Rabi oscillations and Ramsey-type pulses in ultracold bosons: Role of interactions
RRabi oscillations and Ramsey-type pulses in ultracold bosons: Role of interactions
Q. Guan,
1, 2
T. M. Bersano, S. Mossman, P. Engels, and D. Blume
1, 2 Homer L. Dodge Department of Physics and Astronomy,The University of Oklahoma, 440 W. Brooks Street, Norman, Oklahoma 73019, USA Center for Quantum Research and Technology, The University of Oklahoma,440 W. Brooks Street, Norman, Oklahoma 73019, USA Department of Physics and Astronomy, Washington State University, Pullman, Washington 99164-2814, USA (Dated: June 25, 2020)Double-well systems loaded with one, two, or many quantum particles give rise to intriguingdynamics, ranging from Josephson oscillation to self-trapping. This work presents theoretical andexperimental results for two distinct double-well systems, both created using dilute rubidium Bose-Einstein condensates with particular emphasis placed on the role of interaction in the systems.The first is realized by creating an effective two-level system through Raman coupling of hyperfinestates. The second is an effective two-level system in momentum space generated through thecoupling by an optical lattice. Even though the non-interacting systems can, for a wide parameterrange, be described by the same model Hamiltonian, the dynamics for these two realizations differin the presence of interactions. The difference is attributed to scattering diagrams that contributein the lattice coupled system but vanish in the Raman coupled system. The internal dynamicsof the Bose-Einstein condensates for both coupling scenarios is probed through a Ramsey-typeinterference pulse sequence, which constitutes a key building block of atom interferometers. Theseresults have important implications in a variety of contexts including lattice calibration experimentsand momentum space lattices used for quantum analog simulations.
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I. INTRODUCTION
We consider the famous model in which two isolatedstates are resonantly coupled by a monochromatic field,e.g., two energetically separated atomic states that arecoupled by an oscillating electric field. Starting with allparticles populating one of the states, the population os-cillates periodically between the two states under the in-fluence of the external field. In the presence of a posi-tive or negative detuning δ , the population still oscillatesback and forth; however, the oscillation period and themaximum transfer probability (amplitude) of these Rabioscillations are modified. This coupled two-level system,which finds applications in many areas of physics, is dis-cussed in nearly every quantum text book [1–3]. For aninteracting ensemble of particles, the Rabi oscillationsare, in general, further modified. In particular, the pop-ulation oscillations may not be fully periodic and the am-plitude of the oscillations may decrease or drift (dephase)due to many-body effects [4–6].This work considers Rabi oscillations in the contextof ultracold atoms, specifically a degenerate Rb Bose-Einstein condensate (BEC). The two-level system is real-ized in two different ways. In scenario 1, one-dimensionalRaman coupling along the z -direction, realized using twoRaman lasers, generates an effective pseudo-spin-1/2 sys-tem of two coupled internal hyperfine states [7–9]. In sce-nario 2, a one-dimensional moving optical lattice alongthe z -direction couples momentum states with momentaof 2 n (cid:126) k L , where n is an integer and k L the lattice wavevector [10–12]. Considering only the n = 0 and n = 1states, the lattice Rabi coupling case can, in the absence of interactions, be mapped to the same two-state descrip-tion as the Raman coupling case considered in scenario 1,with the velocity of the moving lattice determining theeffective detuning.In the presence of atom-atom interactions, the Rabioscillations for scenarios 1 and 2 are found to differ. Thereason for this is traced back to how the two-level sys-tems are realized. In second quantization, the interactionpotential takes the formˆ v = (cid:90) ˆΨ † ( r ) ˆΨ † ( r ) V ( r , r ) ˆΨ( r ) ˆΨ( r ) d r d r , (1)where ˆΨ † ( r ) creates a particle at position r . Parametriz-ing V ( r , r ) in terms of a contact interaction of strength g , V ( r , r ) = gδ ( r − r ), and assuming that the fieldoperator ˆΨ † ( r ) can be expressed in terms of two states,ˆΨ † ( r ) = ˆ c † a Ψ ∗ a ( r ) + ˆ c † b Ψ ∗ b ( r ) (2)(ˆ c † a and ˆ c † b create a particle in states Ψ a and Ψ b , respec-tively), it can be seen that ˆ v contains 16 terms. Some ofthese vanish in the Raman coupling case but contributeappreciably in the lattice coupling case.We write Ψ a ( r ) = ψ a ( r ) |↑(cid:105) and Ψ b ( r ) = ψ b ( r ) |↓(cid:105) . Inthe Raman coupling case, |↑(cid:105) and |↓(cid:105) represent two dif-ferent hyperfine states and ψ a ( r ) and ψ b ( r ) correspondto states with momentum 0 and 2 (cid:126) k R ( (cid:126) k R is the Ra-man momentum). The resulting non-vanishing interac-tion terms or scattering diagrams are shown in the firstrow of Fig. 1. The first scattering diagram is propor-tional to ˆ c † a ˆ c † a ˆ c a ˆ c a , the second scattering diagram is pro-portional to ˆ c † b ˆ c † b ˆ c b ˆ c b , and the third and fourth scattering a r X i v : . [ c ond - m a t . qu a n t - g a s ] J un FIG. 1: Scattering diagrams. The scattering diagrams in thefirst row contribute in the Raman and lattice coupling cases.The scattering diagrams in the second row vanish in the Ra-man coupling case. In the first diagram of the first row, par-ticle 1 (top incoming arrow) and particle 2 (bottom incomingarrow) are both in the same state (both arrows are blacksolid); the scattering process does not change the state (thearrows are black solid after the scattering has occured). Inthe first diagram of the second row, particle 1 (top incomingarrow) and particle 2 (bottom incoming arrow) are in differentstates (black solid and red dotted arrows, respectively); afterthe scattering process, the states of particle 1 (top outgoingarrow) and particle 2 (bottom outgoing arrow) are changed(red dotted and black solid arrows, respectively). diagrams are proportional to ˆ c † a ˆ c † b ˆ c a ˆ c b . The last two pro-cesses can be written as | Ψ a (cid:105) + | Ψ b (cid:105) → | Ψ a (cid:105) + | Ψ b (cid:105) and | Ψ b (cid:105) + | Ψ a (cid:105) → | Ψ b (cid:105) + | Ψ a (cid:105) , where the notation | Ψ a (cid:105) means that particle 1 occupies state Ψ a . The “mixedscattering diagrams” that are shown in the second rowof Fig. 1, which are also proportional to ˆ c † a ˆ c † b ˆ c a ˆ c b , van-ish in the Raman coupling case due to the orthogonal-ity of the two hyperfine states |↑(cid:105) and |↓(cid:105) . They corre-spond to the processes | Ψ a (cid:105) + | Ψ b (cid:105) → | Ψ b (cid:105) + | Ψ a (cid:105) and | Ψ b (cid:105) + | Ψ a (cid:105) → | Ψ a (cid:105) + | Ψ b (cid:105) .In the lattice system, all the atoms occupy the samehyperfine state and Ψ a ( r ) and Ψ b ( r ) correspond to stateswith momentum ≈ ≈ (cid:126) k L , respectively. In thiscase, |↑(cid:105) and |↓(cid:105) represent two different plane wave statesand Ψ a ( r ) and Ψ b ( r ) are not orthogonal to each other.As a consequence, the scattering diagrams in the secondrow of Fig. 1 are finite, leading to an enhancemenet of theinteraction terms that are proportional to ˆ c † a ˆ c † b ˆ c a ˆ c b . Thisfactor of two enhancement can be thought of as beingdue to an exchange process; it is not specific to bosonsand also exists in fermionic systems. The doubling of themixed scattering diagrams for the lattice coupling casecompared to the Raman coupling case leads—in certainparameter regimes—to distinct Rabi oscillations for sce-narios 1 and 2. Good agreement between experimentaland theoretical results is found and implications for, e.g.,lattice calibration experiments are discussed.Working in a parameter regime where the Rabi os-cillations are noticeably impacted by the interactions,we probe the internal dynamics of condensed atomclouds through a Ramsey-type π/ π/ π/ π/ II. EXPERIMENTAL SETUP
The experiments are performed with a Rb BECconsisting of approximately N = 10 atoms. Nearlypure BECs are confined in an elongated harmonic trapwith trap frequencies of approximately { ω x , ω y , ω z } =2 π { , , } Hz. The spin-independent trapping po-tential is produced by two crossed, optical dipole beamswith a wavelength of 1064 nm. Anharmonic correctionsfor this trapping configuration are estimated to be neg-ligible for the purpose of this work. After preparationof the initial state, we remove the trapping potential attime t = 0. For all practical purposes, the turning off ofthe external confinement is done instantaneously, V trap ( r , t ) = (cid:26) m (cid:0) ω x x + ω y y + ω z z (cid:1) for t <
00 for t ≥ , (3)where m denotes the atom mass. The trap frequenciesand atom number are calibrated daily by fitting dipoleoscillation data and cloud widths during expansion, re-spectively. The relevant values are reported in the figurecaptions.Scenario 1 is realized by applying two 789.1 nm Ra-man lasers with effective coupling strength Ω R and Ra-man detuning δ R to couple the | F, m F (cid:105) = | , − (cid:105) = |↑(cid:105) and | F, m F (cid:105) = | , (cid:105) = |↓(cid:105) hyperfine states of Rb underan external magnetic field of approximately 10 G. Here, F denotes the total angular momentum of the Rb atomand m F the corresponding projection quantum number.The two-photon Raman coupling scheme follows the pro-cedure described in Ref. [19]. In momentum space, thetwo hyperfine states are separated by 2 (cid:126) k R , where k R is determined by the wave number and orientation ofthe Raman lasers. Specifically, the two Raman laserswith wave vectors k and k cross at an angle of θ R .Defining 2 k R = | k − k | and using | k | = | k | , wehave k R = | k | sin( θ R / θ R ≈ π/ k R ≈ | k | / √
