Interacting atomic quantum fluids on momentum-space lattices
Bryce Gadway, Fangzhao Alex An, Eric J. Meier, Jackson Ang'ong'a
IInteracting atomic quantum fluids on momentum-space lattices
Bryce Gadway, ∗ Fangzhao Alex An, Eric J. Meier, and Jackson Ang’ong’a
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA (Dated: August 7, 2017)We study the influence of atomic interactions on quantum simulations in momentum-space lattices(MSLs), where driven atomic transitions between discrete momentum states mimic transport be-tween sites of a synthetic lattice. Low energy atomic collisions, which are short ranged in real space,relate to nearly infinite-ranged interactions in momentum space. However, the distinguishabilityof the discrete momentum states coupled in MSLs gives rise to an added exchange energy betweencondensate atoms in different momentum orders, relating to an effectively attractive, finite-rangedinteraction in momentum space. We explore the types of phenomena that can result from this in-teraction, including the formation of chiral self-bound states in topological MSLs. We also discussthe prospects for creating squeezed states in momentum-space double wells.
Quantum simulation with ultracold atoms [1, 2] hasbeen a powerful tool in the study of many-body physicsand nonequilibrium dynamics. There has been recentinterest in extending quantum simulation studies fromreal-space potentials to synthetic lattice systems com-posed of discrete internal [3, 4] or external [5] states.These synthetic dimensions enable many unique capabil-ities for quantum simulation, including new approachesto engineering nontrivial topology [4, 6], access to higherdimensions [3], and potential insensitivity to finite mo-tional temperature.The recent development of momentum-space lattices(MSLs), based on the use of discrete momentum states aseffective sites, has introduced a fully synthetic approachto simulating lattice dynamics [7–11]. As compared topartially synthetic systems [12, 13], fully synthetic lat-tices offer complete microscopic control of system param-eters. While this level of control is analogous to thatfound in photonic simulators [14, 15], matter waves ofatoms can interact strongly with one another.However, fully synthetic systems also present appar-ent challenges for studying nontrivial atomic interactions.Synthetic systems based purely on internal states sufferfrom limited state spaces, sensitivity to external noise forgeneric, field-sensitive states [16], and possible collisionalrelaxation [17] and three-body losses [18]. Furthermore,for isotropic scattering lengths as in Rb [16] and al-kaline earth atoms [19], interactions in the synthetic di-mension are nearly all-to-all. Similarly, s -wave contactinteractions relate to nearly infinite-ranged momentum-space interactions at low energy, and should naively bedecoupled from particle dynamics in MSLs.Here, we investigate the role of atomic interactions inMSLs, showing that finite-ranged interactions in momen-tum space result from the exchange energy of bosoniccondensate atoms in distinguishable momentum states.We explore potential interaction-driven phenomena thatcan be studied in topological MSLs, showing that chiralpropagating bound states can emerge in the presence ofan artificial magnetic flux. We additionally discuss theuse of momentum-space double wells for the generation of squeezed many-particle states.MSLs provide a bottom-up approach to engineeringdesigner Hamiltonians with field-driven transitions. Thistechnique is based on the coherent coupling of multipleatomic momentum states via two-photon Bragg transi-tions, synthesizing an effective lattice of coupled modesin momentum space. In the general case, the transi-tion frequency associated with each Bragg transition isunique. For free non-interacting particles, this stemsfrom the quadratic dispersion E p = p / m , with mo-mentum p and atomic mass m . Considering atoms ini-tially at rest and driven by laser fields of wavelength λ and wavevector k = 2 π/λ , a discrete set of momentumstates p n = 2 n (cid:126) k may be coupled, having energies 4 n E r ,with E r = (cid:126) k / m being the photon recoil energy. Byindividually addressing the unique Bragg transition res-onances, one may realize MSLs with full “local” andtemporal parameter