$N$ bosons under quasi-one-dimensional confinement

For tightly-confined trapped gases the 1D regime is reached if the transverse motion of the atoms is frozen, with all the particles occupying the ground state of the transverse harmonic oscillator. At zero temperature, this condition requires that the energy per particle is dominated by the trapping potential, $E/N=\hbar\omega_\perp+\epsilon$, where the excess energy $\epsilon$ is much smaller than the separation between levels in the transverse direction, $\epsilon\ll\hbar\omega_\perp$. In the following we consider situations where the Bose gas is in the 1D regime for any value of the 3D scattering length $a_{3D}$. For a fixed trap anisotropy parameter $\lambda$ and a fixed number of particles $N$ the above requirement is satisfied if $N\lambda\ll 1$. For $\lambda =0.01$ and $N=10$ (as considered in Sec. 4.4.1) this condition is fulfilled.

To compare the 3D and 1D energetics of a Bose gas, we consider the 3D and 1D Hamiltonian describing $N$ spin-polarized bosons,

$$\hat H_{3D}= \sum_{i=1}^N \left[-\frac{\hbar^2}{2m}\Delta_i + \fr... ...a_\perp^2 \rho_i^2 + \omega_z^2 z_i^2 \right) \right] + \sum_{i<j}^N V(r_{ij}),$$ (4.18)

and

$$\hat H_{1D}=\sum_{i=1}^N \left(-\frac{\hbar^2}{2m} \frac{\partial... ...ga_z^2 z_i^2 \right) + g_{1D}\sum_{i<j}^N \delta(z_{ij}) + N \hbar\omega_\perp,$$ (4.19)

respectively. The corresponding eigenenergies and eigenfunctions are given by solving the Schrödinger equations,

$$\hat H_{3D} \psi_{3D}(\vec{r}_1,...,\vec{r}_N)=E_{3D}\psi_{3D}(\vec{r}_1,...,\vec{r}_N)$$ (4.20)

and

$$\hat H_{1D} \psi_{1D}(z_1,...,z_N)=E_{1D} \psi_{1D}(z_1,...,z_N),$$ (4.21)

respectively. In contrast to Sec. 4.2, here we do not separate out the center of mass motion since the MC calculations used to solve the 3D and 1D many-body Schrödinger equations can be most conveniently implemented in Cartesian coordinate space (see Sec. 2.3.2). In the following, we refer to eigenstates of the confined Bose gas with energy greater than $N\hbar\omega_\perp$ as gas-like states, and to those with energy smaller than $N\hbar\omega_\perp$ as cluster-like bound states.

Section 4.4.1 compares the energetics of the lowest-lying gas-like state of the 3D Schrödinger equation, Eq. 4.20, obtained using the short-range potential V$^{SR}$, Eq. 4.14, with that obtained using the hard-sphere potential $V^{HS}$ Eq. 1.48. For $V^{HS}$, the $s$-wave scattering length $a_{3D}$ coincides with the range of the potential (see Sec. 1.3.2.2). For $V^{SR}$, in contrast, $R$ determines the range of the potential, while the scattering length $a_{3D}$ is determined by $R$ and $V_0$. For $a_{3D}\ll a_\perp$, both potentials give nearly identical results for the energetics of the lowest-lying gas-like state, which depend to a good approximation only on the value of $a_{3D}$. For $a_{3D}\gtrsim a_\perp$, instead, deviations due to the different effective ranges become visible and only $V^{SR}$ yields results, which do not depend on the short-range details of the potential and which are compatible with a 1D contact potential.

Section 4.4.1 also discusses the energetics of the 1D Hamiltonian, Eq. 4.19. For small $\vert g_{1D}\vert$, the energetics of the many-body 1D Hamiltonian are described well by a 1D mean-field equation with non-linearity. For negative $g_{1D}$, the mean-field framework describes, for example, bright solitons [CCR00a,KSU03], which have been observed experimentally [SPTH02,KSF+02]. For large $\vert g_{1D}\vert$, in contrast, the system is highly-correlated, and any mean-field treatment will fail. Instead, a many-body description that incorporates higher order correlations has to be used. In particular, the limit $\vert g_{1D}\vert \rightarrow \infty$ corresponds to the strongly-interacting TG regime.

