Coherent States

Quantize a single mode of the electromagnetic field and it becomes, mathematically, a quantum harmonic oscillator: the same ladder of Fock states $|n\rangle$ , the same $\hat a$ and $\hat a^\dagger$ , the same zero-point energy $\hbar\omega/2$ that the field-quantization and Fock-state articles built. But a laser beam does not look like a rung of that ladder. A number state $|n\rangle$ has sharp energy and random phase, so its mean field $\langle\hat E\rangle$ vanishes — it carries no oscillating wave at all. The everyday classical wave, a definite amplitude at a definite phase, is nowhere among the $|n\rangle$ . The coherent state puts it back. Roy Glauber introduced coherent states in 1963 to answer a sharp question: which quantum state of a field mode behaves as much like a classical monochromatic wave as the uncertainty principle allows? The answer is simple. Take the vacuum — the ground-state Gaussian of the oscillator, a fuzzy blob of zero-point noise at the origin of phase space — and slide it, without deforming it, out to a point $\alpha$ in the complex amplitude plane. That rigidly displaced vacuum is the coherent state $|\alpha\rangle$ : it carries a full classical-amplitude field, oscillates as a clean cosine, and keeps the minimum, symmetric noise the vacuum had. Two properties then make it foundational: its Poissonian photon number pins $g^{(2)}(0)=1$ , the exact line between classical and nonclassical light, and its vacuum-floor quadrature noise $\Delta X_1=\Delta X_2=1/2$ sets the standard quantum limit squeezed light is built to beat.

Key takeaways

A coherent state $|\alpha\rangle$ is the right-eigenstate of the annihilation operator, $\hat a|\alpha\rangle=\alpha|\alpha\rangle$ , with complex $\alpha=|\alpha|e^{i\phi}$ carrying amplitude and phase.
Equivalently it is the displaced vacuum, $|\alpha\rangle=\hat D(\alpha)|0\rangle$ : the vacuum's noise blob rigidly slid to $\alpha$ , keeping minimum symmetric uncertainty $\Delta X_1=\Delta X_2=1/2$ .
Its photon number is Poissonian, $P(n)=e^{-\bar n}\bar n^{\,n}/n!$ with $\bar n=|\alpha|^2$ , $(\Delta n)^2=\bar n$ , so relative noise falls as $1/\sqrt{\bar n}$ .
It is second-order coherent, $g^{(2)}(0)=1$ and Mandel $Q=0$ — the exact boundary between classical bunched and nonclassical antibunched light.
Coherent states are non-orthogonal and overcomplete, $|\langle\beta|\alpha\rangle|^2=e^{-|\alpha-\beta|^2}$ , yet still resolve the identity and underpin the P, Q and Wigner representations.
Key idea: a coherent state is the displaced vacuum — a minimum-uncertainty Gaussian rigidly translated in phase space to amplitude $\alpha$ , carrying a classical-like field with Poissonian photon number, the most classical of all quantum light states.

The core insight

A coherent state is the displaced vacuum: a minimum-uncertainty Gaussian wavepacket rigidly translated in phase space to amplitude $\alpha$ . It carries a classical-like field with Poissonian photon number, making it the most classical of all quantum light states.

Where this lives today

The coherent state is the working baseline of nearly every precision optical measurement. Balanced homodyne detection mixes a signal with a strong coherent local oscillator to read a quadrature $\hat X_\theta$ ; a coherent signal returns a Gaussian of width exactly $1/2$ , and every claim of squeezing is quoted in decibels below this floor. Gravitational-wave interferometers make the stakes concrete: Advanced LIGO's sensitivity is limited by the shot noise of its coherent laser light — the $\Delta n=\sqrt{\bar n}$ of worked example 1 — and since 2019 the detectors inject squeezed vacuum to push high-frequency noise below the coherent standard quantum limit, buying real astrophysical range. Coherent states are equally the carriers of information. Attenuated laser pulses — weak coherent states like the $\bar n=0.1$ pulse above — are the signals in most deployed quantum-key-distribution systems, where their Poissonian multiphoton tail drives the decoy-state protocols that keep BB84 secure. Their non-orthogonality $e^{-|\alpha-\beta|^2}$ also fixes the error floor for distinguishing symbols, tying the Holevo bound to the geometry of phase space.

