Coherent States in Phase Space

Coherent states are the harmonic-oscillator states whose Wigner functions are ground-state Gaussian ellipses displaced in phase space. Their center follows the classical oscillator trajectory, while their covariance remains fixed. This makes them a natural bridge between Hilbert-space quantum mechanics and phase-space dynamics.

The Hilbert-space construction is canonical in Coherent States. This page focuses on what the same states look like in Wigner phase space and why they are useful in semiclassical reasoning, coherent-state path integrals, and field-theory previews.

Coherent States as Minimum-Uncertainty Gaussians

For the harmonic oscillator, define

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

The canonical page uses

$$\hat x = \frac{\ell}{\sqrt2} (\hat a+\hat a^\dagger), \qquad \hat p = \frac{\hbar}{i\ell\sqrt2} (\hat a-\hat a^\dagger).$$

A coherent state satisfies

$$\hat a|\alpha\rangle = \alpha|\alpha\rangle.$$

Its phase-space center is

$$x_0 = \sqrt2\,\ell\,\operatorname{Re}\alpha, \qquad p_0 = \frac{\sqrt2\,\hbar}{\ell} \operatorname{Im}\alpha.$$

Equivalently,

$$\alpha = \frac{x_0}{\sqrt2\,\ell} + i\frac{\ell p_0}{\sqrt2\,\hbar}.$$

The covariance matrix is the oscillator ground-state covariance:

$$V_0 = \begin{pmatrix} \dfrac{\hbar}{2m\omega} & 0\\ 0 & \dfrac{m\hbar\omega}{2} \end{pmatrix},$$

so

$$\det V_0 = \frac{\hbar^2}{4}.$$

Thus coherent states are minimum-uncertainty Gaussian states. They are not the only minimum-uncertainty states; squeezed states also saturate the bound but have different covariance matrices.

Phase-Space Localization

The Wigner function of a coherent state is

$$W_\alpha(x,p) = \frac{1}{\pi\hbar} \exp\left[ -\frac{m\omega(x-x_0)^2}{\hbar} -\frac{(p-p_0)^2}{m\hbar\omega} \right].$$

It is everywhere nonnegative and localized near $(x_0,p_0)$, but it still represents a quantum state. The phase-space area of the Gaussian is fixed by the uncertainty principle, not by ignorance about a hidden classical point.

In dimensionless oscillator quadratures,

$$Q=\frac{x}{\ell}, \qquad P=\frac{\ell p}{\hbar},$$

the center is

$$Q_0=\sqrt2\,\operatorname{Re}\alpha, \qquad P_0=\sqrt2\,\operatorname{Im}\alpha,$$

and the dimensionless density $\widetilde W_\alpha(Q,P)=\hbar W_\alpha(x,p)$ becomes a circular Gaussian in the $(Q,P)$ plane:

$$\widetilde W_\alpha(Q,P) = \frac{1}{\pi} \exp\left[ -(Q-Q_0)^2-(P-P_0)^2 \right],$$

normalized with respect to $dQ\,dP$. Equivalently, $dx\,dp=\hbar\,dQ\,dP$ in the dimensional variables. The circular shape is one reason the complex number $\alpha$ is the natural phase-space coordinate for the oscillator.

Classical Trajectories

Under the harmonic-oscillator Hamiltonian,

$$H = \hbar\omega \left( \hat a^\dagger\hat a+\frac12 \right),$$

a coherent state stays coherent:

$$\alpha(t) = \alpha(0)e^{-i\omega t},$$

up to an overall phase in the state vector. Therefore the Wigner function stays Gaussian and its center follows

$$x_c(t) = x_c(0)\cos\omega t + \frac{p_c(0)}{m\omega}\sin\omega t,$$

and

$$p_c(t) = p_c(0)\cos\omega t - m\omega x_c(0)\sin\omega t.$$

The covariance matrix remains $V_0$. In phase-space language, the coherent-state Wigner ellipse rotates rigidly along the classical oscillator orbit. There is no spreading because the harmonic oscillator is quadratic, and Wigner evolution is exactly classical for quadratic Hamiltonians.

The exact state-vector phase, linear-drive solution, and preservation criterion are developed in Coherent-State Dynamics. This page retains the canonical Wigner-function geometry.

Overlap in Phase Space

Coherent states are localized but not orthogonal. The Hilbert-space overlap is

$$|\langle\beta|\alpha\rangle|^2 = e^{-|\alpha-\beta|^2}.$$

The Wigner representation gives the same result through the trace formula

$$\operatorname{Tr}(\rho_\alpha\rho_\beta) = 2\pi\hbar \int dx\,dp\, W_\alpha(x,p)W_\beta(x,p).$$

Since $\rho_\alpha=|\alpha\rangle\langle\alpha|$, this trace is the overlap probability. Phase-space localization therefore has a quantum resolution limit: two coherent-state Gaussians can have small overlap when they are far apart, but they are never exactly disjoint at finite separation.

Displacement Versus Squeezing

Coherent states and squeezed states are both Gaussian, but they change different data:

Operation	First moment $\mathbf d$	Covariance $V$
Displacement	changes	unchanged
Squeezing	may change, but not necessarily	changes
Harmonic evolution	rotates	rotates or remains invariant in scaled coordinates

A coherent state is a displaced ground state. A squeezed state reshapes the uncertainty ellipse. The distinction matters because “localized in phase space” and “minimum uncertainty” do not uniquely identify a coherent state unless the covariance is the oscillator ground-state covariance.

