Bose–Einstein condensate

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Schematic Bose–Einstein condensation versus temperature and the energy diagram

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which point microscopic quantum phenomena, particularly wavefunction interference, become apparent. A BEC is formed by cooling a gas of extremely low density, about one-hundred-thousandth the density of normal air, to ultra-low temperatures.

This state 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.

Satyendra Nath 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.^[1] (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 matter in two other papers.^[3][4] 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, which include the photon as well as atoms such as helium-4 (⁴He), 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 BEC as a mechanism for superfluidity in ⁴He and superconductivity.^[5][6]

On June 5, 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).^[7] Shortly thereafter, Wolfgang Ketterle at MIT demonstrated important BEC properties. For their achievements Cornell, Wieman, and Ketterle received the 2001 Nobel Prize in Physics.^[8]

Many isotopes were soon condensed, then molecules, quasi-particles, and photons in 2010.^[9]

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:

${\displaystyle T_{c}=\left({\frac {n}{\zeta (3/2)}}\right)^{2/3}{\frac {2\pi \hbar ^{2}}{mk_{B}}}\approx 3.3125\ {\frac {\hbar ^{2}n^{2/3}}{mk_{B}}}}$

where:

${\displaystyle \,T_{c}}$	is	the critical temperature,
${\displaystyle \,n}$	is	the particle density,
${\displaystyle \,m}$	is	the mass per boson,
${\displaystyle \hbar }$	is	the reduced Planck constant,
${\displaystyle \,k_{B}}$	is	the Boltzmann constant, and
${\displaystyle \,\zeta }$	is	the Riemann zeta function; ${\displaystyle \,\zeta (3/2)\approx 2.6124.}$

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.

Models

Bose Einstein's non-interacting gas

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

If we can tell which particle is which, there are ${\displaystyle 2^{N}}$ different configurations, since each particle can be in ${\displaystyle \scriptstyle |0\rangle }$ or ${\displaystyle \scriptstyle |1\rangle }$ independently. In almost all of the configurations, about half the particles are in ${\displaystyle \scriptstyle |0\rangle }$ and the other half in ${\displaystyle \scriptstyle |1\rangle }$. 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 ${\displaystyle \scriptstyle |1\rangle }$, there are N − K particles in state ${\displaystyle \scriptstyle |0\rangle }$. Whether any particular particle is in state ${\displaystyle \scriptstyle |0\rangle }$ or in state ${\displaystyle \scriptstyle |1\rangle }$ cannot be determined, so each value of K determines a unique quantum state for the whole system.

Suppose now that the energy of state ${\displaystyle \scriptstyle |1\rangle }$ is slightly greater than the energy of state ${\displaystyle \scriptstyle |0\rangle }$ by an amount E. At temperature T, a particle will have a lesser probability to be in state ${\displaystyle \scriptstyle |1\rangle }$ by ${\displaystyle e^{-E/kT}}$. In the distinguishable case, the particle distribution will be biased slightly towards state ${\displaystyle \scriptstyle |0\rangle }$. 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 ${\displaystyle \scriptstyle |0\rangle }$.

In the distinguishable case, for large N, the fraction in state ${\displaystyle \scriptstyle |0\rangle }$ can be computed. It is the same as flipping a coin with probability proportional to p = exp(−E/T) to land tails.

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:

${\displaystyle \,P(K)=Ce^{-KE/T}=Cp^{K}.}$

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

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

${\displaystyle \,p(k)=e^{-k^{2} \over 2mT}.}$

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

Bogoliubov theory for weakly interacting gas

Bogoliubov considered perturbations on the limit of dilute gas,^[11] 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): ${\displaystyle P=g/2n^{2}}$.