The physical picture

Draw the phase-space plane of the two field quadratures: $X_1$ , the in-phase amplitude, horizontal, and $X_2$ , ninety degrees out of phase, vertical. A classical wave is a single point that circles the origin at $\omega$ , its shadow on $X_1$ the cosine you would see on an oscilloscope. Quantum mechanics forbids an exact point — since $[\hat X_1,\hat X_2]=i/2$ , any state occupies a patch of area at least $1/4$ — and the vacuum is the smallest such patch: a Gaussian blob of radius $1/2$ at the origin, all zero-point jitter and no mean field. The coherent state is that same blob, unchanged in size or shape, carried rigidly out to the point $\alpha$ — nothing squeezed or stretched. The disk keeps its vacuum radius $1/2$ while its centre sits at distance $|\alpha|$ , so the state carries a genuine classical amplitude with the least possible fuzz; as it evolves the disk circles the origin at $\omega$ , reproducing classical harmonic motion.

Figure

Phase-space portrait of a coherent state with α = 3 at phase 35°: the vacuum noise disk of radius ΔX = one half is rigidly displaced to amplitude 3, giving mean photon number n̄ = 9 and in-phase projection ⟨X₁⟩ ≈ 2.46; under evolution the disk circles the origin at frequency ω

Why Poissonian? Displacing the vacuum spreads its weight across the number states with weights following the Poisson law $P(n)=e^{-\bar n}\bar n^{\,n}/n!$ . A coherent field is a stream of statistically independent photon arrivals — the memorylessness that produces shot noise, and exactly what makes $g^{(2)}(0)=1$ . With mean $\bar n=|\alpha|^2$ and deviation $\sqrt{\bar n}$ , the fractional width $\Delta n/\bar n=1/\sqrt{\bar n}$ shrinks as the beam brightens.

Figure

Poissonian photon-number distribution of a coherent state with n̄ = 9: P(n) peaks at n = 8 and 9 with P ≈ 0.132 and is spread by one standard deviation Δn = √n̄ = 3, so the fractional width Δn over n̄ is one third

Finally, coherent states are not orthogonal: any two overlap, $|\langle\beta|\alpha\rangle|^2=e^{-|\alpha-\beta|^2}$ , dying off only when far apart in phase space. The family — one state per point of the complex plane — is vastly larger than a basis needs, an overcomplete set, yet it still resolves the identity, and that redundancy is what makes it the natural language for the P, Q and Wigner distributions.

The core equations

The cleanest definition is spectral: a coherent state is a right-eigenstate of the annihilation operator. Since $\hat a$ lowers photon number, this asks for a state that reproduces itself when a photon is removed — and the eigenvalue is complex because $\hat a$ is not Hermitian.

\hat a\,|\alpha\rangle=\alpha\,|\alpha\rangle \qquad (1)

where: $\hat a$ — annihilation operator of the mode, $[\hat a,\hat a^\dagger]=1$ ; $\alpha=|\alpha|e^{i\phi}$ — complex eigenvalue, $|\alpha|$ the amplitude and $\phi$ the phase; $|\alpha\rangle$ — the coherent state. Its real and imaginary parts are the mean quadratures, so $\alpha$ is the phase-space point where the state sits. Solving equation (1) in the Fock basis expands the coherent state over number states with Poisson-square-root amplitudes. This expansion bridges the abstract eigenvalue condition to every photon-counting prediction.

