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Quantum Optics

Field: Optics

The study of the application of quantum mechanics to phenomena involving light and its interactions with matter.

Sequence of Expressions

Let H\mathcal{H} be a separable Hilbert space representing the quantum state ∣ψ⟩|\psi\rangle. The wave aspect is described by the continuous field ψ(r,t)\psi(\mathbf{r}, t), satisfying the time-dependent Schrödinger equation: iℏ∂∂tψ(r,t)=H^ψ(r,t)i\hbar\frac{\partial}{\partial t}\psi(\mathbf{r}, t) = \hat{H}\psi(\mathbf{r}, t). The particle aspect is characterized by the expectation values of position ⟨x^⟩\langle\hat{x}\rangle and momentum ⟨p^⟩\langle\hat{p}\rangle. The duality is formally constrained by the generalized uncertainty principle, which mandates that for any observable pair (A^,B^)(\hat{A}, \hat{B}) with commutator [A^,B^]=iℏC^[\hat{A}, \hat{B}] = i\hbar\hat{C}, the following inequality must hold for any state ∣ψ⟩|\psi\rangle: ΔAΔB≥12∣⟨[A^,B^]/iℏ⟩∣\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}]/i\hbar \rangle| Specifically, for position X^\hat{X} and momentum P^\hat{P}, this yields: ΔXΔP≥ℏ2\Delta X \Delta P \ge \frac{\hbar}{2}. Furthermore, the energy-momentum relation, linking the wave frequency ω\omega and wave number kk, is quantized via the Planck relation: E=ℏω=ℏckE = \hbar\omega = \hbar c k, where EE is the energy operator H^\hat{H} and kk is the wave vector magnitude, confirming the particle energy EE derived from the wave properties.
Let H\mathcal{H} be a separable Hilbert space representing the state space of the quantum system. Assume H\mathcal{H} possesses a complete orthonormal basis set {∣ϕk⟩}k=1N\{|\phi_k\rangle\}_{k=1}^{N} (or N\mathbb{N} for infinite dimensions), where ⟨ϕk∣ϕj⟩=δkj\langle\phi_k|\phi_j\rangle = \delta_{kj}. A quantum state ∣ψ⟩∈H|\psi\rangle \in \mathcal{H} is said to be in a quantum superposition if it can be expressed as a linear combination of these basis states: ∣ψ⟩=∑k=1Nck∣ϕk⟩|\psi\rangle = \sum_{k=1}^{N} c_k |\phi_k\rangle where ck∈Cc_k \in \mathbb{C} are the complex probability amplitudes. The state must be normalized, satisfying the condition: ⟨ψ∣ψ⟩=∑k=1N∣ck∣2=1\langle\psi|\psi\rangle = \sum_{k=1}^{N} |c_k|^2 = 1 The measurement postulate dictates that the probability of observing the system in the state ∣ϕk⟩|\phi_k\rangle is Pk=∣ck∣2P_k = |c_k|^2. This linearity is guaranteed by the structure of the underlying quantum mechanical operators O^\hat{O} acting on H\mathcal{H}, such that O^(a∣ψ1⟩+b∣ψ2⟩)=aO^∣ψ1⟩+bO^∣ψ2⟩\hat{O}(a|\psi_1\rangle + b|\psi_2\rangle) = a\hat{O}|\psi_1\rangle + b\hat{O}|\psi_2\rangle for a,b∈Ca, b \in \mathbb{C}.
Let HA\mathcal{H}_A and HB\mathcal{H}_B be finite-dimensional Hilbert spaces representing two subsystems AA and BB, respectively. The total system Hilbert space is H=HA⊗HB\mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B. A pure state ∣Ψ⟩∈H|\Psi\rangle \in \mathcal{H} is defined as entangled if and only if it cannot be factorized into a product state, i.e., ∣Ψ⟩≠∣ψA⟩⊗∣ϕB⟩|\Psi\rangle \neq |\psi_A\rangle \otimes |\phi_B\rangle for any ∣ψA⟩∈HA|\psi_A\rangle \in \mathcal{H}_A and ∣ϕB⟩∈HB|\phi_B\rangle \in \mathcal{H}_B. Equivalently, the reduced density matrix ρA=TrB(ρ)\rho_A = \text{Tr}_B (\rho) and ρB=TrA(ρ)\rho_B = \text{Tr}_A (\rho) do not satisfy the condition ρ=ρA⊗ρB\rho = \rho_A \otimes \rho_B. Specifically, for a state ρ=∣Ψ⟩⟨Ψ∣\rho = |\Psi\rangle \langle \Psi|, entanglement is detected if the Schmidt rank of ∣Ψ⟩|\Psi\rangle is greater than one, meaning the Schmidt decomposition requires more than one non-zero Schmidt coefficient λk\lambda_k: ∣Ψ⟩=∑k=1Rλk∣ak⟩⊗∣bk⟩,where R>1 and ∑k=1Rλk=1.|\Psi\rangle = \sum_{k=1}^{R} \sqrt{\lambda_k} |a_k\rangle \otimes |b_k\rangle, \quad \text{where } R > 1 \text{ and } \sum_{k=1}^{R} \lambda_k = 1.
