Chapter 27 Exercises: Quantum Optics

Problems are organized by section and graded by difficulty: - [B] Basic — direct application of formulas and definitions - [I] Intermediate — requires multi-step reasoning or combining concepts - [A] Advanced — requires deeper analysis, proof, or synthesis

Section 27.1–27.2: Field Quantization

Problem 27.1 [B]

A single-mode electromagnetic field has frequency $\omega = 3 \times 10^{15}\;\text{rad/s}$ (visible light, $\lambda \approx 630\;\text{nm}$).

(a) What is the energy of the vacuum state $|0\rangle$ for this mode?

(b) What is the energy of the state $|5\rangle$?

(c) How many photons are needed to carry 1 eV of energy in this mode?

Problem 27.2 [B]

Verify the commutation relation $[\hat{a}, \hat{a}^\dagger] = 1$ implies $[\hat{n}, \hat{a}] = -\hat{a}$ and $[\hat{n}, \hat{a}^\dagger] = \hat{a}^\dagger$, where $\hat{n} = \hat{a}^\dagger\hat{a}$.

Problem 27.3 [I]

The electric field operator for a single mode in a cavity of volume $V$ is:

$$\hat{E}(t) = \mathcal{E}_0(\hat{a}e^{-i\omega t} + \hat{a}^\dagger e^{i\omega t}), \qquad \mathcal{E}_0 = \sqrt{\frac{\hbar\omega}{2\epsilon_0 V}}$$

(a) Compute $\langle 0|\hat{E}(t)|0\rangle$ and $\langle 0|\hat{E}^2(t)|0\rangle$. Interpret the results.

(b) Compute $\langle n|\hat{E}^2(t)|n\rangle$ for a general Fock state. Show that the field fluctuations grow with photon number.

(c) For a Fabry-Perot cavity of length $L = 1\;\text{cm}$ and cross-sectional area $A = 10^{-8}\;\text{m}^2$ at $\omega = 3 \times 10^{15}\;\text{rad/s}$, calculate $\mathcal{E}_0$ in V/m. Compare with the electric field of sunlight at Earth's surface ($\sim 900\;\text{V/m}$).

Problem 27.4 [I]

Consider the quadrature operators $\hat{X}_1 = \frac{1}{2}(\hat{a} + \hat{a}^\dagger)$ and $\hat{X}_2 = \frac{1}{2i}(\hat{a} - \hat{a}^\dagger)$.

(a) Show that $[\hat{X}_1, \hat{X}_2] = \frac{i}{2}$.

(b) Compute $\Delta X_1$ and $\Delta X_2$ in the vacuum state $|0\rangle$. Verify that $\Delta X_1 \cdot \Delta X_2 = \frac{1}{4}$ (the minimum allowed by Heisenberg).

(c) Compute $\Delta X_1$ and $\Delta X_2$ in the Fock state $|n\rangle$. Does $|n\rangle$ saturate the uncertainty bound?

Problem 27.5 [A]

Normal ordering and vacuum expectation values.

The normal-ordered product $:\hat{O}:$ of an operator $\hat{O}$ places all creation operators to the left of all annihilation operators (without using the commutation relation).

(a) Show that $\langle 0|:\hat{O}:|0\rangle = 0$ for any operator that is a polynomial in $\hat{a}$ and $\hat{a}^\dagger$ with no constant term.

(b) Express $\hat{E}^2$ in normal-ordered form and identify the vacuum fluctuation term explicitly.

(c) The Casimir energy between two parallel conducting plates separated by distance $d$ is $E_{\text{Casimir}} = -\frac{\pi^2\hbar c}{720 d^3}$ per unit area. For $d = 1\;\mu\text{m}$, compute the Casimir pressure and compare with atmospheric pressure.

Section 27.3: Fock States

Problem 27.6 [B]

Show that the Fock state $|n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}|0\rangle$ is correctly normalized: $\langle n|n\rangle = 1$.

(Hint: Use $\hat{a}|n\rangle = \sqrt{n}|n-1\rangle$ and induction, or evaluate $\langle 0|\hat{a}^n(\hat{a}^\dagger)^n|0\rangle/n!$ directly.)

Problem 27.7 [I]

A quantum state is prepared in the superposition $|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$.

