Bound states of spin-orbit coupled cold atoms in a Dirac delta-function potential
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y Bound states of spin-orbit coupled cold atoms in aDirac delta-function potential
Jieli Qin , Renfei Zheng , Lu Zhou , School of Physics and Electronic Engineering, Guangzhou University, 230 Wai HuanXi Road, Guangzhou Higher Education Mega Center, Guangzhou 510006, China Department of Physics, School of Physics and Electronic Science, East ChinaNormal University, Shanghai 200241, China Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan,Shanxi 030006, ChinaE-mail: [email protected]
November 2019
Abstract.
Dirac delta-function potential is widely studied in quantum mechanicsbecause it usually can be exactly solved and at the same time is useful in modelingvarious physical systems. Here we study a system of delta-potential trapped spin-orbit coupled cold atoms. The spin-orbit coupled atomic matter wave has two kindsof evanescent modes, one of which has pure imaginary wavevector and is an ordinaryevanescent wave; while the other with a complex number wave vector is recognized asoscillating evanescent wave. We identified the eigenenergy spectra and the existence ofbound states in this system. The bound states can be constructed analytically usingthe two kinds of evanescent modes and we found that they exhibit typical featuresof stripe phase, separated phase or zero-momentum phase. In addition to that, theproperties of semi-bound states are also discussed, which is a localized wave packet ona plane wave background.
Keywords : Spin-orbit coupling, cold atoms, Delta-function potential
Submitted to:
J. Phys. B: At. Mol. Phys.
1. Introduction
Spin-orbit (SO) coupling has been widely studied in diverse branches of physics includingnanotechnology [1, 2], nuclear physics [3, 4, 5], optics [7, 6, 8], condensed matterphysics [9, 10] and cold atom physics [11, 12, 13]. For a charged particle with non-zero spin, its spin magnetic momentum will interact with the magnetic field induced byits movement, thus generating a coupling between its orbital motion and spin degree offreedom. For neutral cold atom systems, SO-coupling can be artificially generated via ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential
2a Raman coupling scheme [14]. The occurrence of SO-coupling will greatly enrich thephysics of atomic matter wave [15, 16, 12]. Many emergent phenomena such as spinHall effect [17], topological insulator [18], Zitterbewegung [19, 20, 21, 22], supersolid[23, 24, 25, 26], solitons [27, 28, 29, 30], Beliaev damping [31] and spin-dependent atomoptics [32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43] have been reported.It was found that in the presence of SO-coupling the atomic system can displaya rich phase diagram in which the ground state wavefunction can favor stripe, spinseparated or zero momentum phase [14, 44, 45, 46, 47]. Specifically, the stripe phasecan be regarded as a signature of supersolid [23, 24, 25, 26] which receives much recentattention. Physical insight into the properties of eigenfunction of the SO-coupledultracold atoms can be gained via bound state solutions. Previous work has solvedbound state in a one-dimensional short-range potential, and predicts a type of spin-orbitinduced extra states [48]. More interestingly, bound state in the continuum canexist under appropriate trapping potentials [49].In this article we analytically study the bound states of SO-coupled atomicmatterwave in a δ -function potential. Physical models with δ -function potential playsignificant roles in quantum mechanics. Analytical or partially analytical solutions canbe deduced for these models and in the meanwhile they provide physical insight into realsystems. Typical examples include the Kronig-Penney model with δ -function potential[50], the hydrogen-like atoms and diatomic molecules [51], the inter-atom interaction inultra cold atomic cloud [52, 53], very narrow potential [54, 55, 56, 57], obstacle [58, 59]and impurity [60, 61]. It is well known that a δ -function potential well supports only onebound state which is constructed with free particle evanescent waves. For SO-coupledmatter wave, there are three different types of free particle modes: plane wave, ordinaryevanescent wave and oscillating evanescent waves [63, 36]. We found that there existtwo types of bound states which can be constructed using the oscillating evanescent andordinary evanescent waves, respectively. The bound state constructed with oscillatingevanescent waves has stripe structure on its density profile, while the bound state withordinary evanescent waves is a zero momentum wave packet. Due to the spin-1/2 natureof the system, a δ -function potential well can (but not always) support bound statesboth in ground state and excited state. A separated phase bound state can then beconstructed by superposing the ground and excited bound state. Besides these boundstates, we found that there also exists a kind of semi-bound state, which is a localizedstate on a plane wave background. The interference between the localized state and theplane wave background will produce a dip (bump) on the density profile for a δ -potentialwell (barrier).