|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}\,|n\rangle \qquad (2)

where: $|n\rangle$ — Fock state of $n$ photons; $e^{-|\alpha|^2/2}$ — normalization ensuring $\langle\alpha|\alpha\rangle=1$ . Every number state from the vacuum up appears with amplitude $\propto\alpha^n/\sqrt{n!}$ , so a coherent state has no definite photon number. There is an equivalent, more physical definition. The coherent state is the vacuum acted on by the unitary displacement operator, which rigidly translates the mode's phase-space distribution by $\alpha$ .

|\alpha\rangle=\hat D(\alpha)\,|0\rangle,\qquad \hat D(\alpha)=\exp\!\big(\alpha\hat a^\dagger-\alpha^*\hat a\big) \qquad (3)

where: $\hat D(\alpha)$ — unitary displacement operator, $\hat D^\dagger(\alpha)=\hat D(-\alpha)$ ; $|0\rangle$ — the vacuum. It transforms operators as $\hat D^\dagger\hat a\hat D=\hat a+\alpha$ , exactly a rigid shift of the amplitude by $\alpha$ . Squaring the expansion amplitudes gives the photon-number distribution. It is Poissonian — the fingerprint of independent arrivals — and links coherent light to classical shot noise.

P(n)=|\langle n|\alpha\rangle|^2=e^{-\bar n}\,\frac{\bar n^{\,n}}{n!},\qquad \bar n=|\alpha|^2 \qquad (4)

where: $P(n)$ — probability of counting $n$ photons; $\bar n=\langle\hat n\rangle=|\alpha|^2$ — mean photon number. Variance equals mean, $(\Delta n)^2=\bar n$ , so the Mandel parameter $Q=[(\Delta n)^2-\bar n]/\bar n=0$ . The coherent state saturates the quadrature uncertainty relation symmetrically — equal noise in both quadratures, each at the vacuum floor. That is what "minimum-uncertainty" means, and it sets the standard quantum limit.

\Delta X_1=\Delta X_2=\tfrac12,\qquad \Delta X_1\,\Delta X_2=\tfrac14 \qquad (5)

where: $\hat X_1=\tfrac12(\hat a+\hat a^\dagger)$ — in-phase quadrature; $\hat X_2=\tfrac{1}{2i}(\hat a-\hat a^\dagger)$ — out-of-phase quadrature, with $[\hat X_1,\hat X_2]=i/2$ giving $\Delta X_1\Delta X_2\ge1/4$ . The noise is independent of $\alpha$ — vacuum and bright beam carry identical absolute fuzz. Because $\hat a$ is non-Hermitian its eigenstates overlap, so coherent states are neither orthogonal nor a minimal basis. Their overlap falls as a Gaussian in distance, yet still integrates to the identity.

|\langle\beta|\alpha\rangle|^2=e^{-|\alpha-\beta|^2},\qquad \frac{1}{\pi}\int \mathrm{d}^2\alpha\;|\alpha\rangle\langle\alpha|=\hat{\mathbb 1} \qquad (6)

where: $|\langle\beta|\alpha\rangle|^2$ — overlap probability, never exactly zero at finite separation; $\mathrm{d}^2\alpha=\mathrm{d}(\mathrm{Re}\,\alpha)\,\mathrm{d}(\mathrm{Im}\,\alpha)$ — area element on the complex plane; $\hat{\mathbb 1}$ — identity on the mode. The $1/\pi$ weighting is the overcomplete resolution of identity grounding the P, Q and Wigner representations. The normalized intensity autocorrelation at zero delay measures how photon detections cluster. For every coherent state it equals one, fixing the reference line between classical and nonclassical light.

g^{(2)}(0)=\frac{\langle\hat a^\dagger\hat a^\dagger\hat a\hat a\rangle}{\langle\hat a^\dagger\hat a\rangle^2}=1 \qquad (7)

where: $g^{(2)}(0)$ — second-order coherence at zero delay; unity means no bunching and no antibunching. Classical light has $g^{(2)}(0)\ge1$ , so a value below one certifies nonclassicality. Under free evolution the label simply rotates as $\alpha e^{-i\omega t}$ , so the blob orbits the origin and stays coherent.

