Condensate

Learn: the model

Section numbers (§) refer to C. F. Barenghi, T. Bland and N. G. Parker, Quantum Fluids, Solitons, and Vortices: A Primer with Worked Problems, 2nd edn, Lecture Notes in Physics 1050 (Springer, 2026), doi:10.1007/978-3-032-20171-3, written BBP below. Quantitative predictions are for the 128² grid; the “Set this up” buttons switch to it.

Foundations: the Gross–Pitaevskii equation and how to read the picture

The equation

A dilute Bose gas well below its condensation temperature is described by a single complex field, the condensate wavefunction $ψ (𝐫, t)$ , normalised to the atom number, $\int {| ψ |}^{2} d^{2} r = N$ . It obeys the Gross–Pitaevskii equation (BBP §3.1)

i ℏ \frac{\partial ψ}{\partial t} = [- \frac{ℏ^{2}}{2 m} \nabla^{2} + V (𝐫) + g {| ψ |}^{2}] ψ .

The nonlinear term is the mean field of s-wave contact collisions. The app is two-dimensional: the gas is assumed to be frozen in a narrow Gaussian profile along z, of width $l_{z}$ , so the 3D coupling is reduced to $g_{2 D} = g_{3 D} / (\sqrt{2 π} l_{z})$ (BBP §3.5.4). Everything is in trap units (BBP §3.8.2): $ℏ = m = 1$ , times in $1 / ω$ and lengths in $a_{h o} = \sqrt{ℏ / m ω}$ , where ω is the harmonic trap frequency. Here $g = 1$ , $N = 475$ and $l_{z} = 1.5$ at 128².

Reading the picture

Brightness is the density $n = {| ψ |}^{2}$ . “Show phase” shows the phase θ of $ψ = \sqrt{n} e^{i θ}$ as colour. The Madelung transformation turns the GPE into the equations of an inviscid, irrotational fluid with velocity $𝐯 = (ℏ / m) \nabla θ$ (BBP §3.3):

\frac{\partial n}{\partial t} + \nabla \cdot (n 𝐯) = 0, m \frac{\partial 𝐯}{\partial t} + \nabla (\frac{m v^{2}}{2} + V + g n - \frac{ℏ^{2}}{2 m} \frac{\nabla^{2} \sqrt{n}}{\sqrt{n}}) = 0 .

The last term, the quantum pressure, matters only where the density changes over short distances, such as at the cloud's edge and in vortex cores. Where the phase colours wind once round a point, the density there must vanish: that point is a quantised vortex.

Scales to know

In the Thomas–Fermi limit the kinetic term is negligible in the bulk and $n = (μ - V) / g$ (BBP §3.5.2). For the 2D harmonic trap this gives $μ = ℏ ω \sqrt{g N m / π ℏ^{2}}$ , which is $\sqrt{475 / π} \approx 12.3$ in trap units, and a cloud radius $R = \sqrt{2 μ} / ω \approx 5.0$ . The simulated peak density is 12.8, slightly higher because of the kinetic energy at the edge. Three scales follow from the peak density $n_{0}$ :

the healing length $ξ = ℏ / \sqrt{m g n_{0}} \approx 0.28$ , the shortest distance over which the density can change and the size of a vortex core (BBP §3.4.2, §7.4);
the speed of sound $c = \sqrt{g n_{0} / m} \approx 3.6$ (BBP §4.1.2);
the Bogoliubov dispersion $ϵ (k) = \sqrt{\frac{ℏ^{2} k^{2}}{2 m} (\frac{ℏ^{2} k^{2}}{2 m} + 2 g n_{0})}$ , linear (phonons) for $k ξ ≪ 1$ and free-particle-like for $k ξ ≫ 1$ (BBP §4.1.1).

The visible field is 13.3 trap units across for the harmonic, box and lattice traps, 13.6 for the ring and 17 with no trap. On larger grids the field and the cloud grow by $s = N / 128$ while ξ and c stay fixed.

Quantised vortices

Because ψ is single-valued, the phase can only change by a multiple of 2π around a closed loop, so the circulation is quantised (BBP §7.1–7.3):

\oint 𝐯 \cdot d ℓ = q \frac{h}{m}, v_{θ} = \frac{q ℏ}{m r}, q = \pm 1, \pm 2, \dots

The core has radius of order ξ, and the energy of a vortex grows as $\ln (R / ξ)$ with the system size (BBP §7.4–7.5). In 2D, vortices move like point vortices: each is carried by the flow of all the others (BBP §8.2, §9.1).

How it is computed

The app uses the split-step Fourier method: half a step of the potential and nonlinear terms in real space, a full kinetic step in Fourier space, then the second half-step (BBP Appendix A.1–A.2; Bao, Jaksch and Markowich 2003). Ground states come from imaginary-time evolution, $t \to - i τ$ , which damps every excited state faster than the ground state (BBP Appendix A.3). Real-time runs can include phenomenological damping, replacing $i \partial_{t}$ by $(i - γ) \partial_{t}$ , which models energy loss to a thermal cloud (BBP §3.9.1; Choi, Morgan and Burnett 1998). Implementation details are under “Implementation notes” beside the controls.

