Dynamical preparation of stripe states in spin-orbit coupled gases
EExcited-state quantum phase transitions in spin-orbit coupled Bose gases
J. Cabedo, J. Claramunt, and A. Celi Departament de Física, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain. Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YW, United Kingdom.
In spinor Bose-Einstein condensates, spin-changing collisions are a remarkable proxy to coher-ently realize macroscopic many-body quantum states. These processes have been, e.g., exploited togenerate entanglement, to study dynamical quantum phase transitions, and proposed for realizingnematic phases in atomic condensates. In the same systems dressed by Raman beams, the couplingbetween spin and momentum induces a spin dependence in the scattering processes taking place inthe gas. Here we show that, at weak couplings, such modulation of the collisions leads to an effectiveHamiltonian which is equivalent to the one of an artificial spinor gas with spin-changing collisionsthat are tunable with the Raman intensity. By exploiting this dressed-basis description, we proposea robust protocol to coherently drive the spin-orbit coupled condensate into the ferromagnetic stripephase via crossing an excited-state quantum phase transition of the effective low-energy model.
Artificial spin-orbit coupling (SOC) in ultracold atomgases offers an excellent platform for studying quantummany-body physics [1–3]. The interplay between lightdressing induced by Raman coupling [4] and atom-atominteractions can lead, for instance, to high-order syn-thetic partial waves [5], to the formation of stripe phases[6], a supersolid like phase only very recently unambigu-ously observed [7, 8] (see also [9], for dipolar gases real-ization see [10–12]), or to chiral interactions and density-dependent gauge fields [13]. Here we propose to useSOC for studying dynamical [14] and excited [15] quan-tum phase transitions in spinor Bose-Einstein conden-sates (BECs).In analogy to ground-state quantum phase transitions[16, 17], dynamical and excited-state quantum phasetransitions involve the existence of singularities, respec-tively, in the time evolution and in the energy (or an orderparameter) of an excited energy level, and can extendacross the excitation spectra. Dynamical phase tran-sitions have been demonstrated in quench experimentswith cold atoms in optical lattices [18–20] and cavities[21], trapped ions [22, 23], and with superconductingqubits [24]. At the same time, excited-state quantumphase transitions (ESQPTs) have been shown to occur ina variety of models [25–31], and have been observed in su-perconducting microwave Dirac billiards [32]. Recently,dynamical and ESQPTs have been theoretically [33, 34]and experimentally [35, 36] studied in spin-1 BECs withspin-changing collisions.Several authors have suggested a connection betweenspinor gases with spin-changing collisions and SOC BECs[37–44]. In this work, we show analytically that theRaman-dressed spin-1 SOC gas at low energy is equiv-alent, for weak Raman coupling and interactions, andzero total magnetization, to an artificial spin-1 gas with tunable spin-changing collisions. Under these conditions,the system is well described by a one-axis-twisting Hamil-tonian [45, 46]. Such Hamiltonian explains several quan-tum many-body phenomena in spinor condensates [47],including the generation of macroscopic entanglement [48–61], with potential metrological applications [62], andthe observation of nonequilibrium phenomena such as the -4 -2 0 2 4 (a) (c2.1) (b) (c2.2) (c1) -0.04-0.02 0
0 0.2 0.4 0.6 0.8 1 -0.04-0.02 0
P TFBATF PBA (c2)C F C F P TFBA
Figure 1. (Color online)
Pseudospin dynamics in SOCBECs. (a) Dispersion bands of the dressed Hamiltonian ˆ H k with Ω = 0 . E r , δ = 0 and (cid:15) = Ω / E r . The color tex-ture represents the expected value of the spin of the dressedstates. Dashed lines show the undressed dispersion bands.(b) Schematic representation of resonant collisions mediatedby Raman transitions (represented in wavy lines) which act aseffective spin-changing collisions. For weak Raman couplingand interactions, the dressed-state dynamics is well capturedby the pseudospin Hamiltonian (4). (c1) Phase diagram of(4), as a function of the Raman Rabi frequency Ω and effec-tive quadratic Zeeman shift (cid:15) , for Rb at n = 7 . · cm − .The polar (P), twin-Fock (TF) and broken-axisymmetry (BA)phases meet at the tricritical point C F (black dot). (c2) Cor-responding phase diagram for the highest-excited eigenstate.Inset (c2.1) indicates the energy gap ∆ E between the twomost excited eigenstates of (4) along the red dashed segmentin (c2), for N = 1000 . Inset (c2.2) shows the collective tensormagnetization L zz (solid blue) and the mean squared totalspin L (dashed red), which characterize the different phases. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n formation of spin domains and topological defects [63–73]. Here we exploit this exact map to provide a many-body protocol to access the ferromagnetic stripe phase ofthe SOC gas via crossing an ESQPT of the low-energyHamiltonian. This preparation enhances the accessibil-ity of the phase, which has as ground-state phase a verynarrow region of stability [74] and has not been experi-mentally demonstrated so far. Our mapping identifies theexcited-state stripe phase with the broken-axisymmetry(BA) phase of the effective spinor gas. System.—
