Thermal Light

Thermal light is radiation whose resolved electromagnetic modes are in thermal states. One mode of angular frequency $\omega$ at temperature $T$ has the density operator

$$\rho_{\mathrm{th}} = \frac{ e^{-\beta\hbar\omega\hat N} }{ \operatorname{Tr} e^{-\beta\hbar\omega\hat N} }, \qquad \beta=\frac1{k_{\mathrm B}T}.$$

The zero-point energy cancels between numerator and partition function. In the number basis,

$$\rho_{\mathrm{th}} = \frac1{1+\bar n} \sum_{n=0}^{\infty} \left( \frac{\bar n}{1+\bar n} \right)^n \lvert n\rangle\langle n\rvert,$$

where

$$\bar n = \frac1{ e^{\beta\hbar\omega}-1 }.$$

For nonzero mean occupation, this formula gives the single-mode signatures:

$$\operatorname{Var}(N) = \bar n(1+\bar n),$$ $$g^{(2)}(0)=2,$$

and

$$\langle\hat a\rangle=0.$$

The state is number diagonal, phase insensitive, super-Poissonian, and classical in the positive Glauber–Sudarshan sense. Its bunching is real, but bunching by itself is not a nonclassicality witness.

Required background. Quantized Electromagnetic Modes defines the mode normalization, Photon Number States introduces factorial moments, Bose–Einstein Statistics gives bosonic equilibrium occupations, and Density Operators explains statistical mixtures.

Thermal Equilibrium and Chaotic Fields

Three phrases are often used near one another:

Term	Meaning
thermal equilibrium light	radiation modes in a Gibbs state at a specified temperature
chaotic or Gaussian light	a zero-mean circular complex Gaussian random optical field
pseudo-thermal light	engineered fluctuating light that reproduces selected thermal correlations

Ideal equilibrium radiation is Gaussian and obeys thermal statistics. Chaotic light can obey the same optical correlation laws without being in global thermodynamic equilibrium. Pseudo-thermal light may reproduce $g^{(2)}$ over a chosen bandwidth while differing in spectrum, higher correlations, stationarity, polarization, or modal structure.

The word “thermal” should therefore be attached to a state model and a set of resolved modes, not inferred from a single bunching peak.

One Thermal Electromagnetic Mode

For one lossless radiation mode,

$$\hat H = \hbar\omega \left( \hat N+\frac12 \right), \qquad \hat N=\hat a^\dagger\hat a.$$

The canonical state is

$$\rho = \frac{e^{-\beta\hat H}}{Z}, \qquad Z=\operatorname{Tr}e^{-\beta\hat H}.$$

Define

$$q \equiv e^{-\beta\hbar\omega}.$$

The oscillator partition function is

$$Z = \frac{ e^{-\beta\hbar\omega/2} }{ 1-q }.$$

The common zero-point factor cancels, leaving

$$\rho_{\mathrm{th}} = (1-q) \sum_{n=0}^{\infty} q^n \lvert n\rangle\langle n\rvert.$$

The mean occupation is

$$\bar n = \frac{q}{1-q} = \frac1{ e^{\beta\hbar\omega}-1 }.$$

Solving for $q$ gives

$$q = \frac{\bar n}{1+\bar n}.$$

Hence the photon-number law is geometric:

$$P(n) = \frac1{1+\bar n} \left( \frac{\bar n}{1+\bar n} \right)^n.$$

Unlike a coherent state, a thermal state is mixed. Its purity is

$$\begin{aligned} \operatorname{Tr}(\rho_{\mathrm{th}}^2) &= \frac{(1-q)^2}{1-q^2} \\ &= \frac1{2\bar n+1}. \end{aligned}$$

The state approaches the pure vacuum as $T\to0$, but any nonzero thermal occupation lowers the purity.

Why the photon chemical potential is zero

In ordinary blackbody equilibrium, photon number is not conserved. The cavity walls can absorb and emit photons while conserving total energy, so the equilibrium photon chemical potential is

$$\mu_\gamma=0.$$

The mode occupation is therefore

$$\bar n(\omega,T) = \frac1{ e^{\beta\hbar\omega}-1 }.$$

Driven photonic systems can realize effective nonzero chemical potentials or approximately conserved excitation numbers, but those are nonequilibrium or constrained settings and should not be inserted into the blackbody formula without a physical mechanism.

