Controlling spin motion and interactions in a one-dimensional Bose gas
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t Controlling spin motion and interactions in a one-dimensional Bose gas
P. Wicke, S. Whitlock, and N. J. van Druten*
Van der Waals-Zeeman Institute, University of Amsterdam,Science Park 904, 1098 XH Amsterdam, The Netherlands (Dated: submitted: October 19, 2010)Experiments on ultracold gases offer unparalleled opportunities to explore quantum many-body physics, with excellent control over key parameters including temperature, density,interactions and even dimensionality. In some systems, atomic interactions can be adjustedby means of magnetic Feshbach resonances, which have played a crucial role in realizingnew many-body phenomena. However, suitable Feshbach resonances are not always avail-able, and they offer limited freedom since the magnetic field strength is the only controlparameter. Here we show a new way to tune interactions in one-dimensional quantumgases using state-dependent dressed potentials, enabling control over non-equilibrium spinmotion in a two-component gas of Rb. The accessible range includes the point of spin-independent interactions where exact quantum many-body solutions are available and thepoint where spin motion is frozen. This versatility opens a new route to experiments onspin waves, spin-“charge” separation and the relation between superfluidity and magnetismin low-dimensional quantum gases.
Advances in optical and magnetic trapping of ultracold gases have played an essential rolein opening up novel avenues in quantum many-body physics by providing experimental accessto new physical regimes [1]. In particular, one-dimensional (1D) quantum gases, created usingoptical lattices or atom chips, exhibit a surprisingly rich variety of regimes not present in 2D or3D [2–9]. For example, a 1D Bose gas becomes more strongly interacting as the density decreases .Furthermore, the many-body eigenstates and thermodynamic properties of these 1D systems canoften be described using exact
Bethe Ansatz methods [10–14], and direct comparisons betweentheory and experiment are possible [6, 8, 9, 15–17]. Adding the possibility to dynamically controlthe strength of atomic interactions, for example via Feshbach resonances [16, 18–20], there is now astrong impetus to extend these experimental and theoretical studies to non-equilibrium dynamics.Spinor quantum gases offer the opportunity to study the interplay between internal (spin)and external (motion) degrees of freedom [16, 17, 21–27]. In this context, strong candidates forexperiments are the two magnetically trappable clock states in Rb [23, 25], in part becausethey experience equal trapping potentials and have nearly spin-independent interactions [28–30].The drawback is that no convenient Feshbach resonances are available for these states, preventingprecise control of the three relevant (inter and intra-state) interaction strengths.In the two-component (“spin-1/2”) 1D Bose gas, the presence of spin-independent (symmetric)interactions is of particular interest. For all interaction strengths (weak and strong) the dispersionrelation of spin waves is quadratic here [23, 31], and the low-energy spin velocity vanishes. As aconsequence the usual Luttinger-liquid description [8, 16, 32, 33] cannot be applied. However, it isprecisely the point where exact Bethe Ansatz methods can be used [14, 23]. Furthermore, it is thepoint where buoyancy effects vanish and in the weakly interacting (mean-field) regime it also lieson the border that separates miscible and immiscible regimes of binary superfluids [31].We show that radio-frequency-dressed potentials on atom chips offer a new way to tune theeffective interactions in 1D and to control spin motion. We make use of the fact that, for ellipticalrf polarizations, different hyperfine states experience different dressed potentials, allowing for state-dependent manipulation [34]. Here we exploit the dependence of the 1D coupling strength on thetransverse confinement frequency ω ⊥ [2]. State-dependent optical lattice potentials have previouslyfound use for spin-dependent transport and entanglement of atoms [35, 36]. More recently, state-dependent