Experiment: Bose–Einstein Condensation
Bose–Einstein condensation is the one phase transition that requires no interaction whatsoever: it follows from the counting rule for identical bosons alone, which is why Quantum Statistics can derive it for the ideal gas and why Identical Particles is its real premise. This chapter is the experimental counterpart. It presents the 1995 JILA and MIT condensates in full phenomenological detail [Anderson:1995] [Davis:1995] — apparatus, cooling sequence, measured numbers with their uncertainties — together with the seventy years of technique that separate Einstein's prediction [Einstein:1925] from its realization in a dilute alkali vapour, and with the modern repetitions that have turned a once-heroic measurement into a routine laboratory tool. The delay was not conceptual: a condensate of the kind Einstein described needs a gas held at a temperature six orders of magnitude below anything a cryostat can reach, in a container with no walls, at densities low enough that three-body recombination does not convert it to a solid first. Every element of the apparatus in Section 121.2 exists to meet one of those three constraints.
The chapter sits between the statistics that predict the effect
(Quantum Statistics) and the interacting condensed phases
of Superconductivity and Superfluidity, where the same
macroscopic wavefunction appears in a strongly correlated liquid;
its coherence measurements are the atomic counterpart of the optical
ones in Quantum Optics and the Photon, and its matter-wave interference
closes the argument opened in Matter Waves. When written
out, the chapter will use the structured experiment
environment throughout, with apparatus, procedure, observations,
interpretation and primary references as separate fields, and with all
quantities in SI.
Experiment: Bose–Einstein Condensation: all derivations of this chapter are pending.
The prediction under test
Bose statistics and Einstein's condensation
[Reserved: Bose's re-derivation of Planck's law by counting cells in phase space rather than by assuming a classical field [Bose:1924]; Einstein's extension of the counting to a gas of massive particles [Einstein:1924] and, in the second paper, the conclusion that below a finite temperature a macroscopic fraction of the particles occupies the single-particle ground state [Einstein:1925]; why the effect is purely statistical, arising from the symmetry of the many-body state (Identical Particles) and not from any attraction; the ideal-gas thermodynamics derived in Quantum Statistics.]
The condensation criterion
[Reserved: the thermal de Broglie wavelength \(\lambda_{\mathrm{dB}}=h/\sqrt{2\pi m k_{B}T}\) and the criterion \(n\lambda_{\mathrm{dB}}^{3}\geq\zeta(3/2)\approx2.612\) for the homogeneous ideal gas [Einstein:1925]; the modified criterion \(k_{B}T_{c}=\hbar\bar{\omega}(N/\zeta(3))^{1/3}\) for the harmonic trap that every experiment actually uses [Dalfovo:1999]; the numerical consequence that a vapour at \(n\sim10^{19}\,/\mathrm{m}^{3}\) condenses only below about \(1\,\mu\mathrm{K}\), six decades colder than liquid helium; the standard monograph treatment [Pethick:2008].]
Seventy years of near misses
[Reserved: London's identification of the helium-4 lambda transition as a Bose–Einstein condensation distorted by strong interactions [London:1938], and why a liquid with a condensate fraction near \(10\,\mathrm{\%}\) cannot test Einstein's ideal-gas prediction (the interacting case is Superconductivity and Superfluidity); the spin-polarized atomic hydrogen programme as the deliberate attempt at a weakly interacting Bose gas [Hess:1986] [Masuhara:1988], its recombination-limited densities, and the switch to alkali vapours that could be laser-cooled.]
Apparatus
Laser cooling
[Reserved: radiation pressure as a mechanical force on neutral particles [Ashkin:1970]; the proposal to cool a gas with red-detuned light [Hansch:1975a]; optical molasses — six counter-propagating beams producing a viscous damping force and temperatures near the Doppler limit \(k_{B}T_{D}=\hbar\Gamma/2\) [Chu:1985]; the magneto-optical trap that adds a quadrupole field to make the force position-dependent [Raab:1987]; Zeeman slowing of the atomic beam that loads it; the level structures involved belong to Atoms and Molecules.]
