Transmission phase of a quantum dot and statistical fluctuations of - PowerPoint PPT Presentation

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 e ikx t e ikx r e − ikx V = 2 e 2 G = I h | t | 2 Transmission amplitude t = | t | e i α α transmission phase

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions AB interferometer containing a quantum dot ∼ 200 electrons continuous phase evolution in resonances (Friedel sum rule) ∝ δα abrupt drops of π in valleys Schuster et al. , Nature ’97 subsequent peaks in phase

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Crossover mesoscopic ↔ universal “Mesoscopic”: N � 10; irregular phase evolution Avinum-Kalish et al. , Nature ’05 “Universal”: N > 14; subsequent peaks in phase

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Between resonances: Two-level model ǫ 2 γ r γ l 1 γ r γ l 2 γ r γ l 2 2 1 2 t ( ǫ ) ∼ ǫ − ǫ 1 + i Γ 1 / 2 + ǫ − ǫ 2 + i Γ 2 / 2 γ r γ l 1 1 n | 2 + | γ r ǫ n = ǫ ( 0 ) Γ n = | γ l n | 2 ǫ 1 − V g n γ l 1 γ r 1 = − γ l 2 γ r γ l 1 γ r 1 = γ l 2 γ r opposite parity same parity 2 2 2 π 2 π π π α α 0 0 1 1 | t | 2 | t | 2 0 . 5 0 . 5 0 0 V g V g | t | > 0 ↔ smooth phase increase | 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 γ l 1 γ r 1 = − γ l 2 γ r γ l 1 γ r 1 = γ l 2 γ r opposite parity same parity 2 2 2 π 2 π π π α α 0 0 1 1 | t | 2 | t | 2 0 . 5 0 . 5 0 0 V g V g 1 1 0 . 5 0 . 5 Im t Im t 0 0 − 0 . 5 − 0 . 5 − 1 − 1 − 1 − 0 . 5 0 0 . 5 1 − 1 − 0 . 5 0 0 . 5 1 Re t Re t

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Parity rule ǫ n D n = γ l n γ r n γ l n + 1 γ r n + 1 sgn ( γ l n γ r n ) = ± sgn ( γ l n + 1 γ r n + 1 ) → D n ≷ 0 γ r γ l n n D n < 0 D n > 0 → no transmission zero → | t | = 0 no phase lapse phase lapse [Lee PRL ’99; Taniguchi & B¨ uttiker PRB ’99, Levy-Yeyati & B¨ uttiker PRB ’00, Aharony et al. PRB ’02] Disordered dots: P ( D n < 0 ) = 1 / 2 � irregular phase evolution Experiment: P ( D n < 0 ) = 0 � correlations between γ n and γ n + 1 ?

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Wave-function correlations in chaotic dots y � W γ l ( r ) d y Φ l ( r ) ( y ) ψ n ( x l ( r ) , y ) ∼ ψ n ( x l ( r ) , 0 ) W ∝ n x 0 D n = γ l n γ r n γ l n + 1 γ r L n + 1 V E F = ψ n ( x l , 0 ) ψ n ( x r , 0 ) ψ n + 1 ( x l , 0 ) ψ n + 1 ( x r , 0 ) Random wave model: [M.V. Berry, J. Phys. A ’77] V g N ( r ) = 1 � ψ RWM cos [ k j r + δ j ] n N j = 1 E n = � 2 k 2 random δ j 2 m � randomly oriented k j with | k j | = k n n Correlations over a distance L = x r − x l � ψ n ( r ) ψ n ( r ′ ) � ≃ 1 A J 0 ( k | r − r ′ | ) � � D n � ∼ J 0 ( k n L ) J 0 ( k n + 1 L )

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Universal behavior in the semiclassical limit 1 0 . 5 J 0 ( kL ) � D n � ∼ J 0 ( k n L ) J 0 ( k n + 1 L ) ∆ kL = k n + 1 L − k n L ≃ π 0 kL L ∆ k 2 π − 0 . 5 0 5 10 15 20 25 kL Average-based probability of having no phase lapse ( � D n � < 0) P ( � D n � < 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 | t | 0 . 5 0 86 88 90 92 94 kL y 1 W x 0 . 5 L V E F Im[ t ] V g 0 − 0 . 5 circles ↔ Breit-Wigner peaks − 1 − 1 − 0 . 5 0 0 . 5 1 Re[ t ]

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Counting zeros and resonances some long universal sequences kL 1 0 20 30 40 50 | t | 2 0 . 5 100 0 2 π 75 π α N r 0 N z 50 0 . 034 0 . 0345 0 . 035 V g α ( − ) /π 25 c 0 . 2 ∆ N r − ∆ N z 0 0 . 1 ∆ N r − 0 . 005 0 0 . 005 0 . 01 0 . 015 0 . 02 V g 0 25 50 75 100 “universal” regime at very large kL kL

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Are averaged correlations sufficient? Conclusions were drawn from the sign of the average � D n � = � γ l n γ r n γ l n + 1 γ r n + 1 � and assuming narrow leads γ l / r ∼ ψ n ( x l / r , 0 ) n Questions: What happens in the case of wider leads? Do statistical fluctuations of the γ l / r change the results? n

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Gaussian fluctuations of the partial-width amplitudes Dots with chaotic classical dynamics � Gaussian distribution of ψ n [Voros ’76, Berry ’77, Srednicki ’96] � W γ l ( r ) d y Φ l ( r ) ( y ) ψ n ( x l ( r ) , y ) ∝ n 0 � Gaussian distribution of the PWAs γ l / r with joint density n � − ( γ l n ) 2 +( γ r n ) 2 − 2 ρ n γ l n γ r � p ( γ l n , γ r √ 1 n ) = n exp n 2 σ 2 n ( 1 − ρ 2 2 πσ 2 1 − ρ 2 n ) n Variance and correlator with LR symmetry: σ 2 1 n = � γ l n γ l n � = � γ r n γ r n � ρ n = n � γ l n γ r n � σ 2 � Probability for positive parity n > 0 ) = 1 2 + 1 P ( γ l n γ r π arcsin ( ρ n )

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Probability for no phase lapse P ( D n < 0 ) = P ( γ l n γ r � 1 − P ( γ l n + 1 γ r � + P ( γ l n + 1 γ r � 1 − P ( γ l n γ r � n > 0 ) n + 1 > 0 ) n + 1 > 0 ) n > 0 ) With mean wave-number spacing ∆ k n = π/ k n L 2 : f ( k ) = 1 4 − 1 π 2 arcsin 2 � P ( D n < 0 ) ≃ 2 f ( k n ) + ∆ k n f ′ ( k n ) � ρ ( k ) with 0 . 5 P ( D n < 0) 0 . 4 0 . 3 1 0 . 5 ρ n 0 − 0 . 5 � 2 � L L W W − 1 0 25 50 75 100 125 150 k n L

Experiments Parity rule Mean wave-function correlations Fluctuations Conclusions Numerics for P ( D n < 0 ) averaging over 14 cavities 0 . 5 P ( D n < 0) 0 . 4 0 . 3 40 60 80 100 k n L Blue: P ( D n < 0 ) ; smoothing k n 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)