Number Fluctuations

The probability-generating function of the geometric law is

$$G(z) = \sum_{n=0}^{\infty}P(n)z^n = \frac{ 1-q }{ 1-qz }.$$

Differentiation gives

$$\langle N\rangle = G'(1) = \bar n,$$

and

$$\langle N(N-1)\rangle = G''(1) = 2\bar n^2.$$

Therefore

$$\langle N^2\rangle = \bar n+2\bar n^2,$$

and

$$\operatorname{Var}(N) = \bar n(1+\bar n).$$

The Fano factor is

$$F = \frac{ \operatorname{Var}(N) }{ \bar n } = 1+\bar n,$$

while the Mandel parameter is

$$Q=F-1=\bar n.$$

Thermal light is super-Poissonian for every nonzero $\bar n$.

Relative fluctuations do not vanish

The relative number uncertainty is

$$\frac{\Delta N}{\bar n} = \sqrt{ 1+\frac1{\bar n} }.$$

For a highly occupied single thermal mode,

$$\frac{\Delta N}{\bar n} \longrightarrow 1.$$

Large occupation does not make one thermal mode a deterministic classical wave. It makes it a bright classical stochastic wave with order-one relative intensity fluctuations. Relative fluctuations become small only after averaging many effectively independent modes, coherence cells, or time samples.

Phase-Space Description

Thermal light has no preferred field phase:

$$\langle\hat a\rangle=0.$$

For the quadratures

$$\hat X = \frac{ \hat a+\hat a^\dagger }{ \sqrt2 }, \qquad \hat P = \frac{ \hat a-\hat a^\dagger }{ i\sqrt2 },$$

one finds

$$\langle X\rangle = \langle P\rangle = 0,$$

and

$$(\Delta X)^2 = (\Delta P)^2 = \bar n+\frac12.$$

The Wigner function is the isotropic Gaussian

$$W_{\mathrm{th}}(\alpha) = \frac{ 2 }{ \pi(2\bar n+1) } \exp\left[ -\frac{ 2\lvert\alpha\rvert^2 }{ 2\bar n+1 } \right].$$

The Glauber–Sudarshan function is also an ordinary positive Gaussian:

$$P_{\mathrm{th}}(\alpha) = \frac1{\pi\bar n} \exp\left( -\frac{\lvert\alpha\rvert^2}{\bar n} \right)$$

for $\bar n>0$. Thus

$$\rho_{\mathrm{th}} = \int d^2\alpha\, P_{\mathrm{th}}(\alpha) \lvert\alpha\rangle\langle\alpha\rvert.$$

Thermal light can be modeled as a circular Gaussian ensemble of coherent amplitudes. Both phase and magnitude fluctuate. This positive representation is why ideal thermal bunching admits a semiclassical random-wave interpretation.

Top: a thermal mode and a coherent mode both have $\bar n=1$, but their number laws are geometric and Poisson, respectively. Bottom: the free single-frequency thermal oscillator has $g^{(2)}(\tau)=2$ at every delay. A stationary circular Gaussian traveling field with continuous Lorentzian spectrum has $g^{(2)}(\tau)=1+e^{-2\gamma|\tau|}$ in one selected spatial and polarization channel. The decaying curve involves a continuum of frequencies, not the isolated oscillator of the dashed curve.

Thermal Versus Phase-Randomized Coherent Light

A phase-randomized coherent state with mean $\bar n$ is

$$\rho_{\mathrm{PRC}} = \int_0^{2\pi} \frac{d\phi}{2\pi} \lvert \sqrt{\bar n}e^{i\phi} \rangle \langle \sqrt{\bar n}e^{i\phi} \rvert.$$

It is number diagonal, just like a thermal state, but its number distribution is Poisson:

$$P_{\mathrm{PRC}}(n) = e^{-\bar n} \frac{\bar n^n}{n!}.$$

The comparison is:

Property	Phase-randomized coherent	Thermal
$\langle a\rangle$	$0$	$0$
complex amplitude	fixed radius, uniform phase	circular Gaussian; Rayleigh radius and uniform phase
number law	Poisson	geometric
variance	$\bar n$	$\bar n(1+\bar n)$
$g^{(2)}(0)$	$1$	$2$
positive $P$ model	ring distribution	circular Gaussian

Random phase alone does not create thermal light. Thermal statistics require amplitude fluctuations with the appropriate exponential intensity law.

