Spectral Gaps of Spin-orbit Coupled Particles in Deformed Traps
O. V. Marchukov, A. G. Volosniev, D. V. Fedorov, A. S. Jensen, N. T. Zinner
SSpectral Gaps of Spin-orbit Coupled Particles inDeformed Traps
O V Marchukov, A G Volosniev, D V Fedorov, A S Jensenand N T Zinner
Department of Physics and Astronomy, Aarhus University, DK-8000 Aarhus C,Denmark
Abstract.
We consider a spin-orbit coupled system of particles in an externaltrap that is represented by a deformed harmonic oscillator potential. Thespin-orbit interaction is a Rashba interaction that does not commute with thetrapping potential and requires a full numerical treatment in order to obtainthe spectrum. The effect of a Zeeman term is also considered. Our resultsdemonstrate that variable spectral gaps occur as a function of strength of theRashba interaction and deformation of the harmonic trapping potential. Thesingle-particle density of states and the critical strength for superfluidity varytremendously with the interaction parameter. The strong variations with Rashbacoupling and deformation implies that the few- and many-body physics of spin-orbit coupled systems can be manipulated by variation of these parameters.PACS numbers: 67.85.-d,73.20.At,05.30.Fk,73.22.Dj a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps
1. Introduction
The last decade is associated with breakthroughs in ultracold atomic and state-of-the-art optical lattice experiments, which provided not only extremely important datafor understanding of the fundamentals of quantum mechanics, but also introduce asophisticated set of tools for investigation of exceptionally pure and tunable quantumsystems [1, 2, 3, 4]. Recently, this toolbox has been expanded to also include thepossibility of applying controllable gauge fields to both bosonic [5, 6, 7, 8, 9] andfermionic systems [10, 11] (see references [12] or [13] for short reviews). Spin-orbitcoupling is a prime example of a non-abelian potential that plays an important rolethroughout physics. Examples are the spin-orbit splittings in atomic and nuclearspectra, and the distinct effect it imposes on the band structure of solid-state systemsthat has lead to great recent advances in the exploration of materials with robustmetallic surface states, the so-called topological insulators [14, 15].In many experiment with ultracold atoms, the external trapping potentials, whilepresent, are often quite shallow and can be ignored for many purposes or one caninclude them using a local density approximation. However, some recent experimentshave demonstrated that tight trapping is not only possible but also a very excitingpossibility as this brings the physics one can study close to what is known from atomicor nuclear structure. Some recent theoretical papers have shown that the trappingpotential can play an important role in spin-orbit coupled systems [16, 17, 18, 19].The cited works consider the case where the trapping potential is isotropic in allthree dimensions [16] or isotopic in the plane where the spin-orbit coupling acts[17, 18, 19, 20]. In the current presentation we extend this discussion by consideringalso the case where the trapping potential is deformed. This can be done by adjustingthe optical or magnetic trap geometry [2].Our findings show that deformation of the external potential has a decisiveeffect on the spectral density and can lead to the opening and (near) closing of theenergy gaps in the single-particle spectrum. In the strongly deformed effectively one-dimensional limit the level spacing is completely determined by the shallow trapharmonic frequency at low energy. However, for small deformation one can findregimes where the spectral gaps will almost close and produce a (quasi)-continuum.This implies that deformation can be used as a control parameter for the leveldensity. In the case of a many-body system with non-abelian gauge potentials, thespectral density plays a decisive role in trapped systems when exploring many-bodyphenomena such as exotic pairing and crossover [21, 22, 23, 24, 26], superfluidityand condensation [27, 28, 29, 30, 31], ferromagnetism [32] or quantum Hall states[33, 34, 35, 36, 37, 38, 39, 40]. Our results represent an initial step in exploring theseinteresting questions for spin-orbit coupled systems in deformed traps.
