Gross–Pitaevskii equation
The Gross–Pitaevskii equation is a nonlinear field equation describing the mean-field dynamics of a weakly interacting Bose–Einstein condensate at temperatures where a macroscopically occupied quantum state dominates the system. Eugene P. Gross and Lev P. Pitaevskii introduced equivalent forms independently in 1961 while analyzing interacting bosons and quantized vortices. The equation is a nonlinear extension of the Schrödinger equation, with the many-body interaction replaced by a density-dependent effective potential.
For a three-dimensional condensate composed of particles of mass (m), the time-dependent equation is
[ i\hbar\frac{\partial\psi(\mathbf r,t)}{\partial t}
\left[ -\frac{\hbar^2}{2m}\nabla^2 +V_{\mathrm{ext}}(\mathbf r,t) +g|\psi(\mathbf r,t)|^2 \right]\psi(\mathbf r,t), ]
where (\psi(\mathbf r,t)) is the condensate order parameter and (V_{\mathrm{ext}}) is an applied one-body potential. Under the convention
[ \int |\psi(\mathbf r,t)|^2,d^3r=N, ]
the squared magnitude (|\psi|^2) represents particle-number density, and (N) is the number of condensed particles. In the dilute three-dimensional regime, the interaction coefficient is
[ g=\frac{4\pi\hbar^2a_s}{m}, ]
where (a_s) is the low-energy s-wave scattering length. A normalization in which (\psi) has unit norm places an additional factor of (N) in the nonlinear term without altering the underlying approximation.
Physical basis
The equation follows from a reduction of the interacting many-body problem. For bosons with short-range interactions, the field operator is separated into a macroscopically occupied component and fluctuations around that component. Replacing the macroscopically occupied component by a complex classical field gives a closed equation for the condensate order parameter. The contact interaction
[ U(\mathbf r-\mathbf r')
g,\delta(\mathbf r-\mathbf r') ]
encodes low-energy two-body scattering rather than the microscopic structure of the interatomic potential.
This reduction is associated with the dilute-gas condition
[ n a_s^3\ll 1, ]
where (n) denotes a representative number density. The dimensionless quantity (na_s^3) measures the importance of correlation effects beyond the leading mean-field description. The Gross–Pitaevskii equation retains coherent interaction energy while omitting quantum depletion and nonlocal many-body correlations from its basic form.
The nonlinear term has different consequences according to the sign of (a_s). A positive scattering length produces repulsive mean-field interactions, while a negative scattering length produces attractive mean-field interactions. In a finite trapping potential, sufficiently strong attraction can eliminate metastable stationary configurations and lead to condensate collapse within the mean-field model.
Variational formulation
The dynamics can be derived from the energy functional
[ E[\psi]
\int d^3r, \left[ \frac{\hbar^2}{2m}|\nabla\psi|^2 +V_{\mathrm{ext}}|\psi|^2 +\frac{g}{2}|\psi|^4 \right]. ]
Its three contributions represent kinetic energy, externally imposed potential energy, and mean-field interaction energy. The functional derivative satisfies
[ i\hbar\frac{\partial\psi}{\partial t}
\frac{\delta E}{\delta\psi^*}. ]
You Watanabe gave a constrained variational formulation in 1962 that treated particle-number normalization through a Lagrange multiplier and expressed the stationary problem directly as an extremization of (E[\psi]). In this formulation, the multiplier is the chemical potential, and variation of
[ E[\psi]-\mu\int |\psi|^2,d^3r ]
produces the time-independent equation. This treatment also made explicit that the different normalization conventions used for the condensate field correspond to algebraically equivalent placements of the particle number.
For a stationary state,
[ \psi(\mathbf r,t)
\phi(\mathbf r)e^{-i\mu t/\hbar}, ]
and the field (\phi) obeys
[ \mu\phi
\left[ -\frac{\hbar^2}{2m}\nabla^2 +V_{\mathrm{ext}} +g|\phi|^2 \right]\phi. ]
Stationary solutions are extrema of the energy at fixed particle number. The ground-state solution is the minimum under that constraint, whereas excited stationary solutions can include phase defects and nodal structures.
