(chap:03)= # Foundations of quantum optics :::{admonition} Chapter opening :class: chapter-opening Light received by a telescope is most naturally described in modes. A mode has a frequency, a spatial shape, a polarization, and a time interval. A photon is an excitation of such a mode, and a detector records clicks after the mode has been measured. Fock states, coherent states, thermal states, and squeezed states describe photon number, phase stability, random intensity fluctuations, and quadrature noise. Density matrices keep track of what remains after unobserved degrees of freedom have been traced out. Glauber coherence functions connect these optical states to the joint probabilities of detector clicks {cite:p}`1963PhRv..130.2529G,1963PhRv..131.2766G,1965RvMP...37..231M,1995ocqo.book.....M,1997quop.book.....S,2001qops.book.....S,2013quop.book.....A`. ::: ## Modes, photons, and the field received by a telescope The electric field can be expanded in an orthogonal set of modes: ```{math} :label: eq:ch03-field-expansion \hat E^{(+)}(\bm{r},t)=\sum_k {\cal E}_k u_k(\bm{r})e^{-i\omega_k t}\hat a_k , \qquad {\cal E}_k=\left(\frac{\hbar\omega_k}{2\epsilon_0}\right)^{1/2}. ``` Here $\hat E^{(+)}$ is the positive-frequency electric-field operator. $\bm{r}$ is measured in meters, $t$ in seconds, and $\omega_k=2\pi\nu_k$ in $\mathrm{rad\,s^{-1}}$. The index $k$ is a bundle of labels: central frequency, bandwidth, arrival direction, spatial mode across the aperture, polarization, and time window. With cavity normalization, $u_k$ carries a factor with dimensions $V^{-1/2}$. With continuum traveling-wave modes, the normalization is redistributed between ${\cal E}_k$ and $u_k$, but the algebra of $\hat a_k$ is unchanged. In telescope data, a practical \"mode\" can be thought of as one field degree of freedom within one coherence time $\tau_c$, one diffraction-limited spatial cell, and one polarization channel {cite:p}`2000stop.book.....G,1956Natur.177...27B,1957RSPSA.242..300B,1974iiia.book.....H`. For orthogonal modes, the creation and annihilation operators obey ```{math} :label: eq:ch03-commutator [\hat a_i,\hat a_j^\dagger]=\delta_{ij},\qquad \hat n_i=\hat a_i^\dagger\hat a_i . ``` $\hat a_i^\dagger$ adds one photon to mode $i$, $\hat a_i$ removes one photon from that mode, and $\hat n_i$ is the photon-number operator. Its eigenvalues are $N=0,1,2,\ldots$ and have no units. The Kronecker symbol $\delta_{ij}$ says that the order of creation and annihilation leaves a quantum correction only for the same mode. A telescope does not directly read out $\hat a$ or $\hat E$. It reads out click times, pixel positions, energy channels, and polarization channels. Those discrete records are measurements of $\hat n$, or correlations among several $\hat n$'s, after convolution with efficiency, dark counts, and timing response. Astronomical light is usually multimode. If the detection window has duration $\Delta t$, spectral bandwidth $\Delta\nu$, effective etendue $A\Omega$, and $N_{\rm pol}$ polarization channels, a rough mode count is ```{math} :label: eq:ch03-effective-modes M \simeq \left(\frac{\Delta t}{\tau_c}\right) \left(\frac{A\Omega}{\lambda^2}\right) N_{\rm pol}, \qquad \tau_c\simeq\frac{1}{\Delta\nu}\simeq\frac{\lambda^2}{c\,\Delta\lambda}. ``` $\Delta t$ and $\tau_c$ are in seconds, $A\Omega$ is in $\mathrm{m^2\,sr}$, and $\lambda^2$ is in $\mathrm{m^2}$. A diffraction-limited spatial mode has $A\Omega\sim\lambda^2$. A seeing-limited fiber or a large pixel receives many spatial modes. At visible wavelengths, $\Delta\lambda=1\,\mathrm{nm}$ and $\lambda=500\,\mathrm{nm}$ correspond to $\Delta\nu\simeq1.2\times10^{12}\,\mathrm{Hz}$, so $\tau_c\simeq0.8\,\mathrm{ps}$. A detector response