Bose–Einstein condensate

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero (that is, very near 0 K or −273.15 °C[1]). Under such conditions, a large fraction of the bosons occupy the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale. These effects are called macroscopic quantum phenomena.

Although later experiments have revealed complex interactions, this state of matter was first predicted, generally, in 1924–25 by Satyendra Nath Bose and Albert Einstein.

History

Velocity-distribution data (3 views) for a 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 quantum statistics of light quanta (now called photons). Einstein was impressed, translated the paper himself from English to German and submitted it for Bose to the Zeitschrift für Physik, which published it. (The Einstein manuscript, once believed to be lost, was found in a library at Leiden University in 2005.[2]). Einstein then extended Bose's ideas to material particles (or matter) in two other papers.[3] The result of the efforts of Bose and Einstein is the concept of a Bose gas, governed by Bose–Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4 (4He), are allowed to share quantum states with each other. 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 BEC as a mechanism for superfluidity in 4He and superconductivity.[4][5]

In 1995 the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NISTJILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin (nK) [6] (1.7×10−7 K). For their achievements Cornell, Wieman, and Wolfgang Ketterle at MIT received the 2001 Nobel Prize in Physics.[7] In November 2010 the first photon BEC was observed.[8] In 2012, the theory of the photon BEC was developed.[9][10]

Concept

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:

$T_c=\left(\frac{n}{\zeta(3/2)}\right)^{2/3}\frac{2\pi \hbar^2}{ m k_B} \approx 3.3125 \ \frac{\hbar^2 n^{2/3}}{m k_B}$

where:

 $\,T_c$ is the critical temperature, $\,n$ is the particle density, $\,m$ is the mass per boson, $\hbar$ is the reduced Planck constant, $\,k_B$ is the Boltzmann constant, and $\,\zeta$ is the Riemann zeta function; $\,\zeta(3/2)\approx 2.6124.$ [11]

Einstein's non-interacting model

Consider a collection of N noninteracting particles, which can each be in one of two quantum states, $\scriptstyle|0\rangle$ and $\scriptstyle|1\rangle$. If the two states are equal in energy, each different configuration is equally likely.

If we can tell which particle is which, there are $2^N$ different configurations, since each particle can be in $\scriptstyle|0\rangle$ or $\scriptstyle|1\rangle$ independently. In almost all of the configurations, about half the particles are in $\scriptstyle|0\rangle$ and the other half in $\scriptstyle|1\rangle$. 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 different configurations. If there are K particles in state $\scriptstyle|1\rangle$, there are N − K particles in state $\scriptstyle|0\rangle$. Whether any particular particle is in state $\scriptstyle|0\rangle$ or in state $\scriptstyle|1\rangle$ cannot be determined, so each value of K determines a unique quantum state for the whole system. If all these states are equally likely, there is no statistical spreading out; it is just as likely for all the particles to sit in $\scriptstyle|0\rangle$ as for the particles to be split half and half.

Suppose now that the energy of state $\scriptstyle|1\rangle$ is slightly greater than the energy of state $\scriptstyle|0\rangle$ by an amount E. At temperature T, a particle will have a lesser probability to be in state $\scriptstyle|1\rangle$ by exp(−E/kT). In the distinguishable case, the particle distribution will be biased slightly towards state $\scriptstyle|0\rangle$, and the distribution will be slightly different from half-and-half. 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 $\scriptstyle|0\rangle$.

In the distinguishable case, for large N, the fraction in state $\scriptstyle|0\rangle$ can be computed. It is the same as flipping a coin with probability proportional to p = exp(−E/T) to land tails. The probability to land heads is 1/(1 + p), which is a smooth function of p, and thus of the energy.

