Generalized nonpolynomial Schrödinger equations
for matter waves under anisotropic transverse confinement

At ultralow temperature dilute Bose-Einstein condensates (BECs) can be accurately described by the three-dimensional Gross-Pitaevskii equation (3D GPE) [1]. Some years ago we have found that, starting from the 3D GPE and using a Gaussian variational approach [2, 3], it is possible to derive an effective 1D wave equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry [4, 5]. In this derivation the trapping potential is harmonic and isotropic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent nonpolynomial nonlinear Schrödinger equation (1D NPSE) [4, 5], has been used to model cigar-shaped condensates by many experimental and theoretical groups (see for instance [6, 7, 8, 9]). Moreover, by using the 1D NPSE we have found analytical and numerical solutions for solitons and vortices, which generalize the ones known in the literature [10]. Recently, we have obtained a disrete version of NPSE, the 1D DNPSE, which models Bose-Einstein condensate confined in a combination of a cigar-shaped trap and deep optical lattice acting in the axial direction [11].

A relevant limitation of the 1D NPSE [4, 5] is the fact that the transverse harmonic confinement is isotropic. To overcome this problem in this paper we introduce a 1D generalized nonpolynomial Schrödinger equation (1D g-NPSE) which describes the dynamics of matter waves under the action of a generic potential in the longitudinal axial direction and of an anisotropic harmonic potential in the transverse radial direction. This equation reduces to the familiar 1D NPSE [4, 5] in the case of isotropic transverse harmonic confinement. In the second part of the paper we consider a deep periodic optical lattice in the axial direction. In this case we find that the 3D GPE can be mapped into a 1D generalized discrete nonpolynomial Schrödinger equation (1D g-DNPSE), which becomes the 1D DNPSE obtained in Ref. [11] if the matter waves are confined by a transverse isotropic harmonic trap.

We consider a dilute BEC confined in the axial direction by a generic potential $V(z)$ and in the transverse plane by the anisotropic harmonic potential

$$U(x,y)={m\over 2}\omega_{\bot}^{2}\left(\lambda^{2}x^{2}+y^{2}\right)\;,$$

(1)

where $\lambda$ is the anisotropy of the transverse harmonic trap: $\omega_{x}=\lambda\omega_{\bot}$ and $\omega_{y}=\omega_{\bot}$ are the two harmonic trapping frequencies in the $x$ and $y$ directions. The characteristic harmonic length is $a_{\bot}=(\hbar/(m\omega_{\bot}))^{1/2}$ and so the characteristic length and time units are $a_{\bot}$ and $\omega_{\bot}^{-1}$, and the characteristic energy unit is $\hbar\omega_{\bot}$. The system is well described by the 3D GPE, and in scaled units it reads

$$i{\frac{\partial\psi}{\partial t}}=\left[-{\frac{\hbar^{2}}{2m}}\nabla^{2}+{1\over 2}\left(\lambda^{2}x^{2}+y^{2}\right)+V(z)+2\pi g|\psi|^{2}\right]\psi\;,$$

(2)

where $\psi(\mathbf{r},t)$ is the macroscopic wave function of the condensate normalized to unity, and $g=2Na_{s}/a_{\bot}$, with $N$ the number of atoms and $a_{s}$ the s-wave scattering length of the inter-atomic potential. This equation can be derived from the Lagrangian density

$$\mathcal{L}={i\over 2}\big{(}\psi^{\ast}\frac{\partial\psi}{\partial t}+\psi\frac{\partial\psi^{\ast}}{\partial t}\big{)}-{\frac{1}{2}}|\nabla\psi|^{2}-{\frac{1}{2}}(\lambda^{2}x^{2}+y^{2})|\psi|^{2}-V(z)|\psi|^{2}-\pi g|\psi|^{4}\;.$$

(3)

To reduce the 3D problem to a 1D one, we apply a variational approach. In the case of an anisotropic cigar-shaped geometry, it is natural to extend the variational representation [4, 5] of the 3D GPE which led to the NPSE for the axial wave function in the isotropic case to the present situation. Thus we adopt the following ansatz for the state described by 3D GPE

$$\psi={1\over\sqrt{\pi}\sigma\eta}\exp{\left[-\left({x^{2}\over 2\sigma^{2}}+{y^{2}\over 2\eta^{2}}\right)\right]}\,f\;,$$

(4)

where $\sigma(z,t)$, $\eta(z,t)$ and $f(z,t)$, which account for transverse widths and axial wave function, are the 3 effective fields to be determined variationally.

