---
title: Coherent and Squeezed States
module: The Oscillator Algebraically, and Symmetry
moduleNumber: 5
lessonNumber: 2
order: 502
summary: >
  A single number state never moves — its position expectation is pinned at the
  origin. The superposition that oscillates like a classical particle is the
  eigenstate of the annihilation operator: the coherent state. It is a displaced
  vacuum, carries Poissonian photon statistics, saturates the uncertainty bound,
  and traces a rigid Gaussian orbit in phase space. Squeezing deforms that circle,
  trading precision in one quadrature for noise in the other.
topics: [The Oscillator Algebraically, and Symmetry]
sources:
  - book: Shankar
    ref: "Ch. 7 — The Harmonic Oscillator (problems); Ch. 21 Path Integrals (coherent-state asides)"
  - book: Sakurai & Napolitano
    ref: "Ch. 2 — Quantum Dynamics; §2.3 The Simple Harmonic Oscillator (coherent states)"
  - book: Cohen-Tannoudji, Diu & Laloë
    ref: "Ch. V; Complement G_V — Quasi-classical (coherent) states of the oscillator"
draft: false
---

The [number states](/quantum-mechanics/oscillator-and-symmetry/ladder-operators-and-the-number-states)
diagonalize the Hamiltonian but look nothing like a classical oscillator. Each has
$\langle\hat x\rangle = \langle\hat p\rangle = 0$ for all time, a probability cloud
frozen in place. A pendulum, by contrast, has a definite position that swings
sinusoidally. The quantum states that reproduce that behavior as closely as the
uncertainty principle allows are the **coherent states**, and they are singled out
by one clean condition: they are eigenstates of the annihilation operator.[^ct-gv]

## Eigenstates of the annihilation operator

Ask for a state $\lvert\alpha\rangle$ obeying

$$
\hat a\lvert\alpha\rangle = \alpha\lvert\alpha\rangle,
\qquad \alpha \in \mathbb{C}.
$$

The eigenvalue $\alpha$ is complex because $\hat a$ is not Hermitian. The raising
operator has no such eigenstates — $\hat a^\dagger$ pushes probability up the ladder
without bound, and no normalizable superposition survives — but the lowering
operator does, because it terminates on the vacuum. Expand
$\lvert\alpha\rangle = \sum_{n} c_n\lvert n\rangle$ and impose the eigenvalue
condition. Using $\hat a\lvert n\rangle = \sqrt{n}\,\lvert n-1\rangle$,

$$
\hat a\lvert\alpha\rangle = \sum_{n\ge1} c_n\sqrt{n}\,\lvert n-1\rangle
= \alpha\sum_{n} c_n\lvert n\rangle .
$$

Matching the coefficient of $\lvert n\rangle$ gives the recursion
$c_{n+1}\sqrt{n+1} = \alpha\,c_n$, so

$$
c_n = \frac{\alpha^{n}}{\sqrt{n!}}\,c_0 .
$$

Normalization fixes $c_0$. Since
$\sum_n |c_n|^{2} = |c_0|^{2}\sum_n |\alpha|^{2n}/n! = |c_0|^{2}e^{|\alpha|^{2}} = 1$,

> **Definition (Coherent state).** The normalized eigenstate of the annihilation
> operator with eigenvalue $\alpha$,
> $$
> \lvert\alpha\rangle = e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}\,\lvert n\rangle .
> $$
> The vacuum is the coherent state with $\alpha = 0$. Coherent states are not
> orthogonal: $|\langle\beta\vert\alpha\rangle|^{2} = e^{-|\alpha-\beta|^{2}}$,
> nonzero for every pair, though negligible once the eigenvalues are well separated.