We consider a spin-1 Raman-dressed Bosegas held in an isotropic harmonic potential V t = mω r with the atoms interacting via two-body s-wave colli-sions. In a frame corotating and comoving with thelaser beams, the system is described by the Hamiltonian ˆ H = (cid:82) d r (cid:104) ˆ ψ † (cid:16) ˆ H k + V t (cid:17) ˆ ψ + g ( ˆ ψ † ˆ ψ ) + g (cid:80) j ( ˆ ψ † ˆ F j ˆ ψ ) (cid:105) ,with ˆ ψ = ( ˆ ψ − , ˆ ψ , ˆ ψ ) T being the spinor field opera-tor and { (cid:126) ˆ F x , (cid:126) ˆ F y , (cid:126) ˆ F z } being the spin-1 matrices. Here g = 4 π (cid:126) ( a + 2 a ) / m and g = 4 π (cid:126) ( a − a ) / m ,with a and a being the scattering lengths in the F = 0 and F = 2 channels, respectively. The dressed kineticHamiltonian reads ˆ H k = (cid:126) m (cid:16) k − k r ˆ F z e z (cid:17) + Ω √ ˆ F x + δ ˆ F z + (cid:15) ˆ F z , where Ω is the Raman coupling strength, δ is the Raman detuning and (cid:15) is the effective quadrupoletensor field strength. The latter term can be controlledindependently of δ by employing two different Ramancouplings between the two Zeeman pairs {| , (cid:105) , | , (cid:105)} and {| , (cid:105) , | , − (cid:105)} , and simultaneously adjusting theRaman frequency differences [75]. We label the Ramansingle-photon recoil energy and momentum as E r = (cid:126) k r m and (cid:126) k r , respectively. In the weakly-coupled regime, thelowest dispersion band of ˆ H k presents a triple-well shapealong the direction of the momentum transfer, which wearbitrarily set along the ˆ z axis. Spin texture is present inthe band, with the spin mixture being the largest at thevicinity of the avoided crossings (see Fig. 1(a)). Whilemuch smaller, the spin overlap between states locatedat the vicinity of adjacent minima is nonzero, and in-creases linearly with Ω . This overlap allows collision pro-cesses that exchange large momentum at low energies.These Raman-mediated processes act as spin-changingcollisions, as illustrated in Fig. 1(b). Low-energy effective theory.—
We now consider theregime where δ , (cid:15) , (cid:126) ω t and the interaction energy per par-ticle are all much smaller than the recoil energy E r . Suchlow-energy landscape is well captured by an effective the-ory in which all the dynamics involves only the lowestband modes around each band minima k j ∼ jk r e z , with j ∈ {− , , } . Under these considerations, we re-expressthe spinor field ˆ ψ in terms of the lowest-band dressedfields at the vicinity of each k j , which we label as ˆ ϕ j , andset a cut-off Λ (cid:28) (cid:126) k r to the momentum spread p aroundthem. With this notation, we can identify the operatorsacting in the separated regions as a pseudospinor field ˆ ϕ = ( ˆ ϕ − , ˆ ϕ , ˆ ϕ ) T , with (cid:104) ˆ ϕ i ( p ) , ˆ ϕ † j ( p (cid:48) ) (cid:105) = δ ( p − p (cid:48) ) δ ij . By using perturbation theory up to second order in Ω , thelow-energy Hamiltonian can be written as ˆ H (cid:39) ˆ H S + ˆ H A (see Supplemental Material for more details). Here ˆ H S and ˆ H A include the pseudospin-symmetric and nonsym-metric contributions, respectively, given by ˆ H S = (cid:90) d r (cid:34) (cid:88) i ˆ ϕ † i (cid:18) p m + V t (cid:19) ˆ ϕ i + g (cid:88) ij ˆ ϕ † i ˆ ϕ † j ˆ ϕ j ˆ ϕ i (cid:35) (1) and ˆ H A = (cid:90) d r (cid:34) g (cid:88) j ( ˆ ϕ † ˆ F j ˆ ϕ ) + ˜ g (cid:16) ˆ ϕ † ˆ ϕ + ˆ ϕ †− ˆ ϕ − (cid:17) ˆ ϕ † ˆ ϕ +˜ g (cid:16) ˆ ϕ † ˆ ϕ †− ˆ ϕ ˆ ϕ + ˆ ϕ ˆ ϕ − ˆ ϕ † ˆ ϕ † (cid:17) (cid:35) , (2) with ˜ g = g E r (1 + O ((Λ /k r + (cid:15) + δ E r ) )) . In (2), wehave excluded the terms ∝ g Ω , since typically | g | (cid:28) g . Notice that, even in the case of SU (3) -symmetricinteractions (i.e. g = 0 ), ˆ H A includes SOC-induced spin-changing collision processes with a spin-mixing rate ˜ g . Three-mode model.—
