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Inphysics, specifically inquantum mechanics, acoherent state is the specificquantum state of thequantum harmonic oscillator, often described as a state that hasdynamics most closely resembling the oscillatory behavior of aclassical harmonic oscillator. It was the first example ofquantum dynamics whenErwin Schrödinger derived it in 1926, while searching for solutions of theSchrödinger equation that satisfy thecorrespondence principle.^[1] The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.^[2] For instance, a coherent state describes the oscillating motion of a particle confined in a quadraticpotential well (for an early reference, see e.g.Schiff's textbook^[3]). The coherent state describes a state in a system for which the ground-state wavepacket is displaced from the origin of the system. This state can be related to classical solutions by a particle oscillating with an amplitude equivalent to the displacement.

These states, expressed aseigenvectors of thelowering operator and forming anovercomplete family, were introduced in the early papers ofJohn R. Klauder, e.g.^[4] In the quantum theory of light (quantum electrodynamics) and otherbosonicquantum field theories, coherent states were introduced by the work ofRoy J. Glauber in 1963 and are also known asGlauber states.

The concept of coherent states has been considerably abstracted; it has become a major topic inmathematical physics and inapplied mathematics, with applications ranging fromquantization tosignal processing andimage processing (seeCoherent states in mathematical physics). For this reason, the coherent states associated to thequantum harmonic oscillator are sometimes referred to ascanonical coherent states (CCS),standard coherent states,Gaussian states, or oscillator states.

Inquantum optics the coherent state refers to a state of the quantizedelectromagnetic field, etc.^[2][6][7] that describes a maximal kind ofcoherence and a classical kind of behavior.Erwin Schrödinger derived it as a "minimumuncertainty"Gaussian wavepacket in 1926, searching for solutions of theSchrödinger equation that satisfy thecorrespondence principle.^[1] It is aminimum uncertainty state, with the single free parameter chosen to make the relative dispersion (standard deviation in natural dimensionless units) equal for position and momentum, each being equally small at high energy.

Further, in contrast to theenergy eigenstates of the system, the time evolution of a coherent state is concentrated along the classicaltrajectories. The quantum linear harmonic oscillator, and hence coherent states, arise in the quantum theory of a wide range of physical systems. They occur in the quantum theory of light (quantum electrodynamics) and otherbosonicquantum field theories.

While minimum uncertainty Gaussian wave-packets had been well-known, they did not attract full attention untilRoy J. Glauber, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field.^[8] In this respect, the concurrent contribution ofE.C.G. Sudarshan should not be omitted,^[9] (there is, however, a note in Glauber's paper that reads: "Uses of these states asgenerating functions for the${\displaystyle n}$-quantum states have, however, been made by J. Schwinger^[10]).Glauber was prompted to do this to provide a description of theHanbury-Brown & Twiss experiment, which generated very wide baseline (hundreds or thousands of miles)interference patterns that could be used to determine stellar diameters. This opened the door to a much more comprehensive understanding of coherence. (For more, seeQuantum mechanical description.)

In classicaloptics, light is thought of aselectromagnetic waves radiating from a source. Often, coherent laser light is thought of as light that is emitted by many such sources that are inphase. Actually, the picture of onephoton being in-phase with another is not valid in quantum theory. Laser radiation is produced in aresonant cavity where theresonant frequency of the cavity is the same as the frequency associated with theatomic electron transitions providing energy flow into the field. As energy in the resonant mode builds up, the probability forstimulated emission, in that mode only, increases. That is a positivefeedback loop in which the amplitude in the resonant modeincreases exponentially until somenonlinear effects limit it. As a counter-example, alight bulb radiates light into a continuum of modes, and there is nothing that selects any one mode over the other. The emission process is highly random in space and time (seethermal light). In alaser, however, light is emitted into a resonant mode, and that mode is highlycoherent. Thus, laser light is idealized as a coherent state. (Classically we describe such a state by anelectric field oscillating as a stable wave. See Fig.1)

