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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.

In a parabolic condensate in the Thomas–Fermi approximation for a condensate, the local sound speed is $c(D)={\omega }_r\sqrt{R^2-D^2}/\sqrt{2}$. Equating it to the obstacle speed $\Omega D$ gives the displayed threshold. This is the stipulated local phonon estimate; a finite strongly depleting obstacle can modify flow and nucleate vortices, so the inequality is not a universal exact onset law for every stirring experiment.

For the normalized Hartree product state $|N:\varphi ⟩$, applying the bosonic annihilation operator gives $\psi (y)|N:\varphi ⟩=\sqrt{N}\varphi (y)|N-1:\varphi ⟩$. Taking the inner product proves the displayed factorization of the one-body density matrix. Its diagonal is the total number density in this approximation. For a real positive orbital, $G^{(1)}(0,r)=\sqrt{n(0)n(r)}$. Restoring the kinetic energy in a Gross–Pitaevskii equation can smooth a Thomas–Fermi approximation for a condensate edge while retaining this factorization. Describing quantum depletion of a Bose condensate additionally requires noncondensate mode occupations and their contribution to the one-body density matrix.

Use the condensate wavefunction convention $\int |\psi {|}^2{d}^{3}r=N$, so $n(\mathbf{r})=|\psi (\mathbf{r}){|}^2$ is the number density and $\psi /\sqrt{N}$ is a normalized one-particle orbital. The contact interaction strength is $U_0=4\pi {\hslash }^{2}a/m$ for dilute atoms with s-wave scattering length $a$. The chemical potential $\mu =\partial E/\partial N$ fixes the stationary time dependence $e^{-i\mu t/\hslash }$ and enforces normalization when minimizing the energy. The Gross–Pitaevskii equation follows from an energy functional that contains the pair-counting factor $1/2$:Varying $E-\mu \int |\psi {|}^2{d}^{3}r$ gives the stationary Gross–Pitaevskii equation; the interaction contribution to that equation is $U_{0}n\psi$, without the factor $1/2$.

For the broadening discussed here assume repulsive interactions, $a>0$. The Thomas–Fermi approximation for a condensate neglects the gradient kinetic energy compared with the trapping and interaction energies. At fixed spatial scale, increasing $N{U}_0$ increases the mean-field potential. Expansion then reduces the kinetic energy further: for a cloud radius $R$, its scale per atom is ${\hslash }^2/(m{R}^2)$, whereas the trap energy scale is $m{\omega }^2{R}^2$. Their ratio decreases as $(a_0/R{)}^4$. In the weak-interaction limit the kinetic energy instead balances confinement, giving the Gaussian wave packet ground state of the quantum harmonic oscillator.

The spherical Thomas–Fermi condensate hasIntegrating this density gives $N=8\pi \mu R^3/(15U_0)$. ThusAt fixed $N$, stronger repulsion produces a wider, lower wavefunction: the Gaussian wave packet crosses over towards the square root of an inverted parabola, with an almost flat central region and a steeper outer drop. The density is the inverted parabola; the wavefunction is its square root. The exact wavefunction remains smooth and has a tail, since kinetic energy cannot be neglected near the Thomas–Fermi condensate edge rounding region. The healing length must be small compared with the cloud size in the bulk. Attractive interactions would not follow this large-$N$ expansion argument.

The original plot compares the noninteracting endpoint with two normalized repulsive spherical Thomas–Fermi condensate profiles. The latter are asymptotic shapes, rather than exact intermediate solutions of the Gross–Pitaevskii equation. Each has the same integrated atom number; $\chi =Na/a_0$ labels the illustrative interaction strengths.

Choose $A$ real and positive. The Gaussian integral givesThe three zero-point energies add for each atom. The ideal-gas chemical potential is a one-atom energy; the total energy includes all $N$ atoms.

