Two Bosons under quasi-one-dimensional confinement

Consider two interacting mass $m$ bosons with position vectors $\vec{r}_1$ and $\vec{r}_2$, where $\vec{r}_i=(x_i,y_i,z_i)$, in a waveguide with harmonic confinement in the radial direction. If we introduce the center of mass coordinate $\vec{R}$ and the relative coordinate $\vec{r}=\vec{r}_2-\vec{r}_1$, the problem separates. Since the solution to the center of mass Hamiltonian is given readily, we only consider the internal Hamiltonian $H_{3D}^{int}$, which can be conveniently written in terms of cylindrical coordinates $\vec{r}=(\rho,\phi,z)$,

$$\hat H^{int}_{3D} = -\frac{\hbar^2}{2\mu}\Delta + V_{int}(\vec{r}) + \frac{1}{2} \mu \omega_\perp^2 \rho^2,$$ (4.1)

where $\mu$ denotes the reduced two-body mass, $\mu = m/2$, and $V_{int}(\vec{r})$ denotes the full 3D atom-atom interaction potential.

Considering a regularized zero-range pseudo-potential $V_{int}(\vec{r}) = 2 \pi \hbar^2 a_{3D}/\mu \delta(\vec{r})\frac{\partial}{\partial r}r$, where $a_{3D}$ denotes the 3D scattering length, Olshanii [Ols98] derives an effective 1D Hamiltonian,

$$H^{int}_{1D}= - \frac{\hbar^2}{2 \mu} \frac{d^2}{d z^2} + g_{1D} \delta(z) + \hbar \omega_\perp,$$ (4.2)

and renormalized coupling constant $g_{1D}$,

$$g_{1D}=\frac{2 \hbar^2 a_{3D}}{m a_\perp^2} \left [ 1-\vert\zeta(1/2)\vert \frac{a_{3D}}{\sqrt{2}a_\perp} \right]^{-1},$$ (4.3)

which reproduce the low energy scattering solutions of the full 3D Hamiltonian, Eq. 4.1. Here, $\zeta(\cdot)$ denotes the Riemann zeta function, $\zeta(1/2) = -1.4604$. Alternatively, $g_{1D}$ can be expressed through the effective 1D scattering length $a_{1D}$ [Ols98],

$$g_{1D}= -\frac{2\hbar^2}{ma_{1D}},$$ (4.4)

where

$$a_{1D} = -a_\perp\left(\frac{a_\perp}{a_{3D}}-\frac{\vert\zeta(1/2)\vert}{\sqrt 2}\right)$$ (4.5)

Olshanii's result shows that the waveguide gives rise to an effective interaction, parameterized by the coupling constant $g_{1D}$, which can be tuned to any strength by changing the ratio between the 3D scattering length $a_{3D}$ and the transverse oscillator length $a_\perp$.

The renormalized coupling constant, Eq. 4.3, can be compared with the unrenormalized coupling constant $g_{1D}^0$ (1.124),

$$g_{1D}^0=\frac{2\hbar^2 a_{3D}}{m a_\perp^2},$$ (4.6)

Figure 4.1 shows the unrenormalized coupling constant $g_{1D}^0$ [dashed line, Eq. 4.6] together with the renormalized coupling constant [solid line, Eq. 4.3]. For $\vert a_{3D}\vert\ll a_\perp$, the renormalized coupling constant $g_{1D}$ is nearly identical to the unrenormalized coupling constant $g_{1D}^0$. For large $\vert a_{3D}\vert$, however, the confinement induced renormalization becomes important, and the effective 1D coupling constant $g_{1D}$, Eq. 4.3, has to be used. At the critical value $a_{3D}^c=0.9684 a_\perp$ (indicated by a vertical arrow in Fig. 4.1), $g_{1D}$ diverges. For $a_{3D}\to\pm\infty$, $g_{1D}$ reaches an asymptotic value, $g_{1D}=-1.9368a_\perp \hbar \omega _\perp$ (indicated by a horizontal arrow in Fig. 4.1). Finally, $g_{1D}$ is negative for all negative 3D scattering lengths. The inset of Fig. 4.1 shows the effective 1D scattering length $a_{1D}$, Eq. 4.5, as a function of $a_{3D}$. For small positive $a_{3D}$, $a_{1D}$ is negative and it changes sign at $a_{3D}=a_{3D}^c$ ($a_{1D}=0$ for $a_{3D}=a_{3D}^c$). Moreover, $a_{1D}$ reaches, just as $g_{1D}$, an asymptotic value for $\vert a_{3D}\vert \rightarrow \infty$, $a_{1D}=1.0326a_\perp$ (indicated by a horizontal arrow in the inset of Fig. 4.1). The renormalized 1D scattering length $a_{1D}$ is positive for negative $a_{3D}$, and approaches $+\infty$ as $a_{3D}\rightarrow -0$. Figure 4.1 suggests that tuning the 3D scattering length $a_{3D}$ to large values allows a universal quasi-one dimensional regime, where $g_{1D}$ and $a_{1D}$ are independent of $a_{3D}$, to be entered.