of $100\,\mathrm{ps}$ to $1\,\mathrm{ns}$ has already averaged over hundreds to thousands of time modes. ## Fock, coherent, thermal, and squeezed states A Fock state $|n\rangle$ is an eigenstate of photon number, $\hat n|n\rangle=n|n\rangle$. For a single-mode $n$-photon Fock state, an ideal photon-number-resolving detector gives ```{math} :label: eq:ch03-fock P_{\rm Fock}(N)=\delta_{Nn},\qquad g^{(2)}(0)=\frac{\langle \hat n(\hat n-1)\rangle}{\langle\hat n\rangle^2} =1-\frac{1}{n}. ``` $P(N)$ is a dimensionless probability, and $g^{(2)}$ is also dimensionless. A single-photon state has $g^{(2)}(0)=0$, the limiting case of antibunching. A large-$n$ Fock state has small relative photon-number fluctuations but a completely undefined phase. Astrophysical light rarely arrives as a pure Fock state, because the source, propagation path, and instrument mix many unobserved degrees of freedom. Fock states remain useful because they make the measurement clear: click statistics measure photon-number correlations, not a continuous waveform {cite:p}`1977PhRvL..39..691K,1979OptL....4..205M,1998RvMP...70..101P`. A coherent state $|\alpha\rangle$ satisfies $\hat a|\alpha\rangle=\alpha|\alpha\rangle$. Its mean photon number is $\bar n=|\alpha|^2$, and the single-mode count distribution is Poisson: ```{math} :label: eq:ch03-coherent P_{\rm coh}(N)=e^{-\bar n}\frac{\bar n^N}{N!}, \qquad \operatorname{Var}(N)=\bar n,\qquad g^{(2)}(0)=1. ``` $\alpha$ is a dimensionless complex amplitude. Its squared modulus is the mean photon number per mode, and its argument gives the radiation phase. A stable laboratory laser is close to a coherent state for many measurements. A bright, narrow astronomical line is not automatically a coherent state, because many independent emitting regions, velocity fields, and scattering paths can average the phase away. Thermal light is the usual starting point for astrophysical radiation. The single-mode thermal density matrix is ```{math} :label: eq:ch03-thermal-density \rho_{\rm th}=\sum_{N=0}^{\infty} \frac{\bar n^N}{(1+\bar n)^{N+1}}|N\rangle\langle N|, \qquad \operatorname{Var}(N)=\bar n(1+\bar n),\qquad g^{(2)}(0)=2. ``` It has no fixed phase. It is not a pure state with many random phases added by hand, but a mixed state that is diagonal in the photon-number basis. The $\bar n$ in Eq. {eq}`eq:ch03-thermal-density` is the mean occupation number of one mode, not the total count rate received by a telescope. A bright star can deliver $10^6$--$10^{10}\,\mathrm{s^{-1}}$ detected photons, yet still have $\bar n\ll1$ per optical mode once the optical frequency, picosecond coherence time, and large spatial mode number are included. Multimode thermal light averages down the excess variance in Eq. {eq}`eq:ch03-thermal-density`; the common approximation is $g^{(2)}(0)=1+1/M$ {cite:p}`1965RvMP...37..231M,1995ocqo.book.....M,2000stop.book.....G,2017MNRAS.472.4126G,2024arXiv240418606N`. ```{figure} ../_static/figures/generated/chapter_03/ch03_state_count_distributions.png :name: fig:ch03-count-distributions :width: 82.0% Fock, coherent, and thermal states give different count distributions near the same single-mode mean photon number. A Fock state has fixed photon number, a coherent state has width $\sqrt{\bar n}$, and a thermal state has a longer high-count tail, which produces the bunching signal g(2)(0) > 1. ``` A squeezed state changes the noise in the two field quadratures rather than the mean intensity alone. Define ```{math} :label: eq:ch03-squeezing \hat X_\theta=\frac{1}{2}\left(\hat a e^{-i\theta}+\hat a^\dagger e^{i\theta}\right), \qquad \Delta X_{\rm sq}^2=\frac{1}{4}e^{-2r},\qquad \Delta X_{\rm anti}^2=\frac{1}{4}e^{2r}. ``` $\hat X_\theta$ is a dimensionless