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Bose–Einstein condensate

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 or 0 K). 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 condensate was 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. (The Einstein 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, particles that include the photon and atoms with an even number of neutrons (such as helium-4 (4Helium)), 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 4Helium 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). 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).

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. 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.

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_\text{c} = \left(\frac{n}{\zeta(3/2)}\right)^{2/3} \frac{2\pi\hbar^2}{m k_\text{B}} \approx 3.3125\,\frac{\hbar^2 n^{2/3}}{m k_\text{B}}, $$ where:
 * $$T_\text{c}$$ is the critical temperature,
 * $$n$$ is the particle density,
 * $$m$$ is the mass per boson,
 * $$\hbar$$ is the reduced Planck constant,
 * $$k_\text{B}$$ is the Boltzmann constant,
 * $$\zeta$$ is the Riemann zeta function ($$\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.

Ideal Bose gas
For an ideal Bose gas we have the equation of state

 * $$\frac{1}{v} = \frac{1}{\lambda^3} g_{3/2}(f) + \frac{1}{V}\frac{f}{1 - f},$$

where $$v = V/N$$ is the per-particle volume, $$\lambda$$ is the thermal wavelength, $$f$$ is the fugacity, and
 * $$g_\alpha(f) = \sum\limits_{n=1}^\infty \frac{f^n}{n^\alpha}.$$

It is noticeable that $$g_{3/2}$$ is a monotonically growing function of $$f$$ in $$f \in [0, 1]$$, 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 $$\langle n_0 \rangle$$, the equation of state can be rewritten as

 * $$\frac{1}{v} = \frac{1}{\lambda^3} g_{3/2}(f) + \frac{\langle n_0\rangle}{V} \Leftrightarrow \frac{\langle n_0\rangle}{V} \lambda^3 = \frac{\lambda^3}{v} - g_{3/2}(f).$$

Because the left term on the second equation must always be positive, $$\frac{\lambda^3}{v} > g_{3/2}(f)$$, and because $$g_{3/2}(f) \le g_{3/2}(1)$$, a stronger condition is

 * $$\frac{\lambda^3}{v} > g_{3/2}(1),$$

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:

 * $$\lambda_c^3 = g_{3/2}(1)v = \zeta(3/2) v,$$

 * $$T_\text{c} = \frac{2\pi \hbar^2}{m k_\text{B} \lambda_c^2},$$

recovering the value indicated on the previous section. The critical values are such that if $$T < T_\text{c}$$ or $$\lambda > \lambda_\text{c}$$, 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 $$f = 1$$, obtaining

 * $$\frac{\langle n_0\rangle}{N} = 1 - \left(\frac{\lambda_\text{c}}{\lambda}\right)^3$$ and equivalently $$\frac{\langle n_0\rangle}{N} = 1 - \left(\frac{T}{T_\text{c}}\right)^{3/2}.$$

So, if $$T \ll T_\text{c}$$, the fraction $$\frac{\langle n_0 \rangle}{N} \approx 1$$, and if $$T \gg T_\text{c}$$, the fraction $$\frac{\langle n_0 \rangle}{N} \approx 0$$. At temperatures near to absolute 0, particles tend to condense in the fundamental state, which is the state with momentum $$\vec{p} = 0$$.

Bose Einstein's non-interacting gas
Consider a collection of N non-interacting particles, which can each be in one of two quantum states, $$|0\rangle$$ and $$|1\rangle$$. 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 $$|0\rangle$$ or $$|1\rangle$$ independently. In almost all of the configurations, about half the particles are in $$|0\rangle$$ and the other half in $$|1\rangle$$. 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 $$|1\rangle$$, there are N − K particles in state $$|0\rangle$$. Whether any particular particle is in state $$|0\rangle$$ or in state $$|1\rangle$$ cannot be determined, so each value of K determines a unique quantum state for the whole system.

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

In the distinguishable case, for large N, the fraction in state $$|0\rangle$$ 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:

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 $$N\rightarrow \infty$$, is equal to
 * $$\sum_{n>0} C n p^n=p/(1-p) $$

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 $$|k\rangle$$. 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 (also known as Bose–Einstein 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 $$\mu$$. In Bose–Einstein statistics distribution, $$\mu$$ is actually still nonzero for BECs; however, $$\mu$$ is less than the ground state energy. Except when specifically talking about the ground state, $$\mu$$ can be approximated for most energy or momentum states as $$\mu \approx 0$$.

Bogoliubov theory for weakly interacting gas
Nikolay Bogoliubov 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): $$P = gn^2/2$$.

