In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero, i.e. 0 K (−273.15 °C; −459.67 °F). Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which microscopic quantum-mechanical phenomena, particularly wavefunction interference, become apparent macroscopically. More generally, condensation refers to the appearance of macroscopic occupation of one or several states: for example, in BCS theory, a superconductor is a condensate of Cooper pairs. As such, condensation can be associated with phase transition, and the macroscopic occupation of the state is the order parameter. Bose–Einstein condensates were first predicted, generally, in 1924–1925 by Albert Einstein, crediting a pioneering paper by Satyendra Nath Bose on the new field now known as quantum statistics. In 1995, the Bose–Einstein condensate was created by Eric Cornell and Carl Wieman of the University of Colorado Boulder using rubidium atoms. Later that year, Wolfgang Ketterle of MIT produced a BEC using sodium atoms. In 2001 Cornell, Wieman, and Ketterle shared the Nobel Prize in Physics "for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates".
History
Bose first sent a paper to Einstein on the quantum statistics of light quanta (now called photons), in which he derived Planck's quantum radiation law without any reference to classical physics. Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it in 1924. Einstein's manuscript, once believed to be lost, was found in a library at Leiden University in 2005. Einstein then extended Bose's ideas to matter in two other papers. The result of their efforts is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now called bosons. Bosons are allowed to share a quantum state. Einstein proposed that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter. Bosons include the photon, polaritons, magnons, some atoms and molecules (depending on the number of nucleons, see #Isotopes) such as atomic hydrogen, helium-4, lithium-7, rubidium-87 or strontium-84. In 1938, Fritz London proposed the BEC as a mechanism for superfluidity in helium-4 and superconductivity. The quest to produce a Bose–Einstein condensate in the laboratory was stimulated by a paper published in 1976 by two program directors at the National Science Foundation (William Stwalley and Lewis Nosanow), proposing to use spin-polarized atomic hydrogen to produce a gaseous BEC. This led to the immediate pursuit of the idea by four independent research groups; these were led by Isaac Silvera (University of Amsterdam), Walter Hardy (University of British Columbia), Thomas Greytak (Massachusetts Institute of Technology) and David Lee (Cornell University). However, cooling atomic hydrogen turned out to be technically difficult, and Bose-Einstein condensation of atomic hydrogen was only realized in 1998. On 5 June 1995, the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST–JILA lab, in a gas of rubidium atoms cooled to 170 nanokelvins (nK). Shortly thereafter, Wolfgang Ketterle at MIT produced a Bose–Einstein Condensate in a gas of sodium atoms. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics. Bose-Einstein condensation of alkali gases is easier because they can be pre-cooled with laser cooling techniques, unlike atomic hydrogen at the time, which gave a significant head start when performing the final forced evaporative cooling to cross the condensation threshold. These early studies founded the field of ultracold atoms, and hundreds of research groups around the world now routinely produce BECs of dilute atomic vapors in their labs. Since 1995, many other atomic species have been condensed (see #Isotopes), and BECs have also been realized using molecules, polaritons, and other quasi-particles. BECs of photons can be made, for example, in dye microcavites with wavelength-scale mirror separation, forming a two-dimensional harmonically confined photon gas with tunable chemical potential. BEC of plasmonic quasiparticles (plasmon-exciton polaritons) has been realized in periodic arrays of metal nanoparticles overlaid with dye molecules, exhibiting ultrafast sub-picosecond dynamics and long-range correlations.