quadrature. The vacuum variance is $1/4$, and $r$ is the squeezing parameter. Experiments usually quote the noise variance relative to vacuum in dB. For example, $10\log_{10}(e^{-2r})=-10\,\mathrm{dB}$ means that the variance in that quadrature has been reduced to one tenth of the vacuum value. Modern squeezed light experiments near $1064\,\mathrm{nm}$ have measured noise reductions of roughly 10--15 dB, and squeezed light is now used in gravitational-wave detectors and quantum-efficiency calibration. It is a real resource for instrumental quantum-noise engineering, but it is not the default state of an ordinary stellar continuum {cite:p}`1983Natur.306..141W,1987JMOp...34..709L,2016PhRvL.117k0801V`. ## Phase space, density matrices, and discarded degrees of freedom A phase-space plot represents one mode by two quadratures, usually called $X$ and $P$. A coherent state appears as a vacuum-sized circular patch whose center has been displaced from the origin. A thermal state is still circular, but its radius is enlarged by $\bar n$. A squeezed state is narrowed in one direction and broadened in the conjugate direction. The Wigner function can become negative. The $Q$ function is the distribution smoothed by vacuum noise as in heterodyne detection. The Glauber--Sudarshan $P$ representation writes the density matrix as a mixture of coherent states: ```{math} :label: eq:ch03-phase-space \rho=\int P(\alpha)|\alpha\rangle\langle\alpha|\,\mathrm{d}^2\alpha, \qquad Q(\alpha)=\frac{1}{\pi}\langle\alpha|\rho|\alpha\rangle . ``` If $P(\alpha)$ is an ordinary non-negative probability density, many intensity and coherence functions can be obtained from a classical random field. Thermal and coherent light are often classical in this sense. If $P$ is more singular than a delta function, or if the Wigner function becomes negative, the semiclassical picture fails. Most basic formulae in astronomical intensity interferometry lie in the region where semiclassical and quantum theories agree. That does not make the quantum language redundant. Detection, loss, and the weak-light limit are still most cleanly counted with density matrices and photon number {cite:p}`1963PhRvL..10..277S,1972PhRvA...6.2211A,2001qops.book.....S,1996PhRvL..76.4344B,1999PhRvA..60..674B`. ```{figure} ../_static/figures/generated/chapter_03/ch03_phase_space_states.png :name: fig:ch03-phase-space :width: 72.0% Widths of common single-mode states in phase space. A coherent state has vacuum-sized noise around a displaced center. A thermal state is broader because its intensity and phase fluctuate. A squeezed state reduces the fluctuation in one quadrature and increases it in the conjugate quadrature. ``` In astronomical problems, density matrices are often used to sum over unobserved degrees of freedom: ```{math} :label: eq:ch03-partial-trace \rho_{\rm obs}=\operatorname{Tr}_{\rm unobs}\rho_{\rm full}. ``` $\rho_{\rm full}$ may include source position, frequency, propagation path, polarization, internal detector degrees of freedom, and the environment. $\rho_{\rm obs}$ keeps only the degrees of freedom that remain distinguishable in the data. Finite bandwidth traces over out-of-band frequencies. A seeing-limited aperture mixes spatial modes. Unrecorded polarization averages over two polarization channels. Long time bins add together fluctuations from many coherence times. This is why astrophysical light is usually not a pure state in practice. Quantum mechanics has not failed. The observer has not kept enough degrees of freedom to describe pure-state coherence. Cosmological decoherence, coherence limits in gravitational-lensing paths, and angular-size decoherence in stellar intensity interferometry all ask the same bookkeeping