For weak interactions, the first-order correction in perturbation theory is the energy functional of the contact interaction evaluated on the unperturbed Gaussian wave packet. The integral needed isConsequently the Gaussian interaction energy of a trapped condensate isHere $\hslash \omega ={\hslash }^2/(m{a}_0^2)$. These are the mean-field expressions consistent with the supplied equation and $\int |\psi {|}^2=N$. Exact first-order perturbation theory for precisely $N$ atoms in one orbital replaces $N^2$ by $N(N-1)$ through the number of distinct pairs; this distinction is negligible in the large-$N$ regime of the crossover estimate.

To judge the ideal-gas approximation, compare the interaction energy with the total ideal energy, or compare the interaction energy per atom with $\hslash \omega$:Thus the crossover from Gaussian to Thomas–Fermi condensate is governed by $\chi =Na/a_0$: perturbation theory is reliable for $\chi \ll 1$, while the regime of the Thomas–Fermi approximation for a condensate requires $\chi \gg 1$. The characteristic crossover number is $N_c\sim a_0/a$; there is no sharp critical atom number here. If one specifically defines crossover by $E_I=E_0$ using the weak-interaction expression, it gives $N_c=(3\sqrt{2\pi }/2)a_0/a$. That prefactor is a convention-dependent estimate extrapolated to the edge of perturbation theory, not a phase transition. Comparing $E_I$ itself to a single $\hslash \omega$ would incorrectly omit a factor of $N$.

For the numerical estimate use $m\simeq 87\times 1.66054\times 10^{-27}\mathrm{kg}$ and $\hslash \simeq 1.05457\times 10^{-34}\mathrm{J}\mathrm{s}$. ThenThe particular equality $E_I=E_0$ would instead give about $1.15\times 10^3$ atoms. These quantify different choices within a smooth crossover; a well-developed spherical Thomas–Fermi condensate profile requires going substantially beyond the characteristic scale.

For the Bogoliubov spectrum, define ${ϵ}_k={\hslash }^2{k}^2/(2m)$ and $E_k=\sqrt{{ϵ}_k({ϵ}_k+2{\mu }_0)}$. For repulsive interactions and ${\mu }_0>0$,Thus the low-momentum excitations are gapless phonons and the high-momentum branch approaches free-particle curvature. The crossover occurs when kinetic and interaction energy are comparable. The Landau criterion follows because an obstacle moving at velocity $v$ can supply energy $\mathbf{v}\cdot \mathbf{p}$ to an excitation. HereThis is the homogeneous phonon kinematic threshold, not a guarantee that every perturbation dissipates immediately above it.

In two dimensions, $n=|\psi {|}^2$ has units of inverse area and ${\hslash }^2/m$ has units of energy times area. Therefore $\tilde{g}$ is dimensionless. Neglecting the kinetic term in the Gross–Pitaevskii equation, with positive coupling, giveswith $n=0$ outside the radius in the Thomas–Fermi approximation for a condensate. Multiplication givesThe sharp edge is an approximation; the neglected gradient term rounds it over a healing length scale.

Using the local sound-speed criterion for circular condensate stirring, the moving beam has speed $v=\Omega D$. The local interaction chemical potential is ${\mu }_0-V(D)=m{\omega }_r^2(R^2-D^2)/2$, soThe onset equality is $\Omega D=c(D)$ and the permitted excitation condition isThis uses the beam speed as the local relative-flow scale. A finite hole can distort the surrounding flow or nucleate vortices, so the result is the requested local phonon estimate, not a universal necessary condition for all experimental stirring.

For the loss-driven onset of condensate stirring, first integrate the density:At fixed coupling and trap frequency, ${\mu }_0\propto N^{1/2}$ and $R\propto N^{1/4}$. Hence $R(t)=R(0)e^{-t/(4\tau )}$ while $D$ stays fixed. For $\Omega ={\omega }_r/\sqrt{2}$, threshold requires $D/R=1/\sqrt{2}$. Starting at $D/R(0)=1/2$,At onset, $N=N_0/4$, $R=R(0)/\sqrt{2}$ and the beam is still inside the cloud. Excitations become kinematically allowed just beyond this equality, provided the stipulated adiabatic regime of the Thomas–Fermi approximation for a condensate remains valid.

