Adiabatic Passage and Noise in Quantum Dots Sigmund Kohler - PowerPoint PPT Presentation

Adiabatic Passage and Noise in Quantum Dots Sigmund Kohler Instituto de Ciencia de Materiales de Madrid, CSIC 1 0 1

Adiabatic Passage and Noise 1 Steady-state transfer passage by adiabatic passage shot noise as signal 2 Landau-Zener-(Stückelberg-Majorana) interferometry background fluctuations probed via transport

Adiabatic Passage and Noise Steady-state coherent transfer by adiabatic passage Ω 1 2 Ω 23 | 1 〉 | 2 〉 | 3 〉 Huneke, Platero, SK, PRL 110 , 036802 (2013)

What is “coherent transfer by adiabatic passage” (CTAP)? ? electron transfer from dot 1 to dot 3 without occupying dot 2 Greentree et al. , PRB 2004 cf. STIRAP (stimulated Raman adiabatic passage)

What is “coherent transfer by adiabatic passage” (CTAP)? Hamiltonian  0 Ω 12 0   Ω 23  0 ϕ 0 ∼ 0 H = Ω 12 Ω 23  ;    0 Ω 23 0 − Ω 12 eigenvector with E = 0: occupation 1 ? electron transfer | c 3 | 2 from dot 1 to dot 3 without occupying dot 2 | c 2 | 2 | c 1 | 2 Greentree et al. , PRB 2004 0 Ω 23 / Ω 12 cf. STIRAP (stimulated Raman adiabatic passage) ➔ adiabatic switching Ω ij ( t )

CTAP — dephasing dephasing by phonons  0 Ω 12 ( t ) 0  ➔ small occupation of dot 2 H ( t ) = Ω 12 ( t ) 0 Ω 23 ( t )   Greentree et al. , PRB 2004 0 Ω 23 ( t ) 0 charge monitor increases Gauss pulse: dephasing Rech & Kehrein, PRL 2011 intensity | Ω 23 | 2 | Ω 12 | 2 time problem: experimental evidence for non-occupation (Zeno effect!)

CTAP — dephasing dephasing by phonons  0 Ω 12 ( t ) 0  ➔ small occupation of dot 2 H ( t ) = Ω 12 ( t ) 0 Ω 23 ( t )   Greentree et al. , PRB 2004 0 Ω 23 ( t ) 0 charge monitor increases Gauss pulse: dephasing Rech & Kehrein, PRL 2011 intensity | Ω 23 | 2 | Ω 12 | 2 time problem: experimental evidence for non-occupation (Zeno effect!)

Steady-state CTAP leads Ω 1 2 Ω 23 ➔ current | 1 〉 | 2 〉 | 3 〉 ➔ steady-state transport

Steady-state CTAP leads Ω 1 2 Ω 23 ➔ current | 1 〉 | 2 〉 | 3 〉 ➔ steady-state transport Ω i j time evolution (propagation of ρ ) ρ 11 ρ 22 ρ 33 occupation ➔ direct transition | 1 〉 − → | 3 〉 0.5 ➔ ideally: 1 electron per pulse ? fingerprint: shot noise 0 0 2 4 6 suppression time [ T ]

Noise and propagation method master equation approach: � perturbation theory for weak wire-lead coupling Γ � master equation for reduced density operator: (Bloch-R edfield equation, consistent with equilibrium conditions) d t ρ wire = d d d t tr leads ρ

Noise and propagation method master equation approach: � perturbation theory for weak wire-lead coupling Γ � master equation for reduced density operator: (Bloch-R edfield equation, consistent with equilibrium conditions) d t ρ wire = d d I ∼ d S ∼ d d t tr leads N 2 d t tr leads ρ , d t tr leads N L ρ , L ρ ➔ Fano factor: Elattari & Gurvitz, Phys. Lett. (2002) ; Bagrets & Nazarov, PRB (2003) ; Novotný, Donarini, Flindt & Jauho, PRL (2004) ; Kaiser & SK, Ann. Phys. (2007) ➔ iterative calculation of FCS by numerical propagation ➔ more efficient than N -resolved master equation