microwave dressing was used to generate spin squeezing in 3D Bose-Einstein condensatesby varying the wavefunction overlap for two hyperfine states to control collisions [37, 38], and state-dependent potentials created by combining an optical trap with a magnetic field gradient were usedto obtain record low spin temperatures via spin gradient demagnetization cooling of a quantumgas [39]. By tuning the transverse confinement for the two states independently through the rfpolarization and amplitude, we show that it is possible to control the interactions in a state- andtime-dependent manner. Suddenly changing interactions, combined with the state dependence ofthe axial trapping then results in dynamical evolution in the spin degree of freedom. In particular,we are able to tune to (i) the point where the spin motion is frozen, and (ii) the point where the1D interactions become spin-independent.We first discuss our results on the one-dimensional non-equilibrium dynamics for state- in dependent potentials, highlighting the importance of small differences in interaction param-eters. The starting point of our experiments is a nearly-pure 1D quasi-condensate in the | i = | F = 1 , m f = − i state of Rb in a highly elongated magnetic trap created by an atomchip (see Methods). From this initial state, we induce a sudden transition to a coherent super-position of the | i and | i = | F = 2 , m f = 1 i hyperfine states via a two-photon pulse, effectivelycreating a spin-1/2 system [28, 40]. The resulting non-equilibrium situation is allowed to evolve fora variable hold time. Subsequently, we directly image the longitudinal distributions, and obtainthe linear densities n and n of the two states along the length of the trap. b atoms/ µ m0 50 100 a t i m e ( m s ) atoms/ µ m0255075100 −50 0 50 d position ( µ m) −50 −25 0 25 50 c position ( µ m) t i m e ( m s ) −50 −25 0 25 500255075100 FIG. 1: [Color] One-dimensional spin dynamics and total density after a sudden transfer of internal-statepopulation, for the case of state-independent trapping potentials. Shown are spin polarization ( n − n ,left) and total linear density ( n + n , right), as a function of axial position and time after the transfer.Top: experiments, bottom: corresponding simulations resulting from integration of two coupled 1D Gross-Pitaevskii equations (GPE). The spin polarization data clearly shows how n is focused towards the center(blue), while n moves towards the sides (red); the total density shows little dynamics. Differences betweenexperiment and simulation can be explained by the limited optical resolution of our imaging system and asmall tilt of the trap, leading to a slight spatial asymmetry in the experiments. In figure 1 we present measurements of the evolution of spin polarization ( n − n ) and thetotal linear density ( n + n ) as a function of hold time. The spin pattern shows clear dynamicalevolution [fig. 1(a)] whereas the total density remains approximately constant with no significantdynamics [fig. 1(b)]. The spin dynamics can be interpreted as a “focusing” of state | i in thepresence of state | i , resulting in a negative spin polarization ( n > n ) toward the center of thetrap.We find good agreement with the experimental data using the coupled 1D Gross-Pitaevskii equa-tions (1D-GPE) with solutions also shown in fig. 1(c,d). The 1D-GPE is obtained by integratingthe full 3D-GPE over the transverse ground-state wavefunctions [41], with interaction parametersderived from the intra- and interstate scattering lengths taken from ref. [29]: a = 100 . · a , a = 95 . · a and a = 97 . · a , where a is the Bohr radius. Generalizing for state-dependentharmonic confinement (as will be relevant below) we obtain for the 1D interaction parameters u ij : u = 2 ~ ω ⊥ , a ,u = 2 ~ ω ⊥ , a ,u = 4 ~ ω ⊥ , ω ⊥ , ( ω ⊥ , + ω ⊥ , ) a , (1)with ω ⊥ ,j the transverse trap frequency for state | j i . Similarly we use values for the scaled rateconstants for inelastic two-body and three-body losses derived from the 3D values in ref. [29]. The1D-GPE simulations reproduce the features of the experiment, i.e. absence of dynamics in thetotal density and the overall structure of the spin dynamics including the time of maximum stateseparation around t ≈