Sub-Doppler cooling
[Reserved: the measurement that found sodium at \(43(20)\,\mu\mathrm{K}\), well below the \(240\,\mu\mathrm{K}\) Doppler limit, and so falsified the accepted theory of molasses [Lett:1988]; the polarization-gradient (Sisyphus) mechanism that explains it, in which an atom climbs a light shift and is optically pumped to the bottom of the next well [Dalibard:1989]; the recoil limit \(k_{B}T_{R}=\hbar^{2}k^{2}/m\) as the residual floor; why sub-Doppler temperatures still leave phase-space density a factor \(\sim10^{6}\) short of condensation.]
Laser-cooled sodium in optical molasses is measured at \(43(20)\,\mu\mathrm{K}\), well below the \(240\,\mu\mathrm{K}\) Doppler limit \(k_{B}T_{D}=\hbar\Gamma/2\) that the accepted theory of molasses predicted and that had been treated as a floor [Lett:1988]. The measurement was a falsification, not a refinement: it forced the recognition of a second cooling mechanism that the two-level model does not contain, and the temperature it reaches instead is set by the photon recoil.
The polarization-gradient (Sisyphus) mechanism: the spatially modulated light shifts of the ground-state sublattice in a standing wave of varying polarization, the optical pumping that preferentially transfers an atom from a hill to the bottom of the next well, the resulting friction coefficient, and the recoil energy as the residual floor
Magnetic traps
[Reserved: the Zeeman potential \(U=\mu_{B}g_{F}m_{F}\abs{B}\) and the first magnetic confinement of neutral atoms [Migdall:1985]; Majorana spin flips at the zero of a pure quadrupole field, and the two cures — the JILA time-orbiting potential trap that rotates the zero faster than the atoms can follow [Petrich:1995] and the MIT optical plug that repels atoms from it [Davis:1995]; why only weak-field-seeking states can be trapped, and why Earnshaw's theorem forbids a static maximum.]
Evaporative cooling
[Reserved: forced evaporation — selectively removing the highest-energy atoms with a radio-frequency knife and letting elastic collisions rethermalize the remainder — proposed and demonstrated for spin-polarized hydrogen [Hess:1986] [Masuhara:1988]; the runaway condition relating the ratio of good (elastic) to bad (inelastic) collisions; the price paid in atom number, typically a factor \(10^{3}\) lost for a factor \(10^{6}\) gained in phase-space density; why this last stage, and not laser cooling, is what actually reaches degeneracy.]
Imaging and thermometry
[Reserved: destructive absorption imaging after ballistic expansion, in which the released cloud maps momentum onto position and the optical depth is fitted to a Bose-distributed column density; non-destructive dispersive (phase-contrast) imaging, which permits repeated observation of one condensate [Andrews:1996]; how temperature is extracted from the thermal wings rather than from any thermometer in contact with the gas; systematic uncertainties in magnification, atom number and the resonant cross-section, which dominate the error budget quoted in Section 121.4.2.]
Procedure
The JILA rubidium-87 sequence
[Reserved: the full 1995 cycle — vapour-cell magneto-optical trap, polarization-gradient molasses, optical pumping into \(\ket{F=2,m_{F}=2}\), transfer to the time-orbiting potential trap, and radio-frequency evaporation over about \(70\,\mathrm{s}\) [Anderson:1995] [Petrich:1995]; the endpoint of roughly \(2\times10^{3}\) condensed atoms of rubidium-87 at \(170\,\mathrm{nK}\) and a peak density near \(2.5\times 10^{18}\,/\mathrm{m}^{3}\); the anisotropic expansion after release as the signature that survived every check.]
The MIT sodium-23 sequence
[Reserved: the contrasting design — a Zeeman-slowed sodium beam, a dark-spot magneto-optical trap, and a plugged quadrupole trap — reaching some \(5\times10^{5}\) condensed atoms, two orders of magnitude more than JILA, at a transition temperature near \(2\,\mu\mathrm{K}\) [Davis:1995]; why the larger condensate is what made the coherence experiments of Section 121.4.4 possible; the independence of the two apparatus designs as the strongest argument against an artefact.]
Lithium-7 and attractive interactions
[Reserved: the third 1995 result, in lithium-7, whose negative scattering length makes the condensate mechanically unstable above a critical atom number [Bradley:1995]; the observed limit of order \(10^{3}\) atoms and its later refinement; why this case tests the interaction-dependent part of the theory that rubidium and sodium, both repulsive, cannot.]