## Poissonian photon statistics

The probability of finding $n$ quanta in a coherent state is
$P(n) = |\langle n\vert\alpha\rangle|^{2}$, which reads off the expansion:

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

This is a **Poisson distribution** with mean $\bar n = |\alpha|^{2}$. The mean and
variance of a Poisson law are equal, so

$$
\langle\hat N\rangle = |\alpha|^{2},
\qquad
(\Delta N)^{2} = |\alpha|^{2},
\qquad
\frac{\Delta N}{\langle\hat N\rangle} = \frac{1}{|\alpha|} = \frac{1}{\sqrt{\bar n}} .
$$

The fractional spread in photon number shrinks as the field grows, so a bright
coherent beam has a nearly definite intensity even though the number is not sharp.
The photon-counting fluctuations $\Delta N = \sqrt{\bar n}$ are the **shot noise** of
an ideal laser, whose output is very close to a coherent state.[^sn-shot]

$$
% caption: The photon-number distribution of a coherent state is Poissonian,
% $P(n) = e^{-\bar n}\bar n^{n}/n!$ with $\bar n = |\alpha|^{2}$. The peak sits near
% $\bar n$ and the width scales as $\sqrt{\bar n}$, so larger fields are relatively
% sharper in intensity.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black] (0,0) -- (9.4,0) node[right, font=\scriptsize, text=black] {$n$};
  \draw[black] (0,0) -- (0,3.2) node[above, font=\scriptsize, text=black] {$P(n)$};
  % Poisson mean 4, heights scaled (P(n)*8): 0:.018 1:.073 2:.147 3:.195 4:.195 5:.156 6:.104 7:.060 8:.030 9:.013
  \foreach \n/\h in {0/0.15,1/0.59,2/1.17,3/1.56,4/1.56,5/1.25,6/0.83,7/0.48,8/0.24,9/0.11} {
    \draw[black] (\n*0.9+0.18,0) rectangle (\n*0.9+0.72,\h);
    \node[black, anchor=north, font=\tiny] at (\n*0.9+0.45,0) {\n};
  }
  \draw[acc, thick, dashed] (4*0.9+0.45,0) -- (4*0.9+0.45,2.0);
  \node[acc, anchor=south, font=\scriptsize] at (4*0.9+0.45,2.0) {$\bar n = 4$};
\end{tikzpicture}
$$

## The displacement operator

The coherent state has a compact operator description. Define the **displacement
operator**

$$
\hat D(\alpha) = \exp\!\bigl(\alpha\hat a^{\dagger} - \alpha^{\ast}\hat a\bigr),
$$

which is unitary because its exponent is anti-Hermitian. Its exponent splits by the
Baker–Campbell–Hausdorff identity: with $\hat A = \alpha\hat a^\dagger$ and
$\hat B = -\alpha^\ast\hat a$, the commutator $[\hat A,\hat B] = |\alpha|^{2}$ is a
number, so

$$
\hat D(\alpha) = e^{-|\alpha|^{2}/2}\,e^{\alpha\hat a^{\dagger}}\,e^{-\alpha^{\ast}\hat a} .
$$

Acting on the vacuum, $e^{-\alpha^\ast\hat a}\lvert 0\rangle = \lvert 0\rangle$
because $\hat a\lvert0\rangle = 0$, while
$e^{\alpha\hat a^\dagger}\lvert 0\rangle = \sum_n (\alpha^{n}/n!)(\hat a^\dagger)^{n}\lvert0\rangle
= \sum_n (\alpha^n/\sqrt{n!})\lvert n\rangle$. The two combine into exactly the
coherent-state expansion:

$$
\hat D(\alpha)\lvert 0\rangle = \lvert\alpha\rangle .
$$

A coherent state is a **displaced vacuum**. The name is literal: conjugating the
ladder operators shows the displacement shifts phase space by $\alpha$,

$$
\hat D^{\dagger}(\alpha)\,\hat a\,\hat D(\alpha) = \hat a + \alpha,
\qquad
\hat D^{\dagger}(\alpha)\,\hat x\,\hat D(\alpha) = \hat x + \sqrt{\tfrac{2\hbar}{m\omega}}\,\mathrm{Re}\,\alpha .
$$

$$
% caption: The displacement operator $\hat D(\alpha)$ slides the vacuum Gaussian
% rigidly to the point $(\mathrm{Re}\,\alpha,\ \mathrm{Im}\,\alpha)$ in phase space
% without changing its shape or width. The coherent state is this displaced vacuum.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, ->] (-0.4,0) -- (5.6,0) node[right, font=\scriptsize, text=black] {$x$};
  \draw[black, ->] (0,-0.5) -- (0,3.2) node[above, font=\scriptsize, text=black] {$p$};
  % vacuum blob at origin
  \draw[black, thick] (0,0) circle (0.5);
  \fill[black] (0,0) circle (1.6pt);
  \node[black, anchor=north east, font=\tiny] at (-0.05,-0.05) {vacuum};
  % displaced blob
  \draw[acc, thick, fill=acc!12] (3.6,2.0) circle (0.5);
  \fill[acc] (3.6,2.0) circle (1.8pt);
  \node[acc, anchor=south west, font=\scriptsize] at (3.75,2.15) {coherent};
  % displacement arrow
  \draw[->, black!70, thick] (0.4,0.28) -- (3.25,1.85);
  \node[black!70, anchor=north west, font=\scriptsize] at (1.4,1.05) {displace};
\end{tikzpicture}
$$

