Coherent-State Dynamics

A coherent state of the harmonic oscillator is a minimum-uncertainty Gaussian whose center follows the classical oscillator orbit without changing shape. This exact stability under harmonic evolution makes coherent states a model of classical-like quantum dynamics, but not classical states: they remain nonorthogonal, retain irreducible fluctuations, and can leave the coherent-state family under squeezing or nonlinear interactions.

Coherent States owns the Hilbert-space construction, number-basis expansion, and displacement-operator identities. Coherent States in Phase Space owns the Wigner-function geometry. This page owns the time evolution and the conditions under which coherent dynamics is preserved or lost. Coherent-State Semiclassics Preview compares this exact dynamics with projected one-packet motion, coherent-state saddle propagation, and frozen-Gaussian trajectory sums.

When the chosen coherent or Gaussian family is not exactly invariant, the Time-Dependent Variational Principle supplies the canonical tangent-space projection and residual diagnostics.

Minimal Recap

For an oscillator with

$$H_0 = \hbar\omega \left( a^\dagger a+\frac{1}{2} \right),$$

a coherent state satisfies

$$a\lvert\alpha\rangle = \alpha\lvert\alpha\rangle.$$

Equivalently,

$$\lvert\alpha\rangle = D(\alpha)\lvert0\rangle,$$

where

$$D(\alpha) = \exp\left( \alpha a^\dagger-\alpha^*a \right).$$

The coherent label is related to the first moments by

$$x_c = \sqrt{2}\,\ell\, \operatorname{Re}\alpha,$$

and

$$p_c = \frac{\sqrt{2}\hbar}{\ell} \operatorname{Im}\alpha,$$

with oscillator length

$$\ell = \sqrt{\frac{\hbar}{m\omega}}.$$

The label $\alpha$ is dimensionless. It packages a position and momentum into one complex phase-space coordinate adapted to the oscillator.

Exact Harmonic Evolution

Let

$$U_0(t) = e^{-iH_0t/\hbar}.$$

The annihilation operator evolves as

$$U_0^\dagger(t)aU_0(t) = ae^{-i\omega t}.$$

Act on the evolved state with $a$:

$$\begin{aligned} aU_0(t)\lvert\alpha\rangle &= U_0(t) \left[ U_0^\dagger(t)aU_0(t) \right] \lvert\alpha\rangle \\ &= \alpha e^{-i\omega t} U_0(t)\lvert\alpha\rangle. \end{aligned}$$

Therefore $U_0(t)\lvert\alpha\rangle$ is another coherent state with label $\alpha e^{-i\omega t}$. Including the zero-point phase,

$$U_0(t)\lvert\alpha\rangle = e^{-i\omega t/2} \lvert\alpha e^{-i\omega t}\rangle.$$

The global phase does not affect ordinary expectation values, but it matters when this branch interferes with another amplitude.

The coherent-state family is therefore invariant under harmonic evolution:

$$\alpha(t) = \alpha_0e^{-i\omega t}.$$

Classical Phase-Space Motion

Writing

$$\alpha_0 = \frac{x_0}{\sqrt{2}\ell} + i\frac{\ell p_0}{\sqrt{2}\hbar},$$

the first moments evolve as

$$x_c(t) = x_0\cos\omega t + \frac{p_0}{m\omega}\sin\omega t,$$

and

$$p_c(t) = p_0\cos\omega t - m\omega x_0\sin\omega t.$$

These are exactly the classical harmonic-oscillator equations. In scaled phase-space coordinates, the label rotates on a circle; in dimensional $(x,p)$ coordinates, the center follows the classical energy ellipse.

The mean energy is

$$\langle H_0\rangle_\alpha = \hbar\omega \left( \lvert\alpha\rvert^2+\frac{1}{2} \right).$$

The classical contribution is associated with the action scale

$$J \sim \hbar\lvert\alpha\rvert^2.$$

Large $\lvert\alpha\rvert$ therefore corresponds to large action compared with $\hbar$, while the zero-point contribution remains.

Fixed Covariance and Minimum Uncertainty

For every coherent state,

$$(\Delta x)^2 = \frac{\hbar}{2m\omega},$$

and

$$(\Delta p)^2 = \frac{m\hbar\omega}{2}.$$

The symmetrized covariance is zero in the natural oscillator axes:

$$C_{xp}=0.$$

Hence

$$\Delta x\,\Delta p = \frac{\hbar}{2}.$$

Under harmonic evolution these variances remain constant. The Wigner Gaussian translates around the classical orbit without shearing or changing area.