75 ms. The decay in atom number on a &
100 ms timescale is dominatedby two-body losses in intrastate interactions and between | i atoms ( γ and γ ) [29].The rate of spin focusing/defocusing is critically dependent on the precise differences in 1Dinteraction strengths for the respective internal states, a fact that is readily confirmed by changingthese differences in the simulations. The observed general behavior can be understood as follows:in the initial state (an interacting trapped quantum gas in a single internal state in equilibrium)the repulsive interactions balance the external confining potential. Suddenly transferring a fractionof the population to a second internal state with weaker intra- and interstate interactions resultsin a net contracting force (a confining effective curvature, c > total density aredominated by the (relatively large) average scattering length which remains nearly constant. Hencethe total density shows only weak dynamics; the focusing in n is accommodated by “pushing” n to the sides (red in fig. 1).We now describe the state-dependent radio-frequency-dressed potentials that we use to controlthe spin motion. We consider near-resonant coupling ( ~ ω rf . g F µ B | B | ) of the rf field with tuneablepolarization determined by the relative phase of two independently controlled rf-fields. A crosssection of the wire geometry used is shown in fig. 2(a). The fields originate from direct digitalsynthesis (DDS) supplied currents in two wires neighboring the Z-shaped trapping wire [42]. Withthese two fields we can readily control the ellipticity of the total rf field at the trap position bycontrolling the relative phase φ of the rf currents in the two wires. This includes linear (horizontaland vertical) and circular ( σ ± ) polarizations. z-wireRF RF yz μ m π /22200230024002500260027002800 rf phase φ (rad) t r ap bo tt o m ( k H z ) π π /2 2 π a b Ioffe field
FIG. 2: [Color] State-dependent potentials. (a) wire geometry used for state-dependent radio-frequency-dressed traps. The static quadrupole magnetic field and the two rf fields are indicated by green, orangeand purple arrows respectively. The direction of the bias
Ioffe field which defines the quantization axis isinto the plane of the figure. (b) Trap bottom determined via dressed state rf spectroscopy as a function of φ . Data points correspond to the measured trap bottom for state | i (red) and state | i (blue). Solid anddashed curves are fits to the data. The dash-dotted green line indicates the fitted trap Larmor frequency ω L / π = 2 .
25 MHz.
The corresponding dressed-state potential for state | j i (with j = 1 ,
2) has the form V j ( x, y, z ) =(( V ( x, y, z ) − ~ ω rf ) + ~ Ω j ) / where V ( x, y, z ) is the bare magnetic (harmonic) potential. Thestate-dependent part of the potential enters through the coupling Rabi frequency Ω j [43, 44], whichacts to weaken the overall confinement near the trap bottom by an amount given by the dressingparameter δ j . Taking the second derivative of the potential V j around the origin yields new trapfrequencies, ˜ ω ⊥ , k = δ j ω ⊥ , k , where δ j = ∆ / q Ω j + ∆ , (2)with detuning ∆ = ω L − ω rf and Larmor frequency ω L .The state-dependent rf potential is characterized using dressed-state rf spectroscopy with aweak additional rf probe [45, 46]. Figure 2(b) shows the measured trap bottom as a function of thedressing phase φ , for ω rf = 2 π × .
20 MHz. For φ = 0 . π and φ = 1 . π the potential is maximallystate-dependent, corresponding to the pure circular polarizations σ − and σ + respectively (dressingonly state | i and only state | i , respectively). The potentials are state-independent for linearpolarization at φ = 0 , π (equal dressing of state | i and | i ). The deviation from a simple sin ( φ )behavior is due to the wire geometry, as the two rf fields are not quite orthogonal at the trapposition. A fit to the data (solid and dashed curves) taking into account the wire geometry is usedto precisely calibrate all parameters of the rf field coupling (see Methods).To control the spin motion we turn on the state-dependent dressing, by ramping up the rfcurrents in 2 ms with ω rf = 2 π × .
