## Abstract

We report the observation of a one-dimensional (1D) Tonks-Girardeau (TG) gas of bosons moving freely in 1D. Although TG gas bosons are strongly interacting, they behave very much like noninteracting fermions. We enter the TG regime with cold rubidium-87 atoms by trapping them with a combination of two light traps. By changing the trap intensities, and hence the atomic interaction strength, the atoms can be made to act either like a Bose-Einstein condensate or like a TG gas. We measure the total 1D energy and the length of the gas. With no free parameters and over a wide range of coupling strengths, our data fit the exact solution for the ground state of a 1D Bose gas.

At zero temperature, dense, weakly interacting bosons in a 1D trap form a Bose-Einstein condensate (BEC) (*1*). Dilute, strongly interacting bosons in 1D, however, act in a completely different manner. Rather than condensing into a single quantum state, they are expected to repel each other, as if they were noninteracting fermions (*2*, *3*). Known as a Tonks-Girardeau (TG) gas (*4*, *5*), this system provides a textbook example of the critical role played by coupling strengths in many-body physics (*6*). Recent cold atom experiments have made observations of a TG gas a possibility (*7*), substantially increasing the already extensive body of theoretical research. Strongly coupled 1D atomic gases, and similarly accessible 2D gases, are poised to experimentally elucidate many subtleties of many-body quantum systems (*8*–*17*).

The gradual transition between the BEC and TG regimes is usually characterized by the parameter γ = ϵ_{int}/ϵ_{kin}, where ϵ_{int} and ϵ_{kin} are the average interaction and kinetic energies calculated with mean field theory. A BEC (low γ) in a 1D trap is qualitatively like a BEC in 3D (*18*, *19*). By trapping atoms in red-detuned 2D optical lattices, which create 2D arrays of atoms in tightly confined tubes (*20*), three recent experiments have approached the 1D TG (high γ) regime. For γ = 0.5, a marked suppression of three-body collisional loss was observed (*21*). With γ = 1, changes in the collective excitation spectrum were reported (*22*), and changes in the 1D superfluid to Mott insulator transition were observed by adding an additional 1D optical lattice along the 1D axis (*23*). With the addition of a 1D lattice, the spatially modulated atoms can be described as quasiparticles with increased effective mass (*24*). In this way, a system with γ = 0.5 can be interpreted as having γ_{eff} ∼ 200. The measured momentum distributions in such a system were found to fit well to a modified theoretical result derived for the lattice TG gas (*24*).

We create a TG gas with no periodic potential along the 1D axis. Our observations extend to γ = 5.5. We enter the strong coupling regime by using a combination of two independent light traps. A blue-detuned 2D optical lattice tightly confines nearly zero-temperature ^{87}Rb atoms in an array of parallel tubes, and a red-detuned crossed dipole trap weakly confines them along the tubes. As we explain below, the transverse confinement can thus be made tighter, which increases γ, without strengthening the axial confinement, which decreases γ. By changing trap intensities we scan γ, making the atoms either BEC-like or TG-like. We measure ϵ, the average 1D energy per particle (excluding the trap potential energy) and the equilibrium 1D cloud length. Our data fit well to the exact theory for the ground state of a TG gas in a harmonic trap (*25*), with no free parameters and over a wide range of coupling strengths. Our experiment provides a clear illustration of fermionization in a TG gas.

The interaction regimes of cold bosons in 1D can be understood by comparing the length of single-particle wave functions, ℓ, with the interparticle spacing, *r* = 1/*n*_{1D}, where *n*_{1D} is the linear density. The various regimes are illustrated in the wave function sketches of Fig. 1. In the mean field regime, ℓ ≫ *r*, and ℓ corresponds to the coherence length, which is the deBroglie wavelength associated with the 1D interaction energy (*25*). Long-range phase coherence makes such a trapped 1D gas superfluid. As ℓ approaches *r*, so that the system enters the TG regime, single-particle wave functions become spatially distinct. With less overlap, ℓ is not predominantly determined by the interaction energy, but by the kinetic energy associated with localizing the mutually repelling particles on a line. In the asymptotic TG limit, ℓ = *r*, and the single-particle wave functions are completely distinct. The TG limit can be reached either by increasing the interaction energy (decreasing ℓ) or decreasing *n*_{1D} (increasing *r*). The absence of long-range phase coherence and of number fluctuations in space makes the 1D TG gas resemble a gas of classical hard spheres or noninteracting fermions. With its progressive loss of number fluctuations and phase coherence, the transition from a 1D BEC to a TG is similar to the superfluid to Mott insulator transition of a gas in a periodic potential (*26*), but without the manifest long-range spatial order (*21*).