2. The difference between the angular fre-quencies ω and ω of the two lasers allows one to set theRaman detuning δ R , δ R = 4 E R − (cid:126) ω R + E Zeeman , where ω R = ω − ω and E R = (cid:126) k R m . (4)Here, E Zeeman is the Zeeman splitting between the twocoupled hyperfine states. The hyperfine state | , (cid:105) ,which is off-resonant due to the quadratic Zeeman shift,is not included in our theoretical description. We havechecked that inclusion of this state does not notablychange the dynamics in the parameter regime of inter-est.Scenario 2 is realized by preparing all atoms in the | F, m F (cid:105) = | , − (cid:105) = |↑(cid:105) state and loading the BEC intoa moving optical lattice [20]. Spin changing collisionsplay a negligible role in the magnetic fields used in thiswork. The lattice is created by crossing two 1064 nmlasers at an angle of θ L ( θ L ≈ π/ k and k ( | k | = | k | ), and angular frequencies ω and ω . The resulting lattice is characterized by the effectivecoupling strength Ω L , the wave vector k L , and the de-tuning δ L ( k L ≈ | k | / √ δ L = 4 E L − (cid:126) ω L , where ω L = ω − ω ). Energies and lengths are measured inunits of E L [Eq. (4) with the subscript “ R ” replaced by“ L ”] and 1 /k L , respectively. Specific values are given inthe context of the experiments described below. In allcases, the external harmonic confinement is turned offtime t = 0.In the remainder of this paper, we denote the couplingstrength by Ω( t ) when the discussion is independent ofthe specific scheme, i.e., when the discussion applies toboth the Raman and lattice coupling cases. When thediscussion is specific to one of the scenarios, we use, re-spectively, Ω R ( t ) and Ω L ( t ) for the Raman and latticecoupling cases [correspondingly, Ω in Eqs. (5) and (6)below are replaced by Ω ,R and Ω ,L , respectively]. Thecoupling, which is assumed to be real, is turned on attime t start , where t = 0 is the time at which the trap-ping potential is removed. For t start >
0, the initialBEC expands in the absence of the Raman or latticedrive, thereby reducing the interaction strength duringthe subsequent pulse sequence. For the Rabi oscilla-tion measurements, we keep Ω( t ) on for a time interval t seq = t end − t start ,Ω( t ) = t < t start Ω for t start ≤ t < t end t ≥ t end . (5)For the Ramsey-type pulse sequence of length t seq = τ + t hold + τ , the coupling strength Ω( t ) readsΩ( t ) = t < t start Ω for t start ≤ t < t start + τ t start + τ ≤ t < t start + τ + t hold Ω for t start + τ + t hold ≤ t < t end t ≥ t end . (6)In the experiment, the turning on of the couplingstrength is not quite instantaneous but instead occursover about 75 µ s. To facilitate the comparison betweentheory and experiment, we choose t start to be the time atwhich Ω( t ) has reached half of its maximum. In many ap-plications that involve momentum transfer, a π/ π -wait- π/ π/ π/ π reversalpulse” does not, as in other scenarios, remove the linearphase in our systems due to the presence of interactions.The imaging is done at time t end + t ToF , i.e., afteran additional expansion time of t ToF . In the absenceof the trapping potential, the momentum componentsseparate naturally due to the fact that the states Ψ a andΨ b have different velocities. For the lattice case, e.g.,an expansion time of t ToF ≈
10 ms corresponds to aseparation of the cloud centers by about 85 µ m alongthe z -direction. This distance is larger than the size ofthe clouds after the expansion. For an initial cloud withThomas-Fermi radius 22 µ m, e.g., the size of the cloudat time t seq + t ToF is about 43 µ m.Depending on the observable, the time-of-flight expan-sion plays no role, a negligible role, or an essential rolewhen comparing experimental and theoretical data. Forthe Raman coupling case, the populations of the statesΨ a and Ψ b do not change during the time-of-flight ex-pansion. This implies that theoretical results for thepopulations, calculated by neglecting the time-of-flightexpansion, can be compared directly with experimentallymeasured populations. Correspondingly, we do not sim-ulate the time-of-flight sequence when we compare Rabioscillation data. Experiment-theory comparisons of theRamsey-type pulse sequence, in contrast, require thatthe time-of-flight expansion be simulated to explain theobserved fringe structures.For the lattice case, the situation is slightly different.The populations of the states Ψ a and Ψ b , which are dis-tinguished only by their momentum, can change dur-ing the time-of-flight expansion due to atom-atom col-lisions that involve states with momenta ≈ n (cid:126) k L , where n = − , ± , ± , · · · . However, such population trans-fer is typically small; note that this is the reason whythe two-state model introduced in Sec. IV A provides areliable description for a fairly large parameter window.The small population transfer implies that the dynam-ics during the time-of-flight expansion can, in a first ap-proximation, be neglected when analyzing populations.Understanding the internal dynamics such as the forma-tion of density patterns, in contrast, requires that thetime-of-flight sequence be modeled explicitly. III. RAMAN COUPLING CASEA. General framework
Our theoretical analysis of the Raman-coupled sys-tem is based on the standard mean-field formulation [9], which writes the mean-field spinor in terms of the com-ponents ψ a ( r , t ) and ψ b ( r , t ). Here and in what follows, ψ a and ψ b are time-dependent; note that the discussionin Sec. I adopted a stationary framework for simplicity.The unrotated 2 × H readsˆ H = (cid:18) ˆ p m + V trap ( r , t ) (cid:19) ⊗ I + (cid:18) g aa | ψ a ( r , t ) | + g ab | ψ b ( r , t ) | g ba | ψ a ( r , t ) | + g bb | ψ b ( r , t ) | (cid:19) + (cid:32) Ω R ( t )2 exp( − ık R z + ıω R t ) Ω R ( t )2 exp(2 ık R z − ıω R t ) E Zeeman (cid:33) , (7)where I is the 2 × N a and N b , is N a + N b = 1 (8)with ( j = a or b ) N j = (cid:90) | ψ j ( r , t ) | d r . (9)The interaction strengths g ij between atoms in hyperfinestates i and j are given by g ij = 4 π (cid:126) ( N − a ij m . (10)For Rb, we have a aa = 100 . a B , a ab = a ba = 100 . a B ,and a bb = 100 . a B [21], where a B denotes the Bohrradius. In the arguments presented in Sec. I, the four g ij were assumed to be the same; this simplifying assumption is again made in Sec. III E. The time dynamics of thesystem is governed by ı (cid:126) ∂∂t (cid:18) ψ a ( r , t ) ψ b ( r , t ) (cid:19) = ˆ H (cid:18) ψ a ( r , t ) ψ b ( r , t ) (cid:19) . (11)Defining the rotated states ˜ ψ a ( r , t ) and ˜ ψ b ( r , t ), (cid:18) ˜ ψ a ( r , t )˜ ψ b ( r , t ) (cid:19) = ˆ U ( z, t ) (cid:18) ψ a ( r , t ) ψ b ( r , t ) (cid:19) , (12)in terms of the rotation operator ˆ U ( z, t ),ˆ U ( z, t ) = (cid:18) − ık R z + ıω R t ) (cid:19) , (13)we obtain Eq. (11) with ψ a ( r , t ), ψ b ( r , t ), and ˆ H replacedby ˜ ψ a ( r , t ), ˜ ψ b ( r , t ), and ˆ˜ H , respectively, where the ro-tated Hamiltonian ˆ˜ H is given byˆ˜ H = (cid:18) ˆ p m + V trap ( r , t ) (cid:19) ⊗ I + (cid:18) g aa | ˜ ψ a ( r , t ) | + g ab | ˜ ψ b ( r , t ) | g ba | ˜ ψ a ( r , t ) | + g bb | ˜ ψ b ( r , t ) | (cid:19) + (14) (cid:32) Ω R ( t )2Ω R ( t )2 2 (cid:126) k R ˆ p z m + δ R (cid:33) . To obtain Eq. (14), we used the relation | ˜ ψ j ( r , t ) | = | ψ j ( r , t ) | , where j = a or b . Importantly, the position-and time-dependent phase ˜ γ b ( r , t ) of the rotated compo-nent ˜ ψ b ( r , t ) differs from the phase γ b ( r , t ) of the unro-tated component ψ b ( r , t ). Since the change of the phasesof the unrotated spinor components is dominated by thelaser coupling term, thereby masking the change due tothe internal dynamics, it is more convenient to analyzethe phases of the spinor-components in the rotated basis,whose phase dynamics is governed by “internal effects”as opposed to the laser coupling.For the Rabi oscillation measurements and theRamsey-type pulse sequence, the BEC is initially (i.e.,at t = 0) prepared in the state ψ a ( r , t ) |↑(cid:105) = ˜ ψ a ( r , t ) |↑(cid:105) ,which is characterized by a vanishing average mechanicalmomentum along the z -direction, i.e., (cid:104) ˆ p z (cid:105) initial = 0. Ourcalculations assume an axially symmetric harmonic trapwith ω x = ω y = ω ρ . The trapping potential defines theharmonic oscillator lengths a ho ,z and a ho ,ρ , a ho ,z/ρ = (cid:115) (cid:126) mω z/ρ . (15)The coupled mean-field equations are solved using stan-dard techniques. The initial state is obtained by imag-inary time propagation. The real time dynamics is im-plemented by expanding the time evolution operator interms of Chebychev polynomials [22, 23]. We use equallyspaced grid points in z and ρ . The convergence of theresults presented has been tested with respect to the sizeof the simulation box, the number of grid points, and thetime step. B. Rabi oscillations: Vanishing Raman detuning
This section discusses Rabi oscillation results for theRaman coupling case with δ R = 0. The numerical solu-tions are obtained by solving the time-dependent mean-field equation for ˆ˜ H [see Eq. (14)] with E R /h = 1960 Hz.The Raman coupling is turned on at t = 0, i.e., we have t start = 0. Figs. 2(a)-2(c) show the difference N a − N b between the fractional populations as a function of thedimensionless time t seq Ω ,R /h for different N , ω z , andΩ ,R , respectively.Figure 2(a) shows numerical results for Ω ,R = E R andthree different N , namely N = 1, 3 × , and N = 10 .Even though the Rabi coupling lasers are turned on, attime t = 0, after the trapping potential has been switchedoff, the figure caption quotes the trapping frequenciessince they determine the initial state and thus the distri-bution of the kinetic and potential energy, including themean-field energy, in the system. For the non-interactingsingle-atom system [black solid line in Fig. 2(a)], the Rabioscillation period is nearly constant for the times consid-ered; the amplitude, however, is visibly damped. Whilethis “non-perfect” sinusoidal behavior might be surpris-ing at first sight, it can be explained as follows: Thecenter of mass of the component ˜ ψ b ( r , t ) moves relative -1-0.500.51 N a - N b -1-0.500.51 N a - N b seq Ω / h-1-0.500.51 N a - N b (a)(b)(c) FIG. 2: Rabi oscillations for Raman coupling case (numericalresults). The lines show the difference N a − N b between thefractional populations as a function of the dimensionless time t seq Ω ,R /h for t start = 0, E R /h = 1960 Hz, δ R = 0, and ω ρ = 2 π ×
