Spin Orbit Coupling in Periodically Driven Optical Lattices
SSpin-Orbit Coupling in Periodically Driven Optical Lattices
J. Struck, ∗ J. Simonet,
1, 2 and K. Sengstock
1, 2 Institut f¨ur Laserphysik, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany ZOQ, Universit¨at Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany
We propose a method for the emulation of artificial spin-orbit coupling in a system of ultracold,neutral atoms trapped in a tight-binding lattice. This scheme does not involve near-resonant laserfields, avoiding the heating processes connected to the spontaneous emission of photons. In our case,the necessary spin-dependent tunnel matrix elements are generated by a rapid, spin-dependent,periodic force, which can be described in the framework of an effective, time-averaged Hamiltonian.An additional radio-frequency coupling between the spin states leads to a mixing of the spin bands.
PACS numbers: 67.85.-d, 03.65.Vf, 71.70.Ej, 37.10.Jk
The phenomenon of spin-orbit coupling (SOC) gener-ally describes an interplay between the spin state of aparticle and its motional degrees of freedom. This effectnaturally arises in the framework of relativistic quantummechanics described by the Dirac equation. A spinfulparticle moving through an electric field experiences amagnetic field in the co-moving reference frame. The re-sulting interaction between the spin and the magneticfield depends on the amplitude of the field and thus onthe velocity of the particle, leading to the coupling ofmotion and spin. In solid state materials, SOC can re-sult in exotic phases and phenomena, such as topologicalinsulators [1, 2] or the spin Hall effect [3–7].The experimental realization of synthetic spin-orbit in-teractions with equal Rashba [8] and Dresselhaus [9] con-tributions for bosonic [10, 11] and fermionic quantumgases [12, 13] has raised considerable interest over the lastyears ([14] and references therein). For ultracold atoms –interacting via s-wave scattering – SOC can lead to effec-tive interactions with higher partial wave contributions[15–17]. In a single-component Fermi gas this could leadto stable p-wave interactions and topological superfluids[18–20]. All of the experimentally implemented schemesrely on near-resonant Raman-laser coupling schemes andthus suffer from spontaneous emission, leading to excita-tions and particle loss. Currently a lot of effort is directedtowards novel methods for the creation of artificial SOCavoiding this issue, e.g., by using magnetic field pulses[21, 22] or far-off-resonant light [23].As thoroughly investigated over the last years a rapid,periodic lattice drive coherently manipulates the tunnel-ing processes in an optical lattice. This allows for, e.g.,the coherent destruction of tunneling and sign inversionof the tunnel elements [24–27], photon assisted tunnel-ing [28, 29], and the creation of artificial gauge fields[30–38]. In this manuscript, we propose to engineer a ∗ E-mail address: [email protected];Present address: MIT-Harvard Center for Ultracold Atoms,Research Laboratory of Electronics, and Department ofPhysics, Massachusetts Institute of Technology, Cambridge, Mas-sachusetts 02139, USA B s (t)∂ B s (t)-∂ B s (t)Ω∆JH Kin H RF H Force
