Noisy Quantum Measurements: a nuisance or fundamental physics? - PowerPoint PPT Presentation

Noisy Quantum Measurements: a nuisance or fundamental physics? Wolfgang Belzig Universität Konstanz Conference on Quantum Measurement: Fundamentals, Twists, and ApplicaBons ICTP Triest, 2019

Content • Quantum measurement: projection and weak measurements • Quantum dynamics: Keldysh contour • Facets of weak quantum measurements 1. Time-resolved counting statistics and quantum transport 2. Keldysh-ordered expectations are quasiprobabilities 3. Time-reversal symmetry breaking 4. General non-markovian weak measurement

Order of operators matters! Quantum measurement and correla2ons? ( [ * + & , - . # ] /2 ? +(&) - * . # !(#)%(&) → - . # * +(&) + & , - * . # /2 Textbook (LL Vol. V): Quantum optics: photodetector measures ‚normal ordered‘ expectations (one click) homodyning and heterodyning are highly specific

x Von Neumann measurement: from strong to weak P(x) Idea: couple system ( ! ψ i ⊗ " ) to a P ( x ) pointer wavefunction # $ ψ i = α 1 A 1 + α 2 A 2 ˆ p ˆ U int = e ig ˆ x A ⟩ ⟩ à % & | " & # $ + +" & + % ' | " ' # $ + +" ' ' % & gA 2 Strong measurement (large g): projective measurement on well separated pointer positions P(x) implies projection of system state ψ f = A 2 ψ f = A 1 or % ' ' gA 1 Weak measurement (small g): projective measurement of pointer state gives almost no information, but correct average. x The system state in one measurement is almost unchanged! After reading the pointer P(x) gA 2 ψ f ≈ α 1 A 1 + α 2 A 2 + O ( g 2 ) gA 1 Price to pay for non-invasiveness: large uncertainty of the detection

Quantum dynamics: time evolution of a quantum system !ℏ # #$ Ψ($) = ) Ψ($) = + → ($) Ψ(0) * Ψ($) Forward time-evolution −!ℏ # #$ ⟨Ψ($)| = ⟨Ψ($)| ) ⟨Ψ($)| = ⟨Ψ(0)|+ ← ($) * Backward +me-evolu+on Physical expectations 2($) = ⟨Ψ($)| 3 = ⟨Ψ(0)|+ ← $ 3 2 Ψ $ 2+ → ($) Ψ(0) Backward and Forward time-evolution Quantum dynamics requires Forward and Backward time-evolution à Keldysh contour

The Keldysh contour: expanding the time dimension forward +me ! time initial state measurement Imaginary time ((*) backward +me Thermal state [Schwinger 1961 −"ℏ/% & ' Keldysh 1964]

Keldysh contour and measurements: projec4on forward time # ! initial state time $ ☹ ☹ backward time strong strong measurement measurement Imaginary time −"ℏ/% & ' ( ) + , * - Thermal state

Keldysh contour and measurments: weak and markovian (instantaneous) forward time " ! ini/al state - * time " " weak weak backward time measurement measurement Imaginary /me −"ℏ/% & ' {) * , , - } Thermal state

Keldysh contour and measurements: weak and con4nuous forward time * - " ! time ini/al state " " weak weak backward time measurement measurement with memory with memory Imaginary /me −"ℏ/% & ' {) * , , - } or " [ 0 ) * , 1 , - ] ? Thermal state

1) Full Counting Statistics: C 4 - sharpness Cumulants Probability that a total charge Ne 6 7 = ! is transferred in given time t 0 6 8 = ⟨Δ! 8 ⟩ P 6 < = ⟨Δ! < ⟩ C 2 - width Current & C 3 - skewness N V = const C 1 - mean (Quantum) definition ! = ∫ $% &(%) Definition through Cumulant Generating function (CGF): ) ! = ∫ $*+ ,-. + /0(.) + /0(.) = + , 1 -. = + ,.∫ 23 4 5(3) Is this correct in the quantum case? Levitov, Lesovik, JETPL (1993/94)

How to calculate the CGF quantum mechanically? Quantum mechanical current detection has to account for non-commuting current operators! Belzig, Nazarov, PRL 2001 Microscopic justification: time evolution of ideal current detecto r and projective measurement [Levitov et al. 1997; Kindermann, Nazarov 2003]. (Projection can be problematic for superconductors, due to charge-phase uncertainty [Belzig, Nazarov, PRL 2001]) Important difference to classical definition (see also Levitov, Lesovik 93)

Generalization of FCS to Time-Dependent Counting: “Probability” density functional for given current profile I(t): Inverse transformation classical average [c.f. stochastic path integral Sukhorukov, Jordan, et al. 2003]

Keldysh ordered! Quantum definition of CGF for time-independent FCS Generalization of standard Keldysh functional to time dependent counting [see also e.g. Nazarov and Kindermann, PRL 2004] Can we interpret this as probability density generating functional? No ! Analogous to Wigner function we can have negative probabilities Problem: current operators at different times do not commute The current cannot be measured at all times, but only up to some uncertainty

