Fermi-Dirac Thermal Measurements

• Fermi-Dirac thermal measurements are techniques that infer temperature, chemical potential, and related properties by mapping observables onto fermionic statistical laws.
• They employ time-domain monitoring, quasimomentum-selective spectroscopy, and fluctuation-based methods to extract thermodynamic information from mesoscopic dots, cold-atom lattices, and impurity probes.
• These approaches capture finite-size effects, interaction corrections, and nonequilibrium dynamics, offering precise insights into thermometry in quantum and correlated systems.

Fermi-Dirac thermal measurements are procedures in which temperature, chemical potential, density, or related thermodynamic information is inferred from observables governed by fermionic occupation statistics rather than by a conventional macroscopic thermometer. In the literature, this includes single-electron capture and emission statistics in mesoscopic dots and traps (Prati, 2010), quasimomentum-selective band-excitation spectroscopy in optical lattices (Loida et al., 2015), density- and fluctuation-based thermometry of interacting lattice fermions (Pasqualetti et al., 2023), impurity-dephasing interferometry in degenerate Fermi gases (Mitchison et al., 2020), fluctuation thermometry in finite fermion systems (Zheng et al., 2010), and entropy-based probes of thermal mixing in Fermi-Hubbard states (Pichler et al., 2013). A distinct theoretical extension interprets the spectrum of a binary measurement operator itself as an effective set of fermionic occupations, leading to “Fermi-Dirac thermal measurements” in quantum hypothesis testing and semidefinite optimization (Liu et al., 4 Mar 2026).

1. Statistical basis and scope

The common statistical structure is the grand-canonical description of fermions. For ideal gases, the occupation of a single-particle mode is

$$\langle n_k\rangle=\frac{1}{e^{\beta(\varepsilon_k-\mu)}+1},$$

while the crossover between the degenerate regime $t=T/T_{\rm F}\ll 1$ and the nondegenerate regime $t\gg 1$ controls the temperature dependence of the chemical potential, internal energy, magnetic susceptibility, heat capacity, pressure, compressibility, and thermal expansion in dimension-dependent ways (Johnston, 2020). In quantum-metrological language, the precision of estimating $\mu$ and $T$ in a grand-canonical fermionic gas is governed by fluctuation observables,

$$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$

so particle-number and energy fluctuations are themselves thermometric resources (Marzolino et al., 2013). At the mean-field level, Fermi-Dirac theory serves as the exact noninteracting baseline, while thermal Hartree-Fock replaces fixed orbital energies by self-consistent ones without changing the Fermi-Dirac form of the occupations (Gu et al., 2024).

Different implementations probe different observables, but they share the same statistical logic: a measured signal is mapped to a fermionic occupation law, a thermodynamic equation of state, or a fluctuation relation.

Platform	Measured observable	Inferred quantity
Mesoscopic dot/trap	Random-telegraph dwell times $\tau(1),\tau(2)$	$T$ with finite-$N$ correction (Prati, 2010)
Optical lattice fermions	Excitation rate to higher Bloch bands	$T$ from lowest-band Fermi distribution (Loida et al., 2015)
2D SU($t=T/T_{\rm F}\ll 1$0) Hubbard gas	Density profile and density fluctuations	$t=T/T_{\rm F}\ll 1$1 and $t=T/T_{\rm F}\ll 1$2 (Pasqualetti et al., 2023)
$t=T/T_{\rm F}\ll 1$3 Fermi gas with $t=T/T_{\rm F}\ll 1$4 probe	Boson condensate fraction $t=T/T_{\rm F}\ll 1$5	$t=T/T_{\rm F}\ll 1$6 (Lous et al., 2017)
Impurity qubit in Fermi gas	Ramsey coherence $t=T/T_{\rm F}\ll 1$7	In-situ $t=T/T_{\rm F}\ll 1$8 (Mitchison et al., 2020)
Pump/probe electrons	tr-PES and Raman effective temperatures	Distance from equilibrium (Matveev et al., 2018)

2. Mesoscopic few-electron thermometry

A particularly literal realization of Fermi-Dirac thermal measurement is the finite-$t=T/T_{\rm F}\ll 1$9 quantum grand-canonical description of a quantum dot or localized trap exchanging a single electron with a small two-dimensional electron reservoir (Prati, 2010). The subsystem of interest can contain only $t\gg 1$0 or $t\gg 1$1, so the device forms a $t\gg 1$2 electron system. Because the bath is not macroscopic, terms of order $t\gg 1$3 survive and the usual grand-canonical assumption of a fixed reservoir state fails. The paper therefore replaces spatial ensemble averaging by time averaging, invokes ergodicity so that dwell-time fractions equal occupation probabilities, and retains finite-$t\gg 1$4 corrections through a finite quantum grand partition function.