Traps

Harmonic: $V = \frac{1}{2} (ω_{x}^{2} x^{2} + ω_{y}^{2} y^{2})$ , with $ω_{x} = ω$ and the aspect slider setting $λ = ω_{y} / ω_{x}$ . In two dimensions the Thomas–Fermi atom number is $N = π μ^{2} / (g ω_{x} ω_{y})$ , so the app takes $N = 475 / λ$ : μ, the central density and ξ then match the round trap, and the radii are $R_{i} = \sqrt{2 μ} / ω_{i}$ . As $R_{y}$ falls towards a few ξ the cloud leaves the Thomas–Fermi regime and approaches a quasi-one-dimensional gas (BBP §3.5.4). Box: a flat floor with a smooth wall, as in uniform-gas experiments (Gaunt et al. 2013). Ring: $V = \frac{1}{2} ω_{r}^{2} (r - R)^{2}$ , the geometry of atomtronic persistent-current experiments (Ramanathan et al. 2011). Lattice: $V_{L} [\sin^{2} (π x / a) + \sin^{2} (π y / a)]$ on a harmonic trap, with depth in recoil energies (BBP §3.9.3; Morsch and Oberthaler 2006). None: a periodic box, which approximates an infinite uniform gas.

BBP Chapters 3, 4 and 7, and Appendix A.
F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose–Einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
L. P. Pitaevskii and S. Stringari, Bose–Einstein Condensation and Superfluidity (Oxford University Press, 2016).
C. J. Pethick and H. Smith, Bose–Einstein Condensation in Dilute Gases, 2nd edn (Cambridge University Press, 2008).
W. Bao, D. Jaksch and P. A. Markowich, Numerical solution of the Gross–Pitaevskii equation for Bose–Einstein condensation, J. Comput. Phys. 187, 318 (2003).
S. Choi, S. A. Morgan and K. Burnett, Phenomenological damping in trapped atomic Bose–Einstein condensates, Phys. Rev. A 57, 4057 (1998).
A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. Smith and Z. Hadzibabic, Bose–Einstein condensation of atoms in a uniform potential, Phys. Rev. Lett. 110, 200406 (2013).
A. Ramanathan et al., Superflow in a toroidal Bose–Einstein condensate: an atom circuit with a tunable weak link, Phys. Rev. Lett. 106, 130401 (2011).
O. Morsch and M. Oberthaler, Dynamics of Bose–Einstein condensates in optical lattices, Rev. Mod. Phys. 78, 179 (2006).

A single condensate with contact interactions: the simplest superfluid. Moved slowly, an obstacle passes through it without drag; moved fast enough, it sheds quantised vortices.

The model

(i - γ) \frac{\partial ψ}{\partial t} = [- \frac{1}{2} \nabla^{2} + V + V_{b e a m} + g {| ψ |}^{2} - μ] ψ, V_{b e a m} = V_{0} e^{- | 𝐫 - 𝐫_{b} (t) |^{2} / σ^{2}} .

The beam is a repulsive Gaussian, like a blue-detuned laser, with $V_{0} = 60$ , well above μ, so it bores a hole in the fluid. The slider sets the damping γ; at γ = 0 the dynamics conserve energy.

The physics

Landau's argument says a fluid with excitation spectrum ε(p) can flow past an obstacle without creating excitations below $v_{c} = {min}_{p} ϵ (p) / p$ , which for the Bogoliubov spectrum is the speed of sound c (BBP §6.9). In practice dissipation starts earlier. The flow speeds up around the obstacle, to twice the free-stream speed at its poles for a cylinder in potential flow, and vortex–antivortex pairs nucleate where the local flow becomes supersonic (BBP §9.5). Simulations give a critical velocity of roughly 0.4–0.5c for a large impenetrable obstacle, lower in a trap where the density, and so c, falls towards the edge (Frisch, Pomeau and Rica 1992). Experiments see the same onset (Raman et al. 1999; Neely et al. 2010).

Once shed, the vortices move with the flow. An opposite-sign pair separated by d travels perpendicular to the line joining them at speed $ℏ / (m d)$ ; a same-sign pair orbits its midpoint at angular velocity $2 ℏ / (m d^{2})$ (BBP §8.2, §9.2). In a harmonic trap a single vortex precesses round the centre, and with damping it drifts outwards to lower its energy until it leaves the cloud (Fetter 2009).