Derivation AFrom the eigenvalue equation to Poissonian photon statistics

The goal is to turn the abstract condition $\hat a|\alpha\rangle=\alpha|\alpha\rangle$ into the concrete number-state expansion and the Poisson distribution a photodetector sees. We expand in the Fock basis, match coefficients to a recursion, normalize, and square the amplitudes to read off $P(n)$ — one spectral definition delivering both the structure and the statistics.

Step 1. Write $|\alpha\rangle=\sum_n c_n|n\rangle$ and act with $\hat a$ using $\hat a|n\rangle=\sqrt n\,|n-1\rangle$ : $\hat a|\alpha\rangle=\sum_n c_n\sqrt n\,|n-1\rangle$ . Setting this equal to $\alpha\sum_n c_n|n\rangle$ and matching the coefficient of $|n-1\rangle$ gives $c_n\sqrt n=\alpha\,c_{n-1}$ . Why: the eigenvalue equation read term by term becomes an ordinary recursion — equal kets on both sides must have equal coefficients.

Step 2. Iterate $c_n=(\alpha/\sqrt n)\,c_{n-1}$ down to $c_0$ : $c_n=(\alpha^n/\sqrt{n!})\,c_0$ . Why: each step contributes one $\alpha$ and one $1/\sqrt n$ , stacking to the $\alpha^n/\sqrt{n!}$ that is the signature of a coherent state.

Step 3. Fix $c_0$ by $\langle\alpha|\alpha\rangle=1$ : $|c_0|^2\sum_n|\alpha|^{2n}/n!=|c_0|^2 e^{|\alpha|^2}=1$ , so $c_0=e^{-|\alpha|^2/2}$ up to a phase, reproducing equation (2). Why: the sum is the exponential series, so normalization is one Gaussian factor — the same $e^{-|\alpha|^2/2}$ the displacement route gives.

Step 4. Square $\langle n|\alpha\rangle=e^{-|\alpha|^2/2}\alpha^n/\sqrt{n!}$ to get $P(n)=e^{-|\alpha|^2}|\alpha|^{2n}/n!$ , and identify $\bar n=|\alpha|^2$ : the Poisson law with $(\Delta n)^2=\bar n$ . Why: the probability keeps only the modulus, and what survives is exactly Poisson — hence $Q=0$ and shot noise $\Delta n/\bar n=1/\sqrt{\bar n}$ . *Note: the distribution sums to 1, and direct computation from the amplitudes confirms mean and variance both equal $\bar n$ .*

Go deeper · Derivation B · 1 minSymmetric minimum uncertainty of the quadratures

Derivation BSymmetric minimum uncertainty of the quadratures

The goal is to show the coherent state saturates the Heisenberg bound for the quadratures evenly, $\Delta X_1=\Delta X_2=1/2$ — the precise content of "minimum-uncertainty state" and the origin of the standard quantum limit. We reduce every expectation to a c-number using $\hat a|\alpha\rangle=\alpha|\alpha\rangle$ and its adjoint $\langle\alpha|\hat a^\dagger=\alpha^*\langle\alpha|$ , tracking ordering since $\hat a,\hat a^\dagger$ do not commute.

Step 1. Means: $\langle\hat X_1\rangle=\tfrac12(\alpha+\alpha^*)=\mathrm{Re}\,\alpha$ and likewise $\langle\hat X_2\rangle=\mathrm{Im}\,\alpha$ . Why: the mean quadratures are the real and imaginary parts of $\alpha$ , confirming $|\alpha\rangle$ sits at the phase-space point $\alpha$ — the classical field amplitude.

Step 2. Expand the second moment $\langle\hat X_1^2\rangle=\tfrac14\langle\hat a^2+\hat a^{\dagger2}+\hat a\hat a^\dagger+\hat a^\dagger\hat a\rangle$ , keeping the cross terms ordered. Why: squaring non-commuting operators is not the ordinary binomial — the order of $\hat a\hat a^\dagger$ versus $\hat a^\dagger\hat a$ is where the vacuum term enters.