FIG. 1. Schematic illustration of the relevant processes, whichare captured in the two-component single-particle Hamilto-nian. H Kin represents the spin-independent nearest-neighbourtunneling processes, H RF describes the radio frequency cou-pling of the two spin states and H Force corresponds to thespin-dependent driving of the atoms in the lattice. The driveoriginates from an oscillating magnetic field gradient. time-periodic and spin-dependent drive in order to real-ize spin-dependent tunnel matrix elements and SOC. Weconsider ultracold atoms with (at least) two spin statesconfined in a 1D optical lattice. A time-dependent mag-netic field ( B s ( t ) = B s ( t + T )) periodically drives theatoms at the frequency ω = 2 π/T . For simplicity we as-sume that the two relevant spin states are characterizedby magnetic moments with same magnitude but oppo-site signs so that they are submitted to opposite forces.However all results presented here can be generalized tospin combinations experiencing a different drive.A time-periodic driving can lead to complex valuedtunnel matrix elements if the driving breaks specific sym-metries, resulting in a gauge dependent shift of the dis-persion relation for a 1D lattice [31]. In our case, the dis-persion relations of the two spin states are shifted in op-posite direction due to the inverted drive for both states.An additional radio-frequency coupling between the spinstates leads to a mixing of the spin dispersion relationsand a spin-orbit gap in the band structure.In this scheme, the strength of the SOC can be con-tinuously tuned, simply by adjusting the driving ampli-tude (see Ref. [39] for a Raman coupling scheme withtunable SOC strength). Importantly as no near-resonantlaser fields are involved, the quantum gas is not prone to a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug heating processes induced by the spontaneous emissionof photons.The Hamiltonian can be separated into three terms H = H Kin + H RF + H Force (Fig. 1). Here we neglectedinteractions between the particles. However, we wouldlike to emphasize that spin preserving, on-site Hubbardtype interactions are not renormalized by the drive in theeffective Hamiltonian.First, H Kin describes the next-neighbor tunneling pro-cesses, which are spin-independent and do not couple thespin states: H Kin = − J (cid:88) s (cid:16) ˆ A + s ˆ A s − + ˆ A + s − ˆ A s (cid:17) , (1)where J is the tunnel matrix element between neighbour-ing lattice sites and ˆ A s = (ˆ a s , ˆ b s ) T , ˆ A + s = (ˆ a + s , ˆ b + s ) arevectors with the annihilation operators ˆ a s and ˆ b s on lat-tice site s for the two spin components. For the rest ofthis manuscript we assume that these operators obey thebosonic commutation relations. Nevertheless, the single-particle results derived here are valid for fermionic par-ticles as well.Second, the Rabi coupling of both spin states in therotating-wave approximation is described by H RF = (cid:126) ∆2 (cid:88) s ˆ A + s ˆ σ z ˆ A s − (cid:126) Ω2 (cid:88) s ˆ A + s ˆ σ x ˆ A s , (2)where ∆ = ω A − ω RF is the detuning of the radio-wave( ω RF ) with respect to the atomic resonance ( ω A ), Ω isthe Rabi frequency and ˆ σ x , y , z denote the Pauli matrices.The third term describes the spin-dependent drivingof the atoms by an oscillating magnetic field gradient: H Force = (cid:88) s v s ( t ) ˆ A + s ˆ σ z ˆ A s . (3)Here v s ( t ) = µB s ( t ) is the energy shift due to the pres-ence of the oscillating magnetic field. The resulting forceis proportional to the gradient of the field and acts in op-posite directions for the two different spin components.The time-periodic Hamiltonian H can be treated inthe framework of the Floquet theory, where the energystructure is described by a time-independent, periodicquasi-energy spectrum [40, 41]. In the high driving fre-quency limit – where the coupling elements between dif-ferent Floquet bands can be neglected – the quasi-energyspectrum can be approximated by an effective Hamil-tonian [24, 42]. The effective Hamiltonian H eff can beobtained from the Floquet Hamiltonian H F = H − i (cid:126) ∂∂t by the relation H eff = (cid:68) ˆ U + Q ( t ) H F ˆ U Q ( t ) (cid:69) T , (4)with the unitary operator ˆ U Q ( t ) = exp( − i ˆ Q ( t )) and thenotation (cid:104)· · ·(cid:105) T = T (cid:82) T · · · d t for the time-average. Thehermitian, time-periodic operator ˆ Q ( t ) is chosen such that the driving term (Eq. (3)) is absent in the effectiveHamiltonian [24]:ˆ Q = 1 (cid:126) (cid:88) s W s ( t ) ˆ A + s ˆ σ z ˆ A s . (5)Here, W s ( t ) is defined by, W s ( t ) = (cid:90) tt d t (cid:48) v s ( t (cid:48) ) − (cid:28)(cid:90) tt d t (cid:48) v s ( t (cid:48) ) (cid:29) T , (6)for times t ≥ t , where t denotes the time when theamplitude of the time-dependent magnetic field has beenfully ramped up. Eq. (6) is independent of t for t ≥ t .The effective Hamiltonian (Eq. (4)) then reads H eff = − J (cid:88) s | f s | (cid:16) ˆ A + s e i θ s ˆ σ z ˆ A s − + ˆ A + s − e − i θ s ˆ σ z ˆ A s (cid:17) − (cid:126) Ω2 (cid:88) s | g s | ˆ A + s (cos( χ s )ˆ σ x − sin( χ s )ˆ σ y ) ˆ A s + (cid:126) ∆2 (cid:88) s ˆ A + s ˆ σ z ˆ A s , (7)where the complex variables f s and g s have been de-composed into magnitude and phase ( f s = | f s | exp(i θ s ), g s = | g s | exp(i χ s )). Here, the function f s ≡ (cid:104) exp (i [ W s ( t ) − W s − ( t )] / (cid:126) ) (cid:105) T , (8)describes the spin-dependent renormalization of the tun-neling matrix elements, whereas g s ≡ (cid:104) exp (i 2 W s ( t ) / (cid:126) ) (cid:105) T , (9)describes the renormalization of the Rabi frequency. Ingeneral, both functions depend on the lattice site index.The spatial rotation of the Pauli matrices in the Rabicoupling term of Eq. (7) can be canceled by the trans-formation into dressed spin states, which are defined bythe operators ˆ C s = ˆ T + s ˆ A s and ˆ C + s = ˆ A + s ˆ T s . The unitaryoperator ˆ T s = exp(i χ s ˆ σ z /
2) represents a local rotation inspin space around the z-axis. In the new basis the effec-tive Hamiltonian is given by H eff = − J (cid:88) s | f s | (cid:16) ˆ C + s e i α s ˆ σ z ˆ C s − + ˆ C + s − e − i α s ˆ σ z ˆ C s (cid:17) − (cid:126) Ω2 (cid:88) s | g s | ˆ C + s ˆ σ x ˆ C s + (cid:126) ∆2 (cid:88) s ˆ C + s ˆ σ z ˆ C s , (10)where we have introduced the SOC parameter α s = θ s + ( χ s − − χ s ) /
2, which can be understood as a spin-dependent Peierls phase. The Hamiltonian (10) describesa tight-binding lattice with one-dimensional SOC.Generally, the site dependence of the parameters | f s | , α s and | g s | (see Eq. (10)) breaks the translational sym-metry of the lattice potential and prevents a meaningfuldescription in terms of Bloch states. However, for cer-tain driving functions v s ( t ) these parameters are in goodapproximation site-independent. B s ( t ) ∂ B • s • s T T t (a)(c) | f | T / T=0.68T / T=0.90.5 1.5 2.5 (b) α [ π ] K0 1 2 31.50.5 2.5-0.10.1-0.3-0.5 T / T=0.9 T / T=0.68 FIG. 2. Spin-dependent driving of the atoms with sinusoidalpulses. (a) The amplitude of the magnetic field is modulatedaround zero with trains of sinusoidal pulses. (b) The resultingSOC strength and (c) the renormalization of the tunnelingrate as functions of the forcing parameter K . In (b) and (c)the colored lines correspond to different ratios T /T of thepulse to hold time. This is for instance the case for a sinusoidally pulsedmagnetic field given by (see Fig. 2(a)): B s ( t ) = ∂B · s (cid:40) sin ( ω t ) for 0 < t mod T < T ( (cid:63) )0 for T < t mod T < T ( (cid:63)(cid:63) ) , (11)where ∂B is the magnetic field difference between neigh-bouring lattice sites and ω = 2 π/T . This functionhas two important features: First, it breaks time-reversalsymmetry, leading to a continuously tunable SOC pa-rameter. Second, it vanishes during a finite time interval T ( T = T + T ), which