In a bulk metal, hot nonequilibrium electrons lose their energy mostly during nonradiative collisions with phonons or impurity atoms. Primary photons are emitted as a result of the corresponding Bremsstrahlung processes (at temperatures of our interest, one can omit photons emitted in bound-bound transitions in lattice atoms). Establishment of thermal equilibrium of photons is the consequence of complicated kinetics of free-free electron transitions consisting in emission and absorption Bremsstrahlung processes as well as the Compton effect [53], [54]. In a simplified diffusion approximation, photon emission can be treated through the radiation transfer equation

$$\frac{d{I}_{\omega}}{ds}=-{\alpha}_{\omega}{I}_{\omega}+{\alpha}_{\omega}{B}_{\omega}\mathrm{(}{T}_{\text{e}}\mathrm{)},$$(6)

where *I*_{ω} is the radiation intensity spectrum, *α*_{ω} is the absorption coefficient at the given frequency, and *B*_{ω}(*T*_{e}) is the equilibrium radiation intensity given by Planck’s law. In a bulk metal, when the optical skin depth ${\alpha}_{\omega}^{-1}$ is much smaller than the characteristic dimension, Eq. (6) results in Kirchhoff’s law, and the emissivity is given by *j*_{ω}=*α*_{ω}B_{ω}(*T*_{e}).

In the case of a small interconnection region near the constriction, the region of elevated electron temperature is given by *r*_{N}≈*Nλ*_{F}/4 and is much smaller than the optical skin depth ${\alpha}_{\omega}^{-1}.$ As a result, an equilibrium photon distribution cannot be established within the interconnection region and Kirchhoff’s law is no longer valid. Consequently, the emission of photons is primarily guided by a thermal Bremsstrahlung process. Hot electrons in the interconnection region are quasiballistic: the electron-phonon mean-free path is measured as l_{e−ph}=*ν*_{F}*τ*_{e−ph}~60 nm for Au [55]. Digressing from the presence of impurities and from the two-photon Compton emission in electron-electron collisions, we conclude that the optical activity detected in our experiment is mainly due to a Bremsstrahlung process resulting from hot electrons colliding with the surface potential. We note that within the detected spectral range, a spontaneous emission of an elevated electron temperature discussed in Refs. [36], [45] cannot be distinguished from thermal Bremsstrahlung, as both processes share the same wavelength dependence [56].

To find the photon yield in the electron-wall Bremsstrahlung radiation process, we utilize the conventional quantum mechanical calculation technique, which is analogous to that used in the theory of size-dependent conductivity of thin metal films by Trivedi and Ashcroft [57] as well as in the theory of intersubband transitions in semiconductor quantum wells [58], [59]. We consider a metal slab of the thickness *L*, which is considered to be sufficiently large to provide limiting transition to the continuous spectrum of electron momentum. Let the coordinate axis *z* be transverse to the slab boundary and $\overrightarrow{\rho}$ be the coordinate in the boundary plane. The wall of the slab, at *z*=0, *L*, is modeled by an infinite stepwise potential. Within the jellium model of a noninteracting electron system, the wavefunction of an electron inside the slab is

$$\Psi \mathrm{(}z\mathrm{,}\text{\hspace{0.17em}}\overrightarrow{\rho}\mathrm{,}\text{\hspace{0.17em}}t\mathrm{)}=\sqrt{\frac{2}{{V}_{\text{e}}}}\mathrm{sin}\mathrm{[}{k}_{z}z\mathrm{]}\mathrm{exp}\mathrm{(}i{\overrightarrow{k}}_{\perp}\overrightarrow{\rho}\mathrm{)}\mathrm{exp}\mathrm{(}-i\frac{\epsilon}{\hslash}t\mathrm{)},$$(7)

which satisfies the boundary conditions Ψ(*z*=0)=Ψ(*z*=*L*)=0, *k*_{z}=(*π*/*L*)*j*, *j*=1, 2, 3…, is the longitudinal wavenumber, $\hslash {\overrightarrow{k}}_{\perp}$ is the transverse momentum, and *V*_{e} is the quantization volume for the electron. The energy of electrons $\epsilon =\mathrm{(}{\hslash}^{2}/2m\mathrm{)}\mathrm{(}{k}_{z}^{2}+\mathrm{|}{k}_{\perp}{\mathrm{|}}^{2}\mathrm{)}$ is the eigen value of the unperturbed Hamiltonian, the perturbation Hamiltonian ${H}_{\text{int}}=-\mathrm{(}e\mathrm{/}mc\mathrm{)}\widehat{\overrightarrow{A}}\widehat{\overrightarrow{p}},$ $\widehat{\overrightarrow{p}}=-i\hslash \nabla ,$ describes the spontaneous photon emission in a given mode of the wavevector $\overrightarrow{k},$ the polarization *σ*, and the frequency $\omega =\text{\hspace{0.17em}}\mathrm{|}\overrightarrow{k}\mathrm{|}c,$ with the following vector potential:

