Elementary Excitations of a Bose-Einstein Condensate in an Effective Magnetic Field
aa r X i v : . [ c ond - m a t . o t h e r] J un Elementary Excitations of a Bose-Einstein Condensate in an Effective Magnetic Field
D. R. Murray and Stephen M. Barnett
Dept. of Physics, SUPA, University of Strathclyde, Glasgow G4 0NG, UK
P. ¨Ohberg
Dept. of Physics, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK
Dami´a Gomila
Instituto de F´ısica Interdisciplinar y Sistemas Complejos (IFISC,CSIC-UIB),Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
We calculate the low energy elementary excitations of a Bose-Einstein Condensate in an effectivemagnetic field. The field is created by the interplay between light beams carrying orbital angularmomentum and the trapped atoms [1]. We examine the role of the homogeneous magnetic field,familiar from studies of rotating condensates, and also investigate spectra for vector potentials witha more general radial dependence. We discuss the instabilities which arise and how these may bemanifested.
PACS numbers: 03.75.Ss,42.50.Gy,42.50.Fx
I. INTRODUCTION
Quantum degenerate gases are in many ways the idealquantum many-body system. In an experimental situa-tion they afford an unprecedented level of control overthe system parameters, such as the strength (and evensign) of the interaction between the atoms, the geometryof the external trap and the properties of the lattice inwhich the atoms are loaded. It is no surprise thereforethat Bose-Einstein Condensates (BECs) and degenerateFermi gases are often used as a laboratory to study a hostof phenomena from many different areas of physics. Thisis especially true in condensed matter physics; for exam-ple, ultracold atoms in an optical lattice can be stud-ied using the Hubbard model [2]. Similarly, a system oftrapped fermions tightly confined in one direction invitesobvious comparisons with the 2D electron gas [3].Without doubt, some of the most striking effects insolid state physics are observed when an external mag-netic field is applied to a collection of charged particles.Well known examples include the quantum Hall effectsin 2D electron gases and the Meissner effect in Type IIsuperconductors. As the atoms forming quantum gasesare electrically neutral, it is not obvious at a first glancehow they might be used to study such effects.The solution lies in the ability to create artificialmagnetic fields. For example, rotating the system andstudying it in the rotating frame is analogous to study-ing charged particles in a homogeneous magnetic field[4, 5, 6]. Alternatively, lasers can be used to alter thestate-dependent tunneling amplitudes of atoms in an op-tical lattice to simulate an effective magnetic flux [7, 8].A recent proposal involves the adiabatic motion oflambda-type three level atoms interacting with laser-fields which create a non-degenerate dark state, that isan eigenstate of the atom-laser interaction. It has beenshown that if the atoms interact with a pair of laser beams possessing a relative orbital angular momentum[9, 10], then an effective vector potential appears in theeffective equation for the atomic wavefunction [1, 11].The corresponding effective magnetic field created is en-tirely dependent on the form of the incident light, so thatby appropriately choosing the light’s phase and intensitywe can control both the strength and shape (homoge-neous or inhomogeneous) of the effective magnetic field.The inherent flexibility of the system allows for wide-ranging studies into the magnetic properties of both de-generate Bose and Fermi gases, and could provide insightinto gauge theories in general.It is therefore pertinent to gain an understanding ofhow the fundamental properties of the gas may be mod-ified in the presence of artificial magnetic fields. A com-plete analysis must include the excitations, which deter-mine the dynamical behaviour of the system