control. Specifically, single-particletight-binding models of the form H sp ≈ (cid:88) n t n ( e iϕ n ˆ c † n +1 ˆ c n + h . c . ) + (cid:88) n ε n ˆ c † n ˆ c n , (1)may be realized by a single pair of Bragg lasers, where ˆ c n (ˆ c † n ) is the annihilation (creation) operator for the statewith momentum p n . Here, nearest-neighbor tunneling el-ements are controlled through the amplitude and phaseof individual frequency components of the Bragg laserfield, which drive first-order, two-photon Bragg transi-tions [20]. Similarly, an effective potential landscape ofsite energies ε n is controlled by small frequency detuningsof the laser fields from Bragg resonances.While there have been several demonstrations [8–11] ofthe ability to engineer diverse single-particle Hamiltoni-ans using MSLs, the prospects for studying interactionsand correlated dynamics have not yet been examined. Intypical real-space atomic quantum simulations, two-bodycontact interactions are the dominant mechanism lead-ing to correlated behavior [21]. The two-body contactpotential V ( r , r’ ), being nearly zero ranged in real space,relates to a nearly infinite-ranged interaction potential V ( k , k’ ) in momentum space. At first glance, it appears a r X i v : . [ c ond - m a t . qu a n t - g a s ] A ug p/2ħk -2 -1 0 1 2 -2 -1 0 1 21.00.80.60.40.20.0 p/2ħk p/2ħk E / E r ( E - E ) / μ pp ( E - E ) / μ pp (a) (b)(c) (d) μ / E = 0 r μ / E = 4 r μ / E = 4rμ / E = 0r rr rμ / E = / E = / E = / E = / E = / E = FIG. 1.
Interaction shifts of Bragg tunneling reso-nances. (a)
Energy dispersion of non-interacting massiveatoms E p (red solid line) in units of the recoil energy E r ,and the Bogoliubov dispersion E p of a homogeneous gas withweak repulsive interactions and a mean-field energy µ = 4 E r (blue dashed line). (b) The effective interaction potential(normalized to µ ) experienced by weakly-coupled excitationswith momentum p , shown for the cases µ/E r = 0.1, 1, and4. (c) Semi-log plot of the effective interaction potentials in(b), shown for a larger range of momenta, compared to theform µ − µ / E p (dotted lines) relevant in the free-particlelimit ( E p (cid:29) µ ). (d) Cartoon depiction of effective site en-ergies shifted by central population density, for µ/E r = 0 (nointeractions) and µ/E r = 4. as though only all-to-all interactions should result (con-sidering only mode-preserving interactions), which un-fortunately cannot give rise to correlated behavior for afixed total density.However, we find that finite-ranged, attractive interac-tions arise in the MSL due to atom statistics in the quan-tum fluid. As the landscape of MSL site energies is de-termined by synthesized detunings from the Bragg tran-sition resonances, an interaction potential results fromdensity-dependent modifications to the free-particle en-ergy dispersion. We consider the case of small-amplitudemomentum excitations of a homogeneous bosonic quan-tum gas at rest with uniform particle density n . Thequadratic dispersion E p for a non-interacting gas isshown in Fig. 1(a), along with that of Bogoliubov quasi-particles of a weakly-interacting quantum gas [22–24], E p = (cid:113) E p ( E p + 2 µ ) (ignoring a uniform energy shift of µ for all states). Here, µ = gn is the uniform condensatemean-field energy, with the interaction parameter g re-lated to the s -wave scattering length a as g = 4 π (cid:126) a/m .The interactions mainly modify the shape of the free-particle dispersion at low momenta, with a character-istic linear dispersion near p = 0. At higher mo-menta ( E p (cid:29) µ ), the Bogoliubov quasiparticles arefree-particle-like and have a roughly quadratic disper-sion which is shifted in energy by the chemical poten- tial ( E p ≈ E p + µ − µ / E p ). This extra energy shiftof order µ for high momentum states is a consequence ofexchange interactions with the zero-momentum conden-sate. Indeed, the modification of the energy-momentumdispersion results not from momentum-dependent colli-sional interactions, but rather from the quantum statis-tics of the identical bosons in distinguishable motionalstates [25–27]. This is analogous to the effective mag-netic interactions of electrons in condensed matter thatresult from spin-independent Coulomb interactions andexchange statistics.The shifts of the state energies from their non-interacting