Editing Coherent state (section)

== Quantum mechanical definition ==
Mathematically, a coherent state <math>|\alpha\rangle</math> is defined to be the (unique) eigenstate of the [[annihilation operator]] {{math|''â''}} with corresponding eigenvalue {{mvar|α}}. Formally, this reads,
:<math>\hat{a}|\alpha\rangle=\alpha|\alpha\rangle  ~.</math>

Since {{math|''â''}} is not [[hermitian operator|hermitian]], {{mvar|α}} is, in general, a [[complex number]]. Writing <math>\alpha = |\alpha|e^{i\theta},</math> |{{mvar|α}}| and  {{mvar|θ}} are called the amplitude and phase of the state <math>|\alpha\rangle</math>.

The state <math>|\alpha\rangle</math> 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 charged 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 (<math>|1\rangle</math> [[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'', {{mvar|X}} and {{mvar|P}}, normally called ''field quadratures'' in quantum optics.
(See [[Nondimensionalization]].) These operators are related to the position and momentum operators of a mass {{mvar|m}} on a spring with constant {{mvar|k}},
:<math> {P}=\sqrt{\frac{1}{2\hbar m\omega }}\ \hat{p}\text{,}\quad  {X}=\sqrt{\frac{m\omega }{2\hbar }}\ \hat{x}\text{,}\quad \quad \text{where }\omega \equiv \sqrt{k/m}~.</math>

[[Image:photon numbers coherent state.jpg|thumb|300px|Figure 4:  The probability of detecting n photons, the photon number distribution, of the coherent state in Figure 3. As is necessary for a [[Poissonian distribution]] the mean photon number is equal to the [[variance]] of the photon number distribution. Bars refer to theory, dots to experimental values.]]

For an [[optical field]], 
:<math>~E_{\rm R} =
\left(\frac{2\hbar\omega}{\epsilon_0 V}
\right)^{1/2} \!\!\!\cos(\theta) X \qquad \text{and} \qquad ~E_{\rm I} = 
\left(\frac{2\hbar\omega}{\epsilon_0 V}\right)^{1/2} \!\!\!\sin(\theta) X~</math>
are the real and imaginary components of the mode of the electric field inside a cavity of volume <math>V</math>.<ref>{{cite web |url=https://www.sjsu.edu/faculty/watkins/fieldenergy.htm |title=The Energy Density of Fields |website=www.sjsu.edu |url-status=dead |archive-url=https://web.archive.org/web/20160102194602/http://www.sjsu.edu/faculty/watkins/fieldenergy.htm |archive-date=2016-01-02}} </ref>

With these (dimensionless) operators, the Hamiltonian of either system becomes
:<math>{H}=\hbar \omega \left({P}^{2}+{X}^{2} \right)\text{,}
\qquad\text{with}\qquad
\left[ {X},{P} \right]\equiv {XP}-{PX}=\frac{i}{2}\,{I}.</math>

[[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 {{mvar|X}} and {{mvar|P}} satisfies the equation
:<math>\left( {X}-\langle {X}\rangle \right)\,|\alpha \rangle  = -i\left( {P}-\langle{P}\rangle \right)\, |\alpha\rangle \text{,}</math>
or, equivalently,
:<math> \left( {X}+i{P} \right)\, \left|\alpha\right\rangle  = \left\langle {X}+i{P} \right\rangle \, \left|\alpha\right\rangle ~,</math>
and hence 
:<math> \langle \alpha \! \mid   \left( {X}-\langle X\rangle \right)^2+  \left( {P}-\langle P\rangle \right)^2   \mid \!\alpha\rangle  = 1 ~.</math>

Thus, given {{math| (∆''X''−∆''P'')<sup>2</sup> ≥ 0}}, Schrödinger found that ''the minimum uncertainty states for the linear harmonic oscillator are the eigenstates of'' {{math|(''X'' + ''iP'')}}. 
Since ''â'' is {{math|(''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 name ''coherent state'' took hold after Glauber's work.

If the uncertainty is minimized, but not necessarily equally balanced between {{mvar|X}} and {{mvar|P}}, 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 {{mvar|θ}} 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 {{Fraction|1|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.

[[Image:Coherent state2.png|thumb|300px|Figure 5: Phase space plot of a coherent state. This shows that the uncertainty in a coherent state is equally distributed in all directions. The horizontal and vertical axes are the X and P quadratures of the field, respectively (see text). The red dots on the x-axis trace out the boundaries of the quantum noise in Figure 1. For more detail, see the corresponding figure of the [[phase space formulation]].  ]]

Since the uncertainty (and hence measurement noise) stays constant at  {{Fraction|1|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 <math>|0\rangle</math> is just the coherent state with {{mvar|α}}=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  <math>|\alpha\rangle</math> does not refer to a [[Fock state]]. For example, when {{math|1=''α'' = 1}}, one should not mistake <math>|1\rangle</math> for the single-photon Fock state, which is also denoted <math>|1\rangle</math> in its own notation. The expression <math>|\alpha\rangle</math> with {{math|1=''α'' = 1}} represents a Poisson distribution of number states <math>|n\rangle</math> with a mean photon number of unity.

