Geometric Quantum Pump for Fermion Transport　

Hisao Hayakawa　 (YITP, Kyoto University)

based on collaboration with T. Yuge, T. Sagawa, and A. Sugita

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44 Symposium on Mathematical Physics"New Developments in the Theory of Open Quantum Systems"Toruń, June 20-24, 2012 (June 24)

Collaborators

Tetsuro Yuge (YITP->Osaka Univ.)

Ayumu Sugita (Osaka City Univ.)

Takahiro Sagawa (YITP)

& Ryosuke Yoshii (YITP)

I would like to thank all these collaborators on this problem.

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Introduction Geometric Pump for Fermion Transport

◦Setup◦Main Results◦Special Cases & Example

Application to Entropy Production Discussion Conclusion

Contents

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Introduction Geometric Pump for Fermion Transport

◦Setup◦Main Results◦Special Cases & Example

Application to Entropy Production DIscussion Conclusion

Contents

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 In mesoscopic systems, a current can exist even at zero bias. This effect is called the quantum pumping.

Introduction

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Nano-machine to extract work

Adiabatic quantum pump◦ Geometric effect is important (P. W. Brower,

PRB58, 10135 (1998)).◦ Control of system parameters

Can we get the pump effect by controlling reservoir parameters?

Previous studies and question

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Introduction Geometric Pump for Fermion Transport

◦Setup◦Main Results◦Special Cases & Example

Application to Entropy Production Discussion Conclusion

Contents

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Projection measurement

Counting: ◦ Number of spinless electrons transfer from L to R

Statistics & cumulant generating function

Full counting statistics (FCS)

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We assume that the total Hamilitonian satisfies von-Neumann equation.

We calculate the modified von-Neumann equation via the counting field:

Ref.

How can we compute FCS?9/24

Quantum Master equation

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χ

Adiabatic modulation

Control parameters

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Net current in an adiabatic cycle

where

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Based on FCS Born-Markov approximation + rotational wave approximation (RWA), we obtain

N non-interacting quantum dots

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Explicit calculation on double quantum dots

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Results of double dots

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Results 2: no path dependence

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Introduction Geometric Pump for Fermion Transport

◦Setup◦Main Results◦Special Cases & Example

Application to Entropy Production Discussion Conclusion

Contents

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The method we adopted can be used for the calculation of any other quantities.

We can discuss the path dependence of the nonequilibrium entropy production.

Namely, the entropy is a geometric quantity under a nonequilibrium situation.

Note that the entropy production is a non-conserved quantity.

See Sagawa and HH, PRE 84, 051110 (2011).

Application to entropy production

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heat

Results for entropy production

Path-dependencequasi-static process

parameters space

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Introduction Geometric Pump for Fermion Transport

◦Setup◦Main Results◦Special Cases & Example

Application to Entropy Production Discussion Conclusion

Contents

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Effects of spins and many-body interactions◦ We have already calculated Kondo problem (R.

Yoshii and HH, in preparation).◦ The many-body effect can be absorbed via

Schrieffer-Wolff transform. Without the potential scattering term, the

result is unchanged. If we introduce the term, the symmetry of

evolution matrix is changed. So there is possibility to have the geometric effect.

Discussion 1

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So far, we assume that particles are Fermions.

However, our analysis is based on RWA (quasi-classical) and the result contains only distribution function of reservoirs.

We expect that the geometric effect can appear for Bosons.

See Jie Ren et al., PRL 104, 170601 (2010).

Discussion 2 ; Effect of statistics

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Introduction Geometric Pump for Fermion Transport

◦Setup◦Main Results◦Special Cases & Example

Application to Entropy Production Discussion Conclusion

Contents

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We have analyzed a quantum pump effect on Fermion transport.

We have found that spinless Fermions without interactions do not have any geometric effect if we control reservoir parameters.

We confirm that there exist geometric effects for the control of system parameters.

Such an idea can be used for entropy production.◦ Geometric effects are important.

We are now calculating the Kondo problem.

Summary

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Thank you for your attention.