200 Hz. (a) Changing the particle number N . Theblack solid, red dashed, and blue dotted lines are for N = 1, N = 3 × , and N = 10 , respectively. The weak trappingfrequency is ω z = 2 π ×
40 Hz and the coupling strength isΩ ,R = E R . (b) Changing the angular trapping frequency ω z . The black solid, red dashed, and blue dotted lines are for ω z = 2 π ×
10 Hz, ω z = 2 π ×
20 Hz, and ω z = 2 π ×
60 Hz,respectively. The coupling strength is Ω ,R = E R and thenumber of particles is N = 3 × . (c) Changing the couplingstrength Ω ,R . The black solid, red dashed, and blue dottedlines are for Ω ,R = E R /
2, Ω ,R = 3 E R /
2, and Ω ,R = 5 E R / N = 3 × and theweak trapping frequency is ω z = 2 π ×
40 Hz. to the center of mass of the component ˜ ψ a ( r , t ) duringthe Rabi oscillations. Thus, the two components arenot perfectly overlapping spatially. As a consequence,the relative phase of the spinor components at fixed r ischanging slightly due to the relative motion of the twocomponents with respect to each other. This phase differ-ence is responsible for the non-perfect population trans-fer (“damping”). An alternative but equivalent picture isthat the finite momentum width of the initial state cor-responds to a small effective momentum-dependent de-tuning. This effective detuning decreases with increasingmean-field interactions due to the decrease of the widthof the initial state in momentum space.When mean-field interactions are present [the reddashed and blue dotted lines in Fig. 2(a) are for N =3 × and N = 10 , respectively], the amplitude of theRabi oscillation data changes somewhat while the oscil-lation period is essentially unaffected by the interactions.For these two N -values, the chemical potential (in unitsof h ) of the initial state is ≈ . E R and 2 . E R ), i.e., the chemicalpotential at t = 0 is larger than Ω ,R .To highlight the effect of the interactions, Fig. 2(b)shows the Raman-induced Rabi oscillations for N = 3 × [same atom number as used for the red dashed line inFig. 2(a)] for weaker and stronger confinement along the z -direction than used in Fig. 2(a). Stronger confinementleads to higher density and thus to enhanced interactioneffects. For the largest ω z considered [blue dotted linein Fig. 2(b)], the fractional population difference N a − N b deviates appreciably from a simple sinusoidal curveafter a few oscillations. This indicates that care needs tobe taken when calibrating the effective Raman couplingstrength Ω ,R ; in particular, a fit to a simple sinusoidalfunction, applicable to the ideal two-level model, mightnot yield the correct effective coupling strength.The results presented in Figs. 2(a)-2(b) are for Ω ,R = E R . Figure 2(c) shows the dynamics for smaller andlarger coupling strengths, namely Ω ,R = E R /
2, Ω ,R =3 E R /
2, Ω ,R = 5 E R /
2, and the same trap confinementas in Fig. 2(a). Even though the particle number isquite moderate (namely, N = 3 × ), the oscillationamplitude and period for Ω ,R = E R / ,R . This implies that care needs to beexercised if the calibration of the effective Raman cou-pling strength is done for low coupling strengths. In thisregime, one has to make sure that the particle numberis sufficiently low or that one allows for sufficient time-of-flight expansion prior to turning on the Raman Rabicoupling (Fig. 2 is for t start = 0). If this is not done,the value of the effective Rabi coupling strength, whichenters into the underlying system Hamiltonian, may beimpacted by interaction effects, potentially leading to er-rors in experiments that require reliable precision, suchas quantum analog simulations and many-body studies.Alternatively, explicit comparisons with Gross-Pitaevskiiequation results, as done in this work, would be very use-ful when interactions are present. Last, one may considerperforming the calibration in the “large power” regimeand extrapolating the resulting calibration curve insteadof performing the calibration in the “low power” regime. C. Rabi oscillations: Theory-experimentcomparison
Figure 3 shows a comparison between theory and ex-periment for the Raman-induced Rabi oscillations for aninitial state with chemical potential µ = 0 . E R . Thischemical potential corresponds to a mean-field energy perparticle at t = 0 of 0 . E R . To reduce the mean-fieldenergy in the system, a 0 . t = 0. The resulting Rabi oscillations areslightly damped. Although the experimental data (reddots) are obtained for a small negative detuning δ R , thedetuning is not the only cause for the damping. Extrap-olating the mean-field Gross-Pitaevskii results for finitedetuning to zero detuning, we conclude that even the zerodetuning case displays damping [explicit calculations for δ R = 0 (not shown) confirm this]. Combining the goodagreement between the experimental data and theoreti-cal curves with the discussion of the previous section, weconclude that the damping can be partially attributed tothe mean-field interactions. Indeed, if we let the BECexpand longer prior to turning on the Raman coupling,the damping or dephasing, for the same detuning δ R , isreduced.Interestingly, fitting of the mean-field Gross-Pitaevskiiresults for δ R /h = −
200 Hz (this detuning gives the bestagreement with the experimental data) to a damped pe-riodic function of the form N a − N b = cos(2 πf t ) exp( − t/τ ) (16)yields a frequency f of 2578 Hz. This “fitted Rabi cou-pling strength” is about 1.2 % larger than the Rabi cou-pling strength Ω ,R used in the simulations. This in-dicates that the interactions impact, for the parametercombinations considered, the oscillation frequency muchless than the amplitude. More specifically, for the param-eter combination considered in Fig. 3, the effect of theinteractions on the Rabi oscillations can be described, toa good approximation, phenomenologically by the timeconstant τ . D. Ramsey-type pulse sequence: Theory overview
Throughout this section, the initial BEC ( N = 3 × )is prepared for a confinement with ω ρ = 2 π ×
200 Hzand ω z = 2 π ×
40 Hz. The dynamics are analyzedfor the Ramsey-type pulse sequence with Raman cou-pling strength Ω ,R = E R , detuning δ R = 0, and—asin Sec. III B— E R /h = 1960 Hz and t start = 0. Themain emphasis lies on developing, motivated by numer-ical simulations of the time-dependent mean-field equa-tion for the Hamiltonian given in Eq. (14), a benchmarkand physical picture that provides the motivation for theanalytical treatment presented in Sec. III E.When the Raman coupling is turned on, populationis transferred from the component ˜ ψ a ( r , t ) to the com-ponent ˜ ψ b ( r , t ). As discussed in the previous sections,the interactions can notably impact the Rabi oscillations,in particular for longer times. Despite of this, we mea-sure the lengths of our pulses in terms of the charac-teristic time scale of the non-interacting system, i.e., werefer to a π/ N a - N b FIG. 3: Theory-experiment comparison for Raman Rabi os-cillations for a Rb BEC. The red dots show the experi-mentally determined difference in the fractional populationsas a function of time. The experimental parameters are ω x = 2 π ×
155 Hz, ω y = 2 π ×
179 Hz, ω z = 2 π × . N = 1 . × , E R /h = 1960 Hz, Ω ,R /h = 2548 Hz,and t start = 0 . δ R /h that is estimated to be be-tween −
200 Hz and −
600 Hz, where the uncertainty is dueto fluctuations in the external magnetic field responsible forthe Zeeman splitting. The solutions to the mean-field Gross-Pitaevskii equation (lines) are obtained for an axially sym-metric trap characterized by the experimentally measured ω z and ω ρ = 2 π × . ω ρ is taken to be the mean of theexperimental ω x and ω y ). The black dotted, green solid, andblue dashed lines show results for δ R /h = −
200 Hz, −
400 Hz,and −
600 Hz, respectively; the other parameters are takenfrom experiment. The results for δ R /h = −
200 Hz describethe experimental data the best. The chemical potential µ prior to turning off the trap is 0 . E R . The mean-fieldenergy per particle prior to turning off the trap and after the0 . . E R and 0 . E R , respectively. the state ˜ ψ a ( r , t ) to the state ˜ ψ b ( r , t ). For the param-eters employed in Fig. 4, a π/ πh/ (2Ω ,R ) ≈ . ψ b ( r , t ) after the first π/ .
95 %). In addition, it can beseen that the population in the component ˜ ψ b ( r , t ) movesa tiny bit relative to the population in the component˜ ψ a ( r , t ) during the first π/ ψ b ( r , t ) has an average mechani-cal momentum of about 2 (cid:126) k R along the z -direction whilethe population in state ˜ ψ a ( r , t ) has an average mechani-cal momentum very close to zero along the z -direction.During the variable hold time, no population trans-fer occurs since the Raman coupling lasers are turnedoff. The two key characteristics during the hold timeare: First, the population in state ˜ ψ b ( r , t ) continues tomove relative to that in component ˜ ψ a ( r , t ). Second, theinteracting BEC expands a tiny bit. Figures 4(aii) and4(aiii) show ρ = 0 cuts and are for t hold = 1 ms and t hold = 2 ms, respectively.To “reunite” the populations of the states ˜ ψ a ( r , t ) and˜ ψ b ( r , t ), a second π/ ψ a ( r , t ) after the second π/ .
74 % for t hold = 1 ms and 50 .