2. Model and eigensolution of free SO-coupled cold atomic matter wave
As schematically shown in figure 1, we consider a system of quasi-one-dimensionalSO-coupled cold atoms subjected to a spin-independent δ -function potential. TheSO coupling is realized by a typical Raman scattering scheme [14]. And the spin- ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential cold atoms Figure 1.
Schematic of the considered system. A quasi-one-dimensional spinorcold atomic cloud (two Zeeman levels acts as the pseudo-spin states) is shined by two x -direction counterpropagating laser beams ( L and L ) to realize the SO-coupling[14]. A third far off-resonant tightly focused laser beam ( L ) shining from y -directiongenerates a spin-independent delta-function potential [54, 55]. independent δ -potential is generated by a far off-resonant laser beam [62] which is tightlyfocused at the center of the atomic cloud [54, 55]. Such a system can be described bythe Hamiltonian H = H − V δ ( x ) , (1)with V being the depth of δ -function potential, and H being the free particleHamiltonian of SO-coupled matter wave H = ~ m ( k x − k c ) ~ Ω / ~ Ω / ~ m ( k x + k c ) ! , (2)which can be implemented with a Raman coupling scheme in cold atom system [14].Here ~ is the reduced Planck constant, m is the mass of an atom, p x = ~ k x = − i ~ ∂/∂x is the x -direction momentum operator, k c signals SO-coupling strength and Ω is theRabi coupling strength. For simplicity, we have assumed that the interatomic collisioninteraction is eliminated using the technique of Feshbach resonance [64, 65].Since the total Hamiltonian H equals H except at the point x = 0 , we first give abrief discussion on the properties of free particle Hamiltonian H . The eigenenergy E of H is given by the equation (cid:20) E − ~ ( k x + k c )2 m (cid:21) − (cid:18) ~ k c k x m (cid:19) − (cid:18) ~ Ω2 (cid:19) = 0 . (3)For a given energy E , this quartic algebraic equation has four solutions k x = ± s k c + 2 mE ~ ± p mEk c + m Ω ~ . (4)These four solutions can generally be written as k x = β + iα , with α and β being tworeal numbers. In the wavefunction e ik x x = e − αx e iβx , the real part β of wavevector k x contributes a plane wave factor, while the imaginary part α contributes an exponentialdecay factor. Thus, depending on the values of α and β , the corresponding free particle ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential -1.0-0.50.00.51.01.52.0 − − − − E ± f ( k x ) E ± f ( k x ) Figure 2.
