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

Landau-Zener Transitions with quantum noise Valery Pokrovsky Department of Physics, Texas A&M University and Институт теоретической физики им . Л . Д . Ландау Nikolai A. Sinitsyn Department of Physics and Astronomy UT Austin Participants : 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

I ntroduction LZ theory energy Adiabatic levels 2 1 Diabatic levels time 1 2 Avoided level crossing (Wigner-Neumann theorem) Schrödinger equations − = Ω = = + ∆ h ( ) ( ) ( ); 1 E t E t t & ( ) ia E t a a 2 1 1 1 1 2 & Ω = Ω = ∆ + * & ( ) ia a E t a ( ) t t 2 1 2 2 August 2006 Dresden, 2006

Adiabatic levels: 2 + − ⎛ ⎞ E E E E = ± + ∆ 2 1 2 1 2 ⎜ ⎟ E ± ⎝ ⎠ 2 2 & =− =Ω /2 Center-of mass energy = 0 E E t 2 1 ∆ & µ γ = g B LZ parameter: & Ω = B z & Ω h h γ ฀ 1 γ ฀ 1 August 2006 Dresden, 2006

α β ⎛ ⎞ = ⎜ α + β = ⎟ 2 2 U LZ transition matrix | | | | 1 − β α * * ⎝ ⎠ e πγ − α = 2 Amplitude to stay at the same d-level ⎛ ⎞ π γ π 2 π − + 2 e x p ⎜ ⎟ i ⎝ 2 4 ⎠ β = − Amplitude of transition γ Γ − γ 2 ( ) i ∆ ∆ ⎛ ⎞ 2 τ = Ω 1 & τ = Ω − 1 / 2 LZ transition time: m a x ⎜ , ⎟ & Ω & L Z ⎝ ⎠ L Z τ LZ & && τ τ = Ω Ω ฀ / Condition of validity: 1 2 LZ sat 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 & = + = Ω + ∆ b b η b ˆ ˆ ; z t x tot reg reg ⎛ ⎞ ′ − η ( ) t t t ′ η η = -- Gaussian noise ⎜ ⎟ ( ) ( ) t t f τ i k ik ⎝ ⎠ n & − τ Ω 1/2 ฀ Noise is fast if n I ω ∆ ω = τ 1/ August 2006 Dresden, 2006 n

1 ρ = + ⋅ g s Density matrix: ˆ( ) ( ) t I t 2 t − g ( ) Bloch vector . It obeys Bloch equation: = − × g b g & tot 1 1 ( ) ( ) = − ≡ − g n n n n Difference of populations ↑ ↓ 1 2 z 2 2 ± = ± g g ig Coherence amplitude x y = g 2 const Integral of motion: August 2006 Dresden, 2006

Transitions produced by noise & Ω = Ω ( ) t t t τ acc & Ω = Ω ≤ τ ( ) 1/ t t It produces transitions until n 1 τ = Ω τ ฀ Accumulation time: & τ acc n n ( ) g t is slowly varying z 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. = − η η * & n n Ω Ω Ω 1 ( ) ( ) 1 t t t Transition probability measures the spectrum of noise ⎛ ⎞ π η 2 2 | | ⎜ ⎟ = − Transition probability for infinite time exp P → ⎜ ⎟ & Ω 1 1 2 h ⎝ ⎠ Fermi golden rule is exact for gaussian fast noise! August 2006 Dresden, 2006

Regular and random field act together Separation of times : noise τ is essential only beyond LZ τ acc τ LZ τ acc ⎡ ⎤ π η 2 ⎛ ⎞ 2 | | π ∆ 2 2 − − 1 1 ⎢ ⎥ = − ⎜ − ⎟ & & Ω Ω 2 2 h h 2 1 P e e ⎜ ⎟ → ⎢ ⎥ 1 2 2 ⎝ ⎠ ⎣ ⎦ August 2006 Dresden, 2006

π η 2 2 | | J= ω & & β = Ω Plot of transition probability vs. inverse frequency rate. It is possible to get P larger than ½ at faster sweep rate or stopping the process at some specific field. August 2006 Dresden, 2006

Graph representation τ ฀ n 1 t ′ 2 1 1 2 1 t ′ t ′ t ′ t t t t 1 1 2 2 3 3 ( ) ⎛ ⎞ & ′ Ω − 2 2 i t t G t t ′ t ′ 1 t ′ 2 ′ ⎜ ⎟ = * ( , ) ( , ) exp t t G t t ⎜ ⎟ 2 ⎝ ⎠ t ′ ′ ′ η η = η η † † ( ) ( ) ( ) ( ) t t t t t Chronological time order τ ฀ n t ′ t ′ t t 1 2 1 2 4 times are close August 2006 Dresden, 2006

