Stability of quasi-one-dimensional Bose gases

This section discusses the stability of inhomogeneous quasi-one dimensional Bose gases with negative $g_{1D}$, that is, with $a_{3D}>a_{3D}^c$ and $a_{3D}<0$, against cluster formation. Section 4.4.1 shows that the FN-MC results for the 1D Hamiltonian, Eq. (4.19), are in very good agreement with the FN-MC results for the 3D Hamiltonian. Hence, we carry our analysis out within the 1D model Hamiltonian, Eq. 4.19; we believe that our final conclusions also hold for the 3D Hamiltonian, Eq. 4.18. For the inhomogeneous 1D Hamiltonian $H_{1D}$, Eq. 4.19, the lowest-lying gas-like state is a highly-excited state (see Sec. 4.3). We now address the question whether this state is stable quantitatively using the VMC method.

We solve the 1D many-body Schrödinger equation for the Hamiltonian $H_{1D}$, Eq. 4.19, by the VMC method using the trial wave function $\psi _T$ given by Eqs. 2.64 and 2.65. This many-body wave function has the same nodal constraint as a system of $N$ hard-rods of size $a_{1D}$. However, contrary to hard-rods, for interparticle distances smaller than $a_{1D}$ the amplitude of the wave function increases as $\vert z\vert$ decreases. This effect arises from the attractive nature of the 1D effective potential and gives rise in the many-body framework to the formation of cluster-like bound states as the average interparticle distance is reduced below a certain critical value.

Figure 4.7 shows the resulting VMC energy per particle, $E_{1D}/N-\hbar \omega _\perp$, for $N=5$ and $\lambda =0.01$ as a function of the Gaussian width $\alpha _z$ for four different values of $a_{1D}$. For $a_{1D}/a_\perp =1.0326$ and $2$, Fig. 4.7 shows a local minimum at $\alpha_{z,min} \approx a_{z}$. The minimum VMC energy nearly coincides with the FN-MC energy (see also Fig. 4.8), which suggests that our variational wave function provides a highly accurate description of the quasi-one dimensional many-body system. The energy barrier at $\alpha_z \approx 0.2 a_{z}$ decreases with increasing $a_{1D}$, and disappears for $a_{1D}/a_\perp\approx 3$. We interpret this vanishing of the energy barrier as an indication of instability [BEG98]. For small $a_{1D}$, the energy barrier separates the lowest-lying gas-like state from cluster-like bound states. Hence, the gas-like state is stable against cluster formation. For larger $a_{1D}$, this energy barrier disappears and the gas-like state becomes unstable against cluster formation.

Figure 4.7: VMC energy per particle, $E_{1D}/N-\hbar \omega _\perp$, as a function of the variational parameter $\alpha _z$ for $N=5$, $\lambda =0.01$ and $a_{1D}/a_\perp =1.0326$ (pluses), $2$ (asterisks), $3$ (diamonds) and $4$ (triangles). An energy barrier is present for $a_{1D}/a_\perp =1.0326$ and 2, but not for $a_{1D}/a_\perp =4$.

We stress that our stability analysis should not be confused with that carried out for attractive inhomogeneous 3D systems at the level of mean-field Gross-Pitaevskii theory [BP96]. In fact, a mean-field type analysis of inhomogeneous 1D Bose gases does not predict stability of gas-like states [CKR01]. In our analysis, the emergence of local energy minima in configuration space is due to the structure of the two-body correlation factor $f_2(z)$ entering the VMC trial wave function $\psi _T$, Eqs. 2.64 and 2.65. It is a many-body effect that cannot be described within a mean-field Gross-Pitaevskii framework.

To additionally investigate the dependence of stability on the number of particles, Fig. 4.8 shows the VMC energy for $\lambda =0.01$ as a function of the variational parameter $\alpha _z$ for different values of $N$, $N=5,10$ and $20$. The scattering length $a_{1D}$ is fixed at the value corresponding to the unitary regime, $a_{1D}=1.0326a_\perp$. Figure 4.8 shows that the height of the energy barrier decreases for increasing $N$. Figures 4.7 and 4.8 suggest that the stability of 1D Bose gases depends on $a_{1D}$ and $N$. To extract a functional dependence, we additionally perform variational calculations for larger $N$ and different values of $\lambda$ and $a_{1D}$. We find that the onset of instability of the lowest-lying gas-like state can be described by the following criticality condition

$$\sqrt{N\lambda}\frac{a_{1D}}{a_\perp}\simeq 0.78,$$ (4.33)

or, equivalently, by $\sqrt{N}{a_{1D}}/{a_z} \simeq 0.78$. Our 1D many-body calculations thus suggest that the lowest-lying gas-like state is stable if $\sqrt{N\lambda}a_{1D}/a_\perp\lesssim 0.78$, and that it is unstable if $\sqrt{N\lambda}a_{1D}/a_\perp\gtrsim 0.78$. The stability condition, Eq. (4.33), implies that reducing the anisotropy parameter $\lambda$ should allow stabilization of relatively large quasi-one dimensional Bose gases.

Figure 4.8: VMC energy per particle, $E_{1D}/N-\hbar \omega _\perp$, as a function of the variational parameter $\alpha _z$ for $a_{1D}/a_\perp =1.0326$ (corresponding to the unitary regime), $\lambda =0.01$, and $N=5$ (pluses), $10$ (asterisks), and $20$ (diamonds). (The $N=5$ data are also shown in Fig. 4.7.) The height of the energy barrier decreases with increasing $N$. Horizontal solid lines show the corresponding energies for $N=5$, $10$ and $20$ obtained using the FN-MC technique, which are in excellent agreement with the VMC energy obtained for $\alpha _z=\alpha _{z,min}$.

To express the stability condition, Eq. 4.33, in terms of the 1D gas parameter $n_{1D}a_{1D}$, where $n_{1D}$ denotes the linear density at the trap center, we approximate the density for negative $g_{1D}$ by the TG density, Eq. 4.24. Figure 4.9 compares the TG density with that obtained from the VMC calculations for $N=5,10$ and 20 and values of $a_{1D}/a_\perp$ close to the criticality condition, Eq. 4.33. The density at the center of the trap is described by the TG density to within 10 %. Since the TG density at the trap center is given by $\sqrt{2N}/(\pi a_z)$ (see Eq. 4.24), the stability condition, Eq. 4.33, expressed in terms of the 1D gas parameter reads $n_{1D}a_{1D}\lesssim 0.35$.

Figure 4.9: TG density [Eq. 4.24, dashed lines] as a function of $z$ together with VMC density (solid lines), obtained by solving the 1D many-body Schrödinger equation, Eq. 4.21, for $N=5$ and $a_{1D}/a_\perp =3.6$, for $N=10$ and $a_{1D}/a_\perp =2.6$, and for $N=20$ and $a_{1D}/a_\perp =1.8$. The TG density at the center of the trap, $z=0$, deviates from the VMC density at the center of the trap by less than 10 %.