Free particle energy spectra of SO-coupled atomic matter waves. Leftpanel: strong SO-coupling case with parameters k c > m Ω / (4 ~ ) ( k c = 1 , Ω = 1 ).Right panel: weak SO-coupling case with parameters k c < m Ω / (4 ~ ) ( k c = 1 , Ω = 3 ).The upper branch spectrum E + is plotted in red color, while the lower branch E − isplotted in blue color. The plane, ordinary evanescent and oscillating evanescent wavemodes are plotted with solid, dash-dot and dash lines respectively. For plane wavemode, k x is a real number, x -axis is simply set to f ( k x ) = k x ; for ordinary evanescentwave mode, k x has no real part, x -axis is set to its imaginary part f ( k x ) = Im ( k x ) ;and for oscillating evanescent wave mode, k x = ± β ± iα is a complex number, x -axisis set to f ( k x ) = sgn [ Im ( k x )] · | k x | (note that in such a way the points for + β and − β will overlap with each other, thus the seemingly two curves in the figure are in factfour curves). Other parameters used are m = ~ = 1 . waves can be divided into three types: (1) Plane wave with α = 0 , β = 0 ; (2) Ordinaryevanescent wave with α = 0 , β = 0 ; and (3) Oscillating evanescent wave when α, β = 0 .From equation (3), eigenenergy can be further split into two spectrum branches(referred as upper and lower branch) E ± ( k x ) = ~ ( k x + k c )2 m ± s(cid:18) ~ k c k x m (cid:19) + (cid:18) ~ Ω2 (cid:19) , (5)with the corresponding eigenstates Ψ ± = χ ± ( k x ) e ik x x = C ± ζ ± ! e ik x x , (6)in which ζ ± = − (cid:16) ~ k c k x ∓ p ~ k c k x + m Ω (cid:17) /m Ω and C ± = 1 / q | ζ ± | is thenormalization constant.Equation (4) indicates that there are four states for any energy E , the eigenenergydispersion display two typically different structures depending on the SO-couplingstrength:(i) The strong SO-coupling case with k c > m Ω / ~ . In this case, when E > E + (0) = ~ k c / m + ~ Ω / , both the upper and lower branches support twoplane wave states with k x real. In the region E ∈ [ E − (0) = ~ k c / m − ~ Ω / , E + (0)] , ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential [ E + (0) , + ∞ ] [ E − ( iα ) , E + (0)] [ E − (0) , E − ( iα )] [ E − ( k ) , E − (0)] [ −∞ , E − ( k )] Table 1.
Plane (P), ordinary evanescent (Ev), and oscillating evanescent (OE) modenumbers of SO-coupled atomic matter wave in different energy ranges. Strong couplingmeans k c > m Ω / (4 ~ ) , while weak coupling means k c < m Ω / (4 ~ ) . Letters “U”and “L” are used to label the upper and lower branch of the spectrum. In the table α and k are α = m Ω / (2 ~ k c ) , k = p k c − m Ω / (4 ~ k c ) . The expression of function E ± is given by formulae (5) in the text. there are two plane wave states in the lower spectrum branch while the other two areordinary evanescent states coming from either the upper branch when E > E − ( iα ) =(4 ~ k c − m Ω ) / mk c ( α = m Ω / ~ k c ) or lower branch when E < E − ( iα ) . For E ∈ [ E − ( k ) = − m Ω / k c , E − (0)] with k = p k c − m Ω / (4 ~ k c ) there are four planewave states in the lower branch. Four oscillating evanescent states possess minimumenergies with E < E − ( k ) and they are linked to the energy minimum of the upperenergy spectra at the points | k x | = k .(ii) The weak SO-coupling case with k c < m Ω / ~ . It differs from the strongcoupling case only in the energy region [ E − ( k ) , E − (0)] , in which all four eigenstatescome from the lower branch are ordinary evanescent waves with k x imaginary.In order to better understand the properties of these eigenstates, we plot in figure2 the energy spectra as a function of k x for k c = 1 and two typical values of Ω , signalingthe strong coupling and weak coupling case respectively. The corresponding numbers ofplane, evanescent and oscillating evanescent modes in different energy ranges are alsosummarized in Table 1.