For a paramagnetic point defect or natural quantum dot, the one- and two-electron energies are

$t\gg 1$5

with $t\gg 1$6 the charging energy and $t\gg 1$7 the lattice-relaxation contribution. The finite-bath occupation ratio is

$t\gg 1$8

and ergodicity identifies

$t\gg 1$9

The resulting thermometry formula is therefore

$\mu$0

In the limit $\mu$1, the correction $\mu$2 tends to unity and the usual grand-canonical Fermi-Dirac expression is recovered. The method is explicitly tied to random-telegraph monitoring of single-electron capture and emission events, so temperature is extracted from time-domain occupation statistics rather than from a macroscopic reservoir formula.

A related mesoscopic perspective appears in the exactly solvable calorimetric model of a resonant level coupled to a finite Fermion reservoir (Donvil et al., 2020). There the reservoir plays a dual role as environment and calorimetric detector, its initial populations are

$\mu$3

and the measured temperature change of the reservoir is connected to the deposited energy through

$\mu$4

The exact solution shows that weak coupling gives nearly textbook jump-like calorimetric behavior, while stronger coupling makes interaction energy and hybridization thermodynamically significant. This suggests that in mesoscopic fermionic devices, a measured reservoir heat signal cannot in general be interpreted independently of coupling regime.

3. Cold-atom and lattice implementations

In ultracold lattices, one route to Fermi-Dirac thermometry is modulation spectroscopy into higher Bloch bands (Loida et al., 2015). A periodic lattice perturbation excites atoms from the lowest band to an excited band with resonance condition

$\mu$5

and because the perturbation is quasimomentum selective, the excitation rate samples the occupation of specific initial states. In one dimension, for homogeneous modulation, the excitation rate is

$\mu$6

The measured transferred population therefore directly encodes the Fermi-Dirac distribution in the lowest band. The paper concludes that thermometry of one- and two-dimensional systems is within reach of current experiments and remains valid down to temperatures of a few percent of the hopping amplitude, comparable to the Néel-temperature scale in interacting systems.

For interacting lattice fermions, direct fitting to a free Fermi-Dirac profile is no longer sufficient. In the 2D SU($\mu$7) Fermi-Hubbard model, thermometry is based on the interacting equation of state $\mu$8 and on a local fluctuation-dissipation relation (Pasqualetti et al., 2023). The model-independent fluctuation thermometer uses

$\mu$9

while the model-based thermometer fits the full density distribution to DQMC or NLCE equation-of-state data to extract $T$0. The reported good agreement between $T$1 and $T$2 across the studied interaction strengths supports the interpretation that the gas is in local thermal equilibrium. The paper also emphasizes a recurrent point in fermionic thermal measurements: in a correlated lattice system, thermometry is not a matter of fitting a bare Fermi-Dirac function, but of using the full interacting many-body equation of state.

A third cold-atom strategy uses a weakly interacting bosonic probe species to measure the temperature of a deeply degenerate Fermi gas (Lous et al., 2017). When $T$3 bosons thermalize with a $T$4 Fermi sea, the boson condensate fraction $T$5 determines the common temperature through

$T$6

and, for the Li–K trap geometry used in the experiment,

$T$7

The reported lowest temperature is $T$8. This is not a direct fermionic occupation measurement, but it is still a Fermi-Dirac thermal measurement in the broader sense that the target thermodynamic scale is the fermionic $T$9, while the readout circumvents the weak temperature sensitivity of the fermion density profile deep below $$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$0.

4. Fluctuation, entropy, and probe-based thermometry

A different line of work extracts thermodynamic information from fluctuation observables that are modified by Pauli blocking. For low-temperature fermion systems, the quadrupole-momentum variance and multiplicity variance obey (Zheng et al., 2010)

$$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$1

$$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$2

Because the first relation contains both $$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$3 and $$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$4, while the second directly gives $$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$5, the pair can be inverted to obtain both temperature and density. The paper stresses that the resulting “quantum” temperatures are systematically lower than the corresponding classical ones, because part of the observed momentum spread is intrinsic Fermi motion rather than thermal agitation.

Entropy-based protocols probe thermal mixing rather than temperature directly. For spinful fermions in optical lattices, the order-two Rényi entropy

$$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$6

and subsystem entropy

$$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$7

can be measured using two copies of the many-body state, a beam-splitter operation between copies, and site- and spin-resolved readout with a quantum gas microscope (Pichler et al., 2013). At $$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$8, subsystem Rényi entropies measure entanglement, whereas at finite temperature they acquire an extensive thermal contribution. The protocol therefore distinguishes thermal entropy from entanglement entropy and provides an entropy-based diagnostic of whether a fermionic state is dominated by low-temperature entanglement structure or by thermal mixing.