The formal solution of the eigenvalue equation is the vacuum state displaced to a location {{mvar|α}}  in phase space, i.e., it is obtained by letting the unitary [[displacement operator]] {{math|''D''(''α'')}} operate on the vacuum,
:<math>|\alpha\rangle=e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle</math>,
where {{math|1=''â'' = ''X'' + ''iP''}}  and {{math|1=''â''<sup>†</sup> = ''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,
:<math>|\alpha\rangle =e^{-{|\alpha|^2\over2}}\sum_{n=0}^{\infty}{\alpha^n\over\sqrt{n!}}|n\rangle =e^{-{|\alpha|^2\over2}}e^{\alpha\hat a^\dagger}e^{-{\alpha^* \hat a}}|0\rangle =e^{\alpha \hat a^\dagger - \alpha^*\hat a}|0\rangle = D(\alpha)|0\rangle ~,</math>
where <math> |n\rangle </math> are energy (number) eigenvectors of the Hamiltonian 
:<math>H =\hbar \omega \left( \hat a^\dagger \hat a + \frac 12\right)~,</math>

and the final equality derives from the [[Baker-Campbell-Hausdorff formula]]. For the corresponding [[Poissonian]] distribution, the probability of detecting {{mvar|n}} photons is
:<math>P(n)= |\langle n|\alpha \rangle |^2 =e^{-\langle n \rangle}\frac{\langle n \rangle^n}{n!}    ~.</math>

Similarly, the average photon number in a coherent state is 
:<math>~\langle n \rangle =\langle \hat a^\dagger \hat a \rangle =|\alpha|^2~</math>
and the variance is 
:<math>~(\Delta n)^2={\rm Var}\left(\hat a^\dagger \hat a\right)=
|\alpha|^2~</math>.

That is, the standard deviation of the number detected goes like the square root of the number detected. So in the limit of large {{mvar|α}}, 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 {{mvar|n}}.  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 <math> g^2(0)</math> 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.<ref>Pearsall, Thomas P., "Quantum Photonics, 2nd ed." Springer Nature, Cham, Switzerland, 2020, pp. 287 ff</ref>

:<math>~g^2(0) =1+\frac{{\rm Var}\left(\hat a^\dagger \hat a\right)-\langle \hat a^\dagger \hat a \rangle}{(\langle \hat a^\dagger \hat a \rangle)^2} = 1+\frac{{\rm Var}(n)-\bar{n}}{\bar{n}^2} </math>

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. 
:<math>{\rm Var}(n) =\bar{n}</math>
:<math>g^2(0) = 1</math>.

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 
:<math>{\rm Var(n)}=\bar{n}+\bar{n}^2</math>
:<math>g^2(0) = 2</math>.

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
:<math>{\rm Var}(n)=\bar{n}-\bar{n}^2</math>
:<math>g^2(0) = 0</math>.

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 term ''coherent state'' and showed that they are produced when a classical electric current interacts with the electromagnetic field.

At  {{math|''α'' ≫ 1}},  from Figure 5, simple geometry gives ''Δθ'' |''α'' | = 1/2. 
From this,  it appears that there is a tradeoff between number uncertainty and phase uncertainty,  ''Δθ''  ''Δn''  = 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.<ref>L. Susskind and J. Glogower, Quantum mechanical phase and time operator,''Physics'' '''1''' (1963) 49.</ref> 
<ref>{{cite journal | last1=Carruthers | first1=P. | last2=Nieto | first2=Michael Martin | s2cid=121002585 | title=Phase and Angle Variables in Quantum Mechanics | journal=Reviews of Modern Physics | publisher=American Physical Society (APS) | volume=40 | issue=2 | date=1968-04-01 | issn=0034-6861 | doi=10.1103/revmodphys.40.411 | pages=411–440| bibcode=1968RvMP...40..411C }}</ref> 
<ref>{{cite journal | last1=Barnett | first1=S.M. | last2=Pegg | first2=D.T. | title=On the Hermitian Optical Phase Operator | journal=Journal of Modern Optics | publisher=Informa UK Limited | volume=36 | issue=1 | year=1989 | issn=0950-0340 | doi=10.1080/09500348914550021 | pages=7–19| bibcode=1989JMOp...36....7B }}</ref>   
<ref>{{cite journal | last1=Busch | first1=P. | last2=Grabowski | first2=M. | last3=Lahti | first3=P.J. | title=Who Is Afraid of POV Measures? Unified Approach to Quantum Phase Observables | journal=Annals of Physics | publisher=Elsevier BV | volume=237 | issue=1 | year=1995 | issn=0003-4916 | doi=10.1006/aphy.1995.1001 | pages=1–11| bibcode=1995AnPhy.237....1B }}</ref> 
<ref>{{cite journal | last=Dodonov | first=V V | title='Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years | journal=Journal of Optics B: Quantum and Semiclassical Optics | publisher=IOP Publishing | volume=4 | issue=1 | date=2002-01-08 | issn=1464-4266 | doi=10.1088/1464-4266/4/1/201 | pages=R1–R33}}</ref>
<ref>V.V. Dodonov and V.I.Man'ko (eds),   ''Theory of Nonclassical States of Light'', Taylor \& Francis, London, New York, 2003.</ref> 
<ref>{{cite journal | last=Vourdas | first=A | title=Analytic representations in quantum mechanics | journal=Journal of Physics A: Mathematical and General | publisher=IOP Publishing | volume=39 | issue=7 | date=2006-02-01 | issn=0305-4470 | doi=10.1088/0305-4470/39/7/r01 | pages=R65–R141}}</ref> 
<ref>J-P. Gazeau,''Coherent States in Quantum Physics'', Wiley-VCH, Berlin, 2009.</ref>