$$\widehat{\overrightarrow{A}}={\left(\frac{2\pi \hslash c}{{V}_{\text{ph}}\omega}\right)}^{1/2}{\widehat{a}}_{\overrightarrow{k}\mathrm{,}\sigma}^{+}{\overrightarrow{e}}_{\overrightarrow{k}\mathrm{,}\sigma}^{\mathrm{*}}\mathrm{exp}\mathrm{(}-i\overrightarrow{k}\overrightarrow{r}\mathrm{)}{\text{e}}^{-i\omega t}.$$(8)

Here, ${\widehat{a}}_{\overrightarrow{k}\mathrm{,}\sigma}^{+}$ is the corresponding photon creation operator, ${\overrightarrow{e}}_{\overrightarrow{k}\mathrm{,}\sigma}^{\mathrm{*}}$ is the unit polarization vector, and *V*_{ph} is the quantization volume for photons. Transitions are induced between the initial state |*i*⟩ of an elecron with the energy *ε*_{i} and empty photon state and the final state |*f*⟩ of electron with energy *ε*_{f} and one photon of the above mode. The rate of transition is given by first-order perturbation theory

$${W}_{i\to f}=\frac{2\pi}{\hslash}|\u3008f|{H}_{\text{int}}|i\u3009{\mathrm{|}}^{2}\delta \mathrm{(}{\epsilon}_{f}-{\epsilon}_{i}+\hslash \omega \mathrm{)}d{\rho}_{f}.$$(9)

Here, *dρ*_{f} is the number density of the final states, $d{\rho}_{f}=\mathrm{(}{V}_{\text{e}}/\mathrm{(}2\pi {\mathrm{)}}^{3}\mathrm{)}{d}^{3}{\overrightarrow{k}}_{f}\times \mathrm{(}2{V}_{\text{ph}}/\mathrm{(}2\pi {\mathrm{)}}^{3}\mathrm{)}{d}^{3}\overrightarrow{k},$ in the limit of continuous states. The matrix element can be easily calculated as follows:

$$\u3008f\left|{H}_{\text{int}}\right|i\u3009=i\frac{4e\hslash}{m{V}_{\text{e}}}{\mathrm{(}\frac{2\pi \hslash c}{{V}_{\text{ph}}\omega}\mathrm{)}}^{1/2}\times \frac{{k}_{i\mathrm{,}z}{k}_{f\mathrm{,}z}}{{k}_{i\mathrm{,}z}^{2}-{k}_{f\mathrm{,}z}^{2}}\mathrm{(}{\overrightarrow{e}}_{\overrightarrow{k}\mathrm{,}\sigma}^{\mathrm{*}}\mathrm{,}\text{\hspace{0.17em}}{\overrightarrow{e}}_{z}\mathrm{)}\delta \mathrm{(}{\overrightarrow{k}}_{i\mathrm{,}\perp}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overrightarrow{k}}_{f\mathrm{,}\perp}\mathrm{}\mathrm{)}\mathrm{.}$$(10)

Here, ${\overrightarrow{e}}_{z}$ is the unit vector along the *z*-axis. The delta function in the matrix element [Eq. (10)] demonstrates the conservation of the transverse (parallel to the wall) component of the electron momentum, while its *z*-component changes according to the energy conservation law corresponding to the delta function in the right-hand side of the golden rule [Eq. (9)]. To find the spectrum rate of photon emission by a single electron, we have to sum over all the final electron states as well as over the polarization and solid angles of photon emission. The sum over the electron final states in the continuous limit *L*→∞ is provided through the following relation:

$$\begin{array}{l}{\displaystyle \int}\frac{{k}_{i\mathrm{,}z}^{2}{k}_{f\mathrm{,}z}^{2}}{{\mathrm{(}{k}_{i\mathrm{,}z}^{2}-{k}_{f\mathrm{,}z}^{2}\mathrm{)}}^{2}}{\delta}^{2}\mathrm{(}{\overrightarrow{k}}_{i\mathrm{,}\perp}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\overrightarrow{k}}_{f\mathrm{,}\perp}\mathrm{)}\times \delta \mathrm{(}{\epsilon}_{f}-{\epsilon}_{i}+\hslash \omega \mathrm{)}{d}^{3}{\overrightarrow{k}}_{f}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\mathrm{(}2\pi {\mathrm{)}}^{2}{S}_{\perp}\frac{{m}^{2}{v}_{i\mathrm{,}z}^{2}{v}_{f\mathrm{,}z}}{4{\hslash}^{3}{\omega}^{2}}\mathrm{,}\end{array}$$(11)

