Virtual Cooling Experiments

• Virtual cooling experiments are methodologies that map measured system observables at temperature T to effective states at lower virtual temperatures.
• They enable investigation of complex phenomena in quantum many-body systems, mesoscopic evaporative setups, and active matter through controlled simulation and measurement protocols.
• Key protocols involve collective operations, kinetic simulations, and transient analyses, offering insights into phase transitions and design-of-experiments for cooling technologies.

Virtual cooling experiments comprise a family of methodologies—spanning molecular, quantum, mesoscale, and engineering domains—that leverage physical, computational, or information-theoretic mechanisms to obtain effective system behavior at temperatures lower than those directly accessible or realized in experiment. Emphasizing either measurements or simulation domains, these approaches extract properties corresponding to lower ("virtual") temperatures, enabling investigation of phases and phenomena ordinarily masked by intrinsic thermal limitations. Virtual cooling has notable implementations in nonequilibrium thermodynamics, many-body quantum systems, active matter, and design-of-experiments (DoE) for cooling technologies.

1. Physical Principles and Theoretical Frameworks

Virtual cooling necessitates the mapping of system observables measured at a physical temperature $T$ to those corresponding to a lower temperature $T' < T$. Several classes of mechanisms underlie this mapping:

• Information-theoretic transformation: In quantum systems, collective measurements on multiple copies of a system at temperature $T$ simulate an effective Gibbs state at lower temperature via traces involving permutation operators, as formalized by

$$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$

where $\rho(\beta)$ is the Gibbs state at inverse temperature $\beta$ and $n$ is the number of copies. The protocol exploits the property $$\rho(n\beta) = \rho(\beta)^n / \mathrm{Tr}[\rho(\beta)^n]$$ to construct measurements at a "virtual" temperature $T/n$ (Cotler et al., 2018).

• Nonequilibrium transport effect: In mesoscopic evaporation-condensation problems, interfacial kinetic asymmetries and non-continuum (Knudsen-layer) expansions create local vapor temperatures below that of either solid boundary, yielding a net "evaporative refrigeration" effect (Chen et al., 14 Apr 2025).
• Transient dynamic undershooting: In active or dissipative systems, certain relaxation spectra enable effective system temperatures to undershoot the bath temperature during transient evolution (activity-induced overcooling) (Schwarzendahl et al., 2021).

Virtual cooling experiments thus instantiate nontrivial mappings between realized and effective thermodynamic states, exploiting statistical, quantum, or kinetic asymmetries.

2. Experimental Implementations and Simulation Methods

Quantum Virtual Cooling

Quantum virtual cooling protocols are operationalized by preparing multiple identical copies of a many-body system at finite temperature and performing collective operations—primarily beam-splitter unitaries $F_n$ and site-resolved projective measurements $T' < T$0, as realized in Bose-Hubbard chains with $T' < T$1Rb atoms in optical lattices (Cotler et al., 2018). For $T' < T$2, the fundamental sequence is:

1. State Preparation: Equilibrate two decoupled chains to a Gibbs state at temperature $T' < T$3.
2. Beamsplitter Operation: Couple chains via controlled tunneling, implementing $T' < T$4.
3. Measurement: Perform on-site number-resolved imaging to extract both $T' < T$5 and symmetrized observables $T' < T$6.
4. Averaging: Combine measurement statistics to estimate expectation values at $T' < T$7.

The method generalizes, in principle, to higher $T' < T$8, with exponentially increasing experimental overhead due to shot noise ($T' < T$9), where $T$0 is the entropy density and $T$1 the measurement region.

Kinetic Virtual Cooling in Evaporative Systems

Computational virtual cooling of mesoscopic liquid-vapor systems, as studied in (Chen et al., 14 Apr 2025), employs:

• Coupled Boltzmann-BGK and continuum conduction solvers across two-sided liquid films and a vapor gap.
• Boundary conditions featuring diffuse Maxwellian emission from interface, partial accommodation ($T$2), and mass/energy flux matching.
• Numerical fixed-point cycles to self-consistently determine interfacial temperatures $T$3, $T$4, and the minimum vapor temperature $T$5.
• Verification of virtual refrigeration via observation of $T$6 (the cold plate temperature), commonly reaching differences of up to 5 K at relevant micro-scale plate spacings.

Virtual Cooling in Nonequilibrium and Active Matter

Experiments with active colloids and analogous simulations apply:

• Stochastic integration of Langevin or master equations with activity terms (e.g., run-and-tumble dynamics), temperature quenches, and parametric sweeps of persistence and Péclet number (Schwarzendahl et al., 2021).
• Extraction of effective temperature from time-evolving distributions via either static-matching with precomputed equilibrium families or by mapping time-dependent diffusive lag statistics.
• Diagnosis of the Mpemba effect and overcooling through spectral analysis of relaxation eigenmodes, tracking non-monotonic evolution relative to final target temperature.

