## Introduction

Almost a century ago, Johnson and Nyquist1,2 presented evidence of fluctuating electrical current and the governing fluctuation dissipation theorem (FDT). Whether, likewise, temperature $$T$$ can fluctuate is a controversial topic and has led to scientific debates for several decades3,4,5,6,7. Consider a system with coupling to a heat bath at temperature $$T$$ for which the classical FDT of fluctuations $${S}_{\dot{Q}}^{{\rm{eq}}}$$ of heat current $$\dot{Q}$$ holds in form $${S}_{\dot{Q}}^{{\rm{eq}}}=2{k}_{B}{T}^{2}{G}_{{\rm{th}}}$$ in equilibrium. Here, $${G}_{{\rm{th}}}$$ is the heat conductance to the bath. We can write the energy balance equation $$\dot{Q}={\mathcal{C}}{\mathrm{d}}\widetilde{T}/{\mathrm{d}}t$$ for the temperature of the system $$\widetilde{T}(t)=T+\delta T(t)$$ at time $$t$$, where $${\mathcal{C}}$$ denotes the heat capacity. The heat current is composed of its expectation value $$-{G}_{{\rm{th}}}\delta T$$ and fluctuations $$\delta \dot{Q}$$ around it. There are two origins of noise in this heat current: (1) the standard randomness of transport known for particle current noise (time randomness), and (2) random energies exchanged, leading to enhancement of fluctuations on top of those known for particle current only. We obtain the noise spectrum of temperature of the system by Fourier transformation as $${S}_{T}(\omega )=\int dt{e}^{i\omega t}\langle \delta T(t)\delta T(0)\rangle$$. This yields under steady state conditions

$${S}_{T}(\omega )=\frac{2{k}_{B}{T}^{2}}{{G}_{{\rm{th}}}}\frac{1}{1+{\omega }^{2}{{\mathcal{C}}}^{2}/{G}_{{\rm{th}}}^{2}}.$$
(1)

At low frequencies we have

$${S}_{T}(0)=2{k}_{B}{T}^{2}/{G}_{{\rm{th}}},$$
(2)

and the spectrum has Lorentzian cutoff at $${\omega }_{c}={G}_{{\rm{th}}}/{\mathcal{C}}$$. These results hold also for a system coupled to several equilibrium baths, if one takes $${G}_{{\rm{th}}}$$ to represent the sum of all the individual thermal conductances to these baths. For the root-mean-square (rms) fluctuations we obtain the well-known result3 $$\langle \delta {T}^{2}\rangle =\int_{-\infty }^{\infty}\frac{d\omega }{2\pi }{S}_{T}(\omega )={k}_{B}{T}^{2}/{\mathcal{C}}$$.

Here, we measure the time-dependent temperature of the absorber of a nano-calorimeter at low mK temperatures both under equilibrium and nonequilibrium conditions. We observe that the equilibrium fluctuations follow the fluctuation dissipation theorem (FDT) for temperature. Ideally, the noise of this calorimeter permits measurements of microwave photons in GHz regime at the lowest temperatures that we achieve. This method is then a way to observe calorimetrically, e.g., the quantum trajectories with superconducting circuits8,9,10.

## Results

### The calorimeter

In a fermionic system, like the electrons (about $$1{0}^{8}$$ of them) in the nano-calorimeter in the present experiment, temperature is coded in the Fermi distribution $$f(\epsilon )={[{e}^{(\epsilon -\mu )/{k}_{{\rm{B}}}T}+1]}^{-1}$$, which directly determines the readout signal of our thermometer. Here, $$\epsilon$$ and $$\mu$$ denote the single particle energy and chemical potential of the system, respectively. We illustrate the calorimeter11,12,13,14,15 principle of our experiment and set-up in Fig. 116. The electron system (absorber), is coupled to the phonon heat bath at constant temperature $$T$$ via electron–phonon collisions, which lead to stochastic exchange of heat, as indicated by the many vertical arrows between the two in Fig. 1a. This forms the bottleneck of heat transport in a nano-calorimeter, in contrast to macroscopic calorimeters. The red arrows from the left depict the electronic injection of heat under nonequilibrium conditions, fluctuating due to the stochastic nature of tunneling. By attaching a fast thermometer to the absorber, one records its time $$t$$ dependent temperature fluctuations $$\delta T(t)$$ as shown by a measured time trace. The actual sample (scanning electron micrograph in Fig. 1b) is realized as a $$\ell =1\,$$ µm long copper normal-metal absorber (brown) connected to three superconducting leads (blue). The right one is a tunnel contact of the thermometer and the other tunnel junction on the left the hot electron injector. The third one pointing down and $$50$$ nm away from the thermometer, is a direct clean metal-to-metal contact grounded at the sample stage. It provides a fixed chemical potential for the absorber and induces proximity superconductivity to the thermometer facilitating its proper operation. The measuring set-up for the thermometer junction shown on the right side of Fig. 1b consists of a parallel on-chip $$LC$$ resonator, coupled to input $${V}_{1}$$ and output $${V}_{2}$$ RF  (radio frequency)  lines, operating at frequency $${f}_{0}=620$$ MHz, which also admits DC biasing at voltage $${V}_{{\rm{th}}}$$. The measured signal $${S}_{21}$$ obtained from the ratio of $${V}_{2}/{V}_{1}$$ yields the conductance of the thermometer junction. It is measured at a finite sampling rate in order to acquire statistics of temporal temperature of the absorber.