Try this

Find the critical velocity. Drag the beam slowly through the cloud: the fluid parts and closes with no wake. Repeat faster. Above a threshold, vortex pairs peel off the beam's sides. The central sound speed here is about 3.6 trap units per unit time, so a sweep across the cloud (10 units) in about 6 time units is near the threshold. Mark vortices to count them.
A vortex dipole. In the box trap, tap a +1 vortex and a −1 vortex about 2 units apart. Predict the direction and speed (about $1 / d \approx 0.5$ ) before you watch it go, then try a same-sign pair and watch it orbit.
Dissipation. Tap a single vortex off-centre in the harmonic trap with γ = 0: it precesses indefinitely. Raise γ to 0.03: it spirals out of the cloud, because a vortex nearer the edge, where the density is lower, has less energy.
A cigar-shaped condensate. Set the trap aspect to $λ = ω_{y} / ω_{x} = 2$ , then 6. Thomas–Fermi theory predicts the radii $R_{x} = \sqrt{2 μ} / ω_{x} \approx 5.0$ and $R_{y} = R_{x} / λ$ . At λ = 2 the simulated width ratio is 1.94; at λ = 6 it is only 5.0. Why does the Thomas–Fermi prediction fail as the cloud narrows? Compare $R_{y}$ with ξ ≈ 0.28. Then drag the beam along the cigar.

BBP §3.5.4 (anisotropic traps and reduced dimensionality), §3.9.1 (dissipative GPE), §6.9 (Landau critical velocity), §8.2 (point vortices), §9.2 (vortex pairs), §9.5 (vortex nucleation).
T. Frisch, Y. Pomeau and S. Rica, Transition to dissipation in a model of superflow, Phys. Rev. Lett. 69, 1644 (1992).
C. Raman, M. Köhl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic and W. Ketterle, Evidence for a critical velocity in a Bose–Einstein condensed gas, Phys. Rev. Lett. 83, 2502 (1999).
T. W. Neely, E. C. Samson, A. S. Bradley, M. J. Davis and B. P. Anderson, Observation of vortex dipoles in an oblate Bose–Einstein condensate, Phys. Rev. Lett. 104, 160401 (2010).
A. L. Fetter, Rotating trapped Bose–Einstein condensates, Rev. Mod. Phys. 81, 647 (2009).

A condensate cannot rotate like a solid body, because its flow is irrotational. It rotates by admitting quantised vortices, which arrange themselves into a triangular lattice.

The model

i \frac{\partial ψ}{\partial t} = [- \frac{1}{2} \nabla^{2} + V + g {| ψ |}^{2} - Ω L_{z}] ψ, L_{z} = - i (x \frac{\partial}{\partial y} - y \frac{\partial}{\partial x}),

written in the frame rotating with the trap at angular velocity Ω (the slider gives Ω/ω).

The physics

In the rotating frame the system minimises $E - Ω L_{z}$ , so a vortex lowers the free energy once Ω exceeds a critical value $Ω_{c 1}$ , of order $(ℏ / m R^{2}) \ln (R / ξ)$ (BBP §7.6–7.7). That is a few tenths of ω here. The app seeds vortices before cooling, so at small Ω a single vortex may survive as a metastable state below $Ω_{c 1}$ .

At higher Ω, many vortices mimic solid-body rotation on average. The coarse-grained vorticity must equal 2Ω, which fixes the areal density of vortices (Feynman 1955; BBP §7.6):

n_{v} = \frac{2 m Ω}{h} = \frac{Ω}{π} (trap units), b = {(\frac{2 π}{\sqrt{3} Ω})}^{1 / 2}

where b is the spacing of the triangular (Abrikosov) lattice that minimises the interaction energy. Centrifugal force weakens the trap to $ω_{e f f} = \sqrt{ω^{2} - Ω^{2}}$ , so the cloud swells as Ω approaches ω and deconfines at Ω = ω (Fetter 2009). Feynman's rule fixes the density, not the total: near the edge of a finite cloud there is a vortex-free band, so the number of vortices is well below $n_{v} π R^{2}$ .

Try this

Test Feynman's rule. At Ω = 0.7, mark the vortices. Predict the lattice spacing, $b \approx 2.3$ , about a sixth of the 13.3-unit field. Then count the vortices inside a circle through the outermost ones, of radius $R_{v}$ , and compare with $Ω R_{v}^{2}$ . The simulation gives about 11 inside a radius of 4.2, against a prediction of 12.
The vortex-free edge. With Thomas–Fermi radius $R = (2 μ / ω_{e f f}^{2})^{1 / 2}$ and $μ = ω_{e f f} \sqrt{g N / π}$ , the naive count $Ω R^{2}$ at Ω = 0.7 is about 24, twice what you see. Where is the missing rotation carried?
Towards deconfinement. Raise Ω to 0.85 and watch the cloud spread and the lattice fill it, as $ω_{e f f}$ falls to about 0.53ω.

BBP §7.5 (vortex energy and angular momentum), §7.6 (vortex lattice in a bucket), §7.7 (vortex lattice in a trapped condensate).
R. P. Feynman, Application of quantum mechanics to liquid helium, Prog. Low Temp. Phys. 1, 17 (1955).
K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex formation in a stirred Bose–Einstein condensate, Phys. Rev. Lett. 84, 806 (2000).
J. R. Abo-Shaeer, C. Raman, J. M. Vogels and W. Ketterle, Observation of vortex lattices in Bose–Einstein condensates, Science 292, 476 (2001).
M. Tsubota, K. Kasamatsu and M. Ueda, Vortex lattice formation in a rotating Bose–Einstein condensate, Phys. Rev. A 65, 023603 (2002).
A. L. Fetter, Rotating trapped Bose–Einstein condensates, Rev. Mod. Phys. 81, 647 (2009).

