Part VI, Chapter 6

Quantum Optics

The quantum theory of light reveals phenomena impossible to explain classically: photon antibunching, two-photon interference, squeezed states below the vacuum noise, and secure quantum communication.

6.1 Photon Statistics

Different quantum states of light are distinguished by their photon number statistics. The second-order correlation function quantifies intensity fluctuations:

$$g^{(2)}(\tau) = \frac{\langle I(t)I(t+\tau)\rangle}{\langle I\rangle^2} = \frac{\langle \hat{a}^\dagger \hat{a}^\dagger \hat{a}\, \hat{a}\rangle}{\langle \hat{a}^\dagger \hat{a}\rangle^2}$$

6.1.1 Coherent States

Coherent states |α〉 (produced by lasers) have Poissonian photon number statistics:

$$P(n) = \frac{|\alpha|^{2n}}{n!} e^{-|\alpha|^2}, \qquad \langle n\rangle = |\alpha|^2, \quad \Delta n = \sqrt{\langle n\rangle}$$

For coherent states, g(2)(0) = 1 for all τ. There is no bunching or antibunching.

6.1.2 Thermal (Chaotic) Light

Thermal light (blackbody radiation, LED) follows the Bose-Einstein distribution:

$$P(n) = \frac{\langle n\rangle^n}{(1 + \langle n\rangle)^{n+1}}, \qquad \Delta n = \sqrt{\langle n\rangle(1+\langle n\rangle)}$$

Thermal light exhibits photon bunching: g(2)(0) = 2 > 1. Photons tend to arrive in clusters. This is super-Poissonian statistics.

6.1.3 Fock (Number) States

Fock states |n〉 have exactly n photons with zero uncertainty: Δn = 0. They exhibit photon antibunching:

$$g^{(2)}(0) = 1 - \frac{1}{n} < 1 \quad (n \ge 1)$$

For a single-photon state |1〉, g(2)(0) = 0: the photon cannot be split between two detectors. Antibunching is a purely quantum effect with no classical analog.

6.2 The Quantum Beam Splitter

A 50:50 beam splitter transforms input mode operators according to:

$$\hat{c} = \frac{1}{\sqrt{2}}(\hat{a} + i\hat{b}), \qquad \hat{d} = \frac{1}{\sqrt{2}}(i\hat{a} + \hat{b})$$

This unitary transformation preserves commutation relations and is the fundamental building block of quantum optical circuits.

6.2.1 Single Photon at a Beam Splitter

A single photon |1〉a|0〉b entering port a produces an entangled state:

$$|1\rangle_a |0\rangle_b \to \frac{1}{\sqrt{2}}(|1\rangle_c |0\rangle_d + i|0\rangle_c |1\rangle_d)$$

The photon is in a superposition of being in output c or output d. Each detector clicks with probability 1/2, but they never both click simultaneously (anticorrelation).

6.3 The Hanbury Brown-Twiss Experiment

Hanbury Brown and Twiss (1956) measured intensity correlations of starlight using two detectors. They observed that photons from a thermal source tend to arrive together (bunching). The coincidence rate as a function of time delay τ directly measures g(2)(τ):

$$R_{cc}(\tau) \propto g^{(2)}(\tau) = 1 + |g^{(1)}(\tau)|^2$$

For thermal light, g(2)(0) = 2, decaying to 1 over the coherence time τc. For a laser, g(2)(τ) = 1 for all τ. For a single-photon source, g(2)(0) = 0 (antibunching dip). The HBT experiment is the standard test for quantum light sources.

Classification of Light by g(2)(0)

  • g(2)(0) = 0: Single-photon (Fock) state. Purely quantum.
  • g(2)(0) < 1: Sub-Poissonian (antibunched). Quantum, no classical analog.
  • g(2)(0) = 1: Poissonian (coherent state). Boundary of classical/quantum.
  • g(2)(0) = 2: Super-Poissonian (thermal). Maximum classical bunching.

6.4 The Hong-Ou-Mandel Effect

When two identical single photons enter a 50:50 beam splitter from different ports, they always exit together from the same port. This is a purely quantum interference effect.