3. Bound states and semi-bound states with delta-function potential
The bound state of a δ -function potential well V ( x ) = − V δ ( x ) can be constructed usingthe free particle modes by matching the boundary conditions at x = 0 . Because theplane wave mode extends to infinity, it can not be used to construct a bound state. Whileordinary and oscillating evanescent wave modes decay to zero when x approaches + ∞ or −∞ , they are the candidates for bound state constructing. So based on the discussionin section 2, it is concluded that bound states can exist in energy range [ −∞ , E − ( k )] ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential [ E − ( k ) , E − (0)] for onlyweak SO-coupling, since in these energy ranges there exist the candidate modes. And inenergy range [ E − (0) , E + (0)] , the plane wave and ordinary evanescent wave modes existsimultaneously, this gives a chance to construct a kind of semi-bound state, which showsas a localized wave packet on plane wave background. These states will be discussed inthe rest content of this section one by one.In the energy range E < E − ( k ) , as having been discussed above there exist fouroscillating evanescent modes and bound states can be constructed using them. One canfind that the wave vector (4) of these four modes have symmetrical form k x = ± β ± iα and we label them as k = β + iα , k = − β + iα , k = β − iα , k = − β − iα , so thebound state can be written as Ψ b = ((cid:2) A χ − ( k ) e iβx + A χ − ( k ) e − iβx (cid:3) e − αx , x > , (cid:2) A χ − ( k ) e iβx + A χ − ( k ) e − iβx (cid:3) e αx , x < , (7)with A , A , A , A and eigenenergy E (note that k , , , are determined by E accordingto formula (4)) to be determined by normalization constraint Z ∞−∞ | Ψ b ( x ) | dx = 1 , (8)and boundary conditions at x = 0 : continuity of the wave function Ψ b | = Ψ b | − , (9)with Ψ b | = A χ − ( k ) + A χ − ( k ) , (10) Ψ b | − = A χ − ( k ) + A χ − ( k ) , (11)and jump of the first-order derivation caused by divergence of δ -function potential ∂ Ψ b ∂x (cid:12)(cid:12)(cid:12)(cid:12) − ∂ Ψ b ∂x (cid:12)(cid:12)(cid:12)(cid:12) − = − mV ~ Ψ b ( x = 0) , (12)with ∂ Ψ b ∂x (cid:12)(cid:12)(cid:12)(cid:12) = ik A χ − ( k ) + ik A χ − ( k ) , (13) ∂ Ψ b ∂x (cid:12)(cid:12)(cid:12)(cid:12) − = ik A χ − ( k ) + ik A χ − ( k ) . (14)Solving equations (9) and (12), one can have two solutions (the lower energyone is the ground state and the other one is an excited state) fulfill properties | A | = | A | = | A | = | A | , indicating that the bound state is a spin symmetric statewith | Ψ b, ↑ | = | Ψ b, ↓ | . From equation (7) one can understand that the interferencebetween e iβx and e − iβx terms will produce an interference stripe on the density profile ofbound states. This type of bound state has very similar properties as the stripe phasestate in free space or harmonically trapped SO-coupled matter wave [46]. In figure 3,examples of such bound states are plotted for parameters V = 0 . , k c = 1 , Ω = 1 , clearly ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential -0.2-0.10.00.10.20.30.4 − −
10 10 30 -0.4-0.3-0.2-0.10.00.10.2 − −
10 10 30 -0.2-0.10.00.10.2 − −
10 10 30 -0.2-0.10.00.10.2 − −
10 10 300.000.040.080.120.16 − −
10 10 30 0.000.040.080.120.16 − −
10 10 30 0.000.010.020.030.040.05 − −
10 10 30 0.000.010.020.030.040.05 − −
10 10 30 Ψ x ( G, ↑ ) ReIm Ψ x ( G, ↓ ) Ψ x ( E, ↑ ) Ψ x ( E, ↓ ) | Ψ | x ( G, ↑ ) | Ψ | x ( G, ↓ ) | Ψ | x ( E, ↑ ) | Ψ | x ( E, ↓ ) Figure 3.
