Coherence-Induced Effective Temperatures

• The paper reveals that quantum coherence modifies operational temperature definitions, making them channel-dependent and distinct from equilibrium Gibbs temperatures.
• Utilizing collision models and non-secular master-equation approaches, the work demonstrates how coherent reservoirs can alter heat flow and induce anomalous transport behavior.
• Coherence-induced effective temperatures enable innovative quantum thermometry techniques by introducing metrological sensitivity beyond conventional temperature scales.

Searching arXiv for the cited papers and closely related work on coherence-induced effective temperatures. Coherence-induced effective temperatures denote temperature-like descriptors whose operational meaning is modified by quantum coherence, internal degeneracies, collective correlations, or structured system–environment couplings. In the works surveyed here, the underlying theme is that temperature inferred from populations, heat-flow expressions, or local reduced states need not coincide with the thermodynamic temperature of a Gibbs state once coherence is present. Two complementary constructions recur. One defines an effective temperature from thermal populations while allowing nonzero off-diagonal elements in the energy basis, so that coherence alters transport without changing the diagonal sector (Li et al., 2018). The other defines an “apparent temperature” directly from the operators that govern heat exchange, making the relevant temperature channel-dependent and explicitly sensitive to coherence and correlations (Latune et al., 2018).

1. Definitions and formal structures

In a microscopic collision model with coherent ancillas, a reservoir state can be prepared so that its diagonal is identical to that of a thermal state while its off-diagonal elements are nonzero. For a two-level ancilla with Hamiltonian $\hat H_j=\frac{\omega}{2}\hat\sigma_z$, the thermal populations satisfy

$$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$

and this same relation is used to define the effective temperature of the coherent reservoir from its diagonal populations alone. The crucial point is that coherence modifies only the off-diagonal elements at preparation, yet through the dynamics it affects energy exchange and heat currents, so the steady state and thermodynamic behavior cannot be characterized by $T$ alone (Li et al., 2018).

A more general temperature-like quantity is the apparent temperature. For a transition frequency $\omega$ and ladder operator $A_X(\omega)$, it is defined as

$$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$

This quantity is interaction-defined: it is meaningful only for the heat flow generated by the relevant operators. For equilibrium Gibbs states, $\mathcal{T}_X(\omega)=T_X$ for every Bohr frequency, so the construction reduces to the standard thermodynamic temperature. Out of equilibrium, however, it becomes channel-dependent and need not define a unique global temperature (Latune et al., 2018).

The role of coherence is especially transparent in degenerate systems. For a single-frequency ladder with degeneracies,

$$\mathcal{T}_X = \omega \left( \ln \frac{\sum_{n=1}^N l_n(\rho_{n-1}+c_{n-1})} {\sum_{n=1}^N l_{n-1}(\rho_n+c_n)} \right)^{-1},$$

where $\rho_n$ are populations and $c_n$ are internal coherences within degenerate subspaces. In this expression, the same combination $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$0 appears in numerator and denominator: coherences between degenerate states contribute just like populations. In collective many-body settings the same logic extends to correlations and products of local coherences, which enter the apparent temperature through a single additive term $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$1 (Latune et al., 2018).

2. Microscopic transport with coherent reservoirs

A concrete realization is the collision model for heat transport between two nonthermal reservoirs. The working medium is a bipartite two-qubit system $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$2, each subsystem interacting with a stream of independent ancillas prepared either in thermal states or in coherent states with the same diagonal as the thermal state. Inter-subsystem coupling is implemented by a swap-like unitary with parameter $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$3, while subsystem–ancilla collisions are governed by a swap-like unitary with parameter $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$4. Because the Heisenberg-like interaction is resonant and energy conserving, heat exchanged in a collision can be identified with the change in ancilla energy, and the model is effectively Markovian at the system level (Li et al., 2018).

For complete subsystem–ancilla swap, $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$5, the steady heat current from the hot reservoir takes the form

$$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$6

The first term is the thermal contribution generated solely by the diagonal sector and is negative for $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$7, corresponding to heat flow from hot to cold. The second term is coherence-induced, depends on $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$8, $$\frac{p_1}{p_0}=e^{-\beta\omega},\qquad T=\frac{1}{\beta},$$9, and the relative phase $T$0, and can have either sign (Li et al., 2018).