under weakperturbation and are crucial in determining its superfluidproperties. Of particular interest are the lowest energy(or elementary) excitations, which are collective in na-ture. In this paper we calculate the spectra for a trapped2D BEC in both homogenous and non-homogeneousmagnetic fields, which are created as described in [1].Two-dimensional quantum gases have recently attracteda considerable interest in connection with the Kosterlitz-Thouless transition [12] and the quantum Hall effect inclouds of ultracold atoms (see [3] and references therein).The paper is organised as follows: in section II a briefdescription of the model is given and in section III weoutline how the excitations are calculated. As the inter-action between the light and atoms introduces two ef-fective potentials - a vector potential and also a scalarpotential - and both have a significant role to play inthe dynamics and excitations, we present our results intwo parts. In section IV we assume the external traphas been chosen to counteract the effect of the additionaleffective trap so that the potential felt by the atoms iscompletely harmonic. This allows us to isolate the role
FIG. 1: Method for creating effective magnetic fields in de-generate atomic gases. On the left is the level scheme forthe Λ-type atoms interacting with the resonant probe beamΩ p and control beam Ω c . On the right is a schematic repre-sentation of the experimental setup with the two light beamsincident on the cloud of atoms. The effective magnetic fieldis generated if there is relative angular momentum betweenthe beams. This will occur, for example, if the probe field isof the form Ω p ∼ e i ℓφ , where each probe photon carries anorbital angular momentum ~ ℓ along the propagation axis z ,and Ω c is independent of the azimuthal angle. of the magnetic field alone on the excitations. Then insection V we include the full effective trapping potentialterms and study the excitations numerically. Finally insection VI we discuss and summarise the main results. II. THE MODEL
We consider a system of three-level atoms charecter-ized by two hyperfine ground levels | i and | i and anelectronically excited level | i interacting with two co-propagating resonant laser beams in an EIT configura-tion (figure 1). The probe beam, which has couplingstrength Ω p and is allowed to have angular momentum ℓ ~ per photon along the z-axis, drives the transition | i → | i , whilst the control beam has coupling strengthΩ c and is concerned with the transition | i → | i . Theseabsorption paths destructively interfere to suppress tran-sitions to level | i , driving the atoms to the dark statesuperposition of levels | i and | i : | D i = | i− ζ | i √ | ζ | , where ζ = Ω p Ω c = (cid:12)(cid:12)(cid:12) Ω p Ω c (cid:12)(cid:12)(cid:12) e iS and S is the relative phase betweenthe probe and control beam. If the atoms in the darkstate form a BEC, then the coupling between the lightand the atoms introduces an effective vector potentialinto the mean-field equation for the atomic wavefunctionΨ [1]: i ~ ∂ Ψ ∂t = 12 M ( i ~ ∇ + A ) Ψ + V ( r )Ψ + g | Ψ | Ψ , (1)where A = − ~ | ζ | | ζ | ∇ S (2) and V = V ext + ~ M | ζ | ( ∇ S ) + ( ∇| ζ | ) (1 + | ζ | ) (3)are, respectively, the effective vector and trapping poten-tials. The external trapping potential for the dark stateatoms is V ext = V + | ζ | V (1 + | ζ | ) (4)where V j is the trapping potential for the atoms in hy-perfine state j ( j = 1 , g = g + 2 g | ζ | + g | ζ | (1 + | ζ | ) . (5)Here, g ij = π ~ a ij M , where a ij is the s -wave scatteringlength between atoms in the levels i and j ( i, j = 1 , a jj is the scattering length of atoms in the same electronicstate and a = a corresponds to collisions betweenatoms in different electronic states. Note that in generalthe intractions can depend on position since ζ is positiondependent. However, if ζ is small or alternatively