20 MHz, directly after preparing the equal superposition of | i and | i . The ramp time is slow compared to the inverse Larmor frequency and the inverseradial trap frequency, but sudden with respect to any axial motion. We use the two circular rfpolarizations and various rf amplitudes, corresponding to 0 . < δ < , δ = 1 and 0 . < δ < , δ = 1. For each time step we extract the widths of the axial distributions in both states. w ( μ m ) lllllllll ll l l l l l l llllllllll ll l l l l l l l b c . 0.95 δ δ a c / c j u ij / ω ┴ , ( a ) FIG. 3: [Color] Overview of the possibilities of the state-dependent potentials, as a function of the dressingparameters (left: varying δ , with δ = 1; right: varying δ with δ = 1). (a): 1D interaction strengths, u ij normalised by the bare transverse trap frequency ω ⊥ , . (b) Widths of the distribution at t = 44 ms and(c) scaled effective curvature c j /c at t = 0. Red indicates state | i (and u ) and blue state | i (and u )and in (a) u is indicated in green. The widths in (b) are obtained by a fit to the experimentally measureddensity profiles (dots) and to GPE simulation (shaded regions). The shaded areas in (b) represent the effectof shot-to-shot atom number fluctuations in the experiment. Results for the full range of dressing parameters are depicted in figure 3. Figure 3(a) showsthe calculated interaction strengths taken from equation (1) as a function of δ and δ . We havecompared the measured widths of the distributions as a function of time with solutions of thecoupled 1D-GPE. These widths and the corresponding simulations for one fixed hold time of 44 msare shown in figure 3(b). The measured widths follow the 1D-GPE simulations closely (takinginto account the finite optical resolution), with the biggest uncertainties originating from atomnumber fluctuations which cause the peak linear density to vary between 70 µ m − and 100 µ m − throughout the entire data set (systematic uncertainty shown by shaded regions). The solid verticalline at δ = 0 .
895 indicates the point where the difference in interaction strengths is minimized[fig. 3(a)] with u , u and u differing by less than 0.05% (100 times reduction in differenceswhen compared to the unmodified interactions). These conditions are of interest for comparing toBethe Ansatz solutions which require spin-independent interactions [14, 23].To explain the data we have to consider both the effect of rf-dressing on the collisional interactionstrengths as well as the state-dependent modification to the axial potential. A simple analyticaldescription can be obtained using a Thomas-Fermi description near the cloud center where thecloud shape is an inverted parabola. The combination of the state-dependence of the axial trappingfrequency and of the interactions can then be expressed as a net harmonic potential characterisedby an effective state-dependent curvature c j . We solve for the effective curvatures (Fig. 3c) for t & δ j , and find c c = δ − (1 − β ) p δ − a a β √ δ δ √ δ + √ δ c c = δ − β a a p δ − a a − β ) √ δ δ √ δ + √ δ (3)Here the first term on the right-hand side reflects the modification to the external axial potentialand the second and third terms deal with the modified interactions u ij . The axial curvature of thebare potential is c = mω k / β corresponds to the fraction of the population transferred tostate | i ( β ≈ / δ = 0 .
96 in fig. 3 indicates the point where the difference in interactions iscompensated by the state-dependent longitudinal potential and c = c . This point is characterizedby small equal curvatures of the effective potentials (including interaction energy) for both states[fig. 3(c)], which result in frozen spin dynamics. These conditions are important for applicationswith on-chip atomic clocks, to minimize inhomogeneous broadening due to mean field shifts. For δ < .
96 the difference in interaction strengths is further enhanced and the time evolution of thespin dynamics becomes inverted, with focusing of state | i while state | i is pushed outward, as isvisible in fig. 3(b).Figure 4 shows the full time evolution of the spin polarization for two selected rf-dressingparameters. The selected cases are: dressing of state | i alone ( φ = 0 . π , δ = 0 . | i alone ( φ = 1 . π , δ = 0 .
96) [fig. 4(b)], corresponding to the intersection pointsin figure 3(a) and (c), respectively. Qualitatively state | i focuses faster with rf dressing appliedto state | i , when compared to the case of state independent potentials [fig. 1(a)]. Generally, thesimulated density profiles reveal a rich and dynamic nonlinear evolution of the spin polarization,reminiscent of filament propagation in optical systems with competing nonlinearities [47]. Thisis clearly visible in figure 4(c) for example. This detailed structure depends sensitively on theprecise values of the dressing. The development and propagation of this fine structure in the spinpolarization is partially observed in the experimental data, but is not fully resolved due to thefinite imaging resolution. Convolving the simulated profiles with the point-spread function of ourimaging system yields excellent agreement with all of the data. With weak dressing of state | i ( δ = 0 .
