Charge Pumping of Interacting Fermion Atoms in the Synthetic Dimension
CCharge Pumping of Interacting Fermion Atoms in the Synthetic Dimension
Tian-Sheng Zeng, Ce Wang, and Hui Zhai School of Physics, Peking University, Beijing 100871, China Institute for Advanced Study, Tsinghua University, Beijing, 100084, China (Dated: October 10, 2018)Recently it has been proposed and experimentally demonstrated that a spin-orbit coupled multi-component gas in 1d lattice can be viewed as spinless gas in a synthetic 2d lattice with a magneticflux. In this letter we consider interaction effect of such a Fermi gas, and propose signatures incharge pumping experiment, which can be easily realized in this setting. Using 1 / High spin quantum gas is a unique system of cold atomphysics. The spin of atoms ranges from hyperfine F = 1of alkali atoms like Rb and Na, F = 9 / SU ( W ) symmetry of alkali-earth-(like) atoms such as Yb and Sr, where W can be aslarge as ∼
10. Recently it emerges an interesting ideato use the internal spin degrees of freedom as anotherdimension, named as “synthetic dimension”, which nat-urally extends a D -dimensional system into a ( D + 1)-dimension one [1, 2]. In a 1d lattice system, by applyingtwo counter propagating Raman beams to couple differ-ent spin states, one can create a magnetic flux latticein synthetic 2d [2]. This proposal requires a minimumamount of laser light and therefore minimizes heatingfrom spontaneous emission. It also gives rise to a sharpedge in the synthetic dimension, which can help to visu-alize edge states. Very recently, two experimental groupshave implemented this scheme, in Rb atom [3] and inYb atom [4], respectively, and chiral edge states havebeen observed, for non-interacting (or weakly interact-ing) bosons [3] and fermions [4], respectively. Moreover,it is possible to create more exotic nontrivial geometry[5].The experimental setup and basic idea of synthetic di-mension are briefly illustrated in Fig. 1. For instance,two Raman beams with π and σ polarization can couplespin state | m (cid:105) to | m ± (cid:105) , where m can run from − F to F with totally W = 2 F + 1 components. The Raman cou-pling has a spatial dependent phase factor e i k R x , where k R is the recoil momentum of Raman laser. We considerthe situation γ = 2 k R a = 2 π/q ( a is lattice spacing). Thesingle particle Hamiltonian is therefore written asˆ H = (cid:88) j,m ( − t ˆ c † j +1 ,m ˆ c j,m + Ω e − iγj ˆ c † j,m − ˆ c j,m + h.c.) (1)where j labels the site along the physical dimension ˆ x and m labels internal spin components. t is hopping along ˆ x and Ω is Raman coupling strength. There are two differ-ent but equivalent pictures of this single-particle Hamil-tonian (a) (b) ns S = 1 / L = 0, S / I F = S + Ins S = 0, L = 0, S IN = 2 I + 1 SU ( N ) SU (2) S = ( N B N A ) s s S = 0 S = 1 L = 1 J = 1 J = 0 J = 1 J = 2 P P P P J = 0 J = 2 s · l + s · l SL jj | e i | g i d · E ; d = e r k = A B (1) e i e i ˆ H eff = ˆ H + 1 ! [ ˆ H , ˆ H ] + . . . (2) n = 1 / / C = 1 + 1 = 2 C = 0 Ut = 0 h n i = 1ˆ z † z s z ⌦ e ik x N N W = 2 N + 1 ⌫ = NN flux = NW L/q = N qLW (3) ⌫ = 1 / /q ⌫ d = NL (4) ⌫ d = 1 e i✓ E P = 1 W [ h M z i ( ✓ = 2 ⇡ ) h M z i ( ✓ = 0)] (5) E n ns S = 1 / L = 0, S / I F = S + Ins S = 0, L = 0, S IN = 2 I + 1 SU ( N ) SU (2) S = ( N B N A ) s s S = 0 S = 1 L = 1 J = 1 J = 0 J = 1 J = 2 P P P P J = 0 J = 2 s · l + s · l SL jj | e i | g i d · E ; d = e r k = A B (1) e i e i ˆ H eff = ˆ H + 1 ! [ ˆ H , ˆ H ] + . . . (2) n = 1 / / C = 1 + 1 = 2 C = 0 Ut = 0 h n i = 1ˆ z † z s z ⌦ e ik x N N W = 2 N + 1 ⌫ = NN flux = NW L/q = N qLW (3) ⌫ = 1 / /q ⌫ d = NL (4) ⌫ d = 1 e i✓ E P = 1 W [ h M z i ( ✓ = 2 ⇡ ) h M z i ( ✓ = 0)] (5) E n ChargePumping ns S = 1 / L = 0, S / I F = S + Ins S = 0, L = 0, S IN = 2 I + 1 SU ( N ) SU (2) S = ( N B N A ) s s S = 0 S = 1 L = 1 J = 1 J = 0 J = 1 J = 2 P P P P J = 0 J = 2 s · l + s · l SL jj | e i | g i d · E ; d = e r k = A B (1) e i e i ˆ H ef f = ˆ H + 1 ! [ ˆ H , ˆ H ] + . . . (2) n = 1 / / C = 1 + 1 = 2 C = 0 Ut = 0 h n i = 1ˆ z † z s z ⌦ e ik x N N W = 2 N + 1 ⌫ = NN f lux = NW L/q = N qLW (3) ⌫ = 1 / /q ⌫ d = NL (4) ⌫ d = 1 e i✓ E P = 1 W [ h M z i ( ✓ = 2 ⇡ ) h M z i ( ✓ = 0)] (5) E n ˆ x ns S = 1 / L = 0, S / I F = S + Ins S = 0, L = 0, S IN = 2 I + 1 SU ( N ) SU (2) S = ( N B N A ) s s S = 0 S = 1 L = 1 J = 1 J = 0 J = 1 J = 2 P P P P J = 0 J = 2 s · l + s · l SL jj | e i | g i d · E ; d = e r k = A B (1) e i e i ˆ H eff = ˆ H + 1 ! [ ˆ H , ˆ H ] + . . . (2) n = 1 / / C = 1 + 1 = 2 C = 0 Ut = 0 h n i = 1ˆ z † z s z ⌦ e ik x N N W = 2 N + 1 ⌫ = NN flux = NW L/q = N qLW (3) ⌫ = 1 / /q ⌫ d = NL (4) ⌫ d = 1 e i✓ E P = 1 W [ h M z i ( ✓ = 2 ⇡ ) h M z i ( ✓ = 0)] (5) E n ˆ x FIG. 1: Illustration of two different pictures of this systemand the idea of charge pumping. (a) Two Raman beams alongˆ x are applied to a multi-component quantum gases in opti-cal lattices. The Raman beams couple different spin statesand generate a coupling between spin and momentum k x . (b)Different spin components are viewed as another dimension,the system is mapped into a 2d spinless particle in a mag-netic field. By applying an electric field along the physicaldimension ˆ x , it generates a charge pumping along the syn-thetic dimension, which can be detected by measuring spinpopulations. (a) A 1d system of high spin atoms with spin-orbitcoupling: by applying a spin and site dependent rotationˆ c j,m → e iγjm ˆ c j,m , the Hamiltonian Eq. 1 becomesˆ H = (cid:88) j,m ( − te − iγm ˆ c † j +1 ,m ˆ c j,m + Ωˆ c † j,m − ˆ c j,m + h.c.) (2)The spin dependent hopping term, together with a con-stant spin flipping coupling, gives rise to the spin-orbitcoupling effect, which have been extensively discussed forspin-1 / m asanother synthetic dimension, say, labelled by ˆ y , Eq. 1represents a situation that spinless atoms hops in a 2dspace, with finite number ( W ) of chains and open bound-ary condition along ˆ y . More importantly, hopping alonga close loop around a plaquette accumulates a phase of a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r e iγ , that is equivalent to say, each plaquette has a flux ofΦ /q (Φ is magnetic flux unit), which is gauge coupledto motion of atoms.These two equivalent pictures build up an intriguingconnection between spin-orbit coupled high spin particlesin 1d and spinless but charged particles in a 2d laddergeometry with magnetic flux. For instance, at single-particle level, the chiral edge current from the picture(b), as observed in Ref. [3, 4], is equivalent to spin-momentum locking effect from picture (a), as observedin spin-1 / SU ( W )invariant interaction [11],ˆ H int = U (cid:88) j,m (cid:54) = m (cid:48) ˆ n j,m ˆ n j,m (cid:48) , (3)atoms in any two sites along the synthetic dimension,despite of their separation, interact with the same inter-action strength. In another word, in the 2d lattice theinteraction is very anisotropic.3. For normal FQH effect, the only relevant parameter ν is the ratio between fermion number to flux number.In our case, it will be ν = NN flux = NW L/q = N qW L , (4)where N is total number of fermions, L is the numberof sites along ˆ x -direction. However, from the picture (a)that our system is one-dimensional, it is natural to intro-duce another filling factor ν = NL , (5)and if ν is an integer, one may expect a trivial Mottinsulator rather than a FQH state when interaction U issufficiently large. Thus, when ν is fixed, we still have twoother tunable parameters, i.e. ν and W .Hereafter we shall fix ν = 1 / W (cid:105) (b) (a) (c) FIG. 2: Charge pumping for different ν and W . Numbersin box are charge pumping Q after insertion of one flux. Bluetriangles are points calculated by ED and red circles are pointscalculated by DMRG, with ν fixed at 1 /
3, Ω = t and U = 6 t .(a-c) marked three cases where spectral flow will be shown inFig. 3. is an effective scheme to detect such a state in this coldatom setting. The rest of this paper is devoted to answerthis question. Charge Pumping.