(a) Calculate $\langle\hat{n}\rangle$, $\langle\hat{n}^2\rangle$, and $\Delta n$.

(b) Calculate $\langle\hat{E}(t)\rangle$ and verify that it oscillates sinusoidally (unlike a Fock state).

(c) Calculate $g^{(2)}(0) = \langle\hat{n}(\hat{n}-1)\rangle/\langle\hat{n}\rangle^2$. Is this state classical or nonclassical?

Problem 27.8 [A]

Thermal states. At temperature $T$, the single-mode thermal state has density matrix:

$$\hat{\rho}_{\text{th}} = \sum_{n=0}^{\infty} \frac{\bar{n}^n}{(\bar{n}+1)^{n+1}}|n\rangle\langle n|$$

where $\bar{n} = 1/(e^{\hbar\omega/k_BT} - 1)$ is the Bose-Einstein mean photon number.

(a) Verify $\text{Tr}(\hat{\rho}_{\text{th}}) = 1$.

(b) Compute $\langle\hat{n}\rangle$ and $\langle\hat{n}^2\rangle$. Show that $(\Delta n)^2 = \bar{n}^2 + \bar{n}$ (super-Poissonian).

(c) Compute $g^{(2)}(0)$ and verify $g^{(2)}(0) = 2$ (photon bunching) for the thermal state.

(d) Compute the von Neumann entropy $S = -\text{Tr}(\hat{\rho}_{\text{th}}\ln\hat{\rho}_{\text{th}})$. Express the result in terms of $\bar{n}$.

Section 27.4: Coherent States

Problem 27.9 [B]

For the coherent state $|\alpha\rangle$ with $\alpha = 3$:

(a) Calculate the mean photon number $\bar{n}$ and the standard deviation $\Delta n$.

(b) Compute $P(n)$ for $n = 0, 1, 2, \ldots, 12$. Identify the most probable photon number.

(c) What is the signal-to-noise ratio $\bar{n}/\Delta n$?

Problem 27.10 [B]

Show that coherent states are not orthogonal: compute $|\langle\beta|\alpha\rangle|^2$ for $\alpha = 2$, $\beta = 3$, and for $\alpha = 2$, $\beta = 2 + 5i$. Comment on when the overlap is negligible.

Problem 27.11 [I]

The displacement operator.

(a) Using the Baker-Campbell-Hausdorff formula $e^{\hat{A}+\hat{B}} = e^{\hat{A}}e^{\hat{B}}e^{-[\hat{A},\hat{B}]/2}$ (valid when $[\hat{A},\hat{B}]$ commutes with both $\hat{A}$ and $\hat{B}$), show that:

$$\hat{D}(\alpha) = e^{\alpha\hat{a}^\dagger - \alpha^*\hat{a}} = e^{-|\alpha|^2/2}e^{\alpha\hat{a}^\dagger}e^{-\alpha^*\hat{a}}$$

(b) Use this to derive the Fock-basis expansion of $|\alpha\rangle = \hat{D}(\alpha)|0\rangle$.

(c) Show that $\hat{D}(\alpha)\hat{D}(\beta) = e^{(\alpha\beta^* - \alpha^*\beta)/2}\hat{D}(\alpha + \beta)$.

Problem 27.12 [I]

Time evolution of coherent states.

(a) Starting from $|\psi(0)\rangle = |\alpha\rangle$, show that $|\psi(t)\rangle = e^{-i\omega t/2}|\alpha e^{-i\omega t}\rangle$.

(b) Compute $\langle\hat{x}(t)\rangle$ and $\langle\hat{p}(t)\rangle$ for the coherent state (using $\hat{x} = \sqrt{\hbar/(2m\omega)}(\hat{a} + \hat{a}^\dagger)$, $\hat{p} = i\sqrt{m\omega\hbar/2}(\hat{a}^\dagger - \hat{a})$). Show they follow classical trajectories.

(c) Show that $\Delta x$ and $\Delta p$ remain constant in time — the coherent state does not spread.

Problem 27.13 [A]

Coherent state completeness. Prove the resolution of the identity:

$$\frac{1}{\pi}\int|\alpha\rangle\langle\alpha|\,d^2\alpha = \hat{I}$$

(Hint: Write $|\alpha\rangle$ in the Fock basis, use polar coordinates $\alpha = re^{i\phi}$, perform the angular integral first to extract $\delta_{mn}$, then evaluate the radial integral using $\int_0^\infty r^{2n+1}e^{-r^2}\,dr = n!/2$.)