question: which degrees of freedom are retained, and which have been averaged over {cite:p}`1998RvMP...70..101P,2020MNRAS.491.5789L,1996CQGra..13..377P`. ## Photodetection: from field operators to clicks Glauber detection theory writes photoelectric click probabilities as normally ordered field correlations: ```{math} :label: eq:ch03-glauber G^{(n)}(1,\ldots,n) = \left\langle \hat E^{(-)}(1)\cdots \hat E^{(-)}(n) \hat E^{(+)}(n)\cdots \hat E^{(+)}(1) \right\rangle . ``` Here $1$ denotes $(\bm{r}_1,t_1)$, and may also include polarization and frequency channel. $G^{(1)}$ gives mean intensity and first-order coherence. $G^{(2)}$ gives the joint probability density for two clicks. Normal ordering has a direct physical meaning: the detector must absorb a photon before it can click, so the click probability is proportional to the number of photons available to be absorbed. The normalized second-order coherence is $g^{(2)}=G^{(2)}/(G^{(1)}G^{(1)})$, with no units. In an event table it is usually estimated by dividing a delay histogram by the accidental-coincidence baseline {cite:p}`1963PhRv..130.2529G,1965RvMP...37..231M,1995ocqo.book.....M,2017MNRAS.472.4126G`. Finite efficiency is equivalent to putting a virtual beamsplitter in front of the detector: ```{math} :label: eq:ch03-efficiency-loss \hat a_{\rm det}=\sqrt{\eta}\,\hat a+\sqrt{1-\eta}\,\hat v, \qquad R_{\rm det}\simeq \eta R_{\rm src}+R_{\rm dark}+R_{\rm sky}. ``` $\eta$ is a dimensionless quantum efficiency, $\hat v$ is a vacuum noise mode, and each rate $R$ is measured in $\mathrm{s^{-1}}$. Ideal loss alone does not change the normalized $g^{(2)}$ of a single isolated source, but it reduces the effective count rate and increases the statistical error. Dark counts and sky background mix source photons with unrelated clicks, reducing the visible excess correlation roughly by the square of the source-photon fraction. Avalanche photodiodes, superconducting nanowire detectors, and photomultipliers often have timing jitter from tens of picoseconds to nanoseconds. Dark-count rates can range from $\ll1\,\mathrm{s^{-1}}$ to more than $10^2\,\mathrm{s^{-1}}$, depending on the technology and operating temperature {cite:p}`1996ApOpt..35.1956C,2010NatNa...5..391K,2016PhRvL.117k0801V`. ```{figure} ../_static/figures/generated/chapter_03/ch03_detection_loss_background.png :name: fig:ch03-detection-loss :width: 92.0% How detector efficiency, sky background, and dark counts enter click statistics. The left panel shows that higher efficiency increases the source count rate linearly, while the total count still includes sky and dark counts. The right panel shows that the visible excess in the normalized second-order correlation is multiplied approximately by the square of the source-photon fraction. Background both adds noise and dilutes the signal amplitude. ``` When a click histogram is compared with Eq. {eq}`eq:ch03-glauber`, three time scales must be kept separate. The coherence time $\tau_c$ is set by the spectral bandwidth. The detector response or time bin $\Delta t$ determines how the correlation peak is convolved. The total integration time $T_{\rm int}$ sets the statistical error. If two channels have count rates $R_1$ and $R_2$, the accidental-pair count is roughly $N_{\rm pair}\simeq R_1R_2T_{\rm int}\Delta t$, with Poisson noise $\sqrt{N_{\rm pair}}$. The thermal excess signal is approximately $(g^{(2)}_{\rm obs}-1)N_{\rm pair}$. A very narrow correlation peak can therefore be physically present and still appear with a visible contrast of $10^{-3}$ or less after nanosecond electronics have averaged it down. ## The astronomical weak-light limit and useful scales The single-mode occupation number of thermal radiation is set