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:

$\, P(K)= C e^{-KE/T} = C p^K.$

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 $\scriptstyle N\rightarrow \infty$, is equal to $\scriptstyle \sum_{n>0} C n p^n=p/(1-p)$. 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 $\scriptstyle|k\rangle$. 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):

$\, N = V \int {d^3k \over (2\pi)^3} {p(k)\over 1-p(k)} = V \int {d^3k \over (2\pi)^3} {1 \over e^{k^2\over 2mT}-1}$
$\, p(k)= e^{-k^2\over 2mT}.$

When the integral is evaluated with the factors of kB and 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, μ is actually still nonzero for BEC's; however, μ is less than the ground state energy. Except when specifically talking about the ground state, μ can consequently be approximated for most energy or momentum states as μ ≈ 0.

Interacting models

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 $\psi(\vec{r})$. For a system of this nature, $|\psi(\vec{r})|^2$ is interpreted as the particle density, so the total number of atoms is $N=\int d\vec{r}|\psi(\vec{r})|^2$

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 $\psi(\vec{r})$ is:

$E=\int d\vec{r}\left[\frac{\hbar^2}{2m}|\nabla\psi(\vec{r})|^2+V(\vec{r})|\psi(\vec{r})|^2+\frac{1}{2}U_0|\psi(\vec{r})|^4\right]$

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

$i\hbar\frac{\partial \psi(\vec{r})}{\partial t} = \left(-\frac{\hbar^2\nabla^2}{2m}+V(\vec{r})+U_0|\psi(\vec{r})|^2\right)\psi(\vec{r})$

where:

 $\,m$ is the mass of the bosons, $\,V(\vec{r})$ is the external potential, $\,U_0$ is representative of 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 $\ T= 0$):

${\omega _p} = \sqrt {\frac{{{p^2}}}{{2m}}\left( {\frac{{{p^2}}}{{2m}} + 2{U_0}{n_0}} \right)}$

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 $\ T= 0$. It is not applicable for example for the condensates of excitons, magnons and photons, where the critical temperature is up to room one.

Weaknesses of Gross–Pitaevskii model

The Gross–Pitaevskii model of BEC is a physical approximation valid for certain classes of BEC's only. By construction, GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also it neglects anomalous contributions to self-energy.[12] These assumptions are suitable mostly for the dilute three-dimensional condensates. If one relaxes any of these assumptions, the equation for the condensate wavefunction acquires the terms containing higher-order powers of the wavefunction. Moreover, for some physical systems the amount of such terms turns out to be infinite, therefore, the equation becomes essentially non-polynomial. The examples where this could happen are the Bose–Fermi composite condensates,[13][14][15][16] effectively lower-dimensional condensates,[17] and dense condensates and superfluid clusters and droplets.[18]

Other models

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 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 simply as the superfluidity of Fermi gas).

In the simplest form, the origin of supefluidity can be seen from the weakly interacting bosons model.

Experimental observation

Superfluid He-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 of the properties of superfluid helium also appear in the gaseous Bose–Einstein 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, consisting of fermions instead of bosons, also enters a superfluid phase at low temperature, which can be explained by the formation of bosonic Cooper pairs of two atoms each (see also fermionic condensate).

Atomic condensates

The first "pure" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on 5 June 1995. They did this by cooling a dilute vapor consisting 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 created a condensate made of sodium-23. Ketterle's condensate had about a hundred times more atoms, allowing him to obtain several 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.[19] A group led by Randall Hulet at Rice University announced the creation of a condensate of lithium atoms only one month following the JILA work.[20] Lithium has attractive interactions which causes the condensate to be unstable and to collapse for all but a few atoms. Hulet and co-workers showed in a subsequent experiment that the condensate could be stabilized by the quantum pressure from trap confinement for up to about 1000 atoms.

The Bose–Einstein condensation also applies to quasiparticles in solids. A magnon in an antiferromagnet carries spin 1 and thus obeys Bose–Einstein statistics. The density of magnons is controlled by an external magnetic field, which plays the role of the magnon chemical potential. This technique provides access to a wide range of boson densities from the limit of a dilute Bose gas to that of a strongly interacting Bose liquid. A magnetic ordering observed at the point of condensation is the analog of superfluidity. In 1999 Bose condensation of magnons was demonstrated in the antiferromagnet Tl Cu Cl3.[21] The condensation was observed at temperatures as large as 14 K. Such a high transition temperature (relative to that of atomic gases) is due to the greater density achievable with magnons and the smaller mass (roughly equal to the mass of an electron). In 2006, condensation of magnons in ferromagnets was even shown at room temperature,[22][23] where the authors used pumping techniques.