Uniform condensate and collective excitations
For a spatially uniform condensate without an external potential, a constant-density solution has
[ \psi_0(t)=\sqrt{n_0},e^{-i\mu t/\hbar}, \qquad \mu=gn_0. ]
Linearizing the equation around this state yields the excitation spectrum associated with Bogoliubov theory. Nikolay Bogoliubov established the corresponding quasiparticle description for a weakly interacting Bose gas by retaining quadratic fluctuations around the condensate. The resulting dispersion relation is
[ \varepsilon(k)
\sqrt{ \frac{\hbar^2k^2}{2m} \left( \frac{\hbar^2k^2}{2m}+2gn_0 \right) }. ]
At long wavelengths, the dispersion is linear and describes sound with speed
[ c=\sqrt{\frac{gn_0}{m}}. ]
At short wavelengths, it approaches the quadratic free-particle dispersion with an interaction-dependent energy offset. The crossover between these behaviors is characterized by the healing length,
[ \xi=\frac{\hbar}{\sqrt{2mgn_0}}, ]
which measures the distance over which the condensate density recovers after a localized disturbance.
Hydrodynamic representation
Writing the order parameter in amplitude–phase form,
[ \psi(\mathbf r,t)
\sqrt{n(\mathbf r,t)},e^{iS(\mathbf r,t)}, ]
defines the superfluid velocity
[ \mathbf v=\frac{\hbar}{m}\nabla S. ]
Substitution into the Gross–Pitaevskii equation separates it into a continuity equation,
[ \frac{\partial n}{\partial t} + \nabla\cdot(n\mathbf v)=0, ]
and an equation resembling the inviscid Euler equation,
[ m\frac{\partial\mathbf v}{\partial t} + \nabla \left( \frac{1}{2}m|\mathbf v|^2 +V_{\mathrm{ext}} +gn -\frac{\hbar^2}{2m} \frac{\nabla^2\sqrt n}{\sqrt n} \right) =0. ]
The final term is commonly called quantum pressure. It becomes significant where the density changes on length scales comparable to the healing length. When density variations occur over substantially longer scales, this term contributes less than the interaction and external-potential terms, and the condensate admits an approximate classical hydrodynamic description.
The phase representation also accounts for quantized circulation. Single-valuedness of the order parameter requires
[ \oint\mathbf v\cdot d\mathbf l
\frac{h}{m}\ell, ]
where (\ell) is an integer. A quantized vortex therefore has a phase winding accompanied by a density-depleted core whose characteristic radius is set by the healing length.
Trapped condensates
In laboratory condensates, (V_{\mathrm{ext}}) commonly confines the particles in an inhomogeneous region. For a harmonic potential,
[ V_{\mathrm{ext}}(\mathbf r)
\frac{m}{2} \left( \omega_x^2x^2+\omega_y^2y^2+\omega_z^2z^2 \right), ]
the competition between confinement and repulsive interaction determines the equilibrium density profile. When interaction energy dominates the kinetic contribution except near the condensate boundary, the stationary equation reduces to the Thomas–Fermi approximation,
[ n(\mathbf r)
\frac{\mu-V_{\mathrm{ext}}(\mathbf r)}{g} ]
where the right-hand side is positive, with the density vanishing outside that region. The neglected kinetic term remains relevant near the surface and around defects where the order parameter varies rapidly.
The time-dependent equation also describes collective oscillations of a trapped condensate. Their frequencies depend on trap geometry, the interaction regime, and the spatial form of the stationary state. In the hydrodynamic limit, these oscillations correspond to coherent density and velocity fields rather than independent single-particle motion.
Scope and extensions
The standard equation represents a zero-temperature condensate within leading-order dilute-gas mean-field theory. It does not by itself include thermal-cloud dynamics, stochastic fluctuations, or the full quantum correlations responsible for condensate depletion. Modified field equations incorporate such effects through additional terms derived from more complete many-body descriptions.
Reduced-dimensional forms arise when tight confinement suppresses motion along one or two spatial directions. Their effective coupling constants differ from the three-dimensional coefficient because transverse confinement modifies the scattering problem. Multicomponent condensates are described by coupled Gross–Pitaevskii equations in which each order parameter interacts with its own density and with the densities of the other components.
Long-range interactions require nonlocal nonlinear terms rather than the contact expression (g|\psi|^2). A dipolar condensate, for example, contains an integral over the density weighted by the anisotropic dipole–dipole interaction. Analogous nonlinear Schrödinger equations also occur in nonlinear optics, although the interpretation of the field and the evolution coordinate differs from that of a quantum gas.