Spin-orbit-induced resonances and threshold anomalies in a reduced dimension Fermi gas
aa r X i v : . [ qu a n t - ph ] O c t Spin-orbit-induced resonances and threshold anomalies in a reduced dimension Fermigas
Su-Ju Wang ∗ and Chris H. Greene † Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA (Dated: June 26, 2018)We calculate the reflection and transmission probabilities in a one-dimensional Fermi gas withan equal mixing of the Rashba and Dresselhaus spin-orbit coupling (RD-SOC) produced by anexternal Raman laser field. These probabilities are computed over multiple relevant energy rangeswithin the pseudo-potential approximation. Strong scattering resonances are found whenever theincident energy approaches either a scattering threshold or a quasi-bound state attached to one ofthe energetically closed higher dispersion branches. A striking difference is demonstrated betweentwo very different regimes set by the Raman laser intensity, namely between scattering for the single-minimum dispersion versus the double-minimum dispersion at the lowest threshold. The presence ofRD-SOC together with the Raman field fundamentally changes the scattering behavior and enablesthe realization of very different one-dimensional theoretical models in a single experimental setupwhen combined with a confinement-induced resonance.
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Scattering constitutes the fundamental process used toprobe countless physical systems, ranging from ultracolddilute gases to energetic quark-gluon plasmas. Rich, in-triguing phenomena are found already in one-dimensionalquantum scattering processes. For instance, a quantumparticle can tunnel through a double-barrier structureas if no potential exists when the particle energy (evenwhen classically forbidden inside the barrier regions) isresonant with a quasi-bound state supported by the po-tential. In ultracold atomic systems, a scattering res-onance coupling relative motion of two atoms in one ormore open channels and bound molecular states in closedchannel(s) forms the basis of tunable Fano-Feshbach res-onances [1], now extensively used to tune the scatter-ing lengths and enabling many studies in the unitaryregime [2].Interactions between two particles can be significantlymodified by their external conditions. A recent experi-ment [3] on ultracold atomic collisions in an Rb conden-sate in the presence of a two-photon Raman field has ob-served non-spherical scattering halos at very low temper-ature, where normally only spherically symmetric s -wavescattering is expected. The existence of effective higher-partial waves was attributed to the effects of Raman laserdressing. In short, the presence of laser fields modifiescollisions between two dressed atoms, and creates effec-tive higher partial wave scattering. Similar experimen-tal setups were used to create spin-orbit coupled Bose-Einstein condensates (BECs) [4] and degenerate Fermigases (DFGs) [5][6] with an equal mixing of the Rashbaand the Dresselhaus spin-orbit coupling. Although SOCappears as a single-particle term in the Hamiltonian, thenon-trivial coupling between the internal (spin) with the ∗ Electronic address: [email protected] † Electronic address: [email protected] external (linear momentum) alters the dispersion relationin a fundamental way and thereby changes the intrinsictwo-body scattering process.Of particular interest is the binary collision regime oc-curring when the scattering energy approaches two cross-ing points on the positive and negative sides of the lowestSO band. This paper describes the multi-channel scat-tering resonances in 1D spin-orbit coupled systems con-trolled by Raman laser fields of moderate strength. Ina strong Raman field, the energy band will experience adouble-minimum (DM) to