values ( E p − E p ) are plotted in Fig. 1(b,c).The Bragg transition frequencies, which depend only onthe energy differences between pairs of adjacent momen-tum states, are most strongly perturbed near the initiallypopulated zero-momentum site. The density-dependentperturbations to the transition frequencies result in aneffective interaction potential that is finite ranged andattractive, as depicted in Fig. 1(d). For µ (cid:38) E r , themomentum-space interaction potential has significant off-site contributions, the effective range of which increaseswith larger ratios µ/E r . There is, however, a natural lim-itation on the compatibility of long-ranged interactionswith the scheme for engineering MSLs. This methodbreaks down when unique spectral addressing of the in-dividual Bragg transitions is lost, occurring when multi-ple momentum orders populate the linear phonon branch(occurring roughly when µ exceeds 8 E r , the bare energyspacing of the Bragg resonances).For concreteness, we now focus on the limit µ (cid:28) E r ,where all coupled momentum orders are approximatelydistinguishable and the interaction potential is effectivelylocal in momentum space. This allows us to move beyonda description of weakly-coupled condensate excitations,and describe the more general case where atomic popu-lation is arbitrarily distributed among many momentumorders. We explore new phenomena that may be openedup to investigation by combining this simple local in-teraction with the wide range of tunable lattice modelsenabled by MSLs.The role of momentum-space interactions in the limitof purely distinguishable momentum orders can be sim-ply described by a multimode nonlinear Schr¨odingerequation, where the self- and cross-phase modulationterms describing intra-mode and inter-mode interactionsdiffer by the exchange energy. We make the simplify-ing assumption that all momentum orders share a com-mon spatial wavefunction throughout the dynamics. Thissingle-mode approximation is valid on only relativelyshort timescales, and does not capture the spatial separa-tion of momentum wavepackets. To focus on the uniquecontributions of this exchange-driven momentum-spaceinteraction, we additionally assume single-spin (internalstate) bosonic atoms with effectively one-dimensional dy-namics. In this restricted scenario, four-wave mixing pro- site index n φ φ φ φφ φ φ . . .. . . n = 0 n = 2n = -2n = -4 n = 1n = -1 n = 3n = -3 n = 4 P n P n P n P n 〈 n ⟩ τ [ħ/t] (d) P n max τ [ħ/t] (e) (c) τ [ħ/t] P n P n (a) τ = 12.5τ = 25τ = 50τ = 100 (b) FIG. 2.
Interaction effects in multiply-connected momentum-space lattices. ( a ) Role of interactions on quenchdynamics of atoms in a momentum-space triple well with periodic boundary conditions and an effective magnetic flux of ϕ = 0 . π . Population begins in site n = 0 (red dashed-dotted line), and equal magnitude tunnelings t to sites n = 1 (bluesolid line) and n = − τ = 0. The upper plot shows the dynamics of the sitepopulations P n for µ/t = 0 (no interactions), and the lower plot for the case µ/t = 6. ( b ) Cartoon depiction of atoms on azig-zag ladder with uniform magnetic flux ϕ and population initialized at the central site n = 0 (gray). Tunnelings realizedby first- (solid black lines) and second-order (dashed black lines) Bragg transitions have uniform amplitudes. ( c ) Snapshots ofthe distributions of site populations for increasing evolution times τ = 12 . , , , and 100 (units of (cid:126) /t ), shown for severaldifferent combinations of interaction-to-tunneling ratios ( µ/t ) and flux values ( ϕ ). Solid black denotes { µ/t, ϕ } = { , π/ } ,solid blue for { . , π/ } , dashed red for { , π/ } , solid purple for { . , } , and solid orange for { . , − π/ } . ( d ) Dynamics ofthe average site position (cid:104) n (cid:105) of the atomic distributions, shown for the cases of { µ/t, ϕ } = { , π/ } (black solid line), { . , π/ } (blue solid line), and { , π/ } (red dashed line). The blue and red dots relate to the position of largest site population forthe cases µ/t = 7 . τ = 40 (cid:126) /t results from population reaching theboundary of the 401-site system. ( e ) Dynamics of P maxn , the percentage of atoms in the most highly-populated site, for thesame cases as in (d). The spike of the black line near τ = 40 (cid:126) /t is also due to boundary