Besides describing lasers, coherent states also behave in a convenient manner when describing the quantum action ofbeam splitters: two coherent-state input beams will simply convert to two coherent-state beams at the output with new amplitudes given by classical electromagnetic wave formulas;^[11] such a simple behaviour does not occur for other input states, including number states. Likewise if a coherent-state light beam is partially absorbed, then the remainder is a pure coherent state with a smaller amplitude, whereas partial absorption of non-coherent-state light produces a more complicated statisticalmixed state.^[11] Thermal light can be described as a statistical mixture of coherent states, and the typical way of definingnonclassical light is that it cannot be described as a simple statistical mixture of coherent states.^[11]

The energy eigenstates of the linear harmonic oscillator (e.g., masses on springs, lattice vibrations in a solid, vibrational motions of nuclei in molecules, or oscillations in the electromagnetic field) are fixed-number quantum states. TheFock state (e.g. a single photon) is the most particle-like state; it has a fixed number of particles, and phase is indeterminate. A coherent state distributes its quantum-mechanical uncertainty equally between thecanonically conjugate coordinates, position and momentum, and the relative uncertainty in phase [definedheuristically] and amplitude are roughly equal—and small at high amplitude.

Mathematically, a coherent state${\displaystyle |\alpha ⟩}$ is defined to be the (unique) eigenstate of theannihilation operatorâ with corresponding eigenvalueα. Formally, this reads,

${\displaystyle \hat{a}|\alpha ⟩=\alpha |\alpha ⟩.}$

Sinceâ is nothermitian,α is, in general, acomplex number. Writing${\displaystyle \alpha =|\alpha |e^{i\theta },}$ |α| andθ are called the amplitude and phase of the state${\displaystyle |\alpha ⟩}$.

The state${\displaystyle |\alpha ⟩}$ is called acanonical coherent state in the literature, since there are many other types of coherent states, as can be seen in the companion articleCoherent states in mathematical physics.

Physically, this formula means that a coherent state remains unchanged by the annihilation of field excitation or, say, a charged particle. An eigenstate of the annihilation operator has aPoissonian number distribution when expressed in a basis of energy eigenstates, as shown below. APoisson distribution is a necessary and sufficient condition that all detections are statistically independent. Contrast this to a single-particle state (${\displaystyle |1⟩}$Fock state): once one particle is detected, there is zero probability of detecting another.

The derivation of this will make use of (unconventionally normalized)dimensionless operators,X andP, normally calledfield quadratures in quantum optics.(SeeNondimensionalization.) These operators are related to the position and momentum operators of a massm on a spring with constantk,

${\displaystyle P=\sqrt{\frac{1}{2\hslash m\omega }}\hat{p}\text{,}X=\sqrt{\frac{m\omega }{2\hslash }}\hat{x}\text{,}\text{where }\omega \equiv \sqrt{k/m}.}$

For anoptical field,

${\displaystyle E_{\mathrm{R}}={\left(\frac{2\hslash \omega }{{ϵ}_{0}V}\right)}^{1/2}\cos (\theta )X\text{and}{E}_{\mathrm{I}}={\left(\frac{2\hslash \omega }{{ϵ}_{0}V}\right)}^{1/2}\sin (\theta )X}$

are the real and imaginary components of the mode of the electric field inside a cavity of volume${\displaystyle V}$.^[12]

With these (dimensionless) operators, the Hamiltonian of either system becomes

${\displaystyle H=\hslash \omega \left(P^2+X^2\right)\text{,}\text{with}\left[X,P\right]\equiv XP-PX=\frac{i}{2}I.}$

Erwin Schrödinger was searching for the most classical-like states when he first introduced minimum uncertainty Gaussian wave-packets. Thequantum state of the harmonic oscillator that minimizes theuncertainty relation with uncertainty equally distributed betweenX andP satisfies the equation

${\displaystyle \left(X-⟨X⟩\right)|\alpha ⟩=-i\left(P-⟨P⟩\right)|\alpha ⟩\text{,}}$

or, equivalently,

${\displaystyle \left(X+iP\right)|\alpha ⟩=⟨X+iP⟩|\alpha ⟩,}$

and hence

${\displaystyle ⟨\alpha \mid {\left(X-⟨X⟩\right)}^2+{\left(P-⟨P⟩\right)}^2\mid \alpha ⟩=1.}$

Thus, given (∆X−∆P)² ≥ 0, Schrödinger found thatthe minimum uncertainty states for the linear harmonic oscillator are the eigenstates of(X +iP). Sinceâ is(X +iP), this is recognizable as a coherent state in the sense of the above definition.