single-minimum (SM) transi-tion [7]. These two regimes of physics are found to exhibitqualitatively different 1D models when explored near aconfinement-induced resonance [8] [9].Consider a binary collision as it arises in the experi-mental protocol of the Zhang group [5]. The two-bodyHamiltonian is H = ~ k m + ~ λm k σ x + ~ Ω2 σ z + ~ δ σ x + ~ k m + ~ λm k σ x + ~ Ω2 σ z + ~ δ σ x + V ( x ) , (1)where σ i = x,y,z are Pauli matrices for spin-1/2 particles, λ is the SOC strength, Ω is the Raman coupling strength, δ is the two-photon detuning, and V ( x ) is the two-body in-teraction. Because the 1D Rashba-Dresselhaus spin-orbitcoupling can be gauged away with a unitary transforma-tion, the RD-SOC term simply causes a constant energyshift for different spin states in the absence of the Ra-man field and thus does not grant an interesting result.Therefore, the effective magnetic field from the Ramancoupling, which is perpendicular to the SOC field, is cru-cial in our discussions below because it opens a gap be-tween the energy bands, visible in Fig. 1(a) and 2(a).This is very different from the Rashba SOC in 2D or theWeyl SOC in 3D, where the non-abelian nature of thevector potentials alone makes their effects nontrivial.After defining the relative momentum and the totalmomentum as k = ( k − k ) / K = k + k , we canrecast the Hamiltonian in Eq. (1) as: H = ~ K M ˆ I + ~ k µ ˆ I + V ( x )+ ~ √ kλm − ~ √ kλm ~ √ kλm ~ Ω 0 ~ Kλ √ m + ~ δ √ − ~ √ kλm − ~ Ω ~ Kλ √ m + ~ δ √ ~ Kλ √ m + ~ δ √ ~ Kλ √ m + ~ δ √ , (2)where the matrix is written in the singlet and tripletbasis: {| S i = ( | ↑↓i − | ↓↑i ) / √ , | T i = | ↑↑i , | T i = | ↓↓i , | T i = ( | ↑↓i + | ↓↑i ) / √ } . These vectors forma complete basis for the Hilbert space of two spin-1/2particles. Morover, if we move into the center of massframe of the two colliding atoms, the triplet channel, | T i ,is decoupled from the other spin channels in the caseof zero detuning. Therefore, the dimension of the spinHilbert space is reduced into three. The correspondingeigenstates and eigenvalues from top to bottom along theenergy axis areΨ u = 12 q ~ k + (cid:0) m Ω2 λ (cid:1) −√ ~ k − q ~ k + (cid:0) m Ω2 λ (cid:1) − m Ω2 λ q ~ k + (cid:0) m Ω2 λ (cid:1) − m Ω2 λ e ikx (3)Ψ m = 1 q (cid:2) ~ k + (cid:0) m Ω2 λ (cid:1) (cid:3) − m Ω √ λ ~ k ~ k e ikx (4)Ψ b = 12 q ~ k + (cid:0) m Ω2 λ (cid:1) √ ~ k − q ~ k + (cid:0) m Ω2 λ (cid:1) + m Ω2 λ q ~ k + (cid:0) m Ω2 λ (cid:1) + m Ω2 λ e ikx . (5) E u = ~ ( k + p k λ + m Ω / ~ ) /m, (6) E m = ~ k /m, (7) E b = ~ ( k − p k λ + m Ω / ~ ) /m. (8)In the following, we denote the different curves in thesecomputed dispersion relations as ”states” or ”branches”.The channel structure of the multichannel scatteringin the presence of RD-SOC and the Raman field is de-termined by (1) the incoming scattering energy, E , ofthe relative motion and (2) the relative strength be-tween ( ~ k/m ) λ and Ω. When the Raman couplingstrength is stronger than Ω c = 2 ~ λ /m , the energy bandsare in the single-minimum regime. In this regime, for − ~ Ω < E <
0, there are one open channel and twoclosed channels. For 0 < E < ~ Ω, there are two open channels and one closed channel. For
E > ~ Ω, all chan-nels are open. The channel structure becomes more com-plex when the Raman strength is weaker than Ω c (orstated alternatively, when the RD-SOC strength is sig-nificant). In this double-minimum regime, the channelstructure is the same as in the single-minimum regimefor E > − ~ Ω. However, the double-minimum scatteringthreshold, E DMt = − ~ λ m − m Ω λ , moves below the sin-gle minimum threshold, E SMt = − ~ Ω. Therefore, when E DMt < E < − ~ Ω, the double-minimum structure in-creases the number of open channels into two. The extraopen channel indeed comes from the nonexistence of solu-tion in