Atoms with large magnetic moments, such as chromium, erbium and dysprosium, interact through a long-range, anisotropic dipole–dipole force as well as contact collisions. The competition produces a roton, quantum droplets and supersolids.

The model

Two dipoles aligned along $\hat{𝐞}$ interact through (BBP §12.2)

U_{d d} (𝐫) = \frac{C_{d d}}{4 π} \frac{1 - 3 \cos^{2} ϑ}{r^{3}}, ε_{d d} = \frac{C_{d d}}{3 g},

attractive head-to-tail and repulsive side by side. Averaging over a Gaussian axial profile gives the quasi-2D interaction in Fourier space, for dipoles tilted by α from the z axis, with their in-plane projection along ${\hat{𝐞}}_{∥} = (\cos ϕ, \sin ϕ)$ (Fischer 2006; Ticknor, Wilson and Bohn 2011; BBP §12.4.2):

\tilde{U} (𝐤) = g + g ε_{d d} [\cos^{2} α (2 - 3 F) + \sin^{2} α (\frac{3 F k_{∥}^{2}}{k^{2}} - 1)], k_{∥} = k_{x} \cos ϕ + k_{y} \sin ϕ,

F = \sqrt{π} q e^{q^{2}} e r f c (q), q = \frac{k l_{z}}{\sqrt{2}} .

A Lee–Huang–Yang (LHY) term $γ_{L H Y} (1 + \frac{3}{2} ε_{d d}^{2}) {| ψ |}^{3}$ , from quantum fluctuations, stops collapse at high density (Schützhold et al. 2006; Lima and Pelster 2011; BBP §13.1). The app uses the 3D form of this term with $γ_{L H Y} = 0.02$ .

The physics

At α = 0 the interaction is repulsive at long wavelengths, $\tilde{U} \to g (1 + 2 ε_{d d})$ , but tends to $g (1 - ε_{d d})$ at short wavelengths, where the head-to-tail attraction across the thin axial direction wins. The Bogoliubov spectrum $ϵ (k)^{2} = \frac{k^{2}}{2} (\frac{k^{2}}{2} + 2 n \tilde{U} (k))$ therefore develops a roton minimum at $k ~ 1 / l_{z}$ as $ε_{d d}$ grows (Santos, Shlyapnikov and Lewenstein 2003; BBP §12.7.2). When the roton softens to zero the uniform state gives way to density modulation. The LHY term then stabilises an array of droplets, which, when the droplets stay phase-coherent, is a supersolid (BBP §13.2.2, §13.4; Kadau et al. 2016; Chomaz et al. 2023).

Tilting the dipoles makes the long-wavelength term $g ε_{d d} (3 \cos^{2} α - 1)$ , which vanishes at the magic angle $α_{m} = \arccos (1 / \sqrt{3}) \approx {54.7}^{\circ}$ and is attractive beyond it. The same factor governs dipoles in a rotating field (BBP §12.6; Giovanazzi, Görlitz and Pfau 2002). An in-plane component also makes the interaction anisotropic, so the cloud stretches along the dipoles' in-plane direction φ: magnetostriction (BBP §12.8.3).

Try this

Magnetostriction. At $ε_{d d} = 0.5$ , move the tilt from 0° to 90°. The cloud elongates along x; the simulated aspect ratio $⟨ x^{2} ⟩ / ⟨ y^{2} ⟩$ goes from 1.0 to about 1.5. Now turn the azimuth φ: the elongation follows the dipoles round, reaching y at φ = 90°.
The magic angle. Compare α = 0 with α = 54.7°. At α = 0 the dipoles add repulsion, so the cloud is larger: its mean density ∫n²/∫n falls from 8.0 without dipoles to 7.5. At the magic angle the isotropic part of the interaction cancels and the mean density returns to 8.0, yet the cloud is elongated, with aspect ratio about 1.3, because the anisotropic part of the kernel survives at finite k.
The roton and droplets. At α = 0, raise $ε_{d d}$ slowly towards 2.4. Near 1.9 a ripple appears with a wavelength of a few $l_{z}$ , the roton; near 2.4 the cloud breaks into droplets. Why does the roton wavelength scale with $l_{z}$ rather than with ξ?
Aligned droplets. At large tilt and $ε_{d d} \approx 2$ , the gas forms filaments and droplets lined up along x, the head-to-tail direction. This is slow, because strong tilts need a smaller time step.
Dipoles in a cigar. In a cigar trap (aspect 3) with in-plane dipoles (α = 90°, $ε_{d d} = 1$ ), compare φ = 0, along the cigar, with φ = 90°, across it. Along the cigar the cloud becomes thinner and denser than without dipoles (rms width 0.72 → 0.52, mean density 7.8 → 10.5); across it, wider and more dilute (0.92 and 6.3). Both have the same long-wavelength interaction, so why the difference? Think about which wavenumbers the narrow direction contains, and what $\tilde{U}$ is at large k parallel and perpendicular to the dipoles.

