N-body system

This section presents our many-body study, which investigates the properties of quasi-one dimensional Bose gases over a wide range of scattering lengths $a_{3D}$. We focus specifically on three distinct regimes: 1) $0<a_{3D}<a_{3D}^c$ ($g_{1D}$ is positive), 2) $\vert a_{3D}\vert \rightarrow \infty$ ($g_{1D}$ and $a_{1D}$ are independent of $a_{3D}$; unitary regime); and iii) $a_{3D}\rightarrow -0$ ($a_{1D}$ is large and positive; onset of instability). We discuss the energetics of quasi-one dimensional Bose gases for $N=10$. Our results presented here support our earlier conclusions, which are based on a study conducted for a smaller system, i.e., for $N=5$ [ABGG04b].

For small $\lambda$ ($\lambda =0.01$), the radial angular frequency $\omega _\perp$ dominates the eigenenergies of the 3D and of the 1D Schrödinger equation. The shift of the eigenenergy of the lowest-lying gas-like state as a function of the interaction strength is, however, set by the axial angular frequency $\omega_z$. To emphasize the dependence of the eigenenergies of the lowest-lying gas-like state on $\omega_z$, we report the energy per particle subtracting the constant offset $\hbar\omega_\perp$, that is, we report the quantity $E/N-\hbar\omega_\perp$.

Consider the lowest-lying gas-like state of the 3D Schrödinger equation. Figure 4.6 shows the 3D energy per particle, $E_{3D}/N-\hbar \omega _\perp$, as a function of $a_{3D}$ for $N=10$ under quasi-one dimensional confinement, $\lambda =0.01$, for the hard-sphere two-body potential $V^{HS}$ (diamonds) and the short-range potential $V^{SR}$ (asterisks). The energies for $V^{HS}$ are calculated using the DMC method [with $\psi _T$ given by Eqs. 2.37 and 2.90], while those for $V^{SR}$ are calculated using the FN-MC method [with $\psi _T$ given by Eqs. 2.37 and 2.91]. For small $a_{3D}/a_\perp$, the energies for these two two-body potentials agree within the statistical uncertainty. For $a_{3D}\gtrsim a_\perp$, however, clear discrepancies are visible. The DMC energies for $V^{SR}$ cross the TG energy per particle (indicated by a dashed horizontal line), $E/N-\hbar\omega_\perp=\hbar\omega_\perp \lambda N/2$, very close to the value $a_{3D}^{c}=0.9684a_\perp$ (indicated by a vertical arrow in Fig. 4.6(b)), while the energies for $V^{HS}$ cross the TG energy per particle at a somewhat smaller value of $a_{3D}$.

Figure 4.6: Three-dimensional FN-MC energy per particle, $E_{3D}/N-\hbar \omega _\perp$, calculated using $V^{HS}$ (diamonds) and $V^{SR}$ (asterisks), respectively, together with 1D FN-MC energy per particle, $E_{1D}/N-\hbar \omega _\perp$, calculated using $g_{1D}$ [squares, Eq. 4.3] and $g_{1D}^{0}$ [pluses, Eq. 4.6], respectively, as a function of $a_{3D}$ [(a) linear scale; (b) logarithmic scale] for $N=10$ and $\lambda =0.01$. The statistical uncertainty of the FN-MC energies is smaller than the symbol size. Dotted and solid lines show the 1D energy per particle calculated within the LDA for $g_{1D}^0$, Eq. 4.6 [using the LL equation of state] and for $g_{1D}$, Eq. 4.3 [using the LL equation of state for $g_{1D}>0$, and the hard-rod equation of state for $g_{1D}<0$], respectively. A dotted horizontal line indicates the energy per particle of a non-interacting Bose gas, and a dashed horizontal line indicates the TG energy per particle. A vertical arrow the position where $g_{1D}$, Eq. 4.3, diverges.

For $a_{3D}>a_{3D}^c$, the energy for the short-range potential $V^{SR}$ of the lowest-lying gas-like state increases slowly with increasing $a_{3D}$, and becomes approximately constant for large values of $\vert a_{3D}\vert$. The limit $\vert a_{3D}\vert \rightarrow \infty$ corresponds to the unitary regime (see below). Notably, the 3D energy behaves smoothly as $a_{3D}$ diverges. The 3D energy slowly increases further for increasing negative $a_{3D}$, and changes more rapidly as $a_{3D}\rightarrow -0$. The $\vert a_{3D}\vert \rightarrow \infty$ regime and the $a_{3D}\rightarrow -0$ regime are discussed in more detail below.