To reveal the interaction effect, wepropose to perform a charge pumping experiment utiliz-ing the advantage of synthetic dimension. Let us considerapplying an electric field along ˆ x direction. In our numer-ical calculation below, this is implemented by a periodicboundary condition in ˆ x direction, i.e. Ψ( L ) = e iθ Ψ(0).In cold atom experiment, this can be realized by, for in-stance, applying a field gradient or moving lattice witha constant velocity. Charge pumping here means chargetransfer along the synthetic dimension [12], as schema-tized in Fig. 1. Defining the charge center as Y = 1 W (cid:88) j,m (cid:104) ˆ n j,m (cid:105) m, (6)the charge pumping after inserting one flux is given by Q = Y ( θ = 2 π ) − Y ( θ = 0) . (7)If it is in real space, detecting charge transfer requires in-situ image, while charge transfer in synthetic dimensionmeans changing of spin population, which can be mucheasily detected from the Stern-Gerlach experiment in thetime-of-flight image [13, 14].In Fig. 2, we present the charge pumping value Q for various ν and W , with ν fixed at 1 /
3. This resultis obtained by numerically solving the many-body wave-functions with Hamiltonian ˆ H = ˆ H + ˆ H int , either byexact diagnolization (ED) or density matrix renormal-ization group (DMRG) methods. For ED the maximumnumber of particle is six and the dimension of Hilbertspace is of the order of 3 × . For DMRG, the maxi-mum number of particle is ten, and the truncation error ( E (cid:239) E ) / t (a) K x =0 (cid:239) (cid:239) Y ( (cid:101) ) K x =0 ( E (cid:239) E ) / t (b) K x =0K x =2 (cid:47) /3K x =4 (cid:47) /3 (cid:239) (cid:239) Y ( (cid:101) ) K x =2 (cid:47) /3 (cid:101) /2 (cid:47) ( E (cid:239) E ) / t (c) K x =0K x =2 (cid:47) /3K x =4 (cid:47) /3 (cid:101) /2 (cid:47) Y ( (cid:101) ) K x =0 Q=1/3
FIG. 3: Left column: Spectral flow under the insertion of flux(i.e. changing periodic boundary condition θ from zero to6 π .) Right column: charge center Y as a function of θ . (a-c)correspond to different W and ν as marked in Fig. 2. (a) W = 9, ν = 1 ( N = L = 5); (b) W = 5, ν = 5 / N = 5, L = 6) and (c) W = 14, ν = 2 / N = 4, L = 6). All thesecases are calculated by ED. is of the order of 10 − . Each eigenstate has a well-definedquantum number K that is the center-of-mass momen-tum along ˆ x . We also plot how these eigenstates evolveunder the changing of θ , as shown in Fig. 3, and forground state, we calculate Y as a function of θ with Eq.6 and deduce Q with Eq. 7.We find following features: (i) For ν = 1, Q is identi-cally zero for all W . In this case there is always a uniqueground state which will not interchange with other statesunder flux insertion, as shown in Fig. 3(a). This Mottinsulator phase with commensurate ν is due to one-dimensional nature of the system, or equivalently to say,anisotropic nature of interaction in picture (b). (ii) Sincehere we consider q is an integer, the smallest value for q is q = 2. With q = 2 and ν = 1 / ν = νW/q can at mostbe W/
6. We note that for q = 2, the Hofstadter spec-trum exhibits Dirac cone instead of fully gapped band,and the lowest band does not have well defined Chern-number. Therefore, for ν = W/ W (cid:54)
6) we alsofind Q = 0. A typical spectral flow and charge pumpingis shown in Fig. 3(b). (iii) For small W, or for larger W but ν closer to unity, though Q is generally non-zero,it takes non-universal values. This fluctuating Q indi-cates some Fermi-liquid type states [15]. (iv) For large W and smaller ν , Q gradually approaches a universalfractional value of 1 /
3. A typical spectral flow under fluxinsertion is shown in Fig. 3(c). One can see that thereare three low-lying states (though not exactly degener-ate) exchange one with the other as θ increases, and the (cid:49) /t Q N=4,W=4, (cid:105) =1/3N=4,W=6, (cid:105) =1/3N=4,W=14, (cid:105) =2/3 FIG. 4: Charge pumping Q as a function of Raman couplingstrength Ω /t . The dashed line is 1 / spectrum recovers itself only after θ changes 6 π . Thesefeatures are consistent with a FQH effect. This is becausefor large W , the finite size effect in synthetic dimensionbecomes insignificant, and for smaller ν away from com-mensurate filling, the lattice effect also becomes weaker.Previously, it has been shown by Luttinger liquid theorythat for continuum models, fractional state can emerge ina one-dimensional system with spin-orbit coupling [16].We also study the charge pumping value Q as a func-tion of Raman coupling strength Ω, as shown in Fig.4. As the synthetic magnetic field is resulted from Ra-man coupling, one naturally expects that Q will vanishas Ω →
0. Indeed, we show in Fig. 4 that when Ω /t issmaller than certain value, Q starts to derivates from 1 / W (e.g. green points for W = 14 in Fig. 4).We have also looked at Q for smaller U/t and find when
U < t , Q also takes fluctuating non-universal values. Periodic Boundary Condition in Synthetic Dimension.
