Quantum Refrigeration & Absolute Zero Temperature

1. Quantum Refrigeration & Absolute Zero Temperature Yair Rezek Tova Feldmann Ronnie Kosloff

2. The Third Law of Thermodynamics Heat Theorem: “The entropy change of any process becomes zero when the absolute zero temperature is approached” Walter Nernst 1864-1941 Unattainability Principle: “It is impossible by any procedure, no matter how idealized, to reduce any system to the absolute zero of temperature” P.T. Landsberg, Rev. Mod. Phys. 28, p. 363, 1956. J Phys A: Math. Gen. 22, p. 139, 1989 F. Belgiorno J. Phys. A: Math. Gen. 36, p. 8165, 2003.

3. Adiabatic Compression (cold-to-hot adiabat) Isochoric Cooling (cold isochore) Isochoric Heating (hot isochore) ∆Wch Hot Bath (at Th) Cold Bath (at Tc) ∆Qh ∆Qc ∆Whc Adiabatic Expansion (hot-to-cold adiabat) The Brayton (Otto) Cycle

4. Entropies Von Neumann Entropy: Shannon Entropy of Energy: where Pj is the probability to measure energy eigenvalue Ej The von Neumann entropy is always lower than the Shannon energy entropy (or equal to in a thermal state)

5. S 0 Tc 0 Yet Another Third Law The entropy of the system approaches zero as the absolute temperature approaches zero. Outside of equilibrium, temperature may be defined as:

6. The First Law Quantum dynamical interpretation: Heisenberg equation for Open Quantum System: Applying it to the Hamiltonian: leads to the time-explicit First Law

7. Ideal gas in square (1D) piston • Quantum particles in (1D) harmonic potential • Contact with heat bath • Weak coupling to simple thermalizing environment adiabatic parameter Adiabatic Compression Adiabatic Expansion The Model Equations of motion on the adiabats: Equations of motion on the isochores:

8. Adiabatic Compression Adiabatic Expansion Cooling Rate in Pictures

9. As Tc 0, the heat exchange Qc must diminish to maintain the 2nd law. Unattainability & 2nd Law Entropy Production: Entropy production for a cyclic process is only on the interface.

10. For sufficiently slow change of frequency on the adiabatic segment, adiabatic theorem holds. Isentropic Cycle Closing the cycle, one obtains: In order to maintain cooling at low temperatures, the coth factors necessitate changing the frequency: Unitarity The von Neumann entropy remains constant under unitary evolution. Isentropy in this sense is guaranteed at all temperatures. For a linear frequency change:

11. Cooling per Cycle

12. Adiabatic Compression Isentropic Adiabatic Expansion

13. Isentropic II Dimensionless measure of adiabacity: Compression adiabat is fast Expansion adiabat is slower, but grows faster at low Tc

14. Unattainability

15. Conclusions: • The Brayton model shows that: • The heat theorem does not hold. • Unattainability principle maintained. • Dynamic treatment of the cold bath is required for a more robust analysis.

16. The End

17. “Heat Theorem” Linear Exponential Adiabatic

18. Isentropic III Isochores are long

21. is a completely positive map with generator Second Law relative entropy: Lindblad’s theorem: Entropy change related to energy exchange: Assume steady state: Completely Positive Maps and Entropy Inequalities, Goran Linblad, Commun. Math. Phys. 40, 147-151 (1975)