== The wavefunction of a coherent state ==
[[File:Coherent state gif.gif|thumb|right|450px|Coherent state dynamics for <math>\alpha = \sqrt{10}</math>, in units of the harmonic oscillator length <math>x_0=\sqrt{\hbar/m\omega}</math>, showing the probability density <math>|\psi(x,t)|^2</math> and the quantum phase (color).]]
To find the wavefunction of the coherent state, the minimal uncertainty Schrödinger wave packet, it is easiest to start with the Heisenberg picture of the [[quantum harmonic oscillator]] for the coherent state <math>|\alpha\rangle</math>.  Note that
: <math>~a(t)|\alpha\rangle =e^{-i\omega t}a(0)|\alpha\rangle</math>

The coherent state is an eigenstate of the annihilation operator in the [[Heisenberg picture]].

It is easy to see that, in the [[Schrödinger picture]], the same eigenvalue 
:<math>~ \alpha(t) = e^{-i\omega t}\alpha(0)~</math> 
occurs,
: <math>~a|\alpha(t)\rangle=\alpha(t)|\alpha(t)\rangle</math>.

In the coordinate representations resulting from operating by <math>\langle x|</math>,  this amounts to the  [[differential equation]],
: <math>~\sqrt{\frac{m \omega}{2 \hbar}}\left(x+\frac{\hbar}{m\omega}\frac{\partial }{\partial x}\right)\psi^\alpha(x,t)=\alpha(t)\psi^\alpha(x,t)  ~, </math>
which is easily solved to yield
: <math>~\psi^{(\alpha)}(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \exp \Bigg( -\frac{m\omega}{2\hbar}\left(x-\sqrt{\frac{2\hbar}{m\omega}}\Re[\alpha(t)]\right)^2+i\sqrt{\frac{2m\omega}{\hbar}}\Im[\alpha(t)]x+i\theta(t) \Bigg) ~  ,</math>
where {{math|''θ(t)''}} is a yet undetermined phase, to be fixed by demanding that the wavefunction satisfies the Schrödinger equation.

It follows that
: <math>~\theta(t)=-\frac{\omega t}{2}+\frac{|\alpha(0)|^2\sin(2\omega t-2\sigma)}{2}    ~, \text{where} \qquad        \alpha(0)\equiv|\alpha(0)|\exp(i\sigma)  ~,         </math>
so that {{mvar|σ}} is the initial phase of the eigenvalue.

The mean position and momentum of this "minimal Schrödinger wave packet" {{math| ''ψ<sup>(α)</sup>''}} are thus '''''oscillating just like a classical system''''', 
{{Equation box 1
|indent =:
|equation =  <math>
   \langle \hat{x}(t) \rangle = \sqrt{\frac{2\hbar}{m\omega}}\Re[\alpha(t)]= |\alpha(0)|  \sqrt{\frac{2\hbar}{m\omega}} \cos (\sigma - \omega t)~,  
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
{{Equation box 1
|indent =:
|equation =  <math>
   \langle \hat{p}(t) \rangle = \sqrt{2m\hbar\omega}\Im[\alpha(t)]=  |\alpha(0)|\sqrt{2m\hbar\omega} \sin (\sigma - \omega t)~.
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
The probability density remains a Gaussian centered on this oscillating mean,
:<math>|\psi^{(\alpha)}(x,t)|^2=\sqrt{\frac{m\omega}{\pi\hbar} } e^{-\frac{m\omega}{\hbar}\left(x-  \langle \hat{x}(t) \rangle  \right)^2  }  .</math>