Observations
The bimodal velocity distribution
[Reserved: the primary datum — a narrow, anisotropic peak emerging from a broad isotropic thermal background as the evaporation endpoint is lowered through \(T_{c}\) [Anderson:1995] [Davis:1995]; why the anisotropy is decisive: a thermal cloud expands isotropically whatever the trap shape, while a condensate inherits the inverted aspect ratio of the trapping potential through the uncertainty relation; the three control observations that excluded a cold thermal cloud, an imaging artefact and a mechanical compression.]
As the endpoint of the evaporation is lowered through a critical temperature, the velocity distribution of the released cloud ceases to be a single broad isotropic Gaussian and becomes bimodal: a narrow peak grows out of the broad thermal background and takes over as the temperature is lowered further. The narrow component is anisotropic, and expands fastest along the axis in which the trap was tightest — the inverse of the shape it had in the trap — while the thermal background expands isotropically whatever the trap geometry [Anderson:1995] [Davis:1995].
Derivation. The anisotropy is the discriminating observation, and it follows from the uncertainty relation alone. A thermal cloud in equilibrium at temperature \(T\) has, whatever the shape of the confining potential, an isotropic Maxwell–Boltzmann momentum distribution of width \(\sqrt{mk_{B}T}\) in every direction; after release its size grows as \(\sqrt{k_{B}T/m}\;t\) equally in all three directions, so a long expansion always drives it towards a sphere. A condensate, by contrast, occupies one single-particle state, the ground state of a harmonic trap of frequencies \(\omega_{i}\), whose widths and momentum widths are
The in-trap aspect ratio is therefore \(\sigma_{1}/\sigma_{2}=\sqrt{\omega_{2}/\omega_{1}}\), while after a free expansion long enough that the initial size is negligible the observed size is \(\Delta p_{i}t/m\propto\sqrt{\hbar\omega_{i}/m}\;t\) and the aspect ratio is \(\sqrt{\omega_{1}/\omega_{2}}\) — exactly inverted. The tightly confined direction is the one with the largest momentum spread and so expands fastest. No thermal cloud, at any temperature, can produce that inversion, which is why the anisotropy and not the narrowness of the peak is what identifies the condensate.
∎Condensate fraction and transition temperature
[Reserved: the quantitative test — condensate fraction \(N_{0}/N=1-(T/T_{c})^{3}\) for an ideal gas in a harmonic trap, measured against \(T/T_{c}\) together with the released energy [Ensher:1996]; the measured transition temperature and the several-per-cent negative shift from the ideal-gas value attributable to finite \(N\) and to mean-field repulsion; the tabulated SI values, uncertainties and systematic error budget that will constitute the dataset of this chapter; comparison with the trapped-gas theory [Dalfovo:1999].]
The condensed fraction of a harmonically trapped Bose gas rises from zero at a sharp transition temperature according to
with \(\bar{\omega}\) the geometric mean trap frequency. Both the fraction and the released energy were measured against \(T/T_{c}\) and found to follow Equation (121.2) [Ensher:1996]; the measured \(T_{c}\) lies a few per cent below the ideal-gas value, by an amount attributable to the finite atom number and to the mean-field repulsion between the atoms [Dalfovo:1999]. A prediction made for a gas that does not interact thus survives contact with one that does, to within the corrections that interaction requires.
Derivation. For a three-dimensional harmonic trap with frequencies \(\omega_{i}\) and \(\bar{\omega}=(\omega_{1}\omega_{2}\omega_{3})^{1/3}\), the number of single-particle states with energy below \(E\) is \(E^{3}/6(\hbar\bar{\omega})^{3}\) for \(E\gg\hbar\bar{\omega}\), so the density of states is \(g(E)=E^{2}/2(\hbar\bar{\omega})^{3}\). At and below the transition the chemical potential has reached the ground-state energy, and the number of atoms the excited states can hold is
using \(\int_{0}^{\infty}x^{2}(\ee^{x}-1)^{-1}\dd x=2\zeta(3)\). The right-hand side of Equation (121.3) is finite and depends only on the temperature: the excited states saturate. Any atoms beyond that number have nowhere to go but the ground state, which is the whole content of the effect. Setting \(N_{\mathrm{ex}}=N\) defines \(T_{c}\) as in Equation (121.2); below \(T_{c}\) the excited population is still given by Equation (121.3), that is \(N_{\mathrm{ex}}=N(T/T_{c})^{3}\), and the remainder \(N_{0}=N-N_{\mathrm{ex}}\) gives the condensate fraction. Note that the exponent is \(3\) for the harmonic trap, not the \(3/2\) of the homogeneous gas; the density of states, not the statistics, is what differs.