Critical temperature This transition to BEC occurs below a critical temperature, which for a uniform three-dimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by
T c = ( n ζ ( 3 / 2 ) ) 2 / 3 2 π ℏ 2 m k B ≈ 3.3125 ℏ 2 n 2 / 3 m k B , {\displaystyle T_{\text{c}}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{\text{B}}}}\approx 3.3125\,{\frac {\hbar ^{2}n^{2/3}}{mk_{\text{B}}}},}
where:
T c {\displaystyle T_{\text{c}}} is the critical temperature,
n {\displaystyle n} is the particle density,
m {\displaystyle m} is the mass per boson,
ℏ {\displaystyle \hbar } is the reduced Planck constant,
k B {\displaystyle k_{\text{B}}} is the Boltzmann constant,
ζ {\displaystyle \zeta } is the Riemann zeta function ( ζ ( 3 / 2 ) ≈ 2.6124 {\displaystyle \zeta (3/2)\approx 2.6124} ). Interactions shift the value, and the corrections can be calculated by mean-field theory. This formula is derived from finding the gas degeneracy in the Bose gas using Bose–Einstein statistics. The critical temperature depends on the density. A more concise and experimentally relevant condition involves the phase-space density D = n λ T 3 {\displaystyle {\mathcal {D}}=n\lambda _{T}^{3}} , where
λ T = ℏ 2 π m k B T {\displaystyle \lambda _{T}=\hbar {\sqrt {\frac {2\pi }{mk_{\text{B}}T}}}}
is the thermal de Broglie wavelength. It is a dimensionless quantity. The transition to BEC occurs when the phase-space density is greater than critical value:
D c = ζ ( 3 / 2 ) {\displaystyle {\mathcal {D}}_{\text{c}}=\zeta (3/2)}
in 3D uniform space. This is equivalent to the above condition on the temperature. In a 3D harmonic potential, the critical value is instead
D c = ζ ( 3 ) ≈ 1.202 {\displaystyle {\mathcal {D}}_{\text{c}}=\zeta (3)\approx 1.202}
where n {\displaystyle n} has to be understood as the peak density.
Derivation
Ideal Bose gas For an ideal Bose gas we have the equation of state
1 v = 1 λ 3 g 3 / 2 ( f ) + 1 V f 1 − f , {\displaystyle {\frac {1}{v}}={\frac {1}{\lambda ^{3}}}g_{3/2}(f)+{\frac {1}{V}}{\frac {f}{1-f}},}
where v = V / N {\displaystyle v=V/N} is the per-particle volume, λ {\displaystyle \lambda } is the thermal wavelength, f {\displaystyle f} is the fugacity, and
g α ( f ) = ∑ n = 1 ∞ f n n α . {\displaystyle g_{\alpha }(f)=\sum \limits _{n=1}^{\infty }{\frac {f^{n}}{n^{\alpha }}}.}
It is noticeable that g 3 / 2 {\displaystyle g_{3/2}} is a monotonically growing function of f {\displaystyle f} in f ∈ [ 0 , 1 ] {\displaystyle f\in [0,1]} , which are the only values for which the series converge. Recognizing that the second term on the right-hand side contains the expression for the average occupation number of the fundamental state ⟨ n 0 ⟩ {\displaystyle \langle n_{0}\rangle } , the equation of state can be rewritten as
1 v = 1 λ 3 g 3 / 2 ( f ) + ⟨ n 0 ⟩ V ⇔ ⟨ n 0 ⟩ V λ 3 = λ 3 v − g 3 / 2 ( f ) . {\displaystyle {\frac {1}{v}}={\frac {1}{\lambda ^{3}}}g_{3/2}(f)+{\frac {\langle n_{0}\rangle }{V}}\Leftrightarrow {\frac {\langle n_{0}\rangle }{V}}\lambda ^{3}={\frac {\lambda ^{3}}{v}}-g_{3/2}(f).}
Because the left term on the second equation must always be positive, λ 3 v > g 3 / 2 ( f ) {\displaystyle {\frac {\lambda ^{3}}{v}}>g_{3/2}(f)} , and because g 3 / 2 ( f ) ≤ g 3 / 2 ( 1 ) {\displaystyle g_{3/2}(f)\leq g_{3/2}(1)} , a stronger condition is
λ 3 v > g 3 / 2 ( 1 ) , {\displaystyle {\frac {\lambda ^{3}}{v}}>g_{3/2}(1),}
which defines a transition between a gas phase and a condensed phase. On the critical region it is possible to define a critical temperature and thermal wavelength:
λ c 3 = g 3 / 2 ( 1 ) v = ζ ( 3 / 2 ) v , {\displaystyle \lambda _{c}^{3}=g_{3/2}(1)v=\zeta (3/2)v,}
T c = 2 π ℏ 2 m k B λ c 2 , {\displaystyle T_{\text{c}}={\frac {2\pi \hbar ^{2}}{mk_{\text{B}}\lambda _{c}^{2}}},}
recovering the value indicated on the previous section. The critical values are such that if T < T c {\displaystyle T<T_{\text{c}}} or λ > λ c {\displaystyle \lambda >\lambda _{\text{c}}} , we are in the presence of a Bose–Einstein condensate. Understanding what happens with the fraction of particles on the fundamental level is crucial. As so, write the equation of state for f = 1 {\displaystyle f=1} , obtaining
⟨ n 0 ⟩ N = 1 − ( λ c λ ) 3 {\displaystyle {\frac {\langle n_{0}\rangle }{N}}=1-\left({\frac {\lambda _{\text{c}}}{\lambda }}\right)^{3}} and equivalently ⟨ n 0 ⟩ N = 1 − ( T T c ) 3 / 2 . {\displaystyle {\frac {\langle n_{0}\rangle }{N}}=1-\left({\frac {T}{T_{\text{c}}}}\right)^{3/2}.}
So, if T ≪ T c {\displaystyle T\ll T_{\text{c}}} , the fraction ⟨ n 0 ⟩ N ≈ 1 {\displaystyle {\frac {\langle n_{0}\rangle }{N}}\approx 1} , and if T ≫ T c {\displaystyle T\gg T_{\text{c}}} , the fraction ⟨ n 0 ⟩ N ≈ 0 {\displaystyle {\frac {\langle n_{0}\rangle }{N}}\approx 0} . At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum p → = 0 {\displaystyle {\vec {p}}=0} .
Experimental observation
Superfluid helium-4
In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium-4 became a new kind of fluid, now known as a superfluid, at temperatures less than 2.17 K (the lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly believed that the superfluidity was due to partial Bose–Einstein condensation of the liquid. In fact, many properties of superfluid helium also appear in gaseous condensates created by Cornell, Wieman and Ketterle (see below). Superfluid helium-4 is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong; the original theory of Bose–Einstein condensation must be heavily modified in order to describe it. Bose–Einstein condensation remains, however, fundamental to the superfluid properties of helium-4. Note that helium-3, a fermion, also enters a superfluid phase (at a much lower temperature) which can be explained by the formation of bosonic Cooper pairs of two atoms (see also fermionic condensate).
Dilute atomic gases The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on 5 June 1995. They cooled a dilute vapor of approximately two thousand rubidium-87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT condensed sodium-23. Ketterle's condensate had a hundred times more atoms, allowing important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements. A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work. Lithium has attractive interactions, causing the condensate to be unstable and collapse for all but a few atoms. Hulet's team subsequently showed the condensate could be stabilized by confinement quantum pressure for up to about 1000 atoms. Various isotopes have since been condensed.
Velocity-distribution data graph In the image accompanying this article, the velocity-distribution data indicates the formation of a Bose–Einstein condensate out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: spatially confined atoms have a minimum width velocity distribution. This width is given by the curvature of the magnetic potential in the given direction. More tightly confined directions have bigger widths in the ballistic velocity distribution. This anisotropy of the peak on the right is a purely quantum-mechanical effect and does not exist in the thermal distribution on the left.