We now restrict ourselves to thecase in which ˆ H A can be treated as a perturbation overthe symmetric part ˆ H S . We assume that the dynamics isthen well described by a three-mode model. It includesthree eigenmodes of ˆ H S , labeled as | φ − (cid:105) , | φ (cid:105) and | φ (cid:105) ,which have a quasi-momentum distribution centered atthe vicinity of k − , k and k , respectively. By introduc-ing the associated bosonic operators ˆ b − , ˆ b and ˆ b , wetruncate the field operators to ˆ ϕ † i ( r ) ∼ φ ∗ i ( r )ˆ b † i . We callthe three modes, | φ j (cid:105) , pseudospin states. Finally, drop-ping the terms that only depend on the total number ofparticles, N , we obtain the one-axis-twisting Hamiltonian ˆ H eff = λ N ˆ L − λ − g n N ˆ L z + δ ˆ L z + ˜ (cid:15) ˆ L zz , (3)where we introduce the collective pseudospin operators ˆ L x,y,z = (cid:80) µν ˆ b † µ ( ˆ F x,y,z ) µν ˆ b ν and ˆ L zz = (cid:80) µν ˆ b † µ ( ˆ F z ) µν ˆ b ν .Here, λ = (˜ g + g ) n , where n is the mean density of thegas [76]. The coefficient ˜ (cid:15) parametrizes the energy shiftof the | φ (cid:105) mode, and its value is shifted from (cid:15) as ˜ (cid:15) = (cid:15) + Ω E r (cid:0) − g n E r N + O ((Λ /k r + (cid:15) + δ E r ) ) (cid:1) . To be consistent withthe three-mode approximation, we require that | λ | (cid:28) g n , (cid:126) ω t .Since [ ˆ H eff , ˆ L z ] = 0 , the total magnetization is pre-served by ˆ H eff . Within the zero magnetization subspace(where ˆ L z = 0 ), the effective Hamiltonian (3) reduces to ˆ H = λ ˆ L N + ˜ (cid:15) ˆ L zz . (4)Hamiltonian (4) describes the nonlinear coherent spin dy-namics in a spin-1 BEC, in which the density-dependentspin-symmetric interaction dominates [46].In the SOC-based realization of (4) we propose here,we can control the spin-mixing parameter λ indepen-dently of the density of the gas by adjusting Ω . That is,SOC BECs provide a novel platform for designing entan-glement protocols and studying dynamical phase transi-tions. Quasi-adiabatic driving through ESQPTs.—
The phasediagram of Hamiltonian (4) in the Ω − (cid:15) plane is shown inFig. 1(c1) for Rb, taking g /g = − . [47]. When λ > , the diagram is equivalent to that of an antifer-romagnetic spinor gas without SOC. The ground stateis then either in a polar (P) phase, with practically allthe atoms in the | φ (cid:105) state, or in a twin-Fock (TF)phase, in which the ground state approximates the spin- / balanced Dicke state N/ (ˆ b †− ) N/ (ˆ b † ) N/ | (cid:105) . Thephase transition between the two phases is found along ˜ (cid:15) (Ω) = 0 . At Ω = 4 E r (cid:112) | g | /g , the effective and the in-trinsic spin-mixing dynamics mutually compensate, with ˜ g = − g , setting the onset of effective ferromagneticspin dynamics, i.e. λ < , for lower Ω .Spin interactions then tend to maximize the total spin,giving rise to an additional BA phase [77] in between,with the two phase transitions taking place at ˜ (cid:15) = ± λ inthe thermodynamic limit. The three phases meet at thetricritical point C F , at Ω = 4 E r (cid:112) | g | /g and (cid:15) = g /g .The BA phase of the effective model corresponds to theferromagnetic stripe (FS) phase of the spin-1 SOC gasdiagram, described in detail in [74]. When | g | is small,as for Rb, such phase is only favored in a very nar-row region in parameter space, which makes its experi-mental realization challenging. Alternatively, the ferro-magnetic landscape can be probed in the most-excitedmanifold of ˆ H in the antiferromagnetic regime, giventhat ˆ H ( λ, ˜ (cid:15) ) = − ˆ H ( − λ, − ˜ (cid:15) ) . In Fig. 1(c2), we plot thehighest-energy-state phase diagram of ˆ H , which displaysthe same phases as the ground state diagram, but withthe phase boundaries redefined. In the excited-state di-agram, the BA (or FS) phase occurs for a much broaderrange of parameters. Notably, at the P-BA and BA-TFtransitions, the energy gap between the two most excitedstates scales weakly with the total number of particles as ∝ λN − / . This facilitates the quasi-adiabatic drivingthrough both phase transitions in workable time scaleseven when the number of particles is large. This featurewas exploited in [57] and [58] to generate macroscopic TFand BA states, respectively, in small Rb spinor conden-sates. Following the dressed-spinor description, we pro-pose to probe the most excited phase diagram by drivinga state prepared in the P phase across ESQPTs. Theloading can be easily achieved from an undressed polar-ized condensate in the m f = 0 spin state by adiabaticallyturning up Ω , while setting ˜ (cid:15) < λ . The excited phase di-agram can then be probed by quasi-adiabatically varying (cid:15) and Ω .We assess the validity of the protocol by simulat-ing the preparation with the Gross–Pitaevskii equation(GPE) for the full Hamiltonian ˆ H , i (cid:126) ˙ ψ j = δ E /δψ ∗ j , with E = ψ ∗ (cid:16) ˆ H k + V t (cid:17) ψ + g | ψ | + g (cid:80) j ( ψ ∗ ˆ F j ψ ) , using the -3 -2 -1 0 1 2 3 (a) (b) (c) Figure 2. (Color online)