Derivation: HOM Effect

Step 1. Input state: |1〉a|1〉b. Apply the beam splitter transformation:

$$\hat{a}^\dagger \to \frac{1}{\sqrt{2}}(\hat{c}^\dagger + i\hat{d}^\dagger), \quad \hat{b}^\dagger \to \frac{1}{\sqrt{2}}(i\hat{c}^\dagger + \hat{d}^\dagger)$$

Step 2. The output state is:

$$\hat{a}^\dagger \hat{b}^\dagger |00\rangle = \frac{1}{2}(\hat{c}^\dagger + i\hat{d}^\dagger)(i\hat{c}^\dagger + \hat{d}^\dagger)|00\rangle$$

Step 3. Expanding:

$$= \frac{1}{2}\left[i(\hat{c}^\dagger)^2 + \hat{c}^\dagger\hat{d}^\dagger - \hat{d}^\dagger\hat{c}^\dagger + i(\hat{d}^\dagger)^2\right]|00\rangle$$

Step 4. Since bosonic operators commute (â†câ†d = â†dâ†c), the cross terms cancel:

Result:

$$|\psi_{\text{out}}\rangle = \frac{i}{\sqrt{2}}\left(|2\rangle_c |0\rangle_d + |0\rangle_c |2\rangle_d\right)$$

Both photons exit the same port. The coincidence rate drops to zero (the HOM dip). The visibility of the dip measures the indistinguishability of the photons.

6.5 Squeezed States of Light

A squeezed state has reduced noise in one quadrature below the vacuum level, at the expense of increased noise in the conjugate quadrature. The squeeze operator is:

$$\hat{S}(r) = \exp\!\left[\frac{r}{2}(\hat{a}^{\dagger 2} - \hat{a}^2)\right]$$

where r is the squeezing parameter. The quadrature uncertainties become:

$$\Delta X_1 = \frac{1}{2}e^{-r}, \qquad \Delta X_2 = \frac{1}{2}e^{+r}$$

The uncertainty product remains at the Heisenberg minimum: ΔX1ΔX2 = 1/4. Squeezing is produced by parametric down-conversion or four-wave mixing. Current experiments achieve over 15 dB of squeezing. Applications include gravitational wave detection (LIGO uses squeezed light), precision metrology, and continuous-variable quantum information.

6.6 Quantum Key Distribution: BB84

The BB84 protocol (Bennett and Brassard, 1984) allows two parties (Alice and Bob) to establish a shared secret key with security guaranteed by quantum mechanics. The no-cloning theorem ensures that any eavesdropper (Eve) inevitably introduces detectable errors.

BB84 Protocol Steps

  1. Preparation: Alice randomly chooses a bit (0 or 1) and a basis (rectilinear + or diagonal ×). She encodes the bit as a single-photon polarization state: |↔〉, |&varr;〉 (rectilinear) or |&nearr;〉, |&nwarr;〉 (diagonal).
  2. Transmission: Alice sends the photon to Bob through a quantum channel (e.g., optical fiber).
  3. Measurement: Bob randomly chooses a measurement basis (+ or ×) and records the result.
  4. Sifting: Alice and Bob publicly announce their basis choices (but not the bits). They keep only the bits where they chose the same basis (~50% of the data).
  5. Error estimation: They sacrifice a random subset to estimate the error rate. If the quantum bit error rate (QBER) exceeds ~11%, they abort (an eavesdropper is present).
  6. Privacy amplification: They apply classical post-processing to distill a shorter, perfectly secure key.

The security of BB84 rests on the quantum no-cloning theorem: Eve cannot perfectly copy an unknown quantum state. Any measurement she makes on the photon disturbs it, introducing errors that Alice and Bob can detect. The maximum tolerable QBER for unconditional security is:

$$\text{QBER}_{\max} = 1 - \frac{1}{\sqrt{2}} \approx 11\%$$

6.7 Python Simulation: Photon Statistics & HOM Effect

This simulation compares photon number distributions for coherent, thermal, and Fock states, plots the g(2)(τ) correlation function, visualizes the HOM dip, and shows squeezed-state phase-space distributions.

Quantum Optics: Photon Statistics, HOM & Squeezed States

Python
script.py147 lines

Click Run to execute the Python code

Code will be executed with Python 3 on the server

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