Realization of an Excited, Strongly-Correlated Quantum Gas Phase

In many-body quantum physics the interplay between strong interactions and confinement to a low-dimensional geometry amplifies the effects of quantum fluctuations and correlations. A remarkable example in one dimension is the Tonks-Girardeau (TG) gas, where bosons with strong repulsive interactions minimize their interaction energy by avoiding spatial overlap and acquire fermionic properties Girardeau1960 ; Lieb1963 . Evidence for this ground state phase was found using Bose-Einstein condensates (BEC) loaded into optical lattices Paredes2004 ; Kinoshita2004 . While many-body quantum systems are usually found in their ground state phases, long-lived excited state phases are responsible for some of the most striking physical effects, examples ranging from vortex lattices in superfluids to subtle topological excitations in spin liquids Alet2006 . However, the experimental realization of excited phases is difficult, as these usually quickly decay by intrinsic effects or by coupling to the environment. In this context, cold atoms Petrov2000 ; Moritz2003 ; Paredes2004 ; Kinoshita2004 ; Tolra2004 ; Petrov2004 ; Hofferberth2007 ; Syassen2008 ; Bloch2008 may provide unique opportunities for the realization of long-lived, strongly interacting, excited many-body phases due to the excellent decoupling from the environment and the tunability of interactions via, for example, Feshbach resonances.

For an ultracold one-dimensional (1D) system of bosons, we prepare a highly-excited many-body phase known as the super-Tonks-Girardeau (sTG) gas Astrakharchik2005 . In this highly-correlated quantum phase, interactions are attractive, and rapid decay into a cluster-type ground state is in principle possible. However, a surprising property of this many-body phase is its metastability. Attractive interactions strengthen correlations between particle positions and ensure, similar to an effective long-range repulsive interaction, that particles rarely come together. To realize this exotic phase, we observe and exploit a 1D confinement-induced resonance (CIR) Bergeman2003 ; Olshanii1998 . This resonance allows us to first enter deeply into the repulsive TG regime to establish strong particle correlations and then to switch interactions from strongly repulsive to strongly attractive. The frequency ratio of the two lowest-energy collective modes Menotti2002 provides accurate diagnostics for the crossover from the TG to the sTG regime. In particle loss and expansion measurements we study the time evolution of the system through the crossover.

We tune the strength of the interaction as characterized by the three-dimensional (3D) scattering length $a_{\mathrm{3D}}$ by means of a magnetically-induced Feshbach resonance Inouye1998 . For a 1D system, a CIR arises and strongly modifies the 1D scattering properties when $a_{\mathrm{3D}}$ approaches the harmonic oscillator length $a_{\perp}=\sqrt{\hbar/(m\omega_{\perp})}$ of the transversal confinement with trap frequency $\omega_{\perp}$ Olshanii1998 ; Bergeman2003 . Here, $m$ is the mass of the particles and $\hbar$ is Planck’s constant divided by $2\pi$. More precisely, the coupling constant $g_{\mathrm{1D}}$ of the 1D $\delta$-function contact potential $U_{\mathrm{1D}}(z)=g_{\mathrm{1D}}\delta(z)$ behaves as Bergeman2003

where $a_{\mathrm{1D}}$ is the 1D scattering length defined by this equation and $C=1.0326$ is a constant. Thus, the CIR allows tuning of $g_{\mathrm{1D}}$. For values of $a_{\mathrm{3D}}$ less but close to $a_{\perp}/C$ ($a_{\mathrm{3D}}\lesssim a_{\perp}/C$) the coupling parameter $g_{\mathrm{1D}}$ is large and positive, and for $a_{\mathrm{3D}}\gtrsim a_{\perp}/C$ it is large and negative, leading to an effectively attractive interaction. For homogenous systems with $g_{\mathrm{1D}}>0$, it is customary to characterize the strength of interactions by the Lieb-Liniger parameter $\gamma=g_{\rm 1D}m/(\hbar^{2}n_{\rm 1D})$, where $n_{\rm 1D}$ is the linear 1D density of the system Lieb1963 ; Petrov2000 . The TG gas corresponds to the limit $\gamma\gg 1$ or $g_{\rm 1D}\!\to\!\infty$. As interactions are increased, the system becomes strongly correlated and is fully dominated by its kinetic energy. In previous experiments, without the capability to tune $a_{\mathrm{3D}}$, a maximum of $\gamma\approx 5.5$ was achieved Kinoshita2004 , while an effective strength $\gamma_{\rm eff}\approx 200$ was reached with an additional shallow lattice potential along the longitudinal direction Paredes2004 . In the former experiment, a saturation for the size and energy of the 1D system was observed, and in the latter experiment the momentum distribution was studied.