This phase sensitivity leads directly to anomalous transport. For $T$1, the coherence term reinforces the thermal current. For $T$2, it tends to drive heat from cold to hot, and if $T$3 the total steady current reverses sign, so heat flows from the “cold” effective-temperature reservoir to the “hot” one. In the complete-swap case the threshold for reversal is set by the critical coherence

$T$4

The analysis also clarifies why this does not violate thermodynamics: for coherent reservoirs the entropy-flux term cannot be identified purely with $T$5, because the reservoirs are not Gibbs equilibrium states and the effective temperature captures only the diagonal sector (Li et al., 2018).

3. Near-equilibrium transport and the limits of linear response

When the effective temperature bias is small,

$T$6

the purely thermal case obeys

$T$7

which defines a conductance $T$8. In the high-temperature limit one obtains $T$9, so the current magnitude increases with $\omega$0 at low temperatures and decreases at high temperatures. This is the standard linear-response picture (Li et al., 2018).

With coherent reservoirs the structure changes. For special phases $\omega$1, the steady current remains purely linear in $\omega$2, so a conductance is still well defined, although its magnitude differs from the thermal case and satisfies

$\omega$3

For generic phases, however,

$\omega$4

with a phase-dependent offset $\omega$5. Then a finite current persists even at $\omega$6, and conductance alone ceases to characterize transport (Li et al., 2018).

In the complete-swap limit the small-$\omega$7 expansion reads

$\omega$8

with

$\omega$9

For $A_X(\omega)$0 and $A_X(\omega)$1, the current is dominated by $A_X(\omega)$2, which increases with $A_X(\omega)$3 and saturates. This identifies coherence as an independent driving force beside the temperature gradient and makes explicit that effective temperatures alone do not determine either the direction or the magnitude of heat flow (Li et al., 2018).

4. Open-system realizations beyond coherent ancilla baths

The same basic theme appears in non-equilibrium master-equation settings. For two coupled two-level atoms interacting with two baths at different temperatures, the secular master equation yields a diagonal steady state in the energy basis, whereas the master equation without secular approximation gives a steady state with $A_X(\omega)$4 in the single-excitation manifold. The residual coherence depends on the temperature difference, detuning, and coupling geometry, and disappears in configurations where additional atom–bath channels effectively average the two environments. This suggests that transition-dependent temperature biases can sustain steady-state coherence, but only if non-secular interference terms are retained (Huangfu et al., 2017).

In a collective thermal bath of $A_X(\omega)$5 identical qubits, temperature can either suppress or enhance coherence depending on the initial state. For an incoherent single-excitation initial state, coherence and excitation transfer increase in time and are increasing functions of bath occupation. For a coherent initial state there is a critical temperature: below it coherence decreases, above it coherence increases, and the critical value decreases with system size. The steady states are not full Gibbs states because $A_X(\omega)$6 is conserved; instead they are thermal-like within invariant sectors, which yields local behavior that differs from that of independent qubits at the same bath temperature (Memarzadeh et al., 2017).

A related equilibrium mechanism appears when a target TLS is coupled to $A_X(\omega)$7 interacting source TLSs and the whole composite is in a canonical Gibbs state. Because the interaction is symmetry-breaking, the reduced state of the target in its bare energy basis acquires off-diagonal elements. For ferromagnetic intersource interactions the zero-temperature coherence is

$A_X(\omega)$8

while for antiferromagnetic interactions and even $A_X(\omega)$9 the zero-temperature coherence can vanish on one side of a quantum phase transition and become finite on the other. The temperature dependence then bears signatures of that transition: in some regimes $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$0 decreases monotonically, while in the antiferromagnetic phase it can rise from zero to a finite-temperature maximum before decaying (Hudenko et al., 6 Nov 2025).

5. Thermometric and metrological uses

Coherence-induced effective temperatures are also operational in thermometry. In a probe–ancilla architecture for low-temperature thermometry, a qubit probe does not couple directly to the thermal sample but only through ancilla qubits. A global Born–Markov–secular master equation shows that although the full probe–ancilla system thermalizes with the sample, the probe per se reaches a nonthermal steady state. For one ancilla the probe state in its energy basis is

$$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$1

with

$$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$2

The populations imply a population-based effective temperature different from the bath temperature whenever $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$3, while the nonzero coherence makes clear that no single Gibbs description of the bare probe Hamiltonian is available (Ullah et al., 2022).