Using the notation for multi-photon states, Glauber characterized the state of complete coherence to all orders in the electromagnetic field to be the eigenstate of the annihilation operator—formally, in a mathematical sense, the same state as found by Schrödinger. The namecoherent state took hold after Glauber's work.

If the uncertainty is minimized, but not necessarily equally balanced betweenX andP, the state is called asqueezed coherent state.

The coherent state's location in the complex plane (phase space) is centered at the position and momentum of a classical oscillator of the phaseθ and amplitude |α| given by the eigenvalueα (or the same complex electric field value for an electromagnetic wave). As shown in Figure 5, the uncertainty, equally spread in all directions, is represented by a disk with diameter1⁄2. As the phase varies, the coherent state circles around the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space.

Since the uncertainty (and hence measurement noise) stays constant at1⁄2 as the amplitude of the oscillation increases, the state behaves increasingly like a sinusoidal wave, as shown in Figure 1. Moreover, since the vacuum state${\displaystyle |0⟩}$ is just the coherent state withα=0, all coherent states have the same uncertainty as the vacuum. Therefore, one may interpret the quantum noise of a coherent state as being due to vacuum fluctuations.

The notation${\displaystyle |\alpha ⟩}$ does not refer to aFock state. For example, whenα = 1, one should not mistake${\displaystyle |1⟩}$ for the single-photon Fock state, which is also denoted${\displaystyle |1⟩}$ in its own notation. The expression${\displaystyle |\alpha ⟩}$ withα = 1 represents a Poisson distribution of number states${\displaystyle |n⟩}$ with a mean photon number of unity.

The formal solution of the eigenvalue equation is the vacuum state displaced to a locationα in phase space, i.e., it is obtained by letting the unitarydisplacement operatorD(α) operate on the vacuum,

${\displaystyle |\alpha ⟩=e^{\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}|0⟩=D(\alpha )|0⟩}$,

whereâ =X +iP andâ^† =X -iP.

This can be easily seen, as can virtually all results involving coherent states, using the representation of the coherent state in the basis of Fock states,

${\displaystyle |\alpha ⟩=e^{-\frac{|\alpha {|}^2}{2}}\sum _{n=0}^{\mathrm{\infty }}\frac{{\alpha }^n}{\sqrt{n!}}|n⟩=e^{-\frac{|\alpha {|}^2}{2}}{e}^{\alpha {\hat{a}}^{†}}{e}^{-{\alpha }^{\ast }\hat{a}}|0⟩=e^{\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}|0⟩=D(\alpha )|0⟩,}$

where${\displaystyle |n⟩}$ are energy (number) eigenvectors of the Hamiltonian

${\displaystyle H=\hslash \omega \left({\hat{a}}^{†}\hat{a}+\frac{1}{2}\right),}$

and the final equality derives from theBaker-Campbell-Hausdorff formula. For the correspondingPoissonian distribution, the probability of detectingn photons is

${\displaystyle P(n)=|⟨n|\alpha ⟩{|}^2=e^{-⟨n⟩}\frac{⟨n{⟩}^n}{n!}.}$

Similarly, the average photon number in a coherent state is

${\displaystyle ⟨n⟩=⟨{\hat{a}}^{†}\hat{a}⟩=|\alpha {|}^2}$

and the variance is

${\displaystyle (\mathrm{\Delta }n{)}^2=\mathrm{V}\mathrm{a}\mathrm{r}\left({\hat{a}}^{†}\hat{a}\right)=|\alpha {|}^2}$.

That is, the standard deviation of the number detected goes like the square root of the number detected. So in the limit of largeα, these detection statistics are equivalent to that of a classical stable wave.

These results apply to detection results at a single detector and thus relate to first order coherence (seedegree of coherence). However, for measurements correlating detections at multiple detectors, higher-order coherence is involved (e.g., intensity correlations, second order coherence, at two detectors). Glauber's definition of quantum coherence involves nth-order correlation functions (n-th order coherence) for alln. The perfect coherent state has all n-orders of correlation equal to 1 (coherent). It is perfectly coherent to all orders.