If the classical position amplitude is

$$A_x = \sqrt{2}\,\ell\lvert\alpha\rvert,$$

then

$$\frac{\Delta x}{A_x} = \frac{1}{2\lvert\alpha\rvert}.$$

Thus the packet becomes narrow relative to its orbit for $\lvert\alpha\rvert\gg1$. This is a relative classicality statement. The absolute quantum width does not vanish at fixed $m$ and $\omega$.

Number Statistics and Relative Fluctuations

The occupation-number distribution is Poissonian:

$$P_\alpha(n) = e^{-\lvert\alpha\rvert^2} \frac{\lvert\alpha\rvert^{2n}}{n!}.$$

Therefore

$$\langle n\rangle = \lvert\alpha\rvert^2,$$

and

$$\Delta n = \lvert\alpha\rvert.$$

The relative number fluctuation is

$$\frac{\Delta n}{\langle n\rangle} = \frac{1}{\lvert\alpha\rvert}.$$

Large-amplitude coherent states have small relative number fluctuations even though their absolute fluctuations grow. This scaling is one reason classical fields can emerge from highly occupied coherent modes without eliminating quantum shot noise.

Overlap and Recurrence

Coherent states are not orthogonal:

$$\langle\alpha\vert\beta\rangle = \exp\left[ -\frac{\lvert\alpha\rvert^2}{2} -\frac{\lvert\beta\rvert^2}{2} +\alpha^*\beta \right].$$

The survival probability under harmonic evolution is

$$\begin{aligned} \left\lvert \langle\alpha\vert U_0(t)\vert\alpha\rangle \right\rvert^2 &= \exp\left[ -4\lvert\alpha\rvert^2 \sin^2\left( \frac{\omega t}{2} \right) \right]. \end{aligned}$$

For large $\lvert\alpha\rvert$, the state rapidly becomes nearly orthogonal to its initial state as the packet center moves away, even though it remains perfectly coherent. After one oscillator period,

$$T=\frac{2\pi}{\omega},$$

the label returns and the survival probability is one. Coherent evolution combines classical orbital recurrence with quantum overlap geometry.

Driven Harmonic Oscillator

Add a prescribed linear drive:

$$\begin{aligned} H(t) &= \hbar\omega \left( a^\dagger a+\frac{1}{2} \right) \\ &\quad+ \hbar \left[ \eta(t)a^\dagger +\eta^*(t)a \right], \end{aligned}$$

where $\eta(t)$ has units of angular frequency. A coherent initial state remains coherent up to a global phase:

$$\lvert\psi(t)\rangle = e^{i\phi(t)} \lvert\alpha(t)\rangle.$$

The label obeys the classical linear equation

$$i\dot\alpha = \omega\alpha+\eta(t),$$

or

$$\dot\alpha = -i\omega\alpha-i\eta(t).$$

With initial time $t_0$,

$$\begin{aligned} \alpha(t) &= e^{-i\omega(t-t_0)} \alpha(t_0) \\ &\quad- i \int_{t_0}^{t} ds\, e^{-i\omega(t-s)} \eta(s). \end{aligned}$$

The first term is free rotation. The integral is the displacement produced by the drive. The phase $\phi(t)$ includes the classical action and ordering information but does not change the coherent label.

For a position force $-F(t)x$, one may identify

$$\eta(t) = -\frac{F(t)\ell} {\sqrt{2}\hbar}$$

in the convention used above. The resulting first moments obey the forced classical oscillator equation

$$m\ddot x_c + m\omega^2x_c = F(t).$$

Linear forcing therefore moves the Gaussian center without changing its intrinsic covariance.

Which Hamiltonians Preserve Coherence?

The coherent-state family is preserved by Hamiltonians that generate phase rotations and displacements:

$$H(t) = \hbar\omega(t)a^\dagger a + \hbar \left[ \eta(t)a^\dagger +\eta^*(t)a \right] + c(t)I,$$

with suitable time ordering. The state remains coherent, although its label and global phase change.

A general quadratic Hamiltonian can also contain

$$\frac{\hbar}{2} \left[ \zeta(t)a^{\dagger 2} +\zeta^*(t)a^2 \right].$$

These parametric terms mix $a$ and $a^\dagger$. Gaussianity is preserved, but a coherent state generally becomes a squeezed coherent state. The coherent family is not invariant.

An anharmonic term such as

$$H_{\rm Kerr} \propto (a^\dagger a)^2$$

assigns nonlinear phases to number states. An initial coherent state then shears in phase space and can develop collapse, revival, and cat-like superpositions. Coherence in the technical Glauber sense is not preserved by generic nonlinear dynamics.

