Temp-Dependent Coherent States

Updated 25 December 2025

Temperature-dependent coherent states are quantum or classical states where thermal fluctuations and coherent wave modes interact, resulting in temperature-sensitive statistical and dynamical properties.
They are constructed by methods such as applying the Glauber displacement operator to thermal states and using thermo-field dynamics, which integrates thermal statistics with coherent state formalism.
These states are crucial in applications like quantum thermometry, mesoscopic transport, and nonlinear pattern formation, offering practical insights into decoherence, material instability, and high-temperature quantum coherence.

Temperature-dependent coherent states represent quantum or classical states in which thermal fluctuations and coherent wave behavior are coupled, with the resulting state properties explicitly dependent on temperature. These states appear in both quantum optics—where "thermal coherent states" are displaced thermal Gaussian states—and in classical or semiclassical models of nonlinear material dynamics, such as shear localization in temperature-sensitive fluids. They are central to the understanding of decoherence, quantum thermometry, condensed matter transport, and nonlinear pattern formation, and their mathematical structure integrates canonical coherent-state formalism, thermodynamic ensembles, field-theoretic symmetries, and dissipative system dynamics.

1. Mathematical Formalism of Temperature-Dependent Coherent States

Thermal coherent states in quantum systems are typically constructed by applying the Glauber displacement operator $D(\alpha) = \exp(\alpha a^\dagger - \alpha^* a)$ to a thermal state density operator $\rho_\text{th}(T)$ . The result, $\rho_\text{th,\alpha}(T) = D(\alpha)\rho_\text{th}(T) D^\dagger(\alpha)$, describes a displaced thermal Gaussian state whose width (photon number variance) depends explicitly on temperature via the Bose occupation number $\bar n(T) = [e^{\hbar\omega / k_B T} - 1]^{-1}$ (Ullah et al., 2023). In Thermo-Field Dynamics (TFD), temperature dependence is encoded by doubling the Hilbert space and introducing Bogoliubov-rotated ladder operators $\hat a(\beta), \hat a^\dagger(\beta)$ parametrized by a thermal mixing angle $\theta(\beta)$ , with $\cosh\theta = \sqrt{1+\bar n}$ , $\sinh\theta = \sqrt{\bar n}$ (Popov, 22 Dec 2025, Azuma et al., 2013).

In classical or nonlinear PDE contexts, coherent structures with temperature dependence arise through the explicit dependence of material parameters (such as viscosity) on temperature. For instance, in temperature-dependent fluids, viscosity may follow a law $\mu(\theta) = e^{-\alpha\theta}$ , producing strong spatial and temporal coupling between coherent shear patterns and thermal evolution. Such models generate self-localized solutions—shear bands—whose profile and localization width depend dynamically on temperature (Katsaounis et al., 2014).

2. Physical Mechanisms and Instabilities: Quantum and Classical Regimes

Temperature-dependent coherent states can emerge from different physical mechanisms:

Thermal mixing and reservoir engineering: In quantum thermometry, coupling a resonator longitudinally to a thermally equilibrated qubit yields mixtures of oppositely displaced thermal coherent states, with temperature manifest only in the thermal occupation statistics, but the qubit–resonator interaction also imprints coherent contributions dependent on coupling strength and bath temperature (Ullah et al., 2023).
Nonlinear pattern formation in classical systems: Temperature-dependent viscosity in fluids causes shear localization via Hadamard or Turing-type instabilities. Thermal softening triggers growth of high-frequency modes, momentum diffusion regularizes instability to finite rates, and thermal diffusion stabilizes the system at large times. The nonlinear regime suppresses oscillatory competition and concentrates deformation into narrowing coherent shear bands with a width $\sim (\alpha t + c_0)^{-\lambda/\alpha}$ (Katsaounis et al., 2014).
Dissipation and environment-induced selection: In linearly damped quantum oscillators coupled to dissipative reservoirs, intermediate temperatures can suppress all excited states and collapse the system into a single coherent ground state with imaginary frequency—effectively a temperature-dependent pure state stabilized by dissipative suppression (Cragg, 2014).

