Weak Quantum Measurement Schemes of Mesoscopic Currents

2451-3

Workshop on Interferometry and Interactions in Non-equilibrium Meso- and Nano-systems

8 - 12 April 2013

Weak quantum measurement schemes of mesoscopic currents

BELZIG Wolfgang University of Konstanz Department of Physics M703 Box 703, D-78457 Konstanz Konstanz GERMANY

Weak quantum measurement schemes of mesoscopic currents

Wolfgang Belzig University of Konstanz

Collaboration with Adam Bednorz (University of Warsaw) Kurt Franke (CERN) Christoph Bruder (University of Basel) Bertrand Reulet (University of Sherbrooke) Content

• Quantum measurement, strong and weak, correlations

• Markovian weak quantum measurement and quasiprobabilities, negative Wigner function, quantum violation of classical inequalities

• Weak positivity of the quasiprobability for Markovian measurements; Implications for mesoscopic Bell test and time reversal symmetry.

• Non-Markovian weak measurements and non-classical (squeezed) states of currents in tunnel junction

• Conclusion Quantum measurement

Measuring correlation function AB , ABC , ABCD

Projective measurement Weak measurements allow to measure subsequently A, B, C. Which operator order? Invasive or non-invasive? Quantum description of weak (and strong) measurement (markovian schemes) Von Neumann measurement: from strong to weak

Idea: couple system (Â) to pointer wavefunction (x,p) ψ = α A1 +α 2 A2 1 x igpˆAˆ ˆ gA Uint = e 2 P(x) Strong measurement (large g): Do projective measurement on well separated pointer positions, implies projection of system state gA1 or ψ f = A1 ψ f = A2 Pointer probabilities x Weak measurement (small g): projective measurement of pointer state gives almost no information, but correct average. P(x) The system state in one measurement is almost unchanged! gA2

gA1 After projection of pointer 2 ψ f ≈ α1 A1 +α 2 A2 + O(g ) Generalized measurement by POVMs

Positive operator valued measure (POVM): Measurement with result A: (not necessarily orthogonal) ρˆ → ρˆ(A) = Kˆ (A)ρˆKˆ †(A) Kraus operators ˆ K(A) Probability to ﬁnd A: ˆ † ˆ ∫ dA K (A)K(A) = 1 ρ(A) = Tr ρˆ(A)

E.g. projection operators: ˆ ˆ -> usual projective measurement K(A) = PA Naimarks theorem: Every POVM corresponds to a projective measurement in an extended Hilbert space (including detector)

Concatenation: ρˆ(A, B) = Kˆ (B)ρˆ(A)Kˆ †(B), ρ(A, B) = Tr ρˆ(A, B) ρˆ(A, B,C) = Kˆ (C)ρˆ(A, B)Kˆ †(C), ρ(A, B,C) = Tr ρˆ(A, B,C)

Wiseman, Milburn, Quantum measurement; Notes on Quantum Information by J. Preskill Weak measurement using Gaussian POVM (Kraus operator) = Gaussian detector wave function

(Aˆ − A)2 Phenomenological parameter λ λ −λ ˆ 4 4 λ → ∞ strong, projective measurement Kλ (A) = e 2π λ → 0 weak measurement, large error Measurement Kˆ †(A)ρˆKˆ (A) ρˆ → ρˆ = λ λ ; p(A) = Tr ⎡Kˆ †(A)ρˆKˆ (A)⎤ A p(A) ⎣ λ λ ⎦ Probability density after one measurement: Convolution of quasiprobability with classical Gaussian noise:

2 p (A) = dA′ λ / 2π e−λ(A−A′) /2q (A′) λ ∫ λ

Quasiprobability after the weak measurement λ → 0 dα ˆ ˆ q (A) = e−iα ATr ⎡ei(α /2)A ρˆei(α /2)A ⎤ = Tr ⎡ρˆδ A − Aˆ ⎤ λ=0 ∫ 2π ⎣ ⎦ ⎣ ( )⎦ Corresponds here to usual probability (c.f. cyclic property of the trace) ➠ only additional Gaussian noise Bednorz, Belzig, PRL 2010 Weak measurements and quasiprobability

