Coherent State

In physics, specifically in quantum mechanics, a coherent state is the specific quantum state of the quantum harmonic oscillator, often described as a state which has dynamics most closely resembling the oscillatory behavior of a classical harmonic oscillator. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926, while searching for solutions of the Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator (and hence the coherent states) arise in the quantum theory of a wide range of physical systems.J.R. Klauder and B. Skagerstam, ''Coherent States'', World Scientific, Singapore, 1985. For instance, a coherent state describes the oscillating motion of a particle confined in a quadratic potential well (for an early reference, see e.g.
Schiff's textbook). 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 as ''eigenvectors of the lowering operator'' and forming an '' overcomplete'' family, were introduced in the early papers of John R. Klauder, e. g.
In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories, coherent states were introduced by the work of Roy J. Glauber in 1963 and are also known as Glauber states.

The concept of coherent states has been considerably abstracted; it has become a major topic in mathematical physics and in applied mathematics, with applications ranging from quantization to signal processing and image processing (see Coherent states in mathematical physics). For this reason, the coherent states associated to the quantum harmonic oscillator are sometimes referred to as ''canonical coherent states'' (CCS), ''standard coherent states'', ''Gaussian'' states, or oscillator states.

Coherent states in quantum optics

In quantum optics the coherent state refers to a state of the quantized electromagnetic field, etc.J-P. Gazeau, ''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009. that describes a maximal kind of coherence and a classical kind of behavior. Erwin Schrödinger derived it as a "minimum uncertainty" Gaussian wavepacket in 1926, searching for solutions of the Schrödinger equation that satisfy the correspondence principle. It is a minimum 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 the energy eigenstates of the system, the time evolution of a coherent state is concentrated along the classical trajectories. 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 other bosonic quantum field theories.

While minimum uncertainty Gaussian wave-packets had been well-known, they did not attract full attention until Roy J. Glauber, in 1963, provided a complete quantum-theoretic description of coherence in the electromagnetic field. In this respect, the concurrent contribution of E.C.G. Sudarshan should not be omitted, (there is, however, a note in Glauber's paper that reads: "Uses of these states as generating functions for the $n$-quantum states have, however, been made by J. Schwinger).
Glauber was prompted to do this to provide a description of the Hanbury-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, see Quantum mechanical description.)

In classical optics, light is thought of as electromagnetic waves radiating from a source. Often, coherent laser light is thought of as light that is emitted by many such sources that are in phase. Actually, the picture of one photon being in-phase with another is not valid in quantum theory. Laser radiation is produced in a resonant cavity where the resonant frequency of the cavity is the same as the frequency associated with the atomic electron transitions providing energy flow into the field. As energy in the resonant mode builds up, the probability for stimulated emission, in that mode only, increases. That is a positive feedback loop in which the amplitude in the resonant mode increases exponentially until some non-linear effects limit it. As a counter-example, a light 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 (see thermal light). In a laser, however, light is emitted into a resonant mode, and that mode is highly coherent. Thus, laser light is idealized as a coherent state. (Classically we describe such a state by an electric 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 of beam 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; 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 statistical mixed state. Thermal light can be described as a statistical mixture of coherent states, and the typical way of defining nonclassical light is that it cannot be described as a simple statistical mixture of coherent states.

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. The Fock 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 the canonically conjugate coordinates, position and momentum, and the relative uncertainty in phase efined heuristically">heuristic.html" ;"title="efined heuristic">efined heuristicallyand amplitude are roughly equal—and small at high amplitude.

Quantum mechanical definition

Mathematically, a coherent state $">\alpha\rangle$ is defined to be the (unique) eigenstate of the \alpha\rangle=\alpha, \alpha\rangle ~.

Since is not hermitian operator, hermitian, is, in general, a complex number. Writing $\alpha = , \alpha, e^,$ , , and are called the amplitude and phase of the state $, \alpha\rangle$.

The state $, \alpha\rangle$ is called a ''canonical coherent state'' in the literature, since there are many other types of coherent states, as can be seen in the companion article Coherent states in mathematical physics.