reflection in the non-interacting case. cesses [28–31] are not allowed, and the individual statepopulations are conserved by the atomic interactions.In this single-mode approximation, assuming a ho-mogeneous condensate with fixed total atom number N and thus fixed density, we may represent the con-densate wavefunction simply with appropriately normal-ized (to unity) complex amplitudes φ n of the variousdiscrete plane-wave momentum orders with momenta p n = 2 n (cid:126) k [29]. We furthermore remove a global en-ergy term 2 µ by redefining the φ n , thus transforming themomentum-space interaction into an effectively attrac-tive self-interaction term for atoms residing in the sameorder (valid for µ (cid:28) E R ). Taking into account thecontributions of these interactions to the effective tight-binding models of Eq. 1, the dynamical evolution of theatoms becomes governed by i (cid:126) ˙ φ n = (cid:88) m H spmn φ m − µ | φ n | φ n , (2)a tunable lattice tight-binding model with local attractiveinteractions.Even with only two coupled sites, interactions are ex-pected to significantly alter the system dynamics. For thesimple case of population initialized in one of two equal-energy sites, weak interactions ( µ/t <
4) lead to a slow-down of two-mode Rabi dynamics, giving way to critical slowing for µ/t ≈
4. For stronger interactions ( µ/t > . π ) and uniformtunneling amplitude t . Without interactions (upperplot), an initially localized wavepacket spreads almostevenly to the neighboring sites. For sufficiently large in-teractions ( µ/t = 6, lower plot), however, the initial onsetof a slightly asymmetric chiral current induces the forma-tion of a fully chiral soliton-like mode [43]. For still largervalues of the nonlinear interaction, self trapping occurs.We extend this investigation to a many-site zig-zag lad-der system, shown in Fig. 2(b), with a uniform distribu-tion of effective magnetic fluxes ϕ and tunnelings t . Therole of nonlinear interactions in such a topological lat-tice model is of interest for its connection to emergenttopological phenomena in kinetically frustrated systems.Here we again examine the case of population initializedto a single, central mode ( n = 0), exploring dynamicsfollowing a tunneling quench. The distributions of nor-malized site populations P n at various evolution times τ are shown in Fig. 2(c). With no interactions (blacksolid line), chiral currents are present, but with a rapidballistic spreading of the atomic distribution. We findthat moderate interactions stabilize the atomic distribu-tion, leading to soliton- or breather-like states [43]. Fornon-zero flux values ( ± π/ µ/t (cid:38) n = 0 for all flux values.This interaction-driven stabilization of chiralwavepackets is further summarized in Fig. 2(d,e),through the average position (cid:104) n (cid:105) and largest site pop-ulation P maxn . Figure 2(d) contrasts the dynamics of (cid:104) n (cid:105) for interaction values µ/t = 0 , . , and 12, all for auniform magnetic flux ϕ = π/
6. While some dynamicsof (cid:104) n (cid:105) can be seen even for the self-trapped scenario at µ/t = 12, the position of the most highly populated site(filled circles) never deviates from the initial location.The dynamics of this site population P maxn are shownin Fig. 2(e) for these same cases. Without interactions,ballistic spreading leads to a continuous decrease of themaximum density. For a range of moderate interactions,we find that the distribution self stabilizes at shorttimescales ( τ ≈ (cid:126) /t ). Finally, for very large interactions,behavior analogous to macroscopic quantum self trap-ping inhibits particle transport and population remainslargely localized at the central site.Populating flat energy bands [44–48] of similar topo-logical models with interacting atoms should lead to in-teresting, emergent many-body dynamics. Interactinggases, combined with engineered MSLs having arbitraryand time-fluctuating disorder [11], should also enablehighly controllable explorations into the physics of many-body localization [49, 50]. For both scenarios, the mostinteresting open questions relate to phenomena driven byquantum fluctuations, which are not captured by Eq. 2.The simplest MSL to capture such physics is a sin- (a) (b) θ / π φ/π θ / π φ / π rot ξ [ d B m ] increasing τ sq z FIG. 3.