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

 * {|cellspacing="0" cellpadding="0"

 * $$\,m$$
 * is the mass of the bosons,
 * $$\,V(\vec{r})$$
 * is the external potential, and
 * $$\,U_0$$
 * represents the inter-particle interactions.
 * }
 * $$\,U_0$$
 * 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 $$\ T= 0$$):

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

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 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 and Fourier spectral methods, are used for its solution. There are different Fortran and C programs for its solution for contact interaction and long-range dipolar interaction 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. 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,   effectively lower-dimensional condensates, and dense condensates and superfluid clusters and droplets. It is found that one has to go beyond the Gross-Pitaevskii equation. For example, the logarithmic term $$\psi \ln |\psi|^2 $$ 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.

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.

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. This graph served as the cover design for the 1999 textbook Thermal Physics by Ralph Baierlein.

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, 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 in 2005 on.

Polariton condensation was first detected for exciton-polaritons in a quantum well microcavity kept at 5 K.

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.

Quantized vortices
As in many other systems, vortices can exist in BECs. Vortices can be created, for example, by "stirring" the condensate with lasers, rotating the confining trap, or by rapid cooling across the phase transition. The vortex created will be a quantum vortex with core shape determined by the interactions. Fluid circulation around any point is quantized due to the single-valued nature of the order BEC order parameter or wavefunction, that can be written in 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 quantum number (a.k.a. the "charge" of the vortex). Since the energy of a vortex is proportional to the square of its angular momentum, in trivial topology only $$\ell=1$$ vortices can exist in the steady state; Higher-charge vortices will have a tendency to split into $$\ell=1$$ vortices, if allowed by the topology of the geometry.

An axially symmetric (for instance, harmonic) confining potential is commonly used for the study of vortices in BEC. 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 following analytic form demonstrates the correct behavior, and is a good approximation:

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

Here, $$n$$ is the density far from the vortex and $$x=\rho/(\ell \xi)$$, where $$\xi=1/\sqrt{8\pi a_s n_0}$$ is the healing length of the condensate.

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 vortices 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.

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. 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; 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.

Current research
Compared to more commonly encountered states of matter, Bose–Einstein condensates are extremely fragile. 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.

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, 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. 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, and may be useful in studying Bose–Einstein condensation in fewer than three dimensions, for example the Tonks–Girardeau gas. Further, the sensitivity of the pinning transition of strongly interacting bosons confined in a shallow one-dimensional optical lattice originally observed by Haller has been explored via a tweaking of the primary optical lattice by a secondary weaker one. Thus for a resulting weak bichromatic optical lattice, it has been found that the pinning transition is robust against the introduction of the weaker secondary optical lattice. Studies of vortices in nonuniform Bose–Einstein condensates as well as excitations of these systems by the application of moving repulsive or attractive obstacles, have also been undertaken. Within this context, the conditions for order and chaos in the dynamics of a trapped Bose–Einstein condensate have been explored by the application of moving blue and red-detuned laser beams (hitting frequencies slightly above and below the resonance frequency, respectively) via the time-dependent Gross-Pitaevskii equation.

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

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, working with the same system but outside the molecular regime.

In 1999, Danish physicist Lene Hau led a team from Harvard University which slowed a beam of light to about 17 meters per second using a superfluid. 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.

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 Hannover. The same team demonstrated in 2017 the first creation of a Bose–Einstein condensate in space and it is also the subject of two upcoming experiments on the International Space Station.

Researchers in the new field of atomtronics use the properties of Bose–Einstein condensates in the emerging quantum technology of matter-wave circuits.

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

In 2020, researchers reported the development of superconducting BEC and that there appears to be a "smooth transition between" BEC and Bardeen–Cooper–Shrieffer regimes.

Continuous Bose–Einstein condensation
Limitations of evaporative cooling have restricted atomic BECs to "pulsed" operation, involving a highly inefficient duty cycle that discards more than 99% of atoms to reach BEC. Achieving continuous BEC has been a major open problem of experimental BEC research, driven by the same motivations as continuous optical laser development: high flux, high coherence matter waves produced continuously would enable new sensing applications.

Continuous BEC was achieved for the first time in 2022.

Dark matter
P. Sikivie and Q. Yang showed that cold dark matter axions would form a Bose–Einstein condensate by thermalisation because of gravitational self-interactions. Axions have not yet been confirmed to exist. However the important search for them has been greatly enhanced with the completion of upgrades to the Axion Dark Matter Experiment (ADMX) at the University of Washington in early 2018.

In 2014, a potential dibaryon was detected at the Jülich Research Center at about 2380 MeV. The center claimed that the measurements confirm results from 2011, via a more replicable method. The particle existed for 10−23 seconds and was named d*(2380). This particle is hypothesized to consist of three up and three down quarks. It is theorized that groups of d* (d-stars) could form Bose–Einstein condensates due to prevailing low temperatures in the early universe, and that BECs made of such hexaquarks with trapped electrons could behave like dark matter.