Our experiment can be qualitatively understood as follows (Fig. 1, shaded sketches). Atoms near zero temperature are relatively tightly confined in two transverse dimensions, and loosely confined in an axial dimension. The system is then studied with constant axial confinement and atom number, and successively tighter transverse confinement. At first, the interactions are in the mean field limit, so the Bose gas acts like a 3D fluid. Squeezing the atoms transversely makes them spread out axially (increasing *r*) and increases ϵ (decreasing ℓ). As ℓ and *r* approach each other (γ > 1) and the bosons start to fermionize, the rate at which both ℓ and *r* change decreases. We observe the stabilization of *r* by measuring the 1D cloud length. We also measure ϵ by removing the axial confinement and letting the atoms expand in 1D. With more squeezing, the interaction energy decreases, because although the higher 3D density tends to yield stronger interactions, the reduction in wave function overlap exerts a greater effect. With more localized wave functions, the kinetic energy starts to dominate. The net effect is that ϵ quickly approaches the asymptote to its high γ value. Very far in the TG limit (γ≫ 1), like classical beads on a string or like noninteracting fermions, transverse squeezing of TG atoms would have no effect on either *r* or ϵ.

The starting point for our experiment is a nearly pure ^{87}Rb BEC in the lowest internal energy state, which we produce by all-optical means every 3 s (*27*). The atoms are confined in a horizontal crossed dipole trap made with 1.06-μm yttrium-aluminum-garnet (YAG) laser light (*28*) (Fig. 2). The trap power, *P*, and waist size, *w*_{0}, are dynamically variable, but typically we perform the 1D experiments with *P* = 12 or 320 mW, *w*_{0} = 70 μm, and 2 × 10^{5} BEC atoms (supporting online text). We use high-intensity fluorescent imaging to measure cloud sizes (*29*) (supporting online text). By scanning *P* and progressively reducing the 3D trapped cloud, we determine the resolution of the optical system to be *w*_{ir} = 20 ± 1 μm.

We create an ensemble of parallel 1D traps (*20*) by superimposing on the crossed dipole trap a 2D optical lattice (Fig. 2), made from two horizontal, orthogonal standing waves with slightly different frequencies (*30*). The lattice is generated by a Ti-sapphire laser, 3.2 THz to the blue of the D2 line, with a 600-μm waist and up to 700 mW per beam. The depth of the lattice, *U*_{0}, can thus reach 16 μK·*k*_{B} = 87 *E*_{rec}, where *k*_{B} is the Boltzmann constant and *E*_{rec} is the atom's recoil energy. The maximum transverse oscillation frequency, ω_{⊥}/2π, is 70.7 kHz. The blue-detuned lattice anti-traps in the axial direction, but only very weakly for our atoms, which are in the transverse ground state. The net vertical oscillation frequency, ω_{v}, is reduced from its value in the crossed dipole trap by at most 2.8% by the lattice light, so we can scan the transverse confinement without affecting the axial confinement. The very large lattice beam waist and large *w*_{0} compared with the initial cloud size means that ω_{⊥} and ω_{v} are nearly the same for the whole ensemble of ∼6400 1D traps. The traps differ only in the number of atoms each contains, *N*_{tube}. For *P* = 12 mW (320 mW), *N*_{tube} ∼ 54 (270) for the central tubes.

The 2D lattice is turned on slowly, to avoid nonadiabatic excitation of 1D breathing modes and keep the atoms in the lowest energy axial state. Adiabaticity is achieved by observing the in situ vertical cloud size after the lattice is turned on and keeping residual oscillations below 10% (supporting online text). When turned fully on, the lattice light causes spontaneous emission at a rate of 0.4 Hz, and background gas collisions occur at 0.4 Hz. Either event usually causes an atom to leave the trap, and we observe that 15% of the atoms are lost by the time we make our 1D measurements. To ensure that the remaining atoms are still near zero temperature, we reverse the procedure for turning on the lattice and measure how many atoms return to the BEC in the crossed dipole trap. Of the remaining atoms, 80% return to the 3D BEC, which implies that the thermal energy in 1D is not substantial.

Tunneling between tubes while the lattice is turned on may lead to a redistribution of atoms among the 1D tubes. We do not observe this in the measured horizontal width, *w*_{h}, when the atoms are trapped in both light traps. To the extent that tunneling does occur, it has a minimal effect, after the tubes are averaged, on our 1D calculations.

To measure ϵ, we suddenly turn off only the crossed dipole trap and let the atoms expand ballistically while they are still in the 1D tubes. With *P* = 12 mW, when *U*_{0} < 20 *E*_{rec}, the transverse width of the atomic ensemble (Fig. 3A, squares) increases, ballistically at the lowest *U*_{0} values and like a random walk () above 10 *E*_{rec}, which we attribute to tunneling between tubes. At higher values of *U*_{0}, however, the only expansion is vertical. There are thus no complications from transverse kinetic energy, mean field interactions between tubes, or spreading of 1D energy into other directions. When *U*_{0} ≳ 20 *E*_{rec}, all requirements for observing a TG gas are well realized in this system (*25*).

