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 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

Ω12 Ω23 |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)? ? electron transfer from dot 1 to dot 3 without occupying dot 2

Greentree et al., PRB 2004

• cf. STIRAP (stimulated

Raman adiabatic passage) Hamiltonian H =   Ω12 Ω12 Ω23 Ω23  ; ϕ0 ∼   Ω23 −Ω12   eigenvector with E = 0:

|c1|2 |c3|2 |c2|2 1 Ω23/Ω12

• ccupation

➔ adiabatic switching Ωij(t)

CTAP — dephasing H(t) =   Ω12(t) Ω12(t) Ω23(t) Ω23(t)   Gauss pulse:

|Ω23|2 |Ω12|2 time intensity

dephasing by phonons

➔ small occupation of dot 2

Greentree et al., PRB 2004

charge monitor increases dephasing

Rech & Kehrein, PRL 2011

problem: experimental evidence for non-occupation (Zeno effect!)

CTAP — dephasing H(t) =   Ω12(t) Ω12(t) Ω23(t) Ω23(t)   Gauss pulse:

|Ω23|2 |Ω12|2 time intensity

dephasing by phonons

➔ small occupation of dot 2

Greentree et al., PRB 2004

charge monitor increases dephasing

Rech & Kehrein, PRL 2011

problem: experimental evidence for non-occupation (Zeno effect!)

Steady-state CTAP

Ω12 Ω23 |1〉 |2〉 |3〉

leads

➔ current ➔ steady-state transport

Steady-state CTAP

Ω12 Ω23 |1〉 |2〉 |3〉

leads

➔ current ➔ steady-state transport

2 4 6 0.5 time [T ]

• ccupation

ρ11 ρ22 ρ33 Ωi j

time evolution (propagation of ρ)

➔ direct transition |1〉 −

→ |3〉

➔ ideally: 1 electron per pulse

? fingerprint: shot noise suppression

Noise and propagation method master equation approach:

perturbation theory for weak wire-lead coupling Γ master equation for reduced density operator:

(Bloch-Redfield equation, consistent with equilibrium conditions) d dt ρwire = d dt trleadsρ

Noise and propagation method master equation approach:

perturbation theory for weak wire-lead coupling Γ master equation for reduced density operator:

(Bloch-Redfield equation, consistent with equilibrium conditions) d dt ρwire = d dt trleadsρ, I ∼ d dt trleadsNLρ, S ∼ d dt trleadsN2

Lρ

➔ 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

1 2 3 1 2 3 4 5 6 Γ [Ωmax] pulse distance T [1/Ωmax] 0.25 0.5 ¯ ρ22

average occupation of dot 2 ρ22 ≪ 1/3 if

pulse distance ∆T 2T tunnel rate Γ ≈ 1

2Ωmax 1 2 3 4 5 6 1 2 3 pulse distance T [1/Ωmax] Γ [Ωmax] 0.25 0.5 F

Fano factor shot noise suppression correlates with low Fano factor

➔ Fano factor as fingerprint of

CTAP

Steady-state CTAP — quantitative analysis

0.5 1 1 2 3 4 5 T [1/Ωmax] ¯ ρ22 Fano

for small occupation: Fano factor F ≈ 0.2 (elsewise F ≈ 0.5)

Steady-state CTAP — quantitative analysis

0.5 1 1 2 3 4 5 T [1/Ωmax] ¯ ρ22 Fano

for small occupation: Fano factor F ≈ 0.2 (elsewise F ≈ 0.5)

0.5 1 1.5 0.5 1 1.5 2 2.5 3 Γ [Ωmax] γφ = 0 γφ = 0.1Ωmax 〈F,ρ22〉 Fmax Fmin

CTAP not visible in current correlation 〈F, ¯ ρ22〉 moderate dephasing γφ tolerable ideally: Γ ≈ Ωmax/2

➔ „noise is the signal“

(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 AC-field, H(t)

time energy 1−PLZ PLZ

non-adiabatic transition probability PLZ = e−π∆2/2ħv

Landau, Zener, Stückelberg, Majorana, 1932

AC-driving and Landau-Zener transitions Quantum system in AC-field, H(t)

time energy 1−PLZ PLZ

non-adiabatic transition probability PLZ = e−π∆2/2ħv

Landau, Zener, Stückelberg, 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

ħΩ

photon-assisted tunneling „dipole excitations“ „Conductance is transmission“ (Landauer, 1957)

ǫ ǫ ǫ+ħΩ ǫ−ħΩ ǫ+2ħΩ ǫ−2ħΩ

➔ scattering process ➔ with rf-field: resonances

LMU experiment: interference pattern Experimental LZSM pattern (Ludwig group, LMU Munich)

100 200 −200 200 A [µeV] ǫ [µeV] 475 mK 100 200 A [µeV] 100 I [fA] 18mK

resonance peaks with increasing temperature: pattern blurred

➔ phonons ➔ pattern contains information

about decoherence

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

Hrf(t) ∝ cos(Ωt)

✓ Zeeman splitting

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

Hrf(t) ∝ cos(Ωt)