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

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

${\displaystyle E=\int d{\vec {r}}\left[{\frac {\hbar ^{2}}{2m}}|\nabla \psi ({\vec {r}})|^{2}+V({\vec {r}})|\psi ({\vec {r}})|^{2}+{\frac {1}{2}}U_{0}|\psi ({\vec {r}})|^{4}\right]}$

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

${\displaystyle i\hbar {\frac {\partial \psi ({\vec {r}})}{\partial t}}=\left(-{\frac {\hbar ^{2}\nabla ^{2}}{2m}}+V({\vec {r}})+U_{0}|\psi ({\vec {r}})|^{2}\right)\psi ({\vec {r}})}$

where:

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

${\displaystyle {\omega _{p}}={\sqrt {{\frac {p^{2}}{2m}}\left({{\frac {p^{2}}{2m}}+2{U_{0}}{n_{0}}}\right)}}}$

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

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 ^[12] and Fourier spectral ^[13] methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction ^[14][15] and long-range dipolar interaction ^[16] which can be freely used.

Weaknesses of Gross–Pitaevskii model

The Gross–Pitaevskii model of BEC is a physical approximation valid for certain classes of BECs. By construction, the GPE uses the following simplifications: it assumes that interactions between condensate particles are of the contact two-body type and also neglects anomalous contributions to self-energy.^[17] 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,^{[18][19][20][21]} effectively lower-dimensional condensates,^[22] and dense condensates and superfluid clusters and droplets.^[23]

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 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 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 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).

Gaseous

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.^[24]

A group led by Randall Hulet at Rice University announced a condensate of lithium atoms only one month following the JILA work.^[25] 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.^[26]

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,^[27] at temperatures as large as 14 K. The high transition temperature (relative to atomic gases) is due to the magnons small mass (near an electron) and greater achievable density. In 2006, condensation in a ferromagnetic yttrium-iron-garnet thin film was seen even at room temperature,^[28][29] 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 firstly detected for exciton-polaritons in a quantum well microcavity kept at 5 K.^[30]

Peculiar properties

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 ${\displaystyle |\psi ({\vec {r}})|^{2}}$ term in the GPE.^{[disputed – discuss]} As the vortices must have quantized angular momentum the wavefunction may have the form ${\displaystyle \psi ({\vec {r}})=\phi (\rho ,z)e^{i\ell \theta }}$ where ${\displaystyle \rho ,z}$ and ${\displaystyle \theta }$ are as in the cylindrical coordinate system, and ${\displaystyle \ell }$ 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 ${\displaystyle \phi (\rho ,z)}$, the energy of ${\displaystyle \psi ({\vec {r}})}$ must be minimized, according to the constraint ${\displaystyle \psi ({\vec {r}})=\phi (\rho ,z)e^{i\ell \theta }}$. This is usually done computationally, however in a uniform medium the analytic form:

${\displaystyle \phi ={\frac {nx}{\sqrt {2+x^{2}}}}}$, where:

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

demonstrates the correct behavior, and is a good approximation.

A singly charged vortex (${\displaystyle \ell =1}$) is in the ground state, with its energy ${\displaystyle \epsilon _{v}}$ given by

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

where ${\displaystyle \,b}$ is the farthest distance from the vortices considered.(To obtain an energy which is well defined it is necessary to include this boundary ${\displaystyle b}$.)

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

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

which is greater than that of ${\displaystyle \ell }$ 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.^[31]

Attractive interactions

Experiments led by Randall Hulet at Rice University from 1995 through 2000 showed that lithium condensates with attractive interactions could stably exist up to a critical atom number. Quench cooling the gas, they observed the condensate to grow, then subsequently collapse as the attraction overwhelmed the zero-point energy of the confining potential, in a burst reminiscent of a supernova, with an explosion preceded by an implosion.

Further work on attractive condensates was performed in 2000 by the JILA team, of Cornell, Wieman and coworkers. Their instrumentation now had better control so they used naturally attracting atoms of rubidium-85 (having negative atom–atom scattering length). Through Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, they lowered the characteristic, discrete energies at which rubidium bonds, making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among wave-like condensate atoms.

When the JILA team raised the magnetic field strength further, the condensate suddenly reverted to attraction, imploded and shrank beyond detection, then exploded, expelling about two-thirds of its 10,000 atoms. About half of the atoms in the condensate seemed to have disappeared from the experiment altogether, not seen in the cold remnant or expanding gas cloud.^[24] 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. Most likely they formed molecules of two rubidium atoms;^[32] energy gained by this bond imparts velocity sufficient to leave the trap without being detected.

