Module 6: Coherent States
The Most Classical Quantum States
← Module 5 | ↑ Course Index | Module 7 →
1 Motivation: What is the Most Classical Quantum State?
The energy eigenstates \(|n\rangle\) are weird. For any \(n\):
- \(\langle\hat{x}\rangle = 0\) always (no displacement)
- The probability density is spread out and has nodes
- Nothing seems to oscillate in time
But a real classical oscillator has a definite amplitude and phase. What quantum state most closely resembles a classical oscillating particle?
The answer is the coherent state, introduced by Roy Glauber in 1963 (for which he received the 2005 Nobel Prize in Physics).
Textbook reference: Shankar, Exercise 7.4.9; Cohen-Tannoudji Vol. 1, Complement GV; Griffiths Problem 3.35.
2 Definition: Eigenstates of \(\hat{a}\)
A coherent state \(|\alpha\rangle\) is defined as an eigenstate of the annihilation operator:
\[ \boxed{\hat{a}|\alpha\rangle = \alpha|\alpha\rangle} \]
where \(\alpha\) is a complex number: \(\alpha = |\alpha|e^{i\theta} \in \mathbb{C}\).
\(\hat{a}^\dagger\) has no eigenvalues — there is no normalizable state that satisfies \(\hat{a}^\dagger|\beta\rangle = \beta|\beta\rangle\). (Proof: \(\hat{a}^\dagger\) raises \(n\) without bound, so the series never terminates.)
But \(\hat{a}\) does have eigenvalues — any complex number \(\alpha\) works. The ground state \(|0\rangle\) is the special case \(\alpha = 0\).
3 Expanding Coherent States in the Fock Basis
Let’s write \(|\alpha\rangle = \sum_{n=0}^{\infty} c_n|n\rangle\) and apply \(\hat{a}|\alpha\rangle = \alpha|\alpha\rangle\):
\[ \hat{a}|\alpha\rangle = \sum_n c_n \hat{a}|n\rangle = \sum_n c_n \sqrt{n}|n-1\rangle = \alpha\sum_n c_n|n\rangle \]
Shifting index \(n \to n+1\) on the left:
\[ \sum_n c_{n+1}\sqrt{n+1}|n\rangle = \alpha\sum_n c_n|n\rangle \]
Matching term by term: \(c_{n+1}\sqrt{n+1} = \alpha c_n\)
Recursion: \(c_{n+1} = \frac{\alpha}{\sqrt{n+1}}c_n\)
Solution: \(c_n = \frac{\alpha^n}{\sqrt{n!}}c_0\)
Normalizing: \(\sum_n|c_n|^2 = |c_0|^2\sum_n\frac{|\alpha|^{2n}}{n!} = |c_0|^2 e^{|\alpha|^2} = 1\)
So \(|c_0|^2 = e^{-|\alpha|^2}\) and \(c_n = e^{-|\alpha|^2/2}\frac{\alpha^n}{\sqrt{n!}}\).
\[ \boxed{|\alpha\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle} \]
This is one of the most important equations in quantum optics!
4 The Displacement Operator
An elegant way to construct coherent states: define the displacement operator:
\[ \hat{D}(\alpha) = e^{\alpha\hat{a}^\dagger - \alpha^*\hat{a}} \]
Claim: \(|\alpha\rangle = \hat{D}(\alpha)|0\rangle\)
Proof using Baker-Campbell-Hausdorff (BCH) lemma:
Since \([\alpha\hat{a}^\dagger, -\alpha^*\hat{a}] = |\alpha|^2[\hat{a}^\dagger, \hat{a}] = -|\alpha|^2\) (a c-number), BCH gives:
\[ \hat{D}(\alpha) = e^{\alpha\hat{a}^\dagger - \alpha^*\hat{a}} = e^{-|\alpha|^2/2} e^{\alpha\hat{a}^\dagger} e^{-\alpha^*\hat{a}} \]
Applying to \(|0\rangle\): \(e^{-\alpha^*\hat{a}}|0\rangle = |0\rangle\) (since \(\hat{a}|0\rangle = 0\))
\[ \hat{D}(\alpha)|0\rangle = e^{-|\alpha|^2/2}e^{\alpha\hat{a}^\dagger}|0\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{(\alpha\hat{a}^\dagger)^n}{n!}|0\rangle = e^{-|\alpha|^2/2}\sum_{n=0}^{\infty}\frac{\alpha^n}{\sqrt{n!}}|n\rangle = |\alpha\rangle\ ✓ \]
The displacement operator displaces the ground state to a coherent state:
\[ |\alpha\rangle = \hat{D}(\alpha)|0\rangle \]
Properties of \(\hat{D}(\alpha)\):