Why Coherent States Look Classical

Coherent states combine several classical-like features:

• their first moments follow the classical oscillator equations;
• their covariance remains minimal and fixed under harmonic motion;
• large amplitudes make relative position and number fluctuations small;
• a linear drive transports them according to the forced classical equation;
• normally ordered field correlations factorize for ideal coherent radiation.

They remain quantum because

• $\Delta x\Delta p=\hbar/2$, not zero;
• different coherent states have nonzero overlap;
• number fluctuations and shot noise remain;
• coherent superpositions can interfere;
• nonlinear dynamics can generate squeezing, entanglement, and non-Gaussian states.

The correct phrase is classical-like quantum state, not classical state.

Coherent-State Path-Integral Preview

Canonical coherent states resolve the identity:

$$I = \int_{\mathbb C} \frac{d^2\alpha}{\pi} \lvert\alpha\rangle \langle\alpha\rvert.$$

Inserting this overcomplete identity between short-time evolution factors produces a path integral over complex labels. A common continuum action is

$$\begin{aligned} S[\alpha^*,\alpha] &= \int_{t_i}^{t_f}dt \Bigg[ \frac{i\hbar}{2} \left( \alpha^*\dot\alpha -\dot\alpha^*\alpha \right) \\ &\qquad\qquad- H_{\rm cs}(\alpha^*,\alpha) \Bigg], \end{aligned}$$

together with endpoint terms determined by the boundary convention. The Hamiltonian symbol $H_{\rm cs}$ depends on the operator ordering and time-slicing prescription.

Stationary variation gives

$$i\hbar\dot\alpha = \frac{\partial H_{\rm cs}} {\partial\alpha^*},$$

and

$$-i\hbar\dot\alpha^* = \frac{\partial H_{\rm cs}} {\partial\alpha}.$$

For the harmonic oscillator these are the classical complex phase-space equations. The preview is useful because it explains why coherent-state labels become field variables in many-body and QFT path integrals. It is not a license to ignore boundary terms, overcompleteness, or symbol-ordering corrections.

Why Path Integrals? supplies the general amplitude and action logic. Coherent-State Path Integrals Preview fixes the thermal trace, finite-slice discretization, Grassmann boundary sign, and operator symbol before assigning a continuum action.

Field Coherent States Preview

A free bosonic field decomposes into oscillator modes. A multimode coherent state satisfies

$$a_{\mathbf k} \lvert\{\alpha\}\rangle = \alpha_{\mathbf k} \lvert\{\alpha\}\rangle$$

for every retained mode. Under a free field Hamiltonian,

$$\alpha_{\mathbf k}(t) = \alpha_{\mathbf k}(0) e^{-i\omega_{\mathbf k}t}.$$

Field expectation values built from these amplitudes obey the corresponding classical linear field equations. This is the field analogue of the oscillator center following its classical orbit.

The state is still quantum. Vacuum and shot-noise fluctuations remain, coherent states are not number eigenstates, and interacting evolution generally entangles modes and drives the state away from a simple product of coherent states. Gauge fields also require physical-mode and constraint care.

The oscillator-to-field dictionary continues in Harmonic Oscillator to Fields.

Common Mistakes

• Treating $\alpha$ as a dimensional position rather than a complex oscillator coordinate.
• Dropping the zero-point phase when comparing interfering amplitudes.
• Saying minimum uncertainty means zero uncertainty.
• Assuming every minimum-uncertainty state is an unsqueezed coherent state.
• Calling Poissonian number fluctuations noise-free.
• Assuming every quadratic Hamiltonian preserves coherent states; parametric terms generally squeeze them.
• Assuming a coherent state remains coherent under an anharmonic interaction.
• Writing a coherent-state path integral without specifying endpoint and ordering conventions.
• Treating a large field expectation value as proof that quantum fluctuations and entanglement are absent.

References

• E. Schrödinger, “Der stetige Übergang von der Mikro- zur Makromechanik,” Die Naturwissenschaften 14, 664–666, 1926.
• R. J. Glauber, “Coherent and incoherent states of the radiation field,” Physical Review 131, 2766–2788, 1963, doi:10.1103/PhysRev.131.2766.
• J. R. Klauder and B.-S. Skagerstam, eds., Coherent States: Applications in Physics and Mathematical Physics, World Scientific, 1985.
• A. Perelomov, Generalized Coherent States and Their Applications, Springer, 1986.
• G. A. Hagedorn, “Semiclassical quantum mechanics. I. The $\hbar\to0$ limit for coherent states,” Communications in Mathematical Physics 71, 77–93, 1980.
• W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.