the upper band. The combined fourth-order equa-tion of the most upper and lowest band always gives fouralgebraic solutions at any given real energy. The descrip-tion depends on how these four solutions are distributedamong these two bands. In a more general description,when the branch point, E br = − m Ω / (4 λ ), which ap-pears due to the square root in the non-quadratic en-ergy branches, is located at a higher energy than thelowest threshold, all the four solutions are associatedwith the lowest branch. Fig. 1(a) and Fig. 2(a) showall possible states living in different branches in theSM and DM regimes respectively. For a fixed energy,there are six states available, which are labeled by k i for i = 1 , , ...,
6. The colors are carefully drawn to reflectthe origin of branches for each state. For instance, when E DMt < E < E br , the two states in thin blue come fromthe analytical continuation of the thick blue branch inFig. 2(a).In 1D low-energy collisions, the binary interaction can k k k k k k - - (cid:144) Λ - (cid:144) Λ- - E J Ñ Λ Μ N H a L - - k ' Λ or k " Λ- - Μ E (cid:144) Ñ Λ H b L FIG. 1: (a) The total spectrum of states (the relative en-ergy, E , versus the wave vector, k , where k ′ is the real partand k ′′ is the imaginary part of k ) in the single-minimumregime of the center-of-mass frame, i.e. obtained by set-ting K = 0. The parameters used are ~ Ω = 5 ~ λ / (2 µ )and ~ δ = 0 ~ λ / (2 µ ). The thick curves represent the realwave vector solutions in different dispersion branches shownin thick red, thick green and thick blue. When energy goesbelow the scattering thresholds, application of the analyticalcontinuation gives us the thin curves, which stand for eitherthe purely imaginary wave vectors (thin red, thin green, andthin blue) or the complex ones (thin purple). (b) A compan-ion to plot (a). The solid (dashed) lines represent the real(imaginary) part of the wave vector, k . be well approximated by a contact potential with aneffective coupling strength, g . Assuming a 1D s -wave Fermi pseudo-potential, V ( x ) = g δ ( x ) | S ih S | ,the equations that determine the scattering ampli-tudes can be derived analytically by matching the so-lutions to the asymptotic boundary conditions with thegiven channel structures along with the condition ofthe continuity of the wave function and its derivative.This must be modified in the singlet component, ofcourse, because the contact interaction causes a firstderivative discontinuity. Assuming the scattering so-lutions are Ψ R ( x ) = P α =2 , , c α Ψ α ( x ) and Ψ L ( x ) = P α =1 , , c α Ψ α ( x ), where the states Ψ α =even (odd) arechosen to be either right(left)-moving states or decay-ing states at x → + ∞ ( x → −∞ ). The coefficients canthen be found by matching to the following conditions,Ψ ( i ) L ( x = 0) = Ψ ( i ) R ( x = 0) ∀ i = 1 , , d Ψ ( i ) L dx (cid:12)(cid:12)(cid:12)(cid:12) x =0 − = d Ψ ( i ) R dx (cid:12)(cid:12)(cid:12)(cid:12) x =0 + ∀ i = 2 , d Ψ (1) R dx (cid:12)(cid:12)(cid:12)(cid:12) x =0 + − d Ψ (1) L dx (cid:12)(cid:12)(cid:12)(cid:12) x =0 − = mg ~ Ψ R ( x = 0) , (11)where the superscript, i , labels the component in thespinor wave functions. Moreover, due to the existenceof spin-orbit coupling, all of the non-interacting statescarry a singlet component. Although the states in theclosed channel become evanescent (i.e. the wave vectorshave an imaginary part) when the scattering energy goesbelow any channel threshold, the interaction influencesthe scattering of the propagating states through theircoupling to the singlet contact potential. k k k k k k - (cid:144) Λ - (cid:144)