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 (7lithium, 23sodium, 39potassium, 41potassium, 85rubidium, 87Rubidium, 133caesium, 52chromium, 40calcium, 84strontium, 86strontium, 88strontium, 174ytterbium, 164dysprosium, and 168erbium). Research was finally successful in hydrogen with the aid of the newly developed method of 'evaporative cooling'. In contrast, the superfluid state of 4Helium 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 of the trap near absolute zero, rather than the 100% of a true condensate.

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.

In fiction

 * In the 2016 film Spectral, the US military battles mysterious enemy creatures fashioned out of Bose–Einstein condensates.
 * In the 2003 novel Blind Lake, scientists observe sentient life on a planet 51 light-years away using telescopes powered by Bose–Einstein condensate-based quantum computers.
 * The video game franchise Mass Effect has cryonic ammunition whose flavour text describes it as being filled with Bose–Einstein condensates. Upon impact, the bullets rupture and spray super-cold liquid on the enemy.

title: Bose–Einstein condensate


text:
B (programming language)

B is a programming language developed at Bell Labs circa 1969 by Ken Thompson and Dennis Ritchie.

B was derived from BCPL, and its name may possibly be a contraction of BCPL. Thompson's coworker Dennis Ritchie speculated that the name might be based on Bon, an earlier, but unrelated, programming language that Thompson designed for use on Multics.

B was designed for recursive, non-numeric, machine-independent applications, such as system and language software. It was a typeless language, with the only data type being the underlying machine's natural memory word format, whatever that might be. Depending on the context, the word was treated either as an integer or a memory address.

As machines with ASCII processing became common, notably the DEC PDP-11 that arrived at Bell Labs, support for character data stuffed in memory words became important. The typeless nature of the language was seen as a disadvantage, which led Thompson and Ritchie to develop an expanded version of the language supporting new internal and user-defined types, which became the C programming language.

History
Circa 1969, Ken Thompson and later Dennis Ritchie developed B basing it mainly on the BCPL language Thompson used in the Multics project. B was essentially the BCPL system stripped of any component Thompson felt he could do without in order to make it fit within the memory capacity of the minicomputers of the time. The BCPL to B transition also included changes made to suit Thompson's preferences (mostly along the lines of reducing the number of non-whitespace characters in a typical program). Much of the typical ALGOL-like syntax of BCPL was rather heavily changed in this process. The assignment operator  reverted to the   of Rutishauser's Superplan, and the equality operator   was replaced by.

Thompson added "two-address assignment operators" using  syntax to add y to x (in C the operator is written  ). This syntax came from Douglas McIlroy's implementation of TMG, in which B's compiler was first implemented (and it came to TMG from ALGOL 68's  syntax). Thompson went further by inventing the increment and decrement operators ( and  ). Their prefix or postfix position determines whether the value is taken before or after alteration of the operand. This innovation was not in the earliest versions of B. According to Dennis Ritchie, people often assumed that they were created for the auto-increment and auto-decrement address modes of the DEC PDP-11, but this is historically impossible as the machine didn't exist when B was first developed.

The semicolon version of the for loop was borrowed by Ken Thompson from the work of Stephen Johnson.

B is typeless, or more precisely has one data type: the computer word. Most operators (e.g.,  ,  ,  ) treated this as an integer, but others treated it as a memory address to be dereferenced. In many other ways it looked a lot like an early version of C. There are a few library functions, including some that vaguely resemble functions from the standard I/O library in C. In Thompson's words: "B and the old old C were very very similar languages except for all the types [in C]".

Early implementations were for the DEC PDP-7 and PDP-11 minicomputers using early Unix, and Honeywell GE 645 36-bit mainframes running the operating system GCOS. The earliest PDP-7 implementations compiled to threaded code, and Ritchie wrote a compiler using TMG which produced machine code. In 1970 a PDP-11 was acquired and threaded code was used for the port; an assembler, dc, and the B language itself were written in B to bootstrap the computer. An early version of yacc was produced with this PDP-11 configuration. Ritchie took over maintenance during this period.

The typeless nature of B made sense on the Honeywell, PDP-7 and many older computers, but was a problem on the PDP-11 because it was difficult to elegantly access the character data type that the PDP-11 and most modern computers fully support. Starting in 1971 Ritchie made changes to the language while converting its compiler to produce machine code, most notably adding data typing for variables. During 1971 and 1972 B evolved into "New B" (NB) and then C.

B is almost extinct, having been superseded by the C language. However, it continues to see use on GCOS mainframes and on certain embedded systems for a variety of reasons: limited hardware in small systems, extensive libraries, tooling, licensing cost issues, and simply being good enough for the job. The highly influential AberMUD was originally written in B.