BBP Chapter 12 (§12.2 the dipole–dipole interaction, §12.4.2 lower dimensions, §12.6 tuning, §12.7.2 rotons and maxons, §12.8.3 magnetostriction) and Chapter 13 (§13.1 beyond mean field, §13.2.2 dipolar droplets, §13.3–13.4 supersolids).
T. Lahaye, C. Menotti, L. Santos, M. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys. 72, 126401 (2009).
L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev and T. Pfau, Dipolar physics: a review of experiments with magnetic quantum gases, Rep. Prog. Phys. 86, 026401 (2023).
U. R. Fischer, Stability of quasi-two-dimensional Bose–Einstein condensates with dominant dipole–dipole interactions, Phys. Rev. A 73, 031602(R) (2006).
C. Ticknor, R. M. Wilson and J. L. Bohn, Anisotropic superfluidity in a dipolar Bose gas, Phys. Rev. Lett. 106, 065301 (2011).
L. Santos, G. V. Shlyapnikov and M. Lewenstein, Roton–maxon spectrum and stability of trapped dipolar Bose–Einstein condensates, Phys. Rev. Lett. 90, 250403 (2003).
S. Giovanazzi, A. Görlitz and T. Pfau, Tuning the dipolar interaction in quantum gases, Phys. Rev. Lett. 89, 130401 (2002).
R. Schützhold, M. Uhlmann, Y. Xu and U. R. Fischer, Mean-field expansion in Bose–Einstein condensates with finite-range interactions, Int. J. Mod. Phys. B 20, 3555 (2006).
A. R. P. Lima and A. Pelster, Quantum fluctuations in dipolar Bose gases, Phys. Rev. A 84, 041604(R) (2011).
H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I. Ferrier-Barbut and T. Pfau, Observing the Rosensweig instability of a quantum ferrofluid, Nature 530, 194 (2016).

Two condensates, for example two hyperfine states of one atom, that interact with themselves and with each other. Depending on the cross-interaction they mix, separate, or bind into a self-supporting quantum droplet.

The model

i \frac{\partial ψ_{j}}{\partial t} = [- \frac{1}{2} \nabla^{2} + V + g {| ψ_{j} |}^{2} + g_{12} | ψ_{3 - j} |^{2} + γ_{m} ({| ψ_{1} |}^{2} + {| ψ_{2} |}^{2})^{3 / 2}] ψ_{j}, j = 1, 2,

with equal self-interactions $g_{11} = g_{22} = g$ and an LHY term with $γ_{m} = 0.04$ (BBP §11.1.1, §13.2.1).

The physics

A uniform mixture is miscible when $g_{12}^{2} < g_{11} g_{22}$ and separates otherwise (BBP §11.2; Ao and Chui 1998). Here that is $g_{12} / g = 1$ . The Bogoliubov spectrum has a density branch and a spin branch, with the spin branch's speed of sound proportional to $\sqrt{(g - g_{12}) n}$ . Beyond the threshold the spin mode is unstable and domains grow and coarsen (BBP §11.7.2). Near the threshold the domain walls are wide, because the spin healing length diverges.

For $g_{12} < - g$ mean-field theory predicts collapse. The LHY energy, which grows faster with density, halts it. For a balanced mixture with $δ = - (g + g_{12}) > 0$ the energy density is $- \frac{1}{4} δ n^{2} + \frac{2}{5} γ_{m} n^{5 / 2}$ , and setting the pressure to zero gives a liquid of fixed density (Petrov 2015):

n_{0} = {(\frac{5 δ}{12 γ_{m}})}^{2} .

The result is a self-bound droplet with a flat top at $n_{0}$ , observed with potassium mixtures (Cabrera et al. 2018; Semeghini et al. 2018). In strictly 2D the LHY term becomes logarithmic (Petrov and Astrakharchik 2016); the app keeps the 3D form, which suits a quasi-2D gas whose axial width is large compared with the scattering lengths.

Try this

The miscibility threshold. Raise $g_{12} / g$ from 0.6 through 1.0 and predict where separation begins. Watch the domain walls sharpen as you go further past 1.
Filled vortex cores. At $g_{12} / g = 0.9$ , with “Act on red only” on, tap a vortex into the red species. The teal species flows into the empty core (BBP §11.6).
A quantum droplet. With no trap, set $g_{12} / g = - 1.3$ , so δ = 0.3 and the predicted flat-top density is $n_{0} \approx 9.8$ . Move to −1.5 ( $n_{0} \approx 27$ ) and watch the droplet shrink and densify.