### Principles of the experiment

In order to calibrate the thermometer we measure $${S}_{21}$$ averaged over typically $$1$$ s time interval at different bath temperatures of the cryostat, traceable to primary Coulomb blockade thermometry CBT. An example of dependence of thus obtained averaged $$\langle {S}_{21}\rangle$$ on $${V}_{{\rm{th}}}$$ is shown on a wide bias range in Fig. 2a. The drop of $$\langle {S}_{21}\rangle$$ at about ±$$200\,$$ µV is due to the superconducting gap $$\Delta$$ in aluminum. The main feature, the zero bias anomaly (ZBA) at $${V}_{{\rm{th}}}=0$$, which is indicated by the central red arrow, presents the basis of our thermometer. This dip originates from proximity induced supercurrent due to the presence of clean contact. Now it is placed 50 nm away from the tunnel junction, which is to be contrasted to 500 nm in our earlier work17; this way the sensitivity of the thermometer is enhanced substantially. Quantitatively, the temperature dependence of the average transmission $$\langle {S}_{21}\rangle$$ at this dip is depicted in Fig. 2b. It manifests approximately linear dependence at sub $$200$$ mK temperatures, emphasized by the zoom in the inset of this figure. Owing to the competing quasiparticle tunneling, there is eventually back-bending of the characteristics at temperatures above 300 mK; this leads to loss of sensitivity in the cross-over temperature range. Depending on the range of interest, we employ either linear or nonlinear calibration to convert $$\langle {S}_{21}\rangle$$ to temperature. This calibration needs to be done only once for each cooldown. Fig. 2: The transmission measurement of the RF thermometer at $$-120$$ dBm input power.

### Equilibrium fluctuations

Time domain measurements allow detecting temporal fluctuations of the quantity of interest. In our case we monitor $${S}_{21}(t)$$, yielding the instantaneous temperature of the absorber at $$10$$ kHz sampling rate over a chosen time interval. We collect data under given conditions typically for up to $$1$$ hour. As a result we obtain the total fluctuations (variance) $$\langle \delta {S}_{21,{\rm{tot}}}^{2}\rangle$$ in a bandwidth of $$\Delta f\approx 10$$ kHz. This signal is composed of the amplifier and other instrumental noise $$\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle$$ (“bg” stands for background), in addition to the noise of interest from the actual sample, $$\langle \delta {S}_{21}^{2}\rangle$$ = $$\langle \delta {S}_{21,{\rm{tot}}}^{2}\rangle$$$$\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle$$. Here, we assume uncorrelated noise from the different sources. The way we determine the $$\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle$$ is explained in the Methods section. Our quantitative results depend critically on the precision of determining this background noise. Taking the linear calibration as in the inset of Fig. 2b, with the responsivity $${\mathcal{R}}\equiv | d\langle {S}_{21}\rangle /dT|$$, we have for the temperature noise of the absorber $$\langle \delta {T}^{2}\rangle$$ = $${{\mathcal{R}}}^{-2}\langle \delta {S}_{21}^{2}\rangle$$. We exhibit in Fig. 3 the central quantity in the experiment, low-frequency temperature fluctuations $$\sqrt{{S}_{T}}=\sqrt{\langle \delta {T}^{2}\rangle /2\Delta f}$$ as a function of bath temperature in equilibrium. From now on we denote $${\rm{NET}}\equiv \sqrt{{S}_{T}}$$, which is the noise-equivalent temperature. The data symbols in both panels correspond to the averaged bare noise, where the best guess of the background has been subtracted. The shaded area in Fig. 3a depicts the uncertainty in determining $${\rm{NET}}$$ precisely due to this subtraction. Overall, we observe first increase of $${\rm{NET}}$$ upon lowering $$T$$ and then gradual turn down of it at the lowest temperatures. The dominant contributions to $${G}_{\mathrm{th}}$$ arise from electron–phonon coupling at higher temperatures and radiative heat transfer by thermal photons18 towards low $$T$$ as

$${G}_{\mathrm{th}}=5\Sigma {\mathcal{V}}{T}^{4}+\alpha gT.$$
(3)