Let H\mathcal{H} be a separable Hilbert space, and let ∣Ψ(t)⟩∈H|\Psi(t)\rangle \in \mathcal{H} be the state vector describing the quantum system. Define the Hamiltonian operator H^:H→H\hat{H}: \mathcal{H} \to \mathcal{H} as the generator of time evolution, such that H^=P^22m+V^(r,t)\hat{H} = \frac{\hat{P}^2}{2m} + \hat{V}(\mathbf{r}, t), where P^=−iℏ∇\hat{P} = -i\hbar\nabla is the momentum operator and V^\hat{V} is the potential energy operator. The time evolution of the state vector is governed by the equation:\niℏ∂∂t∣Ψ(t)⟩=H^∣Ψ(t)⟩\mathrm{i\hbar}\frac{\partial}{\partial t} |\Psi(t)\rangle = \hat{H} |\Psi(t)\rangle
Let a^k\hat{a}_k and a^k†\hat{a}_k^{\dagger} be the annihilation and creation operators for the mode kk of the electromagnetic field, satisfying the canonical commutation relation [a^k,a^j†]=δkj[\hat{a}_k, \hat{a}_j^{\dagger}] = \delta_{kj}. Define the total photon number operator N^=∑ka^k†a^k\hat{N} = \sum_k \hat{a}_k^{\dagger} \hat{a}_k. For a quantum state ∣ψ⟩∈H|\psi\rangle \in \mathcal{H}, the probability P(n)P(n) of detecting nn photons is given by the expectation value of the number projection operator Π^n\hat{\Pi}_n: P(n)=⟨ψ∥Π^n∥ψ⟩=1n!⟨ψ∥(N^)n∥ψ⟩P(n) = \langle \psi\| \hat{\Pi}_n \| \psi\rangle = \frac{1}{n!} \langle \psi\| (\hat{N})_n \| \psi\rangle where (N^)n(\hat{N})_n is the nn-th order normally ordered moment of N^\hat{N}. Alternatively, the statistical distribution P(n)P(n) can be derived from the characteristic function χ(λ)=Tr(ρeiλN^)\chi(\lambda) = \text{Tr}(\rho e^{i \lambda \hat{N}}), where ρ=∣ψ⟩⟨ψ∣\rho = |\psi\rangle\langle \psi| is the density matrix, such that P(n) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \text{Re}\left\{ e^{-i n \lambda} \chi(\lambda) \right} d\lambda.
Let the atomic system be described by the Hamiltonian H0H_0, with eigenstates ∣e⟩|e\rangle and ∣g⟩|g\rangle corresponding to energy levels EeE_e and EgE_g. The interaction with the quantized electromagnetic field is given by Hint=−12∇×∇×d∙E(r,t)H_{int} = -\frac{1}{2} \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times \boldsymbol{d} \bullet \boldsymbol{E}(\boldsymbol{r}, t), where d\boldsymbol{d} is the dipole moment operator and E(r,t)\boldsymbol{E}(\boldsymbol{r}, t) is the quantized electric field operator. The transition rate WegW_{eg} for stimulated emission is derived from Fermi's Golden Rule and is given by:\n\nΓe→g=Weg=ω33ϵ0ℏc3∣dge∣2ρ(ω)\Gamma_{e\to g} = W_{eg} = \frac{\omega^3}{3\epsilon_0 \hbar c^3} |\boldsymbol{d}_{ge}|^2 \rho(\omega) \n\nwhere ω=(Ee−Eg)/ℏ\omega = (E_e - E_g)/\hbar is the transition angular frequency, dge=⟨g∣d∣e⟩\boldsymbol{d}_{ge} = \langle g | \boldsymbol{d} | e \rangle is the transition dipole matrix element, and ρ(ω)\rho(\omega) is the spectral energy density of the incident radiation field evaluated at ω\omega. The evolution of the excited state population NeN_e is governed by the rate equation:\n\ndNedt=−WegNe+WgeNg\frac{d N_e}{d t} = -W_{eg} N_e + W_{ge} N_g
Let LQED=−14FμνFμν+ψˉ(iγμDμ−m)ψ\mathcal{L}_{QED} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} + \bar{\psi} (i \gamma^{\mu} D_{\mu} - m) \psi be the classical Lagrangian density for Quantum Electrodynamics. The effective Lagrangian Leff\mathcal{L}_{eff} incorporating vacuum polarization is given by the one-loop correction to the photon propagator. Define the vacuum polarization tensor Πμν(k2)\Pi_{\mu\nu}(k^2) via the electron loop diagram: Πμν(k2)=−e2∫d4k(2π)4Tr[γμSF(k)γνSF(k−k′)]\Pi_{\mu\nu}(k^2) = -e^2 \int \frac{d^4 k}{(2\pi)^4} \text{Tr} \left[ \gamma_{\mu} S_F(k) \gamma_{\nu} S_F(k-k') \right], where SF(k)=iγα(kα+m)k2−m2+iϵS_F(k) = \frac{i \gamma^{\alpha} (k_{\alpha} + m)}{k^2 - m^2 + i\epsilon} is the fermionic propagator. The modified photon propagator Dμν′(k2)D'_{\mu\nu}(k^2) in momentum space is then determined by the Dyson-Schwinger equation: \begin{equation} D'^{-1}(k^2) = D_0^{-1}(k^2) + \Pi(k^2) \end{equation}, where D0−1(k2)=k2gμνD_0^{-1}(k^2) = k^2 g_{\mu\nu} and Π(k2)=Πμν(k2)−k2gμνΠ(k2)k2\Pi(k^2) = \Pi_{\mu\nu}(k^2) - k^2 g_{\mu\nu} \frac{\Pi(k^2)}{k^2}. The resulting effective coupling constant αeff(k2)\alpha_{eff}(k^2) is related to the running coupling constant α(k2)=e24π(1+α3πΠ(k2))\alpha(k^2) = \frac{e^2}{4\pi} \left( 1 + \frac{\alpha}{3\pi} \Pi(k^2) \right), demonstrating the momentum-dependent renormalization of the electromagnetic interaction.