We also find that if one applies a more involved lasersetting to achieve a periodic boundary condition alongthe synthetic dimension, it will greatly help to stabilize a K y +W K x ( E (cid:239) E ) / t (a) 0 1 2 300.010.020.030.04 (cid:101) /2 (cid:47) ( E (cid:239) E ) / t (b) K=(0, (cid:47) )K=(2 (cid:47) /3, (cid:47) )K=(4 (cid:47) /3, (cid:47) )Excited states − (cid:101) /2 (cid:47) ( E (cid:239) E ) / t FIG. 5: Energy diagram with periodic boundary condition insynthetic dimension. (a) Energy levels for different momen-tum K y + W K x . (b) Energy of the lowest three states as afunction of θ . Here W = 4, ν = 1 / N = 4, L = 12). Thiscase is also calculated by ED. q x /2 (cid:47) (cid:108) ( q x ) N=8,W=4, (cid:105) =1/3N=10,W=5, (cid:105) =5/12N=12,W=6, (cid:105) =1/2 FIG. 6: Fourier transformation of density ρ ( q x ) for different W and ν fractional state. For instance, for the cases with W = 4we presented in Fig. 2, we do not find accurate fractionalcharge pumping. While when we apply a periodic bound-ary condition in synthetic dimension, we show the energylevel for different momentum (labelled by K y + W K x asnow both K x and K y are good quantum numbers) in Fig.5. We find a very accurate three-fold degeneracy, withenergy splitting smaller than 10 − t , and these states areseparated from other excited states by a gap ∼ . t . Thetotal momentum of these three states are also consistentwith generalized Pauli-exclusion principle analysis [17].Moreover, these states exchange one and the other underthe flux insertion, and do not intersect with other excitedstates, as shown in Fig. 5(b). By calculating Berry cur-vature with twisted boundary conditions in both physicaland synthetic dimensions [15], we numerically find thattheir many-body Chern number C = C = C = 0 . Density Order.
Finally we look at real-space chargedensity-order. We consider the onsite total density ρ j = (cid:80) m (cid:104) ˆ n j,m (cid:105) . In Fig. 6, we plot the Fourier transform of ρ i as ρ ( q x ) = (1 /L ) (cid:80) j ( ρ j − ¯ ρ ) e iq x j (¯ ρ = ν is the averagedensity). ρ ( q x ) shows a clear peak at q x / (2 π ) = ν and q x / (2 π ) = 1 − ν . This feature exists for both openand periodic boundary condition in synthetic dimension.Similar situation has also been found in several othermodels [18]. This is reminiscent of usual FQH state inthe thin torus limit [19]. Conclusion.
In summary, we have studied interactioneffects in the synthetic dimension picture of high spin lat-tice Fermi gases with Raman-coupling induced spin-orbitcoupling. Our studies are mainly focused on signatures ofcharge pumping experiment, which becomes much easierin this setting, as charging pumping along the syntheticdimension can be visualized by measuring spin popula-tion. For fixed ν = 1 / W , fermion density ν , and Raman coupling Ω /t .We conclude that a universal fractional charge pumping Q = 1 / W (cid:29) ν (cid:28)
1, Ω /t ∼ Acknowledgment : We would like to thank Hong Yao forhelpful discussion. This work is supported by TsinghuaUniversity Initiative Scientific Research Program, NSFCGrant No. 11174176, No. 11325418 and NKBRSFC un-der Grant No. 2011CB921500.
Note Added.
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