3. Mechanisms and Conditions Enabling Virtual Cooling

Two-Copy Quantum Virtual Cooling

The core of the quantum approach is the realization that collective permutations (e.g., swap operators $T$7) extricate higher powers of the Gibbs state, which correspond to effective lower temperatures. Key determinants:

• Permutation symmetry: Enables selection of trace components corresponding to $T$8.
• Fidelity and decoherence: Physical implementations are limited by the accuracy of $T$9 and $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$0 operations, and by the entropy-induced exponential scaling of required repetitions.

Interfacial and Knudsen-Layer Refrigeration

In mesoscopic evaporative systems, two mechanisms dominate:

• Interfacial-asymmetry cooling: The outgoing half-range Maxwellian from the hot interface and the returning energy flux disparity ensure $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$1, enforced quantitatively by

$$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$2

• Knudsen-layer expansion cooling: Non-isenthalpic expansion in the vapor immediately adjacent to the interface reduces internal energy flux, lowering local temperature further, analogous to but distinct from the Joule-Thomson effect.

Amplification of refrigeration is favored by thinner liquid films, higher accommodation coefficients, and lower Knudsen numbers (broader vapor gaps).

Activity-Induced Overcooling and the Mpemba Effect

In overdamped active systems, eigenmode expansions around the cold steady state reveal the possibility of effective temperature undershooting when higher modes contribute destructively ($$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$3 and $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$4 of opposite sign). Criteria for this regime are derived analytically in terms of activity (Péclet number), persistence ratio, and relaxation rates:

$$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$5

Transient thermal inversion arises when $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$6 exceeds a critical threshold determined by ratio of relaxation rates.

4. Practical Protocols and Data Analysis

Essential Steps in Quantum Virtual Cooling

Step	Description	Experimental Handle
Preparation	Thermalize $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$7 copies to $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$8	Atom loading, lattice depth
Coupling	Apply beamsplitter $$\langle X \rangle_{T/n} = \frac{\mathrm{Tr}[\rho(\beta)^n X]}{\mathrm{Tr}[\rho(\beta)^n]}$$9	Controlled tunneling
Measurement	Site-resolved imaging for $\rho(\beta)$0, $\rho(\beta)$1	Quantum gas microscope
Averaging	Compute $\rho(\beta)$2 ratios for given observable $\rho(\beta)$3	Repetition and postprocessing

Virtual Cooling in Kinetic/Evaporation Context

• Define boundary and material parameters.
• Solve 1D conduction for liquid, coupled BGK Boltzmann in the vapor.
• Enforce flux continuity and accommodate interfacial exchange.
• Iterate to convergence for $\rho(\beta)$4 and $\rho(\beta)$5, extract $\rho(\beta)$6.

Nonequilibrium and Active System Protocols

• Initialize ensemble at high $\rho(\beta)$7, sample steady-state distribution.
• Quench to cold $\rho(\beta)$8, propagate trajectories via stochastic equations.
• Extract $\rho(\beta)$9 from distributional or lag-diffusion metrics.
• Fit decay spectrum, assess overcooling and Mpemba regimes via modeled overlaps $\beta$0 and relaxation data.

5. Applications and Extensions

Applications span exploratory and technological domains:

• Many-body quantum simulation: Accessing subthermal regimes and ground-state properties via virtual cooling enables measurement of phase transitions, correlation functions, and entanglement spectra with limited physical cooling resources (Cotler et al., 2018).
• Refrigeration system optimization: Virtual design-of-experiments, leveraging surrogate models and iterative simulation, offers significant speed-up and performance gain for automotive and engineered cooling systems (Splechtna et al., 2024).
• Passive and low-energy cooling technologies: Evaporative refrigeration effects suggest the possibility of micro- and nanoscale passive cooling structures capable of inducing local temperature inversion and sub-environment cooling (Chen et al., 14 Apr 2025).
• Nonequilibrium statistical physics: Simulated or experimental explorations of overcooling guide understanding of relaxation, metastability, and anomalous response in active or dissipative matter (Schwarzendahl et al., 2021).

6. Limitations, Scalability, and Prospects

Current approaches are constrained by exponential scaling of repetition requirements in quantum protocols (Rényi entropy), nonidealities in experimental operations (imperfect beamsplitter, detection), and the requirement of parameter regimes where virtual cooling effects manifest distinctly (e.g., high activity, low Knudsen number).

Advancements include:

• Extension to higher $\beta$1 in quantum systems, enabling deeper virtual cooling at the cost of experimental complexity (Cotler et al., 2018).
• Engineering of tunable boundary and interface conditions in kinetic systems for optimized evaporative refrigeration (Chen et al., 14 Apr 2025).
• Improved surrogate modeling and interactive refinement in engineering DoE for multidimensional, nonlinear cooling process optimization (Splechtna et al., 2024).
• Analytical and simulation exploration of broader parameter regimes in active matter for more general manifestations of overcooling and anomalous relaxation (Schwarzendahl et al., 2021).

Virtual cooling experiments represent a growing toolkit for traversing otherwise inaccessible thermodynamic regimes, with direct impact on quantum simulation, device design, and the understanding of nonequilibrium statistical phenomena.