## Minimum uncertainty and classical motion

Because a coherent state is the vacuum shifted rigidly, its widths equal the
vacuum's. From $\hat a\lvert\alpha\rangle = \alpha\lvert\alpha\rangle$ and its adjoint,

$$
\langle\hat x\rangle = \sqrt{\frac{2\hbar}{m\omega}}\,\mathrm{Re}\,\alpha,
\qquad
\langle\hat p\rangle = \sqrt{2m\hbar\omega}\,\mathrm{Im}\,\alpha,
$$

and a short calculation with $\langle\hat a^{2}\rangle = \alpha^{2}$,
$\langle\hat a^\dagger\hat a\rangle = |\alpha|^{2}$ gives the widths

$$
\Delta x = \sqrt{\frac{\hbar}{2m\omega}},
\qquad
\Delta p = \sqrt{\frac{m\hbar\omega}{2}},
\qquad
\Delta x\,\Delta p = \frac{\hbar}{2} .
$$

Every coherent state saturates the uncertainty bound, and the widths are
independent of $\alpha$: the wave packet keeps the shape of the ground-state
Gaussian while its center sits anywhere in phase space. The vacuum is the special
case centered at the origin.

The center then moves classically. In the
[Heisenberg picture](/quantum-mechanics/oscillator-and-symmetry/ladder-operators-and-the-number-states)
$\hat a(t) = \hat a(0)e^{-i\omega t}$, so the eigenvalue rotates,
$\langle\hat a\rangle_t = \alpha_0 e^{-i\omega t}$, and

$$
\langle\hat x(t)\rangle = \sqrt{\frac{2\hbar}{m\omega}}\,|\alpha_0|\cos(\omega t - \varphi),
\qquad \alpha_0 = |\alpha_0|e^{i\varphi} .
$$

The expectation traces a sinusoid at the classical frequency with amplitude set by
$|\alpha_0|$. Equivalently, under time evolution a coherent state stays coherent,
$\lvert\alpha_0\rangle \to e^{-i\omega t/2}\lvert\alpha_0 e^{-i\omega t}\rangle$: the
Gaussian blob keeps its shape and orbits the phase-space origin on a circle of radius
$|\alpha_0|$, the closest a quantum state comes to a point particle on a classical
trajectory.

$$
% caption: A coherent state is a rigid minimum-uncertainty Gaussian whose center
% orbits the phase-space origin at the classical frequency; its projection onto the
% $x$ axis is the sinusoid $\langle\hat x(t)\rangle$.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  % phase-space panel
  \draw[black, ->] (-2.4,0) -- (2.4,0) node[right, font=\scriptsize, text=black] {$x$};
  \draw[black, ->] (0,-2.4) -- (0,2.4) node[above, font=\scriptsize, text=black] {$p$};
  \draw[black, dashed] (0,0) circle (1.7);
  % blob on the circle
  \draw[acc, thick, fill=acc!12] (1.35,1.03) circle (0.42);
  \fill[acc] (1.35,1.03) circle (1.6pt);
  \draw[->, thick] (1.75,1.45) arc (37:70:2.2);
  \node[anchor=west, font=\tiny] at (1.8,1.75) {orbit};
  % sinusoid panel
  \begin{scope}[xshift=6.2cm]
    \draw[black, ->] (-0.3,0) -- (4.6,0) node[right, font=\scriptsize, text=black] {$t$};
    \draw[black, ->] (0,-1.5) -- (0,1.7) node[above, font=\scriptsize, text=black] {position};
    \draw[acc, thick] (0,1.2)
      .. controls (0.5,1.2) and (0.8,0.0) .. (1.1,0.0)
      .. controls (1.5,0.0) and (1.7,-1.2) .. (2.2,-1.2)
      .. controls (2.7,-1.2) and (2.9,0.0) .. (3.3,0.0)
      .. controls (3.7,0.0) and (3.9,1.2) .. (4.4,1.2);
  \end{scope}
\end{tikzpicture}
$$