Using a Gaussian detector wave functions (or POVM formalism) we find the probability to measure A and B as convolution of a quasiprobability with gaussian noise p(B, A) = dA'dB'g (B − B')g (A − A')q (B', A')  ∫ λ λ  λ probability gaussian noise quasiprobability 2 gλ (A) ~ λ / 2 exp ⎣⎡−λA / 2⎦⎤ Quasiprobability in the weak limit λ → 0 Bednorz, Belzig, PRL 2010 1 ˆ ˆ ˆ q(A, B) = dα dβe−iα A−iβBTr ⎡ρêiα A/2eiβBeiα A/2 ⎤ 4π 2 ∫ ⎣ ⎦ 1 AB = dAdB ABq(A, B) = Tr ⎡ρˆ Aˆ, Bˆ ⎤ ∫ 2 ⎣ { }⎦ • The quasiprobability can be equivalently defined through the quantum cumulants • In the limit of strong measurement λ → ∞ a projective measurement is obtained • In the weak limit λ → 0 we obtain the symmetrized correlation function • No backaction in the weak limit: signal ~ λ , changes to the system are ~ λ 2 Negativity of the quasiprobability can be measured by violating classical inequalities! Cumulants and correlators Quasiprobability of N weak measurements yield (after subtraction of detector noise) λ → 0 ˆ ˆ ˆ ˆ N −i(χ1A1++χN AN ) iχ A /2 iχ A /2 iχ A /2 iχ A /2 q(A ,..., A ) = d χe Tr ⎡e N N e 1 1 ρê 1 1 e N N ⎤ 1 N ∫ ⎣ ⎦

1 Correlators A A A = Tr ⎡ρˆ Aˆ , Aˆ , Aˆ , Aˆ ⎤ 1→2→...→N 1 2 N N−1 ⎢ 1 { 2 { N−1 N }} ⎥ 2 ⎣ { }⎦ • weak measurement requires large detector noise • measured quantities are always “as usual” probabilities • quasiprobability after we subtract the detector noise • order of measurements matters

Continuous measurement ➪ quasiprobability density functional: [Bednorz, Belzig, PRL 08]

i dtχ (t )I (t ) iχ (t )Iˆ(t )dt/2e iχ (t )Iˆ(t )dt/2e q I(t) = Dχe ∫ Tr Te∫ ρˆTe∫ [ ] ∫ C.f. counting statistics via Keldysh orderded cumulant generating function [used e.g. by Nazarov, Kindermann (2003), Golubev, Zaikin (2005)] Negative quasiprobability to violate classical inequalities

Bell inequality: using 2 detector settings for Alice (A,A’) and [Bell 1964, CHSH 1969] Bob (B,B’), all classical correlations fulﬁll the inequality 〈AB〉 + 〈A′B〉 + 〈AB′〉 − 〈A′B′〉 ≤ 2 2 2 2 2 with the additional assumption A = B = A′ = B′ = 1 Leggett-Garg inequality (“Bell inequality in time”) tests the question if a variable has a deﬁnite value if it is not observed (the macrorealism assumption)

〈Q(t2 )Q(t1)〉 + 〈Q(t3 )Q(t2 )〉 − 〈Q(t3 )Q(t1)〉 ≤ 1 [Leggett, Garg, 1988] 2 with the additional assumption Q (t) = 1

Weak values: Measurement protocol [Aharonov, Albert, Vaidman, PRL 88] 1) Measure A weakly (+noisy) Deﬁnes weak value of A (and an upper limit): 2) Measure B projectively 3) Select only results for some B 〈Pˆ Aˆ〉〈Aˆ〉 = B ≤ Max A with the additional assumption B w 〈Pˆ 〉 that Max() is known B Weak positivity of the Markovian quasiprobability

Second-order quantum correlations C = A A = Tr ⎡ρˆ Aˆ , Aˆ ⎤ / 2 ➠ positive deﬁnite matrix ij i j ⎣ { i j }⎦ ⎡ ⎤ A classical Gaussian distribution can reproduce the p ~ exp A C −1A / 2 quantum second-order correlations matrix g ⎢−∑ i ij j ⎥ ➠ all classical inequalities with second-order ⎣ ij ⎦ correlations only are fulﬁlled by the quasiprobability [Bednorz & Belzig, 2011]