Step 3. Use $\hat a\hat a^\dagger=\hat a^\dagger\hat a+1$ and the eigenvalue relation: $\langle\hat X_1^2\rangle=\tfrac14(\alpha^2+\alpha^{*2}+2|\alpha|^2+1)$ . Why: the commutator injects the single "+1", the zero-point term that keeps the variance from vanishing.

Step 4. Subtract the squared mean: $(\Delta X_1)^2=\tfrac14$ , since $\langle\hat X_1\rangle^2=\tfrac14(\alpha+\alpha^*)^2$ cancels every $\alpha$ -dependent term, and by symmetry $(\Delta X_2)^2=\tfrac14$ . Why: all amplitude dependence cancels, leaving the vacuum value $1/4$ in each quadrature, so $\Delta X_1\Delta X_2=1/4$ saturates the bound for every $\alpha$ . *Note: the result is independent of $\alpha$ , and the product $\Delta X_1\Delta X_2=1/4$ exactly saturates the bound $\Delta X_1\Delta X_2\ge1/4$ .*

Go deeper · Derivation C · 1 minThe displacement operator generates the coherent state

Derivation CThe displacement operator generates the coherent state

The goal is to prove the two definitions — eigenstate of $\hat a$ and displaced vacuum $\hat D(\alpha)|0\rangle$ — are the same state, exposing why $\hat D(\alpha)$ acts as a rigid phase-space translation. The Baker–Campbell–Hausdorff identity disentangles $\exp(\alpha\hat a^\dagger-\alpha^*\hat a)$ into a normal-ordered product because the commutator is a c-number; then the annihilation part dies on the vacuum and the creation part builds the Fock series.

Step 1. Apply BCH, $e^{A+B}=e^{A}e^{B}e^{-[A,B]/2}$ (valid when $[A,B]$ commutes with $A,B$ ), with $A=\alpha\hat a^\dagger$ , $B=-\alpha^*\hat a$ . Then $[A,B]=|\alpha|^2$ , so $\hat D(\alpha)=e^{-|\alpha|^2/2}e^{\alpha\hat a^\dagger}e^{-\alpha^*\hat a}$ . Why: the commutator of $\hat a$ and $\hat a^\dagger$ is just $1$ , so BCH terminates immediately and the disentangling factor is the Gaussian $e^{-|\alpha|^2/2}$ .

Step 2. Act on the vacuum: $e^{-\alpha^*\hat a}|0\rangle=|0\rangle$ , since every term beyond the identity contains an $\hat a$ that annihilates $|0\rangle$ . Why: with annihilation operators on the right, the vacuum passes through untouched.

Step 3. Expand using $(\hat a^\dagger)^n|0\rangle=\sqrt{n!}\,|n\rangle$ : $e^{\alpha\hat a^\dagger}|0\rangle=\sum_n(\alpha^n/\sqrt{n!})|n\rangle$ . Why: each power of $\hat a^\dagger$ climbs a rung, and $\sqrt{n!}$ cancels one factorial to leave the coherent-state amplitude.

Step 4. Reassemble: $\hat D(\alpha)|0\rangle=e^{-|\alpha|^2/2}\sum_n(\alpha^n/\sqrt{n!})|n\rangle=|\alpha\rangle$ , identical to equation (2). Why: the two definitions coincide exactly — "eigenstate of $\hat a$ " and "displaced vacuum" are one object. *Note: this reproduces the same normalization factor $e^{-|\alpha|^2/2}$ obtained from the eigenvalue derivation, confirming the two definitions are identical.*

Limiting cases

Coherent states interpolate between quantum and classical and sit on several important boundaries:

Case	Result	Meaning
Vacuum limit	$\alpha\to0$	$\|\alpha\rangle\to\|0\rangle$ : the vacuum is the coherent state of zero amplitude, the disk at the origin.
Classical (bright) limit	$\|\alpha\|\gg1$	$\Delta n/\bar n=1/\sqrt{\bar n}\to0$ , $\Delta\phi\to0$ ; relative noise vanishes, so the field is a noise-free classical wave.
Poissonian boundary	$Q=0,\;g^{(2)}(0)=1$	the line between classical super-Poissonian light ( $Q>0$ , bunched) and nonclassical sub-Poissonian light ( $Q<0$ , antibunched).
Minimum-uncertainty boundary	$\Delta X_1=\Delta X_2=\tfrac12$	symmetric saturation of $\Delta X_1\Delta X_2\ge1/4$ ; squeezed states push below $1/2$ in one quadrature at the other's expense.

Common trapA "coherent state" is the same thing as light being "coherent"

These are different ideas, and conflating them is the classic trap. "Coherence" (first-order, $g^{(1)}$ ) is about phase correlations across space or time — a single-mode Fock state $|n\rangle$ is perfectly first-order coherent yet is emphatically not a coherent state; its mean field is zero. A "coherent state" $|\alpha\rangle$ is one specific state, singled out by $\hat a|\alpha\rangle=\alpha|\alpha\rangle$ and its Poissonian, minimum-uncertainty character. The other half of the trap: $|\alpha\rangle$ is an eigenstate of $\hat a$ but not of $\hat a^\dagger$ or $\hat n$ , so measuring photon number returns a random Poissonian value, never a fixed $n$ .

Worked examples

1 · Shot noise of a laser beam (confidence builder)

A 1 mW helium–neon laser at $\lambda=633$ nm is modelled as a coherent state. In a $\tau=1$ ns detection window, find the mean photon number, the coherent amplitude $|\alpha|$ , the photon-number fluctuation $\Delta n$ , and the fractional intensity noise.

Step 1. Photon energy: $E=hc/\lambda=(6.626\times10^{-34})(2.998\times10^{8})/(633\times10^{-9})=\mathbf{3.14\times10^{-19}\ J}$ . Arrival rate: $R=P/E=(10^{-3})/(3.14\times10^{-19})=\mathbf{3.19\times10^{15}\ s^{-1}}$ .

Step 2. Mean number in the window: $\bar n=R\tau=(3.19\times10^{15})(10^{-9})=\mathbf{3.19\times10^{6}}$ photons. The coherent amplitude is $|\alpha|=\sqrt{\bar n}=\mathbf{1.79\times10^{3}}$ — the blob sits nearly 1800 disk-radii from the origin.

Step 3. For a Poissonian state $(\Delta n)^2=\bar n$ , so $\Delta n=\sqrt{\bar n}=\mathbf{1.79\times10^{3}}$ photons and the fractional intensity noise is $\Delta n/\bar n=1/\sqrt{\bar n}=\mathbf{5.6\times10^{-4}}$ . This is the shot-noise floor — about 0.056 percent relative fluctuation, what a laser-noise measurement hits once technical noise is removed. The blob radius $1/2$ is negligible beside $|\alpha|\approx1790$ : the beam is, for all practical purposes, a classical wave. n̄ = 3.19×10⁶, |α| = 1.79×10³, Δn = 1.79×10³, Δn/n̄ = 5.6×10⁻⁴.

2 · A weak coherent pulse for quantum key distribution

Practical BB84 uses attenuated laser pulses — coherent states with small mean photon number, here $\bar n=0.1$ . Find P(0), P(1), P(2), and the probability that a non-empty pulse contains two or more photons (the multiphoton fraction that opens a security loophole).

Step 1. From $P(n)=e^{-\bar n}\bar n^{\,n}/n!$ with $\bar n=0.1$ : $P(0)=e^{-0.1}=\mathbf{0.905}$ (90% of pulses empty), $P(1)=e^{-0.1}(0.1)=\mathbf{0.0905}$ , $P(2)=e^{-0.1}(0.1)^2/2=\mathbf{0.00452}$ .

