Chapter 27 Quiz: Quantum Optics

Multiple Choice

Q1. Each mode of the quantized electromagnetic field is mathematically equivalent to:

(a) A free particle (b) A quantum harmonic oscillator (c) A hydrogen atom (d) A spin-1/2 system

Q2. The Fock state $|n\rangle$ has the following property for the electric field:

(a) $\langle n|\hat{E}|n\rangle = n\mathcal{E}_0$ — the mean field is proportional to the photon number (b) $\langle n|\hat{E}|n\rangle = \sqrt{n}\,\mathcal{E}_0$ — the mean field scales as $\sqrt{n}$ (c) $\langle n|\hat{E}|n\rangle = 0$ — the mean electric field is zero for all $n$ (d) $\langle n|\hat{E}^2|n\rangle = 0$ — the field fluctuations vanish for all $n$

Q3. A coherent state $|\alpha\rangle$ is best described as:

(a) An eigenstate of the number operator $\hat{n}$ (b) An eigenstate of the creation operator $\hat{a}^\dagger$ (c) An eigenstate of the annihilation operator $\hat{a}$ (d) An eigenstate of the Hamiltonian $\hat{H}$

Q4. The photon number distribution of a coherent state $|\alpha\rangle$ is:

(a) A delta function at $n = |\alpha|^2$ (b) A Gaussian centered at $n = |\alpha|^2$ (c) A Poisson distribution with mean $\bar{n} = |\alpha|^2$ (d) A Bose-Einstein distribution with mean $\bar{n} = |\alpha|^2$

Q5. A squeezed vacuum state with squeezing parameter $r = 2$ has quadrature uncertainties:

(a) $\Delta X_1 = \frac{1}{2}e^{-2} \approx 0.068$, $\Delta X_2 = \frac{1}{2}e^{2} \approx 3.69$ (b) $\Delta X_1 = \frac{1}{2}$, $\Delta X_2 = \frac{1}{2}$ (same as vacuum) (c) $\Delta X_1 = e^{-2} \approx 0.135$, $\Delta X_2 = e^{2} \approx 7.39$ (d) $\Delta X_1 = 0$, $\Delta X_2 = \infty$

Q6. A single photon enters port 1 of a 50:50 beam splitter with vacuum at port 2. The output state is:

(a) $|1\rangle_3|0\rangle_4$ — the photon always goes to port 3 (b) $\frac{1}{2}|1\rangle_3|0\rangle_4 + \frac{1}{2}|0\rangle_3|1\rangle_4$ — a statistical mixture (c) $\frac{1}{\sqrt{2}}(|1\rangle_3|0\rangle_4 - i|0\rangle_3|1\rangle_4)$ — a coherent superposition (d) $\frac{1}{\sqrt{2}}|1\rangle_3|1\rangle_4$ — "half a photon" in each port

Q7. In the Hong-Ou-Mandel effect, two identical photons enter a 50:50 beam splitter (one in each input port). The coincidence probability (one photon in each output) is:

(a) 50% — each photon independently has a 50% chance of going to either output (b) 25% — the product of individual reflection/transmission probabilities (c) 0% — destructive interference eliminates the coincidence term (d) 100% — the photons always exit in different ports

Q8. The Hong-Ou-Mandel effect occurs because:

(a) Photons repel each other through the electromagnetic force (b) Energy conservation forbids the one-photon-per-output configuration (c) The amplitudes for "both transmitted" and "both reflected" cancel exactly (d) The Pauli exclusion principle prevents photons from occupying the same output mode

Q9. A light source has second-order correlation $g^{(2)}(0) = 0.3$. This light is:

(a) Thermal (bunched) (b) Coherent (Poissonian) (c) Nonclassical (antibunched) (d) Impossible — $g^{(2)}(0)$ cannot be less than 1

Q10. For a thermal (blackbody) light source, $g^{(2)}(0)$ equals:

(a) 0 (b) 1 (c) 2 (d) $\infty$

Q11. Squeezed light is used in LIGO to:

(a) Increase the laser power without additional pump lasers (b) Reduce quantum noise in the phase quadrature below the standard quantum limit (c) Cool the mirrors to their quantum ground state (d) Eliminate photon loss in the interferometer arms

Q12. In the KLM scheme for linear optical quantum computing, the key insight is that:

(a) Beam splitters alone can implement any quantum gate (b) Photon-photon interactions provide natural two-qubit gates (c) Measurement and post-selection create effective nonlinear interactions (d) Coherent states can encode error-corrected qubits

True/False

Q13. True or False: A coherent state $|\alpha\rangle$ with $|\alpha|^2 = 100$ has a well-defined photon number (i.e., $\Delta n = 0$).

Q14. True or False: The vacuum state $|0\rangle$ has zero electric field fluctuations: $\langle 0|\hat{E}^2|0\rangle = 0$.

Q15. True or False: Coherent states are orthogonal — $\langle\beta|\alpha\rangle = 0$ whenever $\alpha \neq \beta$.

