Sang Wook Kim (Pusan N. Univ) Collaborators Simone de Liberato - - PowerPoint PPT Presentation

Sang Wook Kim (Pusan N. Univ)

Collaborators

Jung Jun Park (Singapore Natl. U.) Kang-Hwan Kim (KAIST) Takahiro Sagawa (Univ. of Tokyo) Simone de Liberato (Univ. of Southampton) Masahito Ueda (Univ. of Tokyo) Hee Jun Jeon (PNU)

Maxwell’s Demon

Now let us suppose … that a being, who can see the individual molecules,

• pens and closes this hole, so as to

allow only the swifter molecules to pass from A to B, and only the slower ones to pass from B to A. He will thus, without expenditure of work, raise the temperature of B and lower that of A, in contradiction to the 2nd law of thermodynamics.

• J. C. Maxwell (1871)

Szilard’s engine (1929)

T



 pdV W

Flow of entropy

ln 2 S0 S0

increase

ln 2

decrease

ln 2 ln 2 S0 - ln 2 ln 2 S0

ln 2

Information heat engine

) : ( M S kTI F W    

Sagawa & Ueda, PRL (2008)

S

M

Toyabe, Sagawa, Ueda, Muneyuki & Sano, Nature Phys. (2010)

Quantum dynamical demon?

Thermodynamic work in Q-world

n n n

E H   





n n nP

E U Z e P

n

E n  



in equilibrium



 

n n n n n

dE P dP E dU ) (

partition function

dW dU TdS  

 

 

n n n n n n

dE P dU dP E



 

n n n

P P k S ln



 

n n ndE

P dW

q-thermodynamic work Kieu, PRL (2004)

T





n n ndP

E dQ

q-thermodynamic heat

Inserting a wall



 

n n ndE

P dW

Inserting a wall is considered as an isothermal process.

Adiabatic process for inserting a wall

 

n n ndP

E dQ 

n

dP Z e P

n

E n  

 T T 

The final state should be in non-equilibrium, so that the irreversible process inevitably

• ccurs in isothermal expansion.

?  T  

n n ndP

E dQ 

n

dP

T should be changed quantum adiabatic

Q-work in an isothermal process

 

) ( ln ) ( ln ln

1 2

2 1 2 1

X Z X Z kT dE X E E Z kT dE Z e W

n X X n n n X X n En

        

 



Helmholtz free energy difference (Note) Due to isothermal process, we don’t have to consider a full density matrix.

Thermodynamic process

T T Insertion Measurement Expansion T Removal

Q-work of q-Szilard engine





            

N m m m m rem ins tot

f f f kT W W W W

* exp

ln

) ( ) (

* m eq m eq m m

l Z l Z f  1

1 * 



 N m m

f







N n m eq n m eq

l Z l Z ) ( ) (

* m

f

SWK, Sagawa, De Liberato & Ueda, PRL (2011)

m

f

Single particle q-Szilard engine

K.-H. Kim & SWK, J. Korean Phys. Soc. (2012)

T k W

B tot

The 3rd law of thermodynamics

as   T S

) 1 ln( ) 1 ( ln p p p p T k W

B tot

     p p  1

Two particle q-Szilard engine I





         

N m m m m tot

f f f kT W

*

ln

2 L l  1

* 0 

f 1

* 2 

f

1 * 1

f f  f

2

f f 

1

f

Two particle q-Szilard engine II

           T kT W T kT W

tot tot

for 2 ln for 3 ln 3 2

Bosons

     T kT W T W

tot tot

for 2 ln for

Fermions (spinless)

0 ln

2 f kTf Wtot  

   T f as 4 1

and are prohibited due to Pauli exclusion principle in the low T. Both

as 3 1   T f

 

2 ln 2 2 1 kT Wclassical 

(cf) classical work

Two particle q-Szilard engine

Bosons Fermions

kT Wtot

2 ln

Classical

SWK, Sagawa, De Liberato & Ueda, PRL (2011)

Irreversible process I

1

x L

1

x

1

x

1

x

1

x L

Time-forward Time-backward

Inherently irreversible!