2. Formalism
We consider the single-particle problem in a deformed three-dimensional (3D)harmonic trap including a Rashba [41] spin-orbit interaction and an applied magneticfield. Here we use the Rashba form of the spin-orbit interaction but the formalismis completely general and applies to all types of spin-orbit coupling. This includesthe case where the Rashba-type and Dresselhaus-type [42] contributions are equal(amounting to a term, σ x p y , in the notation introduced below which is used in manycurrent cold atomic gas experiments studying spin-orbit effects). In experiments with pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps m experiencing a Rashba and Zeemanterm is H = ˆ p x m + 12 mω x ˆ x + ˆ p y m + 12 mω y ˆ y + ˆ p z m + 12 mω z ˆ z + α R (ˆ σ x ˆ p y − ˆ σ y ˆ p x ) − µ · B , (1)where ˆ σ x are ˆ σ y are 2 × α R is the strength of the Rashba spin-orbitcoupling. We consider a particle with two internal degrees of freedom, which we labelas spin up |↑(cid:105) and spin down |↓(cid:105) . We choose the effective magnetic field as B = (0 , , B )and we can write the Zeeman term as − µB ˆ σ z with ˆ σ z the remaining Pauli matrix and µ is the effective magnetic moment. For the harmonic trapping potentials we assumedifferent trapping frequencies in all directions, ω x , ω y and ω z . However, since we takethe Rashba term to act only in the xy -plane the z -direction effectively decouples fromthe problem. Note that in the case of B = 0, the Hamiltonian is symmetric undertime-reversal and for a single fermion Kramers theorem dictates a two-fold degeneracy.The Hamiltonian can be written in the matrix form (cid:18) ˆ H x + ˆ H y − µB α R (ˆ p y + i ˆ p x ) α R (ˆ p y − i ˆ p x ) ˆ H x + ˆ H y + µB (cid:19) (cid:18) ψ ↑ ψ ↓ (cid:19) = E (cid:18) ψ ↑ ψ ↓ (cid:19) , (2)where we have introduced the short-hand ˆ H x = ˆ p x m + mω x ˆ x and similarly for ˆ H y .This constitutes an effective two-dimensional (2D) problem. The oscillator energy inz-direction can be considered as a parameter that can be included in the total energy, E . The natural basis in which to expand our wave-functions are eigenfunctions of the2D harmonic oscillator. Thus, for a ψ ↑ = (cid:88) n x ,n y a n x ,n y | n x , n y , ↑(cid:105) (3) ψ ↓ = (cid:88) n x ,n y b n x ,n y | n x , n y , ↓(cid:105) , (4)where | n x , n y (cid:105) are vectors in the Hilbert space of the two-dimensional harmonicoscillator solutions.Introducing the standard ladder operators ˆ a x,y and ˆ a † x,y that obeyˆ x = (cid:114) (cid:126) mω x (ˆ a † x + ˆ a x ) , ˆ p x = i (cid:114) m (cid:126) ω x a † x − ˆ a x ) , ˆ y = (cid:115) (cid:126) mω y (ˆ a † y + ˆ a y ) , ˆ p y = i (cid:114) m (cid:126) ω y a † y − ˆ a y ) , (5)the Hamiltonian matrix now reads (cid:32) γ ( ˆ N x + ) + ( ˆ N y + ) − µB (cid:126) ω y β ( i (ˆ a † y − a y ) − √ γ (ˆ a † x − a x )) β ( i (ˆ a † y − a y ) + √ γ (ˆ a † x − a x )) γ ( ˆ N x + ) + ( ˆ N x + ) + µB (cid:126) ω y (cid:33) (cid:18) ψ ↑ ψ ↓ (cid:19) = ε (cid:18) ψ ↑ ψ ↓ (cid:19) , (6) pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps N x = ˆ a † x ˆ a x and ˆ N y = ˆ a † y ˆ a y . We havehere defined the ratio, γ , of the harmonic oscillator frequencies as γ = ω x ω y and thedimensionless Rashba coupling β = α R (cid:113) m (cid:126) ω y . Note that the latter implies that wemeasure the velocity α R in units of the harmonic oscillator velocity (cid:113) (cid:126) ω y m . We willbe using (cid:126) ω y as the unit of energy. The linear system of equations for the coefficients a and b becomes( ε o ( n x , n y ) − ε ) a n x ,n y + β (cid:20) i √ n y b n x ,n y − − i (cid:112) n y + 1 b n x ,n y +1 − √ γn x b n x − ,n y + (cid:112) γ ( n x + 1) b n x +1 ,n y (cid:21) = 0( ε o ( n x , n y ) − ε ) b n x ,n y + β (cid:20) i √ n y a n x ,n y − − i (cid:112) n y + 1 a n x ,n y +1 + √ γn x a n x − ,n y − (cid:112) γ ( n x + 1) a n x +1 ,n y (cid:21) = 0 , (7)where ε o ( n x , n y ) = γ ( n x + ) + ( n y + ). This set of equations cannot be solvedanalytically and one has to resort to numerical methods. In the symmetric case where γ = 1, one could also have used a basis based on the solution of the harmonic oscillatorpotential in cylindrical coordinates [17, 18, 19]. However, this is not appropriate in ourcase since the cylindrical symmetry is broken in the plane for γ (cid:54) = 1 and the problemis therefore better handled using the Cartesian basis expansion presented above.