15 %for t hold = 2 ms. After the second π/ ρ = 0 cuts] show inter-ference fringes in the “central” or “overlap” region, i.e.,in the spatial region where the two clouds overlappedprior to the application of the second π/ | ˜ ψ a ( z, | and | ˜ ψ b ( z, | . We ob-serve analogous fringes for other ρ values. The total den-sity of the two components (not shown), in contrast, ex-hibits oscillations with comparatively small amplitude inthe outer region and no oscillations in the central region.Quite generally, the appearance of fringes such as thosedisplayed in Figs. 4(aiv) and 4(av) suggests the existenceof two interfering pathways, i.e., the existence of a spa-tially dependent phase difference. In the following, weintroduce a theoretical framework that highlights howthe fringe pattern develops for the Ramsey-type pulsesequence with Raman coupling. To this end, it is in-structive to visualize the time-evolving rotated spinor onthe Bloch sphere. Since the two components ˜ ψ a ( r , t ) and˜ ψ b ( r , t ) can each be written in terms of one complex num-ber for each z , ρ , and t (the axial symmetry suggests theuse of cylindrical coordinates), we define (cid:18) ˜ ψ a ( z, ρ, t )˜ ψ b ( z, ρ, t ) (cid:19) = R ( z, ρ, t ) exp [ ı ˜ γ a ( z, ρ, t )] × (cid:32) cos (cid:0) θ ( z,ρ,t )2 (cid:1) exp [ ıφ ( z, ρ, t )] sin (cid:0) θ ( z,ρ,t )2 (cid:1)(cid:33) , (17)where R ( z, ρ, t ) = (cid:113) | ˜ ψ a ( z, ρ, t ) | + | ˜ ψ b ( z, ρ, t ) | , (18) θ ( z, ρ, t ) = 2 arctan (cid:32) | ˜ ψ b ( z, ρ, t ) || ˜ ψ a ( z, ρ, t ) | (cid:33) , (19)and φ ( z, ρ, t ) = ˜ γ b ( z, ρ, t ) − ˜ γ a ( z, ρ, t ) . (20)Here, ˜ γ a ( z, ρ, t ) can be interpreted as an overall spa-tially dependent phase of the spinor wave function. Thisphase has no effect on the physical observables consid-ered in this work. The quantity R ( z, ρ, t ) corresponds toa “weight” at each spatial point. The spinor dynamicsfor a given z and ρ is thus conveniently visualized by avector of length R ( z, ρ, t ) on the Bloch sphere. The di-rection of the vector is given by θ ( z, ρ, t ) and the relativephase φ ( z, ρ, t ) between the components ˜ ψ b ( z, ρ, t ) and˜ ψ a ( z, ρ, t ).To visualize the motion of the spinor on the Blochsphere, we define the local spin expectation values σ j ( z, ρ, t ), where j = x , y , or z , through σ j ( z, ρ, t ) = (cid:18) [ ˜ ψ a ( z, ρ, t )] ∗ [ ˜ ψ b ( z, ρ, t )] ∗ (cid:19) T ˆ σ j (cid:18) ˜ ψ a ( z, ρ, t )˜ ψ b ( z, ρ, t ) (cid:19) , (21) | ~ ψ a / b ( z , , t ) | / k R ( - ) | ~ ψ a / b ( z , , t ) | / k R ( - ) -200 0 200z k R | ~ ψ a / b ( z , , t ) | / k R ( - ) -8-40 σ y / z ( z , , t ) / k R ( - ) -202 σ y / z ( z , , t ) / k R ( - ) -101-202 σ z ( z , , t ) / k R ( - ) -200 0 200z k R -101 024 | ~ ψ a / b ( z , , t ) | / k R ( - ) | ~ ψ a / b ( z , , t ) | / k R ( - ) -200 0 200z k R | ~ ψ a / b ( z , , t ) | / k R ( - ) -8-40 σ y / z ( z , , t ) / k R ( - ) -202 σ y / z ( z , , t ) / k R ( - ) -101-202 σ z ( z , , t ) / k R ( - ) -200 0 200z k R -101 (ai) t= τ (aii) t= τ +1ms(aiii) t= τ +2ms(aiv) t= τ +1ms+ τ (av) t= τ +2ms+ τ (avi)(avii)(aviii)(aix)(ax) (bi) t= τ (bii) t= τ +1ms(biii) t= τ +2ms(biv) t= τ +1ms+ τ (bv) t= τ +2ms+ τ (bvi)(bvii)(bviii)(bix)(bx) mean-field Gross-Pitaevskii results analytical results (Section IIIE) FIG. 4: Density cuts and local spin expectation values for the Ramsey-type pulse sequence with E R /h = 1960 Hz, Ω ,R = E R ,and δ R = 0 (theory results). The Rb BEC consists of N = 3 × atoms and is prepared in an axially symmetric trap with ω ρ = 2 π ×
200 Hz and ω z = 2 π ×
40 Hz. This corresponds to a chemical potential, in units of h , of ≈ t start = 0. The first and second columns are obtained by solving the time-dependent mean-field equation for theHamiltonian given in Eq. (14) numerically. The third and fourth columns show the same observables as the first and secondcolumns but are, instead, calculated using the fully analytical framework developed in Sec. III E; the agreement is very good.The black solid and red dashed lines in panels (ai)-(av) show the density cuts | ˜ ψ a ( z, , t ) | and | ˜ ψ b ( z, , t ) | , respectively. Theblack solid and red dashed lines in panels (avi)-(ax) show the local spin expectation values σ y ( z, , t ) and σ z ( z, , t ), respectively.The time increases from the first row, to the second/third row, to the fourth/fifth row (the value of the time is given in thepanels); the second and fourth row correspond to a hold time of 1 ms, and the third and fifth row correspond to a hold time of2 ms. It can be seen that the agreement between the mean-field Gross-Pitaevskii results and the fully analytical results is quitegood. The figure illustrates that the second π/ σ y ( z, , t ) to σ z ( z, , t ), making theinterference visible in the population difference. where ˆ σ x , ˆ σ y , and ˆ σ z denote the “usual” Pauli matri-ces. Note that we use the term “spin expectation value”for convenience throughout this paper even though ourdefinition in Eq. (21) excludes the conventional (cid:126) / σ z ( z, ρ, t ) corresponds to the local ( z, ρ )-specific population difference at time t . Mathematically,one finds σ x ( z, ρ, t ) = | R ( z, ρ, t ) | cos( φ ( z, ρ, t )) sin( θ ( z, ρ, t )) , (22) σ y ( z, ρ, t ) = | R ( z, ρ, t ) | sin( φ ( z, ρ, t )) sin( θ ( z, ρ, t )) , (23) and σ z ( z, ρ, t ) = | R ( z, ρ, t ) | cos( θ ( z, ρ, t )) . (24)The initial state at t = 0 corresponds to a vectorpointing to the north pole on the Bloch sphere. Fromthe Hamiltonian ˆ˜ H [Eq. (14)], it can be seen that thenon-vanishing Raman coupling term introduces a torquealong the positive x -axis. Thus, neglecting interactions,the first π/ − π/ x -axis on the Bloch sphere. As a re-sult, the spinor points along the negative y -axis on theBloch sphere after the first π/ θ and φ are approxi-mately equal to π/ − π/ π/ σ z ( z, , t ) is approximately zeroand σ y ( z, , t ) has—except for a minus sign—the same z -dependence as the density [red dashed and black solidlines Fig. 4(avi)]. We conclude that mean-field effectscan, in a first-order approximation, be neglected duringthe first π/ ψ b ( r , t ) moves relative to that in state˜ ψ a ( r , t ). Second, the phases of the spinor components˜ ψ a ( z, ρ, t ) and ˜ ψ b ( z, ρ, t ) evolve independently. For each( z, ρ ), the combination of these two effects leads to a ro-tation of the two-component spinor on the Bloch sphere.As an example, Figs. 4(avii)-4(aviii) show σ z ( z, , t ) and σ y ( z, , t ) for two different hold times. For both holdtimes, σ z ( z, , t ) changes approximately linearly with z in the region where the two components overlap. Thisfollows immediately from the approximately parabolicshapes of the two density components, which are offsetfrom each other: the difference leads to a term that is,to leading order, linear in z . The local spin expectationvalue σ y ( z, , t ) develops “wiggles” during the hold timein the region where the two components overlap. Thenumber of wiggles increases with increasing hold time.The wiggles arise from the relative phase dynamics andindicate interference; importantly, the densities do notshow any indication of interference prior to the applica-tion of the second π/ π/ σ y ( z, ρ, t ) to σ z ( z, ρ, t ).This can be seen clearly by comparing Fig. 4(aix) withFig. 4(avii) (both these figures are for a hold time of1 ms) and by comparing Fig. 4(ax) with Fig. 4(aviii)(both these figures are for a hold time of 2 ms). Sincethe relative phase information has been “moved” from σ y ( z, ρ, t ) to σ z ( z, ρ, t ) and since σ z ( z, ρ, t ) is equal to | ˜ ψ a ( z, ρ, t ) | − | ˜ ψ b ( z, ρ, t ) | , the interference is—after thesecond π/ E. Ramsey-type pulse sequence: Analyticaltreatment
The numerical results presented in the previous sectionare obtained using the scattering lengths a ij for Rb.Repeating the numerical calculations for equal scatteringlengths a ij reveals that the effects due to the difference inthe scattering lengths are quite small for the time scalesconsidered in this paper. Motivated by this observation,the analytical treatment presented in this section makesthe simplifying assumption that the scattering lengthsare all equal ( a aa = a ab = a ba = a bb ). Moreover, thetreatment assumes that δ R and t start are equal to zero. Our analytical model is motivated by Refs. [14, 15]; how-ever, the application to the Ramsey-type pulse sequencediscussed here has, to the best of our knowledge, not beendiscussed previously.We assume that the initial state ˜ ψ a ( r , t = 0) is de-scribed well within the Thomas-Fermi approximation.We additionally assume that the population transfer pro-cess commutes with the relative moving and expansionprocesses during the first π/ π/ τ of theactual π/ π/ τ and then treat the population transfer as-sociated with the second π/ ψ a ( z, ρ, t )and ˜ ψ b ( z, ρ, t ) right after the first instantaneous π/ t = 0 + ), during the time 0 + < t < τ + t hold + τ ,and right after the second instantaneous π/ t = t +end ).Motivated by the discussion in Sec. III B, we make theansatz that | ˜ ψ a ( r , t ) | has an inverted parabola-like formduring the “effective” hold time, i.e., for 0 + < t < t end , | ˜ ψ a ( r , t ) | =1 λ z ( t ) λ ρ ( t ) (cid:34) − α z (cid:18) zλ z ( t ) (cid:19) − α ρ (cid:18) ρλ ρ ( t ) (cid:19) + µ g (cid:35) , (25)where µ denotes the chemical potential of the initialstate, i.e., of the system prior to the application of thefirst π/ µ = 12 (cid:34) ω ρ ω z m (cid:18) g π (cid:19) (cid:35) / , (26)and α z/ρ = mω z/ρ g . (27)Compared to the “standard case” [14], α ρ and α z aresmaller by a factor of 2 since the population for t = 0 + isassumed to be equally distributed between the two com-ponents, i.e., | ˜ ψ b ( r , + ) | = | ˜ ψ a ( r , + ) | . In Eq. (25), it isunderstood that | ˜ ψ a ( r , t ) | is zero when the right hand sideof the equation takes negative values. The dimensionlessscaling parameters λ z ( t ) and λ ρ ( t ) obey the initial con-ditions λ z (0 + ) = λ ρ (0 + ) = 1. The differential equationsthat govern the time evolution of λ z ( t ) and λ ρ ( t ) are dis-cussed below. We set ˜ γ a ( r , + ) = 0 and assume that thefirst π/ − π/ γ b ( r , + ) = − π/ + < t < t end , ˜ ψ b ( r , t ) moves with approximatelyconstant velocity v z , v z = 2 (cid:126) k R m , (28)along the z -direction while the center-of-mass of | ˜ ψ a ( r , t ) | remains essentially unchanged. Due to thesymmetry of the system, we enforce˜ ψ b ( r , t ) = ˜ ψ a ( v z t ˆ e z − r , t ) exp (cid:16) − ı π (cid:17) . (29)Thus, once we have expressions for | ˜ ψ a ( v z t ˆ e z − r , t ) | and˜ γ a ( v z t ˆ e z − r , t ), ˜ ψ b ( r , t ) is determined through Eq. (29).To eliminate ˜ ψ b ( r , t ), we insert Eq. (29) into the coupledset of time-dependent mean-field equations. This yields ı (cid:126) ∂∂t ˜ ψ a ( r , t ) = ˆ˜ H hold ˜ ψ a ( r , t ) , (30)whereˆ˜ H hold = ˆ p m + g | ˜ ψ a ( r , t ) | + g | ˜ ψ a ( v z t ˆ e z − r , t ) | . (31)From Eq. (25), we find | ˜ ψ a ( v z t ˆ e z − r , t ) | = | ˜ ψ a ( r , t ) | + 2 α z v z tzλ z ( t ) λ ρ ( t ) − α z v z t λ z ( t ) λ ρ ( t ) . (32)Plugging Eq. (32) into Eq. (31), we obtainˆ˜ H hold = ˆ p m + 2 g | ˜ ψ a ( r , t ) | − F z ( t ) z + C ( t ) . (33)Equation (33) implies that C ( t ), which is independent of r , contributes an overall phase to ˜ ψ a ( r , t ) at each time t . The effective time-dependent force F z ( t ) along thenegative z -direction, F z ( t ) = − gα z v z tλ z ( t ) λ ρ ( t ) , (34)is due to the relative motion of the two components withrespect to each other and the mean-field interactions.We now make the assumption that the effective forceterm in Eq. (33) does not notably affect the time evo-lution of the density | ˜ ψ a ( r , t ) | . Under this assumption,the time evolution of the scaling factors λ z ( t ) and λ ρ ( t ) isgoverned by the differential equations derived by Castinand Dum from the scaling ansatz for a single-componentBEC [14]: d λ z ( t ) dt = ω z λ ρ ( t ) λ z ( t ) , (35)and d λ ρ ( t ) dt = ω ρ λ ρ ( t ) λ z ( t ) . (36) The black solid lines in Fig. 5(a) show λ ρ ( t ) and λ z ( t ),obtained by solving Eqs. (35)-(36) numerically for thesame parameters as those employed in Fig. 4. Using thesesolutions, the solid line in Fig. 5(b) shows the effectiveforce F z ( t ) as a function of the dimensionless time tE R /h .The magnitude of the effective force first increases andthen decreases with increasing time. As shown below,the turn-around time is, to leading order, given by theinverse of the transverse trapping frequency ω ρ .While we assumed that the time dynamics of | ˜ ψ a ( r , t ) | is largely independent of the effective force F z ( t ), we de-duce from Sec. III D that the time evolution of the phase˜ γ a ( r , t ) is non-negligibly impacted by F z ( t ). Accordingto the momentum-impulse relationship, the impulse I z ( t )imparted by the effective force on the system at time t reads I z ( t ) = (cid:90) t F z ( τ ) dτ. (37)The black solid line in Fig. 5(c) shows that the mag-nitude of I z ( t ) increases monotonically with increasingeffective hold time. Using that the change of the momen-tum during the hold time is equal to I z ( t ), we estimatethat the effective force F z ( t ) changes the phase ˜ γ a ( r , t )by φ lin ( z, t ), φ lin ( z, t ) = I z ( t ) (cid:126) z, (38)where 0 + < t < t end . We refer to this phase as “linearphase” since it depends linearly on z . It vanishes in thelimit that the population of state ˜ ψ b ( r , t ) does not moverelative to that in state ˜ ψ a ( r , t ).The expansion of the two components for 0 + < t 5, which corresponds to ω ρ t ≈ . ψ a ( r , t )and ˜ ψ b ( r , t ), we are ready to compare the spin dynamicsobtained within this Thomas-Fermi approximation-likeframework to the spin dynamics obtained by solving theGross-Pitaevskii equation numerically. To this end, thethird and fourth columns of Fig. 4 show the same ob-servables as the first and second columns. While the firstand second columns are obtained—as discussed in de-tail in Sec. III D—by analyzing the solutions to the fullGross-Pitaevskii equation, the third and fourth columnsare obtained using our fully analytical solutions derivedabove. A quick comparison indicates that the overallagreement is strikingly good. This a posteriori justifiesthe assumptions made in developing the analytical frame-work presented in this section. Most importantly, thegood agreement allows us to unambiguously state that λ z , λ ρ -0.002-0.0010 F z ( t ) / ( E R k R ) R / h-0.06-0.04-0.020 I z ( t ) / ( _hk R ) (a)(b)(c) FIG. 5: Characteristics of the analytical framework discussedin Sec. III E: Scaling parameters, effective force, and impulseduring the hold time of the Raman-Ramsey-type pulse se-quence for a Rb BEC. Results are shown for E R /h =1960 Hz, Ω ,R = E R , δ R = 0, N = 3 × , ω ρ = 2 π × 200 Hz,and ω z = 2 π × 40 Hz. (a) The red dashed and blue dash-dotted lines show the scaling parameters λ z ( t ) and λ ρ ( t ), re-spectively, obtained using the formula given in Eqs. (41) and(42); note that the vertical axis employs a logarithmic scale.(b) The red dashed line shows the effective force F z ( t ) calcu-lated using our analytical expressions for the scaling param-eters in Eq. (34). (c) The red dashed line shows the impulse I z ( t ) calculated using our analytical expressions for the scal-ing parameters in Eq. (37). For comparison, the solid linesin panels (a)-(c) show results obtained by numerically solvingthe differential equations for λ ρ ( t ) and λ z ( t ). The excellentagreement between the analytical and numerical results val-idates the use of the analytical expressions for the scalingparameters. both the linear phase and the quadratic phase need tobe accounted for to obtain a faithful description of theinterference fringes.Section IV D returns to the theoretical framework de-veloped in this section. The analytical framework for theRaman Ramsey-type sequence, which relies heavily onthe assumption that all four coupling strengths g ij are(approximately) equal to each other, cannot be applieddirectly to the lattice Ramsey-type sequence since thecorresponding two-state model is described by couplingstrengths g aa = g bb = g ab / g ba / 2. Despite of this,it is argued in Sec. IV D that the model developed hereprovides important insights for the lattice coupling caseas well.2 IV. LATTICE COUPLING CASEA. Two-state model For the lattice case, all atoms occupy the same hyper-fine state; our calculations reported below are for Rbatoms in the | F, m F (cid:105) = | , − (cid:105) state. Assuming a singlemean-field wave function Φ( r , t ), the Hamiltonian ˆ H tobe used in the time-dependent Gross-Pitaevskii equationreads [28]ˆ H = ˆ p m + V trap ( r , t ) + V lat ( r , t ) + g aa | Φ( r , t ) | , (44)where the one-dimensional moving lattice potential V lat ( r , t ) is given by V lat ( r , t ) = 2Ω L ( t ) cos (cid:16) k L z − ω L t (cid:17) (45)and Φ( r , t ) is normalized according to (cid:82) | Φ( r , t ) | d r =1. Figure 6(a) shows the density | Φ( r , t ) | , obtained bysolving the time-dependent mean-field equation for theHamiltonian given in Eq. (44) for typical experimentalparameters, as a function of z for ρ = 0 and a timecorresponding to a π/ t = πh/ (2Ω ,L ). Inthis example, the BEC is prepared in the ground state ofthe harmonic trap. At time t = 0, the trapping potentialis turned off and the lattice with E L /h = 1960 Hz, ω L =4 E L / (cid:126) and Ω ,L = E L is flashed on for 0 . ,L is chosen to be comparable tothe chemical potential µ of the BEC at t = 0 (Ω ,L ≈ . µ ). The size of the BEC does not change notablyduring the duration of the lattice pulse: it extends overapproximately 80 lattice sites. Figure 6(a) shows thatthe lattice pulse “imprints” fine oscillations along the z -direction onto the mean-field density.To facilitate the analysis, it is desirable to bring out theintrinsic dynamics by rotating the lattice induced oscil-lations away. As we discuss in the next paragraphs, thiscan be accomplished within the framework of an approxi-mate two-state model, which assumes that the BEC onlyoccupies momenta along the z -direction near (cid:126) k z = 0 and (cid:126) k z = 2 (cid:126) k L and not near n (cid:126) k L with n = − , ± , ± , · · · .This assumption is well justified for the example shown inFig. 6(a). The density cut in momentum space [Fig. 6(b)]shows peaks centered near (cid:126) k z = 0 and (cid:126) k z = 2 (cid:126) k L ; thepopulations of these peaks are 65 . 46 % and 33 . 79 %, re-spectively. Since the peaks centered near (cid:126) k z = − (cid:126) k L and (cid:126) k z = 4 (cid:126) k L have tiny populations (0 . 665 % and0 . 