Here, $$\Sigma$$, $${\mathcal{V}}$$ are electron–phonon coupling constant19 and volume of the absorber, respectively. For the photonic contribution18, $${G}_{Q}=gT$$ is the quantum of thermal conductance with $$g=\pi {k}_{{\rm{B}}}^{2}/6\hslash$$. We assume the coupling coefficient $$\alpha$$ to have values $$\ll\! 1$$ according to earlier investigations20. Equation (2) predicts then

$${\rm{NET}} =\sqrt{\frac{2{k}_{{\rm{B}}}}{5\Sigma {\mathcal{V}}}}{T}^{-1}\,\,({\rm{high}}\, T)\\ {\rm{NET}} =\sqrt{\frac{2{k}_{{\rm{B}}}}{\alpha g}}{T}^{1/2}\,\,({\rm{low}}\, T),$$
(4)

with cross-over between the two regimes with maximum NET at the temperature $${T}_{{\rm{co}}}={(\frac{\alpha g}{10\Sigma {\mathcal{V}}})}^{1/3}$$. Using the literature value21 $$\Sigma =2\,\times 1{0}^{9}\ {{\rm{WK}}}^{-5}{\rm{m}}^{-3}$$, the measured volume $${\mathcal{V}}=1.0\, \times 1{0}^{-21}\; {{\rm{m}}}^{3}$$ and an educated guess $$\alpha \sim 1{0}^{-4}$$, we obtain a predicted $${\rm{NET}}$$ versus $$T$$. Our simple model above predicts a maximum $${\rm{NET}} \sim\! 60\ \mu {\rm{K}}/\sqrt{{\rm{Hz}}}$$ at $$\sim\! 20$$ mK. This $${\rm{NET}}$$ is within the error bars of the measured signal in Fig. 3a, b at low temperatures. Figure 3b makes a quantitative comparison of the measured sub 50 mK equilibrium noise against the presented model. The solid and dashed red lines indicate $${{\rm{NET}}}_{{\rm{eq}}}=\sqrt{2{k}_{{\rm{B}}}{T}^{2}/{G}_{\mathrm{th}}}$$ with and without the photon contribution using the parameters given above, respectively. The shaded area exhibits the impermissible range due to the fundamental temperature noise in equilibrium. We reach this bound at temperatures well below 30 mK. The rest of the lines in this figure will be discussed later.

The analysis above could be improved, provided the parameters of the system were known precisely. Till now we assumed the absorber to be in the normal state. However, the clean absorber-superconductor contact leads to a proximity induced superconductivity in the absorber. This suppresses the density of states around the Fermi level, on the scale of the Thouless energy $${E}_{{\rm{Th}}}=\hslash D/{\ell}^{2} \sim 10\,$$ µeV, resulting in a decreased electron–phonon coupling. Here, $$D \sim 0.01$$$${{\rm{m}}}^{2}/{\rm{s}}$$ is the diffusion constant of the Cu film. As a consequence, for electron temperatures below $${E}_{{\rm{Th}}}/{k}_{{\rm{B}}} \sim 100$$ mK, the thermal conductance $${G}_{{\rm{th}}}$$ is decreased22 and, hence, the temperature noise $${\rm{NET}}$$ is increased. The experimentally observed $${\rm{NET}} \sim 80\,{\mathrm{\mu}} {\rm{K}}/\sqrt{{\rm{Hz}}}$$ at low $$T$$ can then be obtained using $$D=0.01\,{{\rm{m}}}^{2}/{\rm{s}}$$ and $$\alpha =1{0}^{-3}$$. One should also note that the fluctuations $$\delta T$$ of temperature become non-negligible as compared to $$T$$ based on the estimate $$\delta T/T\simeq \sqrt{{k}_{{\rm{B}}}/\mathcal{C}}\,\gtrsim\, 0.1$$ at $$T=10$$ mK for our absorber.