if theinter and intra-spieces scattering lengths are equal then g is approximately constant throughout the condensate.We shall assume this to be the case for the remainder ofthis paper.For the purposes of our analysis it is convenient rewriteEq. (1) in the form i ~ ∂ Ψ ∂t = (cid:18) − ~ M ∇ + ˜ V + g | Ψ | + i ~ M A · ∇ (cid:19) Ψ , (6)where ˜ V ( r ) = V + | A | M . For the examples considered ∇ · A = 0 so this equation is equivalent to (1).As can clearly be seen from Eq. (2), a non-vanishingeffective magnetic field is created if there is relative or-bital angular momentum between the two light beams,such that the phase of the dimensionless ratio ζ is givenby S = ℓφ , where φ is the azimuthal angle. The shapeof the effective vector potential is controled by the in-tensity ratio of the probe and control beams. We choose | ζ | = (cid:12)(cid:12)(cid:12) Ω p Ω c (cid:12)(cid:12)(cid:12) = α (cid:0) rR (cid:1) ν +1 , where the dimensionless pa-rameter α is the ratio of probe to control beam at acharacteristic radius r = R which is chosen to be largerthan the radius of the trapped cloud. The exact forms ofthe resultant effective vector and trapping potential arethen A = − ~ ℓR α (cid:0) rR (cid:1) ν α (cid:0) rR (cid:1) ν +1 e φ , (7)and˜ V = V ext + ~ M R [ ℓ + ( ν + 1) ] α (cid:0) rR (cid:1) ν − (cid:16) α (cid:0) rR (cid:1) ν +1 (cid:17) + ~ M R ℓ α (cid:0) rR (cid:1) ν (cid:16) α (cid:0) rR (cid:1) ν +1 (cid:17) . (8)In order for the adiabatic dynamics to hold, so thatthe atoms remain in the dark state for a typical BEClifetime, requires, typically, the ratio | ζ | . α . α << A = − ~ ℓR α (cid:16) rR (cid:17) ν e φ , (9)˜ V = V ext + ~ M R [ ℓ + 14 ( ν + 1) ] α (cid:16) rR (cid:17) ν − − ~ M R ℓ α (cid:16) rR (cid:17) ν , (10)and the effective interaction strength (eq. 5) is constantthroughout the cloud: g ≈ g . We obtain Eqs. (9) and(10) simply by expanding eqs (7) and (8) and neglectingall higher order terms in α , except of course the 2nd termon the right hand side of (10), because this is multipliedby an ℓ which can in principle be arbitrarily large.The freedom to choose the form of the spatially-varyingeffective magnetic field relies on the ability to shape theintensities and phases of the incident laser beams. Recentadvances in light beam shaping, using for instance spa-tial light modulators, makes it possible to consider trulyexotic light beams [13]. The tightly confined or two di-mensional gas offers in this respect a clear simplification:shaping a light beam in a plane is much less restrictivethan in three dimensions, although 3D light shaping iscertainly possible. III. CALCULATING THE EXCITATIONS
We calculate the excitations of this system by consid-ering small time-dependent variations of the condensatewavefunction around the ground state Ψ( r ), writing thewavefunction asΨ( r , t ) = [Ψ( r ) + u ( r )e − iωt − v ∗ ( r )e iωt ]e − iµt/ ~ . (11)Substituting this into eq. (6) and keeping only linearterms in u and v we obtain two coupled equations anal-ogous to the Bogoliubov - de Gennes equations [14] (cid:0) L − ~ mM Ar (cid:1) u + g | Ψ | (2 u − v ) = ( µ + E ) u (12) (cid:0) L + ~ mM Ar (cid:1) v + g | Ψ | (2 v − u ) = ( µ − E ) v, (13) where L = − ~ M ∇ + ˜ V and the (non-vortex) ground statesatisfies (cid:16) L + g | Ψ | (cid:17) Ψ = µ Ψ . (14)These are the equations we solve to calculate the eigen-frequencies ω and eigenenergies E = ~ ω of the excitationmodes u and v , which satisfy the normalization condition Z d r ( u a u ∗ b − v a v ∗ b ) = δ ab . (15)In deriving Eqs. (12)-(14) we have allowed the ex-citations to have angular momentum m ~ (where m =0 , ± , ± ... ) by transforming u and v such that u → ue imφ and v → ve imφ . We assume that the ground state ψ = ψ ( r ) only; this is equivalent to saying that ψ does notcorrespond to a vortical state. Note that a more riogor-ous approach for deriving the excitation