084 %, respectively), the two-state model developedbelow is expected to capture the dynamics of this sys-tem semi-quantitatively. More generally, the applicabil-ity of the two-state model requires that the lattice pulseor pulses are sufficiently short and sufficiently weak. Thetwo-state model introduced below can be improved sys-tematically by accounting for successively more “momen-tum components”, i.e., by increasing the number of n values included in Eq. (46). In the limit of an infinite -100 0 100z k L | φ ( z , , τ ) | / k L ( - ) -2 0 2 4k z / k L -6 -4 -2 | φ k z ( k z , , τ ) | / k L (a)(b) FIG. 6: Rb BEC density after the application of a π/ zk L (real space) and k z /k L (momentum space), respectively, for t start = 0, N = 3 × , E L /h = 1960 Hz (corresponding to k L = 5 . µ m − ),Ω ,L = E L , δ L = 0, ω ρ = 2 π × 200 Hz, and ω z = 2 π × 40 Hz.The real space density cut | Φ( z, , τ ) | is governed by fine os-cillations that are related to the fact that the BEC contains,after the application of the lattice pulse, non-zero momen-tum components. The momentum space cut [ | Φ k z ( k z , , τ ) | is obtained by taking the square of the Fourier transform ofΦ( z, , τ )] shows that the BEC density is governed by mo-menta centered around (cid:126) k z ≈ (cid:126) k z ≈ (cid:126) k L . state model that accounts for all n ( n = 0 , ± , · · · ) thedescription is equivalent to that captured by the originalmean-field Hamiltonian [Eq. (44) with V lat ( r , t ) given byEq. (45)].To derive the two-state model, we make the ansatz [32,33] Φ( r , t ) = ˜ ψ a ( r , t ) + ˜ ψ b ( r , t ) exp(2 ık L z ) , (46)where ˜ ψ a ( r , t ) and ˜ ψ b ( r , t ) are assumed to be localized inthe vicinity of the momenta (cid:126) k z = 0 and (cid:126) k z = 2 (cid:126) k L ,respectively. The functions ˜ ψ a ( r , t ) and ˜ ψ b ( r , t ) are nor-malized according to Eq. (8) and Eq. (9) with ψ j ( r , t )replaced by ˜ ψ j ( r , t ). Since the widths of the momen-tum distributions associated with the states ˜ ψ a ( r , t ) and˜ ψ b ( r , t ) are assumed to be narrow compared to 2 (cid:126) k L [thisis, indeed, the case for the example shown in Fig. 6(b)],we demand that the “separation condition” (cid:90) exp( ı k L z ) ˜ ψ a ( r , t ) (cid:104) ˜ ψ b ( r , t ) (cid:105) ∗ d r = 0 (47)3holds.Following the standard mean-field approach, we writethe N -body wave function as a product over single-particle orbitals, namely as Φ( r , t )Φ( r , t ) · · · Φ( r N , t ).Variation of the energy functional with respect to [ ˜ ψ a ( r , t )] ∗ and [ ˜ ψ b ( r , t )] ∗ then yields two coupled non-linear equations, namely Eq. (12) with ˆ H replaced byˆ˜ H , whereˆ˜ H = (cid:18) ˆ p m + V trap ( r , t ) (cid:19) ⊗ I + (cid:18) g | ˜ ψ a ( r , t ) | + 2 g | ˜ ψ b ( r , t ) | 00 2 g | ˜ ψ a ( r , t ) | + g | ˜ ψ b ( r , t ) | (cid:19) + (48) (cid:32) Ω L ( t )2Ω L ( t )2 2 (cid:126) k L ˆ p z m + δ L (cid:33) . In deriving Eq. (48), we assumed that integrals such as (cid:90) | ˜ ψ j ( r , t ) | exp( ± nık L z ) d r (49)( n = 2 , , · · · ), which have rapidly oscillating integrands,vanish. This means that portions of the kinetic energy,lattice potential, trap potential, and mean-field energycontributions are neglected in Eq. (48).Comparison of the approximate two-state latticeHamiltonian [Eq. (48)] and the rotated Raman Hamil-tonian [Eq. (14)] shows that the two Hamiltonians agreeif we enforce that E R , Ω R ( t ), and δ R are equal to E L ,Ω L ( t ), and δ L , respectively and if additionally the fol-lowing holds: g aa = g bb = g and g ab = 2 g . For the F = 1 hyperfine manifold of Rb, the mean-field interac-tions of the two Hamiltonians do not agree since we have g aa ≈ g bb ≈ g ab . Consequently, the dynamics for the Ra-man coupled and lattice coupled systems are expected todiffer even if the single-particle coupling mechanisms arecharacterized by matching parameters. In what follows,we will focus on the interaction-induced differences.Since the population in state ˜ ψ a ( r , t ) [ ˜ ψ b ( r , t )] expe-riences a mean-field interaction due to the populationin state ˜ ψ b ( r , t ) [ ˜ ψ a ( r , t )] that is about two times largerfor the lattice coupled Hamiltonian than for the Ramancoupled Hamiltonian, the lattice coupled system has astronger tendency to phase separate than the Raman cou-pled system (this argument uses the fact that g is positivefor the F = 1 hyperfine manifold of Rb). Phase sepa-ration has been discussed in the literature in the contextof multi-component BECs [29]. The framework devel-oped here may also provide an intuitive understandingof the formation of the ferromagnetic domains observedin Ref. [30].The difference between the Raman and lattice couplingcases can also be interpreted from an alternative view-point. To this end, we rewrite the mean-field terms fromEq. (48) as g | ˜ ψ a ( r , t ) | + 2 g | ˜ ψ b ( r , t ) | = g eff [ | ˜ ψ a ( r , t ) | + | ˜ ψ b ( r , t ) | ] − g | ˜ ψ a ( r , t ) | (50) and g | ˜ ψ b ( r , t ) | + 2 g | ˜ ψ a ( r , t ) | = g eff [ | ˜ ψ a ( r , t ) | + | ˜ ψ b ( r , t ) | ] − g | ˜ ψ b ( r , t ) | , (51)where g eff is defined to be equal to 2 g . The right handsides of Eqs. (50) and (51) suggest that the differencebetween the lattice and Raman coupling cases is due totwo things: First, g aa , g bb , and g ab can be identified to beequal to g eff , suggesting that the lattice coupled system ischaracterized by a two times stronger repulsion than theRaman coupled system. Second, there exists an effectiveon-site attraction in the two-state model of the latticecoupled system of strength − g [31] that has no analog inthe Raman coupled system.We emphasize that the two interpretations introducedabove are consistent with the scattering diagram argu-ments outlined in Sec. I. The “factor of 2” in the second2 × B. Rabi oscillations: Theory overview This section discusses lattice coupling induced Rabi os-cillations. Figure 7 compares the mean-field results forthe full lattice Hamiltonian [Eqs. (44) and (45); solidblack lines] with those obtained using the approximatetwo- and four-state Hamiltonians (red dashed and bluedotted lines, respectively). For all 9 parameter combi-nations considered in Fig. 7, the four-state model re-produces the dynamics obtained for the full mean-fieldlattice Hamiltonian extremely well. While the two-statemodel results deviate somewhat from the results for thefull lattice Hamiltonian, the two-state model captures themain features of the Rabi oscillations such as the changeof the damping of the Rabi oscillations with increasingnumber of particles [Figs. 7(ai)-(aiii)] and with increas-4 -101 N a - N b -101 N a - N b seq Ω 0, L / h-101 N a - N b seq Ω 0, L / h 0 1 2 3t seq Ω 0, L / h (ai) N = 1 (aii) N = 3 x 10 (aiii) N = 10 (bi) ω z = 2 π x 10 Hz (bii) ω z = 2 π x 20 Hz (biii) ω z = 2 π x 60 Hz(ci) Ω = E L /2 (cii) Ω = 3E L /2 (ciii) Ω = 5E L /2 FIG. 7: Rabi oscillations for lattice coupling case (numerical results). The lines show the difference N a − N b between thefractional populations as a function of the dimensionless time t seq Ω ,L /h for t start = 0, E L /h = 1960 Hz, δ L = 0, and ω ρ = 2 π × 200 Hz. The black solid, red dashed, and blue dotted lines show results obtained by solving the time-dependentmean-field equation for the full lattice Hamiltonian [Eqs. (44) and (45)], the approximate two-state Hamiltonian [Eq. (48)], andthe approximate four-state Hamiltonian (this Hamiltonian is not written out explicitly in the text). The black solid and bluedotted lines nearly coincide (in particular, the blue dotted lines are hardly visible on the scale shown). (ai)-(aiii) Changingthe particle number N (the values are given in the panels). The weak angular trapping frequency is ω z = 2 π × 40 Hz and thecoupling strength is Ω ,L = E L . (bi)-(biii) Changing the angular trapping frequency ω z (the values are given in the panels).The coupling strength is Ω ,L = E L and the number of particles is N = 3 × . (ci)-(ciii) Changing the coupling strength Ω ,L (the values are given in the panels). The number of particles is N = 3 × and the weak trapping frequency is ω z = 2 π × 40 Hz. ing strength of the weak trapping frequency [Figs. 7(bi)-(biii)]. Moreover, the rapid reduction of the oscillationamplitude for small coupling strength [Fig. 7(ci) is forΩ ,L = E L / 2] is remarkably well captured by the ap-proximate two-state model. For larger lattice strengths[Figs. 7(bii) and (biii) are for Ω ,L = 3 E L / E L / 2, respectively], in contrast, the two-state model capturesthe period of the Rabi oscillations comparatively poorly.The reason is that larger lattice coupling strengths leadto enhanced and non-negligible occupations of momentacentered near (cid:126) k z ≈ − (cid:126) k L and (cid:126) k z ≈ (cid:126) k L .Since the approximate two-state model provides aqualitatively and for some parameter combinations evena (semi-)quantitatively correct description of the dynam-ics, it is instructive to compare the Rabi oscillations forthe lattice and Raman coupled systems. If the two-statelattice model is exact, the difference between the Rabi os-cillations for the lattice and Raman coupled systems willbe—assuming that the small differences between g aa , g bb ,and g ab do not play a role—solely due to the “factor of 2”discussed in Sec. IV A. The parameters in Figs. 2 and 7are chosen such that the solid line in Fig. 2(a) can be di-rectly compared with the curves in Fig. 7(ai), the dashedline in Fig. 2(a) with the curves in Fig. 7(aii), and the dotted line in Fig. 2(a) with the curves in Fig. 7(aiii).An analogous correspondence exists for Fig. 2(b) andFigs. 