Gross–Pitaevskii equation ($i\hslash {\partial }_t\psi =(-{\hslash }^2{\nabla }^2/(2m)+g|\psi {|}^2+V)\psi$)

The Gross–Pitaevskii equation describes a dilute interacting condensate through a complex order parameter $\psi$, normalized so that $|\psi {|}^2$ is number density. Repulsive contact interactions have $g>0$. In a frame rotating at the condensate frequency, the operator includes $-\mu$, with chemical potential $\mu$. It is generated by a Hamiltonian functional; its homogeneous positive-density state has $n_0=\mu /g$ when $g,\mu >0$.

For a normalized condensate orbital $\varphi ={\pi }^{-3/4}{a}_0^{-3/2}{e}^{-r^2/(2a_0^2)}$ and $\psi =\sqrt{N}\varphi$, the contact interaction functional is $E_I=(U_0/2)\int |\psi {|}^4{d}^{3}r$. The Gaussian integral gives $\int |\varphi {|}^4{d}^{3}r=[(2\pi {)}^{3/2}{a}_0^3{]}^{-1}$. With $U_0=4\pi {\hslash }^{2}a/m$, this becomes $E_I/(N\hslash \omega )=Na/(\sqrt{2\pi }{a}_0)$. An exact number-state pair count replaces $N^2$ by $N(N-1)$.

The density-gradient contribution to the real stationary condensate equation after writing $\psi =\sqrt{\rho }{e}^{i\varphi }$. Neglecting it gives the bulk Thomas–Fermi density relation, but fails near a depleted interface or a vortex core.

Let $D_n={\partial }_t+{\mathbf{v}}_n\cdot \nabla$ be the material derivative of the normal fluid, and $\zeta >0$. The displayed phenomenological density damping preserves the condensate continuity equation because its coefficient is real. Under a Galilean boost, both velocities shift by the boost velocity and the field acquires the Schrödinger phase; $D_n|\psi {|}^2$ is invariant. Thus the normal-fluid derivative, rather than a derivative along the condensate flow, is essential to the model.

With the normal fluid at rest, linearize about the unit-amplitude plane wave $e^{i(\mathbf{k}\cdot \mathbf{x}-k^{2}t/2)}$. For amplitude and phase perturbations proportional to $e^{i\mathbf{l}\cdot \mathbf{x}+\sigma t}$, the continuity equation gives $(\sigma +i\mathbf{k}\cdot \mathbf{l})R=l^2\Theta /2$, and the phase equation gives $(\sigma +i\mathbf{k}\cdot \mathbf{l})\Theta =-(2+l^2/2+2\zeta \sigma )R$. Their determinant is the displayed dispersion relation. The damping contains the laboratory exponent $\sigma$, not the Doppler-shifted exponent, because the reference flow is the normal component.

Write $\psi =\sqrt{\rho }{e}^{i\theta }$ away from zeros, and ${\mathbf{v}}_s=\nabla \theta$. Its Madelung equations give ${\rho }_t+\nabla \cdot (\rho {\mathbf{v}}_s)=0$ and ${\theta }_t+|{\mathbf{v}}_s{|}^2/2=1-\rho +{\nabla }^2\sqrt{\rho }/(2\sqrt{\rho })-\zeta D_n\rho$. For an incompressible normal fluid, $D_n\rho =-\nabla \cdot [\rho ({\mathbf{v}}_s-{\mathbf{v}}_n)]$. Taking a gradient proves the displayed equation when each space-time derivative of the slowly varying amplitudes is $O(ϵ)$: the quantum potential force is $O({ϵ}^3)$, while the bulk term is $O({ϵ}^2)$.

Factorizing a normalized condensate field as ${\Psi }_{3D}(x,y,z)=\psi (x,y)\chi (z)$ with $\int |\chi {|}^{2}dz=1$ reduces its contact interaction energy to $(g_{2D}/2)\int |\psi {|}^{4}dxdy$, with the displayed coupling. Thus $g_{2D}$ has units of energy times area, whereas $g_{3D}=4\pi {\hslash }^{2}a/m$ has units of energy times volume for a physical scattering length $a$. For a normalized axial Gaussian $\chi (z)={\pi }^{-1/4}{a}_z^{-1/2}{e}^{-z^2/(2a_z^2)}$, direct Gaussian integration gives $g_{2D}=g_{3D}/(\sqrt{2\pi }{a}_z)$. A model with density normalized per unit axial length can instead retain $g_{3D}$ with a line-density normalization.