Coherent-State Path-Integral Preview

Coherent states form an overcomplete family. For the harmonic oscillator,

$$I = \int\frac{d^2\alpha}{\pi}\, |\alpha\rangle\langle\alpha|.$$

Inserting this identity repeatedly gives a coherent-state path-integral representation. Schematically, the action contains a phase-space term of the form

$$S[\alpha^*,\alpha] = \int dt\, \left[ i\hbar\,\alpha^*\dot\alpha - H(\alpha^*,\alpha) \right],$$

with important convention and ordering qualifications. The symbol $H(\alpha^*,\alpha)$ depends on whether one uses normal, antinormal, or Weyl ordering. Coherent-State Path Integrals Preview owns the regulated thermal trace, Grassmann construction, determinant checks, and ordering diagnostics.

The reason coherent states are useful here is geometric: paths of $\alpha(t)$ are paths through the oscillator phase plane, and the overcomplete resolution of identity lets one build amplitudes from phase-space labels instead of position eigenstates.

QFT Coherent-State Preview

Free quantum fields decompose into oscillator modes. A many-mode coherent state is specified by mode amplitudes $\alpha_{\mathbf k}$ and satisfies schematically

$$\hat a_{\mathbf k} |\{\alpha\}\rangle = \alpha_{\mathbf k} |\{\alpha\}\rangle.$$

Such states are quantum states with classical-looking field expectation values. They are useful in quantum optics, semiclassical field theory, and discussions of classical radiation. The field-theory story adds infinitely many modes, normalization subtleties, gauge constraints for photons, and renormalization issues. The oscillator-to-field bridge begins in Harmonic Oscillator to Fields.

Common Mistakes

• Treating $\alpha$ as a dimensional position coordinate rather than a dimensionless oscillator phase-space label.
• Saying coherent states are classical states rather than quantum states with classical-looking first moments.
• Confusing displacement with squeezing.
• Forgetting that coherent states are overcomplete and nonorthogonal.
• Assuming every minimum-uncertainty state is a coherent state of the chosen oscillator.
• Ignoring ordering conventions in coherent-state path integrals.

Cross-Links

• Coherent States
• Coherent-State Dynamics
• Coherent-State Path Integrals
• Coherent-State Semiclassics Preview
• Gaussian States and Wigner Functions
• Wigner Function
• Phase-Space Dynamics
• Quantum Harmonic Oscillator
• Squeezed States: First Encounter
• Harmonic Oscillator to Fields
• Second Quantization
• Harmonic-Oscillator Propagator Notebook

References

• R. J. Glauber, “Coherent and Incoherent States of the Radiation Field,” Physical Review 131, 2766-2788, 1963.
• J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics, World Scientific, 1985.
• A. Perelomov, Generalized Coherent States and Their Applications, Springer, 1986.
• W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, 2001.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.

Exercises

1. Derive the relation between $\alpha$ and the phase-space center $(x_0,p_0)$.

Solution

Using

$$\hat x = \frac{\ell}{\sqrt2} (\hat a+\hat a^\dagger), \qquad \hat p = \frac{\hbar}{i\ell\sqrt2} (\hat a-\hat a^\dagger),$$

and $\langle\hat a\rangle=\alpha$, one obtains

$$x_0 = \frac{\ell}{\sqrt2} (\alpha+\alpha^*) = \sqrt2\,\ell\,\operatorname{Re}\alpha,$$

and

$$p_0 = \frac{\hbar}{i\ell\sqrt2} (\alpha-\alpha^*) = \frac{\sqrt2\,\hbar}{\ell} \operatorname{Im}\alpha.$$

Solving gives

$$\alpha = \frac{x_0}{\sqrt2\,\ell} + i\frac{\ell p_0}{\sqrt2\,\hbar}.$$

2. Show that the coherent-state Wigner function is normalized.

Solution

The Wigner function factors into two Gaussians:

$$W_\alpha(x,p) = \frac{1}{\pi\hbar} e^{-m\omega(x-x_0)^2/\hbar} e^{-(p-p_0)^2/(m\hbar\omega)}.$$

The integrals are

$$\int dx\,e^{-m\omega(x-x_0)^2/\hbar} = \left(\frac{\pi\hbar}{m\omega}\right)^{1/2},$$

and

$$\int dp\,e^{-(p-p_0)^2/(m\hbar\omega)} = (\pi m\hbar\omega)^{1/2}.$$

Multiplying them with $1/(\pi\hbar)$ gives $1$.

3. Verify that harmonic evolution rotates $\alpha$.

Solution

The annihilation operator evolves as

$$\hat a_H(t) = e^{-i\omega t}\hat a.$$

Equivalently, a Schrödinger-picture coherent state remains coherent with label

$$\alpha(t) = \alpha(0)e^{-i\omega t},$$

up to an overall phase. Substituting this into the formulas for $x_0$ and $p_0$ gives the classical oscillator trajectory.

4. Use the Wigner trace formula to explain why well-separated coherent states are nearly orthogonal but not exactly orthogonal.

Solution

For pure coherent states,

$$|\langle\beta|\alpha\rangle|^2 = \operatorname{Tr}(\rho_\alpha\rho_\beta) = 2\pi\hbar \int dx\,dp\, W_\alpha(x,p)W_\beta(x,p).$$

The integral of two displaced Gaussians is exponentially small when their centers are far apart, giving

$$|\langle\beta|\alpha\rangle|^2 = e^{-|\alpha-\beta|^2}.$$

For finite $|\alpha-\beta|$, this number is nonzero. It tends to zero only asymptotically as the phase-space separation becomes large.