The second-order correlation coefficient${\displaystyle g^2(0)}$ gives a direct measure of the degree of coherence of photon states in terms of the variance of the photon statistics in the beam under study.^[13]

${\displaystyle g^2(0)=1+\frac{\mathrm{V}\mathrm{a}\mathrm{r}\left({\hat{a}}^{†}\hat{a}\right)-⟨{\hat{a}}^{†}\hat{a}⟩}{(⟨{\hat{a}}^{†}\hat{a}⟩{)}^2}=1+\frac{\mathrm{V}\mathrm{a}\mathrm{r}(n)-\overline{n}}{{\overline{n}}^2}}$

In Glauber's development, it is seen that the coherent states are distributed according to aPoisson distribution. In the case of a Poisson distribution, the variance is equal to the mean, i.e.

${\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}(n)=\overline{n}}$

${\displaystyle g^2(0)=1}$.

A second-order correlation coefficient of 1 means that photons in coherent states are uncorrelated.

Hanbury Brown and Twiss studied the correlation behavior of photons emitted from a thermal, incoherent source described byBose–Einstein statistics. The variance of the Bose–Einstein distribution is

${\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}(\mathrm{n})=\overline{n}+{\overline{n}}^2}$

${\displaystyle g^2(0)=2}$.

This corresponds to the correlation measurements of Hanbury Brown and Twiss, and illustrates that photons in incoherent Bose–Einstein states are correlated or bunched.

Quanta that obeyFermi–Dirac statistics are anti-correlated. In this case the variance is

${\displaystyle \mathrm{V}\mathrm{a}\mathrm{r}(n)=\overline{n}-{\overline{n}}^2}$

${\displaystyle g^2(0)=0}$.

Anti-correlation is characterized by a second-order correlation coefficient =0.

Roy J. Glauber's work was prompted by the results of Hanbury-Brown and Twiss that produced long-range (hundreds or thousands of miles) first-order interference patterns through the use of intensity fluctuations (lack of second order coherence), with narrow band filters (partial first order coherence) at each detector. (One can imagine, over very short durations, a near-instantaneous interference pattern from the two detectors, due to the narrow band filters, that dances around randomly due to the shifting relative phase difference. With a coincidence counter, the dancing interference pattern would be stronger at times of increased intensity [common to both beams], and that pattern would be stronger than the background noise.) Almost all of optics had been concerned with first order coherence. The Hanbury-Brown and Twiss results prompted Glauber to look at higher order coherence, and he came up with a complete quantum-theoretic description of coherence to all orders in the electromagnetic field (and a quantum-theoretic description of signal-plus-noise). He coined the termcoherent state and showed that they are produced when a classical electric current interacts with the electromagnetic field.

Atα ≫ 1, from Figure 5, simple geometry givesΔθ |α | = 1/2. From this, it appears that there is a tradeoff between number uncertainty and phase uncertainty,ΔθΔn = 1/2, which is sometimes interpreted as a number-phase uncertainty relation; but this is not a formal strict uncertainty relation: there is no uniquely defined phase operator in quantum mechanics.^{[14][15][16][17][18][19][20][21]}

To find the wavefunction of the coherent state, the minimal uncertainty Schrödinger wave packet, it is easiest to start with the Heisenberg picture of thequantum harmonic oscillator for the coherent state${\displaystyle |\alpha ⟩}$. Note that

${\displaystyle a(t)|\alpha ⟩=e^{-i\omega t}a(0)|\alpha ⟩}$

The coherent state is an eigenstate of the annihilation operator in theHeisenberg picture.

It is easy to see that, in theSchrödinger picture, the same eigenvalue

${\displaystyle \alpha (t)=e^{-i\omega t}\alpha (0)}$

occurs,

${\displaystyle a|\alpha (t)⟩=\alpha (t)|\alpha (t)⟩}$.

In the coordinate representations resulting from operating by${\displaystyle ⟨x|}$, this amounts to thedifferential equation,

${\displaystyle \sqrt{\frac{m\omega }{2\hslash }}\left(x+\frac{\hslash }{m\omega }\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right){\psi }^{\alpha }(x,t)=\alpha (t){\psi }^{\alpha }(x,t),}$

which is easily solved to yield

${\displaystyle {\psi }^{(\alpha )}(x,t)={\left(\frac{m\omega }{\pi \hslash }\right)}^{1/4}\exp (-\frac{m\omega }{2\hslash }{\left(x-\sqrt{\frac{2\hslash }{m\omega }}\mathrm{\Re }[\alpha (t)]\right)}^2+i\sqrt{\frac{2m\omega }{\hslash }}\mathrm{\Im }[\alpha (t)]x+i\theta (t)),}$

whereθ(t) is a yet undetermined phase, to be fixed by demanding that the wavefunction satisfies the Schrödinger equation.