where *S*_{⊥} is the square of the wall boundary, and the velocities *ν*_{i,z}=*ħk*_{i,z}/m and ${v}_{f\mathrm{,}z}={\mathrm{(}{v}_{i\mathrm{,}z}^{2}-2\hslash \omega \mathrm{/}m\mathrm{)}}^{1/2}$ are introduced. The sum over the polarization of Bremsstrahlung photons can be accounted through the substitution $2|\mathrm{(}{\overrightarrow{e}}_{\overrightarrow{k}\mathrm{,}\sigma}^{\mathrm{*}}\mathrm{,}\text{\hspace{0.17em}}{\overrightarrow{e}}_{z}{\mathrm{)}|}^{2}\to \text{\hspace{0.17em}}\text{\hspace{0.17em}}{{\displaystyle \mathrm{sin}}}^{2}\theta ,$ and after summation over the photon solid angle *d*Ω=sin*θ**d**θ**d**ϕ*, we arrive at the following relation for the Bremsstrahlung emission rate per unit frequency range:

$$\frac{d{N}_{i}\mathrm{(}\omega \mathrm{)}}{d\omega \text{\hspace{0.17em}}dt}=\frac{8{e}^{2}{v}_{i\mathrm{,}z}^{2}{v}_{f\mathrm{,}z}}{3\pi {V}_{\text{e}}\hslash \omega {c}^{3}}{S}_{\perp}.$$(12)

Let *n*_{i} be the number density of electrons with the longitudinal component of velocity *ν*_{i,z}; the rate of collisions with the left wall of the metal slab is $\frac{1}{2}{n}_{i}{S}_{\perp}{v}_{i\mathrm{,}z}$ and the same value holds for the collision rate with the right wall of the slab. Consequently, the total emitted power spectrum per electron (*n*_{i}=1/*V*_{e}) is

$$\frac{d{P}_{\omega}}{d\omega}=\frac{d{N}_{i}\mathrm{(}\omega \mathrm{)}}{d\omega \text{\hspace{0.17em}}dt}\frac{\hslash \omega}{{v}_{i\mathrm{,}z}{S}_{\perp}\mathrm{/}{V}_{\text{e}}}=\frac{8{e}^{2}}{3\pi {c}^{3}}{v}_{i\mathrm{,}z}{v}_{f\mathrm{,}z}.$$(13)

In the classical limit *ħω*→0, i.e. when the energy of scattered electron does not change significantly, one can replace $4{v}_{i\mathrm{,}z}{v}_{f\mathrm{,}z}\approx {\mathrm{(}{v}_{i\mathrm{,}z}+{v}_{f\mathrm{,}z}\mathrm{)}}^{2}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{|}{\overrightarrow{v}}_{i}-{\overrightarrow{v}}_{f}{\mathrm{|}}^{2}$ and arrive at the well-known classical relation *d**P*_{ω}/*d**ω*=(2*e*^{2}/3*π*c^{3})|Δ*ν*|^{2} for the power spectrum emitted by scattered electrons [56], [60].

To find the total photon emission rate per unit surface square, we sum Eq. (12) over all the electron states in the slab, assuming the Fermi distribution *f*_{F}(*ε*)=1/(1+exp[(*ε*–*ε*_{F})/*k*_{B}*T*_{e}]).

$$\begin{array}{c}\frac{d{N}_{\text{ph}}\mathrm{(}\omega \mathrm{)}}{dS\text{\hspace{0.17em}}d\omega \text{\hspace{0.17em}}dt}=\frac{1}{2{S}_{\perp}}{\displaystyle \int}\frac{2{d}^{2}{\overrightarrow{k}}_{i\mathrm{,}\perp}d{k}_{i\mathrm{,}z}}{{\mathrm{(}2\pi \mathrm{)}}^{3}}{V}_{\text{e}}\frac{d{N}_{i}\mathrm{(}\omega \mathrm{)}}{d\omega}{f}_{\text{F}}\mathrm{(}{\epsilon}_{i}\mathrm{)}\mathrm{(}1-{f}_{\text{F}}\mathrm{(}{\epsilon}_{f}\mathrm{)}\mathrm{)}\\ =\frac{8{e}^{2}}{3{\pi}^{3}{\hslash}^{3}{c}^{3}}\frac{{k}_{\text{B}}{T}_{\text{e}}\mathrm{/}\hslash \omega}{\mathrm{exp}\mathrm{(}\hslash \omega \mathrm{/}{k}_{\text{B}}{T}_{\text{e}}\mathrm{)}-1}F\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}{T}_{\text{e}}\mathrm{,}\text{\hspace{0.17em}}{\epsilon}_{\text{F}}\mathrm{)},\end{array}$$(14)