It is impossible to violate a classical inequality using only second order correlation functions without further assumptions! ➠ Weak positivity

Note on dichotomic variables (c.f. Bell or Leggett-Garg inequalities): The additional assumption A = ±1 2 has to be experimentally veriﬁed by 4th-order correlator (A2 − 1) = 0 Quasiprobability and Wigner function Examples

1/2 −1/2 −1/4 −x 2/2a2 ψ1(x)=2 ψ0(x)x/a ψ0(x)=a π e

Quasiprobability and Wigner function Examples πhW− (x,p) 1 − 1 1 πhW0(x,p) 0.5 The Wigner function (for a pure state) 0.5 1 1 0 1 y y 0 -0.5 * ipy 1/2 -0.5 -1 W (x, p) = dyψ−(1x/2+ −)1ψ/4 (−x x−2/2a)2e ψ1(x)=2 ψ0(x)x/a -1 0 2 ψ0(xh)=∫ a π2 e 2 2 1 1 0 pa/h− 0 pa/h− -1 -1 -2 -1 - can take negative values 0 -1 0 1 2 -2 -2 -1 0 -2 1 2 x/a - gives correct expressions for correlators of x or p x/a

Measurements of negativity: W1(0, 0) = 1/π! πhW− (x,p) W0(0, 0) = 1/π! 1 − − 1 1 Reconstruction fromπhW0 (x,p)density matrix, 0.5 0.5 1 0 assuming only 1 photon 1 is present 0 [e.g. Lvovski et al. (2001); Walraff group (2010)] -0.5 -0.5 -1 -1 0 2 2 1 1 0 pa/h− 0 pa/h− -1 -1 -2 -1 0 -1 0 1 2 -2 -2 -1 0 -2 Quasiprobability to measure x and p 1simultaneously 2 x/a x/a (but weakly) 1 −iα x−iβ p iα xˆ/2 iβ pˆ iα xˆ/2 q(x, p) = dα dβe Tr ⎡ρˆe e eW1(0⎤, 0)=  = =1W/π(!x, p) W2 (0∫ , 0) = 1/π! ⎣ ⎦ 4π 0 − Generalizes Wigner function to arbitrary sequential measurements and arbitrary operators. Negativity of the Wigner function by cumulants

Consequence of weak positivity [Bednorz, Belzig, PRA 2011]:

To show the negativity of the Wigner function one needs at least 4th-order cumulants

Prediction tested experimentally using single photons generated by an heralded cavity-enhanced non-degenerate parametric down-conversion. [Eran Kot, Niels Grønbech-Jensen, Bo M. Nielsen, Jonas S. Neergaard-Nielsen, Eugene S Polzik, and Anders S. Sørensen, Phys. Rev. Lett. 108, 233601 (2012)]

N /2 n M = 1+ C x2 + p2 ∑n−1 2n ( ) Data Classical inequality: F = M 2 > 0 2 2 2 σ F = F − F

Violation by almost 20 standard deviations! Maximal order of moment Cumulant based many- body entanglement test Cooperpair splitter 1) Supraleiter Generic entangler (e.g. superconducting junction or tunnel contact): 3) quantumQuanten− dotspunkte Detection of entanglement using spin- barriers Barrieren direkt ﬁltered current-correlation measurements of Alice and Bob? Andreev Energieluecke

N1 N2 N1 Supraleiter N2 Bob Alice     [Experimental setups: Beckmann et al. a(a ') b(b ') (2004), Russo et al. (2006), Hofstetter et 2a) 2b) al. (2009,2011), Herrmann et al. (2009), Das et al. (2012)...] Supraleiter N1 Supraleiter N1 positive negative Korrelation Korrelation N2 N2     Observable: A = σ a , B = σ b ,... Bell test  1 2 Correlation C(a,b) = AB ,...