∎Collective excitations
[Reserved: the low-lying shape oscillations of the trapped condensate, measured to the per-cent level and compared with the hydrodynamic predictions of the Gross–Pitaevskii equation [Jin:1996]; the frequency ratios as a parameter-free test, since they depend on the trap geometry alone; the damping rates and their temperature dependence as the first quantitative probe of the condensate–thermal-cloud coupling.]
Interference of two condensates
[Reserved: two independently prepared sodium condensates, released and overlapped, producing high-contrast straight fringes of period about \(15\,\mu\mathrm{m}\) [Andrews:1997]; the fringes as the direct evidence of long-range phase coherence, and the subtle point that the relative phase is established by the measurement rather than pre-existing; the matter-wave interpretation continuous with Matter Waves and Experiment: Wave Optics; the output coupler that turned the condensate into a coherent atom beam [Mewes:1997].]
Two condensates prepared independently in separate wells of one trap, then released and allowed to overlap, produce high-contrast straight interference fringes of period about \(15\,\mu\mathrm{m}\) [Andrews:1997]. The fringes are matter-wave interference on a macroscopic scale: the phase of the condensate is well defined across the whole cloud, over distances some five orders of magnitude larger than the interparticle spacing. The subtlety the data also establish is that the two clouds never shared a phase — the relative phase is established by the act of detection, and is random from shot to shot while the fringe contrast is not.
Interference of two overlapping expanding condensates: the fringe spacing set by the relative momentum of the two sources, and the emergence of a definite relative phase from repeated measurement on a state of fixed total atom number, in which no relative phase exists before the first detection
Interpretation
The order parameter and the Gross–Pitaevskii equation
[Reserved: the condensate wavefunction \(\psi(\vect{r},t)=\sqrt{n_{0}}\,\ee^{\ii\theta}\) as the order parameter of a broken \(\U(1)\) symmetry, and the nonlinear Schrödinger equation governing it, derived independently for the vortex problem [Gross:1961] [Pitaevskii:1961]; the coupling constant \(g=4\pi\hbar^{2}a/m\) fixed by the measured \(s\)-wave scattering length; the Thomas–Fermi profile and the healing length \(\xi=(8\pi n a)^{-1/2}\), both confirmed by the expansion data [Dalfovo:1999].]
Bogoliubov excitations and superfluidity
[Reserved: Bogoliubov's diagonalization of the weakly interacting Bose gas [Bogoliubov:1947], giving the excitation spectrum \(\varepsilon(k)=\sqrt{(\hbar^{2}k^{2}/2m)(\hbar^{2}k^{2}/2m+2gn)}\), phonon-like at long wavelength with sound speed \(c=\sqrt{gn/m}\); the Landau criterion and the measured critical velocity for a stirred laser beam [Raman:1999]; why condensation and superfluidity are distinct properties, related in Superconductivity and Superfluidity.]
What was and was not shown
[Reserved: the honest ledger — what the 1995 data establish (a phase transition at the predicted temperature, with the predicted condensate fraction and the predicted anisotropic expansion), what they do not (nothing about the interacting many-body ground state beyond mean field), and which later measurements closed each gap; statistical versus systematic uncertainty in atom number, the dominant error throughout; the relation to the general theory of phase transitions in Phase Transitions and Critical Phenomena.]
Repetitions, extensions and precision
Other species, and hydrogen
[Reserved: condensation of atomic hydrogen, the species the programme began with, reached three years after the alkalis and with \(10^{9}\) atoms [Fried:1998]; the subsequent extension to metastable helium, chromium, and the lanthanides; two-photon spectroscopy of the hydrogen condensate as the link to Experiment: Precision Spectroscopy and Atomic Clocks; the present status of species and atom numbers, tabulated with sources.]