Λ - - E J Ñ Λ Μ NH a L - - k ' Λ or k " Λ- - Μ E (cid:144) Ñ Λ H b L FIG. 2: (a) The total spectrum of states in the double-minimum regime in the center-of-mass frame. Parametersused are ~ Ω = 1 ~ λ / (2 µ ) and ~ δ = 0 ~ λ / (2 µ ). The thickcurves represent real wave vector solutions (thick red, thickgreen and thick blue). Besides the thin green curve, which isidentical to the one in Fig. 1(a), the purely imaginary solu-tions (thin red and thin blue) now live between the real solu-tions. The complex solutions are represented by the thin pur-ple lines. (b) A companion to figure (a). The solid (dashed)lines represent the real (imaginary) part of the wave vector, k . The probability fluxes for the three stationary statecomponents are v u = 2 ~ km + 4 ~ λ k/m √ m Ω + 4 ~ λ k , (12) v m = 2 ~ km , (13) v b = 2 ~ km − ~ λ k/m √ m Ω + 4 ~ λ k . (14)Under the assumption that the initial state is always oneof the non-interacting eigenstates with a positive flux,we can recast the scattering information into reflectionprobabilities and transmission probabilities characteris-tic for 1D scattering. For multi-channel scattering, thereflection probability, R if , is defined as the ratio of thereflected flux in the outgoing channel, f , to the incom-ing flux in channel, i . The transmission probability, T if ,is found by evaluating the transmitted flux in outgoingchannel, f , divided by the incoming flux in channel, i .The flux conservation guarantees P f ( R if + T if ) = 1for a given incoming channel, i . The reflection proba-bilities are plotted versus the relative scattering energyin the single-minimum and double-minimum regimes re-spectively in Fig. 3-5 and Fig. 6.When the incident energy increases all the way upto the highest scattering threshold, E = ~ Ω, the statein the highest branch has a vanishing wave vector (i.e. k = 0) and a zero singlet component. If this state consti-tutes the incoming state, then the system can be viewedas a non-interacting system since the effective couplingsamong all scattering states vanish. Therefore, the incom-ing atoms simply transmit freely through each other andthe transmission is 100% as in the red curve of Fig. 5 at E = 5 ~ λ / µ . In K experiments [5], the energy unit, ~ λ / µ , is about 0 . µ K. If the incoming state is anyother state with a non-zero wave vector, then the prop-erty of the zero probability flux of the state | k i causesthe vanishing reflection probabilities of channel | k i inFig. 3, 4, and 6 at the highest threshold even under thecondition of a non-zero coupling between these scatteringstates. Above the highest threshold energy, the reflectionprobability increases with increasing energy because thesinglet component of the state | k i increases.Moving down to the next scattering region, where 0 41 2 FIG. 7: The reflection probability in the double-minimumregime when the incoming state has a negative wave vectorbut a positive flux current ( i = 4). The parameters used inthis example are ~ Ω = ~ λ / µ , δ = 0, and g = − ~ λ/m .Dashed gray lines are the scattering thresholds. solution (or become transparent ) when the energy goesbelow E = − m Ω / (4 λ ), the thin red curve stops at E . Therefore, the occupation probability is replacedby the blue color in Fig. 6 for E < E . In Fig. 8(b), wesee that there is always only a resonance peak and thepeak position is asymptotically approaching E = − ~ Ω as | g | is increased. Due to the completely different topol-ogy of the dispersion relation of the evanescent mode,no two resonance peaks at the same g could be foundsimultaneously in the DM regime.If one inspects the scattering behaviors when the inci-dent energy is at the lowest scattering energy regime inthe SM and DM cases, a striking difference is found. Forthe SM case, there exists a total transmission at the low-est threshold in Fig. 3; however, for the DM case, thereis a partial reflection at E = E DMt in Fig. 6. The single-minimum regime shares the same threshold behavior