Step 2. Non-empty: $P(n\ge1)=1-P(0)=\mathbf{0.0952}$ ; two or more: $P(n\ge2)=1-P(0)-P(1)=\mathbf{0.00468}$ .

Step 3. The multiphoton fraction of non-empty pulses is $P(n\ge2)/P(n\ge1)=0.00468/0.0952=\mathbf{0.049}$ . An eavesdropper can split off the extra photons undetected — which is why real systems run at low $\bar n$ and use decoy states. P(0) = 0.905, P(1) = 0.0905, P(2) = 0.00452, and about 4.9% of non-empty pulses are multiphoton.

Physics of the portrait

Coherent states are where the quantum ladder of the field reconnects with the classical wave — and the person who forged that link was Roy Glauber.

Origins · 1 minRoy Glauber and how this was discovered

Roy Glauber

1925–2018 · the quantum theory of optical coherence

Roy Glauber wrote quantum optics into existence in a pair of 1963 papers. Schrödinger had noticed in 1926 that the harmonic oscillator has special non-spreading wavepackets — the first coherent states — but for the field the question only became urgent after the 1956 Hanbury Brown–Twiss intensity-correlation experiments provoked a debate over whether light needed a quantum description at all. Glauber settled it: he defined $|\alpha\rangle$ as eigenstates of the annihilation operator, showed they are the displaced vacuum with Poissonian statistics and minimum uncertainty, and built from them the hierarchy of coherence functions $g^{(n)}$ that distinguishes a laser from a light bulb by more than brightness. His diagonal P-representation — established at the same moment by E. C. G. Sudarshan, now the Glauber–Sudarshan representation — recast any field state as a quasi-classical distribution over coherent states, giving "nonclassical" a precise meaning: the distribution fails to be a genuine probability. This is the machinery behind $g^{(2)}(0)$ as the classical boundary and the phase-space picture of light — and Glauber received the 2005 Nobel Prize in Physics for it.

History

1926

— Schrödinger constructs the non-spreading oscillator wavepackets, the earliest coherent states

1956

— Hanbury Brown and Twiss measure photon intensity correlations, forcing a quantum theory of coherence

1963

— Glauber defines the coherent states and the hierarchy of optical coherence functions

1963

— Sudarshan proves the diagonal P-representation, the optical equivalence theorem

2005

— Glauber receives the Nobel Prize for the quantum theory of optical coherence

Going further

Texts. Glauber, Phys. Rev. 131, 2766 (1963), "Coherent and Incoherent States of the Radiation Field" · Loudon, The Quantum Theory of Light, 3rd ed., ch. 5 · Gerry and Knight, Introductory Quantum Optics, ch. 3 · Scully and Zubairy, Quantum Optics, ch. 2 · Saleh and Teich, Fundamentals of Photonics, 2nd ed., ch. 12. Every result follows from the single commutator $[\hat a,\hat a^\dagger]=1$ : the number-state expansion by solving $\hat a|\alpha\rangle=\alpha|\alpha\rangle$ recursively; the Poisson form with $(\Delta n)^2=\bar n$ by squaring the amplitudes; $\Delta X_1=\Delta X_2=1/2$ from the quadrature-variance derivation; $g^{(2)}(0)=1$ from normally-ordered moments; the displacement identity and overlap $e^{-|\alpha-\beta|^2}$ from BCH. Key numerical results: 1 mW at 633 nm gives $\bar n=3.19\times10^{6}$ in a 1 ns window and shot noise $5.6\times10^{-4}$ ; a pulse with $\bar n=0.1$ has multiphoton fraction $0.049$ . The P, Q and Wigner distributions, squeezing, and the Mandel-Q formalism are treated in the articles listed below.