Quasiparticles Bose–Einstein condensation also applies to quasiparticles in solids. Magnons, excitons, and polaritons have integer spin which means they are bosons that can form condensates. Magnons, electron spin waves, can be controlled by a magnetic field. Densities from the limit of a dilute gas to a strongly interacting Bose liquid are possible. Magnetic ordering is the analog of superfluidity. In 1999 condensation was demonstrated in antiferromagnetic TlCuCl3, at temperatures as great as 14 K. The high transition temperature (relative to atomic gases) is due to the magnons' small mass (near that of an electron) and greater achievable density. In 2006, condensation in a ferromagnetic yttrium-iron-garnet thin film was seen even at room temperature, with optical pumping. Excitons, electron–hole pairs, were predicted to condense at low temperature and high density by Boer et al., in 1961. Bilayer system experiments first demonstrated condensation in 2003, by Hall voltage disappearance. Fast optical exciton creation was used to form condensates in sub-kelvin Cu2O in 2005 on. Polariton condensation was first detected for exciton-polaritons in a quantum well microcavity kept at 5 K. Quasiparticle BECs have been achieved at room-temperature, for example, in microcavity-coupled organic semiconductors and plasmon-exciton polaritons in periodic arrays of metal nanoparticles coupled to dye molecules.
In zero gravity In June 2020, the Cold Atom Laboratory experiment on board the International Space Station successfully created a BEC of rubidium atoms and observed them for over a second in free-fall. Although initially just a proof of function, early results showed that, in the microgravity environment of the ISS, about half of the atoms formed into a magnetically insensitive halo-like cloud around the main body of the BEC.
Models
Bose Einstein's non-interacting gas
Consider a collection of N non-interacting particles, which can each be in one of two quantum states, | 0 ⟩ {\displaystyle |0\rangle } and | 1 ⟩ {\displaystyle |1\rangle } . If the two states are equal in energy, each different configuration is equally likely. If we can tell which particle is which, there are 2 N {\displaystyle 2^{N}} different configurations, since each particle can be in | 0 ⟩ {\displaystyle |0\rangle } or | 1 ⟩ {\displaystyle |1\rangle } independently. In almost all of the configurations, about half the particles are in | 0 ⟩ {\displaystyle |0\rangle } and the other half in | 1 ⟩ {\displaystyle |1\rangle } . The balance is a statistical effect: the number of configurations is largest when the particles are divided equally. If the particles are indistinguishable, however, there are only N + 1 {\displaystyle N+1} different configurations. If there are K {\displaystyle K} particles in state | 1 ⟩ {\displaystyle |1\rangle } , there are N − K {\displaystyle N-K} particles in state | 0 ⟩ {\displaystyle |0\rangle } . Whether any particular particle is in state | 0 ⟩ {\displaystyle |0\rangle } or in state | 1 ⟩ {\displaystyle |1\rangle } cannot be determined, so each value of K {\displaystyle K} determines a unique quantum state for the whole system. Suppose now that the energy of state | 1 ⟩ {\displaystyle |1\rangle } is slightly greater than the energy of state | 0 ⟩ {\displaystyle |0\rangle } by an amount E {\displaystyle E} . At temperature T {\displaystyle T} , a particle will have a lesser probability to be in state | 1 ⟩ {\displaystyle |1\rangle } by e − E / k T {\displaystyle e^{-E/kT}} . In the distinguishable case, the particle distribution will be biased slightly towards state | 0 ⟩ {\displaystyle |0\rangle } . But in the indistinguishable case, since there is no statistical pressure toward equal numbers, the most-likely outcome is that most of the particles will collapse into state | 0 ⟩ {\displaystyle |0\rangle } . In the distinguishable case, for large N, the fraction in state | 0 ⟩ {\displaystyle |0\rangle } can be computed. It is the same as flipping a coin with probability proportional to exp ( − E / T ) {\displaystyle \exp {(-E/T)}} to land tails. In the indistinguishable case, each value of K {\displaystyle K} is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:
P ( K ) = C e − K E / T = C p K . {\displaystyle \,P(K)=Ce^{-KE/T}=Cp^{K}.}