(cf) Murashita, Funo & Ueda, PRE (2014) Ashida, Funo, Murashita & Ueda, PRE (2014)

T

Irreversible process II

              

 

  N m m m N m m m tot

f f f f kT W

*

ln ln

Horowitz & Parrond, NJP (2011)

(Option 1) Make the protocol reversible

(Option 2) Optimize work via math





         

N m m m m tot

f f f kT W

*

ln

}) { , (

m tot

x l W }) { , (

,

 

m tot x l

x l W

m

Optimal condition:

1 1

m m m m

Z Z Z x Z x     

( ) ( )

m m p m p

F x F x    

tot

W

Physical meaning of optimal condition

1

x

1

x L

1

x L

1

x

1

x

Time-forward

Time-backward

) ( 1

1 x

F







N p m p p p

x F f x F

1 * 1

) ( ) (

) ( 1

* 1 x

f ) ( 1

* 2 x

f ) ( 1

* 3 x

f

) ( 1

* 0 x

f

) ( (cf)   

right left

F F x F



Numerical check I 3 N 

Boson

/ 1

B

k T E 

1

F

1

W

p p

F  

Numerical check II

1)

(

p p

F x  T 

E kT  5E kT 

1

x

1

x L

1

x

1

x

Time-backward

) ( ) (

1 * 1

 

 N p m p p p

x F f x F

Casimir force

Why?

( ) ( )

m m p m p

F x F x   

The optimal condition of the q-SZE with intrinsic irreversibility is achieved once the time-forward force is equivalent to the time back-ward force:

Remark and a new question





         

N m m m m tot

f f f kT W

*

ln

In fact, this equation can be derived from fully classical

• consideration. The point is that the above expression is

mainly ascribed to multi-particle nature of SZE.

K.-H. Kim & SWK, PRE (2011)

Is work extractable from a heat engine by using purely quantum mechanical information? If yes, what is its mathematical formula?

Quantum information demon?

Our Set-up

   

) ( ) ( ) ( ) ( ) ( ) (

exp exp

i R i R i S i S i AB i

Z H Z H         

• Previous works

Oppenheim, Horodeki, Horodeki & Horodeki, PRL (2002) Zurek, PRA (2003) Rio, Aberg, Renner, Dahlsten & Vedral, Nature (2011) Funo, Watanabe & Ueda, PRA (2013) Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

A B

Thermodynamic process

Stage 3 (feedback control)





) 3 ( ) 2 ( ) 3 ( ) (

U U

f

 

Stage 2 (POVM)

 

    

 k BSR A k k k A k A

k k p U U

) 2 ( ) 2 ( ) 1 ( ) 2 ( ) 2 (

      

k A k A k BSR k A k A k

U U U U p      

  ) 2 ( ) 1 ( ) 2 ( A ) 2 ( ) 2 ( ) 1 ( ) 2 (

tr tr   





) 1 ( ) ( ) 1 ( ) 1 (

U U

i

 

Stage 1 (unitary evolution)

) 1 ( ) 1 ( SR AB

U I U  

This can also describe isothermal process.

Entropy consideration

 

   ln tr ) (   S

von Neumann entropy

   

) ( ) 2 ( f SR k k SR k

S S p   



(1) concavity

   

) 2 ( ) (

  S S

i 

(2) POVM

   

) ( can ) ( ) (

ln tr

f SR f SR f SR

S    

(3) Klein inequality

 

kTΔI ΔS ΔS kT ΔF W

B A S

        

) ( ) ( ) 2 ( ) 2 (

: :

i i

B A I B A I I    ) : ( ) ( ) ( ) ( ) : ( A B I S S S B A I

AB B A

      

mutual information Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

Mutual information and Discord

) | ( ) ( ) , ( ) ( ) ( ) : ( A B H B H A B H A H B H A B J     

classical mutual information



 

A

A p A p A H ) ( ln ) ( ) (



 

B A

A B p B A p A B H

,

) | ( ln ) , ( ) | (

Shanon information conditional entropy

??? ) | ( ) ( ) : ( ~ A B S B S A B J  

quantum analogue

 

 

 

i A B B

S S A B J

i A

  



  ) ( ) : ( ~

 

 

) : ( ~

• B)

: I(A min ) | ( A B J A B

i A



 

quantum discord Ollivier & Zurek, PRL (2002)

Final formula

 

 

) ( ) ( i i B A S

A B kT J kT S S kT F W           

Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

Szilard engine containing a heteronuclear diatomic molecule

semi-permeable wall

 