To compare our results obtained for the 3D Hamiltonian, $H_{3D}$, with those for the 1D Hamiltonian, $H_{1D}$, we also solve the Schrödinger equation for $H_{1D}$, Eq. 4.19, for the lowest-lying gas-like state. For positive coupling constants, $g_{1D}>0$, the lowest-lying gas-like state is the many-body ground state, and we hence use the DMC method [with $\psi _T$ given by Eqs. 2.64 and 2.65]. For $g_{1D}<0$, however, the 1D Hamiltonian supports cluster-like bound states. In this case, the lowest-lying gas-like state corresponds to an excited many-body state, and we hence solve the 1D Schrödinger equation by the FN-MC method [with $\psi _T$ given by Eqs. 2.64 and 2.66].

Figure 4.6 shows the resulting 1D energies per particle, $E_{1D}/N-\hbar \omega _\perp$, for the renormalized coupling constant $g_{1D}$ [squares, Eq. 4.3], and the unrenormalized coupling constant $g_{1D}^0$ [pluses, Eq. 4.6], respectively. The 1D energies calculated using the two different coupling constants agree well for small $a_{3D}$, while clear discrepancies become apparent for $a_{3D} \gtrsim a_{3D}^c$. In fact, the 1D energies calculated using the unrenormalized coupling constant $g_{1D}^0$ approach the TG energy (dashed horizontal line) asymptotically for $a_{3D}\rightarrow\infty$, but do not become larger than the TG energy. The 1D energies calculated using the renormalized 1D coupling constant $g_{1D}$ agree well with the 3D energies calculated using the short-range potential $V^{SR}$ (asterisks) up to very large values of the 3D scattering length $a_{3D}$. In contrast, the 1D energies deviate clearly from the 3D energies calculated using the hard-sphere potential $V^{HS}$ (diamonds) at large $a_{3D}$.

The 1D energies calculated using the renormalized coupling constant agree with the 3D energies calculated using the short-range potential $V^{SR}$ also for $\vert a_{3D}\vert \rightarrow \infty$, that is, in the unitary regime, and for negative $a_{3D}$. Small deviations between the 1D energies calculated using the renormalized 1D coupling constant $g_{1D}$ and the 3D energies calculated using the short-range potential $V^{SR}$ are visible; we attribute these to the finite range of $V^{SR}$. The deviations should decrease with decreasing range $R$ of the short-range potential $V^{SR}$. On the other hand, $R$ determines to first order the energy-dependence of the scattering length $a_{3D}$. Thus, usage of an energy-dependent coupling constant $g_{1D}$ should also reduce the deviations between the 1D energies and the 3D energies calculated using the short-range potential $V^{SR}$ [GB04]. Such an approach is, however, beyond the scope of this paper.

We conclude that the renormalization of the effective 1D coupling constant $g_{1D}$ and of the 1D scattering length $a_{1D}$ are crucial to reproduce the results of the 3D Hamiltonian $H_{3D}$ when $a_{3D}\gtrsim a_\perp$ and when $a_{3D}$ is negative.

In addition to treating the 1D many-body Hamiltonian using the FN-MC technique, we solve the 1D Schrödinger equation using the LL equation of state ($g_{1D}>0$) and the hard-rod equation of state ($g_{1D}<0$) within the LDA (see Sec. 4.3). These treatments are expected to be good when the size of the cloud is much larger than the harmonic oscillator length $a_z$, where $a_z=\sqrt{\hbar/m\omega_z}$, that is, when $a_{3D}$ is large and positive or when $a_{3D}$ is negative.

Dotted lines in Fig. 4.6 show the 1D energy per particle calculated within the LDA for $g_{1D}^0$ (using the LL equation of state), while solid lines show the 1D energy per particle calculated within the LDA for $g_{1D}$, Eq. 4.3 (using the LL equation of state for $g_{1D}>0$, and the hard-rod equation of state for $g_{1D}<0$). Remarkably, the LDA energies nearly coincide with the 1D many-body DMC energies calculated using the unrenormalized coupling constant (pluses) and the renormalized coupling constant (squares), respectively. Finite-size effects play a minor role only for $a_{3D}\ll a_\perp$. Our calculations thus establish that a simple treatment, i.e., a hard-rod equation of state treated within the LDA, describes inhomogeneous quasi-one dimensional Bose gases with negative coupling constant $g_{1D}$ well over a wide range of 3D scattering lengths $a_{3D}$.

For $a_{3D}\rightarrow -0$, that is, for large $a_{1D}$, the hard-rod equation of state treated within the LDA, cannot properly describe trapped quasi-one dimensional Bose gases, which are expected to become unstable against formation of cluster-like many-body bound states for $a_{1D} \approx 1/n_{1D}$. Thus, Sec. 4.5 investigates the regime with negative $a_{3D}$ in more detail within a many-body framework.