Landau-Zener Transitions with quantum noise Valery Pokrovsky - - PowerPoint PPT Presentation

August 2006 Dresden, 2006

Landau-Zener Transitions with quantum noise

Valery Pokrovsky

Department of Physics, Texas A&M University and Институт теоретической физики им. Л.Д. Ландау Participants:

Nikolai A. Sinitsyn

Department of Physics and Astronomy UT Austin

Stefan Scheidl

Institut für Theoretishe Physik, Universität zu Köln

Bogdan Dobrescu

Texas A&M Univesity

August 2006 Dresden, 2006

Outline

• Introduction and motivation
• Landau-Zener problem for 2-level crossing
• Fast classical noise in 2-level systems
• Noise and regular transitions work together
• Quantum noise and its characterization
• Transitions due to quantum noise in the LZ system
• Production of molecules from atomic Fermi-gas at

Feshbach resonance

• Conclusions

August 2006 Dresden, 2006

Introduction

LZ theory

energy Adiabatic levels 1 1 2 2 Diabatic levels Avoided level crossing (Wigner-Neumann theorem) Schrödinger equations

1 1 1 2 * 2 1 2 2

( ) ( ) ia E t a a ia a E t a = + ∆ = ∆ + & &

2 1

( ) ( ) ( ); 1 E t E t t − = Ω = h

( ) t t Ω = Ω &

time

August 2006 Dresden, 2006

Adiabatic levels:

2 2 1 2 1 2

2 2 E E E E E ± + − ⎛ ⎞ = ± + ∆ ⎜ ⎟ ⎝ ⎠

Center-of mass energy = 0

2 1

/2 E E t =− =Ω &

LZ parameter:

γ ∆ = Ω & h

1 γ ฀ 1 γ ฀

B z

g B µ Ω = & & h

August 2006 Dresden, 2006

LZ transition matrix

* *

U α β β α ⎛ ⎞ = ⎜ ⎟ − ⎝ ⎠

2 2

| | | | 1 α β + =

Amplitude to stay at the same d-level

2

e πγ α

−

=

Amplitude of transition

2 2

2 e x p 2 4 ( ) i i π γ π π β γ γ ⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠ = − Γ −

LZ transition time:

L Z

τ ∆ = Ω &

1 1 2 2

LZ

τ Condition of validity:

/

LZ sat

τ τ = Ω Ω & && ฀

1 / 2

m a x ,

L Z

τ

−

∆ ⎛ ⎞ = Ω ⎜ ⎟ Ω ⎝ ⎠ & &

August 2006 Dresden, 2006

Nanomagnets: Brief description

August 2006 Dresden, 2006

Spin reversal in nanomagnets

• W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999)

August 2006 Dresden, 2006

Controllable switch between states for quantum computing: The noise introduces mistakes to the switch work.

Transverse noise Longitudinal noise creates decoherence

August 2006 Dresden, 2006

Landau-Zener tunneling in noisy environment

• V. Pokrovsky and N. Sinitsyn, Phys. Rev. B 67, 144303, 2003

Classical fast noise in 2-level system

ˆ ˆ ;

tot reg reg

z t x = + = Ω + ∆ b b η b &

( ) t η

• - Gaussian noise

( ) ( )

i k ik n

t t t t f η η τ ⎛ ⎞ ′ − ′ = ⎜ ⎟ ⎝ ⎠ Noise is fast if

1/2 n

τ

−

Ω & ฀

1/

n

ω τ ∆ =

ω I

August 2006 Dresden, 2006

Density matrix:

1 ˆ( ) ( ) 2 t I t ρ = + ⋅ g s ( ) t − g

Bloch vector. It obeys Bloch equation:

tot

= − × g b g &

( )

( )

1 2

1 1 2 2

z

g n n n n

↑ ↓

= − ≡ −

Difference of populations

x y

g g ig

± =

±

Coherence amplitude Integral of motion:

2

const = g

August 2006 Dresden, 2006

t

( ) t t Ω = Ω &

τacc

Transitions produced by noise

It produces transitions until

( ) 1/

n

t t τ Ω = Ω ≤ &

Accumulation time:

1

acc n n

τ τ τ = Ω ฀ & ( )

z

g t

is slowly varying

August 2006 Dresden, 2006

Transition is produced by a spectral component of noise whose frequency equal to its instantaneous value in the LZ 2-level system. t Ω

* 1 ( ) ( ) 1 t t

n n η η

Ω Ω

= − &

2 1 1 2

2 | | exp P π η

→

⎛ ⎞ ⎜ ⎟ = − ⎜ ⎟ Ω ⎝ ⎠ & h

Fermi golden rule is exact for gaussian fast noise!

Transition probability measures the spectrum of noise Transition probability for infinite time

August 2006 Dresden, 2006

Separation of times: noise is essential only beyond

LZ

τ

Regular and random field act together

2 2 2 2

2 | | 2 1 2

1 1 2 1 2 P e e

π η π∆ − − Ω Ω →

⎡ ⎤ ⎛ ⎞ ⎢ ⎥ = − − ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

& & h h

τLZ τacc τacc

August 2006 Dresden, 2006

It is possible to get P larger than ½ at faster sweep rate

• r stopping the process at some specific field.

Plot of transition probability vs. inverse frequency rate.

J=

2

2 | | π η ω &

β = Ω &

August 2006 Dresden, 2006

Graph representation Chronological time order

1

t′

1

t

2

t′

2

t

n

τ ฀ ( )

2 2

( , ) exp 2 i t t G t t ⎛ ⎞ ′ Ω − ′ ⎜ ⎟ = ⎜ ⎟ ⎝ ⎠ &

*( , )

G t t′

t′

t

† †

( ) ( ) ( ) ( ) t t t t η η η η ′ ′ =

4 times are close t

1

t

1

t′

2

t

2

t′

3

t

3

t′

n

τ ฀

1 2 1 1 1 2

t′ t′

t

1

t′

t

2

August 2006 Dresden, 2006

What was omitted: Debye-Waller factor

exp ( )

t z t

W i t dt η

′

⎡ ⎤ ′′ ′′ = − ⎢ ⎥ ⎣ ⎦

∫

n

t t τ ′ − ฀

if

1, W ≈

2 2

1

z n

η τ ฀

Diagonal noise leads to decoherence for a long time

( ) g± +∞ =

Decoherence time:

2

1

dec n z n

τ τ η τ = ฀

August 2006 Dresden, 2006

Quantum noise and its characterization

† †

( ) ( ') ( ') ( ) t t t t η η η η ≠

Model of noise: phonons

( )

† † int 1 2 2 1 ;

H u a a a a = +

†

1 ; u g b V η η η = + =

∑

k k k

† n

H b b ω = ∑

k k k k

( )

† † 2 1 1 2 2

( ) ; ( ) 2 t H a a a a t t Ω = − Ω = Ω &

August 2006 Dresden, 2006

( )

( )

2 1 † /

; 1

T

N d g N eω

ω ω ω ω

η η δ ω ω

−

= − = −

∫

k k

k

( ) ( )

2 † / †

1

T

N d g eω

ω ω ω ω ω

η η δ ω ω η η = + − =

∫

k k

k

† † ( )

( ) ( ') 2

i t t

d t t e

ω ω ω

ω η η η η π

′ − −

= ∫

Different time scales for induced and spontaneous transitions

1 ni

T τ

−

฀

1 ns D

τ ω − ฀

Noise is fast if

,

D

T ω Ω & ฀

August 2006 Dresden, 2006

( )

† † †

sign( ) ,

z z

ds t s dt η η η η η η

Ω Ω Ω Ω Ω Ω

⎡ ⎤ = − + ⎣ ⎦

Bloch equation for the fast quantum noise

( ) t t Ω = Ω = Ω &

( ) ( )

† † ( ) ( ) ( ) ( ) † † † ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )exp , sign( ) , exp

t z z t t t t t t t t t t t t t t t

s t s t dt dt t dt η η η η η η η η η η

′ ′ ′ ′ Ω Ω Ω Ω ′ ′ ′′ ′′ ′′ ′′ Ω Ω Ω Ω Ω Ω ′

⎛ ⎞ ′ = − + + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎛ ⎞ ⎡ ⎤ ′ ′ ′′ − + ⎜ ⎟ ⎣ ⎦ ⎝ ⎠

∫ ∫ ∫

Solution

Sz turns into zero in the limit of strong noise, but not exponentially

August 2006 Dresden, 2006

Essential graphs

t

1

t

1

t′

2

t

2

t′

3

t

3

t′

n

τ ฀

1 2 1 1 1 2

t′

Golden rule picture

t

Ω

1 1

†

η η

Ω Ω

2 2

†

η η

Ω Ω

August 2006 Dresden, 2006

Saturation of frequency τs

( )

1 acc n s

τ ωτ τ

−

= & ฀

Long time limit

† † †

,

z

s η η η η η η

∞ ∞ ∞ ∞ ∞ ∞

Ω Ω ∞ Ω Ω Ω Ω

⎡ ⎤ ⎣ ⎦ = − + Noise in thermal equilibrium t Ω

† † T

e η η η η

Ω Ω Ω Ω Ω

=

h

tanh 2

z

s T

∞ ∞

Ω = h

∞

Ω

August 2006 Dresden, 2006

Feshbach resonance

40K2 40K

August 2006 Dresden, 2006

Feshbach resonance

6Li2 6Li

August 2006 Dresden, 2006

Feshbach Resonance driven by sweeping magnetic field

B

6Li 6Li

6Li2

hf interaction Hamiltonian:

H H V = +

( )( ) ( )

† † †

( ) 2 H h t a a b b E c c ε µ µ ⎡ ⎤ = − − + + − ⎣ ⎦

∑ ∑

p p p p p q q q p q

( )

† ,

. . ; g V a b c h c V

+

= +

∑

p q p q p q

3 hf m

g a ε ฀

( ) h t ht = &

August 2006 Dresden, 2006

Suggestions of LZ probability for the molecule production

( ) ( , ) (0)

mol LZ a

N t P t N α γ =

2 2 2 3 2 2 hf m

g n na ε γ = Ω Ω ฀ & & h h

? α =

• combinatorial factor.

1/ 2 α =

August 2006 Dresden, 2006

Perturbation theory for molecular production

• B. Dobrescu and VP, Phys. Lett. A 350, 154 (2006)

Keldysh technique for time-dependent field

+

• (

)

†

( ; , ) ( , ) ( , )

c

t t i T a t a t

αβ α β

′ ′ = − p p p A

( )

†

( ; , ) ( , ) ( , )

c

t t i T b t b t

αβ α β

′ ′ = − p p p B

Green functions

( )

†

( ; , ) ( , ) ( , )

c

t t i T c t c t

αβ α β

′ ′ = − p p p C

, α β = ±

Number of molecules

(Fermi sphere)

( ) ( ; , )

m

N t i t t

∈

=

∑

p

p

+,-

C

August 2006 Dresden, 2006

Interaction representation

( )

(0) , ( , , )

( )exp ( )

t F t

i t t i h t dt θ ε ε ε µ

′ + −

⎡ ⎤ ′ = − − − ⎢ ⎥ ⎣ ⎦

∫

p p

p h A

( )

(0) , ( , , )

( )exp ( )

t F t

i t t i h t dt θ ε ε ε µ

′ − +

⎡ ⎤ ′ = − − − − ⎢ ⎥ ⎣ ⎦

∫

p p

p h A

( )

(0) (0)

