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Transmission phase of a quantum dot and statistical fluctuations of partial-width amplitudes

Rodolfo A. Jalabert, Guillaume Weick, Hans A. Weidenm¨ uller Dietmar Weinmann

Rafael A. Molina, Philippe Jacquod

Luchon Superbagn` eres, March 2015

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Transmission phase

Quantum scatterer connected to monochannel leads

eikx r e−ikx t eikx

G = I V = 2e2 h |t|2 Transmission amplitude t = |t|eiα α transmission phase

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

AB interferometer containing a quantum dot

Schuster et al., Nature ’97

∼ 200 electrons continuous phase evolution in resonances (Friedel sum rule) ∝ δα abrupt drops of π in valleys subsequent peaks in phase

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Crossover mesoscopic ↔ universal

Avinum-Kalish et al., Nature ’05

“Mesoscopic”: N 10; irregular phase evolution “Universal”: N > 14; subsequent peaks in phase

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Between resonances: Two-level model

ǫ1 γl

1

γr

1

ǫ2 γl

2

γr

2

t(ǫ) ∼ γl

1γr 1

ǫ − ǫ1 + iΓ1/2 + γl

2γr 2

ǫ − ǫ2 + iΓ2/2

Γn = |γl

n|2 + |γr n|2

ǫn = ǫ(0)

n

− Vg

γl

1γr 1 = −γl 2γr 2

• pposite parity

0.5 1 |t|2 Vg π 2π α

|t| > 0 ↔ smooth phase increase

γl

1γr 1 = γl 2γr 2

same parity

0.5 1 |t|2 Vg π 2π α

|t| = 0 ↔ phase jump

[Lee PRL ’99; Taniguchi & B¨ uttiker PRB ’99, Levy-Yeyati & B¨ uttiker PRB ’00, Aharony et al. PRB ’02]

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Continuous evolution of t

γl

1γr 1 = −γl 2γr 2

• pposite parity

0.5 1 |t|2 Vg π 2π α

−1 −0.5 0.5 1 Im t −1 −0.5 0.5 1 Re t

γl

1γr 1 = γl 2γr 2

same parity

0.5 1 |t|2 Vg π 2π α

−1 −0.5 0.5 1 Im t −1 −0.5 0.5 1 Re t

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Parity rule

γl

n

γr

n

ǫn

Dn = γl

nγr nγl n+1γr n+1

sgn(γl

nγr n) = ±sgn(γl n+1γr n+1) → Dn ≷ 0

Dn < 0 → no transmission zero no phase lapse Dn > 0 → |t| = 0 phase lapse

[Lee PRL ’99; Taniguchi & B¨ uttiker PRB ’99, Levy-Yeyati & B¨ uttiker PRB ’00, Aharony et al. PRB ’02]

Disordered dots: P(Dn < 0) = 1/2 irregular phase evolution Experiment: P(Dn < 0) = 0 correlations between γn and γn+1?

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Wave-function correlations in chaotic dots

x y W L V Vg EF

γl(r)

n

∝ W dy Φl(r)(y) ψn(xl(r), y) ∼ ψn(xl(r), 0)

Dn = γl

nγr nγl n+1γr n+1

= ψn(xl, 0)ψn(xr, 0)ψn+1(xl, 0)ψn+1(xr, 0) Random wave model:

[M.V. Berry, J. Phys. A ’77]

ψRWM

n

(r) = 1 N

N

• j=1

cos[kjr + δj] random δj En = 2k2

n

2m randomly oriented kj with |kj| = kn

Correlations over a distance L = xr − xl ψn(r)ψn(r′) ≃ 1 A J0(k|r − r′|) Dn ∼ J0(knL)J0(kn+1L)

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Universal behavior in the semiclassical limit

Dn ∼ J0(knL)J0(kn+1L) ∆kL = kn+1L−knL ≃ π kL

L∆k 2π −0.5 0.5 1 J0(kL) 5 10 15 20 25 kL

Average-based probability of having no phase lapse (Dn < 0) P(Dn < 0) ∼ 1 kL Tendency towards the universal regime at large kL

R.A. Molina, R.A. Jalabert, DW, Ph. Jacquod, PRL 108, 076803 (2012)

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Numerics for the transmission amplitude

−1 −0.5 0.5 1 Im[t] −1 −0.5 0.5 1 Re[t] 0.5 1 |t| 86 88 90 92 94 kL

x y W L V Vg EF

circles ↔ Breit-Wigner peaks

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Counting zeros and resonances

α(−)

c

/π Nz Nr 25 50 75 100 20 30 40 50 kL −0.005 0.005 0.01 0.015 0.02 Vg

“universal” regime at very large kL

some long universal sequences

π 2π α 0.034 0.0345 0.035 Vg 0.5 1 |t|2 ∆Nr−∆Nz ∆Nr

25 50 75 100 kL 0.1 0.2

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Are averaged correlations sufficient?

Conclusions were drawn from the sign of the average Dn = γl

nγr nγl n+1γr n+1

and assuming narrow leads γl/r

n

∼ ψn(xl/r, 0) Questions: What happens in the case of wider leads? Do statistical fluctuations of the γl/r

n

change the results?

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Gaussian fluctuations of the partial-width amplitudes

Dots with chaotic classical dynamics Gaussian distribution of ψn

[Voros ’76, Berry ’77, Srednicki ’96]

γl(r)

n

∝ W dy Φl(r)(y) ψn(xl(r), y) Gaussian distribution of the PWAs γl/r

n

with joint density p(γl

n, γr n) = 1 2πσ2

n

√

1−ρ2

n exp

• − (γl

n)2+(γr n)2−2ρnγl nγr n

2σ2

n(1−ρ2 n)

• Variance and correlator with LR symmetry:

σ2

n = γl nγl n = γr nγr n

ρn =

1 σ2

n γl

nγr n

Probability for positive parity P(γl

nγr n > 0) = 1

2 + 1 π arcsin (ρn)

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Probability for no phase lapse

P(Dn < 0) = P(γl

nγr n > 0)

• 1 − P(γl

n+1γr n+1 > 0)

• + P(γl

n+1γr n+1 > 0)

• 1 − P(γl

nγr n > 0)

• With mean wave-number spacing ∆kn = π/knL2:

P(Dn < 0) ≃ 2f(kn) + ∆knf ′(kn) with f(k) = 1 4 − 1 π2 arcsin2 ρ(k)

• L

W

• L

W

2 −1 −0.5 0.5 1

ρn

25 50 75 100 125 150

knL

0.3 0.4 0.5

P(Dn < 0)

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Numerics for P(Dn < 0) averaging over 14 cavities

0.3 0.4 0.5 P(Dn < 0) 40 60 80 100 knL

Blue: P(Dn < 0); smoothing kn interval of δ/L with δ = π/4 and π Red: from the statistical model; same smoothing

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions

Conclusions

Wave-function correlations: probability for non-universal evolution ∼ 1/kL in chaotic dots Tendency towards universal behavior at large N Gaussian fluctuations of partial-width amplitudes Reduced tendency towards universal behavior Numerics for ballistic cavities Intermediate tendency towards universal behavior Outlook: More realistic dot models? Beyond Gaussian fluctuations? Correlations n ↔ n + 1?

R.A. Molina et al., PRL 108, 076803 (2012); PRB 88, 045419 (2013) R.A. Jalabert et al., PRE 89, 052911 (2014)