Bose-Einstein condensate ($N_0=O(N)$)

A Bose-Einstein condensate is a state with macroscopic occupation of a one-particle mode. A phase-selected mean-field description can use $⟨\psi ⟩\ne 0$, while a fixed-number finite-volume state need not have a nonzero field expectation. Bose-Einstein condensation and superfluidity are related but distinct notions; long-wavelength phase fluctuations can invalidate a uniform condensate saddle in low dimensions.

Two independent boson species occupy uniform single-particle modes on $L$ sites, with fixed occupation numbers $n_A,n_B$. This differs from a fixed-number two-mode condensate whose particles all occupy one superposition of the two modes. Here $⟨a_i^{†}{a}_j⟩=n_A/L$, $⟨b_i^{†}{b}_j⟩=n_B/L$, and the local total number operator has variance $(n_A+n_B)L^{-1}(1-L^{-1})$. The latter follows from two independent binomial distributions with success probability $1/L$. This state satisfies a prescribed Schwinger boson constraint only in the mean unless the local variance vanishes.

A complex amplitude describing a coherent condensate in mean-field theory. In the convention $\int |\psi {|}^2{d}^{3}r=N$, its squared modulus is the condensate density. Dividing by $\sqrt{N}$ gives a normalized orbital; formulas for interaction strengths and nonlinear terms must use the same normalization convention.

All $N$ bosons occupy the same normalized one-particle superposition with $|C{|}^2+|S{|}^2=1$. Its occupation distribution in the original two modes is binomial, not two independent Poisson distributions. In particular, $⟨N_1⟩=N|S{|}^2$ and $⟨N_0{N}_1⟩=N(N-1)|C{|}^2|S{|}^2$. These factorial moments give finite-particle corrections to interaction energies. The state is also the two-mode realization of a spin coherent state with spin $N/2$.

For $|\chi ⟩=(c{a}_L^{†}+e^{i\chi }s{a}_R^{†}{)}^N|0⟩/\sqrt{N!}$ with real $c,s$ and $c^2+s^2=1$, one has $|\chi ⟩=e^{i\chi n_R}|0_{\mathrm{phase}}⟩$. Differentiating $E=⟨\chi |H|\chi ⟩$ gives ${\partial }_{\chi }E=i⟨[H,n_R]⟩$. If $[H,n_L+n_R]=0$, this equals $-i⟨[H,n_L]⟩$. The Heisenberg equation of motion identifies the left-to-right particle current as $J={\partial }_{\chi }E/\hslash$. Reversing the definition of the relative phase reverses this sign.

For a homogeneous gas, the condensate density is the macroscopic occupation $N_0$ of its condensate mode divided by volume. More generally its spatial density is $N_0|{\phi }_0(x){|}^2$ for a normalized condensate orbital. The mode and occupation are defined by a macroscopic eigenvalue of the one-body density matrix, so a finite fixed-number state need not have a nonzero field expectation. Condensate density is distinct from the superfluid density defined by response to a phase twist; interactions can deplete the former even when the latter equals the total density at zero temperature in a uniform Galilean-invariant gas.

A one-dimensional condensate on a ring of circumference $2\pi R$ has a uniform winding state $\psi =\sqrt{N/(2\pi R)}{e}^{in\theta }$ when the order parameter is normalized to atom number. Its angular momentum per atom is $n\hslash$, while the flow contribution to its total chemical potential is ${\hslash }^2{n}^2/(2m{R}^2)$.

For equal ring halves with potentials $V_0$ and zero, the zero-current Thomas–Fermi bulk solution has $\mu ={\mu }_0+V_0/2$ and populations $N_{L,R}=N(1\mp V_0/2{\mu }_0)/2$ while both halves are occupied. For $V_0>2{\mu }_0$ the raised half is depleted and $\mu =2{\mu }_0$ in this approximation.

For a noninteracting ground state of energy zero, $N_0=(e^{-\beta \mu }-1{)}^{-1}$ gives $\mu =-k_{B}T\ln (1+1/N_0)$. Thus a finite condensate has a slightly negative chemical potential; exact zero pinning is a macroscopic limiting statement.

Occupation outside the macroscopically occupied condensate mode, caused by interactions or thermal excitations. A weak homogeneous three-dimensional Bose gas has zero-temperature depletion fraction $8\sqrt{n{a}_s^3}/(3\sqrt{\pi })$ in the leading dilute-gas approximation. Depletion measures single-particle mode occupation, not directly the superfluid fraction.

Two separately conserved bosonic species have two condensate fields and independent chemical potentials. A contact interaction energy density $U_{11}{n}_1^2/2+U_{22}{n}_2^2/2+U_{12}{n}_1{n}_2$ gives ${\mu }_1=U_{11}{n}_1+U_{12}{n}_2$ and ${\mu }_2=U_{12}{n}_1+U_{22}{n}_2$ in a uniform mixture. Coherent interconversion would change the conservation constraints and cannot be included while retaining independent particle-number constraints unchanged.

In a homogeneous overlapping two-component Bose-Einstein condensate, the long-wavelength energy change at fixed species numbers is one half the integral of the density perturbation quadratic form with matrix $\left(\begin{array}{cc}{U}_{11}& U_{12}\\ U_{12}& U_{22}\end{array}\right)$. Strict positive definiteness gives the displayed criterion. Strong positive cross-repulsion favors separation; sufficiently strong cross-attraction instead produces an unstable joint density mode. Equality is marginal at this order, not strict stability.