Write ${ϵ}_k={\hslash }^2{k}^2/(2m)$, $E_k=\hslash \omega$, and take $\mu >0$. The Bogoliubov spectrum is gapless and positive for $k\ne 0$. Its small- and large-momentum limits areThe low-energy excitations are collective sound modes; high-energy excitations approach particles with a mean-field energy offset. With healing length $\xi =\hslash /\sqrt{2m\mu }$ and $q=|k|\xi$,The crossover occurs at $q$ of order one, not at an additional energy gap.

The critical velocity of a homogeneous condensate follows because the Landau criterion compares an excitation energy with the energy that flow can release by transferring its momentum. Creating a mode is energetically possible when $E_k-\hslash \mathbf{k}\cdot \mathbf{v}\le 0$. For this isotropic spectrum,The infimum is the long-wavelength speed of sound. This tests energetic stability against the stated excitations; it is not a calculation of a dissipation rate or a universal vortex-nucleation threshold.

For an external potential varying slowly on the healing scale, the Thomas–Fermi approximation for a condensate neglects density-gradient kinetic energy in the stationary Gross–Pitaevskii equation. In occupied regions it givesHere $\rho$ is the appropriate local number density; in the ring it is a linear density and $g$ is the one-dimensional coupling. This is the local-density sound speed of a condensate. The formula applies where $\mu >V$, not in an empty region where its square root would be imaginary.

Normalize the condensate order parameter to the atom number:up to an arbitrary constant phase. A wavefunction normalized to one instead has amplitude $1/\sqrt{2\pi R}$, with $N|\psi {|}^2$ then providing density. Acting with $L_z=-i\hslash {\partial }_{\theta }$ gives angular momentum $n\hslash$ per atom and $Nn\hslash$ in total for the uniform ring. The superfluid velocity isThe interaction chemical potential is ${\mu }_0=gN/(2\pi R)$; a flowing uniform state also has the constant kinetic contribution ${\hslash }^2{n}^2/(2m{R}^2)$ to its total chemical potential. Its speed of sound is set by the interaction term. Applying $|v|<\sqrt{{\mu }_0/m}$ givesFor a uniform ring condensate, this is the continuum phonon criterion used here. Quantization of the available excitation wavelengths in a very small ring can modify a finite-size threshold.

For the nonrotating equilibrium with a raised left half, let ${\rho }_L$ and ${\rho }_R$ be the bulk linear densities. Within the same Thomas–Fermi approximation for a condensate,When both halves are occupied, normalization fixesand thereforeThese expressions require $V_0<2{\mu }_0$. Beyond that threshold, extending them would give a negative left density and is unphysical. Instead the left bulk empties, all $N$ atoms occupy a segment of length $\pi R$, andAt equality the left bulk just reaches zero density. kinetic energy rounds the interface and produces tails beyond the Thomas–Fermi approximation for a condensate, so an exactly empty half is the idealized bulk result.

A nonconstant $V(\theta )$ breaks rotational symmetry:Consequently angular momentum is no longer a good quantum number, and an eigenfunction need not be $e^{in\theta }$ times a constant amplitude. Nevertheless, while the order parameter is nonzero around the whole ring, single-valuedness requiresThe integer is a phase winding of a condensate, not an assertion that the inhomogeneous state remains an angular-momentum eigenstate. A superfluid phase slip can change it if the amplitude vanishes locally.

Conservation of particles gives ${\partial }_t\rho +{\partial }_s(\rho v)=0$. For steady density the one-dimensional particle current $j=\rho v$ is constant. Since the two halves have equal lengths,The current bottleneck in an inhomogeneous condensate occurs because a lower-density region has faster flow at the same current; it also has a smaller speed of sound. This makes the raised half the limiting region.