It follows that

${\displaystyle \theta (t)=-\frac{\omega t}{2}+\frac{|\alpha (0){|}^2\sin (2\omega t-2\sigma )}{2},\text{where}\alpha (0)\equiv |\alpha (0)|\exp (i\sigma ),}$

so thatσ is the initial phase of the eigenvalue.

The mean position and momentum of this "minimal Schrödinger wave packet"ψ^(α) are thusoscillating just like a classical system,

The probability density remains a Gaussian centered on this oscillating mean,

${\displaystyle |{\psi }^{(\alpha )}(x,t){|}^2=\sqrt{\frac{m\omega }{\pi \hslash }}{e}^{-\frac{m\omega }{\hslash }{\left(x-⟨\hat{x}(t)⟩\right)}^2}.}$

The canonical coherent states described so far have three properties that are mutually equivalent, since each of them completely specifies the state${\displaystyle |\alpha ⟩}$, namely,

1. They are eigenvectors of theannihilation operator:  ${\displaystyle \hat{a}|\alpha ⟩=\alpha |\alpha ⟩}$.
2. They are obtained from the vacuum by application of a unitarydisplacement operator:  ${\displaystyle |\alpha ⟩=e^{\alpha {\hat{a}}^{†}-{\alpha }^{\ast }\hat{a}}|0⟩=D(\alpha )|0⟩}$.
3. They are states of (balanced) minimal uncertainty:  ${\displaystyle \mathrm{\Delta }X=\mathrm{\Delta }P=\sqrt{\frac{\hslash }{2}}}$.

Each of these properties may lead to generalizations, in general different from each other (see the article "Coherent states in mathematical physics" for some of these). We emphasize that coherent states have mathematical features that are very different from those of aFock state; for instance, two different coherent states are not orthogonal,

${\displaystyle ⟨\beta |\alpha ⟩=e^{-\frac{1}{2}(|\beta {|}^2+|\alpha {|}^2-2{\beta }^{\ast }\alpha )}\ne \delta (\alpha -\beta )}$

(linked to the fact that they are eigenvectors of the non-self-adjoint annihilation operatorâ).

Thus, if the oscillator is in the quantum state${\displaystyle |\alpha ⟩}$ it is also with nonzero probability in the other quantum state${\displaystyle |\beta ⟩}$ (but the farther apart the states are situated in phase space, the lower the probability is). However, since they obey a closure relation, any state can be decomposed on the set of coherent states. They hence form anovercomplete basis, in which one can diagonally decompose any state. This is the premise for theGlauber–Sudarshan P representation.

This closure relation can be expressed by the resolution of the identity operatorI in thevector space of quantum states,

${\displaystyle \frac{1}{\pi }\int |\alpha ⟩⟨\alpha |d^2\alpha =I{d}^2\alpha \equiv d\mathrm{\Re }(\alpha )d\mathrm{\Im }(\alpha ).}$

This resolution of the identity is intimately connected to theSegal–Bargmann transform.

Another peculiarity is that${\displaystyle {\hat{a}}^{†}}$ has no eigenket (whileâ has no eigenbra). The following equality is the closest formal substitute, and turns out to be useful for technical computations,^[22]

${\displaystyle a^{†}|\alpha ⟩⟨\alpha |=\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\alpha }+{\alpha }^{\ast }\right)|\alpha ⟩⟨\alpha |.}$

This last state is known as an "Agarwal state" or photon-added coherent state and denoted as${\displaystyle |\alpha ,1⟩.}$

Normalized Agarwal states of ordern can be expressed as${\displaystyle |\alpha ,n⟩=[{{\hat{a}}^{†}]}^n|\alpha ⟩/\Vert [{{\hat{a}}^{†}]}^n|\alpha ⟩\Vert .}$^[23]

The above resolution of the identity may be derived (restricting to one spatial dimension for simplicity) by taking matrix elements between eigenstates of position,${\displaystyle ⟨x|\cdots |y⟩}$, on both sides of the equation. On the right-hand side, this immediately givesδ(x-y). On the left-hand side, the same is obtained by inserting

${\displaystyle {\psi }^{\alpha }(x,t)=⟨x|\alpha (t)⟩}$

from the previous section (time is arbitrary), then integrating over${\displaystyle \mathrm{\Im }(\alpha )}$ using theFourier representation of the delta function, and then performing aGaussian integral over${\displaystyle \mathrm{\Re }(\alpha )}$.