Exercises

1. Prove that harmonic evolution maps a coherent state to another coherent state.

Solution

Use

$$U_0^\dagger(t)aU_0(t) = ae^{-i\omega t}.$$

Then

$$\begin{aligned} aU_0(t)\lvert\alpha\rangle &= U_0(t) \left[ U_0^\dagger(t)aU_0(t) \right] \lvert\alpha\rangle \\ &= \alpha e^{-i\omega t} U_0(t)\lvert\alpha\rangle. \end{aligned}$$

Thus the evolved state is an eigenstate of $a$ with eigenvalue $\alpha e^{-i\omega t}$. Comparing the vacuum component or using the number expansion gives

$$U_0(t)\lvert\alpha\rangle = e^{-i\omega t/2} \lvert\alpha e^{-i\omega t}\rangle.$$

2. Derive the classical first-moment trajectory from $\alpha(t)=\alpha_0e^{-i\omega t}$.

Solution

Write

$$\alpha_0 = \frac{x_0}{\sqrt{2}\ell} + i\frac{\ell p_0}{\sqrt{2}\hbar}.$$

Using

$$x_c(t) = \sqrt{2}\ell\, \operatorname{Re}\alpha(t)$$

and

$$p_c(t) = \frac{\sqrt{2}\hbar}{\ell} \operatorname{Im}\alpha(t),$$

expand $e^{-i\omega t}=\cos\omega t-i\sin\omega t$. This gives

$$x_c(t) = x_0\cos\omega t + \frac{p_0}{m\omega}\sin\omega t,$$

and

$$p_c(t) = p_0\cos\omega t - m\omega x_0\sin\omega t.$$

3. Compute the harmonic survival probability.

Solution

The global phase cancels from the modulus, so

$$\begin{aligned} \left\lvert \langle\alpha\vert U_0(t)\vert\alpha\rangle \right\rvert^2 &= \left\lvert \langle\alpha \vert \alpha e^{-i\omega t} \rangle \right\rvert^2 \\ &= \exp\left[ -\left\lvert \alpha-\alpha e^{-i\omega t} \right\rvert^2 \right]. \end{aligned}$$

Now

$$\left\lvert 1-e^{-i\omega t} \right\rvert^2 = 4\sin^2\left( \frac{\omega t}{2} \right).$$

Therefore

$$\left\lvert \langle\alpha\vert U_0(t)\vert\alpha\rangle \right\rvert^2 = \exp\left[ -4\lvert\alpha\rvert^2 \sin^2\left( \frac{\omega t}{2} \right) \right].$$

4. A resonant drive has $\eta(t)=\eta_0e^{-i\omega t}$ for $0\leq t\leq T$. Find $\alpha(T)$.

Solution

With $t_0=0$,

$$\begin{aligned} \alpha(T) &= e^{-i\omega T}\alpha_0 \\ &\quad- i \int_0^Tds\, e^{-i\omega(T-s)} \eta_0e^{-i\omega s}. \end{aligned}$$

The integrand is $\eta_0e^{-i\omega T}$, so

$$\alpha(T) = e^{-i\omega T} \left( \alpha_0-i\eta_0T \right).$$

In the frame rotating at $\omega$, the coherent amplitude is displaced linearly with time.

5. Show that the relative number fluctuation vanishes for large coherent amplitude.

Solution

For a Poisson distribution,

$$\langle n\rangle = \lvert\alpha\rvert^2,$$

and

$$(\Delta n)^2 = \lvert\alpha\rvert^2.$$

Hence

$$\frac{\Delta n}{\langle n\rangle} = \frac{\lvert\alpha\rvert} {\lvert\alpha\rvert^2} = \frac{1}{\lvert\alpha\rvert}.$$

This tends to zero as $\lvert\alpha\rvert\to\infty$, although the absolute fluctuation $\Delta n=\lvert\alpha\rvert$ grows.

6. Why does a parametric term proportional to $a^{\dagger2}+a^2$ preserve Gaussianity but not the coherent-state family?

Solution

A quadratic Hamiltonian implements a linear symplectic transformation of the quadratures, so a Gaussian Wigner function remains Gaussian. A parametric term mixes $a$ and $a^\dagger$ through a Bogoliubov transformation. That transformation changes the covariance matrix and generally creates squeezing. An ordinary coherent state is a displaced ground-state Gaussian with the unsqueezed ground covariance, so the evolved squeezed Gaussian is not an eigenstate of the original $a$ operator and is not an ordinary coherent state.