For large N {\displaystyle N} , the normalization constant C {\displaystyle C} is 1 − p {\displaystyle 1-p} . The expected total number of particles not in the lowest energy state, in the limit that N → ∞ {\displaystyle N\rightarrow \infty } , is equal to
∑ n > 0 C n p n = p / ( 1 − p ) {\displaystyle \sum _{n>0}Cnp^{n}=p/(1-p)}
It does not grow when N is large; it just approaches a constant. This will be a negligible fraction of the total number of particles. So a collection of enough Bose particles in thermal equilibrium will mostly be in the ground state, with only a few in any excited state, no matter how small the energy difference. Consider now a gas of particles, which can be in different momentum states labeled | k ⟩ {\displaystyle |k\rangle } . If the number of particles is less than the number of thermally accessible states, for high temperatures and low densities, the particles will all be in different states. In this limit, the gas is classical. As the density increases or the temperature decreases, the number of accessible states per particle becomes smaller, and at some point, more particles will be forced into a single state than the maximum allowed for that state by statistical weighting. From this point on, any extra particle added will go into the ground state. To calculate the transition temperature at any density, integrate, over all momentum states, the expression for maximum number of excited particles, p / ( 1 − p ) {\displaystyle p/(1-p)} :
N = V ∫ d 3 k ( 2 π ) 3 p ( k ) 1 − p ( k ) = V ∫ d 3 k ( 2 π ) 3 1 e k 2 2 m T − 1 {\displaystyle \,N=V\int {d^{3}k \over (2\pi )^{3}}{p(k) \over 1-p(k)}=V\int {d^{3}k \over (2\pi )^{3}}{1 \over e^{k^{2} \over 2mT}-1}}
p ( k ) = e − k 2 2 m T . {\displaystyle \,p(k)=e^{-k^{2} \over 2mT}.}
When the integral (also known as Bose–Einstein integral) is evaluated with factors of k B {\displaystyle k_{B}} and ℏ {\displaystyle \hbar } restored by dimensional analysis, it gives the critical temperature formula of the preceding section. Therefore, this integral defines the critical temperature and particle number corresponding to the conditions of negligible chemical potential μ {\displaystyle \mu } . In Bose–Einstein statistics distribution, μ {\displaystyle \mu } is actually still nonzero for BECs; however, μ {\displaystyle \mu } is less than the ground state energy. Except when specifically talking about the ground state, μ {\displaystyle \mu } can be approximated for most energy or momentum states as μ ≈ 0 {\displaystyle \mu \approx 0} .
Bogoliubov theory for weakly interacting gas
Nikolay Bogolyubov considered perturbations on the limit of dilute gas, finding a finite pressure at zero temperature and positive chemical potential. This leads to corrections for the ground state. The Bogoliubov state has pressure ( T = 0 ) {\displaystyle (T=0)} : P = g n 2 / 2 {\displaystyle P=gn^{2}/2} . The weakly interacting Bose gas can be converted to a system of non-interacting particles with a dispersion law.
Gross–Pitaevskii equation
In some simplest cases, the state of condensed particles can be described with a nonlinear Schrödinger equation, also known as Gross–Pitaevskii or Ginzburg–Landau equation. The validity of this approach is actually limited to the case of ultracold temperatures, which fits well for the most alkali atoms experiments. This approach originates from the assumption that the state of the BEC can be described by the unique wavefunction of the condensate ψ ( r → ) {\displaystyle \psi ({\vec {r}})} . For a system of this nature, | ψ ( r → ) | 2 {\displaystyle |\psi ({\vec {r}})|^{2}} is interpreted as the particle density, so the total number of atoms is N = ∫ d r → | ψ ( r → ) | 2 {\displaystyle N=\int d{\vec {r}}|\psi ({\vec {r}})|^{2}}
Provided essentially all atoms are in the condensate (that is, have condensed to the ground state), and treating the bosons using mean-field theory, the energy (E) associated with the state ψ ( r → ) {\displaystyle \psi ({\vec {r}})} is:
E = ∫ d r → [ ℏ 2 2 m | ∇ ψ ( r → ) | 2 + V ( r → ) | ψ ( r → ) | 2 + 1 2 U 0 | ψ ( r → ) | 4 ] {\displaystyle E=\int d{\vec {r}}\left[{\frac {\hbar ^{2}}{2m}}|\nabla \psi ({\vec {r}})|^{2}+V({\vec {r}})|\psi ({\vec {r}})|^{2}+{\frac {1}{2}}U_{0}|\psi ({\vec {r}})|^{4}\right]}
Minimizing thi