7(bi)-(biii) as well as for Fig. 2(c) and Figs. 7(ci)-(ciii). A careful comparison of Figs. 7 and 2 indicatesthat the most prominent effect of the factor of 2 is to sig-nificantly enhance the damping or dephasing of the Rabioscillations.In what follows we attempt to pinpoint why the factorof 2 (lattice coupling case) enhances the damping com-pared to the case where this factor is equal to 1 (Ra-man coupling case). To start this discussion, we remindthe reader that the analytical treatment in Sec. III E,which assumes vanishing detuning, relies heavily on the5assumption that there exists a symmetry between thecomponents ˜ ψ a ( r , t ) and ˜ ψ b ( r , t ) [see Eq. (29)]. Infact, one can show that this symmetry is—within theThomas-Fermi approximation—an exact symmetry pro-vided g aa = g bb = g ab . Intuitively, this symmetry canbe understood by realizing that the strength of the scat-tering between two atoms in the same hyperfine stateis identical to that of the scattering between two atomsin different hyperfine states. This implies that neitherthe two-body interactions nor the Raman coupling (re-call, we are considering the zero detuning scenario) biaspopulations to one hyperfine state over another. In thelattice coupling case with δ L = 0, the factor of 2 breaksthe symmetry. The effective attractive on-site interac-tions [see the discussion in the context of Eqs. (50) and(51)], which can alternatively be interpreted as effectiverepulsive off-site interactions, favor configurations thatreduce the overlap between the densities | ˜ ψ a ( r , t ) | and | ˜ ψ b ( r , t ) | . Since the effective repulsive off-site interac-tions depend on the density, they vary spatially. Thisspatial dependence can result in a shape of the den-sity | ˜ ψ a ( r , t ) | that is different from that of the density | ˜ ψ b ( r , t ) | . If this occurs, the fractional population dif-ference varies locally, leading to a spatially dependentpopulation transfer and, correspondingly, a damping ordephasing of the Rabi oscillations. In a complementarypicture, the effective repulsive off-site interactions can bethought of as an effective spatially and temporally vary-ing coupling term. In this picture, the damping of theRabi oscillations emerges naturally. Section IV D makesthis discussion concrete for a π/ C. Rabi oscillations: Theory-experimentcomparison The symbols in Figs. 8(a) and 8(b) show experimen-tal data for Rabi oscillations induced by a moving opti-cal lattice with weak and strong coupling, respectively.The excellent agreement between the solutions to theGross-Pitaevskii equations for the full lattice Hamilto-nian (solid lines) and the experimental data indicatesthat the experiments operate in the mean-field regime,i.e., the Gross-Pitaevskii framework captures the popu-lation transfer between the two momentum componentsquantitatively. The approximate two-state model (dot-ted lines) provides, as already discussed in the previoussection, a semi-quantitative description of the lattice-induced Rabi oscillations in the weak coupling regime[Fig. 8(b)]; as such, it provides a meaningful conceptualframework for interpreting the results and contrastingthe lattice- and Raman-induced Rabi oscillations.A fit of the Rabi oscillation data obtained by solvingthe Gross-Pitaevskii equation for the full lattice Hamil-tonian to Eq. (16) yields coupling strengths that are, re-spectively, 5 % and 6 % lower than those used in the sim-ulations. This shows that the interactions do impact the -1-0.500.51 N a - N b N a - N b (a)(b) FIG. 8: Theory-experiment comparison for lattice Rabi os-cillations with “strong” and “weak” coupling strengths fora Rb BEC. The symbols show experimental data and theblack solid lines show results from the Gross-Pitaevskii simu-lations for the full lattice Hamiltonian. For comparison, theblue dotted lines show results obtained for the approximatetwo-state model. The experimental parameters common toboth panels are E L /h = 1080 Hz and t start = 0 . ,L /h = 2646 Hz): The experimentallymeasured parameters are ω x = 2 π × 172 Hz, ω y = 2 π × 139 Hz, ω z = 2 π × . δ L /h = − 264 Hz, and N = 1 . × . Thecalculations set ω ρ equal to the mean of ω x and ω y ; all otherparameters are taken from the experiment. The chemical po-tential µ prior to turning off the trap is 1 . E L . The mean-field energy per particle prior to turning off the trap and afterthe 0 . . E L and 0 . E L , respectively.The red circles show the result from one experimental run.(b) “Weak coupling” (Ω ,L /h = 980 Hz): The experimentallymeasured parameters are ω ρ = 2 π × 146 Hz, ω z = 2 π × 28 Hz, δ L = 0, and N = 2 . × . The transverse trap frequency isdetermined by performing measurements along one axis. Thecalculations use the parameters from the experiment. Thechemical potential µ prior to turning off the trap is 1 . E L .The mean-field energy per particle prior to turning off the trapand after the 0 . . E L and 0 . E L , re-spectively. The red circles and green squares show the resultsfrom two separate experimental runs. Rabi oscillations and that calibration of the experimentallattice strength needs to proceed with care. We note thatthe fit to the data in Fig. 8(a) has a significantly lower χ than the fit to the data in Fig. 8(b). For the experimentaldata shown in Fig. 8, the coupling strength is calibratedby inducing Rabi oscillations of a very dilute Rb BECfor a relative large lattice coupling strength; in this case,the Rabi oscillation data display essentially no damping.This calibration run yields a power-to-coupling-strengthconversion. Assuming that the coupling strength scales6as the square root of the power, the calibration curvecan be used in subsequent science runs that operate atother powers. The outlined approach assumes that thepower fluctuations are negligible over the course of sev-eral hours; we have checked that this is the case in oursetup. D. Ramsey-type pulse sequence: Numerical results This section discusses numerical results for theRamsey-type pulse sequence with lattice coupling (van-ishing detuning, i.e., δ L = 0). Figure 9 shows lattice cou-pling results for the same parameters as used in Fig. 4(recall, Fig. 4 shows results for the Ramsey-type sequence with Raman coupling). The first and second columns inFig. 9 are obtained by solving the time-dependent mean-field equation for the full lattice Hamiltonian while thethird and fourth columns are obtained by solving thetime-dependent mean-field equation for the approximatetwo-state lattice model. It can be seen that the resultsfor the approximate two-state lattice Hamiltonian agreewith those for the full lattice Hamiltonian rather well.Since the approximate two-state lattice model describesthe dynamics faithfully, we use it below to gain insightsinto the results after the first π/ π/ π/ ψ a ( r , t ) has a larger popula-tion than component ˜ ψ b ( r , t ); a 50/50 mixture is realizedfor a pulse of length 0 . . π/ | ˜ ψ a ( z, , τ ) | and | ˜ ψ b ( z, , τ ) | coincide to a good approximation. Inthe central region, in contrast, they differ. While thefirst component profile, | ˜ ψ a ( z, , τ ) | , approximately fol-lows a Thomas-Fermi profile, the second component pro-file, | ˜ ψ b ( z, , τ ) | , is flatter than a Thomas-Fermi pro-file. Correspondingly, the local spin expectation value σ z ( z, , τ ) has a roughly Gaussian shape as opposed tofollowing a linear curve as in the Raman coupled case.This indicates, in agreement with the more general dis-cussion at the end of Sec. IV B, that population from thecenter of the cloud is pushed toward the edge of the clouddue to the larger local effective repulsive off-site interac-tion at the center of the cloud compared to the edge.Using the Bloch-sphere picture, the spatially and tem-porally dependent effective repulsive off-site interactionor coupling leads to a spatially and temporally depen-dent torque along the x -direction during the first π/ σ y ( z, , τ ), whose spatial dependence differs fromthat of the densities of the components.Altogether, the discussion shows that the interactionscan, for the relatively weak lattice coupling strength ofΩ ,L = E L considered in Fig. 9, not be neglected dur-ing the first π/ π/ σ y ( z, , τ ) is not symmetric with respectto z = 0; this asymmetry persists during the hold time.(ii) During the hold time, the shapes of the densities | ˜ ψ a ( z, , t ) | and | ˜ ψ b ( z, , t ) | continue to change appre-ciably. (iii) The densities of the components and the localspin expectation value σ y ( z, , t ) develop spatial modu-lations during the hold time in the region where the twocomponents are not spatially overlapping. These spatialmodulations are more pronounced than in the Ramancoupling case.The second π/ σ y ( r , t ) to the population difference σ z ( r , t ).Since the cloud expands a fair bit during the hold time,the interactions can, to a good approximation, be ne-glected during the second π/ × | ~ ψ a / b ( z , , t ) | / k L ( - ) | ~ ψ a / b ( z , , t ) | / k L ( - ) -200 0 200z k L | ~ ψ a / b ( z , , t ) | / k L ( - ) -6-303 σ y / z ( z , , t ) / k L ( - ) -202 σ y / z ( z , , t ) / k L ( - ) -0.500.5-202 σ z ( z , , t ) / k L ( - ) -200 0 200z k L -0.500.5 0246 | ~ ψ a / b ( z , , t ) | / k L ( - ) | ~ ψ a / b ( z , , t ) | / k L ( - ) -200 0 200z k L | ~ ψ a / b ( z , , t ) | / k L ( - ) -6-303 σ y / z ( z , , t ) / k L ( - ) -202 σ y / z ( z , , t ) / k L ( - ) -0.500.5-202 σ z ( z , , t ) / k L ( - ) -200 0 200z k L -0.500.5 (ai) t= τ (aii) t= τ +1ms(aiii) t= τ +2ms(aiv) t= τ +1ms+ τ (av) τ +2ms+ τ (avi)(avii)(aviii)(aix)(ax) (bi) t= τ (bii) t= τ +1ms(biii) t= τ +2ms(biv) t= τ +1ms+ τ (bv) t= τ +2ms+ τ (bvi)(bvii)(bviii)(bix)(bx) mean-field lattice Hamiltonian numerical two-state model FIG. 9: Density cuts and local spin expectation