Examples
The following examples are from the Users' Reference to B by Ken Thompson:

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: B (programming language)


text:
Beer–Lambert law

The Beer–Lambert law is commonly applied to chemical analysis measurements to determine the concentration of chemical species that absorb light. It is often referred to as Beer's law. In physics, the Bouguer–Lambert law is an empirical law which relates the extinction or attenuation of light to the properties of the material through which the light is travelling. It had its first use in astronomical extinction. The fundamental law of extinction (the process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant) is sometimes called the Beer–Bouguer–Lambert law or the Bouguer–Beer–Lambert law or merely the extinction law.  The extinction law is also used in understanding attenuation in physical optics, for photons, neutrons, or rarefied gases. In mathematical physics, this law arises as a solution of the BGK equation.

History
Bouguer–Lambert law: This law is based on observations made by Pierre Bouguer before 1729. It is often attributed to Johann Heinrich Lambert, who cited Bouguer's Essai d'optique sur la gradation de la lumière (Claude Jombert, Paris, 1729) – and even quoted from it – in his Photometria in 1760. Lambert expressed the law, which states that the loss of light intensity when it propagates in a medium is directly proportional to intensity and path length, in the mathematical form used today.

Lambert began by assuming that the intensity $I$ of light traveling into an absorbing body would be given by the differential equation: $$-\mathrm{d}I=\mu I \mathrm{d}x,$$ which is compatible with Bouguer's observations. The constant of proportionality $μ$ was often termed the "optical density" of the body. Integrating to find the intensity at a distance $d$ into the body, one obtains: $\ln (I_0/I)=\int_{0}^{d} \mu \mathrm{d}x.$   For a homogeneous medium, this reduces to:  $$\ln (I_0/I)= \mu d,$$ from which follows the exponential attenuation law:  $$I=I_0 e^{-\mu d}.$$

Beer's law: Much later, in 1852, the German scientist August Beer studied another attenuation relation. In the introduction to his classic paper, he wrote: "The absorption of light during the irradiation of a colored substance has often been the object of experiment; but attention has always been directed to the relative diminution of the various colors or, in the case of crystalline bodies, the relation between the absorption and the direction of polarization. Concerning the absolute magnitude of the absorption that a particular ray of light suffers during its propagation through an absorbing medium, there is no information available." By studying absorption of red light in colored aqueous solutions of various salts, he concluded that "the transmittance of a concentrated solution can be derived from a measurement of the transmittance of a dilute solution". It is clear that he understood the exponential relationship, as he wrote: "If $${\lambda}$$ is the coefficient (fraction) of diminution, then this coefficient (fraction) will have the value $$\lambda^2$$ for double this thickness." Furthermore Beer stated: "We shall take the absorption coefficient to be the coefficient giving the diminution in amplitude suffered by a light ray as it passes through a unit length of an absorbing material. We then have, according to theory, and as I have found verified by experiment, $$\lambda = \mu^D$$  where $$\mu$$ is the absorption coefficient and $$D$$ the length of the absorbing material traversed in the experiment." This is the relationship that might properly be called Beer's law. There is no evidence that Beer saw concentration and path length as symmetrical variables in an equation in the manner of the Beer-Lambert law.

Beer–Lambert law:  The modern formulation of the Beer–Lambert law combines the observations of Bouguer and Beer into the mathematical form of Lambert. It correlates the absorbance, most often expressed as the negative decadic logarithm of the transmittance, to both the concentrations of the attenuating species and the thickness of the material sample. An early, possibly the first, modern formulation was given by Robert Luther and Andreas Nikolopulos in 1913.

Differences between Bouguer and Beer in application areas
While the observations of Bouguer and Beer have a similar form in the Beer–Lambert law, their areas of observation were very different. For both experimenters, the incident beam was well collimated, with a light sensor which preferentially detected directly transmitted light.

Beer specifically looked at solutions. Solutions are homogeneous and do not scatter light (Ultraviolet, visible, Infrared) of wavelengths commonly used in analytical spectroscopy (except upon entry and exit). The attenuation of a beam of light within a solution is assumed to be only due to absorption. In order to approximate the conditions required for the Beer Lambert law to hold, often the intensity of transmitted light through a reference sample $$(I_R)$$ consisting of pure solvent is measured, and compared to the intensity of light transmitted through a sample $$(I_S)$$, with the absorbance of the sample taken as: $${\log_{10}{\bigl(I_R/I_S\big)}}$$. It is for this case that the common mathematical formulation (see below) applies:  $$\log_{10} (I_R/I_S)=A=\varepsilon \ell c$$