## Squeezed states

A coherent state divides its uncertainty equally between two directions in phase
space. Nothing forces that split. Define the dimensionless **quadrature operators**

$$
\hat X_1 = \frac{\hat a + \hat a^{\dagger}}{2},
\qquad
\hat X_2 = \frac{\hat a - \hat a^{\dagger}}{2i},
$$

the amplitudes of the $\cos\omega t$ and $\sin\omega t$ parts of the motion. Their
commutator $[\hat X_1,\hat X_2] = i/2$ enforces
$\Delta X_1\,\Delta X_2 \ge \tfrac14$. The vacuum and every coherent state are
**isotropic**: $\Delta X_1 = \Delta X_2 = \tfrac12$, a circle of uncertainty. A
**squeezed state** keeps the product at the floor $\tfrac14$ but makes the ellipse
eccentric,

$$
\Delta X_1 = \tfrac12 e^{-r},
\qquad
\Delta X_2 = \tfrac12 e^{+r},
$$

with $r$ the squeeze parameter. It is generated by the **squeeze operator**

$$
\hat S(\xi) = \exp\!\Bigl[\tfrac12\bigl(\xi^{\ast}\hat a^{2} - \xi\,\hat a^{\dagger 2}\bigr)\Bigr],
\qquad \xi = re^{i\theta},
$$

whose quadratic exponent conjugates $\hat a$ into a mixture of $\hat a$ and
$\hat a^\dagger$, stretching one quadrature and compressing the orthogonal one.

> **Definition (Squeezed state).** A minimum-uncertainty state in which one
> quadrature is measured more precisely than the vacuum limit at the cost of the
> conjugate quadrature: $\Delta X_1 = \tfrac12 e^{-r} < \tfrac12$ and
> $\Delta X_2 = \tfrac12 e^{+r}$, with $\Delta X_1\,\Delta X_2 = \tfrac14$ preserved.

$$
% caption: The coherent state's isotropic uncertainty circle (left) becomes an
% ellipse under squeezing (right): one quadrature drops below the vacuum noise
% $\tfrac12$ while the conjugate grows, the area $\Delta X_1\,\Delta X_2 = \tfrac14$
% unchanged.
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  % coherent circle
  \draw[black, ->] (-1.9,0) -- (1.9,0) node[right, font=\tiny, text=black] {$X_1$};
  \draw[black, ->] (0,-1.9) -- (0,1.9) node[above, font=\tiny, text=black] {$X_2$};
  \draw[acc, thick, fill=acc!12] (0,0) circle (1.1);
  \node[acc, anchor=north, font=\tiny] at (0,-2.0) {coherent};
  % squeezed ellipse
  \begin{scope}[xshift=5.4cm]
    \draw[black, ->] (-1.9,0) -- (1.9,0) node[right, font=\tiny, text=black] {$X_1$};
    \draw[black, ->] (0,-1.9) -- (0,1.9) node[above, font=\tiny, text=black] {$X_2$};
    \draw[acc, thick, fill=acc!12] (0,0) ellipse (0.5 and 1.7);
    \node[acc, anchor=north, font=\tiny] at (0,-2.0) {squeezed};
    \draw[<->, black] (-0.5,-0.35) -- (0.5,-0.35);
    \node[black, anchor=west, font=\tiny] at (0.7,-0.35) {reduced};
  \end{scope}
\end{tikzpicture}
$$

Squeezing has a direct experimental payoff. An interferometer limited by the shot
noise of coherent light can beat that limit by injecting squeezed light into its
dark port, aligning the reduced quadrature with the measured phase. Gravitational-wave
detectors run this way, reading out a strain smaller than the vacuum fluctuations of
ordinary laser light would permit.[^sn-sqz] The three regimes of photon statistics
label where a state's number fluctuations sit relative to the coherent benchmark.

