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

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

The Brayton (Otto) Cycle Cold Bath (at T c ) Hot Bath (at T h ) Isochoric Cooling (cold isochore) Adiabatic Compression (cold-to-hot adiabat) Isochoric Heating (hot isochore) Adiabatic Expansion (hot-to-cold adiabat) ∆W ch ∆W hc ∆Q c ∆Q h

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

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: S 0 T c 0

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

Open Quantum Systems Weak coupling limitIsothermal partition

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

Cooling Rate in Pictures Adiabatic Compression Adiabatic Expansion

Lindbladian and applies it to a driven dumped harmonic oscillator Qunatum Statistical Properties of Radiation, Louisell p ; eq on p. 347 Louisell develops the Markov approaximation for a general system and reservoir in Chapter 6.2.

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

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

Cooling per Cycle

Isentropic Adiabatic Compression Adiabatic Expansion

Isentropic II Dimensionless measure of adiabacity: Compression adiabat is fast Expansion adiabat is slower, but grows faster at low T c

Unattainability

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.

“Heat Theorem” Linear Exponential Adiabatic

Isentropic III Isochores are long

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