equations wouldbe to diagonalize the many body Hamiltonian in the Bo-goliubov approximation, expressing the fluctuation op-erator in terms of quasiparticle operators [15]. The re-sultant equations which must be solved are identical to(12)-(13).The excitation spectrum of eqs (12)-(13) can be cal-culated in the spirit of the Thomas-Fermi approxima-tion where the effective trap potential (which we assumeto have a harmonic ( ∼ r ) component) and the repul-sive mean-field interactions provide the dominant en-ergy scales. In this regime µ ≈ n max g >> ~ ω t , where n max is the maximum condensate density. We rescalethe radial coordinates as ρ = rL and ˜ R = RL where L = (cid:16) µmω t (cid:17) is the characteristic length-scale of theharmonically trapped condensate with trap frequency ω t .Introducing the dimensionless eigenenergies ǫ ν = E ν ~ ω t andthe dimensionless density ¯ n = | Ψ | n max , we can re-write ourequations as L d u + (2 u − v )¯ n = h γ ( ǫ + γ mA d ρ ) i u (16) L d v + (2 v − u )¯ n = h − γ ( ǫ + γ mA d ρ ) i v (17) L d ψ + ¯ nψ = ψ, (18)with L d = − γ ∇ + ˜ V d where, in the Thomas-Fermiregime, the parameter γ = ~ ω t µ << d in ˜ V d and A d denotes that dimensionless units are beingused.Equations (16)-(18) are reduced to two fourth or-der differential equations after introducing the functions f ± = u ± v and substituting ¯ n from Eq. (18):1 − ˜ V d ǫ + γmA d ρ (cid:18) −∇ f + + f + ∇ ΨΨ (cid:19) + γ " −∇ ǫ + γmA d ρ ! (cid:18) −∇ f + + f + ∇ ΨΨ (cid:19) + γ " ǫ + γmA d ρ (cid:18) −∇ f + + f + ∇ ΨΨ (cid:19) ∇ ΨΨ = 2( ǫ + γmA d ρ ) f + , (19)and − ∇ − ˜ V d ǫ + γmA d ρ ! f − + 1 − ˜ V d ǫ + γmA d ρ f − ∇ ΨΨ − γ " ∇ − ρ ǫ + γmA d ρ ! (cid:18) −∇ f − + 3 f − ∇ ΨΨ (cid:19) + γ " ∇ − ρ ǫ + γmA d ρ ! (cid:18) −∇ f − + 3 f − ∇ ΨΨ (cid:19) = 2( ǫ + γmA d ρ ) f − , (20)In the Thomas-Fermi approximation, we neglect thekinetic energy term γ ∇ ψ in Eq. (18) to obtain thewavefunctionΨ T F = q n max (1 − ˜ V d ) , Ψ T F ≥ , (21)which can readily be substituted into Eqs. (19) and (20).When considering the low energy excitations, with wave-functions that vary over a scale comparable with the sizeof the condensate, we must for consistency also neglectterms proportional to γ in (19) and (20). Then, applyingthe ansatz f ± = C ± (cid:16) ǫ + γmA d ρ (cid:17) (cid:18) − ˜ V d ǫ + γmAdρ (cid:19) ± Q ( ρ, φ ),we derive in the Thomas-Fermi regime the equation forthe excitations of a 2D condensate with effective mag-netic field:(1 − ˜ V d ) ∇ Q − (1 − ˜ V d ) ∇ Ψ T F Ψ T F Q = 2 (cid:18) ǫ + γmA d ρ (cid:19) Q, (22)The relation between the normalization constants C + and C − can be obtained from Eqs. (16), (17) and (22): C − = γC + . (23) IV. HARMONIC EFFECTIVE TRAPPINGPOTENTIALA. Homogeneous Magnetic Field
By a judicious choice of external trapping potential,a purely harmonic effective trap can be obtained, such A ∝ r α ℓ ω A ∝ r α ℓ ω A ∝ r α ℓ ω A ∝ r α ℓ ω numericanalytic FIG. 2: Numerical results for surface mode (n=0) excitationfrequency as a function of the effective magnetic field strength α ℓ for different vector potentials. The top left panel corre-sponds to the homogeneous magnetic field. The lines cor-respond to the five lowest energy surface excitations, wherethe m=0 case (Goldstone mode) is represented by the low-est frequency mode at α ℓ = 0, and the m = 4 the highest.Each mode exhibits an energetic instability beyond a critical α ℓ . The mode frequencies are purely real, so the system isdynamically stable when the vector and trapping potentialsare radially symmetric. A full description of the numericalmethod is given in section V. that ˜ V = M ω t r [1]. If the ratio of control to probeintensity in the transversal plane is of the form | ζ | ∼ r ,then the exponent ν in eq. (9) is 1. The effective vectorpotential is then A = − ~ ℓα R r ˆ e φ , (24)corresponding to a uniform magnetic field in the z-direction i.e. B = ∇ × A = − ~ ℓα R ˆ e z . In this case, thesolution of (22) is of the form W = x m P ( x )e imφ , where x = ρ and the radial function P ( x ) is the solution of thehypergeometric equation x (1 − x ) d Pdx + (( m + 1) − ( m + 2) x ) dPdx + (cid:18) ǫ − γα ℓmǫ − m (cid:19) P ( x ) = 0 . (25)For a physically well-behaved solution, we require P ( x )to be convergent at x = 0 and converge as x →