Figure 4.1: One-dimensional coupling constants $g_{1D}$ [Eq. 4.3, solid line] and $g_{1D}^{0}$ [Eq. 4.6, dashed line] as a function of the 3D scattering length $a_{3D}/a_\perp$. The vertical arrow indicates the value of $a_{3D}$ for which $g_{1D}$ diverges, $a_{3D}^{c}=0.9684a_\perp$. The horizontal arrow indicates the asymptotic value of $g_{1D}$ as $\vert a_{3D}\vert \rightarrow \infty$, $g_{1D}=-1.9368a_\perp \hbar \omega _\perp$. Inset: One-dimensional scattering length $a_{1D}$, Eq. 4.5, as a function of $a_{3D}/a_\perp$. The vertical arrow indicates the value of $a_{3D}$ for which $a_{1D}$ goes through zero, $a_{3D}^{c}=0.9684a_\perp$. The horizontal arrow indicates the asymptotic value of $a_{1D}$ as $\vert a_{3D}\vert \rightarrow \infty$, $a_{1D}=1.0326a_\perp$. The angular frequency $\omega _\perp$ determines the frequency $\nu _\perp$, $\omega _\perp =2 \pi \, \nu _\perp$ (also, $\hbar \omega _\perp =h \nu _\perp$).

The effective coupling constant $g_{1D}$, Eq. 4.3, has been derived for a wave guide geometry, that is, with no axial confinement. However, it also describes the scattering between two bosons confined to other quasi-one dimensional geometries. Consider, e.g., a Bose gas under harmonic confinement. If the confinement in the axial direction is weak compared to that of the radial direction, the scattering properties of each atom pair are expected to be described accurately by the effective coupling constant $g_{1D}$ and the effective scattering length $a_{1D}$.

The internal motion of two bosons under highly-elongated confinement can be described by the following 3D Hamiltonian

$$\hat H^{int}_{3D} = -\frac{\hbar^2}{2 \mu}\Delta + V_{int}(\vec{r}) +\frac{1}{2} \mu \left(\omega_\perp^2 \rho^2 + \omega_z^2z^2 \right),$$ (4.7)

where $\omega_z$ denotes the angular frequency in the longitudinal direction, $\omega_z= \lambda\,\omega_\perp$ ($\lambda$ denotes the aspect ratio, $\lambda \ll 1$). The eigenenergies $E_{3D}^{int}$ and eigenfunctions $\psi_{3D}^{int}$ of this Hamiltonian satisfy the Schrödinger equation,

$$\hat H^{int}_{3D} \psi_{3D}^{int}(\rho,z)= E^{int}_{3D} \psi_{3D}^{int}(\rho,z).$$ (4.8)

The corresponding 1D Hamiltonian reads

$$H^{int}_{1D}= - \frac{\hbar^2}{2 \mu} \frac{d^2}{dz^2} + g_{1D} \delta(z) + \frac{1}{2}\mu\omega_z^2 z^2 + \hbar\omega_\perp$$ (4.9)

The 1D eigenenergies $E^{int}_{1D}$ of the stationary Schrödinger equation,

$$\hat H^{int}_{1D} \psi_{1D}^{int}(z)=E^{int}_{1D} \psi_{1D}^{int}(z),$$ (4.10)

can be determined semi-analytically by solving the transcendental equation [BEKW98],

$$g_{1D} = 2 \sqrt{2} \frac{\Gamma(\chi_z+1)}{\Gamma(\chi_z+1/2)} \; \mathop{\rm tg}\nolimits (\pi \chi_z) \; \hbar \omega_z \, a_z,$$ (4.11)

self consistently for $\chi_z$ (for a given $g_{1D}$). In the above equation, $\chi_z$ is an effective (possibly non-integer) quantum number, which determines the energy $E_z$,

$$\chi_z=\frac{E_z}{2\hbar \omega_z}-\frac{1}{4}.$$ (4.12)

The energy $E_z$, in turn, determines the internal 1D eigenenergies $E_{1D}^{int}$,

$$E_{1D}^{int} = \lambda E_z + \hbar \omega_\perp.$$ (4.13)

In Eq. 4.11, $a_z$ denotes the characteristic oscillator length in the axial direction, $a_z=\sqrt {\hbar /(m\omega _z)}$.