is essential for a non-vanishingtime-average of the Rabi coupling in the effective Hamil-tonian. Insertion of Eq. (11) into Eq. (6) results in W s ( t ) = (cid:126) K s (cid:40) cos ( ω t ) − T /T ( (cid:63) ) T /T ( (cid:63)(cid:63) ) , (12)with K = µ ∂B/ (cid:126) ω as the dimensionless forcing param-eter. The renormalization function of the tunnel matrixelements (Eq. (8)) f s = e − i KT /T J B0 ( K ) T /T +e i KT /T T /T ≡ f ( K ) , (13)is completely site-independent, with J B0 ( K ) as the ze-roth order Bessel function of the first kind. In con-trast the renormalization function of the Rabi frequency g s = f (2 Ks ) (see Eq. (9)) shows an explicit site depen-dence. However, in the limit 2 Ks → ∞ this site depen-dence only affects the complex phase and we obtain: g s = e i2 KsT /T T /T. (14)If the forcing parameter K is on the order of one and thezero-crossing of the magnetic field is far away (typicallya few thousands of lattice sites) from the center of theatomic cloud, then Eq. (14) is a good approximation forthe exact result of Eq. (9)In this limit the magnitude | g s | ≈ T /T ≡ | g | is site-independent. Note that for T = 0 the effective Rabicoupling between the spin states vanishes. The complexphase of g s is given by χ s ≈ KsT /T and thus we ob-tain a site-independent SOC parameter α = θ − KT /T .This parameter can be continuously tuned via the forcingparameter K and the ratio T /T as shown in Fig. 2(b).For larger T /T , the maximum value of α increases incontrast to the effective Rabi frequency Ω | g | . The effec-tive tunneling strength J | f | , reduced due to the periodicmodulation, presents a minimum value which decreasesto zero as the ratio T /T is increased (Fig. 2(c)). Pleasenote, the forcing parameter cannot become arbitrarilysmall as the condition 2 Ks → ∞ needs to be fulfilled.To estimate the strength of the required magneticfield gradient we assume to work with a bosonic alkali-metal with a nuclear spin of I = 3 / Li, Na or Rb) in the low field Zeeman regime. As the two spinstates we use the hyperfine states | F = 1 , m F = − (cid:105) and | F = 2 , m F = − (cid:105) of the n S / ground state. The re-quired magnetic field gradient for a forcing parameter K would be ∂B/d = K (cid:126) ω / ( dµ B m F g F ) ≈ K ·
31 G / cm , where the Land´e factor is given by g F =1 , = ∓ / ω = 2 π · . d has been set to 0 . µ m.Although this scheme is free of heating arising fromspontaneous emission of photons, the periodic drivingcan create excitations in the system. The heating ratestrongly depends on the specific driving frequency, lat-tice depth and strength of possible interactions betweenthe particles [24, 25, 27, 43, 44]. Therefore the drivingfrequency has to be chosen out of resonance with thedirect and multi-photon transitions to higher bands ofthe lattice. In addition the effective Hamiltonian derivedhere is only a valid approximation if the energy cor-rections due to the coupling between different Floquetbands are negligible. From perturbation theory followsthat the Floquet band coupling elements | (cid:104) ˆ V p,m (cid:105) | andthe eigenvalues of H eff have to be small compared tothe energetic separation (cid:126) ω between the Floquet bands.The coupling between two Floquet bands with indices m and p arises from the perturbation operator ˆ V p,m = (cid:104) e i( p − m ) ωt ˆ U + Q ( t ) H F ˆ U Q ( t ) (cid:105) T [42]. Evaluating the pertur-bation operator for the sinusoidal pulse drive, we arriveat the following constraints: | ξ p,m (1) | · J, | ξ p,m (2 s ) | · (cid:126) Ω (cid:28) (cid:126) ω, (15) (a) (b) (c) (d) FIG. 3. Dispersion relations of the spin-orbit coupled systemfor sinusoidal pulses with T /T = 0 .