For $V_0=2{\mu }_0/3$ the equilibrium approximation givesThe winding condition and equal lengths requireThus the left-hand sound-speed test isThe right-hand condition is only $n^2<3m{R}^2{\mu }_0/{\hslash }^2$, so it is less restrictive.

This local sound-speed estimate for ring-flow stability uses the step-potential equilibrium in a ring condensate densities together with the local sound-speed test, as suggested by the preceding approximations. A quantitatively self-consistent finite-current solution also includes $m{v}^2/2$ and the quantum pressure of a condensate in the stationary Gross–Pitaevskii equation, allowing flow to change the density. Smooth steps suppress reflection but do not alone remove those terms. The $3/8$ result is the stated local-density estimate; it should not be promoted to an exact stability theorem for every finite interacting ring.

Let ${\psi }_0$ be the normalized stationary profile. Multiplying its stationary Gross–Pitaevskii equation by ${\psi }_0^{\ast }$, integrating and integrating the kinetic term by parts yieldsThis is not simply $H/N$: the quartic contribution is twice its contribution to that energy per particle. For an infinite periodic system one uses one cell and the corresponding cell particle number, or equivalently average densities.

At fixed lattice spacing, mean density $n$, coupling and temperature, the commonly used condensate compressibility in an optical lattice isIf compressibility means the thermodynamic volume response instead, the Gibbs-Duhem equation $dp=nd\mu$ at zero temperature gives ${\kappa }_T=[n^2(\partial \mu /\partial n{)}_s{]}^{-1}=\kappa /n$. Stating the convention removes this factor-of-density ambiguity; the lattice-depth derivative has the same sign at fixed $n$.

For the stable repulsive ground-band Bose-Einstein condensate, the intended trend is $\partial \kappa /\partial s<0$ for increasing nonzero lattice depth, with mean density held fixed. Deeper wells localize the field, enhancing the repulsive interaction response. To quantify this, in the weak-interaction band approximation write ${\psi }_0=\sqrt{n}{f}_s$, with the cell average of $|f_s{|}^2$ equal to one. Perturbation in $Un$ givesIn deep wells the localized orbital is approximately a normalized Gaussian of width $a_{\mathrm{ho}}=d/(\pi s^{1/4})$. Neglecting interwell overlap, $I\simeq d/(\sqrt{2\pi }{a}_{\mathrm{ho}})=\sqrt{\pi /2}{s}^{1/4}$, hence $\partial \kappa /\partial s\simeq -\kappa /(4s)<0$. This explicitly establishes the sign in that controlled regime; the repulsive ground-band calculation describes the broader trend.

The sign statement needs the repulsive, stable ground-band interpretation, not merely the literal assumption $U\ne 0$. It also need not be a strictly nonzero derivative at $s=0$: at weak interaction $f_s=1+(s/8)\cos (2\pi x/d)+O(s^2)$, with normalization corrected at second order, gives $I=1+s^2/32+O(s^3)$ and a zero initial slope. A leading density-dominated Thomas–Fermi approximation for a condensate with density positive everywhere gives $\mu =Un+s{E}_R/2$ and no depth dependence of this response at that approximation's order. These limits do not justify a strict universal sign for every unspecified interaction and parameter regime.

Set $A_i=m_i{\omega }_i^2/2$, so $V_i=A_i{r}^2$, and $D=U_{11}{U}_{22}-U_{12}^2$. In the overlap region, the Thomas–Fermi approximation for a condensate neglects the gradient terms and requiresInverting this matrix proves the unambiguous profilesUsing $R_i^2={\mu }_i/A_i$ and introducingthese Thomas–Fermi profiles for unequal condensate trap curvatures becomeThere is a genuine printed error: the PDF puts $\lambda$ rather than $1/\lambda$ in the second bracket as well. In the first bracket its coefficient requires $\lambda =A_2/A_1$, while in the second it requires $\lambda =A_1/A_2$; for nonzero cross-coupling these cannot both hold unless the curvatures are equal. The TeX additionally drops both occurrences of $\lambda$. The matrix-inverse formulas above give the corrected general answer directly.