By inserting this ansatz into the Lagrangian density (3), performing the integration over $x$ and $y$ and neglecting spatial derivatives of transverse widths (adiabatic approximation [4]), we derive the effective Lagrangian density

	$\displaystyle\bar{\mathcal{L}}$	$\displaystyle=$	$\displaystyle{i\over 2}\big{(}f^{\ast}\frac{\partial f}{\partial t}+f\frac{\partial f^{\ast}}{\partial t}\big{)}-\frac{1}{2}\left|{\frac{\partial f}{\partial z}}\right|^{2}-\frac{1}{4}\left(\frac{1}{\sigma^{2}}+\frac{1}{\eta^{2}}\right)|f|^{2}$		(5)
			$\displaystyle-{1\over 4}\left(\lambda^{2}\sigma^{2}+\eta^{2}\right)|f|^{2}-V(z)|f|^{2}-\frac{1}{2}g\frac{|f|^{4}}{\sigma\eta}\;.$

This effective Lagrangian gives rise to a system of 3 Euler-Lagrange equations, obtained by varying $\bar{\mathcal{L}}$ with respect to $f^{\ast}$, $\sigma$ and $\eta$:

	$\displaystyle i\frac{\partial f}{\partial t}$	$\displaystyle=$	$\displaystyle\Big{[}-\frac{1}{2}\frac{\partial^{2}}{\partial z^{2}}+\frac{1}{4}\left(\frac{1}{\sigma^{2}}+\frac{1}{\eta^{2}}+\lambda^{2}\sigma^{2}+\eta^{2}\right)+V(z)+g\frac{|f|^{2}}{\sigma\eta}\Big{]}f\;,$		(6)
	$\displaystyle\lambda^{2}\sigma^{4}$	$\displaystyle=$	$\displaystyle 1+g|f|^{2}{\sigma\over\eta}\;,\quad\quad\quad\quad\quad\quad\quad\;$		(7)
	$\displaystyle\eta^{4}$	$\displaystyle=$	$\displaystyle 1+g|f|^{2}{\eta\over\sigma}\;.\quad\quad\quad\quad\quad\quad\quad$		(8)

We call this system of three coupled equations, which describe the BEC under a transverse anisotropic harmonic confinement, the 1D generalized nonpolynomial Schrödinger equation (1D g-NPSE). Notice that with $g\neq 0$ only for $\lambda=1$ one finds $\sigma=\eta$. In this case of isotropic transverse harmonic confinement ($\lambda=1$) we have

$$\sigma^{4}=\eta^{4}=1+g|f|^{2}\;,$$

(9)

and the g-NPSE becomes

$$i{\partial\over\partial t}f=\Big{[}-\frac{1}{2}\frac{\partial^{2}}{\partial z^{2}}+V(z)\Big{]}f+{1+(3/2)g|f|^{2}\over\sqrt{1+g|f|^{2}}}f\;,$$

(10)

which is the 1D NPSE derived in Ref. [4, 5]. Instead, for an anisotropic transverse harmonic confinement ($\lambda\neq 1$) the transverse widths $\sigma$ and $\eta$ depend on $g$ in a not trivial way.

Let us consider again Eq. (2) where now the axial potential is given by

$$V(z)=V_{0}\cos{(2kz)}\;.$$

(27)

This potential models the optical lattice produced in experiments with Bose-Einstein condensates by using counter-propagating laser beams [15].

To symplify the problem we set

$$\psi(x,y,z,t)=\sum_{j}\phi_{j}(x,y,t)\ W_{j}(z)$$

(28)

where $W_{j}(z)$ is the Wannier function maximally localized at the $j$-th minimum of the axial periodic potential. This tight-binding ansatz is reliable in the case of a deep optical lattice [15, 16]. We insert this ansatz into Eq. (2), multiply the resulting equation by $W_{n}^{*}(z)$ and integrate over $z$ variable. In this way we get

$$i{\partial\over\partial t}\phi_{n}=\left[-\frac{1}{2}\nabla_{\bot}^{2}+\frac{1}{2}\left(\lambda^{2}x^{2}+y^{2}\right)+\epsilon\right]\phi_{n}-J\left(\phi_{n+1}+\phi_{n-1}\right)+2\pi\gamma\left|\phi_{n}\right|^{2}\phi_{n}\>,$$

(29)

where the parameters $\epsilon$, $J$ and $\gamma$ are given by

$$\epsilon=\int W_{n}^{*}(z)\left[-{1\over 2}{\partial^{2}\over\partial z^{2}}+V_{0}\cos\left(2kz\right)\right]W_{n}(z)\ dz\;,$$

(30)

$$J=-\int W_{n+1}^{*}(z)\left[-{1\over 2}{\partial^{2}\over\partial z^{2}}+V_{0}\cos\left(2kz\right)\right]W_{n}(z)\ dz\;,$$

(31)

$$\gamma=g\int|W_{n}(z)|^{4}\ dz\;.$$

(32)

In the tight-binding regime the parameter $J$ is positive definite. In addition the parameters $\epsilon$, $J$ and $\gamma$ are independent on the site index $n$.