For infinitely strong particle interactions ( $\vert g_{1D}\vert \rightarrow \infty$), Girardeau shows [Gir60], using the equivalence between the 1D $\delta$-function potential and a ``1D hard-point potential'', that the energy spectrum of the 1D Bose gas coincides with that of $N$ non-interacting spin-polarized fermions. The lowest eigenenergy per particle of the trapped 1D Bose gas, Eq. 4.21, is, in the TG limit, given by^4.1

$$\frac{E^{TG}_{1D}}{N}=\left(\frac{\lambda N}{2} + 1\right) \hbar \omega_\perp$$ (4.22)

The corresponding gas density is given by the sum of squares of single-particle wave functions

$$n^{TG}_{1D}(z)= \frac{1}{\sqrt{\pi} a_z} \sum_{k=0}^{N-1} \frac{1}{2^k k!} H_k^2(z/a_z) \exp \left\{-(z/a_z)^2\right\},$$ (4.23)

with the normalization $\int_{-\infty}^\infty n^{TG}_{1D}(z)\,dz = N$. In Eq. 4.23, the $H_k$ denote Hermite polynomials, and $z$ denotes the distance measured from the center of the trap. For large numbers of atoms, the density expression in Eq. 4.23 can be calculated using the LDA [DLO01],

$$n_{1D}^{TG}(z)= \frac{\sqrt{2N}}{ \pi a_{z}} \left(1-\frac{z^2}{2Na_z^2} \right)^{1/2} \, .$$ (4.24)

The above result cannot reproduce the oscillatory behavior of the exact density, Eq. 4.23, but it does describe the overall behavior properly (see Sec. 4.5).

To characterize the inhomogeneous 1D Bose gas further, we consider the many-body Hamiltonian of the homogeneous 1D Bose gas,

$$H_{1D}^{hom}=\sum_{i=1}^N-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z_i^2} + g_{1D}\sum_{i<j}^N \delta(z_{ij}) + N \hbar \omega_\perp$$ (4.25)

By introducing the energy shift $N\hbar\omega_\perp$, our classification of gas-like states and cluster-like bound states introduced after Eq. 4.21 remains valid. For positive $g_{1D}$, $H_{1D}^{hom}$ corresponds to the Lieb-Liniger (LL) Hamiltonian. The gas-like states of the LL Hamiltonian, including its thermodynamic properties, have been studied in detail [LL63,Lie63,YY69]. The energy per particle of the lowest-lying gas-like state, the ground state of the system, is given by

$$\frac{E_{1D}^{LL}(n_{1D})}{N}=\frac{\hbar^2}{2m} e(\gamma) n_{1D}^2,$$ (4.26)

where $n_{1D}$ denotes the density of the homogeneous system, and $e(\gamma)$ a function of the dimensionless parameter $\gamma= 2/(n_{1D} \vert a_{1D}\vert)$.

We use the known properties of the LL Hamiltonian to determine properties of the corresponding inhomogeneous system, Eq. 4.19, within the LDA. This approximation provides a correct description of the trapped gas if the size of the atomic cloud is much larger than the characteristic length scale $a_z$ of the confinement in the longitudinal direction [DLO01]. Specifically, consider the local equilibrium condition,

$$\mu(N)= \hbar \omega_\perp + \mu_{local}[n_{1D}(z)]+\frac{1}{2}m \omega_z^2 z^2,$$ (4.27)

where $\mu_{local}(n_{1D})$ denotes the chemical potential of the homogeneous system with density $n_{1D}$,

$$\mu_{local}(n_{1D}) = \frac{\partial\left[ n_{1D} E_{1D}^{LL}(n_{1D})/N \right]}{\partial n_{1D}}.$$ (4.28)

The chemical potential $\mu(N)$, Eq. 4.27, can be calculated using Eq. 4.28 together with the normalization of the density, $\int_{-\infty}^\infty n_{1D}(z)\,dz = N$. Integrating the chemical potential $\mu(N)$ then determines the energy of the lowest-lying gas-like state of the inhomogeneous $N$-particle system within the LDA. The LDA treatment is computationally less demanding than solving the many-body Schrödinger equation, Eq. 4.21, using MC techniques. By comparing with our full 1D many-body results we establish the accuracy of the LDA (see Sec. 4.4.1).