Bell inequality as entanglement criterion [Bell 1964, CHSH 1969]         Classical correlations fulﬁll C(a,b) + C(a ',b) + C(a,b ') − C(a ',b ') ≤ 2 Conditions for derivation of Bell inequality: 1. Correlators described by classical positive probabilities 2. The observables take only the values ±1

    singlet  Quantum mechanics  ⎡ ˆ  ˆ ⎤  C(a,b) = tr σ 1a σ 2b ρˆ = − ab ⎣( )( ) ⎦ Maximal violation: singlet state, coplanar detectors a'  1 ab = 2 2 > 2 b 2 Violation indicates non-classical correlation a b'

[Experiments: Aspect et al. (1982), Tittel et al (1998)] Bell test using current correlations   t0 a a   Nˆ − Nˆ ˆ a ˆa ˆ ↑ ↓ The observables are spin-resolved currents N = I (t)dt A =   σ ∫ σ ˆ a ˆ a 0 N ↑ + N ↓ The correlator is usually deﬁned via     (N a N a )(N b N b ) [Kawabata (2001);   ↑ − ↓ ↑ − ↓ Chtchelkatchev et al. (2001); C(a,b) =     a a b b Samuelsson, Büttiker et al.; Beenakker et al.; ....] (N ↑ + N ↓ )(N ↑ + N ↓ )

The Problem: [e.g. Hannes, Titov (2008)]   * Long detection times ➠ C ( a , b )  1 ➠ violation impossible (and zero temperature)    * High temperature ➠ C ( a , b ) = − a b ➠ always violation

Basic problem: observables are not bounded and due to weak positivity 2nd-order correlators are insufﬁcient without additional assumptions. [Bednorz & Belzig (2011)] Bell-CHSH-type inequality for continuous variables using 4th-order correlators

Using a set of 4 variables A,A’ and B,B’ without simultaneous measurement of AA’ and BB’ K(A, B) = AB(A2 + B2 ) [Bednorz, Belzig, PRB 2011] K(A, B) + K(A', B) + K(A, B') − K(A', B')

4 4 4 4 1 2 ≤ A + B + A' + B' + ∑ C4 D4 (D2−E2 ) 2 D≠C ,E≠C ,D,D ' ∈{A,B,A ',B '} Similar to the Bell inequality, but contains only 4th order correlators

Violation can only be explained by non-classical correlation between A and B

Note: • For spin variables (A2=1,...) the usual Bell inequality is recovered AB + AB' + A'B − A'B' ≤ 2 • The inequality needs no further assumptions Violation at a tunnel Cooper pair splitter with perfectly symmetric spin ﬁlters

3  4  2 Symmetric setup: AB = ab A A2 B2 = A4 − 2 1− ab A2   ( ) ˆ ˆ a ˆ a A = N↑ − N↓ , etc. 2       A Resulting inequality: ab ab ' a'b a'b ' 2 8 (theoretically) + + − ≤ + 2 A4

Maximal value a' in pure singlet Violation only possible if this is small enough! states: b E.g. tunnel contact and short detection 4 2 2 2 a A = A  1 b'

2n eV 〈A 〉q  I ⋅t0 coth( ) Measurement of 4th cumulants is 2kBT necessary to ensure Poisson statistics (I / eV )⋅t0max{eV,kBT}  1 Time reversal symmetry and quantum weak measurement Classical vs. quantum time- resolved measurement

Does the observation of a system in thermal equilibrium show time-reversal symmetry (T)?

Measurement Classical Quantum

T is broken T is broken strong (order of disturbances (order of projections (invasive) inﬂuences the dynamics) inﬂuences the state)

T is observed weak (measurement is completely ? (non invasive) independent of the dynamics) Time reversal symmetry

Classical stochastic movie under time-reversal T

T T T P(a1(t1),…,an (tn )) = P (an (−tn ),…,a1 (−t1)) T In equilibrium and for T-invariant observables ai = ai

a1(t1 )an (tn ) = an (−tn )a1 (−t1 )

Consequence for a 3-point correlation function a(0)a(t)a(t + t ') = a(0)a(t ')a(t + t ')

T +t+t ' 0,t,t + t ' ⎯⎯→−t − t ',−t,0 ⎯⎯⎯→0,t ',t + t ' For a non-invasive measurement, we expect this relation to hold, because the Hamiltonian of the observed system is time-reversal invariant. The weak quantum time paradox

Classical correlation functions of time-reversal-invariant dynamics a(0)a(t)a(t + t ') = a(0)a(t ')a(t + t ')

Quantum correlation functions (measured weakly)

Tr ⎡ρˆ Aˆ, Aˆ(t), Aˆ(t + t ') ⎤ Tr ⎡ρˆ Aˆ, Aˆ(t '), Aˆ(t + t ') ⎤ ⎣ { { }}⎦ ≠ ⎣ { { }}⎦

Classical expectation is not matched: A quantum system observed weakly at equilibrium seemingly breaks time-reversal symmetry Possible consequence: Quantum non-invasive measurement does not exist, since it makes us see things which are not (expected?) there!