1. Thisyields the spectrum E n,m = ~ ω t (cid:0) n + 2 n | m | + 2 n + | m | (cid:1) + ~ M R α ℓm (26)where n is the number of radial nodes and m the angularmomentum quantum number. This spectrum is of thesame form as that found by Ho and Ma [16] for the two-dimensional cloud, except for the shift term proportionalto α ℓm which arises due to the effective magnetic field.The uniform effective magnetic field is thus shown to in-duce a Zeeman-like shift on the energy levels, decreasingthe excitation energy when ℓ and m are opposite in signand increasing it when they are the same. The actualmode observed should be interpreted as a superpositionof the + and - m modes, as the effective magnetic fieldinduces a rotation in modes with m = 0. For example, ifwe excite the m = ± v × B , where v is the velocity.Let us also highlight that as a result of our choice ofexternal trap, the spectrum of Eq. (26) matches that of arotating condensate when studied in the rotating frame,with rotation frequency given by Ω = ~ α ℓMR . The stabilityproperties of this system have been studied extensively(see e.g. [17, 18, 19]).The solutions of Eq. (25) are the Jacobi polynomials P ( | m | , n (1 − x ) and from Eqs. (15) and (23) we obtain f ± = (cid:20) n + 2 | m | + 2 L (cid:21) (cid:20) (1 − ρ ) γ ( ǫ nm − γα ℓm ) (cid:21) ± × ρ m P ( | m | , n (1 − ρ ) . (27) B. Vector Potential for which the Magnetic Fieldis Zero
If instead the ratio of the control and probe beam in-tensities were constant ( ν = 1) we would obtain the vec-tor potential A = − ~ α ℓr ˆe φ (28)in which case B = ∇ × A = 0 so that the effective mag-netic field is zero throughout the cloud (in the same waythat a velocity field proportional to 1 /r around a vor-tex still satisfies the irrotationality criterion for Bose-Einstein Condensation). An approximate energy spec-trum can be derived by treating the effective poten-tial as a small perturbation in eq. (22), noting thatthis treatment breaks down as r →
0. The effecivevector potential plays the role of an additional cen-trifugal potential. The solution of (22) is of the form W = x √ m +4 γα ℓm P ( x )e imφ , where x = ρ and P ( x )is governed by a hypergeometric equation which admitsa physical solution convergent at the origin and as x → ǫ − γα ℓmǫ − ( n + 12 ) p m + 4 γα ℓmǫ = n + n, (29)from which we can obtain the excitation spectrum bysolving for ǫ . In the perturbative regime where ǫγα ℓ << | m | we find E n,m = ~ ω t (cid:18) ǫ (0) + ~ ω t µ α ℓ ( m + (2 n + 1)sgn( m )) (cid:19) , (30)where ǫ (0) = (cid:0) n + 2 n | m | + 2 n + | m | (cid:1) gives the spec-trum when there is no vector potential and sgn( m ) =+1 , − , m positive, negative or 0 respectively. Thisis a somewhat crude approximation but, nevertheless, ityields an insightful result: the effective vector potentialhas significance even if the corresponding effective mag-netic field seen by the atoms is zero, reminiscent of theAharanov-Bohm effect. This significance is manifestedin a shift in excitation energy levels for all modes withangular momentum ( m = 0), and the magnitude of theshift now also depends on n , the number of radial nodes.The origin of the dependence on sgn( m ) may have a topo-logical interpretation. C. Inhomogeneous Magnetic Field
We can also use a perturbative approach on eq. (22)to calculate the first order energy shift due to vectorpotentials with ν ≥