For negative $g_{1D}$, the Hamiltonian given in Eq. 4.25 supports cluster-like bound states. The ground state energy and eigenfunction of the system are [McG64]

$$\frac{E_{1D}^{hom}}{N} =-\frac{\hbar^2}{6m a_{1D}^2} (N^2-1)+\hbar\omega_\perp,$$ (4.29)

and

$$\psi_{1D}^{hom}(z_1,...,z_N)= \prod_{i<j}^N\exp\left\{\frac{-\vert z_i-z_j\vert}{a_{1D}}\right\},$$ (4.30)

respectively. The eigenstate given by Eq. 4.30 depends only on the relative coordinates $z_{ij}$, that is, it is independent of the center of mass of the system. Adding a confinement potential [see Eq. 4.19] with $\omega_z$ such that $a_z \gg a_{1D}$ leaves the eigenenergy $E_{1D}^{hom}$ of this cluster-like bound state to a good approximation unchanged, while the corresponding wave function becomes localized at the center of the trap. This state describes a bright soliton, whose energy can also be determined within a mean-field framework [KSU03]. An excited state of the many-body 1D Hamiltonian with confinement corresponds, e.g., to a state, where $N-1$ particles form a cluster-like bound state, i.e., a soliton with $N-1$ particles, and where one particle approximately occupies the lowest harmonic oscillator state, i.e., has gas-like character. Similarly, molecular-like bound states can form with fewer particles.

The above discussion implies that the lowest-lying gas-like state of the 1D Hamiltonian with confinement, Eq. 4.19 with negative $g_{1D}$, corresponds to a highly-excited state. For dilute 1D systems with negative $g_{1D}$, the nodal surface of this excited state can be well approximated by the following nodal surface: $\psi_{1D}=0$ for $z_{ij}=a_{1D}$, where $i,j=1,...,N$ and $i<j$. As in the two-body case, the many-body energy can then be calculated approximately by restricting the configuration space to regions where the wave function is positive. This corresponds to treating a gas of hard-rods of size $a_{1D}$. In the low density limit, we expect that the lowest-lying gas-like state of the 1D many-body Hamiltonian with $g_{1D}<0$ is well described by a system of hard-rods of size $a_{1D}$.

In addition to treating the full 1D many-body Hamiltonian, we treat the inhomogeneous system with negative $g_{1D}$ within the LDA. The equation of state of the uniform hard-rod gas with density $n_{1D}$ is given by (1.103) [Gir60]:

$$\frac{E_{1D}^{HR}(n_{1D})}{N}= \frac{\pi^2 \hbar^2 n_{1D}^2} {6m \, (1-n_{1D}a_{1D})^2} + \hbar \omega_\perp .$$ (4.31)

We use this energy in the LDA treatment (see Eqs. 4.26 through 4.28 with $E_{1D}^{LL}$ replaced by $E_{1D}^{HR}$). The hard-rod equation of state treated within the LDA provides a good description when $g_{1D}$ is negative, but $\vert g_{1D}\vert$ not too small (see Secs. 4.4.1 and 4.5). To gain more insight, we determine the expansion for inhomogeneous systems with $N\lambda\ll 1$ using the equation of state for the homogeneous hard-rod gas,

$$\frac{E_{1D}}{N} - \hbar\omega_\perp = \hbar\omega_\perp \frac{N\... ...c{128\sqrt{2}}{45\pi^2} \sqrt{N\lambda}\frac{a_{1D}}{a_\perp} + ... \right) \;.$$ (4.32)

The first term corresponds to the energy per particle in the TG regime (see Eq. 4.22), the other terms can be considered as small corrections to the TG energy. In the unitary limit, that is, for $a_{1D}/a_\perp =1.0326$, expression (4.32) becomes independent of $a_{3D}$, and depends only on $N\lambda$. Similarly, the linear density in the center of the cloud, $z=0$, is to lowest order given by the TG result, $n_{1D}^{TG}(0)=\sqrt{2N\lambda}/ (\pi a_\perp)$ (see Eq. 4.24). Section 4.5 shows that the TG density provides a good description of inhomogeneous 1D Bose gases over a fairly large range of negative $g_{1D}$.