Bouguer looked at astronomical phenomena where the size of a detector is very small compared to the distance traveled by the light. In this case, any light that is scattered by a particle, either in the forward or backward direction, will not strike the detector. The loss of intensity to the detector will be due to both absorption and scatter. Consequently, the total loss is called attenuation (rather than absorption). A single measurement cannot separate the two, but conceptually the contribution of each can be separated in the attenuation coefficient. If $$I_0$$ is the intensity of the light at the beginning of the travel and $$I_d$$ is the intensity of the light detected after travel of a distance $$d$$, the fraction transmitted, $$T$$, is given by: $$T=\frac {I_d}{I_0} = \exp(-\mu d)$$, where $$\mu$$ is called an attenuation constant or coefficient. The amount of light transmitted is falling off exponentially with distance. Taking the natural logarithm in the above equation, we get: $$-\ln(T) = \ln \frac {I_0}{I_d} = \mu d$$. For scattering media, the constant is often divided into two parts, $$\mu = \mu_s + \mu_a $$, separating it into a scattering coefficient, $$\mu _s$$, and an absorption coefficient, $$\mu_a$$.

Absorptivity, cross-sections, and units of coefficients
The fundamental law of extinction states that the extinction process is linear in the intensity of radiation and amount of radiatively active matter, provided that the physical state is held constant. (Neither concentration or length are fundamental parameters.) There are two factors that determine the degree to which a medium containing particles will attenuate a light beam:  the number of particles encountered by the light beam, and the degree to which each particle extinguishes the light.

For the case of absorption (Beer), this later quantity is called the absorptivity [$$\epsilon$$], which is defined as "the property of a body that determines the fraction of incident radiation absorbed by the body". The Beer–Lambert law $$[\log_{10} (I_0/I)= A= \epsilon \ell c ]$$ uses concentration and length in order to determine the number of particles the beam encounters. For a collimated beam (directed radiation) of cross-sectional area $$S$$, the number of particles encountered over a distance $$\ell$$ is $$N_{\mathrm{A}} c \sigma \ell S$$, where $$N_{\mathrm{A}}$$ is the Avogadro constant, $$c$$ the molar concentration (in mol/m3), and $$\sigma$$ the particle cross section.

There must be a large number of particles that are uniformly distributed for this relationship to hold. In practice, the beam area is thought of as a constant, and since the fraction [$$I/I_0$$] has the area in both the numerator and denominator, the beam area cancels in the calculation of the absorbance. The units of the absorptivity must match the units in which the sample is described. For example, if the sample is described by mass concentration (g/L) and length (cm), then the units on the absorptivity would be [ L g−1 cm−1], so that the absorbance has no units.

For the case of "extinction" (Bouguer), the sum of absorption and scatter, the terms absorption, scattering, and extinction cross-sections are often used. The fraction of light extinguished by the sample may be described by the extinction cross section (fraction extinguished per particle). the number of particles in a unit distance and the distance in those units. For example: [ (fraction extinguished / particle) (# particles / meter) (# meters / sample) = fraction extinguished / sample ]

Mathematical formulations
A common and practical expression of the Beer–Lambert law relates the optical attenuation of a physical material containing a single attenuating species of uniform concentration to the optical path length through the sample and absorptivity of the species. This expression is:$$\log_{10} (I_0/I)=A=\varepsilon \ell c$$where
 * $A$ is the absorbance
 * $ε$ is the molar attenuation coefficient or absorptivity of the attenuating species
 * $ℓ$ is the optical path length
 * $c$ is the concentration of the attenuating species

A more general form of the Beer–Lambert law states that, for $N$ attenuating species in the material sample,$$\begin{align} T &= \exp\left(-\sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\mathrm{d}z \right) \\[4pt] &= 10^{\;\!\wedge} \!\! \left(-\sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\mathrm{d}z \right), \end{align}$$or equivalently that$$\begin{align} \tau &= \sum_{i = 1}^N \tau_i = \sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\,\mathrm{d}z, \\[4pt] A &= \sum_{i = 1}^N A_i = \sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\,\mathrm{d}z, \end{align}$$where
 * $σi$ is the attenuation cross section of the attenuating species $i$ in the material sample;
 * $ni$ is the number density of the attenuating species $i$ in the material sample;
 * $εi$ is the molar attenuation coefficient or absorptivity of the attenuating species $i$ in the material sample;
 * $ci$ is the amount concentration of the attenuating species $i$ in the material sample;
 * $ℓ$ is the path length of the beam of light through the material sample.