BBP Chapter 11 (§11.1 coupled GPEs, §11.2 miscibility, §11.3 trapped solutions, §11.6 filled vortex cores, §11.7 instabilities) and §13.2.1 (Bose–Bose droplets).
P. Ao and S. T. Chui, Binary Bose–Einstein condensate mixtures in weakly and strongly segregated phases, Phys. Rev. A 58, 4836 (1998).
D. S. Petrov, Quantum mechanical stabilization of a collapsing Bose–Bose mixture, Phys. Rev. Lett. 115, 155302 (2015).
D. S. Petrov and G. E. Astrakharchik, Ultradilute low-dimensional liquids, Phys. Rev. Lett. 117, 100401 (2016).
C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas, P. Cheiney and L. Tarruell, Quantum liquid droplets in a mixture of Bose–Einstein condensates, Science 359, 301 (2018).
G. Semeghini et al., Self-bound quantum droplets of atomic mixtures in free space, Phys. Rev. Lett. 120, 235301 (2018).

Lasers can couple two internal states of an atom so that switching state also kicks its momentum. The atoms behave as if they had spin–orbit coupling, and the condensate can form stripes: a density wave that coexists with superfluidity.

The model

H_{0} = \frac{1}{2} {(p_{x} - k_{s o} σ_{z})}^{2} + \frac{1}{2} p_{y}^{2} + \frac{Ω}{2} σ_{x},

the scheme used at NIST (Lin, Jiménez-García and Spielman 2011), with Raman coupling Ω, recoil momentum $k_{s o}$ , and the two-component interactions of the previous model with $g_{11} = g_{22} = g$ and cross-interaction $g_{12}$ .

The physics

The single-particle energies form two bands,

E_{\pm} (𝐤) = \frac{1}{2} k^{2} + \frac{1}{2} k_{s o}^{2} \pm \sqrt{k_{s o}^{2} k_{x}^{2} + Ω^{2} / 4} .

For $Ω < 2 k_{s o}^{2}$ the lower band has two minima at $\pm k_{1}$ , with $k_{1} = k_{s o} \sqrt{1 - (Ω / 2 k_{s o}^{2})^{2}}$ ; above that there is one minimum at k = 0. Interactions choose how the atoms fill the minima (Ho and Zhang 2011; Li, Pitaevskii and Stringari 2012). When $g_{12} < g$ they occupy both, and the interference gives stripes of period $π / k_{1}$ , a state with both superfluid and crystalline order that is a form of supersolid (Li et al. 2017; BBP §13.3). When $g_{12} > g$ they choose one minimum, a plane wave. For large Ω they sit at zero momentum. Interactions shift these boundaries slightly from the single-particle values.

With $k_{s o} = 0$ this is a coherently (Rabi) coupled mixture. The coupling locks the relative phase of the two species, so a vortex in one species alone drags a domain wall of relative phase behind it (Son and Stephanov 2002).

Try this

Predict the stripe period. At $k_{s o} = 2$ , Ω = 4 and $g_{12} / g = 0.5$ , predict $k_{1} = 2 \sqrt{3 / 4} \approx 1.73$ and a period $π / k_{1} \approx 1.8$ , about seven stripes across the field. Count them.
Find the transition to zero momentum. Raise Ω and watch the stripes widen as $k_{1}$ shrinks. Predict where they vanish: $Ω = 2 k_{s o}^{2} = 8$ .
Stripe or plane wave. Set $g_{12} / g = 1.2$ at Ω = 4. The stripes give way to a single plane wave, with one species dominating.
A confined vortex. With $k_{s o} = 0$ and Ω = 1, tap a vortex into the red species only and watch the relative phase. The phase jump is confined to a line from the vortex: a domain wall.

BBP Chapter 11 (two-component foundations) and §13.3 (supersolids, including spin–orbit-coupled condensates).
Y.-J. Lin, K. Jiménez-García and I. B. Spielman, Spin–orbit-coupled Bose–Einstein condensates, Nature 471, 83 (2011).
V. Galitski and I. B. Spielman, Spin–orbit coupling in quantum gases, Nature 494, 49 (2013).
T.-L. Ho and S. Zhang, Bose–Einstein condensates with spin–orbit interaction, Phys. Rev. Lett. 107, 150403 (2011).
Y. Li, L. P. Pitaevskii and S. Stringari, Quantum tricriticality and phase transitions in spin–orbit coupled Bose–Einstein condensates, Phys. Rev. Lett. 108, 225301 (2012).
J.-R. Li et al., A stripe phase with supersolid properties in spin–orbit-coupled Bose–Einstein condensates, Nature 543, 91 (2017).
D. T. Son and M. A. Stephanov, Domain walls of relative phase in two-component Bose–Einstein condensates, Phys. Rev. A 65, 063621 (2002).

Replace the atoms' contact interactions with Newtonian gravity and the same equation describes a self-gravitating quantum fluid: a model for ultralight (“fuzzy”) dark matter and boson stars, where quantum pressure resists gravitational collapse.