The process of creation of molecular Bose condensate during the sweep of the magnetic field throughout the Feshbach resonance, as well as the reverse process, are described by the exactly solvable model that can explain many experimental observations.^[33]

Current research

Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile.^[34] The slightest interaction with the external environment 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 increase in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave–particle duality,^[35] 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.^[36] 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 laboratory. Experimenters have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential. These have been used to explore the transition between a superfluid and a Mott insulator,^[37] 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.^[38]

Cooling fermions to extremely low temperatures has created degenerate gases, subject to the Pauli exclusion principle. To exhibit Bose–Einstein condensation, the fermions must "pair up" to form bosonic compound particles (e.g. molecules or Cooper pairs). The first molecular 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.^[39]

In 1999, Danish physicist Lene Hau led a team from Harvard University which slowed a beam of light to about 17 meters per second,^{[clarification needed]} using a superfluid.^[40] Hau and her associates have since made a group of condensate atoms recoil from a light pulse such that they recorded the light's phase and amplitude, recovered by a second nearby condensate, in what they term "slow-light-mediated atomic matter-wave amplification" using Bose–Einstein condensates: details are discussed in Nature.^[41]

Another current research interest is the creation of Bose–Einstein condensates in microgravity in order to use its properties for high precision atom interferometry. The first demonstration of a BEC in weightlessness was achieved in 2008 at a drop tower in Bremen, Germany by a consortium of researchers led by Ernst M. Rasel from Leibniz University of Hanover.^[42] The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space^[43] and it is also the subject of two upcoming experiments on the International Space Station.^[44][45]

Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates when manipulating groups of identical cold atoms using lasers.^[46]

In 1970, BECs were proposed by Emmanuel David Tannenbaum for anti-stealth technology.^[47]

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 alkali metal, alkaline earth metal, and lanthanide atoms (⁷Li, ²³Na, ³⁹K, ⁴¹K, ⁸⁵Rb, ⁸⁷Rb, ¹³³Cs, ⁵²Cr, ⁴⁰Ca, ⁸⁴Sr, ⁸⁶Sr, ⁸⁸Sr, ¹⁷⁴Yb, ¹⁶⁴Dy, and ¹⁶⁸Er). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'.^[48] In contrast, the superfluid state of ⁴He below 2.17 K is not a good example, because the interaction between the atoms is too strong. Only 8% of atoms are in the ground state near absolute zero, rather than the 100% of a true condensate.^[49]

The bosonic behavior of some of these alkaline gases appears odd at first sight, because their nuclei have half-integer total spin. It 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 half-integer total spin of the nucleus are coupled by a very weak hyperfine interaction. The total spin of the atom, arising from this coupling, is an integer lower value. The chemistry of systems at room temperature is determined by the electronic properties, which is essentially fermionic, since room temperature thermal excitations have typical energies much higher than the hyperfine values.

D-brane

From Wikipedia, the free encyclopedia

In string theory, D-branes are a class of extended objects upon which open strings can end with Dirichlet boundary conditions, after which they are named. D-branes were discovered by Dai, Leigh and Polchinski, and independently by Hořava in 1989. In 1995, Polchinski identified D-branes with black p-brane solutions of supergravity, a discovery that triggered the Second Superstring Revolution and led to both holographic and M-theory dualities.

D-branes are typically classified by their spatial dimension, which is indicated by a number written after the D. A D0-brane is a single point, a D1-brane is a line (sometimes called a "D-string"), a D2-brane is a plane, and a D25-brane fills the highest-dimensional space considered in bosonic string theory. There are also instantonic D(–1)-branes, which are localized in both space and time.

Theoretical background

The equations of motion of string theory require that the endpoints of an open string (a string with endpoints) satisfy one of two types of boundary conditions: The Neumann boundary condition, corresponding to free endpoints moving through spacetime at the speed of light, or the Dirichlet boundary conditions, which pin the string endpoint. Each coordinate of the string must satisfy one or the other of these conditions. There can also exist strings with mixed boundary conditions, where the two endpoints satisfy NN, DD, ND and DN boundary conditions. If p spatial dimensions satisfy the Neumann boundary condition, then the string endpoint is confined to move within a p-dimensional hyperplane. This hyperplane provides one description of a Dp-brane.