But what happens in the case of strong attractive interactions $g_{\mathrm{1D}}\!\to\!-\infty$, i.e. $a_{\mathrm{1D}}\gtrsim 0$? The ground state for a system of $N$ attractively interacting bosons in 1D is a cluster state McGuire1965 ; Tempfli2008 , which one would expect, in a cold atom system, to decay quickly via molecular channels. However, by crossing the CIR from the TG side, i.e. switching interactions from $g_{\mathrm{1D}}=+\infty$ to $g_{\mathrm{1D}}=-\infty$, an excited gas-like phase, the sTG gas, should be accessible Astrakharchik2005 . Is this excited phase stable, i.e. does it exist at all? The expectation is that the large kinetic energy inherited from the TG gas, in a Fermi-pressure like manner, prevents the gas from collapsing Batchelor05 . This stability can most simply be inferred from a Bethe-ansatz solution to the Lieb-Liniger model with attractive interactions Astracharchik2004 ; Batchelor05 . This ansatz yields for the energy per particle $E/N\approx\hbar^{2}\pi^{2}n_{\rm 1D}^{2}/[6m(1-n_{\rm 1D}a_{\rm 1D})^{2}]$, corresponding to the energy of a gas of hard rods Girardeau1960 , for which $a_{\rm 1D}$ represents the excluded volume. This results in a positive inverse compressibility and also in an increased stiffness of the systems as long as $n_{\rm 1D}a_{\rm 1D}$ is sufficiently small. Interestingly, in this phase the density correlations are even stronger than in the TG gas, as they show a power-law decay that is slower than for a TG gas Astrakharchik2005 , indicating an effective long-range interaction.

We realize the crossover all the way from a non-interacting gas via the 1D mean-field Thomas-Fermi (TF) regime to a TG gas and then to a sTG gas. We exploit the fact that our 1D systems possess weak harmonic confinement along the axial direction characterized by the confinement length $a_{\parallel}$. Whereas the frequency $\omega_{D}$ of the lowest dipole mode depends only on the confinement, the frequency $\omega_{C}$ of the lowest axial compressional mode is sensitive to the various regimes of interaction Menotti2002 . For the non-interacting system one expects $R\equiv\omega_{C}^{2}/\omega_{D}^{2}=4$. This value then changes to $R=3$ for weakly repulsive interactions in a 1D TF regime Moritz2003 . For increasing positive interaction strength, $R$ is expected to change smoothly to 4 when entering the TG regime as the system becomes fermionized and hence effectively non-interacting. A rise beyond the value of 4, after crossing the CIR, would then constitute clear evidence for the sTG regime Astrakharchik2005 . As $a_{\mathrm{1D}}$ is further increased, the system will finally become unstable and $R$ is expected to turn over and drop towards zero. For a harmonically confined system, the point of instability is reached when the overall length of the system of hard rods, $Na_{\mathrm{1D}}$, becomes of the order of the size $\sqrt{N}a_{\parallel}$ for the wave function of N non-interacting fermions, i.e. $A\equiv Na_{\mathrm{1D}}/(\sqrt{N}a_{\parallel})\approx 1$. We use $A^{2}$ as an alternative parameter to $\gamma$ to characterize the strength of the interaction as it accounts for the harmonic confinement.