In the above equations, the transmittance $T$ of material sample is related to its optical depth $τ$ and to its absorbance $A$ by the following definition$$T = \frac{\Phi_\mathrm{e}^\mathrm{t}}{\Phi_\mathrm{e}^\mathrm{i}} = e^{-\tau} = 10^{-A},$$where
 * $$\mathrm{\Phi_e^t}$$ is the radiant flux transmitted by that material sample;
 * $$\mathrm{\Phi_e^i}$$ is the radiant flux received by that material sample.

Attenuation cross section and molar attenuation coefficient are related by$$\varepsilon_i = \frac{\mathrm{N_A}}{\ln(10)}\,\sigma_i,$$and number density and amount concentration by$$\begin{align} \tau &= \ell\sum_{i = 1}^N \sigma_i n_i, \\[4pt] A &= \ell\sum_{i = 1}^N \varepsilon_i c_i. \end{align}$$$$c_i = \frac{n_i}{\mathrm{N_A}},$$where $NA$ is the Avogadro constant. In case of uniform attenuation, these relations become $$\begin{align} T &= \exp\left(-\ell\sum_{i = 1}^N \sigma_i n_i \right) \\[4pt] &= 10^{\;\!\wedge} \!\! \left(-\ell\sum_{i = 1}^N \varepsilon_i c_i \right), \end{align}$$or equivalently$$\begin{align} \tau &= \ell\sum_{i = 1}^N \sigma_i n_i, \\[4pt] A &= \ell\sum_{i = 1}^N \varepsilon_i c_i. \end{align}$$$$\begin{align} \tau &= \ell\sum_{i = 1}^N \sigma_i n_i, \\[4pt] A &= \ell\sum_{i = 1}^N \varepsilon_i c_i. \end{align}$$Cases of non-uniform attenuation occur in atmospheric science applications and radiation shielding theory for instance.

The law tends to break down at very high concentrations, especially if the material is highly scattering. Absorbance within range of 0.2 to 0.5 is ideal to maintain linearity in the Beer–Lambert law. If the radiation is especially intense, nonlinear optical processes can also cause variances. The main reason, however, is that the concentration dependence is in general non-linear and Beer's law is valid only under certain conditions as shown by derivation below. For strong oscillators and at high concentrations the deviations are stronger. If the molecules are closer to each other interactions can set in. These interactions can be roughly divided into physical and chemical interactions. Physical interaction do not alter the polarizability of the molecules as long as the interaction is not so strong that light and molecular quantum state intermix (strong coupling), but cause the attenuation cross sections to be non-additive via electromagnetic coupling. Chemical interactions in contrast change the polarizability and thus absorption.

Expression with attenuation coefficient
The law can be expressed in terms of attenuation coefficient, but in this case is better called the Bouguer-Lambert's law. The (Napierian) attenuation coefficient $$\mu$$ and the decadic attenuation coefficient $$\mu_{10}= \tfrac{\mu}{\ln 10}$$ of a material sample are related to its number densities and amount concentrations as$$\begin{align} \mu(z) &= \sum_{i = 1}^N \mu_i(z) = \sum_{i = 1}^N \sigma_i n_i(z), \\[4pt] \mu_{10}(z) &= \sum_{i = 1}^N \mu_{10,i}(z) = \sum_{i = 1}^N \varepsilon_i c_i(z) \end{align}$$respectively, by definition of attenuation cross section and molar attenuation coefficient. Then the law becomes$$\begin{align} T &= \exp\left(-\int_0^\ell \mu(z)\mathrm{d}z \right) \\[4pt] &= 10^{\;\!\wedge} \!\! \left(-\int_0^\ell \mu_{10}(z)\mathrm{d}z \right), \end{align}$$and$$\begin{align} \tau &= \int_0^\ell \mu(z)\,\mathrm{d}z, \\[4pt] A &= \int_0^\ell \mu_{10}(z)\,\mathrm{d}z. \end{align}$$In case of uniform attenuation, these relations become$$T = e^{-\mu\ell} = 10^{-\mu_{10}\ell},$$or equivalently$$\begin{align} \tau &= \mu\ell, \\[4pt] A &= \mu_{10}\ell. \end{align}$$

In many cases, the attenuation coefficient does not vary with $$z$$, in which case one does not have to perform an integral and can express the law as:$$I(z) = I_0 e^{-\mu z}$$where the attenuation is usually an addition of absorption coefficient $$\alpha$$ (creation of electron-hole pairs) or scattering (for example Rayleigh scattering if the scattering centers are much smaller than the incident wavelength). Also note that for some systems we can put $$1/\lambda$$ (1 over inelastic mean free path) in place of $\mu$.