Bednorz, Franke,Belzig, NJP 2013 Measurable quantities are Time-reversal symmetry cumulants in frequency space Asymmetry means a non- breaking in a quantum dot vanishing (imaginary) third cumulant:

Im S N (ω,ω ) = Im dtdt eiωt +iω ′t ′ 〈δn(t)δn(t )δn(0)〉 ≠ 0 3 ′ ∫ ′ ′

voltage a −¯h(ω + ω!)/Γ ε =Γ/2 δn = n − n 1 current capacitive dot interaction I =aI0 + αN dot 1 1 b

¯hω ¯hω tunneling

Γ Γ tunneling

2 N ! 2 reservoir

8πΓ Im S3 (ω,ω )/¯h reservoir dot energy SreservoirI = S I 0+ α 3S N (ω,ω′) -2 -1 0 1 2 3 3 3 Non-markovian measurements and non- classical current states More general measurment schemes with time-delay and memory

What is the most general Kraus operator to describe weak measurments?

So far: instantaneous Krausoperators ˆ Ka(t ) (A(t)) Consequence: symmetrized correlators a(t)a(t ') Tr ⎡ Aˆ(t), Aˆ(t ') ⎤ / 2 = ⎣ρ{ }⎦ Corresponding quasiprobability: Wigner function (obeying weak positivity)

Now: Kraus operators with memory: ˆ ⎡ ˆ ⎤ ˆ ˆ 2 K ⎣A,a⎦ = k(a)(1+ ∫ dt 'F[a,t]A(t′) + O(A ))

General assumptions: ⎡ ˆ ⎤ and causality lead to the a(t) = Tr ⎣ρA(t)⎦   Correlation functions a (t )…a (t ) = d nt′ Tr ⎡TA (t − t ' )…A (t − t ' )ρ⎤ 1 1 n n ∫ ⎣ n n n 1 1 1 ⎦ Causality Arbitrary real function (depends on F)  Superoperator A(t − t ')X = δ (t − t ') Aˆ(t), X / 2 + f (t − t ')⎡Aˆ(t '), X ⎤ / 2 { } ⎣ ⎦ [Bednorz, Bruder, Reulet, Belzig, arxiv:1211.6056] Fixing the memory function for thermal detectors: equilibrium order

Choices of the memory function  A) Markovian: A(t − t ')X = δ (t − t ') Aˆ(t), X / 2 + f (t − t ')⎡Aˆ(t '), X ⎤ / 2 { }   ⎣ ⎦ --> Wigner quasiprobability =0

B) Non-Markovian: Intuitively in equilibrium no information is transfered

We require: S (ω ) = dteiωt a(t)b(0) = 0 ab ∫ This condition can be fulﬁlled if (derived using ﬂuctuation-dissipation theorem)

f (ω ) = −i coth ω / 2k T ( B ) Further consequence: Arbitrary higher order correlation functions vanish as well

Equilibrium order: Deﬁnition of quasiprobabilities with an arbitrary detector temperature Td≠T f (T ,ω ) = −i coth ω / 2k T d ( B d ) Describes a detector in a thermal state, but not in equilibrium with the measured system Harmonic oscillator and squeezing

Non-Markovian weak measurement scheme applied to harmonic oscillator

1 2 2 ⎛ † 1⎞ H = xˆ + pˆ = ω aˆ aˆ + ( ) ⎝⎜ ⎠⎟ 2 2 Time-dependent correlation functions follow classical e.o.m.; For Td=0 f (ω ) = −isgn(ω )

n *k †k n Initial equal-time correlators α α = Tr ⎡aˆ aˆ ρˆ ⎤ Td =0 ⎣ ⎦ Same result as Glauber- d 2α α nα *k P (α ) ≡ Tr ⎡aˆ†kaˆn ρˆ ⎤ Sudarshan P-quasiprobability ∫ Glauber ⎣ ⎦

Equilibrium order of operators for Td=0 is reminiscent of absorptive photon detection.

