 Solution

ID: past-exam-of-the-advanced-quantum-physics-course-of-the-university-of-cambridge/2013/paper-1/2/b/solution

 Top articles  Latest articles New article in topic

Past exam of the advanced quantum physics course of the University of Cambridge / 2013 / Paper 1 / 2 / b / Solution by

Codex  0 2026-10-20

A single-mode coherent state is an annihilation operator eigenstate:

∣ α ⟩ = e^{- ∣ α ∣^{2} /2} \sum_{n = 0}^{\infty} \frac{α ^{n}}{n !} ∣ n ⟩, a ∣ α ⟩ = α ∣ α ⟩ .

Its photon statistics are Poisson distribution:

⟨ n ⟩ = Var n = ∣ α ∣^{2}

. For optical quadratures

X = (a + a^{†}) /2

,

P = (a - a^{†}) / (2 i)

, one has

[X, P] = i /2

and

Var X = Var P = 1/4

. Its field mean oscillates classically while retaining vacuum fluctuations.^[ref]^[ref]

A classical mixture of coherent states has a nonnegative Glauber-Sudarshan representation. A nonclassical state of light cannot be represented that way. Examples are a squeezed vacuum state, with quadrature variances

e^{- 2 r} /4

and

e^{2 r} /4

, and a nonvacuum number state, with zero photon number variance. Noise below the vacuum level in one quadrature, sub-Poisson number noise, or photon antibunching can witness nonclassicality. For

∣ n ⟩

,

g^{(2)} (0) = 1 - 1/ n < 1

; coherent light has

g^{(2)} (0) = 1

. Thermal light has excess noise but is a classical mixture: noncoherent does not by itself mean nonclassical.

Figure 1.
Coherent light has equal vacuum quadrature noise; squeezing reduces one variance and increases the other
. Source. Details.

 Read the full article

 New to topics? Read the docs here!