✓ Zeeman splitting

two-particle interaction

✗ spin relaxation

(resolves spin blockade)

✗ Coulomb repulsion ✗ coupling to phonons ➔ master equation for

many-body states

Decoherence & slow fluctuations Decoherence: Caldeira-Leggett model

vib vib

HDQD-bath = (nL −nR)ξ Ohmic spectral density J(ω) = π

2 αωexp(−ω/ωcutoff)

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

vib vib

HDQD-bath = (nL −nR)ξ Ohmic spectral density J(ω) = π

2 αωexp(−ω/ωcutoff)

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∗ Central idea comparison experiment/theory

I(ǫ,A) ➔ λ∗ W(τǫ,τA) ➔ α

➔ determine dissipative

parameters

Floquet-Bloch-Redfield master equation Perturbation theory in DQD-environment coupling V d dt ρ = − i ħ

• HDQD(t),ρ
• −

∞ dτ

• [V,[V(t −τ,t),ρ]]
• env

Floquet theory (Bloch theory in time) ➔ rf-field exact

• iħ ∂

∂t −HDQD(t)

• φα(t) = ǫnφn(t),

mit φn(t) = φn(t +2π/Ω) rate equation for occupations ˙ Pn =

• Wn←n′Pn′ −Wn′←nPn
• W = W leads +W spinflip +αW bath

(1,1)T (1,1)S (0,2)S (0,1) ∼

➔ determination of α requires knowledge of W leads and W spinflip

Inhomogeneous broadening

−200 −100 100 200 100 ǫ I [fA] theory experiment

resonance peaks

singlet-triplet mixing inter-dot excitations

Inhomogeneous broadening

−200 −100 100 200 100 ǫ I [fA] theory, λ∗ = 3.5µeV experiment

resonance peaks

singlet-triplet mixing inter-dot excitations

➔

λ∗ = 3.5µeV

in agreement with e.g. Petersson et al. PRB 2010

100 200 −200 200 A [µeV] ǫ [µeV] λ∗ = 3.5µeV 100 200 A [µeV] λ∗ = 0 100 200 A [µeV] 100 I [fA] T = 18mK

theory experiment

Analysis in Fourier space LZSM pattern in Fourier space: decaying arcs

Rudner et al., PRL 2008

−1 1 −1 1 τA [ħ/µeV] τǫ [ħ/µeV] theory −1 1 τA [ħ/µeV] experiment 0.1 0.2 0.3 0.4 0.5 0.6 100 1000 τǫ [ħ/µeV] W (lemon) 18 mK 275 mK 475 mK

Analysis in Fourier space LZSM pattern in Fourier space: decaying arcs

Rudner et al., PRL 2008

−1 1 −1 1 τA [ħ/µeV] τǫ [ħ/µeV] theory −1 1 τA [ħ/µeV] experiment 0.1 0.2 0.3 0.4 0.5 0.6 100 1000 τǫ [ħ/µeV] W (lemon) 18 mK 275 mK 475 mK

arc decay f (τǫ) ∝ e−λτǫ− 1

2 (λ∗τǫ)2

➔ compare λexp and λtheo ➔ determine α

Temperature dependence

100 200 300 400 5 10 15 temperature T [mK] λ [µeV] experiment

λ grows with temperature

10 1·10−4 2·10−4 λ(α) [µeV] dissipation strength α 174 mK 290 mK 406 mK theory

fit parameter: dissipation strength α = 1.5·10−4 (±30%)

Temperature dependence

100 200 300 400 5 10 15 temperature T [mK] λ [µeV] experiment α = 1.5·10−4

λ grows with temperature

➔ Ohmic dissipation consistent

with measured temperature dependence

10 1·10−4 2·10−4 λ(α) [µeV] dissipation strength α 174 mK 290 mK 406 mK theory

fit parameter: dissipation strength α = 1.5·10−4 (±30%)

Application: Charge qubit

(1,1)T (1,1)S (0,2)S (0,1) ∼

1 10 100 1000 100 200 300 400 500 temperature T [mK] T2 [ns] ǫ = 0 ǫ = ∆ ǫ = 2∆ TLS α = 1.5·10−4

(1,1)S –(0,2)S qubit for spin-boson model

Weiss & Wollensak, PRL 1989

T−1

2

= πα ħ 2kTǫ2 E2 +∆2 2E coth E 2kT

• ,

here: T2 from full Bloch-Redfield equation α = 1.5·10−4 ➔ T2 ∼ 200 ns T∗

2 = 200ps ≪ T2

Summary Noise as signal for adiabatic passage

leads allow steady-state operation current noise indicates non-local transport

➔ avoids measurement and its backaction Landau-Zener interference to probe noisy environment

temperature dependence consistent with Ohmic dissipation Floquet approach determine bath coupling strength

(Caldeira-Leggett parameter α)

? propagation method for finite voltage ? charge monitor: controlled decoherence

Thanks to ... Jan Huneke Robert Hussein Gloria Platero (Madrid) Ralf Blattmann Peter Hänggi (Augsburg) Florian Forster Gunnar Petersen Stefan Ludwig (Munich / Berlin)