Quasi-adiabatic drive throughESQPTs. (a) Expected value of ˆ L zz (solid blue) and ˆ L (dashed red) as a function of ˜ (cid:15) for a state initially preparedat ψ = α ( φ − + φ ) + √ N − α φ , with α = √ and N = 10 . The state is evolved under the GPE while driving ˜ (cid:15) from − λ to λ , keeping Ω = 0 . E r , following the reddashed path in Fig. 1(c2). The total drive time is set to τ d =8 h/λ . The corresponding results obtained with simulationsof the three-mode model (4) are shown in light colors. (b)Quasi-momentum density | ˜ ψ ( p z ) | of the driven state at ˜ (cid:15) =0 (solid dark green) and ˜ (cid:15) = 3 λ (dashed light green). (c)Corresponding density profiles at ˜ (cid:15) = 0 (solid purple) and ˜ (cid:15) = 3 λ (dashed pink). XMDS2 library [78]. We label the three self-consistentmodes around k j as φ j , which are calculated via imag-inary time evolution of the GPE. As a reference point,we consider similar conditions to those described in theexperiment from [58], with small Rb condensates inthe F = 1 hyperfine manifold at n ∼ . · cm − ,and take E r / (cid:126) = 2 π · Hz, k r = 7 . · m − and g k r = 1 . E r . Note that in the proposed protocol, thestate is initially prepared in the Fock state √ N ! (ˆ b † ) N | (cid:105) .In these conditions, the dynamics is dominated by quan-tum fluctuations [79, 80], and the mean field descriptionis expected to be inaccurate. Instead, we set the initialstate to a coherent state with a small fraction of atomsoccupying the φ ± modes (See Supplemental Material formore details). In Fig. 2 we show the results for a drivealong the red dashed path drawn in the excited state di-agram from Fig. 1(c2). The drive is obtained with δ = 0 , (cid:126) ω t = 0 . E r / (cid:126) (cid:39) π · Hz and N = 10 . Setting Ω = 0 . E r , the initial state is prepared at ψ = (cid:80) j α j φ j ,with α ± = √ and α = √ N − . In Fig. 2(a) weplot the expected value of ˆ L zz and ˆ L as a function of ˜ (cid:15)/λ . The state is time evolved following the linear ramp ˜ (cid:15) ( t ) = 3 λ (2 t/τ d − , with τ d = 8 h/λ , that crosses bothtransitions at ˜ (cid:15) ∼ ± λ . In the BA phase, the tensormagnetization ˆ L zz increases homogeneously with ˜ (cid:15)/ | λ | ,and the total spin ˆ L peaks at ˜ (cid:15) = 0 , in agreement withthe effective model (see Fig.1(c2.2)). For comparison,the results obtained from the direct simulation of thethree-mode Hamiltonian (4) are shown in light colors.In Fig. 2(b) we plot the momentum-space density at themiddle and at the end of the drive, in which the stateapproaches a BA state and a TF state, respectively. Thecorresponding density profiles are shown in Fig. 2(c). Asexpected, the BA phase exhibits large density modula-tions along the direction of the Raman beams. Remark-ably, being proportional to Ω , the contrast of such modu-lations in this excited-state striped phase is considerablylarger than in its ground-state counterpart. Robust preparation of FS states.—
Now we focus onthe preparation of the excited FS phase (BA phase ofthe dressed spinor gas). In the drive depicted in Fig. 2(along the red dashed path in Fig. 1(c2)), the effectivespin-mixing rate is given by λ/ (cid:126) ∼ π · Hz. The FSstate at ˜ (cid:15) ∼ , with maximal spin and density mod-ulation, is reached in about h/λ ∼ ms, which iscompatible with the lifetime of spin-1 Raman-dressedBECs for Ω < E r [75, 81]. The drive is accurately de-scribed by the Hamiltonian (3), as over of the popu-lation remains within the self-consistent three-mode sub-space. The excited fraction increases with larger λ ’s (or Ω ’s) and depends on the path trajectory (see Supple-mental Material). We can exploit the tunability of theSOC-mediated spin-mixing to reduce further the prepa-ration time while retaining high robustness by varying Ω along the drive. Consider the blue dashed-dotted path inFig. 1(c2). There, λ increases as ˜ (cid:15) approaches , wherethe excitation rate is the lowest along the path.We illustrate a drive along such path in Fig. 3, wherethe initial state is driven from ˜ (cid:15) = − λ to ˜ (cid:15) = 0 in ms.To test the robustness of the drive in experimental con-ditions, we now include fluctuating parameters into theGPE. We model the noise in δ and (cid:15) with Gaussian whitesignals of standard deviation Hz and . Hz, respec-tively. We consider Ω to be stable during the drive, butto have a calibration uncertainty of Hz in each real-ization. These amplitudes are compatible with a mag-netic bias field instability of ∼ . mG and a relativeuncertainty of ± in Ω , within the stabilities reachedin experiments with Rb [6, 75, 82]. The values of thedressing parameters along the drive, together with theirstandard deviations, are represented as a function of timein Fig. 3(a). Due to noise, ∼ of the population is ex-cited out of the three-mode subspace by the end of thedrive. Still, the P-BA ESQPT is captured, as shown inFig. 3(b), where we plot the corresponding expected valueof ˆ L zz and ˆ L and their standard error, averaged over 20drives. The corresponding bare-basis tensor magnetiza-tion (cid:104) ˆ F z (cid:105) and magnetization (cid:104) ˆ F z (cid:105) are shown in Fig. 3(c). -0.04-0.02 0 0.02 0.04-2 -1 0 1 2 (a) (b)(d)(c) -0.2-0.1 0 0.1 0.2 1 1.1 1.2 1.3 Figure 3. (Color online)