We start from a 3D Bose-Einstein condensate (BEC) with up to $2\times 10^{5}$ Cs atoms with no detectable thermal fraction in a crossed-beam dipole trap with magnetic levitation Weber2003 . Depending on the interaction regime to be studied, we then set the number of atoms in the BEC to values in the range of $(1-4)\times 10^{4}$ by means of forced radio-frequency evaporation. To confine the atoms in 1D, i.e. to freeze out transversal motion, we use a two-dimensional optical lattice Bloch2008 , which forms an array of vertically oriented elongated tubes with an aspect ratio that we set to values between $100$ and $1000$ (Fig. 1A). We occupy between $3000-6000$ independent tubes with $8$-$25$ atoms in the center tube. The interaction strength $g_{\mathrm{1D}}$ is controlled by magnetic tuning of $a_{\mathrm{3D}}$ by means of a combination of a broad and a narrow Feshbach resonance (Fig. 1C) with poles at $B=-11.1(6)$ G and $B=47.78(1)$ G and widths of about $29.2$ G and $164$ mG, respectively Lange2009 . The broad resonance provides a slow variation of $a_{\mathrm{3D}}$, allowing us to gently tune $a_{\mathrm{3D}}$ from $0$ a₀ near $17.119$ G to about $1240$ a₀ near $76$ G, while the narrow resonance allows us to tune $a_{\mathrm{3D}}$ to absolute values beyond $4000$ a₀ given our magnetic field control. We convert the applied magnetic field $B$ into $a_{\mathrm{3D}}$ using the fit formula of Ref. 23. A magnetic field gradient, used to levitate the atomic sample onlinematerial , introduces a small spread in the value of $a_{\mathrm{3D}}$ across the sample.

To determine the oscillation frequencies $\omega_{C}$ and $\omega_{D}$ of the fundamental modes (Fig. 1 B), we excite each mode separately at a given value of the magnetic field $B$ onlinematerial and let the atoms evolve for a varying amount of hold time. The distribution is then imaged in momentum space by taking an absorption picture after release and expansion. To avoid possible broadening effects due to interaction during the initial expansion, $a_{\mathrm{3D}}$ is set to zero near $B=17.119$ G at the moment of release. To extract the frequency, we determine for each hold time the axial $1/e$-width of the distribution and then fit a damped sinusoid with linear offset to this data. Typical measurements of $\omega_{C}$ are shown in Fig. 1 D and E. Whereas the atom number remains constant for $g_{\mathrm{1D}}>0$, we observe some atom loss and a slight broadening of the distribution for attractive 1D interactions. In all parameter regimes, the 1D system is sufficiently stable to allow a reliable measurement of $\omega_{C}$.

First, we show that we can tune the system from the non-interacting regime deeply into the repulsive TG regime (Fig. 2). In agreement with expectations, the value for $R=\omega_{C}^{2}/\omega_{D}^{2}$ first drops from 4 to 3 and then increases back to 4 as $\gamma$ is tuned by means of the gently-varying background scattering length. We find that the TG regime is fully reached for $\gamma>50$. A further increase to values up to $\gamma\approx 500$ does not lead to changes for $R$. Note that, as $a_{\mathrm{3D}}$ approaches $a_{\perp}$, the divergence of $g_{\mathrm{1D}}$ according to Eq. 1 has to be taken into account when determining $\gamma$ onlinematerial . Heating of the system can be excluded as we can return to a 3D BEC without significant thermal background when ramping down the lattice potential.

The attractive regime is entered by crossing the CIR on the low-field wing of the 47.78 G Feshbach resonance. $a_{\mathrm{1D}}$ is now small and positive. The central results of this work are summarized in Fig. 3A and compared to the theoretical work of Ref. 13. We plot $R=\omega_{C}^{2}/\omega_{D}^{2}$ as a function of the interaction parameter $A^{2}$. For reference, Fig. 3B plots $a_{\mathrm{3D}}$, $a_{\mathrm{1D}}$, and $g_{\mathrm{1D}}$ in the vicinity of the Feshbach resonance as a function of the magnetic field $B$. As the CIR is crossed and $A^{2}$ is increased, $R$ rises beyond the value of $4$. This provides clear evidence for the sTG regime as $R=4$ is the maximal value for bosons with repulsive contact interaction. This increase is expected from the model of a gas of hard rods, and our data initially follows the prediction from this model. However, as $A^{2}$ is increased, $R$ reaches a maximum and then starts to drop. The maximum of about $4.5$ is reached for $A^{2}\approx 3\times 10^{-2}$. The existence of the maximum is in qualitative agreement with the results obtained from Monte-Carlo simulations Astrakharchik2005 . The theoretical prediction, however, underestimates the measured $R$. This is probably due to the local density approximation, which may not be applicable to our system with low particle numbers. For comparison, the results from Fig. 2 for $\gamma\geq 1$ are shown. Note that $\gamma\approx 500$ corresponds to small values of $A^{2}\approx 10^{-4}$. For this data, at higher particle numbers, there is excellent agreement with the theoretical prediction (solid line) in the entire crossover from the mean-field regime to the TG regimeMenotti2002 .

