This section presents our many-body study, which investigates the properties of quasi-one dimensional Bose gases over a wide range of scattering lengths . We focus specifically on three distinct regimes: 1) ( is positive), 2) ( and are independent of ; unitary regime); and iii) ( is large and positive; onset of instability). We discuss the energetics of quasi-one dimensional Bose gases for . Our results presented here support our earlier conclusions, which are based on a study conducted for a smaller system, i.e., for [ABGG04b].
For small (), the radial angular frequency 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 . To emphasize the dependence of the eigenenergies of the lowest-lying gas-like state on , we report the energy per particle subtracting the constant offset , that is, we report the quantity .
Consider the lowest-lying gas-like state of the 3D Schrödinger equation. Figure 4.6 shows the 3D energy per particle, , as a function of for under quasi-one dimensional confinement, , for the hard-sphere two-body potential (diamonds) and the short-range potential (asterisks). The energies for are calculated using the DMC method [with given by Eqs. 2.37 and 2.90], while those for are calculated using the FN-MC method [with given by Eqs. 2.37 and 2.91]. For small , the energies for these two two-body potentials agree within the statistical uncertainty. For , however, clear discrepancies are visible. The DMC energies for cross the TG energy per particle (indicated by a dashed horizontal line), , very close to the value (indicated by a vertical arrow in Fig. 4.6(b)), while the energies for cross the TG energy per particle at a somewhat smaller value of .
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For , the energy for the short-range potential of the lowest-lying gas-like state increases slowly with increasing , and becomes approximately constant for large values of . The limit corresponds to the unitary regime (see below). Notably, the 3D energy behaves smoothly as diverges. The 3D energy slowly increases further for increasing negative , and changes more rapidly as . The regime and the regime are discussed in more detail below.
To compare our results obtained for the 3D Hamiltonian, , with those for the 1D Hamiltonian, , we also solve the Schrödinger equation for , Eq. 4.19, for the lowest-lying gas-like state. For positive coupling constants, , the lowest-lying gas-like state is the many-body ground state, and we hence use the DMC method [with given by Eqs. 2.64 and 2.65]. For , 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 given by Eqs. 2.64 and 2.66].
Figure 4.6 shows the resulting 1D energies per particle, , for the renormalized coupling constant [squares, Eq. 4.3], and the unrenormalized coupling constant [pluses, Eq. 4.6], respectively. The 1D energies calculated using the two different coupling constants agree well for small , while clear discrepancies become apparent for . In fact, the 1D energies calculated using the unrenormalized coupling constant approach the TG energy (dashed horizontal line) asymptotically for , but do not become larger than the TG energy. The 1D energies calculated using the renormalized 1D coupling constant agree well with the 3D energies calculated using the short-range potential (asterisks) up to very large values of the 3D scattering length . In contrast, the 1D energies deviate clearly from the 3D energies calculated using the hard-sphere potential (diamonds) at large .
The 1D energies calculated using the renormalized coupling constant agree with the 3D energies calculated using the short-range potential also for , that is, in the unitary regime, and for negative . Small deviations between the 1D energies calculated using the renormalized 1D coupling constant and the 3D energies calculated using the short-range potential are visible; we attribute these to the finite range of . The deviations should decrease with decreasing range of the short-range potential . On the other hand, determines to first order the energy-dependence of the scattering length . Thus, usage of an energy-dependent coupling constant should also reduce the deviations between the 1D energies and the 3D energies calculated using the short-range potential [GB04]. Such an approach is, however, beyond the scope of this paper.
We conclude that the renormalization of the effective 1D coupling constant and of the 1D scattering length are crucial to reproduce the results of the 3D Hamiltonian when and when 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 () and the hard-rod equation of state () 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 , where , that is, when is large and positive or when is negative.
Dotted lines in Fig. 4.6 show the 1D energy per particle calculated within the LDA for (using the LL equation of state), while solid lines show the 1D energy per particle calculated within the LDA for , Eq. 4.3 (using the LL equation of state for , and the hard-rod equation of state for ). 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 . 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 well over a wide range of 3D scattering lengths .
For , that is, for large , 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 . Thus, Sec. 4.5 investigates the regime with negative in more detail within a many-body framework.