The model

i \frac{\partial ψ}{\partial t} = [- \frac{1}{2} \nabla^{2} + V + g {| ψ |}^{2} + Φ] ψ, \tilde{Φ} (𝐤) = - G \frac{2 π}{k} e r f c x (\frac{k l_{z}}{\sqrt{2}}) \tilde{n} (𝐤),

the Schrödinger–Poisson system with Newtonian gravity averaged over the thin axial profile ( $e r f c x x = e^{x^{2}} e r f c x$ ). At long wavelengths this is the potential of a thin sheet, $- 2 π G \tilde{n} / k$ . The periodic box removes the uniform background, as cosmological codes do.

The physics

Linearising about a uniform density n gives the Bogoliubov spectrum with gravity included:

ω^{2} = \frac{k^{4}}{4} + g n k^{2} - 2 π G n k e r f c x (\frac{k l_{z}}{\sqrt{2}}) .

Gravity wins at long wavelengths: this is the Jeans instability. Quantum pressure (the $k^{4}$ term) and contact repulsion stabilise short wavelengths, giving a quantum Jeans length below which structure cannot form (Chavanis 2011; Hui et al. 2017). In an infinite sheet any G is unstable at small enough k. The periodic box only allows $k \geq 2 π / L$ , so the uniform gas is stable until G passes a threshold. After collapse, quantum pressure supports dense solitonic cores, the analogue of the cores found in simulations of fuzzy dark-matter haloes (Schive, Chiueh and Broadhurst 2014). Condensates as models of dark-matter haloes are discussed briefly in BBP §1.3.

Try this

Predict the Jeans threshold. With no trap, the uniform density is $n = 475 / 17^{2} \approx 1.64$ and the longest wave has $k = 2 π / 17 \approx 0.37$ . Set ω² = 0 in the dispersion relation and solve for G; you should get about 0.09. Then raise G across it and watch the gas fragment.
Cores and their merging. Well above threshold, at G ≈ 0.2, filaments feed dense cores that orbit and merge. Use the phase view to see the interference between cores.
Gravity in a trap. In the harmonic trap, raise G and watch self-gravity compress the cloud beyond the Thomas–Fermi profile.

BBP §1.3 (quantum fluids as models of dark-matter haloes) and §4.1.1 (the Bogoliubov dispersion relation that gravity modifies).
L. Hui, J. P. Ostriker, S. Tremaine and E. Witten, Ultralight scalars as cosmological dark matter, Phys. Rev. D 95, 043541 (2017).
L. Hui, Wave dark matter, Annu. Rev. Astron. Astrophys. 59, 247 (2021).
H.-Y. Schive, T. Chiueh and T. Broadhurst, Cosmic structure as the quantum interference of a coherent dark wave, Nature Phys. 10, 496 (2014).
P.-H. Chavanis, Mass–radius relation of Newtonian self-gravitating Bose–Einstein condensates with short-range interactions, Phys. Rev. D 84, 043531 (2011).

Exciton-polaritons, part light and part matter, live in semiconductor microcavities. They leak out of the cavity in picoseconds, so a polariton condensate exists only while it is pumped: a superfluid far from equilibrium.

The model

i \frac{\partial ψ}{\partial t} = [- \frac{1}{2} \nabla^{2} + V + g {| ψ |}^{2} + g_{R} n_{R} + \frac{i}{2} (R n_{R} - γ_{c})] ψ, \frac{\partial n_{R}}{\partial t} = P - (γ_{R} + R {| ψ |}^{2}) n_{R},

the driven-dissipative GPE of Wouters and Carusotto (2007). An incoherent reservoir $n_{R}$ , filled by the pump P, feeds the condensate by stimulated scattering at rate R, while polaritons decay at rate $γ_{c}$ . Here $γ_{c} = 1$ , $γ_{R} = 2$ , R = 0.3 and $g_{R} = 1$ . The noise slider adds the truncated-Wigner noise that gain and loss imply (Wouters and Savona 2009), and the disorder slider adds a random potential like the imperfections of a real cavity.

The physics

Gain balances loss when the reservoir is clamped at $n_{R} = γ_{c} / R$ , which requires $P > P_{t h} = γ_{c} γ_{R} / R$ . Above threshold, for a uniform pump, the condensate density is

n = \frac{P - P_{t h}}{γ_{c}} = \frac{γ_{R}}{R} (\frac{P}{P_{t h}} - 1),

so it is set by the balance of gain and loss rather than by a conserved atom number (Carusotto and Ciuti 2013). The reservoir also repels the condensate, which therefore flows outwards from a pumped spot. Vortices form spontaneously where such flows meet or where disorder pins them (Lagoudakis et al. 2008; Keeling and Berloff 2008). Polariton condensation was first observed by Kasprzak et al. (2006); BBP mentions polaritons in §1.3 and polariton supersolids in §13.3.

Try this

Threshold. With no trap (uniform pump), lower P/P_th below 1 and the fluid fades; raise it above 1 and a condensate grows from noise. The predicted density is $n \approx 6.7 (P / P_{t h} - 1)$ , about 10 at the default 2.5.
Outflow from a spot. With the Gaussian spot (the harmonic button), show the phase: the colour cycles outwards from the centre, which is the radial flow driven by reservoir repulsion and gain.
Pinned vortices. Raise the disorder and mark vortices. Do they sit still where they would drift in a clean cavity?