9. The colorcode indicatesthe respective admixture of the bare spin states (Eq. (20)).The forcing parameter, Rabi frequency and detuning are (a) K = 1 . (cid:126) Ω = 3 J and (cid:126) ∆ = 0, (b) K = 1 . (cid:126) Ω = 9 J and (cid:126) ∆ = 0, (c) K = 0 . (cid:126) Ω = 9 J and (cid:126) ∆ = 0 and (d) K = 1 . (cid:126) Ω = 9 J and (cid:126) ∆ = 0 . J . with ξ p,m ( x ) = (cid:104) e i[( p − m ) ωt + W x ( t ) / (cid:126) ] (cid:105) T and W x ( t ) givenby Eq. (12). The function | ξ p,m ( x ) | is always smaller thanone and converges for large x towards a value which neverexceeds | g | . Therefore, a reduced effective Rabi frequency | g | Ω can always be compensated by increasing the bareRabi frequency without violating the constraints given in(15). From the condition for the eigenvalues of the effec-tive Hamiltonian we obtain the additional constraints: | g | · J, | f | · (cid:126) Ω , (cid:126) ∆ (cid:28) (cid:126) ω. (16)Note that, for an interacting system the driving frequencyhas to be also large compared to the energy scale of theinteractions. For the driving frequency, these constraintsresult in a lower bound given by the energy scale of thetunneling strength and an upper bound defined by theband gap. Furthermore, the driving frequency hast bechosen out resonance with multi-photon excitations lyingin between the lower and upper bound.The dispersion relation of the spin-orbit coupled latticecan be obtained by the transformation of the annihilation operator for a particle on site s into the reciprocal space:ˆ C s = 1 √ M (cid:88) q ˆ C q e i q d s , (17)where M is the number of lattice sites and ˆ C q = ( ˆ C q , ˆ C q ) T is the two-component annihilation operator for a particlein the Bloch state with quasimomentum q . The trans-formation of the creation and annihilation operators inthe Hamiltonian (10) into the reciprocal space (Eq. (17))leads to H eff = (cid:88) q ˆ C + q H ( q ) ˆ C q , (18)with the matrix H ( q ) = − J | f | (cos( α ) cos( qd ) + sin( α ) sin( qd ) ˆ σ z )+ (cid:126) ∆ˆ σ z / − (cid:126) Ω | g | ˆ σ x / . (19)The eigenvalues of the matrix H ( q ) represent the two low-est spinful bands of the spin-orbit coupled lattice (Fig. 3).An increase of the Rabi frequency Ω leads to a largersplitting of these bands (Figs. 3(a) and (b)). The forcingparameter K directly influences the SOC parameter α and the bandwidth via the renormalization | f | of the tun-neling rate (Figs. 3(b) and (c)). A finite detuning ∆ liftsthe reflection symmetry of the dispersion relation aroundquasimomentum zero (Fig. 3(d)). Experimentally it istherefore necessary to reduce short-term magnetic fieldfluctuations, directly translating into a time-dependentdetuning and heating, to an energy scale which is smallcompared to the effective tunneling strength. This is-sue can be partially resolved by using a light atomicspecies, where the energy scale connected to the tun-neling is larger than for heavier elements. However, wewould like to point out that this is a generic problemof all SOC schemes using Zeeman states with differentmagnetic moments.The combination of time-of-flight absorption imagingand a Stern-Gerlach spin separation allows to probe theadmixture of the bare spin components in the dispersionrelation. The quasimomentum dependent admixture ofthe bare spin states is related to the transformed spinbasis of Eq. (19) by (cid:104) ˆ A + q ˆ σ z ˆ A q (cid:105) = (cid:104) ˆ C + q − KT /T ˆ C q − KT /T (cid:105)− (cid:104) ˆ C + q + KT /T ˆ C q + KT /T (cid:105) , (20)where ˆ A q is the two-component annihilation operator forthe bare spin states in the Bloch function with quasi-momentum q . It can be obtained from ˆ A s by a Fourierexpansion in analogy to Eq. (17).In conclusion, we have proposed to use a period-ically driven magnetic field gradient to create spin-dependent tunneling matrix elements for bosonic orfermionic species in an optical lattice. The Rabi couplingbetween different spin states leads to a gap in the dis-persion relation and a mixing of the bare spin compo-nents. The proposed scheme avoids the problems associ-ated with the spontaneous emission of photons. An ad-ditional advantage of our driving technique is the simplecontrol over the SOC parameter α , which can be easilytuned in-situ . 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