Coherent States

A coherent state is a harmonic-oscillator state that remains a minimum-uncertainty Gaussian while its center follows the classical oscillator motion. Algebraically, it is an eigenstate of the annihilation operator:

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

Coherent states are the canonical oscillator states closest to classical motion. They are not energy eigenstates; they are special superpositions of number states.

For electromagnetic modes, photon statistics, optical coherence, realistic laser fields, and photodetection, continue to Coherent Light.

Number-Basis Expansion

Expand a coherent state in number states:

$$\lvert\alpha\rangle =\sum_{n=0}^{\infty}c_n\lvert n\rangle.$$

The eigenvalue equation $\hat a\lvert\alpha\rangle=\alpha\lvert\alpha\rangle$ and the ladder action $\hat a\lvert n\rangle=\sqrt n\,\lvert n-1\rangle$ imply

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

Normalization fixes

$$c_0=e^{-\lvert\alpha\rvert^2/2},$$

so

$$\lvert\alpha\rangle =e^{-\lvert\alpha\rvert^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}}\lvert n\rangle.$$

This formula shows that a coherent state contains many oscillator number states unless $\alpha=0$.

Poisson Number Distribution

The probability to measure $n$ oscillator quanta is

$$P(n) =\lvert\langle n\vert\alpha\rangle\rvert^2 =e^{-\lvert\alpha\rvert^2} \frac{\lvert\alpha\rvert^{2n}}{n!}.$$

This is a Poisson distribution with mean

$$\langle \hat N\rangle=\lvert\alpha\rvert^2,$$

and variance

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

The relative number fluctuation is

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

For large $\lvert\alpha\rvert$, the relative fluctuation is small, one reason coherent states can look classical.

Displaced Ground State

Coherent states can also be constructed by displacing the oscillator ground state. Define the displacement operator

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

Then

$$\lvert\alpha\rangle=\hat D(\alpha)\lvert0\rangle.$$

This makes the geometry clear: a coherent state is the ground-state Gaussian shifted in phase space, without changing its minimum-uncertainty width.

Position And Momentum Means

Using

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

with $\ell=\sqrt{\hbar/(m\omega)}$, the coherent-state means are

$$\langle x\rangle_\alpha =\sqrt2\,\ell\,\operatorname{Re}\alpha, \qquad \langle p\rangle_\alpha =\frac{\sqrt2\,\hbar}{\ell}\operatorname{Im}\alpha.$$

Thus the complex number $\alpha$ labels a point in the oscillator phase plane.

The uncertainties are the same as the ground-state uncertainties:

$$\Delta x=\frac{\ell}{\sqrt2}, \qquad \Delta p=\frac{\hbar}{\sqrt2\,\ell}.$$

Therefore

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

Coherent states are minimum-uncertainty states, but they are not the only minimum-uncertainty states.

Time Evolution

Under the harmonic-oscillator Hamiltonian,

$$\hat H=\hbar\omega\left(\hat N+\frac12\right),$$

a coherent state remains coherent. Apart from an overall phase,

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

The mean position and momentum therefore obey

$$\langle x\rangle(t) =\sqrt2\,\ell\, \operatorname{Re}\left(\alpha(0)e^{-i\omega t}\right),$$

and

$$\langle p\rangle(t) =\frac{\sqrt2\,\hbar}{\ell}\, \operatorname{Im}\left(\alpha(0)e^{-i\omega t}\right).$$

These are the classical oscillator equations for the packet center. The probability density does not spread because the harmonic potential refocuses the packet exactly.

Coherent-State Dynamics develops the exact state-vector phase, driven evolution, survival probability, and the criterion separating coherence-preserving motion from squeezing and nonlinear shearing.

Nonorthogonality

Coherent states are normalized but not mutually orthogonal:

$$\langle\beta\vert\alpha\rangle =\exp\left[ -\frac12\lvert\beta\rvert^2 -\frac12\lvert\alpha\rvert^2 +\beta^*\alpha \right].$$

The overlap magnitude is

$$\lvert\langle\beta\vert\alpha\rangle\rvert^2 =e^{-\lvert\alpha-\beta\rvert^2}.$$

Widely separated coherent states are nearly orthogonal, but distinct coherent states are never exactly orthogonal at finite separation in phase space.

The family is nevertheless complete in the overcomplete sense:

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

This resolution of the identity is not an orthonormal-basis expansion. Its continuous coefficients are redundant, which is both useful and a source of care when coherent-state path integrals are constructed.

Displacement Algebra

The displacement operator is unitary and translates the ladder operators:

$$D(\alpha)^\dagger=D(-\alpha), \qquad D(\alpha)^\dagger aD(\alpha)=a+\alpha.$$

Because the commutator in the exponent is a scalar, the Baker–Campbell–Hausdorff formula gives

$$D(\alpha) =e^{-\lvert\alpha\rvert^2/2} e^{\alpha a^\dagger}e^{-\alpha^*a}.$$

Two translations compose with a phase,

$$D(\alpha)D(\beta) =\exp\!\left[ \frac{\alpha\beta^*-\alpha^*\beta}{2} \right]D(\alpha+\beta),$$

so their noncommutativity records the oriented phase-space area enclosed by the translations.

Coordinate Wavefunction and Energy Moments

With $q_\alpha=\langle x\rangle$ and $p_\alpha=\langle p\rangle$, the normalized coordinate wavefunction may be written

$$\psi_\alpha(x) =\left(\frac{1}{\pi\ell^2}\right)^{1/4} \exp\!\left[-\frac{(x-q_\alpha)^2}{2\ell^2} +\frac{ip_\alpha x}{\hbar} -\frac{iq_\alpha p_\alpha}{2\hbar}\right].$$

The last term is an $x$-independent phase fixed by the displacement convention. It cancels from one-state position probabilities but matters in interference and overlap calculations.