To compare the eigenenergies $E^{int}_{3D}$ and $E^{int}_{1D}$, we use, for the 3D atom-atom interaction potential $V(r)$, a short-range (SR) modified Pöschl-Teller potential (1.97) $V^{SR}(r)$ that can support two-body bound states,

$$V^{SR}(r)=-\frac{V_0}{\mathop{\rm ch}\nolimits ^2(r/R)}$$ (4.14)

In the above equation, $V_0$ denotes the well depth, and $R$ the range of the potential. In our calculations, $R$ is fixed at a value much smaller than the transverse oscillator length, $R = 0.1 a_\perp$. To simulate the behavior of $a_{3D}$ near a field-dependent Feshbach resonance, we vary the well depth $V_0$, and consequently, the scattering length $a_{3D}$. It has been shown that such a model describes many atom-atom scattering properties near a Feshbach resonance properly [TWMJ00]. Figure 4.2 shows the dependence of the 3D scattering length $a_{3D}$ on $V_0$. Importantly, $a_{3D}$ diverges for particular values of the well depth $V_0$. At each of these divergencies, a new two-body $s$-wave bound state is created. The inset of Fig. 4.2 shows the range of well depths $V_0$ used in our calculations.

Figure: Three-dimensional $s$-wave scattering length $a_{3D}$ as a function of the well depth $V_0$ for the short-range model potential $V^{SR}$, Eq. 4.14. Each time the 3D scattering length diverges a new two-body $s$-wave bound state is created. Inset: Enlargement of the well depth region used in our calculations.

To benchmark our MC calculations (see Chapter 2 and Sec. 4.4), we solve the 3D Schrödinger equation, Eq. 4.8, with $\lambda =0.01$ for various well depths $V_0$ using a two-dimensional B-spline basis in $\rho$ and $z$. Figure 4.3 shows the resulting 3D eigenenergies $E^{int}_{3D}$ (diamonds) as a function of the 3D scattering length $a_{3D}$. We distinguish between two sets of states:

1) States with $E^{int}_{3D} \ge \hbar \omega_\perp$ are referred to as gas-like states; their behavior is, to a good approximation, characterized by the 3D scattering length $a_{3D}$, and is hence independent of the detailed shape of the atom-atom potential. The energies of the gas-like states are shown in Fig. 4.3(a).

2) States with $E^{int}_{3D}<\hbar \omega_\perp$ are referred to as molecular-like bound states; their behavior depends on the detailed shape of the atom-atom potential. The energies of these bound states are shown in Fig. 4.3(b). The well depth $V_0$ of the short-range interaction potential $V^{SR}$ is chosen such that $V^{SR}$ supports -- in the absence of the confining potential -- no $s$-wave bound state for $a_{3D}<0$, and one $s$-wave bound state for $a_{3D}>0$. Figure 4.3(b) shows that the bound state remains bound for $\vert a_{3D}\vert \rightarrow \infty$ and for negative $a_{3D}$ if tight radial confinement is present. In addition, a dashed line shows the 3D binding energy, $-\hbar ^2/(m a_{3D}^2)$, which accurately describes the highest-lying molecular bound state in the absence of any external confinement if $a_{3D}$ is much larger than the range $R$ of the potential $V^{SR}$.

The B-spline basis calculations yield not only the internal 3D eigenenergies $E_{3D}^{int}$, but also the corresponding wave functions $\psi_{3D}^{int}$. The nodal surface of the lowest-lying gas-like state, which is to a good approximation an ellipse in the $\rho z$-plane, is a crucial ingredient of our many-body calculations. Section 4.4 discuss in detail how this nodal surface is used to parametrize our trial wave function entering the MC calculations.

To compare the energy spectrum for $N=2$ of the effective 1D Hamiltonian with that of the 3D Hamiltonian, Fig. 4.3 additionally shows the 1D eigenenergies $E_{1D}^{int}$ (solid lines) obtained by solving the Schrödinger equation for $H_{1D}^{int}$, Eq. 4.9, semi-analytically [using the renormalized coupling constant $g_{1D}$, Eq. 4.3]. Figure 4.3(a) demonstrates excellent agreement between the 3D and the 1D internal energies for all states with gas-like character. For positive $a_{3D}$, the effective 1D Hamiltonian fails to reproduce the energy spectrum of the molecular-like bound states of the 3D Hamiltonian accurately [see Fig. 4.3(b), and also [BMO03,BTJ03]].