as inzero energy due to the decoupling of the incoming chan-nel | k i with the other states. In the double-minimumregime, one extra channel is open as E < − ~ Ω. Theextra state carries a flux current, which is in the oppo-site direction of the wave vector. In Fig. 6, the scatteringbetween | k i and | k i cancels out exactly the first deriva-tive discontinuity of the wave function at the threshold,and thus the reflection probability is robust against anychange in the interaction strength, g . A partial trans-mission at threshold is explained by the existence of abound state near the continuum [11]. The bound statespectrum has been calculated to support this claim, withevidence shown in Fig. 8(c). A bound state is accessibledue to the enhancement of the density of states at thelowest threshold energy in the double-minimum regime,and it thus contributes to the partial reflection, which isabsent in the normal quadratic dispersion case. In Fig. 7,the incoming channel becomes the state | k i with a pos-itive flux current but a negative wave vector, analogousto the behavior of light in a metamaterial [12]. Partialreflection is also observed at the lowest threshold of theenergy range of E DM t < E < − ~ Ω as in Fig. 6. How-ever, at E = ~ Ω, the reflection disappears at Fig. 7 sincethe channel | k i soon becomes closed after crossing thatenergy.A quasi-1D system can be realized when the longitu-dinal kinetic energy is small compared with 2 ~ ω ⊥ , where ω ⊥ is the transverse trapping frequency. In this limit,the low-energy scattering process has only a total reflec-tion [8][13] at the scattering threshold as g → ∞ whichleads to the experimental realization of the theoretical 1DTonks-Girardeau gas [14][15] in a strongly repulsive Bosegas. The confinement-induced resonances in the simul-taneous presence of the Rashba-Dresselhaus spin-orbitcoupling and nonzero Raman coupling are predicted tooccur with a less stringent condition of the trapping fre-quency than the case without RD-SOC upon increasingthe magnitude of the Raman coupling strength [9]. Ournew discoveries of the threshold behaviors in differentregimes of Raman coupling in the spin-orbit coupled sys-tem extends the capability of using the cold atoms toperform quantum simulations, where very different 1Dmodels could possibly be realized.This work was supported in part by the National Sci-ence Foundation PHY-1607180 and in part by fundingfrom the Purdue University EVPRP. Helpful discussionswith Panagiotis Giannakeas and Francis Robicheaux areacknowledged. Note : While preparing the manuscript, the authorsnoticed that Ref. [16] developed a scattering frameworkwhich is applicable all the way to the negative energies foran isotropic spin-orbit coupling in 3D. Their treatmentshould also be applicable to our 1D RD-SOC case and we - - - - - - - - - g D (cid:144)H Ñ Λ (cid:144) Μ L Μ E (cid:144) H Ñ Λ L H a L Ñ W= Ñ W= Ñ W= - - - - - È g D È(cid:144)H Ñ Λ (cid:144) Μ L Μ E (cid:144) H Ñ Λ L H b L Ñ W= Ñ W= - - - - - - - - - - - - È g D È(cid:144)H Ñ Λ (cid:144) Μ L Μ E b (cid:144) H Ñ Λ L H c L Ñ W= Ñ W= FIG. 8: Resonance position as a function of g in the en-ergy range [ − ~ Ω , ~ Ω=2 (orange),4 (green) and 8 (blue) in the unit of ~ λ / µ . (b) Parametersused are ~ Ω = 1 (blue) and 0 . ~ λ / µ .(c) Bound state spectra for ~ Ω = 1 (blue) and 4 (green) inthe unit of ~ λ / µ . expect solutions from these two different methods shouldagree. [1] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev.Mod. Phys. , 1225 (2010).[2] M. Randeria and E. Taylor, Ann. Rev. Condens. MatterPhys. , 209 (2014).[3] R. A. Williams, L. J. LeBlanc, K. Jim´enez-Garc´ıa, M. C.Beeler, A. R. Perry, W. D. Phillips, and I. B. Spielman,Science , 314 (2012).[4] Y.-J. Lin, K. 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