3. Single-particle spectra
We now present the results of our study of the single-particle spectral structure inthe presence of a Rashba spin-orbit term and with an external trap that can bedeformed. All the results presented here are in the regime 0 ≤ β ≤ .
5. It is asimple matter to go to even higher Rashba couplings but it requires the use of abigger single-particle basis in order for all states to properly converge. The results wepresent below have all been obtained by using a basis with about 700 single-particlestates. However, due to deformation the maximum number of quanta, max( n x ) andmax( n y ), is generally not equal to each other since we cut the basis in energy space,i.e. max( ε ( n x , n y )) = γ (max( n x )+1 / n y )+1 /
2) is our cut-off that restrictsthe values of n x and n y . In the left panel in figure 1 we show the single-particle energy spectrum in a two-dimensional spherical ( γ = 1) harmonic oscillator as function of the coupling strength, β . The external magnetic field is not included here ( µB = 0). The oscillatordegeneracies that are well-known for β = 0 are lifted as β increases and the oscillatorshells become more and more mixed. Each level is still doubly degenerate due to timereversal symmetry mentioned above. The lowest levels decrease and for sufficientlylarge strengths the behavior approaches a parabolic dependence on β that can be foundin a semi-classical treatment [16]. The fact that we recover this quadratic behaviorfor the lowest states imply that our calculations capture the large β limit for these pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps E [ − h ω ] β E [ − h ω y ] β Figure 1.
Energy as function of the dimensionless spin-orbit coupling parameter β for the case of equal frequencies ω = ω x = ω y with no Zeeman shift (left panel)and including a Zeeman shift of magnitude µB = (cid:126) ω . states. Intuitively, we can understand the behavior by taking a large α R limit wherewe neglect all but the Rashba term in the Hamiltonian in (1). The energy shouldthen be proportional to α R | p | , where p is the momentum. However, since the onlyavailable velocity is α R itself, we can infer that the energy must scale like mα R . Insection 3.4 we find this same scaling in perturbation theory. The perturbation theorybelow shows that the scaling is − mα R , i.e. the energy decreases with β in general.This can be understood again by neglecting the trap in which case the minimum ofthe free particle dispersion is at − mα R . This is discussed in reference [21] with anargument that is the same as the one we use for the one-dimensional case in section 3.3(completing the square and obtaining equation (11)).The many levels are spread out over energies and overall they cover rather denselythe energy space. In many regions the spectrum resembles a continuous distributionwhere the individual levels cannot be distinguished. The deviations from the regularpicture seen in the top right-hand corner on both panels in figure 1 is due to thenumerics and the use of a finite basis set. We have numerically checked that this canbe remedied by systematic expansion of the basis size.We note the many avoided crossings that can be seen in the spectrum. The factthat the Hamiltonian does not preserve the usual spherical symmetry (or cylindricalsymmetrty in the plane) means that one cannot decouple sectors of given angularmomentum or parity. Only time-reversal symmetry remains (in the absence of aZeeman term). Other studies have used cylindrical expansions [18, 19] but at theexpensive of coupling different angular momenta. Here we use the Cartesian basisfrom the start since we find this more convenient. The lack of symmetry is reflected in pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps β = 0 of one (cid:126) ω is strongly reduced at β ∼ . ∼ (cid:126) ω . Indeed these sorts of changes are reflected on theproperties of a many-body system in a symmetric trap as discussed for the case ofcondensates in references [17], [18] and [19]. A recent study has shown that similarspectra can be obtained by mapping the current problem to a quantum Rabi model[44]. On the right panel in figure 1 we show again the case of equal frequencies, ω = ω x = ω y , but this time with the inclusion of a Zeeman field of magnitude µB = (cid:126) ω . The Zeeman field will not influence the oscillator levels but only displacethe spin components. Note that the absolute ground state of the spectrum now startsat zero energy and then decreases. We have opted to keep the same vertical scale onboth panels in figure 1 in order to make a comparison of the overall spectral densityat higher energies and therefore the Zeeman shifted ground state level cannot beseen. What we find is that the Zeeman split will tend to counteract the effect of theRashba term at small β so that we now find a spectrum with large gaps at β ∼ . β ∼ . We now consider the case of an oscillator potential that is non-spherical correspondingto different frequencies in the x and y directions. In this situation, high degeneracy isfound for special frequency ratios of deformed harmonic oscillators. These specialconfigurations occur when the frequency ratios equal ratios of small integers as2 /
1, 3 /
2, etc. This is due to the harmonic oscillator spectrum being linear inboth the frequencies and quantum numbers in the different spatial directions. Theharmonic oscillator has especially high degeneracy compared to deformations of(almost) all other radial shapes. This means that gaps are more likely to occur inthe spectrum. The implication for pure oscillators with no Rashba or Zeeman termsis that degeneracies are periodically occurring as function of the frequency ratio, andthat degeneracies are washed out in between the highly degenerate points. In figure 2we show the spectrum for frequency ratios γ = 2 (left panel) and γ = 3 (right panel) asa function of Rashba coupling strength, β , with no Zeeman field. For these ratios weclearly see that the spectrum is going towards a more degenerate form with stronger pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps E [ − h ω y ] β E [ − h ω y ] β Figure 2.