For example, $U_{11}=U_{22}=1$, $U_{12}=1/2$, ${\mu }_1={\mu }_2=2$, $A_1=1$, $A_2=2$ give the physical overlap profiles $n_1=4/3$, $n_2=4/3-2r^2$. Using $\lambda =2$ in both printed brackets would instead give $n_2=4/3$, which fails the second equilibrium equation at every $r\ne 0$ in the overlap region.

Only the region where both computed densities are positive belongs to this overlap solution. Beyond its edge, a remaining species has $n_i=({\mu }_i-A_i{r}^2)/U_{ii}$ where positive, and both vanish outside the support; simply retaining negative values from the coupled inverse is unphysical. The actual chemical potentials are determined by the two normalization integrals. If the reference uncoupled radii are defined at the coupled solution's chemical potentials, the identities $R_i^2={\mu }_i/A_i$ are algebraic scales, not the actual interacting-cloud edges; holding particle numbers fixed and removing cross-coupling generally changes those chemical potentials. The approximation also fails in the healing regions at the edges.

Write $g=V_0$ for the effective coupling and keep $N$ for the number obtained by the stated two-dimensional normalization. A repulsive interaction, $g>0$, permits the Thomas–Fermi approximation for a condensate: density-gradient kinetic energy is small compared with the trapping and interaction terms over most of the cloud. The stationary Gross–Pitaevskii equation then givesand $n_{\mathrm{TF}}=0$ outside the cloud, up to the narrow edge region. HereNormalize the parabolic number density directly:Consequently the two-dimensional Thomas–Fermi condensate parameters areWith $a_{\mathrm{ho}}=\sqrt{\hslash /(m\omega )}$ and central healing length $\xi =\hslash /\sqrt{2m\mu }$, the bulk approximation requiresIndeed $R/\xi =2\mu /(\hslash \omega )\gg 1$. In the coupling convention printed in the paper, formal substitution of $g=4\pi {\hslash }^{2}a/m$ givesThis is the requested relationship in that convention; $\omega$ sets the physical size and chemical potential, while the interaction-strength ratio to $\hslash \omega$ is independent of $\omega$.

There is a dimensional convention to make explicit. A physical three-dimensional scattering length $a$ makes $4\pi {\hslash }^{2}a/m$ a three-dimensional contact coupling, whereas a genuinely two-dimensional particle-number normalization requires coupling units of energy times area. For an axial factor $\chi (z)$ normalized to one, the effective two-dimensional contact coupling is $g_{2D}=(4\pi {\hslash }^{2}a/m)\int |\chi {|}^{4}dz$. For a Gaussian of width $a_z$ this is $4\pi {\hslash }^{2}a/(m\sqrt{2\pi }{a}_z)$, and the physical TF condition becomes $Na/a_z\gg 1$ up to numerical constants. Alternatively the printed coupling can be retained if $N$ is a line density. The paper supplies no axial scale, so its $Na$ expressions require that effective-unit or line-density convention; the formulas in terms of $g$ above are unambiguous. The approximation fails inside vortex cores and at the cloud edge, and it is not a repulsive TF ground-state construction for $g\le 0$.

Take a singly quantized vortex of positive circulation. The superfluid velocity is $\mathbf{u}=(\hslash /m)\nabla S$, where $S$ is the complex argument of the condensate. Since $S=\theta$, quantized circulation gives $u_{\theta }=\hslash /(mr)$. Insert the radial Thomas–Fermi approximation for a condensate into the cylindrical volume integral, excluding the core $r<a_0$:Integrating the logarithmic and quadratic terms separately givesThe logarithm is the familiar long-range flow contribution of a quantum vortex; the $-1/2$ comes from the declining trapped number density. If the bucket truncates the cloud, put $R=R_0<R_r$ and the same integration gives $E_{\mathrm{kin}}=\pi H_0{\hslash }^2{n}_0[\log (R/a_0)-(R^2-a_0^2)/(2R_r^2)]/m$, provided $a_0<R$. A vortex of integer winding number $q$ has $u_{\theta }=q\hslash /(mr)$ and multiplies this flow energy by $q^2$. The core radius acts as a cutoff of order the local healing length; its internal gradient and interaction energies are not computed by this shell estimate.