by the brightness temperature $T_b$: ```{math} :label: eq:ch03-occupation \bar n_\nu=\frac{1}{\exp(h\nu/k_{\rm B}T_b)-1}. ``` Both $h\nu$ and $k_{\rm B}T_b$ are energies in joules, and $\bar n_\nu$ is dimensionless. For a stellar surface with $T_b=5800\,\mathrm{K}$, $h\nu/k_{\rm B}T_b\simeq5$ at $\lambda=500\,\mathrm{nm}$, giving $\bar n_\nu\simeq7\times10^{-3}$. At $1\,\mu\mathrm{m}$, $\bar n_\nu\simeq0.09$. At $10\,\mu\mathrm{m}$, it can exceed unity. Radio and millimeter wavelengths often lie in the Rayleigh--Jeans regime, $h\nu\ll k_{\rm B}T_b$, where the occupation number is large and the field behaves more like a classical wave. This scale explains why optical astronomy can have high total photon rates while remaining in the weak-light limit per mode. ```{figure} ../_static/figures/generated/chapter_03/ch03_blackbody_mode_occupation.png :name: fig:ch03-mode-occupation :width: 82.0% Single-mode occupation number as a function of wavelength for several blackbody brightness temperatures. In the visible band, even a solar-temperature surface has n̄ν ≪ 1. Infrared, millimeter, and high-brightness maser regimes can reach n̄ν ≳ 1. ``` Weak thermal light is often written, for each coherence-time window, as the approximate density matrix ```{math} :label: eq:ch03-weak-thermal \rho\simeq (1-\epsilon)\rho_0+\epsilon\rho_1+O(\epsilon^2), \qquad \epsilon\equiv\langle N\rangle_{\rm mode}\ll1. ``` $\rho_0$ is the vacuum term, $\rho_1$ is the one-photon term, and $\epsilon$ is the mean photon number in that mode and time window. This is the expansion used by Tsang, Nair, and Lu in the theory of weak-thermal-light superresolution: most coherence-time windows contain no photon, a small fraction contain one photon, and the information is carried mainly by the spatial modes of the detected photons {cite:p}`1969JSP.....1..231H,2015Optic...2..646T,2016PhRvX...6c1033T`. The same structure appears in astronomical event tables. Empty time windows are usually not stored. The data file keeps the time-tagged clicks. The visible contrast of thermal photon bunching is controlled by both mode number and timing resolution: ```{math} :label: eq:ch03-time-dilution g^{(2)}_{\rm obs}(0)-1 \simeq \frac{1}{M} \min\left(1,\frac{\tau_c}{\Delta t}\right). ``` $M$ is the effective number of spatial, polarization, and otherwise unresolved modes. $\Delta t$ is the electronic response or time bin. The 2017 stellar photon-bunching experiment worked near $\lambda_0=780\,\mathrm{nm}$ with $\Delta\lambda=10\,\text{\AA}$, giving a coherence time of about $1.6\,\mathrm{ps}$. The APD response was about $500\,\mathrm{ps}$, so the single-mode thermal peak, whose intrinsic value is $g^{(2)}(0)-1=1$, was diluted to a contrast of order $2\times10^{-3}$. The thermal light had not stopped bunching. The peak was simply much faster than the detector {cite:p}`2017MNRAS.472.4126G`. ```{figure} ../_static/figures/generated/chapter_03/ch03_coherence_time_dilution.png :name: fig:ch03-coherence-dilution :width: 82.0% Contrast of the thermal second-order correlation peak after finite timing resolution and multimode averaging. The horizontal axis is the detector time window divided by the coherence time. When Δt ≫ τc, the peak area remains, but the height is spread out by τc/Δt. The effective mode number M divides the amplitude again. ``` ## Nonclassical boundaries, masers, and natural lasers The most common nonclassical boundary is $g^{(2)}(0)<1$. It means that the probability of detecting two photons together is below the Poisson value, which cannot be explained by a positive classical intensity fluctuation. Laboratory resonance fluorescence and single-photon sources can produce clear antibunching. Astronomical observations face