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: since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width. This width is given by the curvature of the magnetic trapping 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.[24]

Peculiarities of a condensate

Vortices

As in many other systems, vortices can exist in BECs. These can be created, for example, by 'stirring' the condensate with lasers, or rotating the confining trap. The vortex created will be a quantum vortex. These phenomena are allowed for by the non-linear $|\psi(\vec{r})|^2$ term in the GPE. As the vortices must have quantized angular momentum the wavefunction may have the form $\psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}$ where $\rho, z$ and $\theta$ are as in the cylindrical coordinate system, and $\ell$ is the angular number. This is particularly likely for an axially symmetric (for instance, harmonic) confining potential, which is commonly used. The notion is easily generalized. To determine $\phi(\rho,z)$, the energy of $\psi(\vec{r})$ must be minimized, according to the constraint $\psi(\vec{r})=\phi(\rho,z)e^{i\ell\theta}$. This is usually done computationally, however in a uniform medium the analytic form

$\phi=\frac{nx}{\sqrt{2+x^2}}$

where:

 $\,n^2$ is density far from the vortex, $\,x = \frac{\rho}{\ell\xi},$ $\,\xi$ is healing length of the condensate.

demonstrates the correct behavior, and is a good approximation.

A singly charged vortex ($\ell=1$) is in the ground state, with its energy $\epsilon_v$ given by

$\epsilon_v=\pi n \frac{\hbar^2}{m}\ln\left(1.464\frac{b}{\xi}\right)$

where:

 $\,b$ is the farthest distance from the vortex considered.

(To obtain an energy which is well defined it is necessary to include this boundary $b$.)

For multiply charged vortices ($\ell >1$) the energy is approximated by

$\epsilon_v\approx \ell^2\pi n \frac{\hbar^2}{m}\ln\left(\frac{b}{\xi}\right)$

which is greater than that of $\ell$ singly charged vortices, indicating that these multiply charged vortices are unstable to decay. Research has, however, indicated they are metastable states, so may have relatively long lifetimes.

Closely related to the creation of vortices in BECs is the generation of so-called dark solitons in one-dimensional BECs. These topological objects feature a phase gradient across their nodal plane, which stabilizes their shape even in propagation and interaction. Although solitons carry no charge and are thus prone to decay, relatively long-lived dark solitons have been produced and studied extensively.[25]

Attractive interactions

The experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist, but only up to a certain critical atom number. Beyond this critical number, the attraction overwhelmed the zero-point energy of the harmonic confining potential, causing the condensate to collapse in a burst reminiscent of a supernova explosion where an explosion is preceded by an implosion. By quench cooling the gas of lithium atoms, they observed the condensate to first grow, and subsequently collapse when the critical number was exceeded.

Further experimentation on attractive condensates was performed in 2000 by the JILA team, consisting of Cornell, Wieman and coworkers. They originally used rubidium-87, an isotope whose atoms naturally repel each other, making a more stable condensate. Their instrumentation now had better control over the condensate so experimentation was made on naturally attracting atoms of another rubidium isotope, rubidium-85 (having negative atom–atom scattering length). Through a process called Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which the rubidium atoms bond into molecules, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among condensate atoms which behave as waves.