96) it is possible to freeze spin dynamics altogether such that the two states maintain theiroverlap and the widths remain constant (apart from a small in-phase quadrupole oscillation anddecay from state | i ), see Figure 4(b,d). A more quantitative representation of the data, showingexcellent agreement between experiment and simulation, is given in figure 5, where the widths ofthe two states are shown for different evolution times. Clearly, the focus point can be identified infigure 5(a) around t = 75 ms and in figure 5(b) around t = 30 ms, whereas no focussing is presentin figure 5(c).We have shown that by introducing a small state-dependence to the radial trapping potentialusing rf dressing we can precisely tune the 1D interaction parameters in a two-component quantumgas by more than 10%, over an experimentally significant range. In our experiments this modi-fication competes with the state dependence of the axial trapping and provides a new “knob” tocontrol spin motion, leading to tuneable nonlinear behavior.Our method can be naturally extended in several ways. For instance, control over the interac-tions without the accompanying state-dependence of the axial trapping can be obtained by usingone-dimensional box-shaped potentials [42]. By introducing an additional displacement of thetransverse potential in a state-dependent way it is possible to further reduce u , allowing all threeinteraction parameters to be tuned independently, something that is not generally possible with amagnetically controlled Feshbach resonance.The observed spin dynamics depend critically on the precise differences in interaction strengths.For Rb, the three relevant scattering lengths are nearly equal and therefore weak dressing issufficient to tune the system parameters to the point of symmetric interactions or to where the spindynamics become frozen. Since the rf parameters can be precisely known, such experiments could
FIG. 4: [Color] Spatiotemporal behavior of the spin polarization ( n − n ) following the sudden transferto the state-dependent potentials. States | i and | i are indicated red and blue, respectively. (a) showsthe evolution with rf parameters corresponding to equal inter-atomic interactions ( δ = 0 . , δ = 1) and(b) equal effective potentials ( δ = 1 , δ = 0 . also allow precision determination of the scattering length differences. More generally, tuning thesystem parameters around the point of spin-independent interactions strongly affects the dispersionrelation of the spin excitations [31]. In particular this allows the spin velocity to be tuned aroundzero, providing a new handle for the study of spin waves in one-dimensional atomic gases.Tunable interactions in two-component quantum gases have important applications in the areasof spin-squeezing and quantum metrology [38, 48], and the ability to control spin motion opensnew avenues for future studies of quantum coherence in interacting quantum systems [16, 29,30, 40, 49]. Our current experiments are performed in the weakly interacting 1D regime andat low temperature, and we find that a description based on two coupled 1D Gross-Pitaevskiiequations is sufficient to describe our data. The methods presented here to tune interactions arenot limited to this regime, however. In particular, we plan to apply these methods to systems withstronger interactions (e.g., by lowering the 1D density) and with higher temperatures. This will0 w ( µ m ) a w ( µ m ) b t (ms) w ( µ m ) c FIG. 5: [Color] Widths of the atomic distributions as function of hold time, for the parameters indicated bythe three vertical lines in figure 3. Dots are fits to experimental data, lines are results of 1D GPE solutions.States | i and | i are indicated red and blue, respectively. In (a) no rf dressing is applied δ , = 1, asin figure 1. (b) and (c) correspond to the data in figure 4. (b) shows the evolution with rf parameterscorresponding to equal inter-atomic interactions ( δ = 0 . , δ = 1), and (c) equal effective potentials( δ = 1 , δ = 0 . provide experimental tests of (and challenges to) more sophisticated theoretical methods, for bothequilibrium and non-equilibrium phenomena. For instance, it will be possible to experimentallyexplore predictions of the thermodynamic Bethe Ansatz for the two-component Bose gas [14] and1to explore quantum quenches in strongly interacting 1D systems by dynamical control over thespin-dependent interactions. Finally, we expect that the experimental control over spin motionand interactions, as demonstrated here, will benefit the realization of spin-“charge” separation ina Bose gas [23, 25]. Acknowledgments
We thank R. J. C. Spreeuw and J. T. M. Walraven for valuable discussions. We are grateful toFOM and NWO for financial support. SW acknowledges support from a Marie-Curie fellowship(PIIF-GA-2008-220794).