Q16. True or False: Photon antibunching ($g^{(2)}(0) < 1$) can be explained by classical electromagnetic theory.

Short Answer

Q17. A coherent state has $\alpha = 5e^{i\pi/4}$. Calculate (a) the mean photon number, (b) the standard deviation of the photon number, and (c) the probability of detecting zero photons.

Q18. Explain in 2–3 sentences why the Hong-Ou-Mandel effect vanishes (the coincidence rate returns to the classical value) when the two photons arrive at the beam splitter at different times (i.e., with a time delay $\tau$ much larger than the coherence time $\tau_c$).

Q19. A squeezed state has 6 dB of squeezing. Convert this to the squeezing parameter $r$ and calculate the reduction factor $e^{-r}$ for the squeezed quadrature. By what factor is the anti-squeezed quadrature amplified?

Q20. Name two advantages and two challenges of using photons as qubits for quantum computing, compared to superconducting qubits.

Answer Key

Q1. (b) — Each mode of the quantized EM field is a QHO with $\hat{H} = \hbar\omega(\hat{a}^\dagger\hat{a} + 1/2)$.

Q2. (c) — $\langle n|\hat{E}|n\rangle = 0$ because $\hat{E} \propto (\hat{a} + \hat{a}^\dagger)$ and $\langle n|n\pm 1\rangle = 0$.

Q3. (c) — $\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$. The annihilation operator is not Hermitian, so its eigenvalues are complex.

Q4. (c) — $P(n) = e^{-|\alpha|^2}|\alpha|^{2n}/n!$, a Poisson distribution with mean and variance both equal to $|\alpha|^2$.

Q5. (a) — $\Delta X_1 = \frac{1}{2}e^{-r}$, $\Delta X_2 = \frac{1}{2}e^{r}$. With $r = 2$: $\Delta X_1 \approx 0.068$, $\Delta X_2 \approx 3.69$.

Q6. (c) — The photon is in a superposition, not a mixture. It is never "split" into fractions.

Q7. (c) — The amplitudes for the two paths leading to one-photon-per-output cancel by destructive interference.

Q8. (c) — The transmitted-transmitted amplitude is $+1/2$ and the reflected-reflected amplitude is $-1/2$ (from the $i^2 = -1$ phase shifts). They sum to zero.

Q9. (c) — $g^{(2)}(0) < 1$ is impossible for any classical field (which must satisfy $g^{(2)}(0) \geq 1$). This is a signature of nonclassical, antibunched light.

Q10. (c) — Thermal light has $g^{(2)}(0) = 2$, indicating photon bunching (Hanbury Brown-Twiss effect).

Q11. (b) — Squeezed vacuum injected into the antisymmetric port reduces phase-quadrature noise, improving displacement sensitivity.

Q12. (c) — KLM showed that measurement and post-selection (plus teleportation) can substitute for the deterministic photon-photon interactions that nature does not provide.

Q13. False — A coherent state has Poissonian statistics: $\Delta n = \sqrt{|\alpha|^2} = 10$. Only Fock states have $\Delta n = 0$.

Q14. False — The vacuum has zero mean field ($\langle\hat{E}\rangle = 0$) but nonzero field fluctuations ($\langle\hat{E}^2\rangle = \mathcal{E}_0^2 \neq 0$). These vacuum fluctuations are responsible for spontaneous emission and the Casimir effect.

Q15. False — Coherent states are not orthogonal: $|\langle\beta|\alpha\rangle|^2 = e^{-|\alpha-\beta|^2}$. The overlap is small but never zero for finite $|\alpha - \beta|$.

Q16. False — Antibunching ($g^{(2)}(0) < 1$) violates the classical Cauchy-Schwarz inequality $\langle I^2\rangle \geq \langle I\rangle^2$ and has no classical explanation.

Q17. (a) $\bar{n} = |\alpha|^2 = 25$. (b) $\Delta n = \sqrt{\bar{n}} = 5$. (c) $P(0) = e^{-25} \approx 1.4 \times 10^{-11}$ — essentially zero.

Q18. When the time delay $\tau \gg \tau_c$, the two photons arrive at the beam splitter at distinguishable times. Since they no longer overlap temporally, they behave as independent classical particles rather than interfering quantum amplitudes. The which-photon information encoded in the arrival time destroys the destructive interference that produces the HOM dip.

Q19. $6\;\text{dB} = 10\log_{10}(e^{2r})$, so $r = 6/(20\log_{10}e) \approx 0.691$. The squeezed quadrature is reduced by $e^{-r} \approx 0.501$ (factor of 2). The anti-squeezed quadrature is amplified by $e^{r} \approx 1.995$ (also factor of 2).

Q20. Advantages: (1) Low decoherence — photons interact weakly with the environment and maintain coherence over long distances; (2) Room-temperature operation — no cryogenic cooling required. Challenges: (1) Photon loss — absorbed or undetected photons are irreversible errors; (2) Deterministic two-qubit gates are difficult — linear optics requires measurement-based approaches with probabilistic gates and significant overhead.