Landau-Zener Transitions with quantum noisecoqusy06/SLIDES/pokrovsky.pdf · August 2006 Dresden, 2006 Landau-Zener Transitions with quantum noise Valery Pokrovsky Department of Physics,

August 2006 Dresden, 2006

Landau-Zener Transitions with quantum noise

Valery PokrovskyDepartment of Physics, Texas A&M University

andИнститут теоретической физики им. Л.Д.

Ландау

Participants:

Nikolai A. SinitsynDepartment of Physics and Astronomy UT Austin

Stefan ScheidlInstitut für Theoretishe Physik, Universität zu Köln

Bogdan DobrescuTexas A&M Univesity


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


Introduction

LZ theory energy Adiabatic levels

1

12

2

Diabatic levels

Avoided level crossing (Wigner-Neumann theorem)

Schrödinger equations

1 1 1 2*

2 1 2 2

( )( )

ia E t a aia a E t a

= + ∆

= ∆ +

&

&

2 1( ) ( ) ( ); 1E t E t t− = Ω =h

( )t tΩ = Ω&

time


Adiabatic levels: 221 2 1 2

2 2E E E EE ±

+ −⎛ ⎞= ± + ∆⎜ ⎟⎝ ⎠

Center-of mass energy = 02 1 /2E E t=− =Ω&

LZ parameter: γ ∆=

Ω&h

1γ

1γ

B zg BµΩ =

&&

h


LZ transition matrix * *Uα ββ α

⎛ ⎞= ⎜ ⎟−⎝ ⎠

2 2| | | | 1α β+ =

Amplitude to stay at the same d-level 2

e πγα −=

Amplitude of transition

2

2

2 e x p2 4

( )

i

i

π γ ππβ

γ γ

⎛ ⎞− +⎜ ⎟

⎝ ⎠= −Γ −

LZ transition time: L Zτ ∆=

Ω&

1

12

2LZτ

Condition of validity: /LZ satτ τ = Ω Ω& &&

1 / 2m a x ,L Zτ −∆⎛ ⎞= Ω⎜ ⎟Ω⎝ ⎠&

&


Nanomagnets: Brief description


Spin reversal in nanomagnets

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


Controllable switch between statesfor quantum computing:

The noise introducesmistakes to the switchwork.

Transverse noise

Longitudinal noise creates decoherence


Landau-Zener tunneling in noisy environment

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

Classical fast noise in 2-level system

ˆˆ;tot reg reg z t x= + = Ω + ∆b b η b &

( )tη -- Gaussian noise ( ) ( )i k ikn

t tt t fη ητ

⎛ ⎞′−′ = ⎜ ⎟⎝ ⎠

Noise is fast if 1/ 2nτ

−Ω&

1/ nω τ∆ =ω

I


Density matrix: 1ˆ ( ) ( )2

t I tρ = + ⋅g s

( )t −g Bloch vector. It obeys Bloch equation:

tot= − ×g b g&

( ) ( )1 21 12 2zg n n n n↑ ↓= − ≡ − Difference of populations

x yg g ig± = ± Coherence amplitude

Integral of motion:2 const=g


t

( )t tΩ = Ω&

τacc

Transitions produced by noise

It produces transitions until ( ) 1/ nt t τΩ = Ω ≤&

Accumulation time:1

acc nn

τ ττ

=Ω&

( )zg t is slowly varying


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 ( ) ( ) 1t tn nη ηΩ Ω= −&

2

1 1 2

2 | |expP

π η→

⎛ ⎞⎜ ⎟= −⎜ ⎟Ω⎝ ⎠

&h

Fermi golden rule is exact for gaussian fast noise!

Transition probability measuresthe spectrum of noise

Transition probability for infinite time


Separation of times: noise is essential only beyond LZτ

Regular and random field act together

2 2

2 2

2 | | 2

1 21 1 2 12

P e eπ η π∆

− −Ω Ω

→

⎡ ⎤⎛ ⎞⎢ ⎥= − −⎜ ⎟

⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

& &h h

τLZ τaccτacc


It is possible to get P larger than ½ at faster sweep rateor stopping the process at some specific field.