Figure 4.3: Internal eigenenergies $E^{int}$ as a function of the 3D scattering length $a_{3D}/a_\perp$ for two bosons under highly-elongated confinement with $\lambda =0.01$. (a) 3D $s$-wave eigenenergies $E^{int}_{3D}$ (diamonds) of gas-like states obtained using the short-range model potential $V^{SR}$, Eq. 4.14, in a B-spline basis set calculation together with internal 1D eigenenergies $E^{int}_{1D}$ (solid lines). Excellent agreement between the 3D and 1D energies is found. Horizontal dotted lines show the lowest internal eigenenergies for two non-interacting spin-polarized bosons, while horizontal dashed lines show those for two non-interacting spin-polarized fermions (indicated respectively by ``B'' and ``F'' on the right hand side). (b) Binding energy of molecular-like bound states. In addition to the 3D and 1D energies [diamonds and solid lines, respectively; see (a)], a dashed line shows the 3D binding energy $-\hbar ^2/(m a_{3D}^2)$. (c) Enlargement of the lowest-lying gas-like state. In addition to the 3D and 1D energies shown in (a), asterisks show the 3D energies for the interaction potential $V^{SR}$ calculated using the FN-MC technique, and squares the 1D energies for the contact interaction potential calculated using the FN-MC technique. The statistical uncertainty of the FN-MC energies is smaller than the symbol size. Good agreement between the FN-MC energies (asterisks and squares) and the energies calculated by alternative means (diamonds and solid lines) is found.

Our main focus is in the lowest-lying energy level with gas-like character. This energy branch is shown in Fig. 4.3(c) on an enlarged scale. A horizontal dashed line shows the lowest internal 3D eigenenergy for two non-interacting spin-polarized fermions (where the anti-symmetry of the wave function enters in the $z$ coordinate). Our numerical calculations confirm [BMO03] that for $a_{3D}=a_{3D}^c$ ( $g_{1D} \rightarrow \infty$) the two boson system behaves as if it consisted of two non-interacting spin-polarized fermions (TG gas). The energy $E^{int}_{3D}$ is larger than that of two non-interacting fermions for $a_{3D}>a_{3D}^c$, and approaches the first excited state energy of two non-interacting bosons for $a_{3D}\rightarrow -0$ [indicated by a dotted line in Fig. 4.3(a)].

For positive $g_{1D}$, the 1D Schrödinger equation, Eq. 4.10, does not support molecular-like bound states. Consequently, the wave function of the lowest-lying gas-like state is positive definite everywhere. For negative $g_{1D}$, however, one molecular-like two-body bound state exists. If $a_{1D}\ll a_z$ the bound-state wave function is approximately given by the eigenstate $\psi_{1D}^{int}$ of the 1D Hamiltonian without confinement, Eq. 4.2,

$$\psi_{1D}^{int}(z)=\exp\left(-\frac{\vert z\vert}{a_{1D}}\right),$$ (4.15)

with eigenenergy $E_{1D}^{int}$,

$$E_{1D}^{int}= -\frac{\hbar^2}{ma_{1D}^2} + \hbar \omega_\perp.$$ (4.16)

For the highly-elongated trap with $\lambda =0.01$ shown in Fig. 4.3(b) and positive $a_{1D}$ the above binding energy nearly coincides with the exact eigenenergy of the molecular-like bound state obtained from the solution of the transcendental equation (4.11) (solid line). The two-body binding energy, Eq. 4.16, is largest for $a_{1D} \rightarrow +0$ ( $g_{1D} \rightarrow -\infty$); in this case, the molecular-like bound state wave function is tightly localized around $z=0$, where $z=z_2-z_1$. Consider a system with $a_{1D}\ll a_z$. For negative $g_{1D}$ (positive $a_{1D}$), the nodes along the relative coordinate $z$ of the lowest-lying gas-like wave function (in this case, the first excited state) are then approximately given by $\pm a_{1D}$. Thus, imposing the boundary condition $\psi_{1D}^{int}=0$ at $\vert z\vert=a_{1D}$ and restricting the configuration space to $z > a_{1D}$ allows one to obtain an approximation to the eigenenergy of the first excited eigen state. Furthermore, imposing the boundary condition $\psi_{1D}^{int}=0$ at $z=a_{1D}$ is identical to solving the 1D Schrödinger equation for a hard-rod interaction potential $V^{HR}(z)$ (1.79),

$$V^{HR}(z) = \left\{ \begin{array}{cll} \infty & \mbox{ for } & z<a_{1D} \\ 0 & \mbox{ for } & z \ge a_{1D}\;. \end{array}\right.$$ (4.17)

For $N=2$, asterisks in Fig. 4.3(c) show the fixed-node diffusion Monte Carlo (FN-MC) results obtained using the above fictitious hard-rod potential (see Sec. 4.4.2). Good agreement is found with the exact 1D eigenenergies obtained from Eqs. 4.11-4.12. For $N>2$ bosons, our 1D FN-MC algorithm and our usage of the hard-rod equation of state both take advantage of a reduction of configuration space similar to that discussed here for two bosons (see Sec. 4.3 and Chapter 2).