When the JILA team raised the magnetic field strength still further, the condensate suddenly reverted to attraction, imploded and shrank beyond detection, and then exploded, expelling off about two-thirds of its 10,000 or so atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not being seen either in the cold remnant or the expanding gas cloud.[19] Carl Wieman explained that under current atomic theory this characteristic of Bose–Einstein condensate could not be explained because the energy state of an atom near absolute zero should not be enough to cause an implosion; however, subsequent mean field theories have been proposed to explain it. The atoms that seem to have disappeared almost certainly still exist in some form, just not in a form that could be accounted for in that experiment. Most likely they formed molecules consisting of two bonded rubidium atoms.[26] The energy gained by making this transition imparts a velocity sufficient for them to leave the trap without being detected.

Current research

 How do we rigorously prove the existence of Bose-Einstein condensates for general interacting systems?

Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile. The slightest interaction with the outside world can be enough to warm them past the condensation threshold, eliminating their interesting properties and forming a normal gas.[citation needed]

Nevertheless, they have proven useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an explosion in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality,[27] the study of superfluidity and quantized vortices, the creation of bright matter wave solitons from Bose condensates confined to one dimension, and the slowing of light pulses to very low speeds using electromagnetically induced transparency.[28] Vortices in Bose–Einstein condensates are also currently the subject of analogue gravity research, studying the possibility of modeling black holes and their related phenomena in such environments in the lab. Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential for the condensate. These have been used to explore the transition between a superfluid and a Mott insulator,[29] and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas.

Bose–Einstein condensates composed of a wide range of isotopes have been produced.[30]

Related experiments in cooling fermions rather than bosons to extremely low temperatures have created degenerate gases, where the atoms do not congregate in a single state due to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form compound particles (e.g. molecules or Cooper pairs) that are bosons. The first molecular Bose–Einstein condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate composed of Cooper pairs.[31]

In 1999, Danish physicist Lene Hau led a team from Harvard University which succeeded in slowing a beam of light to about 17 meters per second.[clarification needed] She was able to achieve this by using a superfluid.[32] Hau and her associates at Harvard University have since successfully made a group of condensate atoms recoil from a "light pulse" such that they recorded the light's phase and amplitude, which was recovered by a second nearby condensate, by what they term "slow-light-mediated atomic matter-wave amplification" using Bose–Einstein condensates: details of the experiment are discussed in an article in the journal Nature, 8 February 2007.[33]

Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.[34] Further, Bose–Einstein condensates have been proposed by Emmanuel David Tannenbaum to be used in anti-stealth technology.[35]

Isotopes

The effect has mainly been observed on alkaline atoms which have nuclear properties particularly suitable for working with traps. As of 2012, using ultra-low temperatures of 10−7 K or below, Bose–Einstein condensates had been obtained for a multitude of isotopes, mainly of alkaline, alkaline earth, and lanthanoid atoms (7Li, 23Na, 39K, 41K, 85Rb, 87Rb, 133Cs, 52Cr, 40Ca, 84Sr, 86Sr, 88Sr, 174Yb, 164Dy, and 168Er ). Condensation research was finally successful even with hydrogen with the aid of special methods. In contrast, the superfluid state of the bosonic 4He at temperatures below 2.17 K is not a good example of Bose–Einstein condensation, because the interaction between the 4He bosons is too strong. Only 8% of the atoms are in the single-particle ground state near zero temperature, rather than the 100% expected of a true Bose–Einstein condensate.

The spin-statistics theorem of Wolfgang Pauli states that half-integer spins (in units of $\scriptstyle \hbar$) lead to fermionic behavior, e.g., the Pauli exclusion principle forbidding that more than two electrons possess the same energy, whereas integer spins lead to bosonic behavior, e.g., condensation of identical bosonic particles in a common ground state.

The bosonic, rather than fermionic, behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. The bosonic behavior arises from a subtle interplay of electronic and nuclear spins: at ultra-low temperatures and corresponding excitation energies, the half-integer total spin of the electronic shell and the half-integer total spin of the nucleus of the atom are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer value leading to the bosonic ultra-low temperature behavior of the atom. The chemistry of the systems at room temperature is determined by the electronic properties, which is essentially fermionic, since at room temperature, thermal excitations have typical energies much higher than the hyperfine values.

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