3. Experimental Realizations and Observable Effects

Temperature-dependent coherent states have been engineered and studied in several platforms:

Quantum optics (OPO and thermometry): In optical parametric oscillators (OPO), the Lie–Trotter product formula realizes thermal coherent states by symmetrically applying thermalizing (squeezing) and displacement operators in TFD formalism. The observable photon statistics, coherence functions, and quantum Fisher information in thermometry can be analytically computed, showing temperature-dependent enhancements in both sensitivity and operational temperature range (Azuma et al., 2013, Ullah et al., 2023).
Transport in mesoscopic solid-state systems: Coherent current peaks in double quantum dots (DQDs) display strong temperature-dependent broadening due to quantum dissipation from bosonic baths—primarily substrate phonons. The linewidth grows as $\Delta(T) = \Gamma_R + 4\pi\alpha k_B T$ , directly relating decoherence rates to thermal phonon occupations. Magnetic field dependence further helps identify coherent states and their thermal evolution (Dani et al., 2022).
Bulk-boundary quantum field systems and Casimir energy: Thermal coherent states in a confined quantum field theory with dynamical boundary conditions are constructed by dressing the thermal state with Weyl displacement operators. All local observables (field squares, Casimir energy density) acquire explicit classical coherent shifts, and temperature impacts local energy and force, sometimes inverting Casimir force sign (Juárez-Aubry et al., 2020).

4. Thermodynamic Consistency and Quantum–Thermal Analogs

Traditional squeezed coherent states (Bogoliubov vacua) saturate the Schrödinger uncertainty relation but remain pure (zero entropy), thus failing to reproduce thermal equilibrium properties at $T>0$ unless mixedness is introduced. Correlated coherent states (CCS), defined as pure Gaussian states with temperature-dependent squeezing parameter $r(T)$ and phase $\phi = \pi/2$ , reproduce the full covariance structure and mean energy of Planck's law, providing a legitimate quantum analog of thermal states (Sukhanov et al., 2012). Thermal coherent states constructed in doubled Hilbert spaces via TFD preserve correct thermal statistics in both expectation values and variances (Popov, 22 Dec 2025).

Contrastingly, temperature-dependent kinetic and thermodynamic temperatures in non-equilibrium quantum oscillators—especially in Poissonian-coherent-state ensembles—can diverge sharply, highlighting the duality of statistical disorder versus dynamical energy content (Gagliardi et al., 2013).

5. Applications and Implications in Quantum Technologies and Materials Science

Quantum thermometry: Mixtures of thermal coherent states outperform bare thermal oscillators and simple qubit probes in simultaneous temperature precision and sensitivity range. Displacement amplitude and system parameters are tunable, allowing sensors to operate optimally at multiple temperatures (Ullah et al., 2023).
Materials instability and localization: Coherent shear bands in temperature-dependent fluids and metals explain the emergence of localized deformations in high strain-rate processes, with potential relevance to failure dynamics, pattern formation, and instability theory (Katsaounis et al., 2014).
Quantum coherence at high temperature: Environment-mediated suppression of excited states leading to high- $T$ quantum coherence has implications for photosynthetic complexes and high- $T_c$ superconductivity in granular materials (Cragg, 2014).

6. Open Problems and Theoretical Limitations

A persistent misconception is the equivalence of ordinary squeezed coherent states to thermal states: despite formal matching of variances and covariance structure, the lack of entropy and mixedness prevents a consistent thermodynamic interpretation unless the state construction is modified (correlated coherent states, thermo-field doubling) (Sukhanov et al., 2012). Genuine temperature-dependent coherent states in equilibrium require either mixture or explicit environmental modeling—often via doubled Hilbert spaces or noncanonical field-theoretic symmetries (Popov, 22 Dec 2025, Cirilo-Lombardo, 2014).

In classical nonlinear systems, coherent structures can have well-defined temperature dependence only when coupling between thermal fields and nonlinear dynamics is strong and explicitly modeled in the governing equations, as in exponential viscosity fluids (Katsaounis et al., 2014).

7. Summary Table: Representative Models and Contexts

Context/Platform	Mathematical Formulation	Temperature Dependence
Quantum optics: OPO, thermometry	$\rho_\text{th,\alpha}(T)$, TFD states	Squeezing parameter or occupation number $\bar n(T)$ (Azuma et al., 2013, Popov, 22 Dec 2025)
Nonlinear fluids: shear bands	PDEs with $\mu(\theta) = e^{-\alpha \theta}$	Viscosity, instability growth rate, zone width (Katsaounis et al., 2014)
Mesoscopic transport: DQDs	Lorentzian peak width $\Delta(T)$	Direct $\propto k_B T$ broadening, dephasing rates (Dani et al., 2022)
Quantum oscillator (Feynman–Vernon)	Path integral, imaginary frequency $\Omega_\text{im}$	Intermediate- $T$ regime supports single coherent ground state (Cragg, 2014)
Correlated coherent states (CCS)	Pure Gaussian states, Bogoliubov $\xi_T$	Thermal covariance and Planck mean energy (Sukhanov et al., 2012)

Temperature-dependent coherent states bridge nonlinear classical pattern formation, quantum statistical mechanics, and field-theoretic approaches, with rich applications in precision metrology, condensed matter, and fundamental studies of decoherence, stability, and emergent order.