A local two-body and three-body mean-field energy density $V_0{n}^2/2+W_0{n}^3/3$ generates cubic and quintic terms in the Gross–Pitaevskii equation. Variation with respect to ${\psi }^{\ast }$ gives $(V_{0}n+W_0{n}^2)\psi$, with $n=|\psi {|}^2$. A uniform stationary state's chemical potential is $\mu =V_0{n}_0+W_0{n}_0^2$. Its positive-compressibility condition is $V_0+2W_0{n}_0>0$. If $\mu >0$, scaling amplitude by $\sqrt{n_0}$ and distance by $\hslash /\sqrt{2m\mu }$ gives coefficients $\alpha =V_0{n}_0/\mu$, $\beta =W_0{n}_0^2/\mu$ and $\alpha +\beta =1$. Negative chemical potential requires a different sign in the dimensionless linear term.

For $i{\psi }_t=-D{\nabla }^2\psi +[V(n)+i\Gamma (n)]\psi$, with real $D,V,\Gamma$ and $\psi =\sqrt{n}{e}^{iS}$, the continuity equation isMultiply the equation by ${\psi }^{\ast }$ and subtract its complex conjugate multiplied by $\psi$ to obtain this identity; real interaction and potential terms cancel. Thus the current velocity is $\mathbf{j}/n=2D\nabla S$. With dimensional kinetic operator $-{\hslash }^2{\nabla }^2/(2m)$ and left side $i\hslash {\psi }_t$, this gives the physical superfluid velocity $(\hslash /m)\nabla S$. In normalized equations with $D=1/2$ or $D=1$, the current velocity is respectively $\nabla S$ or $2\nabla S$. The distinction fixes the radial-flow slope of a driven quantum vortex.

For the pumped Gross–Pitaevskii equation with gain term $i(\gamma -\Gamma n)\psi$, put $\psi =\sqrt{n}\exp (im\varphi /\hslash )$. The physical superfluid velocity is $\nabla \varphi$. The imaginary part gives the displayed number balance, while the real part gives $m{\varphi }_t+m|\nabla \varphi {|}^2/2+gn-{\hslash }^2{\nabla }^2\sqrt{n}/(2m\sqrt{n})=0$. These Madelung equations hold away from density zeros. The source term permits stationary density with nonzero current divergence; local gain need not balance local loss pointwise.

Neglecting density-gradient kinetic energy in the stationary Gross–Pitaevskii equation gives $n=(\mu -V{)}_{+}/g$ for repulsive interactions. For a radial harmonic trap $V=m{\omega }_r^2{r}^2/2$, this is $n=n_0(1-r^2/R_r^2{)}_{+}$ with $R_r=\sqrt{2g{n}_0/(m{\omega }_r^2)}$. Hard boundaries can truncate this natural support. The approximation fails in vortex cores and boundary healing regions and requires a cloud much larger than its healing length.

Repulsive trapped condensates change smoothly from a nearly ideal quantum harmonic oscillator ground state for $\chi \ll 1$ to a broad Thomas–Fermi approximation for a condensate for $\chi \gg 1$. Comparing the Gaussian interaction energy of a trapped condensate with the total ideal energy gives $E_I/E_0=2\chi /(3\sqrt{2\pi })$. Thus $N\sim a_0/a$ is a crossover scale, not a sharply defined critical particle number. Total interaction energy must be compared with total ideal energy; comparing it with one oscillator quantum confuses total and per-particle scales.

For repulsive contact interactions in a three-dimensional isotropic harmonic trap, neglecting the kinetic term in the Gross–Pitaevskii equation gives $n(r)=(\mu -m{\omega }^2{r}^2/2)/U_0$ inside $R=\sqrt{2\mu /(m{\omega }^2)}$ and zero outside. Normalization yields $N=8\pi \mu R^3/(15U_0)$, hence $\mu =(\hslash \omega /2)(15Na/a_0{)}^{2/5}$ and $R=a_0(15Na/a_0{)}^{1/5}$, with oscillator length $a_0=\sqrt{\hslash /(m\omega )}$. The density is parabolic but the wavefunction is its square root. Neglected kinetic energy rounds the edge and supplies a tail.

In occupied bulk regions where density-gradient energy is negligible, the stationary relation $g\rho =\mu -V$ gives speed of sound $c=\sqrt{g\rho /m}$. Empty regions do not have an imaginary physical speed of sound; the bulk condensate formula is simply inapplicable there.