We take images 7 ms and 17 ms after the crossed dipole trap is turned off. All 1D strips in the 2D charge-coupled device image are summed together to obtain the linear distributions, which fit well to a Gaussian. At all values of *U*_{0}, expansion in 1D exactly resembles ballistic expansion. We extract ϵ from the measured 1D temperature, *T*_{1D}, by the simple relation, ϵ = *k*_{B}*T*_{1D}/2. We measure *T*_{1D} with *P* = 12 mW as a function of *U*_{0}, which changes the transverse confinement (Fig. 3A). We also calculate *T*_{1D} as a function of *U*_{0} and *N*_{tube}, using the exact theory (solid curve, valid for all values of γ), mean field theory (long-dashed curve, valid for γ ≪ 1), and TG theory (short-dashed curve, valid for γ ≫ 1) (*25*). *N*_{tube} values are assigned to tubes according to the initial distribution in the crossed dipole trap, and a weighted average is taken (supporting online material). The theory curves have no free parameters. Agreement between experiment and the exact theory is excellent over the full range so that the system is reliably 1D. When γ_{avg}, the weighted average γ for all the tubes, is 5.5, *T*_{1D} is already quite close to its value in the TG limit. The insensitivity of *T*_{1D} to transverse squeezing is manifest in the data.

Figure 3B shows *T*_{1D} as a function of transverse confinement with the same lattice parameters as in Fig. 3A, but with *P* = 320 mW, where both *N*_{tube} and ω_{v} are larger, and γ is nine times smaller. Here the exact theory (solid curve) more closely resembles mean field theory (long-dashed curve), and *T*_{1D} does not level off to the same degree. The reliably 1D data (*U*_{0} ≳ 20 *E*_{rec}) in this limit also agree well with the exact theory (*25*), with no free parameters. The two limiting and the exact theory curves in Fig. 3, A and B, are each part of a universal family of curves parameterized by γ. The curves in Fig. 3B thus have the same shape as the leftmost part of the curves in Fig. 3A.

Unlike in Fig. 3A, where *P* = 12 mW, there is no transverse expansion between *U*_{o} = 10 and 20 *E*_{rec} when *P* = 320 mW (Fig. 3B, squares). Because the lattice is the same in the two cases (and the other potential has been turned off), we know that the atoms are not isolated in their respective tubes. That suggests that the suppression of tunneling between tubes in this region is due to intertube mean field interactions. Although this is in itself an interesting effect (*15*), for the present purposes it means that the tubes are not independent 1D systems at those lattice depths.

Figure 4 shows the root mean square (rms) full length, *w*_{rms}, of the 1D trapped atom cloud as a function of *U*_{0}. With full lattice power, the aspect ratio of these 1D systems is 350. The density distribution in individual tubes is expected to vary between an inverted parabola (low γ) and the square root of an inverted parabola (high γ) (*25*). When either of these distributions, or the sum over tubes with these distributions, is numerically convolved with *w*_{ir}, the resulting distribution is nearly Gaussian. We experimentally observe Gaussian distributions of rms full length, *w*_{ex}. To compare experiment with theory, we deconvolve *w*_{ir} from Gaussian fits of the data, assuming that both the atomic distribution and the instrumental resolution are Gaussians, so . Numerical calculations show that *w*_{m} – *w*_{rms} ranges from 2 to 4% for our parameters. Because this difference is smaller than the dominant systematic error (our knowledge of *w*_{ir}), we do not adjust our measured widths to account for it.

Without free parameters, the measured lengths accord well with the exact 1D Bose gas theory (Fig. 4, solid curve) (*25*), applied to our ensemble of tubes (supporting online text). Mean field theory (Fig. 4, long-dashed curve) predicts larger lengths and a larger rate of increase with γ. By γ_{avg} = 5.7 (there are 12% fewer atoms in this scan than in the scan in Fig. 3A), the size is still ∼20% short of the asymptotic size in the TG limit (Fig. 4, short-dashed curve). The remaining wave function overlap at γ_{avg} = 5.7 is reduced as γ grows mostly because *r* (and hence *w*_{rms}) increases. At γ_{avg} = 5.7, the total linear extent of the atoms is actually smaller in TG theory than in mean field theory. However, we plot the rms length, and because the TG distribution is flatter, its rms length is larger.

Fermionization is evident from the combination of Figs. 3A and 4. Figure 4 shows that the cloud length is smaller than the mean field value, so the 1D density is larger. But even with a larger 1D density, Fig. 3A shows that the total energy of the atoms is 2.5 times smaller than the mean field value. This can only result from substantially reduced wave function overlap.

In summary, we have constructed a set of parallel 1D Bose gases where axial and transverse trapping are mutually independent. We can therefore change the 1D coupling strength without changing the axial trapping, and realize a free Bose gas in the 1D TG regime. Reminiscent of fermions in a superconductor acting like bosons, strongly coupled 1D bosons act like fermions. By measuring the total 1D energy and the axial size of the atomic distribution, with both strong and weak coupling, we illustrate the large difference between the two coupling regimes.