Coupling constant in quasi one- and two- dimensional systems

The mean-field relation of the coupling constant in 1D, $g_{1D}$ and in 2D, $g_{2D}$, to the three dimensional scattering length $a_{3D}$ in restricted geometries can be found by repeating the derivation given in Sec. 1.5.1 while assuming that the order parameter $\psi$ can be factorized. We start from the energy functional (1.110)

$$E[\psi] = \int \left(\frac{\hbar^2}{2m} \vert\nabla \psi\vert^2+V_{ext}\vert\psi\vert^2 + \frac{g_{3D}}{2} \vert\psi\vert^4 \right) {\bf dr},$$ (1.116)

where, according to (1.86), $g_{3D} = 4\pi\hbar^2a/m$ is the three dimensional coupling constant. The variational procedure

$$i\hbar \frac{\partial \psi}{\partial t} = \frac{\delta E}{\delta \psi^*}$$ (1.117)

gives time-dependent Gross-Pitaevskii equation

$$i\hbar \frac{\partial\psi({\vec r},t)}{\partial t}=\left(-\frac{\hbar^2\triangle}{2m}+ g_{3D}\vert\psi({\vec r},t)\vert^2\right) \psi({\vec r},t)$$ (1.118)

In the presence of an external confinement along one direction (disk-shaped condensate) $V_{ext}(\r) = m\omega^2x^2/2$ we assume a gaussian ansatz for the wave function $\psi({\bf r},t) = \psi_{osc}(x)\varphi(y,z,t)$ with $\psi_{osc}(x) = \pi^{-1/4} a_{osc}^{-1/2} \exp\left(-x^2/2a_{osc}^2\right)$ being ground state wave function of a harmonic oscillator. The integration over $x$ in (1.116) can be easily done by using following properties of the gaussian function $\psi_{osc}$:

1. Normalization properties
   

   $$\int \psi^2_{osc}(x)\, dx = 1,\qquad \int \psi^4_{osc}(x)\, dx = \frac{1}{\sqrt{2\pi}a_{osc}}$$ (1.119)

   
   

2. The function $\psi_{osc}$ is a stationary solution of a one-dimensional Schrödinger equation in a trap
   

   $$\left(-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}+ \frac{m\omega^2 x^2}{2}\right) \psi_{osc}(x) = \frac{\hbar\omega}{2} \psi_{osc}(x)$$ (1.120)

   
   

Integrating out $x$ from the GP energy functional (1.116) and doing the variational procedure (1.117) we obtain the Gross-Pitaevskii equation in a quasi two dimensional system

$$i\hbar \frac{\partial \varphi(y, z, t)}{\partial t} =\left(-\frac... ... g_{2D} \vert\varphi(y,z,t)\vert^2+\frac{\hbar \omega}{2}\right)\varphi(y,z,t),$$ (1.121)

where the two dimensional coupling constant is given by

$$g_{2D} = \frac{g_{3D}}{\sqrt{2\pi}a_{osc}}= \frac{2\sqrt{2\pi}\hbar^2a}{ma_{osc}}$$ (1.122)

If the external potential restricts the motion in two dimensions (i.e. in a cigar-shaped condensate) and the confinement is so strong that no excitations in the radial direction are possible, the wave function gets factorized in the following way: $\psi({\vec r},t) = \psi_{osc}(x)\psi_{osc}(y)\phi(z)$. The explicit integration in (1.116) over $x$ and $y$ leads to one-dimensional Gross-Pitaevskii equation

$$i\hbar \frac{\partial \varphi(z, t)}{\partial t} =\left(-\frac{\h... ...\partial z^2} + g_{1D} \vert\varphi(z,t)\vert^2+\hbar\omega\right) \varphi(z,t)$$ (1.123)

Here $g_{1D}$ denotes effective one-dimensional coupling constant

$$g_{1D} = \frac{g_{3D}}{2\pi a^2_{osc}} = \frac{2\hbar^2a}{ma_{osc}^2}$$ (1.124)

Comparing it with the definition of the 1D coupling constant $g_{1D} = - 2\hbar^2/(ma_{1D})$ (1.69) we find the mean-field relation of one-dimensional scattering length $a_{1D}$ to the three-dimensional scattering length $a$ and oscillator length $a_{osc}$:

$$a_{1D} = -\frac{a_{osc}^2}{a}$$ (1.125)