- Unitary: \(\hat{D}^\dagger(\alpha) = \hat{D}(-\alpha) = \hat{D}^{-1}(\alpha)\)
- Displacement of \(\hat{a}\): \(\hat{D}^\dagger(\alpha)\hat{a}\hat{D}(\alpha) = \hat{a} + \alpha\)
- Product: \(\hat{D}(\alpha)\hat{D}(\beta) = e^{(\alpha\beta^* - \alpha^*\beta)/2}\hat{D}(\alpha + \beta)\)
5 Physical Properties of Coherent States
5.1 Expectation Values
\[ \langle\alpha|\hat{a}|\alpha\rangle = \alpha, \quad \langle\alpha|\hat{a}^\dagger|\alpha\rangle = \alpha^* \]
Therefore: \[ \langle\hat{x}\rangle = \frac{x_0}{\sqrt{2}}\langle\hat{a} + \hat{a}^\dagger\rangle = \frac{x_0}{\sqrt{2}}(\alpha + \alpha^*) = x_0\sqrt{2}\,\text{Re}(\alpha) \]
\[ \langle\hat{p}\rangle = \frac{i\hbar}{\sqrt{2}x_0}\langle\hat{a}^\dagger - \hat{a}\rangle = \frac{\sqrt{2}\hbar}{x_0}\,\text{Im}(\alpha) \]
Writing \(\alpha = |\alpha|e^{i\phi}\):
\[ \langle\hat{x}\rangle = \sqrt{2}x_0|\alpha|\cos\phi, \qquad \langle\hat{p}\rangle = \sqrt{2}\frac{\hbar}{x_0}|\alpha|\sin\phi \]
5.2 Minimum Uncertainty
The uncertainties in a coherent state:
\[ (\Delta x)^2 = \langle\hat{x}^2\rangle - \langle\hat{x}\rangle^2 = \frac{x_0^2}{2} \]
\[ (\Delta p)^2 = \langle\hat{p}^2\rangle - \langle\hat{p}\rangle^2 = \frac{\hbar^2}{2x_0^2} \]
Therefore:
\[ \Delta x\,\Delta p = \frac{x_0}{{\sqrt{2}}}\cdot\frac{\hbar}{\sqrt{2}x_0} = \frac{\hbar}{2} \]
Coherent states saturate the uncertainty principle: \[\Delta x\,\Delta p = \frac{\hbar}{2}\]
This is the same as the ground state \(|0\rangle\), which is a special case (\(\alpha = 0\)) of a coherent state! All coherent states are minimum uncertainty states.
6 Time Evolution of Coherent States
Computing \(|\alpha(t)\rangle = \hat{U}(t)|\alpha\rangle\):
\[ \hat{U}(t)|\alpha\rangle = e^{-i\omega t/2}\sum_n e^{-|\alpha|^2/2}\frac{\alpha^n}{\sqrt{n!}}e^{-in\omega t}|n\rangle \]
\[ = e^{-i\omega t/2}\sum_n e^{-|\alpha|^2/2}\frac{(\alpha e^{-i\omega t})^n}{\sqrt{n!}}|n\rangle \]
\[ = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle \]
\[ \boxed{\hat{U}(t)|\alpha\rangle = e^{-i\omega t/2}|\alpha(t)\rangle, \quad \alpha(t) = \alpha e^{-i\omega t}} \]
A coherent state remains a coherent state under time evolution — only the parameter \(\alpha\) rotates in the complex plane at frequency \(\omega\).
The expectation value of position:
\[ \langle\hat{x}\rangle(t) = \sqrt{2}x_0|\alpha|\cos(\omega t - \phi) = A\cos(\omega t - \phi) \]
This is exactly classical motion! The coherent state oscillates without spreading, with amplitude \(A = \sqrt{2}x_0|\alpha|\) and phase \(\phi = \arg(\alpha)\).
7 Photon Number Distribution
The probability of finding \(n\) quanta in state \(|\alpha\rangle\):
\[ P(n) = |\langle n|\alpha\rangle|^2 = e^{-|\alpha|^2}\frac{|\alpha|^{2n}}{n!} \]
This is a Poisson distribution with mean \(\bar{n} = |\alpha|^2\):
\[ \boxed{P(n) = e^{-\bar{n}}\frac{\bar{n}^n}{n!}, \quad \bar{n} = |\alpha|^2 = \langle\hat{N}\rangle} \]
Properties of the Poisson distribution: - Mean: \(\langle\hat{N}\rangle = |\alpha|^2\) - Variance: \(\langle(\Delta N)^2\rangle = |\alpha|^2 = \langle\hat{N}\rangle\) - Standard deviation: \(\Delta N = |\alpha|\)
The Mandel Q parameter (measure of sub/super-Poissonian statistics):
\[ Q = \frac{\langle(\Delta N)^2\rangle - \langle N\rangle}{\langle N\rangle} = 0 \quad \text{for coherent states} \]
Coherent states have Poissonian statistics — they are on the boundary between classical and non-classical light.