Vortices and quantized circulation
[Reserved: the first vortex, imprinted by a coherent two-component coupling and imaged interferometrically [Matthews:1999]; vortex nucleation by mechanical stirring above a critical rotation frequency [Madison:2000]; the Abrikosov lattice of up to 130 vortices observed in a rotating sodium condensate [AboShaeer:2001]; the circulation quantum \(h/m\) measured directly, and the parallel with flux quantization in Superconductivity and Superfluidity.]
A condensate does not rotate as a fluid body. Set into rotation it remains at rest until a critical angular frequency is exceeded, and then admits angular momentum only in the form of discrete vortex lines, each carrying one quantum of circulation \(h/m\) and a density node at its core [Matthews:1999] [Madison:2000]. Above threshold the lines multiply and arrange themselves in a regular triangular array, up to about 130 of them counted in one rotating sodium condensate [AboShaeer:2001]. The array is the direct image of a quantity that classical hydrodynamics allows to take any value whatever.
Derivation. The condensate is described by a single complex field \(\psi=\sqrt{n_{0}}\,\ee^{\ii\theta}\), and the current it carries identifies the superfluid velocity as \(\vect{v}=(\hbar/m)\nabla\theta\). Let \(C\) be any closed contour along which \(\psi\) is non-zero. Since \(\psi\) is a single-valued function of position, \(\theta\) can change round \(C\) only by an integer multiple of \(2\pi\), and therefore
The circulation is thus quantized in units of \(h/m\), with a value that depends on nothing — not the contour, not the density, not the strength of the interaction — but the integer \(n\). Two consequences follow at once. Since the flow is irrotational wherever \(\psi\neq0\), non-zero circulation requires \(\psi\) to vanish somewhere inside \(C\): a vortex has a core. And since \(n\) cannot vary continuously, a rotating condensate cannot take up angular momentum smoothly and must instead nucleate whole vortices one at a time above a threshold rotation rate, which is what the observations show.
∎Optical lattices
[Reserved: the condensate loaded into a standing-wave potential as a defect-free realization of the lattice models of Phonons and Lattice Dynamics and Electrons in Solids: Band Theory; the observed superfluid-to-Mott-insulator transition, a quantum phase transition driven by the ratio of tunnelling to on-site repulsion [Greiner:2002]; the loss and revival of interference fringes as its order parameter; the review of the field [Bloch:2008].]
A condensate loaded into a three-dimensional optical standing wave loses its interference pattern abruptly as the lattice depth is increased, and recovers it on lowering the depth again. The loss is reversible, occurs at a definite lattice depth, and is not caused by heating: it marks a transition from a superfluid, in which the atoms are delocalized and the phase is shared, to an insulating state with a fixed integer number of atoms per lattice site and a gap for adding one more [Greiner:2002]. The control parameter is the ratio of the tunnelling amplitude between neighbouring sites to the energy cost of double occupancy, a purely quantum-mechanical competition that survives at zero temperature.
The Bose–Hubbard description of atoms in a deep optical lattice, the tunnelling and on-site interaction parameters expressed through the lattice depth and the scattering length, and the mean-field estimate of the ratio at which the compressible superfluid gives way to the gapped, integer-filled insulator
Degenerate Fermi gases and the BEC–BCS crossover
[Reserved: the fermionic counterpart — onset of Fermi degeneracy in trapped potassium-40 [DeMarco:1999]; molecular condensates formed from fermion pairs on a Feshbach resonance [Greiner:2003]; condensation of fermionic atom pairs on the BCS side of the resonance [Regal:2004]; vortex lattices as the unambiguous proof of superfluidity in the strongly interacting regime [Zwierlein:2005]; the crossover as a controlled interpolation between Bose condensation and BCS pairing (Superconductivity and Superfluidity).]
Recognition
[Reserved: the 1997 award for laser cooling and trapping, with its three Nobel lectures as authoritative first-hand accounts of the apparatus [Chu:1998] [CohenTannoudji:1998] [Phillips:1998]; the 2001 award for the condensates themselves [Cornell:2002] [Ketterle:2002]; a note on what the prize citations record about priority among the JILA, MIT and Rice groups, and on the hydrogen programme that supplied the evaporation technique without reaching the transition first.]