In particular, the Gaussian Schrödinger wave-packet state follows from the explicit value

${\displaystyle ⟨x|\alpha ⟩=\frac{1}{{\pi }^{1/4}}{e}^{-\frac{1}{2}(x-\sqrt{2}\mathrm{\Re }(\alpha ){)}^2+ix\sqrt{2}\mathrm{\Im }(\alpha )-i\mathrm{\Re }(\alpha )\mathrm{\Im }(\alpha )}.}$

The resolution of the identity may also be expressed in terms of particle position and momentum. For each coordinate dimension (using an adapted notation with new meaning for${\displaystyle x}$),

${\displaystyle |\alpha ⟩\equiv |x,p⟩x\equiv ⟨\hat{x}⟩p\equiv ⟨\hat{p}⟩}$

the closure relation of coherent states reads

${\displaystyle I=\int |x,p⟩⟨x,p|\frac{\mathrm{d}x\mathrm{d}p}{2\pi \hslash }.}$

This can be inserted in any quantum-mechanical expectation value, relating it to some quasi-classical phase-space integral and explaining, in particular, the origin of normalisation factors${\displaystyle (2\pi \hslash {)}^{-1}}$ for classicalpartition functions, consistent with quantum mechanics.

In addition to being an exact eigenstate of annihilation operators, a coherent state is anapproximate common eigenstate of particle position and momentum. Restricting to one dimension again,

${\displaystyle \hat{x}|x,p⟩\approx x|x,p⟩\hat{p}|x,p⟩\approx p|x,p⟩}$

The error in these approximations is measured by theuncertainties of position and momentum,

${\displaystyle ⟨x,p|{\left(\hat{x}-x\right)}^2|x,p⟩={\left(\mathrm{\Delta }x\right)}^2⟨x,p|{\left(\hat{p}-p\right)}^2|x,p⟩={\left(\mathrm{\Delta }p\right)}^2.}$

A single mode thermal coherent state^[24] is produced by displacing a thermal mixed state inphase space, in direct analogy to the displacement of the vacuum state in view of generating a coherent state. Thedensity matrix of a coherent thermal state in operator representation reads

${\displaystyle \rho (\alpha ,\beta )=\frac{1}{Z}D(\alpha )e^{-\hslash \beta \omega a^{†}a}{D}^{†}(\alpha ),}$

where${\displaystyle D(\alpha )}$ is thedisplacement operator, which generates the coherent state${\displaystyle D(\alpha )|0⟩=|\alpha ⟩}$ with complex amplitude${\displaystyle \alpha }$, and${\displaystyle \beta =1/(k_{B}T)}$ . Thepartition function is equal to

${\displaystyle Z=\text{tr}\left\{{\displaystyle e^{-\hslash \beta \omega a^{†}a}}\right\}=\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\beta \hslash \omega }=\frac{1}{1-e^{-\hslash \beta \omega }}.}$

Using the expansion of the identity operator inFock states,${\displaystyle I\equiv \sum _{n=0}^{\mathrm{\infty }}|n⟩⟨n|}$, thedensity operator definition can be expressed in the following form

${\displaystyle \rho (\alpha ,\beta )=\frac{1}{Z}\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\hslash \beta \omega }D(\alpha )|n⟩⟨n|D^{†}(\alpha )=\frac{1}{Z}\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\hslash \beta \omega }|\alpha ,n⟩⟨\alpha ,n|,}$

where${\displaystyle |\alpha ,n⟩}$ stands for the displacedFock state. We remark that if temperature goes to zero we have