Handling non-projective (weak) measurements: Non-projective measurements: Orthogonal measurements Kraus operators P ositive O perator V alued Probability to find A M easure State after measurement Neumarks Theorem: Every POVM corresponds to a projective measurement in some extended Hilbert space See e.g. Milburn & Wiseman, Quantum Measurement and Control (Cambridge, 2009)

Proposed solution: weak Markovian measurement a la POVM Kraus operator (instead of projection operator) for Markovian measurement Noise of the detector Causality + uncertainty Positive operator valued probability measure (=projection in extended space) Neumarks theorem Positive definite probability distribution: A. Bednorz and W. Belzig, Phys. Rev. Lett. 101 , 206803 (2008)

Final result for current generating functional Generalized Keldysh functional Current generating functional with additional backaction and noise due to detector Gaussian noise of the detector Backaction of the detector (partial projection) Limiting cases: full projection Strong backaction large detector noise Weak measurement 9 → 0 : Large Gaussian with Φ & = ' 1 ,,3 4∫ *+, 5 (+)/78 ! " = ∫ %& ' ()∫ *+, + -(+) Φ & noise substracted!

Generalized Wigner functional WB and Y. V. Nazarov, Phys. Rev. Le7. 87 , 197006 (2001) Phys. Rev. Le7. 87 , 067006 (2001) A. Bednorz and WB, PRL (2008,2010) The generating function of a markovian quantum measurement is Keldysh-ordered: 4(2) . $ /∫ 12& 2 3 Φ " = $ % &,( )% *+, = . $ /∫ 12& 2 3 - 4(2) Quasiprobability density generating functional! Analogously to Wigner function we can have negative probabilities The generaOng funcOon of a non-markovian quantum measurement is ... ... (even) more complicated The answer to the quesOon of operator order: 7~9 : Φ " /9" < 9" = C = , B B Markovian: @ < @(=) → C < /2 G B I, B C, 3 Higher order Markovian: @EF → 7 H C(<), B B + ? ⊗ [ B C(<), B Non-Markovian: @ < @(=) → > ⊗ C(=) C(=)] >, ? depend on the detector, but arbitrary ordering possible ( à engineering)

2) Keldysh-ordered expectations are quasiprobabilities 1-Photon-Fock state Quasiprobability? posi8on Example: Wigner-function !(#, %) momentum = “Probability” for x and p Negative! Cannot be measured directly, but through a noisy and weak measurement Signatures of nega8vity (=non-classicality)? Viola2on of classical inequali8es, e.g. Bell, CHSH, LeggeQ-Garg, weak values.... Bednorz and WB, Phys. Rev. LeQ. 2008

We Weak posi*vity of of the Wigner-Ke Keldysh quas quasipr probabi bability [Bednorz & Belzig, PRB 2011] Weak markovian measurement scheme: { } C ij = A i A j = 1 A i , ˆ ˆ = positive definite correlation matrix A j 2 C can be simulated by classical probability distribution, e.g. ∑ − 1 A j − A i C ij /2 p ( A 1 , A 2 , … ) ~ e ≥ 0 ij With symmetrized second order correlaHon funcHons a violaHon of classical inequaliHes is impossible à the corresponding quasiprobability is weakly posi0ve 2 = 0 ( ) A 2 − 1 Note: does not assume dichotomy, corresponding e.g. to

Possible inequality à Cauchy-Bunyakowski-Schwarz (CBS) inequality ! " # " ≥ !# " à Fullfilled for all posi<ve probabili/es %(!, #)

Test of CBS with Wigner functional for current fluctuations " # = ∫ &'( )#* ! Current operator in frequency space: ! "(') Typical experimental setup # / 41 / /3 &56 ! We choose: - " # 6 ! " 0# and - . = ∫ 7 = ⋯ . # / 01 / /3 à measurement bandwidth Δ ;/< centered at 5 =/> .7 = Δ = Δ K 6 ! " # / 6 ! " 0# / 6 ! " # L 6 ! " 0# L 3 + Δ = 6 ! . 3 = Δ = 3 6 ! " # / 6 ! " # / 6 ! 3 " 0# / " 0# / 2 nd and 4 th -order correlators from tunnel Hamiltonian Forgues, Lupien, Reulet, PRL (2014) 4 F EH + ℎ. F. A B = C ' DE F DG See also Zakka-Bajjani et al. PRL (2010) DE Violation of CBS would be a proof of negativity of Wigner functional! . 3 7 3 ≥ .7 3 ? ? Bednorz and WB, Phys. Rev. Le@. 105 , (2010) Phys. Rev. B 81 , 125112 (2010)

Violation of CBS for a tunnel junction Maximally extended non-overlapping frequency intervals ! " ≈ 2Δ & + Δ " , ! & ≈ Δ & 5 = 5-Ω *+ = - . / = 0 Negative Wigner functional 23 4 2 4 3 4 *+ = 2ℏ! & - . / = 0 1 *+ = 0 - . / = ℏ! & ! " /! & [Bednorz, WB, PRB 2010, PRL 2010] ViolaEon: Quantum many-body entanglement of electrons in different dynamical modes E.g. nonequilibrium many-body wave funcEon, Vanevic, Gabelli, Belzig, Reulet, PRB 2016