Steady-state CTAP 3 0.5 ρ 22 ¯ average occupation of dot 2 2 Γ [ Ω max ] 0.25 ρ 22 ≪ 1/3 if 1 � pulse distance ∆ T � 2 T � tunnel rate Γ ≈ 1 2 Ω max 0 0 0 1 2 3 4 5 6 pulse distance T [1/ Ω max ] 3 Fano factor F 2 0.5 shot noise suppression Γ [ Ω max ] correlates with low Fano 1 factor 0. 25 ➔ Fano factor as fingerprint of 0 0 1 2 3 4 5 6 CTAP pulse distance T [ 1 / Ω max ]

Steady-state CTAP — quantitative analysis ρ 22 ¯ 1 Fano for small occupation: Fano factor F ≈ 0.2 0 .5 (elsewise F ≈ 0.5) 0 0 1 2 3 4 5 T [1/ Ω max ]

Steady-state CTAP — quantitative analysis ρ 22 ¯ 1 Fano for small occupation: Fano factor F ≈ 0.2 0 .5 (elsewise F ≈ 0.5) 0 0 1 2 3 4 5 T [1/ Ω max ] CTAP not visible in current 1.5 correlation 〈 F , ¯ ρ 22 〉 γ φ = 0 γ φ = 0 .1 Ω max 1 moderate dephasing γ φ tolerable F max 0 .5 〈 F , ρ 22 〉 ideally: Γ ≈ Ω max /2 F min 0 0.5 1 1.5 2 2.5 3 ➔ „noise is the signal“ Γ [ Ω max ] (Landauer)

Adiabatic Passage and Noise Landau-Zener-Stückelberg-Majorana Interferometry with Quantum Dots Forster, Petersen, Manus, Hänggi, Schuh, Wegscheider, SK, Ludwig PRL 112 , 116803 (2014)

AC-driving and Landau-Zener transitions Quantum system in A C-field, H ( t ) non-adiabatic transition energy P LZ probability P LZ = e − π ∆ 2 /2 ħ v 1 − P LZ Landau, Zener, Stückelberg, time Majorana, 1932

AC-driving and Landau-Zener transitions Quantum system in A C-field, H ( t ) non-adiabatic transition energy P LZ probability P LZ = e − π ∆ 2 /2 ħ v 1 − P LZ Landau, Zener, Stückelberg, time Majorana, 1932 ➔ beam splitter, interference ➔ Landau-Zener-(Stückelberg- Majorana) interferometry

LZSM interference and photon-assisted tunneling LZSM interference „avoided crossings“

LZSM interference and photon-assisted tunneling photon-assisted tunneling ħ Ω „dipole excitations“ „Conductance is transmission“ (Landauer, 1957) ǫ + 2 ħ Ω ǫ +ħ Ω ➔ scattering process ǫ ǫ ➔ with rf -field: resonances ǫ −ħ Ω ǫ − 2 ħ Ω

LMU experiment: interference pattern Experimental LZSM pattern (Ludwig group, LMU Munich) I [ fA ] 0 100 200 resonance peaks A [ µ eV] 100 with increasing temperature: pattern blurred 1 8 mK 0 200 ➔ phonons A [ µ eV] ➔ pattern contains information about decoherence 100 475 mK 0 − 200 0 200 ǫ [ µ eV]

LMU experiment: realistic modelling (1,1) T (1,1) S (0,2) S (0,1) ∼ single-particle terms ✓ dot-lead tunneling ✓ detuning ✓ AC gate voltage H rf ( t ) ∝ cos( Ω t ) ✓ Zeeman splitting

LMU experiment: realistic modelling (1,1) T (1,1) S (0,2) S (0,1) ∼ single-particle terms two-particle interaction ✓ dot-lead tunneling ✗ spin relaxation (resolves spin blockade) ✓ detuning ✗ Coulomb repulsion ✓ AC gate voltage H rf ( t ) ∝ cos( Ω t ) ✗ coupling to phonons ✓ Zeeman splitting ➔ master equation for many-body states