Although rigid in the limit of zero coupling, the spectrum of open strings ending on a D-brane contains modes associated with its fluctuations, implying that D-branes are dynamical objects. When ${\displaystyle N}$ D-branes are nearly coincident, the spectrum of strings stretching between them becomes very rich. One set of modes produce a non-abelian gauge theory on the world-volume. Another set of modes is an ${\displaystyle N\times N}$ dimensional matrix for each transverse dimension of the brane. If these matrices commute, they may be diagonalized, and the eigenvalues define the position of the ${\displaystyle N}$ D-branes in space. More generally, the branes are described by non-commutative geometry, which allows exotic behavior such as the Myers effect, in which a collection of Dp-branes expand into a D(p+2)-brane.

Tachyon condensation is a central concept in this field. Ashoke Sen has argued that in Type IIB string theory, tachyon condensation allows (in the absence of Neveu-Schwarz 3-form flux) an arbitrary D-brane configuration to be obtained from a stack of D9 and anti D9-branes. Edward Witten has shown that such configurations will be classified by the K-theory of the spacetime. Tachyon condensation is still very poorly understood. This is due to the lack of an exact string field theory that would describe the off-shell evolution of the tachyon.

Braneworld cosmology

This has implications for physical cosmology. Because string theory implies that the Universe has more dimensions than we expect—26 for bosonic string theories and 10 for superstring theories—we have to find a reason why the extra dimensions are not apparent. One possibility would be that the visible Universe is in fact a very large D-brane extending over three spatial dimensions. Material objects, made of open strings, are bound to the D-brane, and cannot move "at right angles to reality" to explore the Universe outside the brane. This scenario is called a brane cosmology. The force of gravity is not due to open strings; the gravitons which carry gravitational forces are vibrational states of closed strings. Because closed strings do not have to be attached to D-branes, gravitational effects could depend upon the extra dimensions orthogonal to the brane.

D-brane scattering

When two D-branes approach each other the interaction is captured by the one loop annulus amplitude of strings between the two branes. The scenario of two parallel branes approaching each other at a constant velocity can be mapped to the problem of two stationary branes that are rotated relative to each other by some angle. The annulus amplitude yields singularities that correspond to the on-shell production of open strings stretched between the two branes. This is true irrespective of the charge of the D-branes. At non-relativistic scattering velocities the open strings may be described by a low-energy effective action that contains two complex scalar fields that are coupled via a term ${\displaystyle \phi ^{2}\chi ^{2}}$. Thus, as the field ${\displaystyle \phi }$ (separation of the branes) changes, the mass of the field ${\displaystyle \chi }$ changes. This induces open string production and as a result the two scattering branes will be trapped.

Gauge theories

The arrangement of D-branes constricts the types of string states which can exist in a system. For example, if we have two parallel D2-branes, we can easily imagine strings stretching from brane 1 to brane 2 or vice versa. (In most theories, strings are oriented objects: each one carries an "arrow" defining a direction along its length.) The open strings permissible in this situation then fall into two categories, or "sectors": those originating on brane 1 and terminating on brane 2, and those originating on brane 2 and terminating on brane 1. Symbolically, we say we have the [1 2] and the [2 1] sectors. In addition, a string may begin and end on the same brane, giving [1 1] and [2 2] sectors. (The numbers inside the brackets are called Chan-Paton indices, but they are really just labels identifying the branes.) A string in either the [1 2] or the [2 1] sector has a minimum length: it cannot be shorter than the separation between the branes. All strings have some tension, against which one must pull to lengthen the object; this pull does work on the string, adding to its energy. Because string theories are by nature relativistic, adding energy to a string is equivalent to adding mass, by Einstein's relation E = mc². Therefore, the separation between D-branes controls the minimum mass open strings may have.

Furthermore, affixing a string's endpoint to a brane influences the way the string can move and vibrate. Because particle states "emerge" from the string theory as the different vibrational states the string can experience, the arrangement of D-branes controls the types of particles present in the theory. The simplest case is the [1 1] sector for a Dp-brane, that is to say the strings which begin and end on any particular D-brane of p dimensions. Examining the consequences of the Nambu-Goto action (and applying the rules of quantum mechanics to quantize the string), one finds that among the spectrum of particles is one resembling the photon, the fundamental quantum of the electromagnetic field. The resemblance is precise: a p-dimensional version of the electromagnetic field, obeying a p-dimensional analogue of Maxwell's equations, exists on every Dp-brane.