Oscillating evanescent wave bound states. Ground (“G”) and Excited (“E”)states of SO-coupled matter wave in δ -function potential well are plotted for parameters V = 0 . , k c = 1 , Ω = 1 . Top panels are real (solid) and imaginary (dashed) parts ofthe wave function Ψ with different spins (labeled with “ ↑ ” and “ ↓ ”). Bottom panelsare the corresponding densities | Ψ | . The energies of the ground and excited states are − . and − . respectively. Natural unit m = ~ = 1 is applied. demonstrating spin mixed stripe phase structure. We also note that no-node theoremdoes not hold here because of SO-coupling [66, 67].The bound states can also be constructed via the linear superposition of the groundand excited states. In figure 4, the superposition state Ψ b,G + E = (Ψ b,G + Ψ b,E ) / √ isplotted which is a spin- ↑ component dominated state. Similarly a spin- ↓ componentdominated state can also be constructed by superposing Ψ b,G and Ψ b,E with oppositephase Ψ b,G − E = (Ψ b,G − Ψ b,E ) / √ . This resembles the separated phase discussed in [46]with nonzero spin polarization. But, it should be noted that because the ground andexcited states have different eigenenergies, these superposition states are not stationarystates of the system.The bound states can also exist in the energy region [ E − ( k ) , E − (0)] for a weakSO-coupling ( k c < m Ω / (4 ~ ) ). In such a case, there exist four ordinary evanescentmodes for a given energy E , the corresponding wavevectors (4) are pure imaginary andhave form k , = ± iκ , k , = ± iκ . The bound state can then be written in the followingform Ψ b = ( A χ − ( k ) e − κ x + A χ − ( k ) e − κ x , x > ,A χ − ( k ) e κ x + A χ − ( k ) e κ x , x < , (15)which decays exponentially and is very similar to the bound states in the SO-uncoupledcase. Applying the boundary conditions and normalization constraints (similar toequaitons (8, 9, 12)) one can solve the bound states. An example is given in figure5, the bound state is spin symmetric and can be viewed as the “zero momentum” statesdiscussed in [46].We also examined the spectrum with δ -function potential well. In the top twoand bottom left panels of figure 6, the binding energies of the ground and excited ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential -0.3-0.10.10.3 − −
10 10 30 -0.20-0.15-0.10-0.050.000.050.10 − −
10 10 300.000.030.060.090.120.15 − −
10 10 30 0.000.010.020.03 − −
10 10 30 Ψ x ( ↑ ) ReIm Ψ x ( ↓ ) | Ψ | x ( ↑ ) | Ψ | x ( ↓ ) Figure 4.
Separated phase bound state as a superposition of ground and excitedstates, (Ψ b,G + Ψ b,E ) / √ . Top panels are the real (solid) and imaginary (dashed)parts of the spin- ↑ and spin- ↓ wave functions. Bottom panels are the correspondingdensities | Ψ | . Parameters are the same as in figure 3. -0.10.00.10.20.30.40.5 − −
10 10 30 -0.5-0.4-0.3-0.2-0.10.00.1 − −
10 10 300.000.050.100.150.200.25 − −
10 10 30 0.000.050.100.150.200.25 − −
10 10 30 Ψ x ( ↑ ) ReIm Ψ x ( ↓ ) | Ψ | x ( ↑ ) | Ψ | x ( ↓ ) Figure 5.
Ordinary evanescent wave bound state. Ground state of SO-coupled matterwave in δ -function potential well is plotted for parameters V = 0 . , k c = 1 , Ω = 3 .Top panels are the real (solid) and imaginary (dashed) parts of the spin- ↑ and spin- ↓ wave functions. Bottom panels are the corresponding densities | Ψ | . The energy ofthis state is − . . Natural unit m = ~ = 1 is applied. bound states are plotted as a function of δ -function potential well depth V for Rabicoupling strength Ω = 0 , and . When Ω = 0 , the two spin components are notcoupled with each other, each component can be separately treated as a usual δ -functionpotential problem (except that the momentum is shifted by ± ~ k c ), thus there exist twobound states with degenerate energy E b ; G = E b ; E = − mV / (2 ~ ) . When Ω = 0 , thisdegeneracy is eliminated. For a small value of Ω = 1 (or in other words, a strong SO-coupling since k c > m Ω / ~ is fulfilled), both the ground and excited states can existregardless of the value of V . Even when V → (approaching the free particle limit),the system has two solutions corresponding to the two minimums of the lower dispersionbranch. However, for a large value of Ω = 3 (weak SO-coupling since k c < m Ω / ~ ), ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential -2.0-1.5-1.0-0.50.00.0 0.5 1.0 1.5 2.0 -2.0-1.5-1.0-0.50.00.0 0.5 1.0 1.5 2.0-4.0-3.0-2.0-1.00.00.0 0.8 1.6 2.4 3.2 0.01.02.03.04.00.0 0.5 1.0 1.5 2.0 E b V Ω = 0GE E b V Ω = 1 E b V Ω = 3 Ω c V G and EOnly G
