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Weak measurement of cotunneling time Alessandro Romito (FU Berlin) Yuval Gefen (WIS) Quy Nhon, Vietnam ,August 2013 Weak values – introduction for pedestrians WV good to observe virtual states Weak measurement of cotunneling time quantum dot I QPC=Quantum Point Contact WEAK MEASUREMENT = WEAK SYSTEM/DETECTOR ENTANGLEMENT Weak measurements FQMC13 weak measurement weak value: Aharonov, Albert, Vaidman 1988 time post selection (eigenvalue of Aˆ ) [ Aˆ , Bˆ ] 0 weak measuremet of Bˆ preparation in preselected state post-selected state 0 fin Bˆ weakly measured observable Bˆ weak 0 fin | Bˆ | in 0 fin | in can be complex ! Weak measurements projective measurement weak measurement [Von Neumann (1932)] FQMC13 Postselection and WV FQMC13 weak value weak time [Y. Aharonov et al. (1988)] What for? CQOX - QIM Foundations of quantum mechanics: counterfactual statements measurement of simultaneous observables access new observables measure wave function Experimental verification violation of classical inequalities quantum optics ... solid state (2013!) Precision measurements quantum spin Hall effect of light spacial displacements charge sensing ... Application quantum feedback control state discrimination WHAT ARE WEAK VALUES GOOD FOR? Foundations of Quantum Mechanics counterfactual correlations simultaneous measurement of non-commuting observables access to new observables measure wave function Bell-type inequalities Precision Measurement / Amplification small spatial displacements charge sensing Quantum spin-Hall effect of light Applications quantum feedback control quantum state discrimination Realizations optics Josephson junction / optical cavity solid state? accessing a quantum virtual state accessing a MANY-BODY quantum virtual state Cotunneling [D. Averin, Yu. V. Nazarov (1990)] Cotunneling Can you detect the extra charge in the dot? For how long there is an extra charge in the dot in a successful cotunneling event? Charging energy Thouless energy Level spacing [D. Averin, Yu. V. Nazarov (1990)] How to detect? Simpler case: sequential tunneling strong measurement Classical probabilities Experimentally available Sukhorukov, Ihn, Ensslin et al. (2007)] sequential tunneling Simpler case: (1) seq lim T T t 0 0 (0) dt ds P ( s )[ J ( t s ) J ] I (t ) weak measurement current-current correlations (self-consistent) T I ( J (1) J (0) ) quantum correlations cotunneling microscopic model weak measurement N N 1 system change in transmission of QPC: detector Amplitude Amplitude ( iu ) Tunneling time FQMC13 long-lasting question Mac Coll (1932), Condon (1933), Wigner (1955) [delay time]..., Smith (1960) [dwell time],,,, Baz’ (1966) [Larmor precession]..., Buttiker & Landauer (1982) [time dep. modulation], Sokolovski & Connor (1993) [path integral], Steinberg (1995) [weak values].... E / EC Cotunneling time <I>, elastic SIJ, elastic x 64 FQMC13 Re{ WV } dwell inelastic Im{ WV } cot 0 / EC 0 1 / EC 2 elastic ETh EC L2 ETh ETh EC | X S X D |2 0 2d L2 ETh ETh EC 2 | XS XD | 0 2 L ETh 3 1 ln ( ) 2 2 EC | X S X D | EC / EC 2d NOTE: cot / EC Summary FQMC13 ga eneralizing the measurement scheme of sequential tunneling , the cotunneling time can be defined via weak a values , The 3e cotunneling time may deviate from the prediction of the uncertainty principle