Size Effects in Quasi-Static Energy Transport for Microscopic Quantum Systems

1. Air Force Research Laboratory • Size Effects in Quasi-Static Energy Transport for Microscopic Quantum Systems MRS-Spring 2014, April 25, San Francisco CA George Y. Panasyuk Timothy J. Haugan Kirk L. Yerkes Aerospace Systems Directorate Wright-Patterson Air Force Research Laboratory Distribution A: Approved for public release; distribution unlimited.

2. Introduction Study of size effects on response to electromagnetic field: F. Hache, D. Ricard, and C. Flytzanis, J. Opt. Soc. Am. B, 1986 S.G. Rautian, Sov. Phys. JETP, 1997; G.Y. Panasyuk, J.C. Schotland, and V.A. Markel, Phys. Rev. Lett., 2008; A.A. Govyadinov, G.Y. Panasyuk, J.C. Schotland, and V.A. Markel, Phys. Rev. B, 2011; G.Y. Panasyuk, J.C. Schotland, and V.A. Markel, Phys. Rev. B, 2011; Study of static properties of small bodies: S.P. Adiga, V. P. Adiga, R.W. Carpick, and D.W. Brenner, J. Phys. Chem. C, 2011; D. Sopu, J. Kotakoski, and K. Albe, Phys. Rev. B, 2011; J. Pohl, C. Stahl, and K. Albe, Beilstein, J. Nanotechnol., 2012; Study of heat transport in finite quantum systems: E.C. Cuansing, H. Li, and J.S. Wang, Phys. Rev. E, 2012; G.Y. Panasyuk and K.L. Yerkes, Phys. Rev. E, 2013.

3. Outline We consider the HEAT CURRENT between thermal reservoirs consisting of finite number of modes mediated by a quantum system - expression for the quasi-static heat current; - time dependencies of mode temperatures; - validity of Fourier’s law for a chain of finite identical subsystems

4. Heat Transport T1 > T2 T2 QS Δ Heat current

5. Total Hamiltonian ν

6. Drude-Ullersma Model Δ Δ is the (finite) mode spacing in a heat bath -“Heisenberg time” is a characteristic (microscopic) time for establishing the steady state current

7. Energy Balance Rate of energy change in the νth heat reservoir: where - is the work the quantum system performs on the νth bath per unit of time or the heat current

8. where Ensemble Averaging , we allow (slow) temperature variations for each mode of both reservoirs:.

9. Finite Size Heat Current = ν = 1, 2, = k

10. G-factor Right figure: = 10-4 Left figure: = 10-3, - Each figure: comparison between accurate and approximate G t-dependence of the G-factor for = 0.1, = 0.8, and = 0.5. tn= 2πn/Δ, integer n > 0

11. Mode Temperatures ) , = , [] where

12. Temporal Relaxation of the Mode Temperatures = Temperature relaxation: = 0.1, = 0.8, = 0.0001. (a) , (b) , (c) , and (d) tn= 2πn/Δ, integer n > 0

13. Temporal Relaxation of the Thermal Current Thermal current at = 0.1, = 0.8, = , and . tn= 2πn/Δ, integer n > 0

14. Fourier’s law d , n = 1, 2, … P x = n d , in and ; averaging over modes c ,(t) thermal equilibrium and Fourier’s law break

15. Conclusions • We considered size effects onheat transport between (finite size) thermal • reservoirs mediated by a quantum particle. As a result of this study, • - We derived, solved, and analyzed equations for the quasi-static energy current and • reservoirs’ mode temperatures; - as is shown, temporal behavior of the reservoirs’ mode temperatures demonstrates peculiarities at tn = 2πn/Δ, where an integer n > 0; - quasi-static heat current vanishes non-monotonically demonstrating peculiarities at the same time moments tn. - Fourier’s law in a chain of identical subsystems (nanoparticles) is correct only on a short time scale but cannot be validated for a time scale t ~ 1/Δ

16. We acknowledge support from: • Air Force Office of Scientific Research • Thank you!