### Nonequilibrium fluctuations

Let us finally consider the nonequilibrium fluctuations23,24,25,26. In the measurements presented up to now the injector junction with tunnel resistance $${R}_{{\rm{T}}}=20$$ k$$\Omega$$ on the left in Fig. 1b has been unbiased in order to ensure equilibrium. By applying a voltage $$V$$ to it, the system can be driven into nonequilibrium. The well-known influence of such biasing of a superconductor-normal-metal junction is that it serves as a local refrigerator of the normal-metal absorber thanks to the energy gap of the superconductor, i.e., it acts as an evaporative cooler27. This effect is manifested in the bias dependence of the average temperature of the absorber, obtained from the values of $$\langle {S}_{21}\rangle$$ in Fig. 4a.

Injecting electrons does not only change the average temperature of the absorber but, due to the stochastic nature of tunneling, it leads to noise of heat current as well28,29. Quantitatively this noise at low frequencies is given by

$${S}_{\dot{Q}}^{{\rm{in}}}= \;\frac{1}{{e}^{2}{R}_{{\rm{T}}}}\int dE{(E-eV)}^{2}{n}_{{\rm{S}}}(E)\\ \times\{{f}_{{\rm{N}}}(E-eV)[1-{f}_{{\rm{S}}}(E)]+{f}_{{\rm{S}}}(E)[1-{f}_{{\rm{N}}}(E-eV)]\},$$
(5)

where $${f}_{{\rm{N}}},\,{f}_{{\rm{S}}}$$ are the energy distribution functions for the normal-metal and superconductor electrons, respectively, and $${n}_{{\rm{S}}}(E)=| E| /\sqrt{{E}^{2}-{\Delta }^{2}}\theta (| E| -\Delta )$$ denotes the density of states for the superconductor, with $$\theta (x)$$ being the Heaviside step function. For typical voltages and temperatures in the regime well below the superconducting gap, the injection noise $$\sqrt{{S}_{\dot{Q}}^{{\rm{in}}}}$$ is exponentially suppressed16. In contrast, the equilibrium noise due to phonons, $$\sqrt{{S}_{\dot{Q}}^{{\rm{eq}}}}$$, is of a roughly constant magnitude $$\sim\! 1{0}^{-20}\ {\rm{W}}/\sqrt{{\rm{Hz}}}$$. Therefore, it is not surprising that the temperature noise in Fig. 4b does not change much at sub-gap voltages $$V\,<\, 200\,$$ µV, in particular as the temperature of the absorber is not changing dramatically in this bias range. For these uncorrelated sources the temperature noise is predicted to obey $${S}_{T}=({S}_{\dot{Q}}^{{\rm{eq}}}+{S}_{\dot{Q}}^{{\rm{in}}})/{G}_{{\rm{th}}}$$. The sudden decrease of temperature noise NET at $$V\,> \, 200$$ µV is natural since $${G}_{{\rm{th}}}$$ increases rapidly when the absorber heats up in this regime (see Fig. 4a). The sharp peak at the gap (Fig. 4b) is possibly an artifact arising from unavoidable voltage noise of the injector, which directly transforms to temperature noise due to the strong voltage dependence of temperature at this point. Yet we find close resemblance of our measured bias-dependent noise and the theoretical predictions by Laakso et al.26 calculated for a SINIS (superconductor-insulator-normal metal-insulator-superconductor) device.

## Discussion

The temperature that fluctuates is given by the energy distribution of the electrons in the absorber. It qualifies as temperature for the following reasons. (i) Number of particles is large, about $$1{0}^{8}$$. (ii) Owing to fast electron–electron internal relaxation over a time scale of $$\sim\! 1{0}^{-9}$$ s30, the carriers form a local Fermi–Dirac distribution: all other relaxation times, most notably the electron–phonon time ($$\sim\! 1{0}^{-5}$$ s) are much slower31. Furthermore, the temperature of the absorber is spatially uniform, since the heat diffusion time of electrons in the absorber, $${\tau }_{{\rm{diff}}}$$ = $$\gamma \rho {\ell }^{2}/{{\mathcal{L}}}_{0} \sim 1{0}^{-10}$$ s is very short. Here, $$c=\gamma T$$ is the specific heat due to conductance electrons with $$\gamma \sim 1{0}^{2}\ {{\rm{Jm}}}^{-3}\ {{\rm{K}}}^{-2}$$, $$\rho \sim 1{0}^{-8}\ \Omega$$m is the resistivity of the Cu, and $${{\mathcal{L}}}_{0}=2.44\,\times 1{0}^{-8}\ {\rm{W}}\Omega {{\rm{K}}}^{-2}$$ is the Lorenz number.

