Superradiant phase transition
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quantum optics Quantum optics is a branch of atomic, molecular, and optical physics dealing with how individual quanta of light, known as photons, interact with atoms and molecules. It includes the study of the particle-like properties of photons. Photons have ...
, a superradiant phase transition is a
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states o ...
that occurs in a collection of
fluorescent Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation. It is a form of luminescence. In most cases, the emitted light has a longer wavelength, and therefore a lower photon energy, ...
emitters (such as atoms), between a state containing few electromagnetic excitations (as in the electromagnetic vacuum) and a superradiant state with many electromagnetic excitations trapped inside the emitters. The superradiant state is made thermodynamically favorable by having strong, coherent interactions between the emitters. The superradiant phase transition was originally predicted by the
Dicke model Dicke is a surname. Notable people with the surname include: * Amie Dicke (born 1978), Dutch artist * Finn Dicke (born 2004), Dutch footballer * Pien Dicke (born 1999), Dutch field hockey player * Robert H. Dicke (1916–1997), American physicis ...
of
superradiance In physics, superradiance is the radiation enhancement effects in several contexts including quantum mechanics, astrophysics and relativity. Quantum optics In quantum optics, superradiance is a phenomenon that occurs when a group of ''N'' emit ...
, which assumes that atoms have only two energetic levels and that these interact with only one mode of the electromagnetic field. The phase transition occurs when the strength of the interaction between the atoms and the field is greater than the energy of the non-interacting part of the system. (This is similar to the case of superconductivity in
ferromagnetism Ferromagnetism is a property of certain materials (such as iron) which results in a large observed magnetic permeability, and in many cases a large magnetic coercivity allowing the material to form a permanent magnet. Ferromagnetic materials ...
, which leads to the dynamic interaction between ferromagnetic atoms and the spontaneous ordering of excitations below the critical temperature.) The collective
Lamb shift In physics, the Lamb shift, named after Willis Lamb, is a difference in energy between two energy levels 2''S''1/2 and 2''P''1/2 (in term symbol notation) of the hydrogen atom which was not predicted by the Dirac equation, according to which th ...
, relating to the system of atoms interacting with the
vacuum fluctuation In quantum physics, a quantum fluctuation (also known as a vacuum state fluctuation or vacuum fluctuation) is the temporary random change in the amount of energy in a point in space, as prescribed by Werner Heisenberg's uncertainty principle. ...
s, becomes comparable to the energies of atoms alone, and the vacuum fluctuations cause the spontaneous self-excitation of matter. The transition can be readily understood by the use of the Holstein-Primakoff transformation applied to a two-level atom. As a result of this transformation, the atoms become Lorentz
harmonic oscillators In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive consta ...
with frequencies equal to the difference between the energy levels. The whole system then simplifies to a system of interacting
harmonic oscillators In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force ''F'' proportional to the displacement ''x'': \vec F = -k \vec x, where ''k'' is a positive consta ...
of atoms, and the field known as Hopfield dielectric which further predicts in the normal state
polaron A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was proposed by Lev Landau in 1933 and Solomon Pekar in 1946 to describe an electro ...
s for photons or
polariton In physics, polaritons are quasiparticles resulting from strong coupling of electromagnetic waves with an electric or magnetic dipole-carrying excitation. They are an expression of the common quantum phenomenon known as level repulsion, also ...
s. If the interaction with the field is so strong that the system collapses in the harmonic approximation and complex polariton frequencies (soft modes) appear, then the physical system with nonlinear terms of the higher order becomes the system with the Mexican hat-like potential, and will undergo ferroelectric-like phase transition. In this model, the system is mathematically equivalent for one
mode Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...
of excitation to the Trojan wave packet, when the circularly polarized field intensity corresponds to the electromagnetic coupling constant. Above the critical value, it changes to the unstable motion of the
ionization Ionization, or Ionisation is the process by which an atom or a molecule acquires a negative or positive charge by gaining or losing electrons, often in conjunction with other chemical changes. The resulting electrically charged atom or molecul ...
. The superradiant phase transition was the subject of a wide discussion as to whether or not it is only a result of the simplified model of the matter-field interaction; and if it can occur for the real physical parameters of physical systems (a no-go theorem). However, both the original derivation and the later corrections leading to nonexistence of the transition – due to Thomas–Reiche–Kuhn sum rule canceling for the harmonic oscillator the needed inequality to impossible negativity of the interaction – were based on the assumption that the quantum field operators are commuting numbers, and the atoms do not interact with the static Coulomb forces. This is generally not true like in case of Bohr–van Leeuwen theorem and the classical non-existence of Landau diamagnetism. The return of the transition basically occurs because the inter-atom dipole-dipole interactions are never negligible in the superradiant matter density regime and the Power-Zienau unitary transformation eliminating the quantum vector potential in the minimum-coupling Hamiltonian transforms the Hamiltonian exactly to the form used when it was discovered and without the square of the vector potential which was later claimed to prevent it. Alternatively within the full quantum mechanics including the electromagnetic field the generalized Bohr–van Leeuwen theorem does not work and the electromagnetic interactions cannot be fully eliminated while they only change the \mathbf \cdot \mathbf vector potential coupling to the electric field \mathbf \cdot \mathbf coupling and alter the effective electrostatic interactions. It can be observed in model systems like
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 −459.6 ...
s and artificial atoms.