Plot of transition probability vs. inverse frequency rate.

J=22 | |π η

ω&

β = Ω&


Graph representation

Chronological time order

1t′ 1t2t′ 2tnτ

( )2 2

( , ) exp2

i t tG t t

⎛ ⎞′Ω −′ ⎜ ⎟=

⎜ ⎟⎝ ⎠

&*( , )G t t′

t′ t † †( ) ( ) ( ) ( )t t t tη η η η′ ′=

4 times are close

t1t1t′ 2t2t′ 3t3t′

nτ1 2 1 1 12t′

t′ t1 t′ t2


What was omitted: Debye-Waller factor exp ( )t

zt

W i t dtη′

⎡ ⎤′′ ′′= −⎢ ⎥

⎣ ⎦∫

nt t τ′− if 1,W ≈ 2 2 1z nη τ

Diagonal noise leads to decoherence for a long time

( ) 0g± +∞ =

Decoherence time:2

1dec n

z n

τ τη τ

=


Quantum noise and its characterization† †( ) ( ') ( ') ( )t t t tη η η η≠

Model of noise: phonons

( )† †int 1 2 2 1 ;H u a a a a= +

† 1;u g bV

η η η= + = ∑ k kk

†nH b bω=∑ k k k

k

( )† †2 1 1 2 2

( ) ; ( )2tH a a a a t tΩ

= − Ω = Ω&


( ) ( )2 1† /; 1TN d g N eωω ω ω ωη η δ ω ω−

= − = −∫ k kk

( ) ( )2† / †1 TN d g eωω ω ω ω ωη η δ ω ω η η= + − =∫ k kk

† † ( )( ) ( ')2

i t tdt t e ωω ω

ωη η η ηπ

′− −= ∫

Different time scales for induced and spontaneous transitions

1ni Tτ − 1

ns Dτ ω−

Noise is fast if , DT ω Ω&


( )† † †sign( ) ,zz

ds t sdt

η η η η η ηΩ Ω Ω Ω Ω Ω⎡ ⎤= − +⎣ ⎦

Bloch equation for the fast quantum noise

( )t tΩ = Ω = Ω&

( )

( )0

0

† †0 ( ) ( ) ( ) ( )

† † †( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) exp ,

sign( ) , exp

t

z z t t t tt

t t

t t t t t tt 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


Essential graphs

t1t1t′ 2t2t′ 3t3t′

nτ1 2 1 1 12t′

Golden rule picture

t

Ω

1 1

†η ηΩ Ω

2 2

†η ηΩ Ω


Saturation of frequency

τs( ) 1

acc n sτ ωτ τ−= &

Long time limit

†

† †

,zs

η η

η η η η∞ ∞

∞ ∞ ∞ ∞

Ω Ω

∞Ω Ω Ω Ω

⎡ ⎤⎣ ⎦= −+

Noise in thermal equilibrium

t

Ω

† †Teη η η ηΩ

Ω Ω Ω Ω=h

tanh2zsT∞

∞

Ω=

h

∞Ω


Feshbach resonance 40K240K


Feshbach resonance 6Li26Li


Feshbach Resonance driven by sweeping magnetic field

B

6Li

6Li6Li2

hf interaction

Hamiltonian: 0H H V= +

( )( ) ( )† † †0 ( ) 2H h t a a b b E c cε µ µ⎡ ⎤= − − + + −⎣ ⎦∑ ∑p p p p p q q q

p q

( )†

,

. . ;gV a b c h cV += +∑ p q p q

p q

3hf mg aε

( )h t ht= &


Suggestions of LZ probability for the molecule production

( ) ( , )(0)

molLZ

a

N t P tN

α γ=22

2 32 2

hfm

g n naε

γ =Ω Ω& &h h

?α = - combinatorial factor. 1/ 2α =


Perturbation theory for molecular productionB. Dobrescu and VP, Phys. Lett. A 350, 154 (2006)