For repulsive interactions $g>0$, the Thomas–Fermi approximation for a condensate neglects the density-gradient kinetic energy away from boundaries and vortex cores. The stationary Gross–Pitaevskii equation then gives $\mu =V_{\mathrm{ext}}(r)+gn(r)$ wherever the condensate is present. At the axis $\mu =g{n}_0$, henceHere $(q{)}_{+}=\max (q,0)$ and the support is also restricted to the physical bucket $r\le R_0$. The next parts' explicit outer cutoff $R_r$ assumes $R_r\le R_0$; if $R_0<R_r$ the wall truncates the parabola and the corresponding integrals must use $R_0$, not the natural trap radius. A hard-wall boundary layer or the smooth trap-edge healing region is beyond the Thomas–Fermi approximation for a condensate. For an untruncated radial profile, the total particle number is $N=\pi H_0{n}_0{R}_r^2/2$.

In the large-$N$ Gross–Pitaevskii equation, dropping the kinetic energy gives $[\kappa r^2/2+gn(r)-\mu ]\Psi (r)=0$. The physical nonnegative Thomas–Fermi approximation for a condensate is thereforewhere $(z{)}_{+}=\max (z,0)$ and $R$ is the cloud radius. Its particle number isThus the two-dimensional Thomas–Fermi condensate obeysHere $N=\int n{d}^{2}r$ is a total particle number. Keeping the exact pair count in the fixed-$N$ Hartree product state would replace $g$ by $g(1-1/N)$ in these formulas expressed in terms of $n=N|\varphi {|}^2$.

The approximation is best at the centre, $r=0$, and worst near the edge, $r=R$. For example its omitted quantum pressure of a condensate, evaluated on the interior profile, isIt is smallest at the centre and diverges at the artificial sharp edge. The central healing length ${\xi }_0=\hslash /\sqrt{2m\mu }$ is much smaller than $R$ when $\mu \gg \hslash \omega$, since $R/{\xi }_0=2\mu /(\hslash \omega )$. This is the bulk validity condition; the full Gross–Pitaevskii equation smooths the edge and gives a decaying tail outside $R$.

The Hartree product state underlying the Gross–Pitaevskii equation has the factorized one-body coherence of a Hartree condensate. Choosing its nodeless ground state orbital real givesIn particular, the Thomas–Fermi approximation for a condensate from part (d) predictsThe curve starts at $\mu /g$, decreases with zero slope at the origin, and reaches zero with an infinite slope at $R$. Restoring the kinetic energy in the full Gross–Pitaevskii equation rounds that edge and produces a nonzero tail. This corrects the Thomas–Fermi approximation for a condensate, but the resulting one-body density matrix still factorizes.

The additional missing physics is quantum depletion. Interactions scatter pairs of bosons into orbitals outside the Bose-Einstein condensate even at zero temperature. In terms of the true Bose-Einstein condensate orbital ${\phi }_0$ the exact one-body density matrix decomposes asThe single-orbital Hartree product state sets $N_0=N$ and omits ${\rho }_{\mathrm{ex}}$ altogether. In a weakly interacting Bose gas, Bogoliubov theory adds a comparatively short-range noncondensed contribution near the origin, together with reduced weight in the broad Bose-Einstein condensate component. The central value remains the total number density, not just the condensate density. Its numerical shift relative to the original Gross–Pitaevskii equation solution also depends on the interaction-induced redistribution of number density; it is not fixed by the condensate fraction alone.