Theory


Criticality of linearized Jaynes-Cummings model

A superradiant phase transition is formally predicted by the critical behavior of the resonant Jaynes-Cummings model, describing the interaction of only one atom with one mode of the electromagnetic field. Starting from the exact Hamiltonian of the Jaynes-Cummings model at resonance :\hat_ = \hbar \omega \hat^\hat +\hbar \omega \frac +\frac \left(\hat\hat_+ +\hat^\hat_- +\hat\hat_- +\hat^\hat_+\right), Applying the Holstein-Primakoff transformation for two spin levels, replacing the spin raising and lowering operators by those for the harmonic oscillators :\hat_- \approx \hat :\hat_+\approx \hat^ :\hat_z\approx 2 \hat^\hat one gets the Hamiltonian of two coupled harmonic-oscillators: :\hat_ = \hbar \omega \hat^\hat +\hbar \omega \hat^\hat +\frac \left(\hat\hat^ +\hat^\hat +\hat\hat +\hat^\hat^ \right), which readily can be diagonalized. Postulating its normal form :\hat_=\Omega_+ \hat^\hat+\Omega_- \hat^\hat+C where :\hat=c_ \hat + c_ \hat^ + c_ \hat + c_ \hat^ one gets the eigenvalue equation : hat,\hat_\Omega_A with the solutions :\Omega_=\omega \sqrt The system collapses when one of the frequencies becomes imaginary, i.e. when :\Omega>\omega or when the atom-field coupling is stronger than the frequency of the mode and atom oscillators. While there are physically higher terms in the true system, the system in this regime will therefore undergo the phase transition.


Criticality of Jaynes-Cummings model

The simplified Hamiltonian of the Jaynes-Cummings model, neglecting the counter-rotating terms, is :\hat_ = \hbar \omega \hat^\hat +\hbar \omega \frac +\frac \left(\hat\hat_+ +\hat^\hat_-\right), and the energies for the case of zero detuning are :E_(n) = \hbar\omega \left(n+\frac\right) \pm \frac \hbar\Omega(n), :\Omega(n) = \Omega \sqrt where \Omega is the Rabi frequency. One can approximately calculate the
canonical partition function The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical exampl ...
: Z = \sum_ \mathrm^ \approx \sum_ \int \mathrm^ dn=\int \mathrm^ dn, where the discrete sum was replaced by the integral. The normal approach is that the latter integral is calculated by the Gaussian approximation around the maximum of the exponent: : \frac =0 : \Phi(n)=-\beta \hbar\omega \left(n+\frac\right)+\log 2 \cosh \frac This leads to the critical equation : \tanh \frac = 4 \frac\sqrt This has the solution only if : \Omega>4 \omega which means that the normal, and the superradiant phase, exist only if the field-atom coupling is significantly stronger than the energy difference between the atom levels. When the condition is fulfilled, the equation gives the solution for the order parameter n depending on the inverse of the temperature 1/\beta, which means non-vanishing ordered field mode. Similar considerations can be done in true thermodynamic limit of the infinite number of atoms.


Instability of the classical electrostatic model

The better insight on the nature of the superradiant phase transition as well on the physical value of the critical parameter which must be exceeded in order for the transition to occur may be obtained by studying the classical stability of the system of the charged classical harmonic oscillators in the 3D space interacting only with the electrostatic repulsive forces for example between electrons in the locally harmonic oscillator potential. Despite of the original model of the superradiance the quantum electromagnetic field is totally neglected here. The oscillators may be assumed to be placed for example on the cubic lattice with the lattice constant a in the analogy to the crystal system of the condensed matter. The worse scenario of the defect of the absence of the two out-of-the-plane motion-stabilizing electrons from the 6-th nearest neighbors of a chosen electron is assumed while the four nearest electrons are first assumed to be rigid in space and producing the anti-harmonic potential in the direction perpendicular to the plane of the all five electrons. The condition of the instability of motion of the chosen electron is that the net potential being the superposition of the harmonic oscillator potential and the quadratically expanded Coulomb potential from the four electrons is negative i.e. : \frac- \frac \times 4 \times \frac \frac <0 or : \frac\frac >1 Making it artificially quantum by multiplying the numerator and the denominator of the fraction by the \hbar one obtains the condition : \frac \frac \left( \frac \right)>1 where : , D_, ^2 = \frac is the square of the dipole transition strength between the ground state and the first excited state of the quantum harmonic oscillator, : E_=\hbar \omega is the energy gap between consecutive levels and it is also noticed that : \frac =\frac is the spatial density of the oscillators. The condition is almost identical to this obtained in the original discovery of the superradiant phase transition when replacing the harmonic oscillators with two level atoms with the same distance between the energy levels, dipole transition strength, and the density which means that it occurs in the regime when the Coulomb interactions between electrons dominate over locally harmonic oscillatory influence of the atoms. It that sense the free
electron gas An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. T ...
with \omega=0 is also purely superradiant. The critical inequality rewritten yet differently : \omega<\sqrt\approx\sqrt expresses the fact that superradiant phase transition occurs when the frequency of the binding atomic oscillators is lower than so called electron gas
plasma frequency Plasma oscillations, also known as Langmuir waves (after Irving Langmuir), are rapid oscillations of the electron density in conducting media such as plasmas or metals in the ultraviolet region. The oscillations can be described as an instability i ...
.


References

{{reflist Quantum mechanics Phase transitions