For $H=\hbar\omega(N+1/2)$,

$$\langle H\rangle =\hbar\omega\left(\lvert\alpha\rvert^2+\frac12\right), \qquad (\Delta H)^2=(\hbar\omega)^2\lvert\alpha\rvert^2.$$

Thus a nonvacuum coherent state is never an energy eigenstate.

Normally Ordered Moments

For a normally ordered function,

$$\langle\alpha\rvert{:}f(a^\dagger,a){:}\lvert\alpha\rangle =f(\alpha^*,\alpha).$$

Normal ordering is essential: $\langle aa^\dagger\rangle =\lvert\alpha\rvert^2+1$, not $\lvert\alpha\rvert^2$. The extra term is the commutator contribution.

Physical Interpretation

A coherent state is often the right first model for a classical-looking oscillator with quantum width. It has:

• a localized Gaussian wave packet;
• minimum position-momentum uncertainty;
• a phase-space center following classical motion;
• a Poisson distribution of oscillator quanta;
• fixed shape under harmonic time evolution.

In quantum optics, coherent states model idealized laser-like field modes. In mechanical and molecular settings, they model displaced oscillator motion. Field-theory coherent states require additional many-mode structure and belong in the relevant field and quantum-optics treatments.

Common Mistakes

• Treating a coherent state as an energy eigenstate.
• Treating $\alpha$ as a position rather than a dimensionless phase-space label.
• Forgetting the Poisson number distribution.
• Assuming coherent states with different $\alpha$ are exactly orthogonal.
• Confusing coherent states with squeezed states; coherent states have ground-state variances in both quadratures.
• Saying coherent states are fully classical. They are quantum states with particularly classical-looking first moments.

Where This Is Used

• Number States gives the basis used in the coherent-state expansion.
• Quantum Harmonic Oscillator gives the oscillator Hamiltonian and natural scales.
• Displaced Oscillator explains how a linear force shifts the oscillator Hamiltonian and makes the shifted ground state coherent relative to the old oscillator.
• Coherent States in Phase Space shows the Wigner-function ellipse, phase-space center, and path-integral bridge.
• Coherent-State Dynamics treats harmonic and linearly driven evolution and explains when an initially coherent state leaves the coherent-state family.
• Coherent-State Semiclassics Preview distinguishes exact coherent transport, one-packet variational motion, coherent-state saddle propagation, and initial-value representations.
• Phase Space gives the classical comparison language.
• Position-Momentum Uncertainty explains the minimum-uncertainty bound.
• Squeezed States: First Encounter contrasts displacement with reshaping of quadrature uncertainties.
• Correspondence Principle explains why large-amplitude coherent states support classical approximations.
• Harmonic Oscillator to Fields previews how oscillator modes become field modes.
• Harmonic-Oscillator Propagator Notebook verifies coherent evolution by spectral reconstruction and exact phase-space motion.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• R. J. Glauber, ‘Coherent and Incoherent States of the Radiation Field,’ Physical Review 131, 2766-2788 (1963).
• E. C. G. Sudarshan, ‘Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams,’ Physical Review Letters 10, 277-279 (1963).
• J. R. Klauder and B.-S. Skagerstam, Coherent States: Applications in Physics and Mathematical Physics, World Scientific, 1985.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.

Exercises

1. Derive the Poisson number distribution from the number-basis expansion.

Solution

The expansion coefficient is

$$\langle n\vert\alpha\rangle =e^{-\lvert\alpha\rvert^2/2} \frac{\alpha^n}{\sqrt{n!}}.$$

Therefore

$$P(n) =\lvert\langle n\vert\alpha\rangle\rvert^2 =e^{-\lvert\alpha\rvert^2} \frac{\lvert\alpha\rvert^{2n}}{n!}.$$

This is a Poisson distribution with mean $\lvert\alpha\rvert^2$.

2. Show that $\langle x\rangle_\alpha=\sqrt2\,\ell\,\operatorname{Re}\alpha$.

Solution

Use

$$\hat x =\frac{\ell}{\sqrt2} \left(\hat a+\hat a^\dagger\right).$$

Since $\hat a\lvert\alpha\rangle=\alpha\lvert\alpha\rangle$, one has

$$\langle\hat a\rangle=\alpha, \qquad \langle\hat a^\dagger\rangle=\alpha^*.$$

Therefore

$$\langle x\rangle_\alpha =\frac{\ell}{\sqrt2}(\alpha+\alpha^*) =\sqrt2\,\ell\,\operatorname{Re}\alpha.$$

3. Explain why a coherent state with large $\lvert\alpha\rvert$ can look classical even though it is not an energy eigenstate.

Solution

The state remains a localized Gaussian whose center follows the classical oscillator trajectory. Its mean number is $\lvert\alpha\rvert^2$, while its number uncertainty is $\Delta N=\lvert\alpha\rvert$. The relative fluctuation is therefore

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

For large $\lvert\alpha\rvert$, relative fluctuations are small, and the first moments follow classical motion. The state is still a quantum superposition of number states.

4. Explain why replacing $a\to\alpha$ and $a^\dagger\to\alpha^*$ in $aa^\dagger$ gives the wrong expectation value, and compute the correct one.

Solution

The substitution rule applies to normally ordered products. Since

$$aa^\dagger=a^\dagger a+I,$$

one obtains

$$\langle aa^\dagger\rangle =\langle a^\dagger a\rangle+1 =\lvert\alpha\rvert^2+1.$$

Direct substitution before normal ordering misses the commutator term.