several dilution mechanisms. The source usually contains many independent emitters. Propagation and the aperture mix many spatial modes. Background and dark counts pull any antibunching signal towards unity. Even if a microscopic process emits nonclassical light, the measured $g^{(2)}$ will be close to 1 unless the observation isolates one emitter and one mode {cite:p}`1977PhRvL..39..691K,1979OptL....4..205M,1995ocqo.book.....M`. ```{figure} ../_static/figures/generated/chapter_03/ch03_nonclassical_mixing_boundary.png :name: fig:ch03-nonclassical-mixing :width: 74.0% Background and source mixing pull the observed g(2)(0) toward 1. The blue curve represents ideal single-emitter antibunching, and the orange curve represents single-mode thermal bunching. As the source-photon fraction f decreases, the departure from 1 falls approximately as f2. The dashed curves show that multimode thermal light with M > 1 also has a shallower bunching peak. ``` Stimulated emission, masers, and astrophysical lasers are easy places to say too much. Stimulated emission can produce enormous brightness temperatures and narrow lines. Radio masers can have very large single-mode occupation numbers. High brightness temperature, however, is not the same as a coherent state, and a narrow line does not automatically imply $g^{(2)}=1$. Unsaturated masers, saturated masers, multiple maser spots, velocity gradients, and pump noise can produce different time statistics. To use photon-correlation spectroscopy to identify a natural laser component, one must specify the line width, spatial coherence, polarization, continuum background, and instrumental timing response. The phrase \"non-Poisson\" alone is not a diagnosis of new physics {cite:p}`1972ApJ...174..517G,2005NewA...10..361J,2008AIPC..984..216D,2017MNRAS.472.4126G`. | Optical state | Single-mode counts | $g^{(2)}(0)$ | Astronomical context | |:------------|:-----------------|:----------------|:-----------------| | Fock state $|n\rangle$ | Fixed $N=n$ | $1-1/n$ | Ideal model for a single emitter or quantum source; extremely hard to isolate astrophysically. | | Coherent state | Poisson | 1 | Good approximation for a stable laser; astronomical narrow lines need not be coherent states. | | Thermal state | Bose--Einstein | 2, or multimode $1+1/M$ | Basic model for stellar continua and many incoherent sources. | | Squeezed state | Depends on the measured quadrature | Can be above, below, or near 1 | Important for instrumental quantum-noise engineering; not assumed for ordinary astrophysical light. | ## Exercises and computational project Exercise 1. Use Eq. {eq}`eq:ch03-occupation` to compute $\bar n_\nu$ for $T_b=5800\,\mathrm{K}$ at $\lambda=0.5\,\mu\mathrm{m}$, $1\,\mu\mathrm{m}$, and $10\,\mu\mathrm{m}$. Explain why \"high total photon rate\" and \"high single-mode occupation\" are not the same statement. Exercise 2. A stellar intensity-interferometry experiment uses $\lambda=780\,\mathrm{nm}$, $\Delta\lambda=1\,\mathrm{nm}$, and $\Delta t=500\,\mathrm{ps}$, with one spatial mode and one selected polarization. Use Eq. {eq}`eq:ch03-time-dilution` to estimate $g^{(2)}_{\rm obs}(0)-1$, then repeat the estimate for $M=10$. Exercise 3. Let $\bar n=0.05$. Compute $P(0)$, $P(1)$, and $P(2)$ for a coherent state and for a thermal state. Compare $P(2)$ with $\bar n^2/2$, and explain why second-order correlations require long integrations. Computational project. Use Python to generate count sequences with $10^6$ time bins for Poisson light and single-mode thermal light. Then sum $M=2,10,100$ independent thermal modes in adjacent bins, and measure how the sample $g^{(2)}(0)$ approaches 1. Plot the simulation results and the theoretical curve $1+1/M$ on the same axes.