Squeezing: If one of the quadrature variances is 2 1 below the Heisenberg boundary, e.g. Δx ≤ 2 Probe of the negativity of the equlibrium order quasiprobability -> Possibility to violate classical Δx2 ≤ 0 inequalities with 2nd-order correlators! Td =0 [see e.g. Shchukin, Richter, Vogel, 2005] Test with ac-excited current correlations at a tunnel junction

t0 Current quadratures Classical inequality iωt Iω = dt e I(t) Aˆ (i / 2) Iˆ Iˆ ∫ = ( ω − −ω ) 0 Bˆ = (1/ 2) Iˆ + Iˆ 2 2 2 ( ω −ω ) I − I ≥ 0 ⇔ δ I ≥ Re δ I (*) ω −ω ω ω Vacuum ﬂuctuations (detection time t0) AC-excited tunnel junction with frequency Ω = ω ⎡Aˆ, Bˆ ⎤ = ωGt 4 ⎣ ⎦ 0 Violation of (*) is equivalent to the p ˆ ] througha ˆ† =(ˆx + ipˆ)/ 2 with [ˆa, aˆ†]=1. This leads 0 2 Markovian order c q c q t 2.0 Re I W squeezing condition

toa ˇ =ˇa f(⌦)ˇa /2 anda ˇ† =ˇa† + f(⌦)ˇa† /2. In the Ω ` ` zero-temperature case, f(⌦)=i (a perfect photodetec- I -W I W