MethodsInitial-state preparation and coherent spin mixing Rb atoms in state | i = | F = 1 , m f = − i are evaporatively cooled to quantum degeneracy in ahighly elongated Ioffe-Pritchard microtrap with trap frequencies of ω ⊥ / π = 1 . ω k / π =26 Hz. The peak linear atomic density is n . µ m − . In this system, both the temperatureand chemical potential are small compared to the radial excitation energy ( µ, k B T < ~ ω ⊥ ) and thedynamics are restricted to the axial dimension (1D regime). A coherent superposition of the | i and | i = | F = 2 , m f = 1 i hyperfine states is prepared using a resonant two-photon rf and microwave(mw) coupling [28, 40]. The microwave frequency is introduced via an external antenna while therf-field is applied directly to the atom chip wires. The measured two-photon Rabi frequency isΩ c / π = 1 .
14 kHz, corresponding to a π/ .
22 ms. This is fast compared to thetimescale for axial dynamics, but sufficiently slow to prevent radial excitations. Coherence timesin excess of 1 second have been measured in this setup via Ramsey spectroscopy of dilute thermalclouds.
State-dependent imaging
The time evolution of the spin distribution is measured by varying the hold time after the π/ | i are imaged directly using absorption on the F =22 , F ′ = 3 transition with an exposure time of 30 µ s and an optical resolution of 4 . µ m. State | i is measured by first removing | i atoms with a resonant light pulse followed by a 1 ms repumpingpulse from F = 1 to F = 2. The remaining atoms are then imaged in the same way as for state | i . Due to the extra repumping step we find a 20% lower detection efficiency for state | i and apoorer resolution of ∼ µ m due to photon recoil, visible in fig. 5. The resulting absorption imagesare integrated along the radial direction to obtain the linear densities n and n of the two states. Characterizing state-dependent potentials
The potential energy at the trap bottom is characterized by the onset of loss as a function ofprobe frequency which we fit to extract V j (0 , , j / π = 450 kHz.A fit to the data (solid lines in figure 2(b)) taking into account the wire geometry results inan accurate calibration of the key experimental parameters, in particular the Larmor frequency ω L = 2 .
25 MHz, rf field amplitudes b = b = 0 .
53 G from the two rf wires and the trap-surfacedistance of 80 µ m. [1] Bloch, I., Dalibard, J. & Zwerger, W. Many-body physics with ultracold gases. Rev. Mod. Phys. ,885–964 (2008).[2] Olshanii, M. Atomic scattering in the presence of an external confinement and a gas of impenetrablebosons. Phys. Rev. Lett. , 938 (1998).[3] Petrov, D. S., Shlyapnikov, G. V. & Walraven, J. T. M. Regimes of quantum degeneracy in trapped1D gases. Phys. Rev. Lett. , 3745 (2000).[4] Recati, A., Fedichev, P. O., Zwerger, W. & Zoller, P. Spin-charge separation in ultracold quantumgases. Phys. Rev. Lett. , 020401 (2003).[5] Kheruntsyan, K. V., Gangardt, D. M., Drummond, P. D. & Shlyapnikov, G. V. Pair correlations in afinite-temperature 1D Bose gas. Phys. Rev. Lett. , 040403 (2003).[6] Kinoshita, T., Wenger, T. & Weiss, D. S. Observation of a one-dimensional Tonks-Girardeau gas. Science , 1125–1128 (2004).[7] Paredes, B. et al.