1, which correspond to inhomo-geneous effective magnetic fields. The eigenfunctionsand eigenvalues of the unperturbed Hermitian Hamil-tionian ˆ H (0) = (cid:0) − ρ (cid:1) ∇ − ρ ∂∂ρ are those of theharmonically trapped BEC when no effective magneticfield is present: Q (0) = ρ m P ( | m | , n (1 − ρ )e imφ and ǫ (0) = (cid:0) n + 2 n | m | + 2 n + | m | (cid:1) , where ǫ (0) is degen-erate with respect to the sign of m . The first order en-ergy corrections due to the perturbation H ′ = γm | A | = γmα ℓ ˜ R (cid:16) ρ ˜ R (cid:17) ν − are the solutions of the secular equation (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) H ′ ++ − ǫ (1) r H ′ + − H ′− + H ′−− − ǫ (1) r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0with H ′±± = κ r h Q (0) ± | H ′ | Q (0) ± i , where + or − denotesthe sign of m , and the constant κ r can be determinedusing Eqs. (15) and (23). The off-diagonal terms vanish,allowing us to express the first order energy shift ∆ E = E (1) − E (0) of the surface modes ( n = 0) as∆ E = ~ α ℓmM R | m | + 1)(2 | m | + ν + 1) , (31)which is plotted in figure 3. The perturbative resultsagree well with our numerical calculations for the lowestenergy excitations but breaks down as we increase n and m as would be expected since these excitations are nolonger slowly varying in space.The observed shift on the excitation energy levels isreduced as we increase the exponent term ν in the vec-tor potential from 1. This is not surprising as the radial analyticnumerical1 2 3 4 5 6 7 800.010.020.030.040.050.06 Frequency shift as a function of ν ( α ℓ = 1) ν F r e q u e n c y s h i f t ∆ ω Ω m=1 (perturbative)m=1 (numerical)m=2 (perturbative)m=2 (numerical) FIG. 3: Frequency shifts for the m = 1 and m = 2 modesdue to the presence of the effective magnetic field as a func-tion of ν , the radial exponent in the vector potential (Eq.(9)). The dotted lines correspond to results obtained solvingEqs. (16)-(18) numerically and the solid lines to the resultsof the perturbative calculation (Eq. (31). All frequencies arein units of the harmonic trap frequency ω t , and the parame-ters used were α ℓ = − γ = 0 . coordinate ρ is ≤ ν the ef-fective magnetic field becomes more concentrated at theedge and permeates less to the centre of the cloud. As aconsequence, the critical α ℓ for energetic instabilities tooccur increases as the exponent ν is increased. However,the perturbative treatment is valid only for E (1) ∆ E << ν = 2 , ,
4, presented in figure2. A full description of the numerical method is given inthe next section. In contrast to the case of homogeneouseffective magnetic fields we also note that the magnitudeof the energy shift depends on the radial node index n .The effective magnetic field does not, however, affect thepurely compressional ( m = 0) modes with zero angularmomentum. V. FULL EFFECTIVE TRAPPING POTENTIALINDUCED BY THE LIGHT
We now present the results of numerical calculationsused to determine the excitations when the only assump-tion we make of our external trap is that it is harmonic,and include the additional effective trapping potentialdue to the interaction with the light and atoms. For re-alistic parameter values, with ℓ > α <
1, the effectof the trapping potential induced by light eclipses that ofthe effective magnetic field, with profound implicationsfor the excitation spectra.The exception is the case where ν = 1. If α is small,then the vector potential approximated by Eq. (9) cor-responds to a homogeneous magnetic field. The secondterm in Eq. (10) represents a uniform shift in the chem- A ∝ r ℓ ω A ∝ r ℓ ω A ∝ r ℓ ω A ∝ r ℓ ω FIG. 4: Numerical results for surface mode ( n = 0) excitationfrequency as a function of the effective magnetic field strength α ℓ for different vector potentials with radial exponent ν . Thelines correspond to the five lowest energy surface excitations,where the m = 0 case is represented by the lowest frequencymode at α ℓ = 0, and the m = 4 the highest. Only the ν = 1 case exhibits energetic instabilities. Here α = 0 . γ = 0 .