For a miscible two-component Bose-Einstein condensate with $V_i=A_i{r}^2$, $D=U_{11}{U}_{22}-U_{12}^2>0$, invert the local interaction matrix. The displayed result and $n_2=[U_{11}({\mu }_2-A_2{r}^2)-U_{12}({\mu }_1-A_1{r}^2)]/D$ hold in the overlap region. Factoring by $R_i^2={\mu }_i/A_i$ puts the ratio $\lambda =A_2/A_1$ in the first profile and ${\lambda }^{-1}$ in the second. Outside overlap, a negative formal density must be replaced by the appropriate single-component solution, not retained as a physical density.

In a two-dimensional isotropic harmonic trap with repulsive effective coupling $g>0$, neglecting density-gradient kinetic energy gives the displayed parabolic number density. Normalization to $N$ particles yields $\mu =\sqrt{Ngm{\omega }^2/\pi }$, radius $R^2=2\mu /(m{\omega }^2)$ and central density $n_c=\mu /g$. The bulk approximation requires $\mu \gg \hslash \omega$, equivalently $mgN/(\pi {\hslash }^2)\gg 1$. It fails near the outer edge and inside quantum vortex cores. The coupling must have two-dimensional units; an unreduced three-dimensional contact coupling cannot be inserted without a normalization convention.

For the normalized conservative equation $-2i{\psi }_t={\nabla }^2\psi +(1-|\psi {|}^2)\psi$, a localized disturbance translating at speed $v$ obeys ${\nabla }^2\psi +(1-|\psi {|}^2)\psi -2iv{\psi }_x=0$ in translating coordinates. Its complex-field boundary is $\psi \to 1$. This permits zero-net-winding vortex pairs, but excludes a lone quantum vortex. The usual subsonic far field has $\psi -1=O(r^{-1})$ in two dimensions.

For a finite-energy Gross–Pitaevskii solitary wave, write $E=E_z+E_{\perp }+E_{\mathrm{pot}}$ with $E_z=\frac{1}{2}\int |{\psi }_z{|}^2$. The longitudinal dilation ${\psi }_{\lambda }(x_{\perp },z)=\psi (x_{\perp },\lambda z)$ leaves its renormalized momentum of a condensate invariant and changes energy to $\lambda E_z+{\lambda }^{-1}(E_{\perp }+E_{\mathrm{pot}})$. Stationarity of $E-Up$ at $\lambda =1$ implies $E_z=E_{\perp }+E_{\mathrm{pot}}$ and hence the displayed identity. This is a directional Pohozaev identity, valid when the integrals and variation boundary terms converge.

With $E=\frac{1}{2}\int [|\nabla \psi {|}^2+\frac{1}{2}(1-|\psi {|}^2{)}^2]$ and the renormalized momentum of a condensate, a Gross–Pitaevskii solitary wave is a critical point of $E-v{p}_x$. Along a differentiable solution family, fixed bulk normalization and vanishing variation boundary terms give $dE/d\lambda =vd{p}_x/d\lambda$. Therefore $dE/d{p}_x=v$ wherever $d{p}_x/d\lambda \ne 0$. At a cusp or turning point the parametrized identity is the appropriate statement; one must not assume a globally single-valued dispersion branch.

For a condensate with uniform bulk complex field one, its convergent localized momentum isThe subtraction removes the uniform-background phase-gradient contribution. If $\psi -1=O(r^{-1})$ and $\nabla \psi =O(r^{-2})$, its integrand is $O(r^{-3})$, integrable in two dimensions. Under decaying variations its functional derivative is $\delta p_j/\delta {\psi }^{\ast }=-i{\partial }_j\psi$; the subtraction is a boundary correction and does not change the bulk equation.

The healing length of a repulsive homogeneous condensate is the length at which density-gradient kinetic energy and interaction energy balance. The displayed convention sets the dimensionless Laplacian coefficient to one in the stationary Gross–Pitaevskii equation. A hard-wall number density profile has width $\sqrt{2}{\ell }_h$; definitions without the factor $\sqrt{2}$ occur, so the equation and convention must be specified.

A repulsive condensate next to an impenetrable planar wall has Dirichlet boundary condition $\psi (0)=0$. Writing $\psi =\sqrt{n_0}f$ reduces its stationary Gross–Pitaevskii equation to ${\ell }_h^2{f}^{\prime \prime }+(1-f^2)f=0$. The bulk conditions $f\to 1,f^{\prime }\to 0$ give ${\ell }_h^2(f^{\prime }{)}^2=(1-f^2{)}^2/2$ and hence the displayed number density. Omitting $n_0$ describes normalized number density, not dimensional number density.