Photon Bunching

The normalized equal-time second-order coherence is

$$g^{(2)}(0) = \frac{ \langle \hat a^\dagger\hat a^\dagger\hat a\hat a \rangle }{ \langle\hat a^\dagger\hat a\rangle^2 } = \frac{ \langle N(N-1)\rangle }{ \bar n^2 }.$$

For one ideal thermal mode,

$$g^{(2)}(0)=2.$$

This is a factorial moment of a selected oscillator or packet mode. An isolated monochromatic thermal oscillator has $g^{(2)}(\tau)=2$ at every delay under free evolution: it does not acquire a decorrelation time from the Gibbs distribution alone.

For a stationary chaotic traveling field in one spatial/polarization channel, with a continuous spectrum and decaying field correlation, a conditional detection rate can instead be twice its long-delay value at zero delay. That temporal bunching peak is derived next.

The phrase photon bunching describes this enhanced coincidence probability. It does not mean that photons exert an attractive force or travel in rigid packets. In a positive-$P$ description, the field intensity fluctuates: detections are more likely during bright fluctuations, and a first detection updates the observer toward such a bright interval.

Higher orders

For one thermal mode,

$$\left\langle (\hat a^\dagger)^m\hat a^m \right\rangle = m!\,\bar n^m.$$

Therefore

$$g^{(m)}(0)=m!.$$

Thermal fluctuations become increasingly distinct from coherent factorization at higher order. Measuring only $g^{(2)}$ does not establish the entire hierarchy.

The Siegert Relation

For one selected scalar field component with zero-mean, stationary, circular complex Gaussian statistics, fourth-order moments factor into products of second-order moments. This gives the Siegert relation

$$g^{(2)}(x_1,x_2) = 1+ \left| g^{(1)}(x_1,x_2) \right|^2.$$

At equal spacetime points,

$$\left| g^{(1)}(x,x) \right|=1,$$

so

$$g^{(2)}(x,x)=2.$$

At separations beyond the mutual coherence region,

$$g^{(1)}(x_1,x_2)\to0,$$

and

$$g^{(2)}(x_1,x_2)\to1.$$

The relation requires circular Gaussian statistics (no anomalous pair correlator) and zero coherent displacement. An unresolved sum of independent polarizations or spatial channels has reduced contrast. The scalar relation need not hold for arbitrary bunched fields, non-Gaussian sources, displaced thermal states, or detector records dominated by technical artifacts.

Temporal coherence

For a stationary field with nonnegative spectral density $S(\omega)$,

$$g^{(1)}(\tau) = \frac{ \int d\omega\, S(\omega)e^{-i\omega\tau} }{ \int d\omega\,S(\omega) }.$$

The first-order coherence time is set by spectral width. Under the Gaussian thermal assumption,

$$g^{(2)}(\tau) = 1+ \left| g^{(1)}(\tau) \right|^2.$$

For a Lorentzian spectrum with

$$g^{(1)}(\tau) = e^{-i\omega_0\tau} e^{-\gamma\lvert\tau\rvert},$$

where $\gamma$ is the angular-frequency spectral HWHM, the intensity correlation is

$$g^{(2)}(\tau) = 1+ e^{-2\gamma\lvert\tau\rvert}.$$

The bunching feature is narrower than the first-order field-coherence envelope under this convention. The Lorentzian Fourier pair uses a carrier envelope with detunings extended to the real line; it approximates an optical positive-frequency spectrum when $\gamma\ll\omega_0$.

Spatial coherence

At equal times,

$$g^{(2)}(\mathbf r_1,\mathbf r_2) = 1+ \left| g^{(1)}(\mathbf r_1,\mathbf r_2) \right|^2.$$

Intensity interferometry uses this relation to infer spatial coherence even when optical phase fluctuates too rapidly for direct amplitude interferometry. For a distant incoherent source, spatial coherence contains information about the source’s angular intensity distribution.

The full interferometer geometry and source-reconstruction problem belong to Hanbury Brown–Twiss Interferometry.

What an HBT Measurement Records

An optical Hanbury Brown–Twiss arrangement splits a field and time-tags detections at two outputs. A normalized coincidence histogram estimates

$$g^{(2)}(\tau)$$

after accounting for detector efficiencies, backgrounds, timing response, dead time, and normalization windows.

Thermal bunching is seen as excess coincidences near the delay over which the field remains coherent. The beam splitter does not create the thermal correlation; it routes the incident fluctuations to two detectors so they can be compared without one detector’s dead time dominating the same-channel record.