~ hG tor), and defining ↵ =(x+ip)/p2 we get the single-time [ 1.5 ` ` Aˆ 2 ≤ ⎡Aˆ, Bˆ ⎤ / 2 n k n k I -W ,I W 2 ⎣ ⎦ quasiprobabilistic average ↵ ↵⇤ =Trâ ⇢âˆ† .Onthe X H L\ h i Equilibrium order other hand, this is a property of the Glauber-Sudarshan 1.0 X H L H L\ function P (↵), defined by⇢ ˆ = d2↵ P (↵) ↵ ↵ for nor- | ih | X8 H L H L<\ê malized coherent statesa ˆ ↵ = ↵ ↵ , ↵ ↵ = 1 [21]. 0.5 AC-excitation of a tunnel n k 2 n| i kR | i h |n i k Since ↵ ↵⇤ = d ↵↵ ↵⇤ P (↵)=Trâ ⇢âˆ† ,wefind junction creates a current state h iP that the quasiprobability for a zero-temperature detector noise Current eVac resembling squeezed light R is identical to P (↵). It is interesting to note that revers- 0.5 1.0 1.5 2.0 2.5 3.0 —W ing the sign of f leads to the Husimi-Kano Q function instead of P [2], while f = 0 gives the Wigner function FIG. 1: (color online) Quantum correlation functions (in units [7, 27]. of 2⇡G~⌦t0) for a tunnel junction at zero temperature T =0. The fact that we obtain the P -function shows the deep The emission noise Iˆ( ⌦)Iˆ(⌦) (red line) violates the classi- h i connection between the non-Markovian weak measure- cal inequality (13) for a certain range of eVac (shaded region). ment formalism and the quantum-optical detector the- This violation is equivalent to the squeezing condition (15) for the symmetrized noise Iˆ( ⌦), Iˆ(⌦) /2(blueline). ory. One of the interesting consequences is that zero- h{ }i temperature equilibrium ordering is consistent with pho- toabsorptive detection schemes, in which the P -function appears naturally [2]. It is also interesting to draw a where Jn are the Bessel functions. In the case of equi- link between the violation of weak positivity in equilib- librium ordering at Td = 0 one only has to subtract 2⇡~G ! (! + !0) from the above result. As shown in rium ordering and the properties of squeezed states. The | | ground state of a harmonic oscillator fulfills xˆ2 =1/2, Fig. 1, the classical inequality is violated for ! = ⌦ in 2 h i a certain range of eVac/~⌦, but only in equilibrium or- which corresponds to x P = 0. A squeezed state can be such that xˆ2 < 1/2,h stilli minimizing the Heisenberg un- dering. This can be reinterpreted in terms of the exis- certainty principle.h i This translates into a negative vari- tence of squeezing in the quantum shot noise : consider ance of the position described by the (quasiprobability) the two quadratures associated with the finite-frequency 2 current operator: Aˆ = i[Iˆ(!) Iˆ( !)]/2 and Bˆ = P -function, i.e. x P < 0 [28] and is therefore equivalent ˆ ˆ ˆ ˆ h i [I(!)+I( !)]/2. Using [I(!), I( !)] =2t0G~!,we to a violation of weak positivity. ˆˆ h i find [31] [A, B] = it0G~! (with the total detection time Let us now consider how our results apply to the case h i of current fluctuations in mesoscopic conductors. The t0). Thus the squeezing condition [2], quantum description of the noise in the junction, SI (!)= Aˆ2 < [A,ˆ Bˆ] /2 , (15) dtei!t I(t)I(0) ,whereI(t)=I(t) I(t) ,willde- h i |h i| h i h i 2 pend on the choice of f in (1). For f = 0, we get sym- is related to the violation of weak positivity, A w < 0 R s h i metrized noise SI = G~ w(! eV/~,T)/2, where in equilibrium ordering with Td = 0 and allows to violate ± ± G is the conductance, V is the constant bias voltage Eq. (13). Hence, according to Fig. 1, quantum shot noise P and w(↵,T)=↵ coth(~↵/kBT ) [19]. For f given by with AC-driving creates current states, which resemble equilibrium ordering with an arbitrary Td, we obtain squeezed light for a certain range of the AC-voltage. s SI = S G~w(!,Td). Hence, the detection schemes In conclusion, we have presented a theory of a generic I di↵er by a term that is independent of the voltage and weak-measurement scheme that includes emission noise. the temperature of the system, making it impossible to It requires a non-Markovian POVM with a specially cho- detect non-classicality in this scheme. sen memory function f, which has no analog in the An experimentally feasible test of squeezing and vi- Markovian picture. The scheme is consistent with the olation of weak positivity is possible using a coherent absence of information flow between system and detec- conductor (e.g. a tunnel junction for the sake of simplic- tor in equilibrium at a given temperature. Hence any ity) subject to an AC voltage bias V (t)=Vac cos ⌦t[29]. detection requires a nonequilibrium situation. Another Consider the classical inequality direct consequence is that even the simple Markovian detection process must involve a nonequilibrium detec- I(!) I( !) 2 0 I(!) 2 Re I2(!) . (13) | | )h| | i h i tor state. Finally, nonsymmetrized ordering leads to a For symmetrized ordering one gets [30] violation of weak positivity, which can be tested experimentally by violation of suitable inequalities, equivalent Iˆ(!), Iˆ(!0) /2=2⇡~G (! + !0 2m⌦) to the squeezing condition in some cases. h{ }i m Note added: Our prediction about AC-driven squeez- X ing has been recently confirmed experimentally [32]. Jn(eVac/~⌦)Jn 2m(eVac/~⌦)w(! n⌦) , (14) n We acknowledge useful discussions with A. Klenner, X Conclusion

• Measuring quantum correlation requires to study weak measurments • Weak measurement correlations are conveniently described by a quasiprobability (sometimes negative + Gaussian noise) • The Markovian quasiprobability obeys weak positivity and second-order correlation functions are not sufﬁcient to violate classical inequalities. • The negativity of the Wigner function can be tested by measuring higher cumulants (experimentally veriﬁed) • Entanglement test in mesoscopic Cooper pair splitters requires a violation of an inequality with forth-order current correlators. • Weak quantum measurements apparently violate the time-reversal symmetry although the system is time-reversal symmetric • The non-Markovian detection scheme describes a set of quasiprobabilities interpolating between emission and absorption • Emission noise can show non-classical current states analog to squeezing

Bednorz&Belzig, PRL 08, PRL 10, PRB 11, PRA 11 Bednorz, Franke, Belzig, NJP 13; Bednorz, Bruder, Reulet, Belzig, arxiv 12