Energy as function of dimensionless spin-orbit coupling parameter β for the case where the oscillator potential is deformed. The left panel has γ = ω x ω y = 2 and the right panel has γ = 3. gaps for increasing γ in comparison to the γ = 1 case shown in figure 1. Indeed for γ = 3 we have to go up to energies of 15 (cid:126) ω y and above before we can see the samebehavior as for lower energies in the left panel of figure 1. When including a Zeemanterm for these ratios (not shown here) we see the same trends as seen from left toright in figure 1, i.e. the spectrum becomes slightly more dense overall and there iscompetition between Rashba and Zeeman terms.A completely different scenario is displayed in the case where γ is not a ratioof small integers. In figure 3 we show the single-particle spectrum for γ = 1 . γ = 2 .
17 (right panel). For these ratios the states are more evenlydistributed and for γ = 1 .
57 we see an almost constant spectral density for energiesof 5 (cid:126) ω y and above, while for γ = 2 .
17 this is not seen until about 10 (cid:126) ω y and above.Comparing the left and right-hand panels in figure 3, we see an overall tendency forthe larger γ to have a smaller overall density of levels since this is closer to the one-dimensional limit that we will return to momentarily. The overall quadratic decreasewith β is still seen as in figures 1 and 2. Comparing the results presented in figure 2for the ratios γ = 2 and γ = 3 to those of figure 3 with γ = 1 .
57 and γ = 2 . pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps E [ − h ω y ] β E [ − h ω y ] β Figure 3.
Same as figure 2 but with ratios γ = 1 .
57 in the left panel and γ = 2 . Increasing the frequency ratio towards infinity separates the Hamiltonian in a low-energy one-dimensional part very weakly coupled to the high-energy one-dimensionalpart in the other direction. With our conventions, the limit γ (cid:29) x -direction and a shallow confinement along the y -direction. These decoupled one-dimensional equations can be solved analytically withthe corresponding Rashba couplings. The Pauli matrix for either σ x or σ y implies thatone should use a basis of eigenfunctions for the operator with the small frequency, i.e. ω y , since the other direction has a very large oscillator frequency that effectivelyfreezes the motion. This linear combination of equal amplitude spin-up and spin-down spinors immediately decouple the equations of motion into separate oscillatoreigenvalue problems. The eigenvectors of σ x areΦ ± = (cid:18) ± (cid:19) , (8)and these can now be used as the two-component basis instead of the spin-up andspin-down spinors. The one-dimensional Schr¨odinger equation then becomes (cid:18) p y m + 12 mω y y ± α R p y − E (cid:19) Φ ± f ± ( y ) = 0 , (9)where f ± ( y ) is the radial wave function. We rewrite these equations as (cid:20) m ( p y ± mα R ) + 12 mω y y − mα R − E (cid:21) Φ ± f ± ( y ) = 0 , (10) pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps E n y = − mα R + (cid:126) ω y ( n y + 12 ) (11)The first term is simply the quadratic decrease of the energy with β that we alreadynoted above. It can be interpreted as a Galilean boost by the Rashba velocity, α R . Wecan compare this dispersion to the one-dimensional solution with no external trappingpotential in the y -direction which is simply E k y = (cid:126) k y m ± α R (cid:126) k y , (12)where k y is the one-dimensional momentum along the y -direction. These twodispersion relations look very different but can be reconciled by recalling that thevirial theorem tells us that (cid:104) ˜ p y (cid:105) ∝ m (cid:126) ω y n y where ˜ p y = p y ± mα R . From the secondterm in equation 11 we would thus get linear, α R (cid:104) p y (cid:105) , and