Robust preparation of FS states. (a) In solid lines, expected values of Ω , (cid:15) and δ as a function oftime, for a drive along the blue dashed-dotted path depicted inthe excited phase diagram from Fig. 1(c2), with Ω = 0 . Er and (cid:15) = − . E r . The shadowed regions represent thewidth of the associated standard errors, which we set to , and . Hz for δ , Ω and (cid:15) , respectively. (b) Expectedvalue of ˆ L zz (solid blue) and ˆ L (dash-dotted red) along thedrive for a state prepared at ψ = α ( φ − + φ )+ √ N − α φ ,with α = √ and N = 10 , averaged over 20 realizations.The shadowed region indicates the associated variance. (c)Corresponding expected values and variances of the bare-basisoperators (cid:104) ˆ F z (cid:105) (solid blue) and (cid:104) ˆ F z (cid:105) (dash-dotted red). (d)Longitudinal density | ψ | (solid blue), spin density F x (dash-dotted red) and nematic density N xx (dashed green) at t =250 ms in a single realization of the drive. The longitudinal density | ψ | , spin density F x = ψ ∗ ˆ F x ψ and nematic density N xx = ψ ∗ (2 / − ˆ F x ) ψ are shownin Fig. 3(d) for the state prepared in a single realizationof the drive. As predicted, it exhibits the characteristicproperties of FS states. The FS phase can be distin-guished from antiferromagnetic stripe phases from theperiodicity of the spatial modulations, with the particledensity and the spin densities having periodicity π/ | k | ,and the nematic densities containing harmonic compo-nents both with period π/ | k | and π/ | k | . As a finalremark, we note that the preparation could be optimizedfurther by employing reinforcement learning techniques,as recently demonstrated in [83]. Conclusions.—
In summary, we have shown that, forweak Raman coupling and interactions, a Raman-dressedspin-1 BEC is equivalent to an artificial spinor BEC withtunable nonsymmetric spin interactions. A gas with fer-romagnetic interactions like Rb can be turned to anti-ferromagnetic by light dressing, and the stability of theFS phase understood in these terms. We have used suchinsight to propose the preparation of FS phases by driv-ing an initially polarized state through an ESQPT of theRaman-dressed gas. In the excited-state phase diagram,the FS phase is broader, and both the energy gap andthe density modulation contrast are larger. These fea-tures enable a robust quasi-adiabatic preparation of thestate and ease the detection of its supersolid properties,e.g. by probing its spectrum of excitations [84].Our dressed-base description of Raman-coupled spinorgases suggests new directions for exploring nonequilib-rium experiments, as in [69, 73], with light-dressed spinorgases of alkali and non-alkali [85] atoms. Remarkably, theFS phase corresponds to the BA entangled phase of theartificial spinor gas: its preparation may thus lead to thegeneration of macroscopic entanglement in momentumspace, cf. [86].We thank J. Mompart and V. Ahufinger for usefuldiscussions and L. Tarruell for insightful discussions onexperimental aspects of the Raman coupled BEC. A.C.thanks G. Juzeliunas for discussions on Raman coupledspinor BEC during his stay at Institute of TheoreticalPhysics and Astronomy of University of Vilnius, sup-ported by the COST action 16221, The Quantum Tech-nologies with Ultracold atoms. J. Cabedo and A.C.acknowledge support from the Ministerio de Economíay Competividad MINECO (Contract No. FIS2017-86530-P), from the European Union Regional Devel-opment Fund within the ERDF Operational Programof Catalunya (project QUASICAT/QuantumCat), andfrom Generalitat de Catalunya (Contract No. SGR2017-1646). A.C. acknowledges support from the UAB Tal-ent Research program. J. Claramunt acknowledges par-tial support from the research funding Brazilian agencyCAPES and the European Research Council (ERC) un-der the European Union’s Horizon 2020 research and in-novation programme (Grant agreement No. 805495). [1] J. Dalibard, F. Gerbier, G. Juzeli¯unas, and P. Öh-berg, “Colloquium : Artificial gauge potentials for neutralatoms,”
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In this supplementary document we include the detailed derivation of the low-energy Hamiltonian introduced in themain text. We also provide additional insights on the approach taken to assess the validity of the three-state modelderived, and on its robustness.