where ${\epsilon}_{i}=\mathrm{(}{\hslash}^{2}/2m\mathrm{)}\mathrm{(}{k}_{i\mathrm{,}z}^{2}+\mathrm{|}{\overrightarrow{k}}_{i\mathrm{,}\perp}{\mathrm{|}}^{2}\mathrm{)},$ *ε*_{f}=*ε*_{i}–ħω, and the factor 1/2*S*_{⊥} before the integral takes into account the doubled scattering surface in the slab. The function *F*(*ω*, *T*_{e}, *ε*_{F}) is given by the following integral:

$$\begin{array}{l}F\mathrm{(}\omega \mathrm{,}\text{\hspace{0.17em}}{T}_{\text{e}}\mathrm{,}\text{\hspace{0.17em}}{\epsilon}_{\text{F}}\mathrm{)}={\displaystyle {\int}_{\hslash \omega}^{\infty}}\text{d}u\sqrt{u\mathrm{(}u-\hslash \omega \mathrm{)}}\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{1em}}\times \left\{\mathrm{ln}\left[1+\mathrm{exp}\mathrm{(}\frac{{\epsilon}_{\text{F}}-u+\hslash \omega}{{k}_{\text{B}}{T}_{\text{e}}}\mathrm{)}\right]-\mathrm{ln}\left[1+\mathrm{exp}\mathrm{(}\frac{{\epsilon}_{\text{F}}-u}{{k}_{\text{B}}{T}_{\text{e}}}\mathrm{)}\right]\right\}.\end{array}$$(15)

The total Bremsstrahlung photon number spectrum emission rate given by Eqs. (14) and (15) is a complicated function, which we will analyze in detail elsewhere. To our particular purpose here, we will restrict ourselves by the conditions of our experiment, where the maximum attainable temperature is well below the energy of the collected photons, and we have the following relation between the parameters:

$${k}_{\text{B}}{T}_{\text{e}}\ll \hslash \omega \ll {\epsilon}_{\text{F}}\mathrm{.}$$(16)

One can easily see that under these conditions, the logarithms in braces in Eq. (15) vanish when the argument exceeds *u*>*ε*_{F}+*ħω* and *u*>*ε*_{F}, respectively. At *ħω*<*u*<*ε*_{F}, the term in braces is approximately constant and equals *ħω*/*k*_{B}*T*_{e}≫1. At *ε*_{F}<*u*<*ε*_{F}+*ħω*, it almost linearly decreases to zero. As a result, we arrive at the following approximation: $F\approx \mathrm{(}{\epsilon}_{\text{F}}^{2}\hslash \omega /2{k}_{\text{B}}{T}_{\text{e}}\mathrm{)}\mathrm{(}1+O\mathrm{(}\hslash \omega \mathrm{/}{\epsilon}_{\text{F}}\mathrm{)}\mathrm{)},$ and finally find

$$\frac{d{N}_{\text{ph}}\mathrm{(}\omega \mathrm{)}}{dS\text{\hspace{0.17em}}d\omega \text{\hspace{0.17em}}dt}\approx \frac{4{e}^{2}{\epsilon}_{\text{F}}^{2}}{3{\pi}^{3}{\hslash}^{3}{c}^{3}}\frac{1}{\mathrm{exp}\mathrm{(}\hslash \omega \mathrm{/}{k}_{\text{B}}{T}_{\text{e}}\mathrm{)}-1}.$$(17)

Compared to the Planck formula for blackbody radiation

$${B}_{\omega}\mathrm{(}{T}_{\text{e}}\mathrm{)}=\frac{{\omega}^{2}}{2\pi {c}^{2}}\frac{1}{\mathrm{exp}\mathrm{(}h\omega \mathrm{/}{k}_{\text{B}}T\mathrm{)}-1},$$(18)

the rate of Bremsstrahlung emission is less by the factor

$$\beta =\frac{8}{3{\pi}^{2}}\frac{{e}^{2}}{\hslash c}{\left(\frac{{\epsilon}_{F}}{\hslash \omega}\right)}^{2}\mathrm{,}$$(19)

which under the experimental condition can be estimated to be β~0.1.