In this sense, then, one can say that string theory "predicts" electromagnetism: D-branes are a necessary part of the theory if we permit open strings to exist, and all D-branes carry an electromagnetic field on their volume.

Other particle states originate from strings beginning and ending on the same D-brane. Some correspond to massless particles like the photon; also in this group are a set of massless scalar particles. If a Dp-brane is embedded in a spacetime of d spatial dimensions, the brane carries (in addition to its Maxwell field) a set of d - p massless scalars (particles which do not have polarizations like the photons making up light). Intriguingly, there are just as many massless scalars as there are directions perpendicular to the brane; the geometry of the brane arrangement is closely related to the quantum field theory of the particles existing on it. In fact, these massless scalars are Goldstone excitations of the brane, corresponding to the different ways the symmetry of empty space can be broken. Placing a D-brane in a universe breaks the symmetry among locations, because it defines a particular place, assigning a special meaning to a particular location along each of the d - p directions perpendicular to the brane.

The quantum version of Maxwell's electromagnetism is only one kind of gauge theory, a U(1) gauge theory where the gauge group is made of unitary matrices of order 1. D-branes can be used to generate gauge theories of higher order, in the following way:

Consider a group of N separate Dp-branes, arranged in parallel for simplicity. The branes are labeled 1,2,...,N for convenience. Open strings in this system exist in one of many sectors: the strings beginning and ending on some brane i give that brane a Maxwell field and some massless scalar fields on its volume. The strings stretching from brane i to another brane j have more intriguing properties. For starters, it is worthwhile to ask which sectors of strings can interact with one another. One straightforward mechanism for a string interaction is for two strings to join endpoints (or, conversely, for one string to "split down the middle" and make two "daughter" strings). Since endpoints are restricted to lie on D-branes, it is evident that a [1 2] string may interact with a [2 3] string, but not with a [3 4] or a [4 17] one. The masses of these strings will be influenced by the separation between the branes, as discussed above, so for simplicity's sake we can imagine the branes squeezed closer and closer together, until they lie atop one another. If we regard two overlapping branes as distinct objects, then we still have all the sectors we had before, but without the effects due to the brane separations.

The zero-mass states in the open-string particle spectrum for a system of N coincident D-branes yields a set of interacting quantum fields which is exactly a U(N) gauge theory. (The string theory does contain other interactions, but they are only detectable at very high energies.) Gauge theories were not invented starting with bosonic or fermionic strings; they originated from a different area of physics, and have become quite useful in their own right. If nothing else, the relation between D-brane geometry and gauge theory offers a useful pedagogical tool for explaining gauge interactions, even if string theory fails to be the "theory of everything".

Black holes

Another important use of D-branes has been in the study of black holes. Since the 1970s, scientists have debated the problem of black holes having entropy. Consider, as a thought experiment, dropping an amount of hot gas into a black hole. Since the gas cannot escape from the hole's gravitational pull, its entropy would seem to have vanished from the universe. In order to maintain the second law of thermodynamics, one must postulate that the black hole gained whatever entropy the infalling gas originally had. Attempting to apply quantum mechanics to the study of black holes, Stephen Hawking discovered that a hole should emit energy with the characteristic spectrum of thermal radiation. The characteristic temperature of this Hawking radiation is given by

${\displaystyle T_{\rm {H}}={\frac {\hbar c^{3}}{8\pi GMk_{B}}}\;\quad (\approx {1.227\times 10^{23}\;kg \over M}\;K)}$,

where G is Newton's gravitational constant, M is the black hole's mass and k_B is Boltzmann's constant.