Derivation
Assume that a beam of light enters a material sample. Define $z$ as an axis parallel to the direction of the beam. Divide the material sample into thin slices, perpendicular to the beam of light, with thickness $dz$ sufficiently small that one particle in a slice cannot obscure another particle in the same slice when viewed along the $z$ direction. The radiant flux of the light that emerges from a slice is reduced, compared to that of the light that entered, by $$\mathrm{d\Phi_e}(z) = -\mu(z)\Phi_\mathrm{e}(z) \mathrm{d}z,$$ where $μ$ is the (Napierian) attenuation coefficient, which yields the following first-order linear, ordinary differential equation:$$\frac{\mathrm{d}\Phi_\mathrm{e}(z)}{\mathrm{d}z} = -\mu(z)\Phi_\mathrm{e}(z).$$ The attenuation is caused by the photons that did not make it to the other side of the slice because of scattering or absorption. The solution to this differential equation is obtained by multiplying the integrating factor$$\exp\left(\int_0^z \mu(z')\mathrm{d}z' \right)$$throughout to obtain$$\frac{\mathrm{d}\Phi_\mathrm{e}(z)}{\mathrm{d}z}\, \exp\left(\int_0^z \mu(z')\mathrm{d}z' \right) + \mu(z)\Phi_\mathrm{e}(z)\, \exp\left(\int_0^z \mu(z')\mathrm{d}z' \right) = 0,$$which simplifies due to the product rule (applied backwards) to$$\frac{\mathrm{d}}{\mathrm{d}z}\left[\Phi_\mathrm{e}(z) \exp\left(\int_0^z \mu(z')\mathrm{d}z' \right)\right] = 0.$$

Integrating both sides and solving for $Φ_{e}$ for a material of real thickness $ℓ$, with the incident radiant flux upon the slice $$\mathrm{\Phi_e^i} = \mathrm{\Phi_e}(0)$$ and the transmitted radiant flux $$\mathrm{\Phi_e^t} = \mathrm{\Phi_e}(\ell)$$ gives$$\mathrm{\Phi_e^t} = \mathrm{\Phi_e^i} \exp\left(-\int_0^\ell \mu(z)\mathrm{d}z \right),$$and finally$$T = \mathrm{\frac{\Phi_e^t}{\Phi_e^i}} = \exp\left(-\int_0^\ell \mu(z)\mathrm{d}z \right).$$

Since the decadic attenuation coefficient $μ_{10}$ is related to the (Napierian) attenuation coefficient by $$\mu_{10} = \tfrac{\mu}{\ln 10},$$ we also have$$\begin{align} T &= \exp\left(-\int_0^\ell \ln(10)\,\mu_{10}(z)\mathrm{d}z \right) \\[4pt] &= 10^{\;\!\wedge} \!\! \left( -\int_0^\ell \mu_{10}(z)\mathrm{d}z \right). \end{align}$$

To describe the attenuation coefficient in a way independent of the number densities $n_{i}$ of the $N$ attenuating species of the material sample, one introduces the attenuation cross section $$\sigma_i = \tfrac{\mu_i(z)}{n_i(z)}.$$ $σ_{i}$ has the dimension of an area; it expresses the likelihood of interaction between the particles of the beam and the particles of the species $i$ in the material sample:$$T = \exp\left(-\sum_{i = 1}^N \sigma_i \int_0^\ell n_i(z)\mathrm{d}z \right).$$

One can also use the molar attenuation coefficients $$\varepsilon_i = \tfrac{\mathrm{N_A}}{\ln 10}\sigma_i,$$ where $N_{A}$ is the Avogadro constant, to describe the attenuation coefficient in a way independent of the amount concentrations $$c_i(z) = n_i \tfrac{z}{\mathrm{N_A}}$$ of the attenuating species of the material sample:$$ \begin{align} T &= \exp\left(-\sum_{i = 1}^N \frac{\ln(10)}{\mathrm{N_A}}\varepsilon_i \int_0^\ell n_i(z)\mathrm{d}z \right) \\[4pt] &= \exp\left(-\sum_{i = 1}^N \varepsilon_i \int_0^\ell \frac{n_i(z)}{\mathrm{N_A}}\mathrm{d}z\right)^{\ln(10)} \\[4pt] &= 10^{\;\!\wedge} \!\! \left(-\sum_{i = 1}^N \varepsilon_i \int_0^\ell c_i(z)\mathrm{d}z \right). \end{align} $$

Validity
Under certain conditions the Beer–Lambert law fails to maintain a linear relationship between attenuation and concentration of analyte. These deviations are classified into three categories:
 * 1) Real—fundamental deviations due to the limitations of the law itself.
 * 2) Chemical—deviations observed due to specific chemical species of the sample which is being analyzed.
 * 3) Instrument—deviations which occur due to how the attenuation measurements are made.