${\displaystyle \underset{\beta \to \mathrm{\infty }}{lim}\rho (\alpha ,\beta )=\underset{\beta \to \mathrm{\infty }}{lim}\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\hslash \beta \omega }(1-e^{-\hslash \beta \omega })|\alpha ,n⟩⟨\alpha ,n|=\sum _{n=0}^{\mathrm{\infty }}{\delta }_{n,0}|\alpha ,n⟩⟨\alpha ,n|=|\alpha ,0⟩⟨\alpha ,0|,}$

which is thedensity matrix for a coherent state. The average number ofphotons in that state can be calculated as below

${\displaystyle ⟨n⟩=\text{Tr}\{\rho a^{†}a\}=\frac{1}{Z}\text{Tr}\{D^{†}(\alpha )a^{†}D(\alpha )D^{†}(\alpha )aD(\alpha )e^{-\beta \hslash \omega a^{†}a}\}=\frac{1}{Z}\text{Tr}\{(a^{†}+{\alpha }^{\ast })(a+\alpha )e^{-\beta \hslash \omega a^{†}a}\}=}$

${\displaystyle =|\alpha {|}^2\frac{1}{Z}\text{Tr}\{e^{-\beta \hslash \omega a^{†}a}\}+\frac{1}{Z}\text{Tr}\{a^{†}a{e}^{-\beta \hslash \omega a^{†}a}\}=|\alpha {|}^2+\frac{1}{Z}\sum _{n=0}^{\mathrm{\infty }}n{e}^{-n\beta \hslash \omega },}$

where for the last term we can write

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}n{e}^{-n\beta \hslash \omega }=-\frac{\mathrm{\partial }}{\mathrm{\partial }(\beta \hslash \omega )}\left(\sum _{n=0}^{\mathrm{\infty }}{e}^{-n\beta \hslash \omega }\right)=\frac{e^{-\beta \hslash \omega }}{(1-e^{-\beta \hslash \omega }{)}^2}.}$

As a result, we find

${\displaystyle ⟨n⟩=|\alpha {|}^2+⟨n{⟩}_{\text{th}},}$

where${\displaystyle ⟨n{⟩}_{\text{th}}}$ is the average of thephoton number calculated with respect to the thermal state. Here we have defined, for ease of notation,

${\displaystyle ⟨O{⟩}_{\text{th}}=\frac{1}{Z}\text{tr}\{e^{-\beta \hslash \omega a^{†}a}O\},}$

and we write explicitly

${\displaystyle ⟨n{⟩}_{\text{th}}=\frac{1}{e^{\beta \hslash \omega }-1}.}$

In the limit${\displaystyle \beta \to \mathrm{\infty }}$ we obtain${\displaystyle ⟨n⟩=|\alpha {|}^2}$, which is consistent with the expression for thedensity matrix operator at zero temperature. Likewise, the photon numbervariance can be evaluated as

${\displaystyle {\sigma }^2=⟨n^2⟩-⟨n{⟩}^2={\sigma }_{\text{th}}^2+|\alpha {|}^2\left(1+2⟨a^{†}a{⟩}_{\text{th}}\right),}$

with${\displaystyle {\sigma }_{\text{th}}^2=⟨n^2{⟩}_{\text{th}}-⟨n{⟩}_{\text{th}}^2}$. We deduce that the second moment cannot be uncoupled to the thermal and the quantum distribution moments, unlike the average value (first moment). In that sense, the photon statistics of the displaced thermal state is not described by the sum of thePoisson statistics and theBoltzmann statistics. The distribution of the initial thermal state in phase space broadens as a result of the coherent displacement.