Using this expression for the Hawking temperature, and assuming that a zero-mass black hole has zero entropy, one can use thermodynamic arguments to derive the "Bekenstein entropy":

${\displaystyle S_{\rm {B}}={\frac {k_{B}4\pi G}{\hbar c}}M^{2}.}$

The Bekenstein entropy is proportional to the black hole mass squared; because the Schwarzschild radius is proportional to the mass, the Bekenstein entropy is proportional to the black hole's surface area. In fact,

${\displaystyle S_{\rm {B}}={\frac {Ak_{B}}{4l_{\rm {P}}^{2}}},}$

where ${\displaystyle l_{\rm {P}}}$ is the Planck length.

The concept of black hole entropy poses some interesting conundra. In an ordinary situation, a system has entropy when a large number of different "microstates" can satisfy the same macroscopic condition. For example, given a box full of gas, many different arrangements of the gas atoms can have the same total energy. However, a black hole was believed to be a featureless object (in John Wheeler's catchphrase, "Black holes have no hair"). What, then, are the "degrees of freedom" which can give rise to black hole entropy?

String theorists have constructed models in which a black hole is a very long (and hence very massive) string. This model gives rough agreement with the expected entropy of a Schwarzschild black hole, but an exact proof has yet to be found one way or the other. The chief difficulty is that it is relatively easy to count the degrees of freedom quantum strings possess if they do not interact with one another. This is analogous to the ideal gas studied in introductory thermodynamics: the easiest situation to model is when the gas atoms do not have interactions among themselves. Developing the kinetic theory of gases in the case where the gas atoms or molecules experience inter-particle forces (like the van der Waals force) is more difficult. However, a world without interactions is an uninteresting place: most significantly for the black hole problem, gravity is an interaction, and so if the "string coupling" is turned off, no black hole could ever arise. Therefore, calculating black hole entropy requires working in a regime where string interactions exist.

Extending the simpler case of non-interacting strings to the regime where a black hole could exist requires supersymmetry. In certain cases, the entropy calculation done for zero string coupling remains valid when the strings interact. The challenge for a string theorist is to devise a situation in which a black hole can exist which does not "break" supersymmetry. In recent years, this has been done by building black holes out of D-branes. Calculating the entropies of these hypothetical holes gives results which agree with the expected Bekenstein entropy. Unfortunately, the cases studied so far all involve higher-dimensional spaces — D5-branes in nine-dimensional space, for example. They do not directly apply to the familiar case, the Schwarzschild black holes observed in our own universe.

History

Dirichlet boundary conditions and D-branes had a long "pre-history" before their full significance was recognized. Mixed Dirichlet/Neumann boundary conditions were first considered by Warren Siegel in 1976 as a means of lowering the critical dimension of open string theory from 26 or 10 to 4 (Siegel also cites unpublished work by Halpern, and a 1974 paper by Chodos and Thorn, but a reading of the latter paper shows that it is actually concerned with linear dilation backgrounds, not Dirichlet boundary conditions). This paper, though prescient, was little-noted in its time (a 1985 parody by Siegel, "The Super-g String," contains an almost dead-on description of braneworlds). Dirichlet conditions for all coordinates including Euclidean time (defining what are now known as D-instantons) were introduced by Michael Green in 1977 as a means of introducing point-like structure into string theory, in an attempt to construct a string theory of the strong interaction. String compactifications studied by Harvey and Minahan, Ishibashi and Onogi, and Pradisi and Sagnotti in 1987–89 also employed Dirichlet boundary conditions.

The fact that T-duality interchanges the usual Neumann boundary conditions with Dirichlet boundary conditions was discovered independently by Horava and by Dai, Leigh, and Polchinski in 1989; this result implies that such boundary conditions must necessarily appear in regions of the moduli space of any open string theory. The Dai et al. paper also notes that the locus of the Dirichlet boundary conditions is dynamical, and coins the term Dirichlet-brane (D-brane) for the resulting object (this paper also coins orientifold for another object that arises under string T-duality). A 1989 paper by Leigh showed that D-brane dynamics are governed by the Dirac–Born–Infeld action. D-instantons were extensively studied by Green in the early 1990s, and were shown by Polchinski in 1994 to produce the e^​–1⁄_g nonperturbative string effects anticipated by Shenker. In 1995 Polchinski showed that D-branes are the sources of electric and magnetic Ramond–Ramond fields that are required by string duality,^[1] leading to rapid progress in the nonperturbative understanding of string theory.