A central question is the projected energy resolution of the presented calorimeter. The objective is to use it for observing quanta in the microwave regime. Unlike some of the previously published works on THz calorimetry32,33, here we aim into the GHz regime common in circuit QED (quantum electrodynamics) experiments. Here, we demonstrated that its resolution is as good as nature can allow, limited only by thermal fluctuations and illustrated by the red lines in Fig. 3b. Indeed, as we present by the blue lines in the figure, the necessary NET of the detector to observe microwave photons, e.g., those emitted by a standard superconducting qubit with $$0.5-1\,{\rm{K}}\times {k}_{{\rm{B}}}$$ energy is well above the fundamental fluctuations at sub 30 mK temperatures.

## Methods

### Background measurements

We measure the instrumental noise dominated by that of the low-temperature Caltech CITLF2 cryogenic SiGe low-noise amplifier $$\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle$$ by carefully off-tuning the interesting fluctuations from the sample itself. This is achieved by simultaneously (i) biasing the thermometer junction away from the ZBA regime ($${V}_{{\rm{th}}}\simeq 85\,$$ µV), and (ii) measuring at either below or above the resonance at frequency $${f}_{0}$$. An example of the corresponding parametric background noise measurement, in form $$\sqrt{\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle }$$ versus $$\langle {S}_{21}\rangle$$ is presented in Fig. 5. We see a typical increase of noise when the attenuation increases towards left. This dependence can be understood quantitatively by assuming constant voltage noise independent of $$\langle {S}_{21}\rangle$$. The measured transmission can be written as

$${S}_{21}=20\,{\rm{lg}}(v/\widetilde{v}),$$
(6)

where $$v$$ is the output of the last stage amplifier, $$\widetilde{v}=\sqrt{50\,{\rm{\Omega }}\times 1\,{\rm{mW}}}\simeq 224$$ mV. Noise of $$v$$ translates then into variations of $${S}_{21}$$ in linear regime as

$$\delta {S}_{21}=\frac{20}{{\rm{ln}}10}\frac{\delta v}{v},$$
(7)

and can be written with the help of Eq. (6) for the rms values as

$$\sqrt{\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle }=\frac{20}{{\rm{ln}}10}\frac{\sqrt{\langle \delta {v}^{2}\rangle }}{\widetilde{v}}1{0}^{-\langle {S}_{21}\rangle /20}.$$
(8)

Based on the fit parameter $$a$$ in Fig. 5a and the total gain of 60 dB of the amplifier chain, we find the input voltage noise to be $$\sim\! 12$$ nV corresponding to the noise temperature of the amplifier of $${T}_{{\rm{n}}} \sim \,5$$ K, which is in line with its specifications by the manufacturer.

Figure 5b presents background measurements at frequencies both below and above the resonance over a wide range of attenuation $$\langle {S}_{21}\rangle$$. We observe two features that we need to consider when making an accurate evaluation of the $$\langle \delta {S}_{21,{\rm{bg}}}^{2}\rangle$$. First, at large attenuations, due to the fact that the changes are not fully linear in the sense of Eq. (7), the exponential dependence of Eq. (8) is not obeyed strictly. Therefore, we resort to polynomial fits in two regimes, to capture the dependence over the full range. Second, there is a weak dependence of the amplifier noise on frequency; thus the data taken below and above the resonance differ from each other slightly. What we do then, e.g., in Fig. 3, is that we take the mean between the two background measurements as the reference and indicate by the shaded area the uncertainty incurred due to the difference between the two extremes. We thus assume that the frequency dependence of the noise is more or less smooth in the narrow range of $$\sim\! 10$$ MHz around $${f}_{0}$$, and interpolate the data accordingly.
The sample (Fig. 1b) was fabricated on standard oxidized Si substrate using Ge process for achieving robust deposition mask34,35. The electron-beam lithography was used to pattern the structure for three-angle shadow evaporation of metals. First we deposit 20 nm of Al making the leads followed by oxidation in pure $${O}_{2}$$ (1 min at 1 mbar). Next another Al layer of 20 nm thickness again provides the clean superconducting contact at the distance of 50 nm from the thermometer junction, and finally we deposit 35 nm Cu to form the absorber. In the main text we give an estimate of the volume of the absorber based on this thickness; the effective thickness may be somewhat smaller due to the partial oxidation of the film. The resonator is a spiral on a separate chip made of 100 nm thick Al by simple one angle evaporation. The heart of the measuring set-up is shown in Fig. 1b with inductance $$L=100$$ nH, $${C}_{1}=10.3$$ fF and $${C}_{2}=59.3$$ fF as coupling capacitors, and $$C=0.2$$ pF. The rest of the RF circuitry follows closely to what is presented in ref. 31. All measurements were performed in a carefully shielded and filtered set-up described in ref. 36.