Section 27.5: Squeezed States

Problem 27.14 [I]

For a squeezed vacuum state with squeezing parameter $r = 1$:

(a) Calculate $\Delta X_1$ and $\Delta X_2$. Compare with the vacuum values.

(b) Express the squeezing in decibels: $\text{dB} = -10\log_{10}(\Delta X_1^2/\Delta X_{1,\text{vac}}^2)$.

(c) Calculate the mean photon number $\bar{n} = \sinh^2 r$.

(d) Verify that $\Delta X_1 \cdot \Delta X_2 = 1/4$ (minimum uncertainty is preserved).

Problem 27.15 [I]

The squeezed vacuum contains only even photon numbers. Show this by computing $\langle n|\hat{S}(\xi)|0\rangle$ for $\xi = r$ (real squeezing).

(Hint: Expand $\hat{S}(r) = e^{\frac{r}{2}(\hat{a}^2 - (\hat{a}^\dagger)^2)}$ and show that $(\hat{a}^\dagger)^2$ always creates pairs.)

Problem 27.16 [A]

LIGO sensitivity estimate.

A Michelson interferometer with arm length $L = 4\;\text{km}$ and laser wavelength $\lambda = 1064\;\text{nm}$ has input power $P = 200\;\text{W}$ (power-recycled).

(a) Estimate the shot-noise-limited displacement sensitivity $\delta x_{\text{shot}} = \frac{1}{2k}\sqrt{\frac{\hbar\omega}{P\tau}}$, where $k = 2\pi/\lambda$ and $\tau = 1\;\text{s}$ is the measurement time.

(b) If 10 dB of squeezing is injected ($e^{-r} = 10^{-10/20} \approx 0.316$), by what factor is $\delta x$ improved?

(c) Estimate the minimum detectable gravitational wave strain $h = \delta x / L$ at 100 Hz with and without squeezing.

Section 27.6: Beam Splitter

Problem 27.17 [B]

Verify that the 50:50 beam splitter matrix $\hat{U}_{\text{BS}} = \frac{1}{\sqrt{2}}\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}$ is unitary.

Problem 27.18 [I]

A single photon enters port 1 of a beam splitter with reflectivity $R = \sin^2\theta$, with vacuum at port 2.

(a) Write the output state in terms of $\theta$.

(b) Calculate the probability of detecting the photon in output port 3 and in output port 4.

(c) For what value of $\theta$ is the probability exactly 70% in port 3?

Problem 27.19 [I]

Two-photon input. Show that the state $|2\rangle_1|0\rangle_2$ at a 50:50 beam splitter produces:

$$|2,0\rangle \to \frac{1}{2}|2,0\rangle - \frac{i}{\sqrt{2}}|1,1\rangle - \frac{1}{2}|0,2\rangle$$

(Hint: Write $|2,0\rangle = \frac{(\hat{a}_1^\dagger)^2}{\sqrt{2}}|0,0\rangle$ and apply the beam splitter transformation to $\hat{a}_1^\dagger$.)

Problem 27.20 [A]

Beam splitter as entangler. An $N$-photon Fock state enters port 1 with vacuum at port 2.

(a) Show that the output state is:

$$|N,0\rangle \to \sum_{k=0}^{N}\binom{N}{k}^{1/2}\frac{t^k (ir)^{N-k}}{\sqrt{2^N}}\,|k, N-k\rangle$$

for a 50:50 beam splitter (using the binomial theorem for operators).

(b) Compute the output state explicitly for $N = 3$.

(c) For what input states does the beam splitter produce a product (non-entangled) output? (Answer: coherent states only.)

Section 27.7: Hong-Ou-Mandel Effect

Problem 27.21 [B]

Verify the HOM calculation: starting from $|1,1\rangle = \hat{a}_1^\dagger\hat{a}_2^\dagger|0,0\rangle$, apply the 50:50 beam splitter transformation and confirm:

$$|1,1\rangle \to \frac{-i}{\sqrt{2}}(|2,0\rangle + |0,2\rangle)$$

Problem 27.22 [I]

Distinguishable photons. Suppose the two photons at the beam splitter have orthogonal polarizations: one is $|H\rangle$ and the other is $|V\rangle$. The input state is $|1_H\rangle_1|1_V\rangle_2$.