Figure 6. δ -function potential well bounded spectrum. The binding energies of theground(“G”) and excited(“E”) states are plotted as a function of potential well depth V for parameters Ω = 0 , , in the top two and bottom left panels respectively. Inthe bottom right panel, the critical value of Rabi coupling Ω c for the disappearing ofthe excited state is plotted as a function of potential well depth V , the solid blackline. Below this line, in the light blue color filled area, there exist both the ground andexcited states. While, above this line, in the light red color filled area, only the groundstate can exist. In all the panels, SO-coupling strength is set to k c = 1 and naturalunit m = ~ = 1 is applied. the excited state can be lifted so high that a shallow potential well can no longer trapit. Thus, we see that for V smaller than a critical value, the excited state disappears.This also coincides with the fact that for weak SO-coupling the lower dispersion branchof the free particle spectrum only has one minimum. And in the bottom right panel, weshow the critical value of Rabi coupling strength Ω c for the disappearing of the excitedstate as a function of V , see the solid black line. Below this critical line, both theground and excited states can exist. While above this line, the potential well can nolonger support an excited bound state, only the ground state can exist. In this panel,we also noticed that when V → , the critical Rabi coupling Ω c → which is just thedemarcation point between strong and weak SO-coupling strength ( p ~ k c /m = 2 ).This also agrees with the above free particle limit discussion.In the energy region [ E − (0) , E + (0)] , for a given energy E there are two ordinaryevanescent modes and two plane wave modes, the corresponding wavevectors (4) haveform k , = ± iκ and k , = ± k . One can then construct a semi-bound state as follows Ψ sb = Ψ P + Ψ E , (16)where Ψ P is a plane wave background consisting of incident, transmission and reflectionwaves (their amplitudes are , t, r respectively) Ψ P = ( tχ − ( k ) e ikx , x > ,χ − ( k ) e ikx + rχ − ( k ) e − ikx , x < , (17) ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential − − − − − − − − | Ψ | x ( W, ↑ ) | Ψ | x ( W, ↓ ) | Ψ | x ( B, ↑ ) | Ψ | x ( B, ↓ ) Figure 7.
Semi-bound states of SO-coupled matter wave in δ -function potential well(“W”, top panels) and barrier (“B”, bottom panels). Density profiles of spin- ↑ andspin- ↓ components are plotted. Parameters used are E = − . , V = ± . ( +0 . for a potential well, while − . for a potential barrier), k c = 1 , Ω = 3 and m = ~ = 1 . and Ψ E is a localized wave packet constructed using the two ordinary evanescent modes Ψ E = ( A χ ± ( k ) e − κx , x > ,A χ ± ( k ) e κx , x < . (18)Here t, r and A , A are parameters to be determined by boundary conditions. We notethat such a semi-bound state is actually a scattering state, thus it can not only exist for a δ -potential well but also a δ -potential barrier. In figure 7, examples of such semi-boundstates are shown for parameters k c = 1 , Ω = 3 and V = ± . ( +0 . for a potentialwell, while − . for a potential barrier). The density profiles of both spin componentsare plotted in this figure. In the left-half space ( x < ), the interference between incidentand reflected waves produces an interference fringe. In the right-half space ( x > ), thetransmission wave produces a flat density background. And around the location of δ -potential ( x = 0 ), a dip of density on plane wave ground can be observed for a potentialwell, while a bump is observed for a potential barrier. This semi-bound state representsthe coupling between the bound state and the plane wave propagating state, thus boundstates in the continuum don’t exist in the present single-particle system [68].
4. Summary
In summary, we studied the bound and semi-bound states of SO-coupled matter wave ina δ -function potential. We found that there are two kinds of bound state in the system,one of which is a stripe one constructed using oscillating evanescent wave, while the otherone constructed using ordinary evanescent wave is an ordinary one having the similarfeature as the SO-uncoupled case. For SO-coupled matter wave, a δ -potential well can(but not always) support both a ground and an excited bound state. By superposingthese two states, a separated phase state can also be constructed. Besides the bound ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential δ -function potential well, a dip emerges on the plane wavebackground. While for a δ -function potential barrier, a bump is formed on the planewave background. Acknowledgments
This work is supported by the National Natural Science Foundation of China (GrantNos. 11904063, 11847059, 11374003, and 11574086).