Laser light is well-described by a coherent state. When you count photons from a laser, you get Poisson statistics — meaning the arrivals are random and uncorrelated. This is the minimal randomness imposed by quantum mechanics on the field amplitude.
Sub-Poissonian (\(Q < 0\)) statistics would indicate squeezed light (non-classical) — the topic of Module 7!
8 Non-Orthogonality of Coherent States
Unlike Fock states, coherent states are not orthogonal:
\[ \langle\beta|\alpha\rangle = e^{-|\alpha|^2/2 - |\beta|^2/2 + \beta^*\alpha} \]
\[ |\langle\beta|\alpha\rangle|^2 = e^{-|\alpha - \beta|^2} \]
Two coherent states with \(|\alpha - \beta| \gg 1\) are approximately orthogonal. Two with \(|\alpha - \beta| \ll 1\) are nearly identical.
This makes sense physically: coherent states with very different amplitudes or phases are distinguishable, while those with nearly equal \(\alpha\) are hard to tell apart.
8.1 Over-Completeness
Despite being non-orthogonal, coherent states form an over-complete basis:
\[ \frac{1}{\pi}\int |\alpha\rangle\langle\alpha|\, d^2\alpha = \hat{I} \]
where \(d^2\alpha = d\,\text{Re}(\alpha)\,d\,\text{Im}(\alpha)\) is the integration measure over the complex plane.
This resolution of identity (over-completeness relation) is fundamental to the Glauber-Sudarshan P-representation of quantum optics.
9 Phase Space Picture: The Wigner Function of Coherent States
The Wigner function of a coherent state \(|\alpha\rangle\):
\[ W_\alpha(x,p) = \frac{1}{\pi\hbar}\exp\!\left(-\frac{(x - \langle\hat{x}\rangle)^2}{x_0^2} - \frac{(p - \langle\hat{p}\rangle)^2\,x_0^2}{\hbar^2}\right) \]
This is a Gaussian centered at \((\langle\hat{x}\rangle, \langle\hat{p}\rangle)\) with:
- Width in \(x\): \(\Delta x = x_0/\sqrt{2}\)
- Width in \(p\): \(\Delta p = \hbar/\sqrt{2}x_0\)
- Phase-space area: \(\pi\Delta x\,\Delta p = \pi\hbar/2\) (minimum area, Planck’s constant/2)
Time evolution of the Wigner function: the Gaussian blob translates around an ellipse (circular for \(m=1\), \(\omega=1\)) in phase space — exactly the classical orbit!
This is the phase-space picture of why coherent states are the most classical quantum states: their Wigner function looks like a classical phase-space distribution, and it moves classically.
10 Summary: Properties of Coherent States
| Property | Value |
|---|---|
| Eigenvalue of \(\hat{a}\) | \(\hat{a}|\alpha\rangle = \alpha|\alpha\rangle\) |
| Fock-state expansion | \(|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n\frac{\alpha^n}{\sqrt{n!}}|n\rangle\) |
| Generated by | \(|\alpha\rangle = \hat{D}(\alpha)|0\rangle\) |
| \(\langle\hat{x}\rangle\) | \(\sqrt{2}x_0\,\text{Re}(\alpha)\) |
| \(\langle\hat{p}\rangle\) | \(\sqrt{2}\hbar x_0^{-1}\,\text{Im}(\alpha)\) |
| \(\Delta x\) | \(x_0/\sqrt{2}\) (independent of \(\alpha\)!) |
| \(\Delta p\) | \(\hbar/\sqrt{2}x_0\) (independent of \(\alpha\)!) |
| \(\Delta x\,\Delta p\) | \(\hbar/2\) (minimum uncertainty) |
| Mean photon number | \(\bar{n} = |\alpha|^2\) |
| Photon number distribution | Poisson: \(P(n) = e^{-\bar{n}}\bar{n}^n/n!\) |
| Time evolution | \(|\alpha(t)\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle\) |
| Orthogonality | Non-orthogonal: \(|\langle\beta|\alpha\rangle|^2 = e^{-|\alpha-\beta|^2}\) |
Before moving on, make sure you can:
- Define a coherent state as an eigenstate of \(\hat{a}\) with eigenvalue \(\alpha\).
- Derive the Fock-state expansion \(|\alpha\rangle = e^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{\sqrt{n!}}|n\rangle\).
- Prove that \(\Delta x\,\Delta p = \hbar/2\) for all coherent states.
- Show that time evolution sends \(|\alpha\rangle \to e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle\).
- Derive the Poisson distribution \(P(n) = e^{-|\alpha|^2}|\alpha|^{2n}/n!\).
- Explain the physical meaning of \(|\alpha|\) and \(\arg(\alpha)\) in terms of classical amplitude and phase.
Conceptual: Why do coherent states not violate the uncertainty principle, even though they look classical?