Squeezing in a momentum-space doublewell. ( a ) Visualization of many-particle ( N = 100) spinstates | Ψ (cid:105) through their overlap with different coherent spinstates |(cid:104) θ, ϕ | Ψ (cid:105)| . Shown are the cases of an initial coherentspin state | π/ , π (cid:105) (upper plot), and the transformed stateafter evolution under H sq for a time κτ sq / (cid:126) = 0 . π (lowerplot). ( b ) Squeezing along the ˆ z -axis, ξ z , for different evo-lution times κτ sq / (cid:126) = { . , . , . , . , } × . π (solidlines, with colors varying from blue to red) and for differentangles of rotation ϕ rot of the final distribution about ˆ J x . gle momentum-space double well. For fixed total par-ticle number N = N + N , one can define effectiveangular momentum operators relating to the coherencesand macroscopic occupations N and N of two Bragg-coupled momentum orders (ignoring thermal and quan-tum depletion). These are given by ˆ J x = (ˆ c † ˆ c + ˆ c † ˆ c ) / J y = i (ˆ c † ˆ c − ˆ c † ˆ c ) /
2, and ˆ J z = (ˆ c † ˆ c − ˆ c † ˆ c ) /
2, whereˆ c n (ˆ c † n ) is the annihilation (creation) operator for mode n [51]. Here, Dicke states | j, m (cid:105) with total spin j = N/ z -projection m = ( N − N ) / | θ, ϕ (cid:105) = (cid:80) jm = − j f jm ( θ, ϕ ) e − i ( j + m ) ϕ | j, m (cid:105) ,for f jm ( θ, ϕ ) = (cid:0) jj + m (cid:1) / cos( θ/ j − m sin( θ/ j + m , resultfrom a global rotation of spin-polarized states | j, j (cid:105) aboutthe spin vector ˆ n ϕ = cos( ϕ ) ˆ J x + sin( ϕ ) ˆ J y by an amount θ [52]. In this description, the momentum-space interac-tion relates to an effective nonlinear squeezing Hamilto-nian H sq = κ ˆ J z , for κ = − µ/ N [51].We examine the case of the CSS | π/ , π (cid:105) initiallyaligned along − ˆ J x . For short evolution times, the non-linear Hamiltonian H sq leads to a “shearing” of such co-herent states. This is depicted in Fig. 3(a), for the initialCSS (upper plot) and the sheared, non-classical squeezedstate after a time κτ sq / (cid:126) = 0 . π (lower plot), throughthe overlap of these states with CSSs of varying θ and ϕ values. For ease of calculation, dynamics are shown forthe case of only N = 100 atoms ( j = 50).Figure 3(b) shows, for sheared distributions relating tovarious evolution times τ sq , the ˆ z -axis squeezing param-eter ξ z = 2 j (cid:104) ∆ ˆ J z (cid:105) j −(cid:104) ˆ J z (cid:105) as a function of rotation angle ϕ rot about the ˆ J x spin axis. For typical experimental param-eter values ( N = 10 atoms, µ/ (cid:126) = 2 π × ξ minz ≈ (3 j ) − / , relating to −
33 dBm(after rotation), would be expected after a total duration τ sq ≈ . j / ( (cid:126) /µ ) ≈ . π (echo) pulses or twist-and-turn squeezing schemes [53]can be used to mitigate decoherence due to spatial sep-aration of the momentum orders. Looking beyond thissimple two-mode case, the straightforward extension tomultiple momentum states should also allow for uniqueinvestigations into multi-mode squeezing and quantumphase transitions in multi-mode analogs of the Lipkin-Meshkov-Glick model [38, 40].We have shown that local real-space interactions ofatomic gases can give rise to finite-ranged effective in-teractions in momentum space. These momentum-spaceinteractions can lead to correlated dynamics of atoms inhighly-tunable MSLs, opening up new possibilities forexploring interacting topological and disordered fluids.To focus on this effectively attractive momentum-spaceinteraction, we have restricted our investigations to thecase of mode-preserving collisions. 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