Timing resolution

If the true bunching feature is narrower than detector jitter or the correlation bin, the measured peak is averaged down:

$$g_{\mathrm{obs}}^{(2)}(0)<2.$$

A small observed excess can therefore be consistent with single-mode thermal statistics at the source. The comparison requires convolution with the instrument response and integration over the detected optical bandwidth.

Uncorrelated background

Let a thermal signal contribute mean rate $S$ and an independent Poissonian background contribute $B$, with their detected intensities added without field interference (for example, orthogonal unresolved modes or independent detector dark counts). At zero delay, the excess normalized correlation is diluted:

$$g_{\mathrm{obs}}^{(2)}(0)-1 = \left( \frac{S}{S+B} \right)^2.$$

Hence

$$g_{\mathrm{obs}}^{(2)}(0) = 1+ \left( \frac{S}{S+B} \right)^2$$

for an ideal single-mode thermal signal and ideal uncorrelated background. Background subtraction without propagated uncertainty can substantially overstate bunching. A coherent displacement in the same detected optical mode instead interferes with the thermal amplitude. Its excess is $(S^2+2SB)/(S+B)^2$, not $S^2/(S+B)^2$. The background model must specify whether fields or independent counts are being combined.

Multimode Thermal Light

Real detectors usually collect several temporal, spectral, spatial, or polarization modes. Here the detector measures the sum $N=\sum_j a_j^\dagger a_j$, not a coherent projection onto one superposed mode. For time-integrated counts, the appropriate independent thermal modes diagonalize the detected first-order coherence kernel. Suppose $M$ such modes have equal mean occupation and total mean

$$\mu = \sum_{j=1}^{M}\bar n_j.$$

For equal modes,

$$\bar n_j=\frac{\mu}{M}.$$

The total-count generating function is

$$G_M(z) = \left[ 1+ \frac{\mu}{M} (1-z) \right]^{-M}.$$

The resulting negative-binomial distribution is

$$P_M(n) ={} \binom{n+M-1}{n} \left( \frac{M}{M+\mu} \right)^M \times \left( \frac{\mu}{M+\mu} \right)^n.$$

Its mean and variance are

$$\langle N\rangle=\mu,$$ $$\operatorname{Var}(N) = \mu+\frac{\mu^2}{M}.$$

Therefore

$$g^{(2)}(0) = 1+\frac1M.$$

As more independent modes are averaged,

$$g^{(2)}(0)\to1.$$

This is averaging in the summed intensity, not conversion of each mode to a coherent state. At fixed total $\mu$, the full negative-binomial law approaches Poisson as $M\to\infty$. At fixed occupation per mode, $g^{(2)}$ still approaches one but the Fano factor stays $1+\bar n_j>1$ as the total mean grows. Near-unity normalized correlation therefore does not always imply shot-noise-limited counts.

Unequal modes and effective mode number

For independent thermal modes with means $\bar n_j$,

$$\langle N\rangle = \sum_j\bar n_j,$$

and

$$\operatorname{Var}(N) = \sum_j \bar n_j(1+\bar n_j).$$

The normalized correlation is

$$g^{(2)}(0) = 1+ \frac{ \sum_j\bar n_j^2 }{ \left( \sum_j\bar n_j \right)^2 }.$$

Define

$$M_{\mathrm{eff}} = \frac{ \left( \sum_j\bar n_j \right)^2 }{ \sum_j\bar n_j^2 }.$$

Then

$$g^{(2)}(0) = 1+ \frac1{M_{\mathrm{eff}}}.$$

The effective mode number is an inverse participation ratio. It equals the integer $M$ only for equally weighted modes.

Polarization must be counted. Detecting two uncorrelated polarization modes with equal intensity gives

$$g^{(2)}(0)=\frac32,$$

not $2$. In contrast, a single coherent projection $c=\sum_j u_j a_j$ with $\sum_j|u_j|^2=1$ is itself one thermal Gaussian mode with mean $\sum_j|u_j|^2\bar n_j$ and $g_c^{(2)}=2$ when that mean is nonzero. Which modes the apparatus sums or projects determines the measured statistic.