001 and ˜ R = 1. The number of grid points used was128 and the system size was L=2.56 ical potential throughout the cloud - an effect we canignore by setting the effective trap minimum to zero -while the third term shifts the harmonic trap frequencydownwards. The spectrum is then the same as Eq. (26)except that we replace the trap potential ω t such that ω t → ω t − (cid:18) ~ α ℓM R (cid:19) ! . (32)The spectrum exhibits energetic instabilities above acritical angular momentum of the light l , as in sectionIV. However, this is not the case for ν >
1. In figure 4 weplot the dispersion curves obtained for ν = 1 , , ,
4. Ananalytical approach is no longer possible when ν >
1, cor-responding to an inhomogeneous effective magnetic field,and/or if the intensity ratio of the control and probebeam is such that the inequality α ≪ r = 0 and r = L , with L the domain size. To do so wediscretize the space and solve the set of coupled ordinarydifferential equations using a Newton method in whichthe derivatives are computed in Fourier space [20, 21].Solving Eqs. (16) and (17) then reduces to finding theeigenvalues of the discretized matrix associated to thelinear problem (16-17). For ν ≥ ℓ , above which they increase monotonicallywith ℓ . VI. SUMMARY AND DISCUSSION OF MAINRESULTS
In the studies of BECs subject to an external rotation -equivalent to applying a homogeneous effective magneticfield - the presence of instabilities in the excitation spec-tra has been shown to play a crucial role in determiningthe evolution of the condensate, particularly with respectto the nucleation of vortices. There are two types of in-stability which ought to be considered. Dynamic insta-bilities are associated with a complex excitation energyand departure from the initial configuration due to inter-action effects. The energetic instability relates to the ex-istence of excitations with negative energy, so that in thepresence of dissipation the system can lower its energy bygoing into an ‘anomolous’ mode, and is a prerequisite fordynamical instability [22]. The dynamical instability hasbeen credited as the primary mechanism for vortex nu-cleation in the rotating trap experiments [17]. It has alsobeen argued that while the dynamic instability helps in-duce vortex nucleation, the actual penetration of vorticesin to the bulk is a consequence of the energetic instability[19].The excitation spectra studied in this paper, both an-alytically and numerically do not exhibit any dynamicinstabilities in that we do not observe a critical effectivemagnetic field at which the excitation energy becomescomplex. Due to the increased complexity of adding aninhomogeneous effective magnetic field, however, we haverestricted ourselves to consider trapping configurationswhich are radially symmetric. For the homogeneous ef-fective magnetic field dynamical instabilities are only ob-served with the introduction of an anisotropic trap [17],and this is likely to be the case for inhomogeneous mag-netic fields as well.Energetic instabilities, by contrast, occur readily. Forall the vector potentials considered in section IV the nu-merical results display energetic instabilities beyond acritical field strength proportional to α ℓ . As we increasethe radial exponent ν in the vector potential we in turnmust significantly increase the probe beam angular mo-mentum before the instability can be observed. Whenthe full effective trapping potential is included the ν = 1case exhibits an energetic instability but in the vast ma-jority of parameter space those for ν > α ap-proaches 1 do we observe instabilities in the low-energymodes. However, it is in this region the adiabatic dynam-ics most easily breaks down [1] and the model describedin section II may become invalid.Another important feature of the spectra calculated insections IV and V is the accidental degeneracies, whichoccur where the mode frequencies intersect in figures 1 and 3. These degeneracies are likely to be manifestedthrough transfer of excitations from higher order modesto lower order modes at the point of degeneracy. Thiskind of phenomenon could have implications for the con-densate evolution depending on whether the effectivemagnetic field is switched on instantly or its strengthis adiabatically ramped up to a final value. For ex-ample, an anisotropically trapped condensate in a ho-mogeneous effective magnetic field naturally undergoes m = 2 quadrupole oscillations, but the higher ordermodes exhibit dynamic instability [17], eventually lead-ing to vortex nucleation. To properly account for themode-coupling we would need to move beyond Bogoli-ubov - de Gennes theory which does not account for in-teractions between the degenerate modes [15].With inclusion of the full effective trap induced by theinteraction with the light we reach a scenario where thereare no accidental degeneracies as ν is increased beyond1. The vector potential appearing in the Bogoliubov-deGennes equations is less important in determining theexcitation frequencies in comparison with the effectivetrap. A direct measure of the contribution of the vectorpotential is the splitting of the + and - m modes. For thepotentials with ν > <
5% of the actual mode fre-quency if no effective magnetic field were present. Thisis in stark contrast to section IV where the splitting caneasily exceed twice the mode frequency causing an ener-getic instability for realistic parameter values.In this paper we have deliberately paid particular at-tention to the surface modes ( n = 0, m = 0), which arenaturally excited by adding the effective magnetic field toan anisotropically trapped condensate [23]. We predictthat for inhomogeneous effective magnetic fields vortexnucleation will be inhibited due to the dominance of thetrapping potential induced by the light. In general, it isthe ratio of the cyclotron frequency to the effective trap-ping frequency which drives the dynamics of the systemand both of these depend on α ℓ . It is therefore oftendesirable to introduce a counter potential to act againstthe additional trapping terms due to the interaction withthe light, as described in section IV. Acknowledgments
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