Eq. (29) can be seen as the Euler-Lagrange equation of the following Lagrangian density

	$\displaystyle{\cal L}$	$\displaystyle=$	$\displaystyle\sum_{n}\Big{\{}\phi_{n}^{*}\left[i{\partial\over\partial t}+{1\over 2}\nabla_{\bot}^{2}-{1\over 2}(\lambda^{2}x^{2}+y^{2})-\epsilon\right]\phi_{n}$		(33)
			$\displaystyle+J\phi_{n}^{*}\left(\phi_{n+1}+\phi_{n-1}\right)-\pi\gamma|\phi_{n}|^{4}\Big{\}}\;.$

To further simplify the problem we set

$$\phi_{n}={1\over\sqrt{\pi}\sigma_{n}\eta_{n}}\exp{\left[-\left({x^{2}\over 2\sigma_{n}^{2}}+{y^{2}\over 2\eta_{n}^{2}}\right)\right]}\,f_{n}\;,$$

(34)

where $\sigma_{n}(t)$, $\eta_{n}(t)$ and $f_{n}(t)$, which account for discrete transverse widths and axial wave function, are the effective generalized coordinates to be determined variationally.

We insert this ansatz into the Lagrangian density (33) and integrate over $x$ and $y$ variables. In this way we obtain an effective Lagrangian for the fields $f_{n}(t)$ and $\sigma_{n}(t)$. This Lagrangian reads

	$\displaystyle L$	$\displaystyle=$	$\displaystyle\sum_{n}\Big{\{}f_{n}^{*}\left[i{\partial\over\partial t}-{1\over 4}\left({1\over\sigma_{n}^{2}}+\lambda^{2}\sigma_{n}^{2}+{1\over\eta_{n}^{2}}+\eta_{n}^{2}\right)-\epsilon\right]f_{n}$		(35)
			$\displaystyle+Jf_{n}^{*}\left(f_{n+1}+f_{n-1}\right)-{\gamma\over 2\sigma_{n}\eta_{n}}|f_{n}|^{4}\Big{\}}\;.$

To obtain this expression we have supposed that the transverse widths $\sigma_{n}(t)$ of nearest-neighbor sites are practically equal (this is the equivalent to the adiabatic approximation made for the continuous model). The Euler-Lagrange equation of (35) with respect to $f_{n}^{*}$ is

	$\displaystyle i{\partial\over\partial t}f_{n}$	$\displaystyle=$	$\displaystyle\left[{1\over 4}\left({1\over\sigma_{n}^{2}}+\lambda^{2}\sigma_{n}^{2}+{1\over\eta_{n}^{2}}+\eta_{n}^{2}\right)+\epsilon\right]f_{n}$		(36)
			$\displaystyle-J\left(f_{n+1}+f_{n-1}\right)+{\gamma\over\sigma_{n}\eta_{n}}|f_{n}|^{2}f_{n}\;.$

while the Euler-Lagrange equations of (35) with respect to $\sigma_{n}$ and $\eta_{n}$ give

	$\displaystyle\lambda^{2}\sigma_{n}^{4}=1+\gamma|f_{n}|^{2}{\sigma_{n}\over\eta_{n}}\;,$		(37)
	$\displaystyle\eta_{n}^{4}=1+\gamma|f_{n}|^{2}{\eta_{n}\over\sigma_{n}}\;.$		(38)

We call this system of coupled equations, which describe the BEC under a transverse anisotropic harmonic confinement and an axial optical lattice, the 1D generalized and discrete nonpolynomial Schrödinger equation NPSE (1D g-DNPSE) In the case of isotropic transverse harmonic confinement ($\lambda=1$) we have

$$\sigma_{n}^{4}=\eta_{n}^{4}=1+\gamma|f_{n}|^{2}\;,$$

(39)

and the 1D g-DNPSE becomes

$$i{\partial\over\partial t}f_{n}=\epsilon f_{n}-J\left(f_{n+1}+f_{n-1}\right)+{1+(3/2)\gamma|f_{n}|^{2}\over\sqrt{1+g|f_{n}|^{2}}}f_{n}\;,$$

(40)

which is substantially the 1D DNPSE derived in Ref. [11].

The 1D g-DNPSE given by Eqs. (36), (37) and (38) is the discrete version of 1D g-NPSE given by Eqs. (6), (7) and (8). It is then strightforward to repeat with the 1D g-DNPSE the analysis of weak-coupling, strong-coupling and quasi-2D regimes we have performed with the 1D g-NPSE.

We have obtained new nonpolynomial Schrödinger equations for matter waves under anisotropic transverse harmonic confinement. In the case of isotropic transverse confinement these continuous and discrete equations reduce to the ones derived in previous papers. The generalized 1D continuous nonpolynomial Schrödinger equation can be used to study the static and dynamics of Bose condensates with a generic longitudinal axial potential. The generalized 1D discrete nonpolynomial Schrödinger equation can be insead used to investigate the effect of a deep axial periodic potential. The comparison with the full 3D Gross-Pitaevskii equations show that these effective 1D equations are indeed quite accurate and much simpler for numerical and analytical computations.

The author thanks Alberto Cetoli, Boris Malomed, Giovanni Mazzarella, Dmitry Pelinovsky and Flavio Toigo for useful comments and suggestions. This work has been partially supported by Fondazione CARIPARO.