Prerequisites. Quantization of the Electromagnetic Field · Fock States and the Photon Number Basis · Vacuum State and Zero-Point Energy · Quadrature Operators and the Uncertainty Relation. Leads to. Squeezed States of Light · Photon Statistics and the Mandel Q Parameter · Phase-Space Quasiprobability Distributions · Coherence Functions and the Degree of Coherence · Photon Antibunching.

Ask the tutor: ask me to squeeze this state — to show which noise gets traded away when $|\alpha\rangle$ becomes a squeezed state, and how homodyne detection sees the difference.

Why is a coherent state an eigenstate of â but not of â† or n̂

Because $\hat a$ lowers photon number and $|\alpha\rangle$ contains every number state from the vacuum up with Poisson-square-root weights: removing one photon rescales the whole superposition by $\alpha$ , reproducing the state. The creation operator $\hat a^\dagger$ raises number with no top rung to anchor a reproducing eigenvalue, so it has no normalizable eigenstates. And $\hat n$ would require a definite photon number, which $|\alpha\rangle$ lacks — its distribution is a spread-out Poissonian of width $\sqrt{\bar n}$ , so "how many photons are in a coherent state" has only a statistical answer.

In what sense is the coherent state the "most classical" light

In two precise senses. First, its mean field oscillates as a clean cosine of amplitude $\propto|\alpha|$ , and the fractional noise $\Delta X/\langle X\rangle\sim1/\sqrt{\bar n}$ vanishes as the beam brightens — a bright coherent beam is a deterministic wave, which is why Maxwell's description of laser light works. Second, its Glauber–Sudarshan P-distribution is a single positive delta function, the borderline case of a classical probability; states whose P-function goes negative or singular are nonclassical, so the coherent state sits exactly on the classical edge.

Where does the Poisson distribution actually come from

From displacing the vacuum. The displacement operator spreads the vacuum's weight across all number states with amplitudes $\alpha^n/\sqrt{n!}$ ; squaring gives $|\alpha|^{2n}/n!$ times a normalization — the Poisson law. Physically, a coherent field is a stream of statistically independent photon arrivals, and independent point events in a fixed interval always follow Poisson statistics. That independence is the same fact expressed by $g^{(2)}(0)=1$ : the arrivals neither bunch nor antibunch.

If ΔX is fixed at 1 over 2, how can squeezed light do better

The coherent state spends its minimum uncertainty symmetrically, $\Delta X_1=\Delta X_2=1/2$ , wasting half its "noise budget" on a quadrature you may not care about. Squeezing redistributes: it pushes one quadrature below $1/2$ while the other rises above, still respecting $\Delta X_1\Delta X_2\ge1/4$ . If your measurement reads only the squeezed quadrature — as a gravitational-wave interferometer does — you beat the coherent standard quantum limit there. Nothing beats the uncertainty product; squeezing only moves where the fuzz lives, and the coherent state is the reference because its allocation is the even one.

What does non-orthogonality cost, and why keep such a basis

Non-orthogonality means you can never perfectly distinguish two coherent states — the overlap $e^{-|\alpha-\beta|^2}$ is small but never zero — which sets fundamental error floors in optical communication. So why use them? Because the family is overcomplete and resolves the identity, $\frac{1}{\pi}\int\mathrm{d}^2\alpha\,|\alpha\rangle\langle\alpha|=\hat{\mathbb 1}$ , enough to expand any state or operator. That redundancy makes them the natural grid for the P, Q and Wigner representations, where operator problems become c-number integrals over the plane.

Does a coherent state stay coherent as it evolves or passes through loss

Yes to both. Under free evolution with $\hat H=\hbar\omega(\hat n+\tfrac12)$ , $|\alpha\rangle\to e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle$ — still coherent, its label just rotating, so the phase-space blob orbits the origin and reproduces classical harmonic motion. Linear loss is equally benign: a channel of transmission $\eta$ maps $|\alpha\rangle\to|\sqrt\eta\,\alpha\rangle$ , still coherent, only dimmer. Surviving the two commonest things that happen to light — propagation and attenuation — makes coherent states the default carriers in optical communication.