Pure Loss and Thermal Light

A pure-loss channel with a vacuum environment and transmissivity $0<\eta\le1$ transforms the number generating function by

$$G_{\mathrm{out}}(z) = G_{\mathrm{in}} \left( 1-\eta+\eta z \right).$$

For one thermal mode,

$$G_{\mathrm{in}}(z) = \frac1{ 1+\bar n(1-z) }.$$

Therefore

$$G_{\mathrm{out}}(z) = \frac1{ 1+\eta\bar n(1-z) }.$$

The output is another thermal state with

$$\bar n_{\mathrm{out}} = \eta\bar n.$$

Ideal uniform loss preserves

$$g^{(2)}(0)=2$$

for a single resolved mode, even though brightness and signal-to-background ratio decrease. Mode-dependent loss can change the effective mode number and therefore the observed bunching. Mode-by-mode thermal form is not the same as a common equilibrium temperature: attenuating a Planck spectrum by constant $\eta<1$ generally does not produce a new Planck spectrum at one lower temperature. Only the Rayleigh–Jeans limit suggests $T_{\rm eff}\simeq\eta T$.

Relation to Blackbody Radiation

In an ideal cavity at temperature $T$, each electromagnetic normal mode has

$$\bar n(\omega,T) = \frac1{ e^{\hbar\omega/(k_{\mathrm B}T)}-1 }.$$

The number of free-space modes per volume per angular-frequency interval, including two transverse polarizations, is

$$\frac{dM}{V\,d\omega} = \frac{\omega^2}{\pi^2c^3}.$$

Multiplying mode density by the mean excitation energy $\hbar\omega\bar n$ gives the Planck energy density

$$u(\omega,T) = \frac{ \hbar\omega^3 }{ \pi^2c^3 } \frac1{ e^{\hbar\omega/(k_{\mathrm B}T)}-1 }.$$

This expression omits the formal zero-point contribution $\hbar\omega/2$ per mode because blackbody emission and ordinary thermal energy measurements concern the temperature-dependent excitation energy.

In ordinary frequency $\nu=\omega/(2\pi)$,

$$u(\nu,T) = \frac{ 8\pi h\nu^3 }{ c^3 } \frac1{ e^{h\nu/(k_{\mathrm B}T)}-1 }.$$

Low- and high-frequency limits

When

$$\hbar\omega\ll k_{\mathrm B}T,$$

the mean occupation is

$$\bar n \approx \frac{ k_{\mathrm B}T }{ \hbar\omega }.$$

The mean mode energy approaches $k_{\mathrm B}T$, recovering the Rayleigh–Jeans limit.

When

$$\hbar\omega\gg k_{\mathrm B}T,$$

the occupation is exponentially small:

$$\bar n \approx e^{-\hbar\omega/(k_{\mathrm B}T)}.$$

Planck’s law describes the mean spectral energy density. The geometric one-mode distribution and correlation hierarchy provide additional information not visible in the mean spectrum alone.

Why blackbody bunching is hard to see optically

Broadband blackbody light has a very short coherence time. Slow detectors and wide timing bins average over many independent temporal modes, pushing the measured $g^{(2)}(0)$ toward one. Narrow spectral filtering lengthens the coherence time but discards flux.

Spatial collection likewise matters. A large detector aperture or extended source may combine many coherence areas. Observing near-ideal bunching requires resolving a small enough set of spatiotemporal and polarization modes.

Physical Sources

Equilibrium cavities and blackbodies

A cavity weakly coupled to walls at temperature $T$ approaches a product of thermal states in its independent normal modes under the ideal equilibrium model. Radiation escaping a small aperture samples that field.

Real surfaces have frequency- and angle-dependent emissivity. For a passive opaque graybody channel of emissivity $\epsilon$, the outgoing occupation includes both emission and reflected incident noise: $\bar n_{\rm out}=\epsilon\bar n(T)+(1-\epsilon)\bar n_{\rm env}$ in a simple uncoupled-channel model. A vacuum environment gives the attenuated thermal result; an environment at the same temperature restores the equilibrium outgoing occupation. Selective emissivity alone does not specify the entire outgoing field.

Independent spontaneous emitters

Light from many independent atoms, molecules, or microscopic current sources can approach a zero-mean Gaussian field through addition of many random complex amplitudes. Gas-discharge lamps and fluorescence can exhibit chaotic statistics after suitable mode selection.

The field need not be in thermodynamic equilibrium with a single temperature. Gaussian chaotic statistics concern amplitude correlations; thermal equilibrium is the stronger thermodynamic statement.

