Bose–Einstein condensate

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Schematic Bose–Einstein condensation versus temperature of the energy diagram[clarify]

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 (−273.15 °C or −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

order parameter
.

Bose–Einstein condensate was first predicted, generally, in 1924–1925 by

quantum statistics.[3] 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".[4]

History

Velocity-distribution data (3 views) for gas of rubidium atoms, confirming the discovery of a new phase of matter, the Bose–Einstein condensate. Left: just before the appearance of a Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate.

Bose first sent a paper to Einstein on the

identical particles with integer spin, now called bosons. Bosons, particles that include the photon and atoms such as helium-4 (4
He
), 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.

In 1938, Fritz London proposed the BEC as a mechanism for superfluidity in 4
He
and superconductivity.[9][10]

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).[11] 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).[12]

On 5 June 1995, the first gaseous condensate was produced by

University of Colorado at Boulder NISTJILA lab, in a gas of rubidium atoms cooled to 170 nanokelvins (nK).[13] 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.[14] 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, and BECs have also been realized using molecules, quasi-particles, and photons.[15]

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

where:

is the critical temperature,
is the particle density,
is the mass per boson,
is the reduced Planck constant,
is the Boltzmann constant,
is the Riemann zeta function ([16]).

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.

Derivation

Ideal Bose gas

For an ideal Bose gas we have the equation of state

where is the per-particle volume, is the thermal wavelength, is the fugacity, and

It is noticeable that is a monotonically growing function of in , 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 , the equation of state can be rewritten as

Because the left term on the second equation must always be positive, , and because , a stronger condition is

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:

recovering the value indicated on the previous section. The critical values are such that if or , 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 , obtaining

and equivalently

So, if , the fraction , and if , the fraction . At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum .

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, and . If the two states are equal in energy, each different configuration is equally likely.

If we can tell which particle is which, there are different configurations, since each particle can be in or independently. In almost all of the configurations, about half the particles are in and the other half in . 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 different configurations. If there are K particles in state , there are N − K particles in state . Whether any particular particle is in state or in state cannot be determined, so each value of K determines a unique quantum state for the whole system.

Suppose now that the energy of state is slightly greater than the energy of state by an amount E. At temperature T, a particle will have a lesser probability to be in state by . In the distinguishable case, the particle distribution will be biased slightly towards state . 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 .

In the distinguishable case, for large N, the fraction in state can be computed. It is the same as flipping a coin with probability proportional to p = exp(−E/T) to land tails.

In the indistinguishable case, each value of K is a single state, which has its own separate Boltzmann probability. So the probability distribution is exponential:

For large N, the normalization constant C is (1 − p). The expected total number of particles not in the lowest energy state, in the limit that , is equal to

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 . 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):

When the integral (also known as

Bose–Einstein integral
) is evaluated with factors of and ℏ 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 . In Bose–Einstein statistics distribution, is actually still nonzero for BECs; however, is less than the ground state energy. Except when specifically talking about the ground state, can be approximated for most energy or momentum states as .

Bogoliubov theory for weakly interacting gas

Nikolay Bogoliubov considered perturbations on the limit of dilute gas,[17] 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): .

The original interacting system 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 . For a system of this nature, is interpreted as the particle density, so the total number of atoms is

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 is:

Minimizing this energy with respect to infinitesimal variations in , and holding the number of atoms constant, yields the Gross–Pitaevski equation (GPE) (also a non-linear Schrödinger equation):

where:

 is the mass of the bosons,
 is the external potential, and
 represents the inter-particle interactions.

In the case of zero external potential, the dispersion law of interacting Bose–Einstein-condensed particles is given by so-called Bogoliubov spectrum (for ):

The Gross-Pitaevskii equation (GPE) provides a relatively good description of the behavior of atomic BEC's. However, GPE does not take into account the temperature dependence of dynamical variables, and is therefore valid only for . It is not applicable, for example, for the condensates of excitons, magnons and photons, where the critical temperature is comparable to room temperature.

Numerical solution

The Gross-Pitaevskii equation is a partial differential equation in space and time variables. Usually it does not have analytic solution and different numerical methods, such as split-step

Crank-Nicolson[18]
and and long-range dipolar interaction[22] which can be freely used.

Weaknesses of Gross–Pitaevskii model

The Gross–Pitaevskii model of BEC is a physical

superfluid clusters and droplets.[29]
It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term found in the Logarithmic Schrödinger equation must be added to the Gross-Pitaevskii equation along with a Ginzburg-Sobyanin contribution to correctly determine that the speed of sound scales as the cubic root of pressure for Helium-4 at very low temperatures in close agreement with experiment.[30]

Other

However, it is clear that in a general case the behaviour of Bose–Einstein condensate can be described by coupled evolution equations for condensate density, superfluid velocity and distribution function of elementary excitations. This problem was solved in 1977 by Peletminskii et al. in microscopical approach. The Peletminskii equations are valid for any finite temperatures below the critical point. Years after, in 1985, Kirkpatrick and Dorfman obtained similar equations using another microscopical approach. The Peletminskii equations also reproduce Khalatnikov hydrodynamical equations for superfluid as a limiting case.

Superfluidity of BEC and Landau criterion

The phenomena of superfluidity of a Bose gas and superconductivity of a strongly-correlated Fermi gas (a gas of Cooper pairs) are tightly connected to Bose–Einstein condensation. Under corresponding conditions, below the temperature of phase transition, these phenomena were observed in helium-4 and different classes of superconductors. In this sense, the superconductivity is often called the superfluidity of Fermi gas. In the simplest form, the origin of superfluidity can be seen from the weakly interacting bosons model.

Experimental observation

Superfluid helium-4

In 1938,

Cooper pairs of two atoms (see also fermionic condensate
).

Dilute atomic gases

The first "pure" Bose–Einstein condensate was created by

interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize in Physics for their achievements.[31]

A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work.[32] 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. This graph served as the cover design for the 1999 textbook Thermal Physics by Ralph Baierlein.[33]

Quasiparticles