quadratic, (cid:104) p y (cid:105) , dependenceon the momentum. This is the same structure as equation 12 and we see that theharmonically trapped system behaves similarly to the free-space case when consideringthe expectation value of the momentum.In figure 4 we show the spectral structure as one approaches the one-dimensionallimit. In the left panel of figure 4 we have γ = 5 and in the right panel γ = 10. Thelow-energy eigenvalues reduce to the equidistant harmonic oscillator spectrum withthe same frequency but shifted quadratically with the spin-orbit coupling strength.The remarkably simple emerging feature is that the lowest part of the spectrum foran even modest deformation already resembles the equidistant one-dimensional limit.This can be clearly seen by making a comparison of figure 4 to figures 2 and 3. Asthe deformation increases an increasing part of the low-energy spectrum approachesthe one-dimensional limit. This feature can be understood from the weak coupling oftwo oscillators with very different frequencies. The perturbation on the lowest energystates in the spectrum from the lowest of the large frequency ( ω x ) states is proportionalto the square of the coupling strength divided by the energy difference by second orderperturbation theory. This tells us that the one-dimensional limit is approached in thebottom of the spectrum with increasing deformation. This low-energy spectrum ismuch simpler and much less dense than that of the two-dimension spherical oscillator.Note that the spectrum is still two-fold degenerate due to the time-reversal symmetry,but now it is practically back to the standard equidistant scheme typical of harmonicconfinement. The weak coupling limit, β →
0, can be investigated in detail using perturbationtheory. In the absence of deformation and external magnetic field, the system is highlydegenerate (Kramers and oscillator degeneracies) and requires degenerate perturbationtheory to second order (first order vanishes as we discuss momentarily). The easiestway to avoid the complicated expressions of the degenerate formalism is to consider adeformed oscillator trap with an external magnetic field which lifts all degeneracies.By continuity, we get the non-deformed and zero external field by taking this limit atthe end of the calculation.The Hamiltonian is ˆ H = ˆ H + ˆ V R , whereˆ H = ˆ p x m + 12 mω x x + ˆ p y m + 12 mω y y − µB ˆ σ z (13) pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps E [ − h ω y ] β E [ − h ω y ] β Figure 4.
Same as figure 2 and 3 for γ = 5 (left panel) and γ = 10 (rightpanel). These results approach the limit of an effectively one-dimensional system,i.e. γ (cid:29) is the unperturbed part of the Hamiltonian and the perturbation isˆ V R = α R (ˆ σ x ˆ p y − ˆ σ y ˆ p x ) . (14)Then the Schr¨odinger equation for the unperturbed Hamiltonian isˆ H ψ (0) ( x, y, σ ) = E (0) ψ (0) ( x, y, σ ) , (15)with the well-known solutions: ψ (0) ( x, y, ↑ ) = φ n x ,n y ( x, y ) (cid:0) (cid:1) and ψ (0) ( x, y, ↓ ) = φ n x ,n y ( x, y ) (cid:0) (cid:1) , with eigenenergies E (0) n x ,n y ,σ = (cid:126) ω x ( n x + ) + (cid:126) ω y ( n y + ) − σµB ,where σ = ± φ n x ,n y ( x, y ) are2D harmonic oscillator solutions.Now we can directly find corrections to the energy. One can see that the diagonalmatrix element for V R is zero, thus the first-order correction will be zero as well. Thesecond-order correction is[47] E (2) n x ,n y ,σ = (cid:88) (cid:48) m x ,m y ,σ (cid:48) |(cid:104) n x , n y , σ | ˆ V R | m x , m y , σ (cid:48) (cid:105)| E (0) n x ,n y ,σ − E (0) m x ,m y ,σ (cid:48) . (16)We can explicitly write down the matrix elements. Due to the properties of Paulimatrices ˆ σ x and ˆ σ y only the matrix elements with different spin