Amplified spontaneous emission

Amplified spontaneous emission can be bright and bunched. Gain, filtering, saturation, and propagation set its spectrum and mode number. It may approximate thermal or Gaussian light in selected modes while remaining a nonequilibrium source.

Pseudo-thermal light

A coherent laser scattered from a rotating diffuser can produce a fluctuating speckle pattern. At a fixed point, the sum of many random phasors may be approximately circular Gaussian, giving

$$g^{(2)}(0)\approx2$$

and a tunable coherence time set by diffuser motion.

Such light is useful because it can provide much higher brightness and longer correlation times than a blackbody at optical frequencies. It is called pseudo-thermal because its spectrum, stationarity, higher correlations, or global mode structure need not equal an equilibrium Gibbs field.

Blackbody Radiation develops the historical spectrum. Thermal and Vacuum Noise connects thermal field correlations to damping and detailed balance.

Exercises

1. Condition on a thermal photon subtraction

An ideal weak-tap, single-photon detection implements the conditional map $\rho\mapsto a\rho a^\dagger/\bar n$ on one thermal mode. Derive the remaining-mode number distribution and its mean. Why can the mean increase after subtracting a photon?

Solution

With $q=\bar n/(1+\bar n)$, the output probability is

$$P_-(m)=\frac{(m+1)(1-q)q^{m+1}}{\bar n} =(m+1)(1-q)^2q^m.$$

Its generating function is $(1-q)^2/(1-qz)^2$, so its mean is $2q/(1-q)=2\bar n$. The detection selects brighter members of the initial ensemble strongly enough to outweigh the removed photon. This is conditioning, not amplification of the unconditioned source; the success probability must also be retained when calculating average energy.

2. Distinguish coherent background from independent counts

A thermal field has mean occupation $S$ and is displaced by a coherent amplitude $\beta$ with $B=|\beta|^2$. Expand $\langle(a^\dagger+\beta^*)^2(a+\beta)^2\rangle$ and compare its $g^{(2)}$ with the sum of independent thermal and Poisson counts.

Solution

Thermal odd and anomalous moments vanish, leaving $2S^2+4SB+B^2$. The mean occupation is $S+B$, hence

$$g^{(2)}_{\rm displaced}-1=\frac{S^2+2SB}{(S+B)^2}.$$

For independent summed counts the factorial moment is $2S^2+2SB+B^2$, so the excess is $S^2/(S+B)^2$. At $S=B>0$, the two values are $1.75$ and $1.25$, respectively. Interference with the thermal amplitude produces the additional beat-noise term in the same-mode displacement experiment.

3. Thermal quadratures and purity

Show that

$$(\Delta X)^2 = (\Delta P)^2 = \bar n+\frac12$$

and verify

$$\operatorname{Tr}(\rho_{\mathrm{th}}^2) = \frac1{2\bar n+1}.$$

Solution

Because the state is number diagonal,

$$\langle a\rangle = \langle a^2\rangle = 0.$$

Also,

$$\langle a^\dagger a\rangle=\bar n, \qquad \langle aa^\dagger\rangle=\bar n+1.$$

Thus

$$\begin{aligned} \langle X^2\rangle &= \frac12 \left\langle a^2+aa^\dagger+a^\dagger a+(a^\dagger)^2 \right\rangle \\ &= \bar n+\frac12. \end{aligned}$$

Since $\langle X\rangle=0$, this is the variance. The same calculation gives the $P$ variance.

For purity,

$$\begin{aligned} \operatorname{Tr}(\rho^2) &= (1-q)^2 \sum_{n=0}^{\infty}q^{2n} \\ &= \frac{(1-q)^2}{1-q^2} \\ &= \frac{1-q}{1+q}. \end{aligned}$$

Using $q=\bar n/(1+\bar n)$ gives

$$\operatorname{Tr}(\rho^2) = \frac1{2\bar n+1}.$$

4. Thermal light through loss

Use Bernoulli thinning to show that a thermal input with mean $\bar n$ remains thermal after pure loss of transmissivity $\eta>0$ and nonzero input occupation. Find the output mean and $g^{(2)}(0)$.