| State | $(\Delta N)^{2}$ vs. $\langle\hat N\rangle$ | Statistics | Physical example |
| --- | --- | --- | --- |
| Number state $\lvert n\rangle$ | $(\Delta N)^{2} = 0$ | sub-Poissonian | single-photon source |
| Coherent $\lvert\alpha\rangle$ | $(\Delta N)^{2} = \langle\hat N\rangle$ | Poissonian | ideal laser |
| Thermal | $(\Delta N)^{2} > \langle\hat N\rangle$ | super-Poissonian | blackbody / chaotic light |

$$
% caption: Three photon-number distributions at the same mean $\bar n$: the number
% state is a single spike (sub-Poissonian), the coherent state is the Poisson
% envelope (variance equal to the mean), and thermal light decays monotonically
% (super-Poissonian, widest).
\begin{tikzpicture}[>=stealth, font=\scriptsize]
  \definecolor{acc}{HTML}{4A6FA5}
  % number state: single spike at n=4
  \draw[black] (0,0) -- (3.2,0) node[right, font=\tiny, text=black] {$n$};
  \fill[acc!40] (1.7,0) rectangle (2.0,2.4);
  \draw[acc] (1.7,0) rectangle (2.0,2.4);
  \node[acc, anchor=south, font=\tiny] at (1.6,2.45) {number};
  % coherent: Poisson envelope
  \begin{scope}[xshift=4.2cm]
    \draw[black] (0,0) -- (3.4,0) node[right, font=\tiny, text=black] {$n$};
    \foreach \x/\h in {0/0.1,1/0.4,2/0.8,3/1.05,4/1.05,5/0.85,6/0.55,7/0.32,8/0.16} {
      \fill[acc!30] (\x*0.36+0.1,0) rectangle (\x*0.36+0.34,\h);
      \draw[acc] (\x*0.36+0.1,0) rectangle (\x*0.36+0.34,\h);
    }
    \node[acc, anchor=south, font=\tiny] at (1.5,1.25) {coherent};
  \end{scope}
  % thermal: monotonic decay
  \begin{scope}[xshift=8.6cm]
    \draw[black] (0,0) -- (3.4,0) node[right, font=\tiny, text=black] {$n$};
    \foreach \x/\h in {0/1.4,1/1.1,2/0.86,3/0.67,4/0.52,5/0.41,6/0.32,7/0.25,8/0.19} {
      \fill[acc!30] (\x*0.36+0.1,0) rectangle (\x*0.36+0.34,\h);
      \draw[acc] (\x*0.36+0.1,0) rectangle (\x*0.36+0.34,\h);
    }
    \node[acc, anchor=south, font=\tiny] at (1.7,1.5) {thermal};
  \end{scope}
\end{tikzpicture}
$$

The coherent state is the boundary case: the most classical pure state of the field,
and the reference against which quantum-enhanced measurement is defined.

[^ct-gv]: **Cohen-Tannoudji, Diu & Laloë**, _Quantum Mechanics_ Vol. I (Wiley, 1977), Complement $G_V$ — quasi-classical (coherent) states: the eigenvalue condition $\hat a\lvert\alpha\rangle = \alpha\lvert\alpha\rangle$, the number-state expansion, minimum uncertainty, and classical-limit oscillation. Original construction: R. J. Glauber, "Coherent and Incoherent States of the Radiation Field," _Phys. Rev._ **131**, 2766 (1963), https://journals.aps.org/pr/abstract/10.1103/PhysRev.131.2766; and E. Schrödinger, _Naturwissenschaften_ **14**, 664 (1926).
[^sn-shot]: **Shankar**, _Principles of Quantum Mechanics_ (2nd ed., Springer, 1994), Ch. 7 problems and Ch. 21 — coherent states as minimum-uncertainty wave packets tracking the classical trajectory; https://link.springer.com/book/10.1007/978-1-4757-0576-8. Poissonian counting statistics and shot noise: **Sakurai & Napolitano**, _Modern Quantum Mechanics_ (3rd ed., Cambridge, 2021), §2.3.
[^sn-sqz]: Squeezed light below the vacuum limit and its use in precision interferometry: the LIGO Collaboration, "A gravitational wave observatory operating beyond the quantum shot-noise limit," _Nature Physics_ **7**, 962 (2011), https://www.nature.com/articles/nphys2083. Formal treatment of the squeeze operator $\hat S(\xi)$ and quadrature variances: **Cohen-Tannoudji, Diu & Laloë**, Vol. I, Complement $G_V$.