values for the Ramsey-type pulse sequence with E L /h = 1960 Hz, Ω ,L = E L ,and δ L = 0 (numerical results); these are the same parameters as used in Figs. 6 and 7(aii). The Rb BEC consists of N = 3 × atoms and is prepared in an axially symmetric trap with ω ρ = 2 π × 200 Hz and ω z = 2 π × 40 Hz (these are thesame parameters as those used in Fig. 4). All results are obtained for t start = 0. The first and second columns are obtainedby solving the time-dependent mean-field equation for the full lattice Hamiltonian [Eq. (44) with V lat ( r , t ) given by Eq. (45)]numerically. The third and fourth columns show the same observables as the first and second columns but are, instead, obtainedusing the approximate two-state model introduced in Sec. IV A; the agreement is quite good. The black solid and red dashedlines in panels (ai)-(av) show the density profiles | ˜ ψ a ( z, , t ) | and | ˜ ψ b ( z, , t ) | , respectively. The black solid and red dashedlines in panels (avi)-(ax) show the local spin expectation values σ y ( z, , t ) and σ z ( z, , t ), respectively. The time increases fromthe first row, to the second/third row, to the fourth/fifth row (the value of the time is given in the panels); the second andfourth row correspond to a hold time of 1 ms, and the third and fifth row correspond to a hold time of 2 ms. Unlike in Fig. 4,the first π/ π/ a factor of 2. E. Ramsey-type pulse sequence:Theory-experiment comparison Figures 10(a) and 10(b) compare experimentally de-termined integrated densities (red circles) and theoreticalresults (black solid lines) for the Ramsey-type pulse se-quence with lattice coupling for hold times of 0 . . 207 ms does not correspond to a 50 % popu-lation transfer in the absence of interactions and vanish-ing momentum spread along the z -direction of the initialstate. The second pulse was taken to be 0 . 200 ms. To cal-ibrate the coupling strength, we performed calculationsfor different Ω ,L and picked the value that yields, usinga 0 . 207 ms pulse, a 50/50 population distribution afterthe first pulse.While the agreement between the symbols and solidlines in Fig. 10 is not perfect, the theoretical and exper-imental data share several key characteristics: (i) Thenumber of fringes increases with increasing hold time.(ii) The density pattern is not characterized by a sin-gle fringe spacing; rather, the fringe spacings seem tovary across the expanded cloud. (iii) The density dis-plays a small amplitude for z -values around − µ m and150 µ m; these peaks correspond to momentum spacecomponents centered around − (cid:126) k L and 4 (cid:126) k L , respec-tively. (iv) The density distributions centered around z ≈ z -direction of ≈ 0) and centered around z ≈ µ m (corresponding to the component with mo-mentum of ≈ (cid:126) k L ) have fairly distinct shapes, i.e., theyare not mirror images of each other. All these observa-tions are consistent with the discussion presented in theprevious section. If the interaction effects played less ofa role, the interference pattern would be “cleaner”, i.e.,more regular.As already alluded to earlier, Ref. [25] measured thelinear and quadratic phases using a Ramsey-type Braggpulse sequence. Their analysis assumed equally spacedfringes. While the fringe pattern in Fig. 2 of Ref. [25]is more “regular” than the fringe pattern displayed inFig. 10, the density peaks in Fig. 2(f) of Ref. [25] are,just as in our case, not fully symmetric with respect tothe midpoint. We speculate that this might be due tothe structural dynamics that is driven by mean-field ef-fects (“factor of 2”) discussed in our work for the latticecoupling case.While the overall agreement between the experimentaland theoretical data in Fig. 10 is satisfactory, the exper-imental data hint at the presence of beyond mean-fieldphysics. In particular, we consistently observe a signif-icant fraction of atoms “between” the two clouds, i.e.,with a momentum of around (cid:126) k L . It is presently unclearif this is due to quantum correlations that are not cap-tured by the mean-field Gross-Pitaevskii equation or if,possibly, the thermal cloud plays a non-negligible role.A detailed investigation of these questions is beyond thescope of this work. V. SUMMARY AND OUTLOOK This paper investigated two realizations of a two-statemodel; in both realizations, the two states are repre-sented by a spatially- and time-dependent mean-fieldwave function or orbital. The description goes beyond n ( z ) ( µ m - ) -200 -100 0 100 200z ( µ m)00.0050.010.015 n ( z ) ( µ m - ) (a)(b) FIG. 10: Theory-experiment comparison for lattice Ramsey-type pulse sequence for a Rb BEC for two different holdtimes t hold and particle numbers N : (a) t hold = 0 . N = 3 . × ) and (b) t hold = 1 ms ( N = 4 . × ).The red symbols show the experimentally measured inte-grated density n ( z ), n ( z ) = (cid:82) | Φ( r , t ) | dxdy , as a functionof z for t start = 0 . t ToF = 12 ms, δ L = 0, and E L /h = 1080 Hz; the results shown are from a single ex-perimental run. The solid black lines show results obtainedby solving the Gross-Pitaevskii equation for the full latticeHamiltonian for Ω ,L /h = 1372 Hz. For comparison, the bluedotted and green dashed lines show results obtained fromusing the two-state model (see text). Both sets of theorydata are convolved using a Gaussian with the experimen-tally measured resolution width of 2 µ m. The pulse sequenceis τ = 0 . 207 ms (first π/ t hold (seeabove), and τ = 0 . 200 ms (second π/ ω x = 2 π × 119 Hz, ω y = 2 π × 163 Hz, and ω z = 2 π × . ω ρ equal to the mean of ω x and ω y . a class of simpler mean-field models, where the dynam-ics of each mode is described by one complex numberthat encodes the population and phase of the mode,thereby assuming that the spatial dynamics of the modesplay a negligible role [32–34]. Our work demonstratesthat time-dependent deformations of the spatial profileof the mean-field wave functions play an important rolewhen the two-state model is realized by loading a single-component Rb BEC into a moving one-dimensional op-tical lattice that introduces a coupling between two dis-tinct momentum states of the atom. When the two-statemodel is, instead, realized by coupling two different hy-perfine states of Rb BEC atoms through a two-photonRaman process, time-dependent deformations of the spa-tial profile of the mean-field wave functions are notablyless pronounced.The difference in the dynamics for the two physical re-alizations (lattice and Raman coupling, respectively) of9the two-mode model was traced back to the contributionof different scattering diagrams; in particular, there existtwo scattering diagrams (these are depicted in the sec-ond row in Fig. 1) that contribute in the lattice couplingcase but not in the Raman coupling case (due to the “fac-tor of 2”). Said differently, the mean-field interactions forthe lattice and Raman coupling cases differ: The effectivetwo-state model for the lattice coupling case contains tworepulsive “off-site” interaction terms that are absent inthe Raman coupling case. As a consequence, the latticecoupled system is characterized by an enhanced tendencyfor phase separation, which “competes” with the latticecoupling term that has a tendency to keep the compo-nents together. This competition gives rise to the internalmean-field dynamics in the lattice coupled system that isdifferent from the mean-field dynamics displayed by thetwo-state Hamiltonian for the Raman coupling case.While the discussion throughout this paper focused on Rb BECs, the lattice coupling results, which rely on theoccupation of a single hyperfine state, apply to any BECwith positive two-body s -wave scattering length. TheRaman coupling results were obtained assuming that thefour coupling strengths g aa , g bb , g ab , and g ba are approx-imately equal or equal to each other; this assumptionholds for the F = 1 states of Rb but not necessarily forother elements.The present work has a number of practical and con-ceptual implications: • The Rabi oscillation data (see Figs. 2, 3, 7, and 8)show, especially for weak coupling strengths, pro-nounced non-sinusoidal behavior. This indicatesthat the analysis of experimental Rabi oscillationdata, taken to calibrate the coupling strength, hasto proceed with care. A simple fit to a sinusoidalfunction (or damped sinusoidal function) may yieldan imprecise coupling strength due to interactioneffects. Such data can be used for calibration pur-poses if compared with mean-field simulations thataccount for the interaction effects. Alternatively,experiments can operate in the dilute regime whereinteraction effects are negligible. Related discus-sions of lattice potential calibrations can be foundin Refs. [28, 35–37]. • In the “weak” lattice coupling case—this is the regime where, as discussed in Sec. IV, the effec-tive two-state model Hamiltonian provides a reli-able description of the system dynamics—the inter-nal dynamics leads to a deformation of the densityprofiles of the components. We argued that thesedensity deformations can be interpreted as corre-sponding to an effective position-dependent detun-ing. For example, starting with all population inone of the two states, a π/ • Integrating out the spatial degrees of freedom, thedynamics of the two-state Hamiltonian consideredin this work reduces to coupled mean-field equa-tions that are characterized by two complex num-bers, representing the populations and phases ofthe two modes [32–34, 38]. 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