( , , ) 0; ( , , ) exp ( ) 2 ( )

c

i t t t t t t ε µ ⎡ ⎤ ′ ′ ′ = = − − ⎢ ⎥ ⎣ ⎦ p p p h

+,-

• ,+

C C

( , , ) ( , , ) t t t t

α β α β

′ ′ = p p

, ,

B A

( , , ) ( ) ( , , ) ( ) ( , , ) ( , , ) ( ) ( , , ) ( ) ( , , ) t t t t t t t t t t t t t t t t t t t t θ θ θ θ

+ + − + + − − − + − − +

′ ′ ′ ′ ′ = − + − ′ ′ ′ ′ ′ = − + − p p p p p p

, , , , , ,

G G G G G G

Vertices:

g V

August 2006 Dresden, 2006

Graphs Second order: Fourth order:

+

∞

−

∞

p

+ p q + p q α β

q

β

+

∞

p q

+ p q + p q α

−

∞

′ p ′ q

+ p q α′ β′

+

∞

−

∞

+

∞

−

∞

August 2006 Dresden, 2006

( )

2 4

( ) 88 ( ) 105

m a

N t N t = ∞ = Γ − Γ + Γ = −∞

Results

2 2 a

g n Γ = Ω & h

( )

S B

g B h t µ Ω = = h h

In first two orders of perturbation theory

( ) ( ) ( )

m a

N t f N t = ∞ = Γ = −∞

Compare to effective LZ theory: ( )

2 4

( ) 1 ( ) 2

m a

N t N t = ∞ = Γ − Γ + Γ = −∞

Collective effects suppress the transition probability During the transition process atoms can perform an exchange.

Renormalizability?

August 2006 Dresden, 2006

Conclusions

• LZ transition proceeds during time interval

LZ

/ τ = ∆ Ω &

• Fast transverse noise produces transitions during the time

interval . Condition is assumed

( )

1 acc n

τ τ

−

= Ω &

n acc

τ τ ฀

• Transitions at any moment of time are due to the spectral

component of noise which is in the resonance with the current frequency of the LZ system

• Classical noise tends to establish equal population of levels. Strong

quantum noise also leads to the equal population on the timescale between τacc and τsat and equilibrates the LZ system on longer time scale.

• Transformation of the Fermi gas of cooled alkali atoms into

molecules driven by sweeping magnetic field is a collective process which can not be described by the LZ formula. Collective effects suppress the transitions.

• The decoherence time due to longitudinal noise is

2

1

dec n z n

τ τ η τ = ฀

August 2006 Dresden, 2006

What else was done Transitions of spin S>1/2 in the sweeping magnetic field Bz

energy

;

s B

H gµ = − = bS b B

( )

, 0,

x z

b b t = b &

VLP and N.A. Sintsyn, Phys. Rev. B 69, 104414 (2004)

August 2006 Dresden, 2006

Correlations in the noisy LZ transitions

Response to a weak pulse signal t’ δh t δg

( ) ( ) ( ) ( )

t

g t g t g t h t dt

α α β β

δ δ

−∞

′ ′ ′ = ∫

( ) ( ) ( , ) g t g t K t t

α β αβ

′ ′ =

VLP and S, Scheidl, Phys. Rev. B 70, 014416 (2004) Solution of the Bethe-Salpeter equations

August 2006 Dresden, 2006

Solution of the Bloch equations

( )

1 ; 2

z z

g g g g i tg i g i η η η

+ − − + ± ± ±

= − = ±

Ω

& & m &

Integral equation for

z

g

2 2

( ' ) 2

1 ( ) ( ) . . ( ) 2

t i t t z z

g e t t c c g t dt η η

Ω − + − −∞

⎡ ⎤ ′ ′ ′ = − + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

∫

&

&

Complete initial decoherence Averaging procedure: ( ) ( ') ( ') ( ) ( ') ( ')

z z

t t g t t t g t η η η η

+ − + −

= Precision:

2

/

n acc

τ τ τ = Ω &