Tonks-Girardeau gas of ultracold atoms in an optical lattice.
Nature , 277–281(2004).[8] Hofferberth, S. et al.
Probing quantum and thermal noise in an interacting many-body system.
NaturePhys. , 489–495 (2008). [9] van Amerongen, A. H., van Es, J. J. P., Wicke, P., Kheruntsyan, K. V. & van Druten, N. J. Yang-Yangthermodynamics on an atom chip. Phys. Rev. Lett. , 090402 (2008).[10] Lieb, E. H. & Liniger, W. Exact analysis of an interacting Bose gas. I. the general solution and theground state.
Phys. Rev. , 1605 (1963).[11] Yang, C. N. & Yang, C. P. Thermodynamics of a one-dimensional system of bosons with repulsivedelta-function interaction.
J. Math. Phys. , 1115 (1969).[12] Korepin, V. E., Bogoliubov, N. M. & Izergin, A. G. Quantum Inverse Scattering Method and CorrelationFunctions . Cambridge University Press,, Cambridge, England (1993).[13] Takahashi, M.
Thermodynamics of One-Dimensional Solvable Models . Cambridge University Press,,Cambridge, England (1999).[14] Caux, J.-S., Klauser, A. & van den Brink, J. Polarization suppression and nonmonotonic local two-bodycorrelations in the two-component Bose gas in one dimension.
Phys. Rev. A , 061605(R) (2009).[15] Kinoshita, T., Wenger, T. & Weiss, D. S. Local pair correlations in one-dimensional Bose gases. Phys.Rev. Lett. , 190406 (2005).[16] Widera, A. et al. Quantum spin dynamics of mode-squeezed luttinger liquids in two-component atomicgases.
Phys. Rev. Lett. , 140401 (2008).[17] Liao, Y.-a. et al.
Spin-imbalance in a one-dimensional fermi gas.
Nature , 567–569 (2010).[18] Haller, E. et al.
Realization of an excited, strongly correlated quantum gas phase.
Science , 1224(2009).[19] Haller, E. et al.
Confinement-induced resonances in low-dimensional quantum systems.
Phys. Rev.Lett. , 153203 (2010).[20] Chin, C., Grimm, R., Julienne, P. & Tiesinga, E. Feshbach resonances in ultracold gases.
Rev. Mod.Phys. , 1225–1286 (2010).[21] Stenger, J. et al. Spin domains in ground-state Bose-Einstein condensates.
Nature , 345 (1998).[22] Schmaljohann, H. et al.
Dynamics of F=2 spinor Bose-Einstein condensates.
Phys. Rev. Lett. ,040402 (2004).[23] Fuchs, J. N., Gangardt, D. M., Keilmann, T. & Shlyapnikov, G. V. Spin waves in a one-dimensionalspinor Bose gas. Phys. Rev. Lett. , 150402 (2005).[24] Zvonarev, M. B., Cheianov, V. V. & Giamarchi, T. Spin dynamics in a one-dimensional ferromagneticBose gas. Phys. Rev. Lett. , 240404 (2007).[25] Kleine, A., Kollath, C., McCulloch, I. P., Giamarchi, T. & Schollw¨ock, U. Spin-charge separation intwo-component Bose gases. Phys. Rev. A , 013607 (2008).[26] Kronj¨ager, J., Becker, C., Soltan-Panahi, P., Bongs, K. & Sengstock, K. Spontaneous pattern formationin an antiferromagnetic quantum gas. Phys. Rev. Lett. , 090402 (2010).[27] Vengalattore, M., Guzman, J., Leslie, S. R., Servane, F. & Stamper-Kurn, D. M. Periodic spin texturesin a degenerate F=1 Rb spinor Bose gas.
Phys. Rev. A , 053612 (2010).[28] Harber, D. M., Lewandowski, H. J., McGuirk, J. M. & Cornell, E. A. Effect of cold collisions on spin coherence and resonance shifts in a magnetically trapped ultracold gas. Phys. Rev. A , 053616(2002).[29] Mertes, K. M. et al. Nonequilibrium dynamics and superfluid ring excitations in binary Bose-Einsteincondensates.
