Jie Ren

1. Geometric Phase Effect in Heat Transport: additional heat pumping and its impact on fluctuation theorems Jie Ren NUS Graduate School for Integrative Science & Engineering Department of Physics & CCSE Collaborate with: Prof. P. Hanggi and Prof. B. Li Physical Review Letters 104, 170601 (2010)

2. Phononics • Art of heat control at low dimensional nano-systems. • Various functional thermal devices: Theory: thermal diode, thermal transistor, thermal logic gate, thermal memory… Experiments: nanotubethermal rectifier, quantum dot diode, nanotube phonon waveguide… (based on temperature bias)

3. Molecular heat pumps: control heat flux againstthermal gradients For example: modulating energy levels in a molecular model, (Segal et al, PRE 2006; PRL,2008) time varying external field in spin system, electronic nano-devices. (Marathe et al, PRE 2007; Rey et al, PRB 2007; Arrachea et al, PRB,2007) oscillating temperatures… (Li et al, EPL 2008; PRE 2009; Zhan et al, PRE 2009; Ren et al, PRE,2010)

4. Emergence and control of heat current from strict zero thermal bias Frenkel-Kontorova Model J. Ren and B. Li, PRE 81, 021111 (2010)

5. Three conditions for the emergence of heat current at zero thermal bias: • nonequilibrium source; • oscillating baths although isothermal • 2) symmetry breaking; • two asymmetric segments defining directionality • 3) nonlinearity. • set V=0,the heat current vanishes However, if the correlation between two baths is introduced, symmetry breaking is sufficient to create heat current. Reminiscent of the quantum coherence

6. Molecular heat pumps:control heat flux againstthermal gradients modulating energy levels (Segal et al, PRE 2006; PRL,2008) time varying external (Rey et al, PRB 2007; Arrachea et al, PRB,2007) oscillating temperatures… (Li et al, EPL 2008; PRE 2009; Zhan et al, PRE 2009; Ren et al, PRE,2010) Key: Time-dependent manipulation! One may speculate: anynon-vanishing geometric phase effect?

7. Introduction: Geometric phase (1) Geometric phase: anholonomy angle An example in elementary geometry parallel transport of a vector along a loop on a sphere compass needle traveling on the earth The rotation angle is connected to the intrinsic curvature of the sphere. http://www.mi.infm.it/manini/berryphase.html

8. Introduction: Geometric phase (2) Geometric phase: adiabatic Berry phase • Transport a closed path C in parameter space: • The final state differs from the initial one only by a phase factor • Dynamic phase • Berry phase Berry considered a Hamiltonian which depends on a set of parameters M. V. Berry (1984)

9. Outline • Introduction of the molecular model • Results in our work • Geometric phase induced heat pumping • Fractional quantized phonon response • Its impact on the Fluctuation Theorems 3. Conclusions J. Ren, P. Hanggi, B. Li, Phys. Rev. Lett. 104, 170601 (2010)

10. Introduction of the molecular model (1) Nonlinear molecular junction Harmonic thermal bath: Bilinear system-bath interaction: Single mode assumed: (Kubo oscillator) two-level, N=2; strong nonlinearity fast dephasing; weak system-bath coupling

11. Introduction of the molecular model (2) Master Equation: Activation rate: Relaxation rate: The steady-state heat flux at the R contact: anharmonic rectification D. Segal, PRB 73, 205415 (2006)

12. Geometric phase contribution in generating function (1) The generating function with phonon counting fields 1 0 The probability that having q phonons transferred into bath R, within time t, while the system is dwelling on the low “0” (high “1”) energy at time t. The time evolution of the system can be described as:

13. Geometric phase contribution in generating function (2) the time evolution of generating functions: where Denote Introduce the characteristic function: Cumulants generating function:

14. Geometric phase contribution in generating function (3) Time-independent The long time behavior is governed by the eignmode whose eigenvalue possesses the smallest real part. the same as the previous solution.

15. Geometric phase contribution in generating functions (4) an analog of the Berry phase only depends on thegeometry of the modulation contour in the parameter space u. two-parameter modulation Stokes theorem an analog of the Berry curvature Total heat flux has two contributions: Could modulate arbitrary two parameters:

16. Geometric Phase induced Heat Pumping Modulating two bath temperatures TL(t), TR(t): Total heat flux has two contributions: The contor map of

17. Fractional Quantized Phonon Response (1) The contor map of Robust fractional quantized phonon response: Red: transitions related tobath L. Blue: transitions related tobath R. Black: stay at original energy level. Gray: transitions related to either L and R pump maximally on average ¼ phonon per cycle.

18. Fractional Quantized Phonon Response (2) Red: transitions related tobath L. Blue: transitions related tobath R. Black: stay at original energy level. Gray: transitions related to either L and R

19. Fluctuation Theorem (1) No parameter modulations: where: Gallavotti-Cohen (GC) symmetry: Detailed balance condition Recalling: Heat-Flux Fluctuation Theorem: inverse Fourier transform finite probability of at least c phonons transporting against the thermal bias

20. Fluctuation Theorem (2) The impact of Geometric Phase effect on the validity of FT For the time-modulated system, the GC symmetry ceases to hold when (I) (II) GC symmetry at every instant is NOT guaranteed (III) NO violation of the FT, no matter how are modulated.

21. Conclusions • Through cyclic two-parameter modulations, we find a geometric phase induced heat pumping, additional to the usual dynamic heat flux. • This geometric contribution exhibits a robust fractional quantized phonon response. • A fluctuation theorem of the Gallavotti-Cohen type for nonlinear quantum heat transfer is identified for modulation-free cases. • Geometric phase contribution causes a breakdown of the heat-flux fluctuation theorem, which can be restored only if • the geometric phase contribution vanishes and if • the cyclic protocol preserves the detailed balance symmetry. J. Ren, P. Hanggi, B. Li, Phys. Rev. Lett. 104, 170601 (2010)

22. Thanks!