − U s L (cid:39) N U s L , (26)where we have assumed N (cid:29)
1. In the thermodynamicequilibrium the chemical potential of the system is µ = ∂E/∂N . Therefore, below the critical point, i.e., for y In order to investigate the stationary properties of thesystem fluctuations one should consider the steady-statecondition governed by Eq.(22) which is reached when thesystem is stable. It occurs if and only if all the eigenvaluesof the matrix A have negative real parts. These stabilityconditions can be obtained by using the Routh-Hurwitzcriterion [34]. Due to the linearized dynamics of the fluc-tuations and since all noises are Gaussian the steady stateis a zero-mean Gaussian state which is fully characterizedby the 6 × V , withcomponents V ij = (cid:104) δZ i ( ∞ ) δZ j ( ∞ ) + δZ j ( ∞ ) δZ i ( ∞ ) (cid:105) / V fulfills the Lya-punov equation AV + V A T = − D, (31)where D = Diag[ κ, κ, γ c , γ c , γ c , γ c ] , (32)is the diffusion matrix. Here, we have also assumed thatthe mean number of thermal excitations of both the con-densate and the Bogoliubov modes are zero which isreasonable for the BEC with temperature about nanoKelvin. Equation(31) is linear in V and can be straight-forwardly solved. However, the explicit form of V is com-plicated and is not reported here.After calculating the mean-field values we can obtainthe elements of the drift matrix A and calculate its eigen-values. In Fig.4 we have shown the dynamical stabil-ity regions (blue region) and instability regions (red re-gion) of the system in terms of the normalized pumping (cid:72) a (cid:76) stable region unstable region y (cid:144) Ω R Ω s w (cid:144) Ω R (cid:72) b (cid:76) y (cid:144) Ω R Ω s w (cid:144) Ω R FIG. 4. (Color online) The regions of dynamical stability(blue) and dynamical instability (red) versus the normalizedpumping strength ( y/ω R ) and the normalized s -wave scatter-ing frequency of atomic collisions ( ω sw /ω R ) for two values ofthe BEC damping constant: (a) γ c = 0 . κ and (b) γ c = 0The other parameters are the same as those of Fig.2. strength ( y/ω R ) and the normalized s -wave scatteringfrequency of atomic collisions ( ω sw /ω R ) for two values of γ c = 0 . κ [Fig.3(a)] and γ c = 0 [Fig.3(b)]. The dynam-ical stable regions are those points of the space y − ω sw where all the eigenvalues of the drift matrix A have neg-ative real parts and the dynamical unstable regions arethose points where at least one of the eigenvalues has apositive real part. As is seen, when γ c = 0 the systemis unstable even for low values of ω sw below the criticalpoint. However, considering a nonzero value for γ c evenas small as 0 . κ preserves the stability of the systemfor wider domains of y and ω sw .On the other hand, by solving the Lyapunov equation[Eq. (31)] we can obtain the correlation matrix V whichgives us the second-order correlations of the fluctuations.In this way we can calculate the effective number of in- (cid:72) a (cid:76) 16 18 20 22 24 260.000.050.100.150.200.25 y (cid:144) Ω R (cid:88) ∆ (cid:97) † ∆ (cid:97) (cid:92) (cid:72) b (cid:76) 16 18 20 22 24 2601020304050 y (cid:144) Ω R (cid:88) ∆ N (cid:92) FIG. 5. (Color online) (a) The effective number of incoherentphotons of the cavity, and (b) the average of quantum fluctua-tions of the particle number operator of the BEC (condensatedepletion) versus the normalized pumping strength y/ω R forthree values of the s -wave scattering frequency: ω sw = 0 (solidlines), ω sw = 0 . ω R (dashed lines), and ω sw = 0 . ω R (dottedlines).The decay rate of the matter field is γ c = 0 . κ andthe other parameters are the same as those of Fig.2. coherent photons of the cavity, (cid:104) δa † δa (cid:105) = 12 ( V + V − , (33)and the effective number of incoherent atoms in the con-densate and the Bogoliubov modes, (cid:104) δc † δc (cid:105) = 12 ( V + V − , (34a) (cid:104) δc † δc (cid:105) = 12 ( V + V − . (34b)The particle number operator, Eq.(18), can be writtenas ˆ N = N + δ ˆ N where δ ˆ N = (cid:90) δ Ψ † ( x ) δ Ψ( x ) dx = δc † δc + δc † δc , (35)is the quantum fluctuations in the total particle num-ber. The average value of this operator ( (cid:104) δ ˆ N (cid:105) ) can bedefined as the depletion of the BEC, i.e., the total occu-pancy of motional modes other than the macroscopicallypopulated BEC wave function [35].In Fig.5 we have plotted the effective number of inco-herent photons of the cavity and the average of quantum TABLE I. The critical values and the critical exponents forthe curves of Fig.5(b) ω sw /ω R y c /ω R ν c fluctuations of the particle number operator of the BEC(condensate depletion) versus the normalized pumpingstrength y/ω R for three different values of atom-atom in-teraction. The results have been shown in that range ofthe pumping