• ABose–Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state.^[25] In a thermodynamic system, the ground state becomes macroscopically occupied below a critical temperature — roughly when the thermal de Broglie wavelength is longer than the interatomic spacing. Superfluidity in liquid Helium-4 is believed to be associated with the Bose–Einstein condensation in an ideal gas. But⁴He has strong interactions, and the liquid structure factor (a 2nd-order statistic) plays an important role. The use of a coherent state to represent the superfluid component of⁴He provided a good estimate of the condensate / non-condensate fractions in superfluidity, consistent with results of slow neutron scattering.^[26][27][28] Most of the special superfluid properties follow directly from the use of a coherent state to represent the superfluid component — that acts as a macroscopically occupied single-body state with well-defined amplitude and phase over the entire volume. (The superfluid component of⁴He goes from zero at the transition temperature to 100% at absolute zero. But the condensate fraction is about 6%^[29] at absolute zero temperature, T=0K.)
• Early in the study of superfluidity,Oliver Penrose andLars Onsager proposed a metric ("order parameter") for superfluidity.^[30] It was represented by a macroscopic factored component (a macroscopic eigenvalue) in the first-order reduced density matrix. Later,C. N. Yang^[31] proposed a more generalized measure of macroscopic quantum coherence, called "Off-Diagonal Long-Range Order" (ODLRO),^[31] that included fermion as well as boson systems. ODLRO exists whenever there is a macroscopically large factored component (eigenvalue) in a reduced density matrix of any order. Superfluidity corresponds to a large factored component in the first-order reduced density matrix. (And, all higher order reduced density matrices behave similarly.) Superconductivity involves a large factored component in the 2nd-order ("Cooper electron-pair") reduced density matrix.
• The reduced density matrices used to describe macroscopic quantum coherence in superfluids are formally the same as the correlation functions used to describe orders of coherence in radiation. Both are examples of macroscopic quantum coherence. The macroscopically large coherent component, plus noise, in the electromagnetic field, as given by Glauber's description of signal-plus-noise, is formally the same as the macroscopically large superfluid component plus normal fluid component in the two-fluid model of superfluidity.
• Every-day electromagnetic radiation, such as radio and TV waves, is also an example of near coherent states (macroscopic quantum coherence). That should "give one pause" regarding the conventional demarcation between quantum and classical.
• The coherence in superfluidity should not be attributed to any subset of helium atoms; it is a kind of collective phenomena in which all the atoms are involved (similar to Cooper-pairing in superconductivity, as indicated in the next section).

• Electrons are fermions, but when they pair up intoCooper pairs they act as bosons, and so can collectively form a coherent state at low temperatures. This pairing is not actually between electrons, but in the states available to the electrons moving in and out of those states.^[32] Cooper pairing refers to the first model for superconductivity.^[33]
• These coherent states are part of the explanation of effects such as theQuantum Hall effect in low-temperaturesuperconducting semiconductors.

• For a quantum system withangular momentum operator${\displaystyle \mathbf{J}=(J_x,J_y,J_z)}$ and angular momentum quantum number${\displaystyle j,}$angular momentum coherent states can be defined as those states which minimize the quantity$${\displaystyle (\mathrm{\Delta }{J}_x{)}^2+(\mathrm{\Delta }{J}_y^2)+(\mathrm{\Delta }{J}_z{)}^2,}$$ i.e. the sums of the uncertainties of the three Cartesian components of angular momentum.^[34] A state${\displaystyle |\psi ⟩}$ minimizes this quantity if and only if there exists aunit vector${\displaystyle \mathbf{n}}$ such that$${\displaystyle (\mathbf{n}\cdot \mathbf{J})|\psi ⟩=\hslash j|\psi ⟩.}$$

• According to Gilmore and Perelomov, who showed it independently, the construction of coherent states may be seen as a problem ingroup theory, and thus coherent states may be associated to groups different from theHeisenberg group, which leads to the canonical coherent states discussed above.^{[35][36][37][38]} Moreover, these coherent states may be generalized toquantum groups. These topics, with references to original work, are discussed in detail inCoherent states in mathematical physics.

• Inquantum field theory andstring theory, a generalization of coherent states to the case where infinitely manydegrees of freedom are used to define avacuum state with a differentvacuum expectation value from the original vacuum.
• In one-dimensional many-body quantum systems with fermionic degrees of freedom, low energy excited states can be approximated as coherent states of a bosonic field operator that creates particle-hole excitations. This approach is calledbosonization.
• The Gaussian coherent states of nonrelativistic quantum mechanics can be generalized torelativistic coherent states of Klein-Gordon and Dirac particles.^[39][40][41]
• Coherent states have also appeared in works onloop quantum gravity or for the construction of (semi)classical canonical quantum general relativity.^[42][43]

• Coherent states in mathematical physics
• Quantum field theory
• Quantum optics
• Quantum amplifier
• Electromagnetic field
• Degree of coherence
• Glauber multiple scattering theory

• Quantum states of the light field
• Glauber States: Coherent states of Quantum Harmonic Oscillator
• Measure a coherent state with photon statistics interactive

• Quantum states

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