? = E Eigenstate properties p e 1 i i Z essential 2. - - PowerPoint PPT Presentation

Roland Ketzmerick

Statistical physics of time-periodic systems + Time-Periodic Driving

• 1. Canonical ensemble

Asymptotic probability to be in state i

?

−β

=

i

1 Z E i

p e

• 2. Ideal quantum gas

Bose condensation? Bose selection

• f multiple states

(odd number) Eigenstate properties essential

Equilibrium vs. Non-Equilibrium Steady States

heat bath system heat bath system driving q p

X X

p q

Equilibrium vs. Non-Equilibrium Steady States

heat bath system q p p q heat bath system driving

• Reduced density operator
• Steady state:

(diagonal in eigenstates)

• Master equation:

Weak coupling to heat bath

∞ =

ρ = ∑

i M i 1

i i

p

=

= − =

∑

j j j i M i i i j 1

p (R p R p ) 

i j

R

γ

= + γ +

system coupl bath

1

H H H H



j i

Blümel et al.; PRA 1991 Kohler, Dittrich, Hänggi; PRE 1997 Breuer et al.; PRE 2000 Kohn; J. Stat. Phys. 2001 Hone, RK, Kohn; PRE 2009 Langemeyer, Holthaus; PRE 2014

• Floquet-Born-Markov Approach

heat bath system driving

Rates

τ

= γ ⋅ ⋅ −

2 2 coupl j i i j

R H g (E j E i )

coupling strength coupling matrix element bath correlation function

= ∑

j j i m i m

R R

− ω τ

= γ ε − ε − ω

∫

2 T 2 i t 1 cou m m i i j i j l j p T

R dt e u (t) H u (t) ( m g ) 

heat bath system driving

ω m

Quartic Oscillator:

= +

2 4

p H(x,p,t) x 2

= ω = A 0.2 0.83

x p

+ ω xcos( A t)

x p

π = ω 2 T

= t 0,T,2T,

Semiclassical Eigenfunction Hypothesis

Percival 1973 Berry 1977

regular chaotic regular

⇒

i j

R

Floquet-Born-Markov Approach

Occupation Probabilities: Quartic Oscillator

i

p < >

i

E

RK, Wustmann PRE 2010

( )

=

− =

∑

i i j j j j i M 1

R p R p

i j

R

Occupation Probabilities: Kicked Rotor

= K 2.9

i j

R

RK, Wustmann PRE 2010 i

p < >

i

E

< >

i

E

i

p

Occupation Probabilities: Kicked Rotor

RK, Wustmann PRE 2010

= K 2.35 ⇒ Eigenstate properties essential

Many-body + periodic driving (no heat bath)

… on to many-body systems …

N=700 Experiment: Tomkovic, Müssel, Oberthaler Theory: Schlagheck, Löck, R.K.

• Poincare-Birkhoff scenario
• Spreading in regular vs. chaotic region

Vorberg, Wustmann, R.K., Eckardt, PRL 2013

Generalization of Bose-Einstein Condensation in Non-Equilibrium Steady States

Bose Selection Which states? How many?

N

i

n

Bose-Einstein Condensation

i

n N

ground state

• dd

André Eckardt

• Reduced density operator
• Steady state:

(diagonal in eigenstates)

• Master equation:

Weak coupling to heat bath

∞ =

ρ = ∑

i M i 1

i i

p

=

= − =

∑

j j j i M i i i j 1

p (R p R p ) 

i j

R

γ

= + γ +

system coupl bath

1

H H H H



j i

Ideal Bose gas: +

j j i i

R n (n 1)

( )

=

= − + =

∑

ji

n i i M j j j n , i 1 n j i

p R p R p n (n 1)

  



=

1 M i j

(n , , n , , n , ,n n )    

quantum statistics

j i

solution: exists, unique

Occupation of single particle states

(mean-field approximation)

=

  = + − + =  

∑

i i M i i i j j 1 j j j

n R n (n 1) R n (n 1) 

Asymptotic theory for :

→ ∞ n

Which states selected?

• existence and uniqueness (H. Schomerus)
• algorithm
• MS=1 (Bose condensation)

iff ฀ „ground state“ k with for all i

• therwise MS ≥ 3

>

i ki k

R R

Odd number MS of states selected

Zero determinant of antisymmetric matrix when dimension odd.

−

i ji j

R R

i

n

mean

• ccupation

Random rate matrix:

from exponential distribution

i j

R

= n N M n

= M 10 = M 200

+ quantum jump Monte-Carlo simulation ─ mean field approximation

Examples for Bose selection

Periodically driven quartic oscillator Kicked rotor

K=10

Examples for Bose selection

driving strength always MS ≤ 3 ! Tight-Binding Chain (10 sites): periodically driven

i j

R

N=10000

Examples for Bose selection

↑

• nly bath 1

↑

• nly bath 2

relative bath coupling Tight-Binding Chain (10 sites): 2 baths

⇒ Quantum switch for heat

N=10000

+ Time-Periodic Driving

• 1. Canonical ensemble

Asymptotic probability to be in state i

?

−β

=

i

1 Z E i

p e

• 2. Ideal quantum gas

Bose condensation? Bose selection

• f multiple states

(odd number) Eigenstate properties essential Open directions: ⇒ anomalous long-range order in 1D (A. Schnell) ⇒ particle reservoir gives even number (D. Vorberg)

• interaction
• experiment (cold atom, quantum dot)
• connection to lasers (J. Wiersig)