Let S\mathcal{S} be the space of all continuous paths x:[ta,tb]→Rn\mathbf{x}: [t_a, t_b] \to \mathbb{R}^n such that x(ta)=xa\mathbf{x}(t_a) = \mathbf{x}_a and x(tb)=xb\mathbf{x}(t_b) = \mathbf{x}_b. Define the classical action functional S[x(t)]S[\mathbf{x}(t)] by the Lagrangian L(x,x˙,t)L(\mathbf{x}, \dot{\mathbf{x}}, t): S[x(t)]=∫tatbL(x(t),x˙(t),t)dtS[\mathbf{x}(t)] = \int_{t_a}^{t_b} L(\mathbf{x}(t), \dot{\mathbf{x}}(t), t) dt The quantum mechanical propagator K(xb,tb;xa,ta)K(\mathbf{x}_b, t_b; \mathbf{x}_a, t_a) is then defined by the Feynman path integral: K(xb,tb;xa,ta)=∫SDx(t)exp⁡(i1ℏS[x(t)])K(\mathbf{x}_b, t_b; \mathbf{x}_a, t_a) = \int_{\mathcal{S}} \mathcal{D}\mathbf{x}(t) \exp\left(i \frac{1}{\hbar} S[\mathbf{x}(t)]\right) where Dx(t)\mathcal{D}\mathbf{x}(t) represents the path integral measure, which is formally defined by the limit of the discretized product of Gaussian integrals over small time steps ϵ=(tb−ta)/N\epsilon = (t_b - t_a)/N: Dx(t)=lim⁡N→∞(1A)N/2∏j=1Ndxj\mathcal{D}\mathbf{x}(t) = \lim_{N \to \infty} \left( \frac{1}{A} \right)^{N/2} \prod_{j=1}^{N} d\mathbf{x}_j and AA is a normalization constant dependent on the system's mass and dimensionality.
Let Aμ(x)A_{\mu}(x) be a massive vector field, where x=(ct,x)∈R1,3x = (ct, \mathbf{x}) \in \mathbb{R}^{1,3}. Define the field strength tensor FμνF_{\mu\nu} as the exterior derivative of AμA_{\mu}: Fμν=∂μAν−∂νAμ.F_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}. The equation of motion governing the propagation of this field, known as Proca's Law, is given by the covariant form:\n∂νFνμ+m2Aμ=0\partial_{\nu} F^{\nu\mu} + m^2 A^{\mu} = 0 \nwhere mm is the rest mass of the vector boson, and the equation is derived from the Lagrangian density LProca\mathcal{L}_{Proca}: \nLProca=−14FμνFμν−12m2AμAμ.\mathcal{L}_{Proca} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu} - \frac{1}{2} m^2 A_{\mu} A^{\mu}.
Let H\mathcal{H} be the Hilbert space of the quantum system, and let ∣ψ⟩∈H|\psi\rangle \in \mathcal{H} be the state vector. Define the position operator x^\hat{x} and the momentum operator p^\hat{p} acting on H\mathcal{H} such that their commutator is [x^,p^]=iℏ[\hat{x}, \hat{p}] = i\hbar. The uncertainty in an observable A^\hat{A} is defined by σA2=⟨A^2⟩−⟨A^⟩2\sigma_A^2 = \langle \hat{A}^2 \rangle - \langle \hat{A} \rangle^2. The Heisenberg Uncertainty Principle states that for any normalized state ∣ψ⟩|\psi\rangle, the product of the variances satisfies: σx2σp2≥14∣⟨[x^,p^]/iℏ⟩∣2 \sigma_x^2 \sigma_p^2 \ge \frac{1}{4} |\langle [\hat{x}, \hat{p}] / i\hbar \rangle|^2 Substituting the canonical commutation relation yields the fundamental bound: σxσp≥ℏ2 \sigma_x \sigma_p \ge \frac{\hbar}{2}