Impurity-based in-situ thermometry converts the non-equilibrium response of a Fermi gas into a temperature estimator (Mitchison et al., 2020). A localized impurity qubit is placed in a spin-polarized non-interacting Fermi gas and interrogated by a Ramsey sequence. The gas temperature is encoded in the impurity coherence

$$F_{\mu,\mu}=\beta^2\Delta^2 N,\qquad F_{\beta,\beta}=\Delta^2(\mu N-H),\qquad F_{\mu,\beta}=\beta\,\mathrm{Cov}(N,\mu N-H),$$9

which can be evaluated exactly for the non-interacting gas using a determinant formula. The quantum Fisher information of the qubit state is

$\tau(1),\tau(2)$0

with $\tau(1),\tau(2)$1. The paper reports that weak impurity-gas coupling gives the greatest sensitivities because decoherence remains slow enough for temperature-sensitive phase information to accumulate; the trade-off is a longer optimal interrogation time.

5. Nonequilibrium and indirect thermal characterization

A recurring methodological warning is that an apparently Fermi-Dirac-like distribution does not by itself establish thermal equilibrium. In pump/probe spectroscopy, the nonequilibrium single-particle distribution

$\tau(1),\tau(2)$2

can often be fit well by a Fermi-Dirac form, but this is not a sufficient test of thermalization (Matveev et al., 2018). The proposed diagnostic compares a fermionic effective temperature extracted from time-resolved photoemission with a collective-mode temperature extracted from the Stokes/anti-Stokes asymmetry of nonresonant electronic Raman scattering. In equilibrium the two temperatures must agree; out of equilibrium they generally differ. The paper’s main conclusion is that the discrepancy provides a more stringent measure of the distance from thermal equilibrium than a photoemission Fermi-Dirac fit alone.

Some approaches are inferential rather than directly experimental. The parametrized partition-function framework

$\tau(1),\tau(2)$3

interpolates continuously between bosons ($\tau(1),\tau(2)$4), distinguishable particles ($\tau(1),\tau(2)$5), and fermions ($\tau(1),\tau(2)$6) (Xiong et al., 2022). By computing thermodynamics only in the numerically accessible $\tau(1),\tau(2)$7 sector and reconstructing the $\tau(1),\tau(2)$8 branch through constant-energy contours in $\tau(1),\tau(2)$9 space, the method infers fermionic internal energy and heat capacity at any temperature. The paper explicitly states that it is not a new direct measurement protocol; rather, it is a computational route to fermionic thermal observables when direct finite-temperature fermion simulation is obstructed by the sign problem.

Temperature-dependent optical spectroscopy supplies another indirect Fermi-Dirac probe. In epitaxial Cd$T$0As$T$1, the interband optical conductivity is modeled as

$T$2

so the Pauli-blocking threshold $T$3 tracks the chemical potential (Chorsi et al., 2019). From infrared reflectivity fits, the paper reports $T$4 meV at 80 K and $T$5 meV at 400 K, implying that the Fermi level moves closer to the Dirac point as temperature rises. This is not direct thermometry, but it is a thermal measurement of Fermi-Dirac occupation effects through the temperature dependence of Pauli blocking and Drude response.

6. Abstract measurement theory and broader extensions

The most abstract use of the phrase appears in a framework for quantum hypothesis testing and semidefinite optimization (Liu et al., 4 Mar 2026). A binary quantum measurement operator satisfies

$T$6

so each eigenvalue $T$7 lies in $T$8. The paper interprets these eigenvalues as effective fermionic occupation numbers and replaces a linear objective $T$9 by a free-energy objective

$N$0

where

$N$1

The optimal “thermal measurement” is then

$N$2

so if $N$3 has eigenvalues $N$4, the corresponding measurement eigenvalues are

$N$5

In the limit $N$6, this reduces to the usual threshold measurement,

$N$7

The framework thereby treats thermalization of the measurement spectrum itself as an optimization device and motivates parameterized “Fermi-Dirac machines” as an alternative to thermal-state-based quantum Boltzmann machines.

A broader, more analogical extension appears in Kerr-black-hole ringdown (Oshita, 2022). There the absolute square of a gravitational-wave spectral amplitude is numerically fit by a Fermi-Dirac form,

$N$8

with fitted temperatures close to the Hawking temperature for rapid and near-extremal rotation. This does not constitute thermometry in the condensed-matter or cold-atom sense; it is an effective thermal characterization of a spectral envelope. Even so, it illustrates how the language of Fermi-Dirac thermal measurement can migrate from literal fermionic occupancies to operator spectra and effective thermal fits of experimentally accessible signals.

Across these settings, the unifying principle is not a single apparatus but a shared inferential structure: one measures an observable whose dependence on occupation, fluctuation, or threshold behavior is controlled by Fermi-Dirac statistics, then inverts that dependence to obtain thermodynamic information. The main methodological divide is between direct thermometry, such as dwell-time or fluctuation measurements, and indirect or computational characterization, such as surrogate partition-function reconstruction or thermalized measurement design. A consistent theme is that finite-size effects, interactions, probe back-action, and nonequilibrium dynamics all matter quantitatively; the most reliable Fermi-Dirac thermal measurements are therefore those for which the relevant occupation law, equation of state, or fluctuation relation is specified at the same level of approximation as the underlying experiment or model.