Von Neumann, A. Peres, V. Vedral, L. B. Levitin

 

can ) 1 (

2 1

R S S AB

R R L L        

 

 

B A B A AB

       2 1

 

can ) 12 (

2 1

R S S

R R L L           

 

 

can ) (

2 1 2 1

R S S f

R R L L           

 

R R L L U        

 

2 1

) 2 (

 

,      

A A k A

 

 

) ( ) ( i i B A S

A B kT J kT S S kT F W           

   

B A

S S  J

2 ln 2 2 ln 2 ln ) ( ) ( ) ( ) : (

) ( ) ( ) ( ) ( ) (

      

i AB i B i A i i

S S S B A I   

 

 

2 ln 2 ln 2 ln 2 ) : ( ~

• B)

: I(A min ) | (

) ( ) (

   



A B J A B

i A

i i



 

S

F

2 ln 2 ln 2 1 2 ln 2 1 kT kT kT W   

Thermodynamic 2nd law

 

 

can ) (

2 1 2 1

R S S f

R R L L           

 

can ) 1 (

2 1

R S S AB

R R L L        

 

   

2 ln

) 1 ( ) (

    S S

f

To prepare the initial state of memory, we need to pay kTln2.

Park, K.-H. Kim, Sagawa & SWK, PRL (2013)

References

• S. W. Kim and M.-S. Choi,

Decoherence driven quantum transport (SZE in atomic systems)

• Phys. Rev. Lett. 95, 226802 (2005)
• S. W. Kim, T. Sagawa, S. De Liberato, and M. Ueda

Quantum Szilard engine Phys. Rev. Lett. 106, 070401 (2011) Parrondo & Horowitz, Physics 4, 12 (2011) “Maxwell’s Demon in the Quantum World” K.-H. Kim and S. W. Kim Information from time-forward and time-backward processes in Szilard engines

• Phys. Rev. E 84, 102101 (2011)

K.-H. Kim and S. W. Kim Szilard's Information Heat Engines in the Deep Quantum Regime

• J. Korean Phys. Soc. 61, 1187 (2012)
• J. J. Park, K.-H. Kim T. Sagawa and S. W. Kim

Heat engine driven by purely quantum information Phys. Rev. Lett. 111, 230402 (2013) Phys.org, 18 Dec 2013 “Maxwell's demon can use quantum information to generate work”

• H. J. Jeon and S. W. Kim

Optimal work of quantum Szilard engine with isothermal processes arXiv:1401.1685

Summary

• We have studied quantum dynamical SZE.
• We have found optimal condition of quantum

SZE with irreversibility

• We have devised Maxwell demon utilizing

quantum information (q-discord)

Non-equilibrium thermodynamics

• Jarzynski equality (1997)

F W

e e

   



  m equilibriu non

Kawai, Parrando & Van den Broeck, PRL (2007)



  ~ ln kT F W W

diss

   

• The dissipative work for

non-equilibrium process

• The dissipative work for

non-equilibrium process with filtering or feedback control

F W p p kT kT

B A

D D

    ln ~ ln



 

t t 

1

t t 

A

D

B

D

Parrando,Van den Broeck & Kawai, New J. Phys. (2009)

Physical meaning of –Σfln(f/f*)

F W p p kT kT

B A

D D

    ln ~ ln



  quasi-static process cyclic engine

 

         

m m m m m m m tot

f f f kT W f W

*

ln

* m D m D

f p f p

B A

 

forward filtering backward filtering

m

f

* m

f

Remark I

 



    

A m

X S X S dX dX e m Q

   



, 2 * 1 2 1

Im ) (

Adiabatic Q-pump

) sin( ) ( , sin ) (

2 2 1 1

          t X t X t X t X

 

) ' ( ) ' , ( ) ( ) , ' ( ' 2 E f E E t E f E E t dEdE h e I

R L  

 



                                         ' ' ' ' ' ' ' '

11 10 11 10 01 00 01 00 11 10 11 10 01 00 01 00

r r t t r r t t t t r r t t r r S

 

  

00

t

10

t

20

t

01

t

11

t

21

t

) (t V

Classical/Quantum ratchet

Classical ratchet http://www.uoregon.edu/~linke/