projections willcontribute in the sum. (cid:104) n x , n y , ↑ | ˆ V R | m x , m y , ↓(cid:105) = − α R (cid:114) m (cid:126) √ ω x ( √ m x + 1 δ n x ,m x +1 −− √ m x δ n x ,m x − ) δ n y ,m y − i √ ω y ( (cid:112) m y + 1 δ n y ,m y +1 − √ m y δ n y ,m y − ) δ n x ,m x ] (17) pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps (cid:104) n x , n y , ↓ | ˆ V R | m x , m y , ↑(cid:105) = α R (cid:114) m (cid:126) √ ω x ( √ m x + 1 δ n x ,m x +1 −− √ m x δ n x ,m x − ) δ n y ,m y + i √ ω y ( (cid:112) m y + 1 δ n y ,m y +1 − √ m y δ n y ,m y − ) δ n x ,m x ] . (18)We see that only transitions between the nearest states contribute to the sum. It isstraightforward now to write down the second-order corrections for the ground stateenergy E (2)0 , , ↑ = − mα R (cid:2) (cid:126) ω x (cid:126) ω x + 2 µB + (cid:126) ω y (cid:126) ω y + 2 µB (cid:3) . (19)For a case of µB (cid:28) (cid:126) ω y we get a simple term, which appears natural considering thatthe dimension of α R is the velocity E (2)0 , , ↑ = − mα R . (20)This equation is valid for coupling parameter values smaller than the oscillator velocity, α R (cid:28) (cid:113) (cid:126) ωm . For a regime of larger α R the contribution of all states becomes crucial,therefore one has to consider the higher-order perturbation corrections to energy.However, for the case of α R (cid:38) (cid:113) (cid:126) ω y m the behavior of the ground state energy is stillquadratic in α R [19].
4. Average spectral properties
The single-particle spectra reveal regions of stability for specific finite particle numbers.The stability is closely related to the density of states at the Fermi level. To examinethese properties for finite systems, we first calculate the average behavior of thedensity of states. Subsequently we relate to the critical strength for superfluiditywhich becomes strongly dependent on the spectral properties.
The density of single-particle states g ( ε ) is a sum of delta functions for a system witha discrete energy spectrum. To emphasize the underlying structures we broaden eachof the eigenvalues, ε i , by a normalized gaussian. Then we get the average density ofstates, g δ ( ε ) = 1 δ √ π (cid:88) i ( 32 − ( ε − ε i δ ) ) exp [ − ( ε − ε i δ ) ] , (21)where the second order polynomial is introduced to guarantee that a smoothly varyingset of eigenenergies produce the same smooth behavior [45].If the smearing parameter, δ , is equal to or larger than the shell spacing of thespherical oscillator only average properties are left in g δ ( ε ). By ’shell’ we mean theset of states, which corresponds to one oscillator level for the β = 0 case. To exhibitshell structure δ has to be less than about half of the shell spacing. For a sphericaltwo-dimensional oscillator the density of states g δ ( ε ) for δ (cid:38) (cid:126) ω becomes independentof the smearing parameter. In this case the index δ can be omitted g δ ( ε ) −−−→ δ (cid:38) (cid:126) ω ˜ g ( ε ) = 2 ε ( (cid:126) ω ) , (22) pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps β = 1 δ = 0.3 δ = 0.5 δ = 1 0 20 40 g δ ( ε ) , / [ − h ω ] β = 0.5 δ = 0.3 δ = 0.5 δ = 1 0 20 40 60 0 2.5 5 7.5 10 12.5 15 17.5 20 ε , [ − h ω ] β = 0 δ = 0.3 δ = 0.5 δ = 1 0 20 40 β = 1 δ = 0.3 δ = 0.5 δ = 1 0 20 40 g δ ( ε ) , / [ − h ω ] β = 0.5 δ = 0.3 δ = 0.5 δ = 1 0 20 40 60 0 2.5 5 7.5 10 12.5 15 17.5 20 ε , [ − h ω ] β = 0 δ = 0.3 δ = 0.5 δ = 1 0 10 20 30 β = 1 δ = 0.3 δ = 0.5 δ = 1 0 10 20 30 g δ ( ε ) , / [ − h ω ] β = 0.5 δ = 0.3 δ = 0.5 δ = 1 0 10 20 30 0 2.5 5 7.5 10 12.5 15 17.5 20 ε , [ − h ω ] β = 0 δ = 0.3 δ = 0.5 δ = 1 Figure 5.
Single-particle density of states as function of energy for the spherical γ = 1 case (upper panel) and deformed cases with the frequency ratios γ = 2(middle panel) and γ = 1 .