strength where the system is stable for thethree different values of the s -wave scattering frequencyas depicted in Fig.4(a) where the decay rate of the matterfield is γ c = 0 . κ .As is seen, below the critical point the fluctuationsof photons and atoms are zero. However, as the trans-verse pumping strength reaches near the critical value y c a sharp pick appears. Increasing the s -wave scatteringfrequency from zero (solid lines) to 0 . ω R (dashed lines)and 0 . ω R (dotted lines) shifts the pick (the phase transi-tion threshold) to higher values of the pumping strengthjust like what was happened for the mean value solutions[Figs.2, 3]. Besides, the greater the strength of atom-atom interaction is, the higher the pick of fluctuations.Near the critical point and in the limit γ c → y − y c ) ν c , near the critical point where ν c < γ c = 0 . κ not only provides the stability conditionsof the system [Fig.4(a)] but also is small enough to showus the scaling law behavior of the system near the criti-cal point. The physical origin of this damping constantis the coupling of the two modes of the BEC to the othermodes ( C n,m ) due to the atomic collisions or through theharmonic trapping potential of the condensate [25].In Table I, we have listed the critical values of trans-verse pumping strength ( y c ) and the critical exponents( ν c ) for the power law behavior of the condensate deple-tion (cid:104) δ ˆ N (cid:105) near the critical point for three different valuesof the s -wave scattering frequency. For this purpose wehave fitted functions like r ( y − y c ) ν c to the three curvesof Fig.5(b) when y gets near to y c from the left hand sideand have obtained the corresponding critical exponents.As is seen from this Table, the increase of ω sw not onlyshifts y c to the larger values but also leads to the increaseof the absolute value of the critical exponents.The last point we are going to investigate is the casethat has been shown in Fig.6 where we have set thestrength of the transverse pumping to a fixed value, say y = 24 . ω R , and have examined the variation of themean fields and quantum fluctuations of the system in (cid:89) ∆ (cid:97) † ∆ (cid:97) (cid:93) Α (cid:180) (cid:72) a (cid:76) Ω sw (cid:144) Ω R (cid:72) b (cid:76) Β Β (cid:88) ∆ N (cid:92) (cid:144) Μ (cid:144) (cid:209) Ω R Ω sw (cid:144) Ω R FIG. 6. (Color online) (a) The mean photon number of thecavity normalized to √ N magnified by 100 (blue solid line)and the effective number of incoherent photons of the cav-ity (red dashed line) versus the s -wave scattering frequencynormalized to ω R . (b) The mean value fraction of atoms inthe condensate mode (red dashed line) and in the Bogoli-ubov mode (blue dashed-dotted line), the chemical potentialnormalized to (cid:126) ω R (black dotted line), and the condensatedepletion divided by 50 (brown solid line) versus the s -wavescattering frequency normalized to ω R . The strength of thetransverse pumping has been set to y = 24 . ω R , the decayrate of the matter field is γ c = 0 . κ and the other parame-ters are the same as those of Fig.2. terms of the variation of the s -wave scattering frequency.As is seen, here the s -wave scattering frequency acts asthe control parameter for the quantum phase transition.There is a critical value ω ( c ) sw = 0 . ω R where below it the system is in the superradiant phase and above that valuethe system goes to the normal phase. It is in accordancewith the results given in the last line of Table I.Besides, near the critical point where ω sw → ω ( c ) sw the quantum fluctuations in photons [red dashed line inFig.6(a)] and also in the total number of atoms (conden-sate depletion) [Brown solid line in Fig.6(b)] get maxi-mum. In this way for each value of the transverse pump-ing strength there is a critical value for the s -wave scat-tering frequency where below it the system is in the su-perradiant phase and above that value the system goesto the normal phase. IV. CONCLUSION In this paper, we have studied the effect of atomic col-lisions on the phase transition form the normal to thesuperradiant phase in a one-dimensional BEC inside anoptical cavity which is pumped from the transverse side.In the first part of the paper, we have investigated ex-plicitly which modes of the BEC are excited when theatoms are driven from the transverse side of the cavityand have shown how the BEC can be described approxi-mately through a two-mode model.In the second part of the paper, we have made useof the two-mode model of the BEC and showed that in-creasing the atom-atom interaction strength causes thethreshold of the phase transition (the critical transversepumping strength) to be shifted to the larger values. 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