Physically, this formula means that a coherent state remains unchanged by the annihilation of field excitation or, say, a particle. An eigenstate of the annihilation operator has a Poissonian number distribution when expressed in a basis of energy eigenstates, as shown below. A Poisson distribution is a necessary and sufficient condition that all detections are statistically independent. Contrast this to a single-particle state ($, 1\rangle$ 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'', and , normally called ''field quadratures'' in quantum optics.
(See Nondimensionalization.) These operators are related to the position and momentum operators of a mass on a spring with constant ,
:$=\sqrt\ \hat\text\quad =\sqrt\ \hat\text\quad \quad \text\omega \equiv \sqrt~.$

For an optical field,
:$~E_ = \left(\frac \right)^ \!\!\!\cos(\theta) X \qquad \text \qquad ~E_ = \left(\frac\right)^ \!\!\!\sin(\theta) X~$
are the real and imaginary components of the mode of the electric field inside a cavity of volume $V$.

With these (dimensionless) operators, the Hamiltonian of either system becomes
:=\hbar \omega \left(^+^ \right)\text
\qquad\text\qquad
\left , \rightequiv -=\frac\,.

Erwin Schrödinger was searching for the most classical-like states when he first introduced minimum uncertainty Gaussian wave-packets. The quantum state of the harmonic oscillator that minimizes the uncertainty relation with uncertainty equally distributed between and satisfies the equation
:$\left( -\langle \rangle \right)\,, \alpha \rangle = -i\left( -\langle\rangle \right)\, , \alpha\rangle \text$
or, equivalently,
:$\left( +i \right)\, \left, \alpha\right\rangle = \left\langle +i \right\rangle \, \left, \alpha\right\rangle ~,$
and hence
:$\langle \alpha \! \mid \left( -\langle X\rangle \right)^2+ \left( -\langle P\rangle \right)^2 \mid \!\alpha\rangle = 1/2 ~.$

Thus, given , Schrödinger found that ''the minimum uncertainty states for the linear harmonic oscillator are the eigenstates of'' .
Since ''â'' is , 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 name ''coherent state'' took hold after Glauber's work.

If the uncertainty is minimized, but not necessarily equally balanced between and , the state is called a squeezed 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 phase and amplitude , ''α'', given by the eigenvalue ''α'' (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 diameter . 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 at 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 $, 0\rangle$ is just the coherent state with =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 $, \alpha\rangle$ does not refer to a Fock state. For example, when , one should not mistake $, 1\rangle$ for the single-photon Fock state, which is also denoted $, 1\rangle$ in its own notation. The expression $, \alpha\rangle$ with represents a Poisson distribution of number states $, n\rangle$ with a mean photon number of unity.

The formal solution of the eigenvalue equation is the vacuum state displaced to a location in phase space, i.e., it is obtained by letting the unitary displacement operator operate on the vacuum,
:$, \alpha\rangle=e^, 0\rangle = D(\alpha), 0\rangle$,
where and .

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,
:$, \alpha\rangle =e^\sum_^, n\rangle =e^e^e^, 0\rangle ~,$
where $, n\rangle$ are energy (number) eigenvectors of the Hamiltonian
:$H =\hbar \omega \left( \hat a^\dagger \hat a + \frac 12\right)~.$

For the corresponding Poissonian distribution, the probability of detecting photons is
:$P(n)= , \langle n, \alpha \rangle , ^2 =e^\frac ~.$

Similarly, the average photon number in a coherent state is
:$~\langle n \rangle =\langle \hat a^\dagger \hat a \rangle =, \alpha, ^2~$
and the variance is
:$~(\Delta n)^2=\left(\hat a^\dagger \hat a\right)= , \alpha, ^2~$.

That is, the standard deviation of the number detected goes like the square root of the number detected. So in the limit of large , 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 (see degree 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 all . 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 $g^2(0)$ 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.

:$~g^2(0) =1+\frac = 1+\frac$

In Glauber's development, it is seen that the coherent states are distributed according to a Poisson distribution. In the case of a Poisson distribution, the variance is equal to the mean, i.e.
:$(n) =\bar$
:$g^2(0) = 1$.

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 by Bose–Einstein statistics. The variance of the Bose-Einstein distribution is
:$=\bar+\bar^2$
:$g^2(0) = 2$.

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 obey Fermi-Dirac statistics are anti-correlated. In this case the variance is
:$(n)=\bar-\bar^2$
:$g^2(0) = 0$.

Anti-correlation id 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