EFFECTIVE LOW-ENERGY THEORY
Here we detail the derivation of the effective low-energy theory presented in the main text for weakly-coupledRaman-dressed spin-1 BECs, with Rabi frequency Ω < (in units of recoil energy). We restrict ourselves to a regimein which the linear and quadratic Zeeman terms, denoted by δ and (cid:15) respectively, are also small, and set | δ | , | (cid:15) | (cid:28) .In this regime, the low-energy landscape only involves the dressed states located around the three minima of thedispersion band. Thus, we set a cut-off Λ (cid:28) (in units of k r ) to the momentum spread p around each minimum, sothat | p | < Λ . Under these conditions, we use second order perturbation theory to express the bare fields ˆ ψ i in termsof the lowest-band dressed-state fields ˆ ϕ j around the center band minimum ˆ ψ ( p ) = (cid:18) − Ω (cid:18) − (cid:15) O ((Λ + (cid:15) + δ ) (cid:19)(cid:19) ˆ ϕ ( p ) + O (cid:18) ( Ω8(1 − Λ) ) (cid:19) , ˆ ψ ± ( p ) = − Ω8 (cid:18) − (cid:15) ± δ ∓ p O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ ( p ) + O (cid:18) ( Ω8(1 − Λ) ) (cid:19) , (S1)and in right/left band minima ˆ ψ ± ( ± p ) = (cid:32) − (cid:18) Ω8 (cid:19) (cid:18) (cid:15) ± δ ∓ p O ((Λ + (cid:15) + δ ) (cid:19)(cid:33) ˆ ϕ ± ( p ) + O (cid:18) ( Ω8(1 − Λ) ) (cid:19) , ˆ ψ ( ± p ) = − Ω8 (cid:18) (cid:15) ± δ ∓ p O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ ± ( p ) + O (cid:18) ( Ω8(1 − Λ) ) (cid:19) , ˆ ψ ∓ ( ± p ) = Ω / p ± δ ± p ) (cid:16) − (cid:15) ∓ δ ± p (cid:17) ˆ ϕ ± ( p ) + O (cid:18) ( Ω8(1 − Λ) ) (cid:19) , (S2)respectively. We made explicit only the dependence on momentum along the direction of the recoil momentum transfer.Notice that the positions of the edge band minima are actually shifted from ± by a small amount proportional to Ω . However, up to second order in Ω , these shifts do not contribute to expressions (S2), and hence are not included.Note that the last term of the above expressions can be neglected since it contributes to the interactions at fourthorder in Ω8(1 − Λ) . As shown below, due to momentum conservation, the nontrivial contributions to the interactingHamiltonian involve only the first order terms in the above expressions, while the second order just renormalize thesymmetric interactions.We adopt the notation short cuts (cid:90) (cid:90) ˆ ψ † a ˆ ψ † b ˆ ψ a ˆ ψ b ≡ g (cid:90) d r (cid:90) (cid:89) j =1 d k j (2 π ) e i r · ( k + k − k − k ) ˆ ψ † a ( k ) ˆ ψ † b ( k ) ˆ ψ a ( k ) ˆ ψ b ( k ) , (S3)and (cid:90) (cid:90) ˆ ϕ † a ˆ ϕ † b ˆ ϕ a ˆ ϕ b ≡ g (cid:90) d r (cid:90) Λ − Λ 4 (cid:89) j =1 d p j (2 π ) e i r · ( p + p − p − p ) ˆ ϕ † a ( p ) ˆ ϕ † b ( p ) ˆ ϕ a ( p ) ˆ ϕ b ( p ) . (S4)When the interaction operators are evaluated on the low-energy states, it follows that (cid:90) (cid:90) ˆ ψ †± ˆ ψ †± ˆ ψ ± ˆ ψ ± = (cid:90) (cid:90) (cid:18) − Ω (cid:18) (cid:15) ± δ ∓ p + p + p + p O ((Λ + (cid:15) + δ ) (cid:19)(cid:19) ˆ ϕ †± ˆ ϕ †± ˆ ϕ ± ˆ ϕ ± + Ω (cid:90) (cid:90) (cid:18) − (cid:15) ± δ ± ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ †± ˆ ϕ † ˆ ϕ ± ˆ ϕ , (S5) (cid:90) (cid:90) ˆ ψ † ˆ ψ † ˆ ψ ˆ ψ = (cid:90) (cid:90) (cid:18) − Ω (cid:18) − (cid:15) O ((Λ + (cid:15) + δ ) (cid:19)(cid:19) ˆ ϕ † ˆ ϕ † ˆ ϕ ˆ ϕ + Ω (cid:90) (cid:90) (cid:18) (cid:15) + δ − ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ † + ˆ ϕ ˆ ϕ + + Ω (cid:90) (cid:90) (cid:18) (cid:15) − δ p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ †− ˆ ϕ ˆ ϕ − + Ω (cid:90) (cid:90) (cid:18) (cid:15) − ( p − p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † + ˆ ϕ †− ˆ ϕ ˆ ϕ + Ω (cid:90) (cid:90) (cid:18) (cid:15) − ( p − p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ † ˆ ϕ + ˆ ϕ − , (S6) (cid:90) (cid:90) ˆ ψ †± ˆ ψ † ˆ ψ ± ˆ ψ = (cid:90) (cid:90) (cid:18) − Ω (cid:18) − (cid:15) ∓ δ ∓ ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19)(cid:19) ˆ ϕ †± ˆ ϕ † ˆ ϕ ± ˆ ϕ + Ω (cid:90) (cid:90) (cid:18) − (cid:15) ± δ ± ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ † ˆ ϕ ˆ ϕ + Ω (cid:90) (cid:90) (cid:18) (cid:15) ± δ ∓ ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ †± ˆ ϕ †± ˆ ϕ ± ˆ ϕ ± + Ω (cid:90) (cid:90) (cid:18) (cid:15) ∓ + δ ± ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ †± ˆ ϕ †∓ ˆ ϕ ± ˆ ϕ ∓ + Ω (cid:18) ∓ δ/ ± ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ †± ˆ ϕ †∓ ˆ ϕ ˆ ϕ + Ω (cid:18) ∓ δ/ ± ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ † ˆ ϕ ± ˆ ϕ ∓ , (S7) (cid:90) (cid:90) ˆ ψ †± ˆ ψ †∓ ˆ ψ ± ˆ ψ ∓ = (cid:90) (cid:90) (cid:18) − Ω (cid:18) (cid:15) ∓ p − p + p − p O ((Λ + (cid:15) + δ ) (cid:19)(cid:19) ˆ ϕ †± ˆ ϕ †∓ ˆ ϕ ± ˆ ϕ ∓ + Ω (cid:90) (cid:90) (cid:18) − (cid:15) ± δ ± ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ †∓ ˆ ϕ ˆ ϕ ∓ + Ω (cid:90) (cid:90) (cid:18) − (cid:15) ∓ δ ∓ ( p + p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ †± ˆ ϕ † ˆ ϕ ± ˆ ϕ + Ω (cid:90) (cid:90) (cid:18) − (cid:15) ± ( p − p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ † ˆ ϕ † ˆ ϕ ± ˆ ϕ ∓ + Ω (cid:90) (cid:90) (cid:18) − (cid:15) ± ( p − p ) + O ((Λ + (cid:15) + δ ) (cid:19) ˆ ϕ †± ˆ ϕ †∓ ˆ ϕ ˆ ϕ . (S8)Inserting (S5)-(S8) into the symmetric contribution to the interacting Hamiltonian ˆ V s , we get ˆ V s = (cid:90) (cid:90) (cid:32) (cid:88) a = − , , +1 (cid:32) ˆ ψ † a ˆ ψ † a ˆ ψ a ˆ ψ a + 2 (cid:88) b>a ˆ ψ † a ˆ ψ † b ˆ ψ a ˆ ψ b (cid:33)(cid:33) = (cid:90) (cid:90) (cid:32) (cid:88) a = − , , +1 (cid:32) ˆ ϕ † a ˆ ϕ † a ˆ ϕ a ˆ ϕ a + 2 (cid:88) b>a ˆ ϕ † a ˆ ϕ † b ˆ ϕ a ˆ ϕ b (cid:33)(cid:33) + Ω (cid:90) (cid:90) (cid:18)(cid:16) ˆ ϕ † +1 ˆ ϕ +1 + ˆ ϕ †− ˆ ϕ − (cid:17) ˆ ϕ † ˆ ϕ + (cid:16) ˆ ϕ † +1 ˆ ϕ †− ˆ ϕ ˆ ϕ + H.c. (cid:17) + O ((Λ + (cid:15) + δ ) (cid:19) + Ω (cid:90) (cid:90) (cid:18) ( p − p ) ˆ ϕ † +1 ˆ ϕ †− ˆ ϕ ˆ ϕ + H.c. + O ((Λ + (cid:15) + δ ) (cid:19) . (S9)The last term in (S9) contains a correction to the spin-mixing contribution that depends linearly on the momentum.However, its value is bounded by the cutoff in the momentum spread around the wells. Since | p − p | < (cid:28) , forsimplicity we neglect such correction to the interacting Hamiltonian.0Finally, considering that, for | p | > Λ , the fields ˆ ϕ j ( p ) vanish when acting on the low energy subspace, we canformally remove the cut-off in the integration and perform the Fourier transform. By doing so, we obtain theexpression introduced in the main text for the symmetric interacting Hamiltonian in the dressed basis, namely ˆ V s = (cid:90) d r g (cid:88) ij ˆ ϕ † i ˆ ϕ † j ˆ ϕ j ˆ ϕ i + ˜ g (cid:16) ˆ ϕ † ˆ ϕ + ˆ ϕ †− ˆ ϕ − (cid:17) ˆ ϕ † ˆ ϕ + ˜ g (cid:16) ˆ ϕ † ˆ ϕ †− ˆ ϕ ˆ ϕ + ˆ ϕ ˆ ϕ − ˆ ϕ † ˆ ϕ † (cid:17) , (S10)with ˜ g = g (cid:0) O ((Λ + (cid:15) + δ ) ) (cid:1) . Proceeding analogously with the nonsymmetric part of the interaction potential, ˆ V a = g (cid:82) d r (cid:80) j ( ˆ ψ † ˆ F j ˆ ψ ) , yields corrections to Hamiltonian (2) in the main text of the order g Ω , which are safelyneglected since | g | (cid:28) g for Rb.