References [1] Majumdar S, Majumdar H S and Österbacka R 2011 Organic spintronics
Comprehensivenanoscience and technology ed D Andrews, G Scholes and G Wiederrecht (Academic Press)pp 109-142[2] Tan S G and Jalil M B A 2012 Spintronics and spin Hall effects in nanoelectronics
Introduction tophysics of nanoelectronics ed S G Tan and M Jalil (Woodhead) pp 141-197[3] Hope J 1957 Nuclear spin-orbit energy for oscillator wave functions
Phys. Rev. , 771[4] Bell J S and Skyrme T H R 1994 CVIII: The nuclear spin-orbit coupling
Selected papers withcommentary of Tony Hilton Royle Skyrme ed G E Brown (World Scientific) pp 71-84[5] Kaiser N 2007 Σ -nuclear spin-orbit coupling from two-pion exchange Phys. Rev. C , 068201[6] Cardano F and Marrucci L 2015 Spin-orbit photonics Nat. Photonics , 776[7] Bliokh K Y, Rodríguez-Fortuño F J, Nori F and Zayats A V 2015 Spin-orbit interactions of light Nat. Photonics , 796[8] Sala V G, Solnyshkov D D, Carusotto I, Jacqmin T, Lemaître A, Terças H, Nalitov A, AbbarchiM, Galopin E, Sagnes I,Bloch J, Malpuech G and Amo A 2015 Spin-orbit coupling for photonsand polaritons in microstructures Phys. Rev. X , 011034[9] Bihlmayer G, Rader O and Winkler R 2015 Focus on the Rashba effect New J. Phys. J. Phys. Condens. Matter Nature
Reports Prog. Phys. Front. Phys. Nature
Rev. Mod. Phys. Reports Prog. Phys. Rev.Mod. Phys. Rev. Mod. Phys. Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. , 418[20] Hestenes D 2010 Zitterbewegung in quantum mechanics Found. Phys. New J. Phys. ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential [22] Qu C, Hamner C, Gong M, Zhang C and Engels P 2013 Observation of Zitterbewegung in aspin-orbit-coupled Bose-Einstein condensate Phys. Rev. A Nature
J. Phys. B At. Mol. Opt. Phys. Phys. Rev. Lett.
Phys. Rev. A
Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. Lett.
J. Phys. B At. Mol. Opt. Phys. Phys.Rev. Lett.
Appl. Phys. B Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys.Rev. A Phys. Rev. Lett.
J. Phys. B At. Mol. Opt. Phys. Phys. Rev. A J. Phys.B At. Mol. Opt. Phys. Phys. Rev. A Nat. Commun. Phys. Rev. Lett.
Phys. Rev. Lett. ound states of spin-orbit coupled cold atoms in a Dirac delta-function potential coupled Bose-Einstein condensates Phys. Rev. Lett.
J. Phys. B At. Mol. Opt. Phys. J. Math. Phys. Phys. Rev. A Introduction to solid state physics (Wiley).[51] Frost A A 1956 Delta-function model I: Electronic energies of Hydrogen-like atoms and diatomicmolecules
J. Chem. Phys. Phys. Rev.
Phys. Rev. A δ functions Phys. Rev. A , 013618[55] Garrett M C, Ratnapala A, van Ooijen E D, Vale C J, Weegink K, Schnelle S K, Vainio O,Heckenberg N R, Rubinsztein-Dunlop H and Davis M J 2011 Growth dynamics of a Bose-Einstein condensate in a dimple trap without cooling Phys. Rev. A Phys. Rev. Lett.
Phys.Rev. Lett.
Phys. Rev. E Phys. Rev. A Phys. Rev. A Phys. Rev. A Adv. At. Mol. Opt. Phys. Phys. Rev. B Rev.Mod. Phys. Phys. Rep.
Mod. Phys.Lett. B J. Phys. B At. Mol. Opt. Phys. Nat. Rev. Mater.1