57 (lower panel). The smearing parameter, δ , and thedimensionless Rashba coupling paremeters, β , are given in the panels. Here weset ω = ω y . pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps particle. Moregenerally, for a deformed oscillator [46]˜ g ( ε ) ≈ ε (cid:126) ω x ω y = 2 εγ ( (cid:126) ω y ) (23)We can then relate the particle number, N , and the Fermi energy, ε F , which is theenergy of the highest occupied state, by N = ε F (cid:82) ˜ g ( ε ) dε . This gives N ≈ ε F (cid:126) ω x ω y (24)or by inversion ε F ≈ (cid:126) (cid:112) N ω x ω y . (25)For zero temperature, the chemical potential, µ , is equal to the Fermi energy for N particles.In figure 5, we show g δ ( ε ) for a number of β -values and deformations of theexternal field. The regular behavior is seen for a spherical oscillator for β = 0. Theamplitudes of the oscillations decrease as the smearing parameter, δ , increases. When δ approaches zero the original discrete spectrum is recovered, and when δ is larger than (cid:126) ω only ˜ g ( ε ), the smooth (linear) average behavior remains. This regularity changeswhen β assumes non-zero values as seen in the middle and upper panels of figure 5.The amplitudes of the oscillations still decrease with increasing δ . However, theseamplitudes are now irregular functions of ε . Small amplitudes produce the smoothaverage background behavior for moderate δ -values, and correspond to a dense single-particle spectrum. The opposite applies to large amplitudes where the single-particlestructure is more pronounced corresponding to a dilute spectrum. This structuralvariation changes as function of energy. The other deformation in figure 5 reveals thesame overall qualitative behavior, that is regular behavior for β = 0 and variation ofthe amplitudes as function of ε for non-zero β -values. The correspondence betweenlarge-small amplitudes and white-black regions are clearly seen by comparing figure 5and figure 1.The similarities between results of a spherical shape and a small integer ratio ofthe directional frequencies, ω x ω y = 2, are due to large degeneracies arising from manycoinciding levels. Irrational frequency ratios remove this regularity in the density ofstates even for β = 0. The most direct implication of small (large) amplitudes atspecific energies is low (high) stability for a particle number corresponding to a Fermilevel at that energy [45]. The calculated spectra are single-particle energies for asystem of non-interacting fermions. However, similar qualitative behavior would bepresent for any mean-field approximation of interacting fermions. For a finite numberof particles, the spectral properties around the Fermi energy are decisive for correctionsbeyond the mean-field structure. Superfluid behavior is possible in the presence of a small residual interaction. Fora system of a finite number of particles the microscopic properties of the eigenvalue pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps G c , of the attractive interaction necessaryfor creation of a superfluid state in a fermionic system as we now discuss.To extract the overall behavior we assume the residual pairing interaction H R = − (cid:88) i,k G ik a k † a k † a i a i , (26)where ( a k , a † k ) are annihilation and creation operators for single-particle states | k (cid:105) .The time reversed state of | k (cid:105) is denoted | k (cid:105) . The BCS approximation provides theequations which determine the gaps2∆ k = (cid:88) i ∆ i G ik (cid:113) ( ε i − µ ) + ∆ i . (27)The states that provide the largest contribution for pairing solutions are those in theinterval, ± S , around the chemical potential, µ , which is equal to the Fermi energy forzero temperature. For a constant pairing matrix element, G , the gap, ∆ i , is also aconstant, ∆, which is determined by the states around µ [48]. The magnitude of ∆ isthen determined by 2 = G (cid:88) i (cid:112) ( ε i − µ ) + ∆ , (28)where the summation now only extends over states with | ε i − µ | < S . However, to geta superfluid solution with finite ∆ we must have G > G c , where the critical strength G c is given by 2 G c = (cid:88) i | ε i − µ | . (29)This critical strength arises only due to the finite number of particles. When N → ∞ ,we have G c → µ is almost inevitably between the lastoccupied, ε occ , and the first unoccupied level, ε un . It is then a reasonableapproximation to choose µ = ( ε occ + ε un ) which depends on the particle number. As µ increases in steps corresponding to increase of the single-particle levels ( ε occ , ε un ) wecan for a single-particle spectrum (specified by fixing β and γ ) compute G c as functionof µ , or equivalently particle number (through the usual BCS number equation [46]).The approximations to Eqs. (28) and (29) are quickly formed by substituting thesummation by an integral. We immediately get1 G c (cid:39) g δ ( µ ) ln( 2 S ∆ ) , (30)1 G c (cid:39) g δ ( µ ) ln( 2 Sε un − ε occ ) (cid:39) g δ ( µ ) ln(2 Sg δ ( µ )) , (31)where we used that ( ε un − ε occ ) g δ ( µ ) (cid:39)