BBP §1.3 (condensates of polaritons and photons) and §13.3 (polariton supersolids); §3.9.1 for damped GPEs in general.
I. Carusotto and C. Ciuti, Quantum fluids of light, Rev. Mod. Phys. 85, 299 (2013).
M. Wouters and I. Carusotto, Excitations in a nonequilibrium Bose–Einstein condensate of exciton polaritons, Phys. Rev. Lett. 99, 140402 (2007).
M. Wouters and V. Savona, Stochastic classical field model for polariton condensates, Phys. Rev. B 79, 165302 (2009).
J. Kasprzak et al., Bose–Einstein condensation of exciton polaritons, Nature 443, 409 (2006).
K. G. Lagoudakis et al., Quantized vortices in an exciton–polariton condensate, Nature Phys. 4, 706 (2008).
J. Keeling and N. G. Berloff, Spontaneous rotating vortex lattices in a pumped decaying condensate, Phys. Rev. Lett. 100, 250401 (2008).

At finite temperature the condensate is surrounded by thermally excited atoms. In two dimensions thermal fluctuations destroy true long-range order, and superfluidity is lost when vortex–antivortex pairs unbind: the Berezinskii–Kosterlitz–Thouless (BKT) transition.

The model

d ψ = 𝒫 {- (i + γ) (L - μ) ψ d t + d W}, L = - \frac{1}{2} \nabla^{2} + V + g {| ψ |}^{2}, ⟨ d W^{*} d W ⟩ = 2 γ T d t,

the simple-growth stochastic projected GPE (Gardiner and Davis 2003; Bradley, Gardiner and Davis 2008). The highly occupied low-energy modes, $| 𝐤 | < k_{c} = 6.5$ , are treated as a classical field ψ. They exchange atoms and energy with a thermal reservoir at temperature T and chemical potential μ = 12, through damping γ and noise dW obeying a fluctuation–dissipation relation. The projector $𝒫$ removes modes above the cutoff.

The physics

In 2D, long-wavelength phase fluctuations destroy true condensation at any T > 0, but at low T phase correlations decay only as a power law: a quasi-condensate that is still superfluid. Thermally excited vortices appear in bound vortex–antivortex pairs. At $T_{B K T}$ the pairs unbind, free vortices proliferate, and the superfluid density jumps to zero, with $n_{s} λ_{T}^{2} = 4$ just below the transition (Berezinskii 1972; Kosterlitz and Thouless 1973). The transition was seen in a 2D atomic gas by Hadzibabic et al. (2006), and classical-field simulations show the pair unbinding directly (Simula and Blakie 2006). Classical-field results depend on the cutoff $k_{c}$ , so the transition temperature in the app is a property of this model rather than a universal number (Blakie et al. 2008). BBP does not cover BKT; its §10.9–10.10 describe 2D vortex gases and how vortices cluster and annihilate.

Try this

Pairs, then free vortices. With no trap, mark vortices and raise T from 4 to 15. Below about 10 there are few vortices, in tight opposite-sign pairs. By 15 they are numerous and unpaired.
Phase coherence. Show the phase at T = 4 and T = 15. At low T the phase varies smoothly over the whole box; above the transition it is scrambled on short scales.
A trapped gas. In the harmonic trap, raise T and watch a thermal cloud grow at the edge as the condensate shrinks. The density shown is averaged over 0.25 time units, as an absorption image would blur it.

BBP §3.9.1 (dissipative GPE), §10.9 (quantum turbulence in 2D), §10.10 (clustering and negative temperatures).
P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh and C. W. Gardiner, Dynamics and statistical mechanics of ultra-cold Bose gases using c-field techniques, Adv. Phys. 57, 363 (2008).
C. W. Gardiner and M. J. Davis, The stochastic Gross–Pitaevskii equation: II, J. Phys. B 36, 4731 (2003).
A. S. Bradley, C. W. Gardiner and M. J. Davis, Bose–Einstein condensation from a rotating thermal cloud: vortex nucleation and lattice formation, Phys. Rev. A 77, 033616 (2008).
V. L. Berezinskii, Destruction of long-range order in one-dimensional and two-dimensional systems possessing a continuous symmetry group. II. Quantum systems, Sov. Phys. JETP 34, 610 (1972).
J. M. Kosterlitz and D. J. Thouless, Ordering, metastability and phase transitions in two-dimensional systems, J. Phys. C 6, 1181 (1973).
Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier and J. Dalibard, Berezinskii–Kosterlitz–Thouless crossover in a trapped atomic gas, Nature 441, 1118 (2006).
T. P. Simula and P. B. Blakie, Thermal activation of vortex–antivortex pairs in quasi-two-dimensional Bose–Einstein condensates, Phys. Rev. Lett. 96, 020404 (2006).

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