To model the total yield of Bremsstrahlung photons detected in our experiment, we integrate the spectrum rate [Eq. (17)] with the transmission function *Q*(*ω*) of the detection path, which includes the spectral sensitivity of the detector. The APD response restricts the detection efficiency to overbias photon energy tailing in the visible part of the spectrum. We model the spectral response of the APDs by the following function:

$$Q\mathrm{(}\omega \mathrm{)}\approx \left\{\begin{array}{rr}\hfill 0;& \hfill \omega \mathrm{<}{\omega}_{1}\\ \hfill 0.65\frac{\omega -{\omega}_{1}}{{\omega}_{2}-{\omega}_{1}}\mathrm{;}& \hfill {\omega}_{2}\mathrm{>}\omega \mathrm{>}{\omega}_{1}\\ \hfill 0.65;& \hfill {\omega}_{3}\mathrm{>}\omega \mathrm{>}{\omega}_{2}\end{array}\right\}.$$(20)

Here, *ω*_{1} corresponds to the detection threshold of the detector at a wavelength *λ*_{1}=1070 nm, *ω*_{2} corresponds to the peak of detection efficiency at *λ*_{2}=740 nm, and *ω*_{3} is taken at *λ*_{3}=600 nm.

One can easily check that for the domain of interest, i.e. for peak temperatures below 3.5×10^{3} K (see Figure 5), the integrated Bremsstrahlung photon yield is well approximated by the relation *d**N*_{ph}/*d**S**d**t*≈6.13×10^{22}×* ζ*^{2}exp(–1/*ζ*)cm^{−2}s^{−1}, where the normalized temperature is *ζ*=*k*_{B}*T*_{e}/*ħω*_{1}. The results of our calculation of the Bremsstrahlung photon yield rate dependence on the quantum channel number are shown in Figure 6 both for limiting values of the parameter *h* governing the heat transfer at the side wall of the system and for varying lengths *L*_{c} and radii *R*_{0} of the interconnection region. The data correspond to the calculated peak temperatures shown in Figure 5. One can find that these dependences at sufficiently small values of length *L*_{c} and radius *R*_{0} qualitatively recover the experimental data shown in Figure 3A, notably the exponential decay of the photon counts with the number of channels opened. We use the model described above to match the experimental dependence of the photon counts versus electrical power delivered in the contact displayed in Figure 3B, again considering the two extreme heat exchange scenarios at the side wall. The open blue and magenta triangles in Figure 3B are the results of the models considering a short cylinder of length *L*_{c}=*λ*_{F}/4 and *R*_{0}=2*λ*_{F}. We estimate the total radiation area as $S\approx \pi {R}_{0}^{2}+2\pi {R}_{0}{L}_{\text{c}}=4.84\times {10}^{-14}{\text{\hspace{0.17em}cm}}^{2}.$ The overall detection efficiency is experimentally unknown, and we leave this as a free parameter *η*. To fit the maximum calculated yield with the experimental value for the fourth quantum channel, one should put *η*≃0.43, which means that the collection efficiency of the microscope is about 43%. Considering the detection NA and the efficiency of the APDs in the spectral window, this value of *η* looks reasonable. The experimental photon counts are bounded by the two limiting cases of the model, indicating the qualitative agreement with the model used and that electron thermalization at the side wall is an important process to consider.

Figure 6: Thermal Bremsstrahlung radiation photon rates versus the number of open quantum channels in the contact.

(A, B) Semilogarithmic plots at a fixed radius *R*_{0}=2.5*λ*_{F}. Squares represent data at large parameter *h*≫1 (A), while circles correspond to *h*≪1 (B). Data are plotted at different lengths of the interconnection region: *L*_{c}=*R*_{0}/6 (red), *L*_{c}=*R*_{0}/4 (green), *L*_{c}=*R*_{0}/3 (blue), *L*_{c}=*R*_{0}/2 (cyan), and *L*_{c}=3*R*_{0}/4 (violet). (C, D) Semilogarithmic plots at fixed cylinder length *L*_{c}=*λ*_{F}/2. Data are plotted at different radii of the cylinder interconnection region: *R*_{0}=2*λ*_{F} (red), *R*_{0}=2.5*λ*_{F} (green), *R*_{0}=3*λ*_{F} (blue), and *R*_{0}=4*λ*_{F} (black).

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