⎡ ⎤ t ∫ ′′ ′′ = − η What was omitted: Debye-Waller factor ⎢ ⎥ exp ( ) W i t dt z ⎣ ⎦ ′ t ′ η τ ฀ W ≈ − τ 2 2 1 ฀ 1, t t if z n n Diagonal noise leads to decoherence for a long time g ± +∞ = ( ) 0 1 τ = τ ฀ Decoherence time: η τ dec n 2 z n August 2006 Dresden, 2006

Quantum noise and its characterization η η ≠ η η † † ( ) ( ') ( ') ( ) t t t t Model of noise: phonons ( ) = + † † 1 ; H u a a a a int 1 2 2 1 ∑ = η + η η = † ; u g b k k V k = ∑ ω † H b b k k k n k Ω ( ) ( ) t & = − Ω = Ω † † ; ( ) H a a a a t t 2 1 1 2 2 2 August 2006 Dresden, 2006

( ) − ( ) 2 ∫ 1 e ω η η = δ ω − ω = − † k / T ; 1 N d g N ω ω ω ω k k ( ) ( ) 2 ∫ η η = + δ ω − ω = e ω η η † k / † T 1 N d g ω ω ω ω ω k k ω = ∫ d ′ − ω − η η η η † † ( ) i t t ( ) ( ') t t e ω ω π 2 Different time scales for induced and spontaneous transitions τ − τ ω − 1 1 ฀ ฀ T ni ns D & T ω Ω ฀ , Noise is fast if D August 2006 Dresden, 2006

Bloch equation for the fast quantum noise ( ) ds ⎡ ⎤ = η η − η η + η η † † † z sign( ) , t s ⎣ ⎦ Ω Ω Ω Ω Ω Ω z dt & Ω = Ω = Ω ( ) t t Solution ⎛ ⎞ ( ) t ∫ ′ = ⎜ − η η + η η ⎟ + † † ( ) ( )exp , s t s t dt ⎜ ⎟ ′ ′ ′ ′ Ω Ω Ω Ω 0 z z ( ) ( ) ( ) ( ) t t t t ⎝ ⎠ t 0 ( ) ⎛ ⎞ t t ⎡ ⎤ ∫ ∫ ′ ′ ′′ η η − η η + η η † † † ⎜ ⎟ sign( ) , exp dt t dt ⎣ ⎦ ′ ′ ′′ ′′ ′′ ′′ Ω Ω Ω Ω Ω Ω ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t ⎝ ⎠ ′ t t 0 Sz turns into zero in the limit of strong noise, but not exponentially August 2006 Dresden, 2006

Essential graphs τ ฀ n 1 t ′ 2 1 1 2 1 t ′ t ′ t ′ t t t t 1 1 2 2 3 3 Golden rule picture Ω t η η † Ω Ω 2 2 η η † Ω Ω 1 1 August 2006 Dresden, 2006

Saturation of frequency Ω Ω ∞ t τ s ( ) − 1 τ = ωτ τ & ฀ acc n s ⎡ ⎤ η η † , ⎣ ⎦ Ω Ω = − ∞ ∞ s Long time limit ∞ z η η + η η † † Ω Ω Ω Ω ∞ ∞ ∞ ∞ Ω h η η = η η † † Noise in thermal equilibrium e T Ω Ω Ω Ω Ω h = ∞ tanh 2 s ∞ z T August 2006 Dresden, 2006

Feshbach resonance 40 K 2 40 K August 2006 Dresden, 2006

Feshbach resonance 6 Li 6 Li 2 August 2006 Dresden, 2006

Feshbach Resonance driven by sweeping magnetic field B 6 Li 6 Li 2 hf interaction 6 Li = + H H V Hamiltonian: 0 ( ) ( ) ( ) ∑ ∑ ⎡ ⎤ = ε − µ − + + − µ † † † ( ) 2 H h t a a b b E c c ⎣ ⎦ p p p p p q q q 0 p q = & ( ) h t ht ( ) g ∑ = + † . . ; V a b c h c ε 3 ฀ g a + p q p q hf m V p q , August 2006 Dresden, 2006

Suggestions of LZ probability for the molecule production ( ) N t ε 2 2 = α γ g n mol ( , ) P t γ = 2 hf 3 ฀ na LZ & & (0) Ω Ω N m 2 2 h h a α = α = 1/ 2 ? - combinatorial factor. 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 + - Green functions ( ) ′ ′ = − p p p † A ( ; , ) ( , ) ( , ) t t i T a t a t αβ α β c ( ) ′ ′ = − p p † p B ( ; , ) ( , ) ( , ) t t i T b t b t αβ α β c α β = ± , ( ) ′ ′ = − p p p † C ( ; , ) ( , ) ( , ) t t i T c t c t αβ α β c Number of molecules ∑ = p C ( ) ( ; , ) N t i t t +,- m ∈ p (Fermi sphere) August 2006 Dresden, 2006