The metrological consequence is that the quantum Fisher information can split into a population term and a coherence-induced term. In the weak-coupling limit,

$$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$4

with $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$5 peaked near $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$6 and $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$7 peaked near $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$8. Increasing coherence or the number of ancillas creates multiple low-temperature sensitivity peaks and broadens the measurable range. In this setting, coherence does not merely accompany thermometry; it reshapes the temperature scales to which the probe is sensitive (Ullah et al., 2022).

A distinct thermometric use appears in multipartite dephasing models. In a tripartite spin–boson model under local non-Markovian dephasing, the relative entropy of coherence decays monotonically and faster at higher temperature for all states considered. Under a common structured reservoir, by contrast, the behavior is non-universal: $$\mathcal{T}_X(\omega) = \omega \left( \ln \frac{\langle A_X(\omega)A_X^\dagger(\omega)\rangle} {\langle A_X^\dagger(\omega)A_X(\omega)\rangle} \right)^{-1}.$$9 and $\mathcal{T}_X(\omega)=T_X$0 states undergo temperature-enhanced degradation, while the $\mathcal{T}_X(\omega)=T_X$1 state exhibits stationary coherence that is largely insensitive to temperature, and $\mathcal{T}_X(\omega)=T_X$2 states show partial decay followed by a temperature-independent plateau. This makes coherence a sensitive probe of structured finite-temperature environments only for states that are not protected by collective dephasing symmetries (Perumalsamy et al., 11 Mar 2026).

6. Counterintuitive thermal behavior and broader significance

Several models sharpen the point that temperature does not always reduce coherence. In a bosonic Mott insulator, the nearest-neighbor correlator can increase substantially with increasing temperature because thermally produced defects tunnel with ease. The effect is traced to a gapless contribution from defect motion,

$\mathcal{T}_X(\omega)=T_X$3

which grows as defects become thermally populated, even though the zero-temperature coherence is only the virtual contribution $\mathcal{T}_X(\omega)=T_X$4. This gives a practical counter-example to the conventional expectation that heating always suppresses coherence (Toth et al., 2010).

An even more explicit reversal of the standard intuition occurs in driven two-level systems swept through a Landau–Zener avoided crossing. There, a high-frequency environment can produce a Lamb shift that reduces the effective gap,

$\mathcal{T}_X(\omega)=T_X$5

and the coherent oscillation amplitude can grow approximately as

$\mathcal{T}_X(\omega)=T_X$6

Because $\mathcal{T}_X(\omega)=T_X$7 can increase with the bath temperature for super-Ohmic or high-frequency environments, the oscillation amplitude may grow exponentially with temperature. A plausible implication is that an effective-temperature interpretation must sometimes be attached to environment-dressed energy scales rather than to bare bath temperature alone (Whitney et al., 2011).

A broader complexity-based perspective appears in the notion of coherence dispersion,

$\mathcal{T}_X(\omega)=T_X$8

evaluated for partially coherent Gibbs states. In a quantum-biology toy model, $\mathcal{T}_X(\omega)=T_X$9 vanishes at low and high temperature and has a single maximum at an intermediate temperature. Using the ATP–ADP energy scale, $$\mathcal{T}_X = \omega \left( \ln \frac{\sum_{n=1}^N l_n(\rho_{n-1}+c_{n-1})} {\sum_{n=1}^N l_{n-1}(\rho_n+c_n)} \right)^{-1},$$0, the temperature window selected by maximal coherence dispersion,

$$\mathcal{T}_X = \omega \left( \ln \frac{\sum_{n=1}^N l_n(\rho_{n-1}+c_{n-1})} {\sum_{n=1}^N l_{n-1}(\rho_n+c_n)} \right)^{-1},$$1

is compatible with reported temperatures for unicellular life, and the position of the maximum is essentially insensitive to very low coherence levels. This suggests that coherence can define preferred temperature windows even when populations are Gibbsian and only remanent off-diagonal structure is added (Parisio, 13 Dec 2025).

Across these constructions, a common misconception is that a single scalar temperature remains sufficient once coherence is present. The literature instead supports a stricter statement: on Gibbs states, temperature, effective temperature, and apparent temperature coincide; out of equilibrium they generally separate. Coherence can act as an additional thermodynamic resource, can modify or even reverse heat currents, can produce finite currents at zero temperature bias, can generate nonthermal local steady states useful for thermometry, and can make the relevant temperature intrinsically channel-dependent, basis-dependent, or interaction-dependent (Latune et al., 2018).