MEAN-FIELD SIMULATIONS OF THE THREE-MODE MODEL
In the protocol described in the main text, the state approaches the Fock states √ N ! (ˆ b † ) N | (cid:105) and N/ (ˆ b †− ) N/ (ˆ b † ) N/ | (cid:105) while being in the P and TF phases, respectively. The mean field description of the evolu-tion away from the BA phase is therefore expected to be inaccurate, with the dynamics being dominated by quantumfluctuations. Expressing Hamiltonian (4) explicitly in terms of the mode operators ˆ b j yields ˆ H = λN (cid:34) (ˆ b †− ˆ b † ˆ b ˆ b + H.c. ) + ˆ N ( ˆ N + ˆ N − ) (cid:21) − ˜ (cid:15) ˆ N . (S11)From eq. (S11), the corresponding three-mode mean-field equations read i (cid:126) ˙ b = λN (cid:2) b ∗− b b + b ∗ b b (cid:3) ,i (cid:126) ˙ b = λN (cid:2) b b − b ∗ + b ∗ b b + b ∗− b − b (cid:3) − ˜ (cid:15)b ,i (cid:126) ˙ b − = λN [ b ∗ b b + b ∗ b b − ] , (S12)where we have identified (cid:104) ˆ b ± , (cid:105) = b ± , . Initially setting b ± = 0 or b = 0 into eqs. (S12) results in a stationarystate, independently of ˜ (cid:15) , in contradiction with the dynamics predicted by Hamiltonian (S11). To address this issue, (a1)(a2) (b1)(b2) Figure S1. (Color online)
Comparison between full quantum and mean-field simulations.
A state initially preparedat | N, α, (cid:105) is driven from ˜ (cid:15) = − λ to ˜ (cid:15) = 3 λ with τ d = 8 h/λ . The relative occupation of the state | φ (cid:105) , N /N , along the driveis plotted in (a1) and (b1) for α = 0 and α = √ , respectively. In both cases N = 1000 . The corresponding spinor phase θ s is plotted in (a2) and (b2). Blue solid lines show the results from full quantum simulations of Hamiltonian (S11). Red dashedlines show the results obtained with the mean-field equations (S12). (a) (b) Figure S2. (Color online)
Validity of the three-mode approximation. (a) Expected value of ˆ F z as a function of ˜ (cid:15) for astate initially prepared at ψ = α ( φ − + φ ) + √ N − α φ , with α = 25 and N = 10 . The state is evolved under the GPEwhile driving ˜ (cid:15) from − λ to λ , and keeping Ω = 0 . E r (dashed-red), Ω = 0 . E r (dashed-dotted green) and Ω = 0 . E r (solid blue). The total drive time is set to τ d = 8 h/λ . (b) Relative occupation of the three self-consistent modes φ − , φ and φ , along the drive depicted in (a). we test the effective model with the GPE of the full gas by simulating an analogous drive across the P-TF-BA exciteddiagram in a slightly lower lying family of excited states. As shown in reference [34] in the main text, the propertiesof the excited phases of Hamiltonian (4) vary smoothly across the energy spectrum. Therefore, we instead preparethe initial state in a coherent state | N, α, θ s (cid:105) = √ N ! ( α e − iθ s / ˆ b †− + √ − α ˆ b † + α e − iθ s / ˆ b † ) N | (cid:105) , averaging α > atoms in the pseudospin ± states. In these conditions, mean-field computations quickly converge to full quantumsimulations as α is increased, as exemplified in Fig. S1, while the energy gap and the location of the phase boundariesdo not vary significantly as long as α (cid:28) N . VALIDITY OF THE THREE-MODE APPROXIMATION