We study the stability of the system in the crossover from the TG to the sTG regime and find further evidence for the existence of the CIR by recording particle loss and measuring the axial width of the atomic cloud after release from the tubes. The axial width is a measure for the kinetic energy of the system as interactions are instantly switched off upon release. Similar conditions are used as for the measurements on the sTG regime presented in Fig. 3. The TG regime is entered adiabatically to avoid the excitation of collective modes. The system is prepared at $a_{\mathrm{3D}}=887(1)$ a₀ at a magnetic field of $B=42.77(2)$ G with about $11$ atoms in the central tube. The magnetic field is then ramped to a specific value within $0.2$ ms and the sample is held at this value for a variable hold time $\tau$ from 10 to 200 ms. $a_{\perp}$ is set to $1523(6)$ a₀. The results (Fig. 4) for different hold times $\tau$ in the tubes show that, for $\tau=10$ ms, corresponding to the timescale of the measurements in the sTG regime shown in Fig. 3, the transition from the TG to the sTG regime appears very smooth. There is essentially no particle loss when the system is deep in the TG regime and close to the CIR. The loss gradually increases in the attractive regime as one moves to larger values of $B$ and towards the pole for a_1D. Correspondingly, the width of the sample exhibits a smooth behavior across the CIR, showing a slight increase for larger $B$. This behavior is consistent with the expectation of an increased energy in the sTG regime Astrakharchik2005 .

For longer hold times, the data for the atom number and the sample width develop distinct features at the calculated position of the CIR. Evidently, the system is in a transient state. For $\tau=50$ ms, the number of remaining atoms shows a dip that correlates with a peak in the kinetic energy of the sample. Both features become more prominent and asymmetric for longer hold times ($\tau=100$ and $200$ ms). Note that, in comparison, no pronounced effects are visible at the pole of the Feshbach resonance for $a_{\mathrm{3D}}$. Our results must be connected to the fact that the energy spectrum of the system changes dramatically across the CIR, from the TG to the sTG regime Tempfli2008 . The system acquires a deeply lying ground state together with a family of lower lying many-body excited states, potentially opening up new decay channels. Also, the CIR strongly modifies the two-body scattering problem, making formation of confinement-induced molecules in transversally excited trap states Bergeman2003 possible.

The non-trivial time evolution observed in this system raises intriguing questions on possible coupling and decay mechanisms for strongly interacting excited many-body systems, in particular in the context of integrability of 1D systems Kinoshita2006 . Our results offer an example of the counter-intuitive effects that occur in many-body systems, and open up the possibility to study the dynamical properties of strongly-correlated systems with effective long-range interactions Bockrath1999 ; Steinberg08 under conditions where all parameters are tunable and, in fact, can be changed dynamically. Similar to magnetic Feshbach resonances in atomic scattering, we expect the confinement-induced resonance demonstrated here to serve as a general tool to tailor interactions in 1D and possibly also in 2D systems Petrov2000b , allowing for the further investigation of strongly correlated phases in the context of cold atomic gases.

We thank S. Giorgini and C. Menotti for helpful discussions and for providing the theory curves shown in Fig. 3A. We are indebted to R. Grimm for generous support and to H. Häffner and his group for the loan of a CCD camera. We gratefully acknowledge funding by the Austrian Ministry of Science and Research (Bundesministerium für Wissenschaft und Forschung) and the Austrian Science Fund (Fonds zur Förderung der wissenschaftlichen Forschung) in form of a START prize grant and by the European Union through the STREP FP7-ICT-2007-C project NAME-QUAM (Nanodesigning of Atomic and MolEcular QUAntum Matter) and within the framework of the EuroQUASAR collective research project QuDeGPM. R.H. is supported by a Marie Curie International Incoming Fellowship within the 7th European Community Framework Programme.