Decoherence & slow fluctuations Decoherence: Caldeira-Leggett model H DQD-bath = ( n L − n R ) ξ Ohmic spectral density J ( ω ) = π 2 αω exp( − ω / ω cutoff ) vib vib dissipation strength α Slow fluctuations time scale < dwell time ǫ Gauss distributed w ( ǫ ) ∝ e − 1 2 ( ∆ ǫ / λ ∗ ) 2 ➔ convolution of I ( ǫ , A ) with Gauss inhomogeneous broadening λ ∗

Decoherence & slow fluctuations Decoherence: Caldeira-Leggett model H DQD-bath = ( n L − n R ) ξ Ohmic spectral density J ( ω ) = π 2 αω exp( − ω / ω cutoff ) vib vib dissipation strength α Slow fluctuations Central idea comparison time scale < dwell time experiment/theory ǫ Gauss distributed � I ( ǫ , A ) ➔ λ ∗ w ( ǫ ) ∝ e − 1 2 ( ∆ ǫ / λ ∗ ) 2 � W ( τ ǫ , τ A ) ➔ α ➔ convolution of I ( ǫ , A ) with Gauss ➔ determine dissipative inhomogeneous broadening λ ∗ parameters

Floquet-Bloch-Redfield master equation Perturbation theory in DQD-environment coupling V � ∞ d t ρ = − i d � � � � H DQD ( t ), ρ − d τ [ V ,[ V ( t − τ , t ), ρ ]] ħ env 0 Floquet theory (Bloch theory in time) ➔ rf- field exact i ħ ∂ � � ∂ t − H DQD ( t ) φ α ( t ) = ǫ n φ n ( t ), mit φ n ( t ) = φ n ( t + 2 π / Ω ) ( 1 , 1 ) T rate equation for occupations (1,1) S (0,2) S � � ˙ P n = W n ← n ′ P n ′ − W n ′ ← n P n (0,1) W = W leads + W spinflip + α W bath ∼ ➔ determination of α requires knowledge of W leads and W spinflip

Inhomogeneous broadening theory experiment I [fA] 100 0 − 200 − 100 0 100 200 ǫ resonance peaks � singlet-triplet mixing � inter-dot excitations

Inhomogeneous broadening I [ fA] 0 100 theory , λ ∗ = 3.5 µ eV experiment 200 experiment I [fA] 100 A [ µ eV] 100 T = 1 8mK 0 0 200 − 200 − 100 0 100 200 ǫ A [ µ eV] resonance peaks 100 � singlet-triplet mixing theory λ ∗ = 0 � inter-dot excitations 0 200 λ ∗ = 3.5 µ eV A [ µ eV] ➔ 100 λ ∗ = in agreement with e.g. 3.5 µ eV 0 − 200 0 200 Petersson et al. PRB 2010 ǫ [ µ eV]

Analysis in Fourier space LZSM pattern in Fourier space: decaying arcs Rudner et al. , PRL 2008 1 1 000 18 mK 275 mK W (lemon) τ A [ ħ / µ eV] 475 mK 0 1 00 e x p e r im en t − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 1 τ ǫ [ ħ / µ eV] τ A [ ħ / µ eV] 0 th e ory − 1 − 1 0 1 τ ǫ [ ħ / µ eV]

Analysis in Fourier space LZSM pattern in Fourier space: decaying arcs Rudner et al. , PRL 2008 1 1 000 18 mK 275 mK W (lemon) τ A [ ħ / µ eV] 475 mK 0 1 00 e x p e r im en t − 1 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 1 τ ǫ [ ħ / µ eV] τ A [ ħ / µ eV] arc decay 0 f ( τ ǫ ) ∝ e − λτ ǫ − 1 2 ( λ ∗ τ ǫ ) 2 th e ory − 1 ➔ compare λ exp and λ theo − 1 0 1 τ ǫ [ ħ / µ eV] ➔ determine α

Temperature dependence 15 λ grows with temperature 10 [ µ eV] 5 λ experiment 0 0 100 200 300 400 t em perature T [mK] 174 mK 290 mK [ µ e V ] 406 mK 10 fit parameter: dissipation strength λ ( α ) theory α = 1.5 · 10 − 4 ( ± 30%) 0 1 · 10 − 4 2 · 10 − 4 0 dissipation strength α