Solution

For an input generating function $G_{\mathrm{in}}(z)$, independent survival with probability $\eta$ replaces

$$z \longmapsto 1-\eta+\eta z.$$

The thermal generating function is

$$G_{\mathrm{in}}(z) = \frac1{ 1+\bar n(1-z) }.$$

Hence

$$\begin{aligned} G_{\mathrm{out}}(z) &= \frac1{ 1+\bar n \left[ 1-(1-\eta+\eta z) \right] } \\ &= \frac1{ 1+\eta\bar n(1-z) }. \end{aligned}$$

This is thermal with

$$\bar n_{\mathrm{out}} = \eta\bar n.$$

Its factorial second moment is

$$\langle N_{\mathrm{out}} (N_{\mathrm{out}}-1)\rangle = 2(\eta\bar n)^2.$$

Therefore

$$g_{\mathrm{out}}^{(2)}(0)=2.$$

Uniform loss changes brightness but not the normalized ideal single-mode bunching.

5. Sum two modes or select one superposition

Two independent thermal modes each have mean $\bar n>0$. Compare $g^{(2)}$ for the total number $N_a+N_b$ with that of the selected output $c=(a+b)/\sqrt2$ of a balanced beam splitter.

Solution

For the total, the mean is $2\bar n$ and the variance is $2\bar n(1+\bar n)$, giving $g^{(2)}_{\rm total}=3/2$. The selected mode has zero mean amplitude and variance $\langle c^\dagger c\rangle=\bar n$. A linear combination of independent circular Gaussian amplitudes remains circular Gaussian, so its reduced state is thermal and $g_c^{(2)}=2$. The other output contains the rest of the intensity. Selecting one mode and summing both output counts are different measurements.

6. Unequal effective mode number

Three independent thermal modes have detected means $4$, $2$, and $1$. Compute $M_{\mathrm{eff}}$ and $g^{(2)}(0)$.

Solution

The total mean is

$$\sum_j\bar n_j = 4+2+1 = 7.$$

The squared-weight sum is

$$\sum_j\bar n_j^2 = 16+4+1 = 21.$$

Therefore

$$M_{\mathrm{eff}} = \frac{7^2}{21} = \frac73.$$

The normalized correlation is

$$\begin{aligned} g^{(2)}(0) &= 1+\frac1{M_{\mathrm{eff}}} \\ &= 1+\frac37 \\ &= \frac{10}{7}. \end{aligned}$$

The effective mode number is not an integer because the three modes are unequally weighted.

7. Integrate a finite coherence time

A stationary Gaussian field has $g^{(2)}(\tau)=1+e^{-2\gamma|\tau|}$. An ideal detector counts all arrivals in a gate of duration $T$ at constant mean flux $\Phi$. Find $\langle K(K-1)\rangle/\langle K\rangle^2$.

Solution

The normally ordered count covariance integrates over both detection times. With $\langle K\rangle=\Phi T$, the normalized excess is

$$g_K^{(2)}-1 =\frac{2}{T^2}\int_0^T(T-\tau)e^{-2\gamma\tau}d\tau =\frac1{\gamma T}-\frac{1-e^{-2\gamma T}}{2(\gamma T)^2}.$$

It tends to one for $\gamma T\ll1$, giving $g_K^{(2)}\to2$. For long gates it falls as $1/(\gamma T)$, even though the underlying pointwise value remains $g^{(2)}(0)=2$. At $\gamma T=1$, the integrated value is about $1.568$. The integration kernel is triangular; averaging a single flat delay bin would be a different measurement.

8. Random phase is not thermal

At mean photon number $\bar n=1$, compare one thermal mode with a phase-randomized coherent state. Compute $P(0)$, $P(1)$, and $g^{(2)}(0)$ for each.

Solution

For the thermal state,

$$P_{\mathrm{th}}(n) = \frac1{1+\bar n} \left( \frac{\bar n}{1+\bar n} \right)^n.$$

At $\bar n=1$,

$$P_{\mathrm{th}}(0)=\frac12, \qquad P_{\mathrm{th}}(1)=\frac14,$$

and

$$g_{\mathrm{th}}^{(2)}(0)=2.$$

The phase-randomized coherent state has the Poisson law

$$P_{\mathrm{PRC}}(n) = e^{-1}\frac1{n!}.$$

Thus

$$P_{\mathrm{PRC}}(0)=e^{-1}, \qquad P_{\mathrm{PRC}}(1)=e^{-1},$$

and

$$g_{\mathrm{PRC}}^{(2)}(0)=1.$$

Both states have zero mean amplitude after phase averaging, but their amplitude-radius and number fluctuations are different.

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