1, following from the basic definition of thedoubly degenerate single-particle density of states.By working with constant matrix elements and critical values, G c , we only includeeffects of the appropriate residual interaction. In particular, we have ignored thereduction due to the fact that the interaction in cold atoms for two-component fermionsis in the spin singlet channel [46], while the spin-orbit coupling mixes triplet and pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps G c obtained fromEqs. (29)) and (31) for different β -values and deformations. The horizontal axis is theenergy ε , and the discrete points are chosen as energies precisely between two single-particle levels. These points represent chemical potential values, µ , and correspondto some uniquely given particle numbers (via the standard number equation). Fordegenerate levels this implies that at least one single-particle energy equals the valueof the chemical potential, and consequently the critical strength must vanish accordingto Eq. (31). The spherical oscillator with β = 0 is therefore not a suitable illustration,and we avoided this example in figure 6. The same problem arises for deformationswhere the frequency ratios have small integer values. The global behavior of G c is a decrease with µ . This is most clearly seen by the average curves arising fromEq. (31) with two different values of δ . For δ = (cid:126) ω the oscillations have essentially alldisappeared, leaving only the decreasing average trend. For a smaller δ value the stillsmooth oscillations tend to pick up and sometimes even reproduce the microscopicbehavior from Eq. (29). This is remarkable due to the simplicity in both derivationand resulting expression, Eq. (31). The dependence on the interaction interval S islogarithmic and hence rather insignificant unless the absolute value is much smallerthan the shell distance.The microscopic behavior is seen in the discrete points where large variationsappear corresponding to critical values from almost zero and up to about (cid:126) ω y / β -values and deformations.However, the particle numbers corresponding to small and large values of G c arecompletely different and strongly depending on the features of the underlying single-particle spectra. An approximate average relation between µ and N can be seen inEq. (25) with µ = ε F .The curves from Eq. (31) are overall at the upper limit of a genuine average.This is due to the systematical underestimate of the density of states at the chemicalpotential chosen to be exactly between two levels. The tendency of increasing thecritical strength with β is due to the overall increase of the density of states forany given particle number. In general, the critical strengths are very fluctuating.This suggests that fine-tuning of particle number, Rashba coupling, and shape of theexternal field can lead to desirable but qualitatively different structures.
5. Discussion
We have shown that the single-particle spectrum for particles that are subject to aRashba spin-orbit coupling in a deformed harmonic trap is very dependent on theparameters and that spectral gaps can be tuned by varying the Rashba couplingstrength, a perpendicular Zeeman field and by changing the deformation of thetrapping potential in the plane where the Rashba coupling acts. In particular, wefind that the effect of deformation can be very different depending on whether theratio of the trapping frequencies in the plane is equal to ratios of small integers or pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps β = 1.5 G Σ G G G , / [ − h ω ] β = 1 G Σ G G µ , [ − h ω ] β = 0.5 G Σ G G β = 1.5 G Σ G G G , / [ − h ω ] β = 1 G Σ G G µ , [ − h ω ] β = 0.5 G Σ G G β = 1.5 G Σ G G G , / [ − h ω ] β = 1 G Σ G G µ , [ − h ω ] β = 0.5 G Σ G G Figure 6.
The critical coupling constant, G c (in units of (cid:126) ω y ), as a function ofthe chemical potential, µ , for the spherical γ = 1 case (upper panel) and deformedcases with the frequency ratios γ = 2 (middle panel) and γ = 1 .
57 (lower panel).The smearing parameter, δ , and the dimensionless Rashba coupling parameter, β , are given in the panels. In the legende we indicate by G Σ results obtainedby summation over the single-particle states, while G . and G are obtainedby smearing using the parameters δ = 0 . δ = 1 respectively. Here we set ω = ω y . pectral Gaps of Spin-orbit Coupled Particles in Deformed Traps Acknowledgements
This work was supported by the Danish Agency for Science,Technology, and Innovation under the Danish Council for Independent Research -Natural Sciences.
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