Landau-Zener Transitions with quantum noisecoqusy06/SLIDES/pokrovsky.pdf · August 2006 Dresden,...
Transcript of Landau-Zener Transitions with quantum noisecoqusy06/SLIDES/pokrovsky.pdf · August 2006 Dresden,...
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
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
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
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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⎝ ⎠&
&
August 2006 Dresden, 2006
Spin reversal in nanomagnets
W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999)
August 2006 Dresden, 2006
Controllable switch between statesfor quantum computing:
The noise introducesmistakes to the switchwork.
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
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
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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
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 ( ) ( ) 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
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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&
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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
=
August 2006 Dresden, 2006
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&
August 2006 Dresden, 2006
( ) ( )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&
August 2006 Dresden, 2006
( )† † †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
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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= &
August 2006 Dresden, 2006
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 =
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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
August 2006 Dresden, 2006
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 ′
+∞ −∞
+∞ −∞
August 2006 Dresden, 2006
( )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?
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 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
=
August 2006 Dresden, 2006
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)
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 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
August 2006 Dresden, 2006
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&