(a) Argue that the two photons are now distinguishable and the HOM interference does not occur.

(b) Calculate the output state and show that $P(1,1) = 1/2$, $P(2,0) = 1/4$, $P(0,2) = 1/4$ — classical statistics.

(c) What happens if the photons have the same polarization but slightly different frequencies $\omega$ and $\omega + \Delta\omega$? Describe qualitatively how the HOM dip visibility depends on $\Delta\omega$.

Problem 27.23 [A]

HOM with unequal beam splitter. Two identical photons enter a beam splitter with transmission amplitude $t = \cos\theta$ and reflection amplitude $r = i\sin\theta$.

(a) Show that the coincidence probability is $P_{1,1}(\theta) = \cos^2(2\theta)$.

(b) For what values of $\theta$ is the coincidence probability zero? For what values is it maximized?

(c) Plot $P_{1,1}(\theta)$ for $\theta \in [0, \pi/2]$ and interpret the result.

Section 27.8: Photon Statistics

Problem 27.24 [I]

Compute $g^{(2)}(0)$ for each of the following states:

(a) Fock state $|n\rangle$ with $n = 0, 1, 2, 5, 100$.

(b) Coherent state $|\alpha\rangle$ with $|\alpha|^2 = 10$.

(c) The superposition $|\psi\rangle = \frac{1}{\sqrt{3}}(|0\rangle + |1\rangle + |2\rangle)$.

(d) Classify each as bunched, random, or antibunched.

Problem 27.25 [A]

Cauchy-Schwarz inequality for classical light. For a classical field with intensity $I(t) \geq 0$:

(a) Show that $\langle I^2\rangle \geq \langle I\rangle^2$ (this is just the statement that variance is non-negative).

(b) Conclude that $g^{(2)}_{\text{classical}}(0) = \langle I^2\rangle/\langle I\rangle^2 \geq 1$.

(c) Explain why $g^{(2)}(0) < 1$ is a sufficient condition for nonclassicality. Is it also necessary?

Section 27.9: Photonic Quantum Computing

Problem 27.26 [I]

In dual-rail encoding, a photon in one of two spatial modes encodes a qubit: $|0\rangle_L = |1,0\rangle$, $|1\rangle_L = |0,1\rangle$.

(a) Show that a 50:50 beam splitter implements a Hadamard gate (up to a global phase) on the logical qubit.

(b) Show that a phase shifter $e^{i\phi}$ in one arm implements a logical $Z$-rotation $R_Z(\phi)$.

(c) Can you implement an arbitrary single-qubit gate with beam splitters and phase shifters? (Hint: Euler decomposition.)

Problem 27.27 [A]

The KLM CNOT gate.

The KLM scheme uses ancilla photons and post-selection to implement a CNOT gate. Consider a simplified version: two photonic qubits (dual-rail encoded) interact via a shared beam splitter on one mode of each.

(a) Explain why a beam splitter alone cannot implement a deterministic CNOT gate on photonic qubits.

(b) If we add an ancilla photon and a detector, and post-select on detecting exactly one photon in the ancilla mode, argue heuristically that the resulting transformation is a nontrivial two-qubit gate.

(c) What is the success probability of the simplest such gate? Why does this necessitate teleportation-based boosting?

Problem 27.28 [A]

Boson sampling. $N$ identical single photons enter an $M$-mode linear optical network (a mesh of beam splitters and phase shifters, $M \gg N$).

(a) Show that the probability of detecting one photon in each of $N$ specified output modes is $|\text{Perm}(U_S)|^2 / (n_1! n_2! \cdots n_M!)$, where $U_S$ is an $N \times N$ submatrix of the $M \times M$ unitary $U$ and Perm denotes the permanent.

(b) The permanent of an $N \times N$ matrix is believed to be #P-hard to compute. Explain why this makes boson sampling a candidate for demonstrating quantum computational advantage.

(c) Xanadu's Borealis machine (2022) uses squeezed states instead of single photons. What advantage does this offer in terms of scalability?