Phys. Rev. Lett. , 190402 (2007).[30] Anderson, R. P., Ticknor, C., Sidorov, A. I. & Hall, B. V. Spatially inhomogeneous phase evolution ofa two-component Bose-Einstein condensate. Phys. Rev. A , 023603 (2009).[31] Timmermans, E. Phase separation of Bose-Einstein condensates. Phys. Rev. Lett. , 5718–5721(1998).[32] Giamarchi, T. Quantum physics in one dimension . Internat. Ser. Mono. Phys. Clarendon Press, Oxford(2004).[33] Hofferberth, S., Lesanovsky, I., Fischer, B., Schumm, T. & Schmiedmayer, J. Non-equilibrium coherencedynamics in one-dimensional Bose gases.
Nature , 324 (2007).[34] Hofferberth, S., Lesanovsky, I., Fisher, B., Verdu, J. & Schmiedmayer, J. Radio-frequency-dressed-statepotentials for neutral atoms.
Nature Phys. , 710–716 (2006).[35] Mandel, O. et al. Controlled collisions for multiparticle entanglement of optically trapped atoms.
Nature , 937–940 (2003).[36] Lee, P. J. et al.
Sublattice addressing and spin-dependent motion of atoms in a double-well lattice.
Phys. Rev. Lett. , 020402 (2007).[37] B¨ohi, P. et al. Coherent manipulation of Bose-Einstein condensates with state-dependent microwavepotentials on an atom chip.
Nature Phys. , 592–597 (2009).[38] Riedel, M. F. et al. Atom-chip-based generation of entanglement for quantum metrology.
Nature ,1170–1173 (2010).[39] Medley, P., Weld, D. M., Miyake, H., Pritchard, D. E. & Ketterle, W. Spin gradient demagnetizationcooling of ultracold atoms. arXiv:1006.4674 (2010).[40] Treutlein, P., Hommelhoff, P., Steinmetz, T., H¨ansch, T. W. & Reichel, J. Coherence in microchiptraps.
Phys. Rev. Lett. , 203005 (2004).[41] Salasnich, L., Parola, A. & Reatto, L. Effective wave equations for the dynamics of cigar-shaped anddisk-shaped Bose condensates. Phys. Rev. A , 043614 (2002).[42] van Es, J. J. P. et al. Box traps on an atom chip for one-dimensional quantum gases.
J. Phys. B: At.Mol. Opt. Phys. , 155002 (2010).[43] Lesanovsky, I. et al. Adiabatic radio-frequency potentials for the coherent manipulation of matterwaves.
Phys. Rev. A , 033619 (2006).[44] Fernholz, T., Gerritsma, R., Kr¨uger, P. & Spreeuw, R. J. C. Dynamically controlled toroidal andring-shaped magnetic traps. Phys. Rev. A , 063406 (2007).[45] Hofferberth, S., Fischer, B., Schumm, T., Schmiedmayer, J. & Lesanovsky, I. Ultracold atoms inradio-frequency dressed potentials beyond the rotating-wave approximation. Phys. Rev. A , 013401(2007). [46] van Es, J. J. P., Whitlock, S., Fernholz, T., van Amerongen, A. H. & van Druten, N. J. Longitudinalcharacter of atom-chip-based rf-dressed potentials. Phys. Rev. A , 063623 (2008).[47] Couairona, A. & Mysyrowicz, A. Femtosecond filamentation in transparent media. Phys. Rep. ,47–189 (2007).[48] Gross, C., Zibold, T., Nicklas, E., Est`eve, J. & Oberthaler, M. K. Nonlinear atom interferometersurpasses classical precision limit.
Nature , 1165–1169 (2010).[49] Deutsch, C. et al.
Spin self-rephasing and very long coherence times in a trapped atomic ensemble.
Phys. Rev. Lett.105