There are at least six conditions that need to be fulfilled in order for the Beer–Lambert law to be valid. These are:
 * 1) The attenuators must act independently of each other.
 * 2) The attenuating medium must be homogeneous in the interaction volume.
 * 3) The attenuating medium must not scatter the radiation—no turbidity—unless this is accounted for as in DOAS.
 * 4) The incident radiation must consist of parallel rays, each traversing the same length in the absorbing medium.
 * 5) The incident radiation should preferably be monochromatic, or have at least a width that is narrower than that of the attenuating transition. Otherwise a spectrometer as detector for the power is needed instead of a photodiode which cannot discriminate between wavelengths.
 * 6) The incident flux must not influence the atoms or molecules; it should only act as a non-invasive probe of the species under study. In particular, this implies that the light should not cause optical saturation or optical pumping, since such effects will deplete the lower level and possibly give rise to stimulated emission.

If any of these conditions are not fulfilled, there will be deviations from the Beer–Lambert law.

Chemical analysis by spectrophotometry
The Beer–Lambert law can be applied to the analysis of a mixture by spectrophotometry, without the need for extensive pre-processing of the sample. An example is the determination of bilirubin in blood plasma samples. The spectrum of pure bilirubin is known, so the molar attenuation coefficient $ε$ is known. Measurements of decadic attenuation coefficient $μ_{10}$ are made at one wavelength $λ$ that is nearly unique for bilirubin and at a second wavelength in order to correct for possible interferences. The amount concentration $c$ is then given by $$c = \frac{\mu_{10}(\lambda)}{\varepsilon(\lambda)}.$$

For a more complicated example, consider a mixture in solution containing two species at amount concentrations $c_{1}$ and $c_{2}$. The decadic attenuation coefficient at any wavelength $λ$ is, given by $$\mu_{10}(\lambda) = \varepsilon_1(\lambda) c_1 + \varepsilon_2(\lambda) c_2.$$

Therefore, measurements at two wavelengths yields two equations in two unknowns and will suffice to determine the amount concentrations $c_{1}$ and $c_{2}$ as long as the molar attenuation coefficients of the two components, $ε_{1}$ and $ε_{2}$ are known at both wavelengths. This two system equation can be solved using Cramer's rule. In practice it is better to use linear least squares to determine the two amount concentrations from measurements made at more than two wavelengths. Mixtures containing more than two components can be analyzed in the same way, using a minimum of $N$ wavelengths for a mixture containing $N$ components.

The law is used widely in infra-red spectroscopy and near-infrared spectroscopy for analysis of polymer degradation and oxidation (also in biological tissue) as well as to measure the concentration of various compounds in different food samples. The carbonyl group attenuation at about 6 micrometres can be detected quite easily, and degree of oxidation of the polymer calculated.

Application for the atmosphere
The Bouguer–Lambert law may be applied to describe the attenuation of solar or stellar radiation as it travels through the atmosphere. In this case, there is scattering of radiation as well as absorption. The optical depth for a slant path is $τ' = mτ$, where $τ$ refers to a vertical path, $m$ is called the relative airmass, and for a plane-parallel atmosphere it is determined as $m = sec θ$ where $θ$ is the zenith angle corresponding to the given path. The Bouguer-Lambert law for the atmosphere is usually written $$T = \exp \big( -m(\tau_\mathrm{a} + \tau_\mathrm{g} + \tau_\mathrm{RS} + \tau_\mathrm{NO_2} + \tau_\mathrm{w} + \tau_\mathrm{O_3} + \tau_\mathrm{r} + \cdots) \bigr),$$ where each $τ_{x}$ is the optical depth whose subscript identifies the source of the absorption or scattering it describes:
 * $a$ refers to aerosols (that absorb and scatter);
 * $g$ are uniformly mixed gases (mainly carbon dioxide (CO2) and molecular oxygen (O2) which only absorb);
 * $NO2$ is nitrogen dioxide, mainly due to urban pollution (absorption only);
 * $RS$ are effects due to Raman scattering in the atmosphere;
 * $w$ is water vapour absorption;
 * $O3$ is ozone (absorption only);
 * $r$ is Rayleigh scattering from molecular oxygen (O2) and nitrogen (N2) (responsible for the blue color of the sky);
 * the selection of the attenuators which have to be considered depends on the wavelength range and can include various other compounds. This can include tetraoxygen, HONO, formaldehyde, glyoxal, a series of halogen radicals and others.

$m$ is the optical mass or airmass factor, a term approximately equal (for small and moderate values of $θ$) to $\tfrac{1}{\cos \theta},$ where $θ$ is the observed object's zenith angle (the angle measured from the direction perpendicular to the Earth's surface at the observation site). This equation can be used to retrieve $τ_{a}$, the aerosol optical thickness, which is necessary for the correction of satellite images and also important in accounting for the role of aerosols in climate.

title: Beer–Lambert law