Keldysh technique for time-dependent field+-

( )†( ; , ) ( , ) ( , )ct t i T a t a tαβ α β′ ′= −p p pA

( )†( ; , ) ( , ) ( , )ct t i T b t b tαβ α β′ ′= −p p pB

Green functions

( )†( ; , ) ( , ) ( , )ct t i T c t c tαβ α β′ ′= −p p pC,α β = ±

Number of molecules

(Fermi sphere)

( ) ( ; , )mN t i t t∈

= ∑p

p+,-C


Interaction representation

( )(0), ( , , ) ( ) exp ( )

t

Ft

it t i h t dtθ ε ε ε µ′

+ −

⎡ ⎤′ = − − −⎢ ⎥

⎣ ⎦∫p pp

hA

( )(0), ( , , ) ( ) exp ( )

t

Ft

it t i h t dtθ ε ε ε µ′

− +

⎡ ⎤′ = − − − −⎢ ⎥

⎣ ⎦∫p pp

hA

( )(0) (0)( , , ) 0; ( , , ) exp ( ) 2 ( )cit t t t t tε µ⎡ ⎤′ ′ ′= = − −⎢ ⎥⎣ ⎦

p p ph

+,- -,+C C

( , , ) ( , , )t t t tα β α β′ ′=p p, ,B A

( , , ) ( ) ( , , ) ( ) ( , , )( , , ) ( ) ( , , ) ( ) ( , , )

t t t t t t t t t tt t t t t t t t t t

θ θθ θ

+ + − + + −

− − + − − +

′ ′ ′ ′ ′= − + −

′ ′ ′ ′ ′= − + −

p p pp p p

, , ,

, , ,

G G G

G G G

Vertices: gV


Graphs

Second order:

Fourth order:

+∞ −∞

p+p q +p q

α βq

β+∞

p

q

+p q +p qα −∞

′p

′q

+p q

α′ β ′

+∞ −∞

+∞ −∞


( )2 4( ) 88 0( ) 105

m

a

N tN t

= ∞= Γ − Γ + Γ

= −∞

Results

2

2ag n

Γ =Ω&h

( )S Bg B h tµΩ = =

h h

In first two orders of perturbation theory( ) ( )

( )m

a

N t fN t

= ∞= Γ

= −∞

Compare to effective LZ theory: ( )2 4( ) 1 0( ) 2

m

a

N tN t

= ∞= Γ − Γ + Γ

= −∞

Collective effects suppress the transition probabilityDuring the transition process atoms can perform

an exchange.Renormalizability?


Conclusions• LZ transition proceeds during time interval LZ /τ = ∆ Ω&

• Fast transverse noise produces transitions during the time interval . Condition is assumed( ) 1

acc nτ τ−

= Ω& n accτ 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 τaccand 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. Collectiveeffects suppress the transitions.

• The decoherence time due to longitudinal noise is 2

1dec n

z n

τ τη τ

=


What else was done

Transitions of spin S>1/2 in the sweeping magnetic field

Bz

energy

;s BH gµ= − =bS b B

( ), 0,x zb b t=b &

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


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 dtα α β βδ δ−∞

′ ′ ′= ∫

( ) ( ) ( , )g t g t K t tα β αβ′ ′=

VLP and S, Scheidl, Phys. Rev. B 70, 014416 (2004)

Solution of the Bethe-Salpeter equations


Solution of the Bloch equations

( )1;

2z zg g g g i tg i giη η η+ − − + ± ± ±= − = ±Ω& & m &

Integral equation for zg2 2( ' )21 ( ) ( ) . . ( )

2

t i t t

z zg e t t c c g t dtη ηΩ −

+ −−∞

⎡ ⎤′ ′ ′= − +⎢ ⎥

⎢ ⎥⎣ ⎦∫

&

& Complete initial decoherence

Averaging procedure: ( ) ( ') ( ') ( ) ( ') ( ')z zt t g t t t g tη η η η+ − + −=

Precision:2/n accτ τ τ= Ω&

Documents

Landau-Zener Transitions with quantum noisecoqusy06/SLIDES/pokrovsky.pdf · August 2006 Dresden, 2006 Landau-Zener Transitions with quantum noise Valery Pokrovsky Department of Physics,