Crossover between strong and weak measurement in interacting many-body systems

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Crossover between strong and weak measurement in interacting many-body systems

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PAPER

Crossover between strong and weak measurement in interacting

many-body systems

Iliya Esin1,4

, Alessandro Romito2

, Ya M Blanter3

and Yuval Gefen1

1 _{Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel}

2 _{Dahlem Center for Complex Quantum Systems and Fachbereich Physik, Freie Universität Berlin, D-14195 Berlin, Germany} 3 _{Kavli Institute of Nanoscience Delft, University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands}

4 _{Author to whom any correspondence should be addressed.} E-mail:iliya.esin@weizmann.ac.il

Keywords: weak measurement, strong measurement, weak value, Mach–Zehnder interferometer

Abstract

Measurements with variable system–detector interaction strength, ranging from weak to strong, have

been recently reported in a number of electronic nanosystems. In several such instances many-body

effects play a significant role. Here we consider the weak-to-strong crossover for a setup consisting of

an electronic Mach–Zehnder interferometer, where a second interferometer is employed as a detector.

In the context of a conditional which-path protocol, we define a generalized conditional value (GCV),

and determine its full crossover between the regimes of weak and strong

(projective) measurement.

We

find that the GCV has an oscillatory dependence on the system–detector interaction strength.

These oscillations are a genuine many-body effect, and can be experimentally observed through the

voltage dependence of cross current correlations.

1. Introduction

Measurement in quantum mechanics is inseparable from the dynamics of the system involved. The formal framework to describe quantum measurement, introduced by von Neumann[1], allows to consider two limits:

in the limit of strong system(S)–detector (D) coupling, the detector’s final states are orthogonal. This is associated with the evasive notion of quantum collapse. In the other limit, that of weak(continuous) measurement of an observable(reflecting weak coupling between S and D [2]), the system is disturbed in a

minimal way, and only partial information on the state of the latter is provided[3]. We note that this hindrance

can be overcome, by resorting to a large number of repeated measurement(or a large ensemble of replica on which the same weak measurement is carried out).

Weak measurements, due to their vanishing back-action, can be exploited for quantum feedback schemes [4,5] and conditional measurements. The latter is especially interesting for a two-step measurement protocol

(whose outcome is called weak value (WV) [6]), which consists of a weak measurement (of the observableAˆ),

followed by a strong one(ofBˆ), A B[ ˆ, ˆ]¹0.The outcome of thefirst is conditional on the result of the second (postselection). WVs have been observed in experiments [7–12]. Their unusual expectation values [6,13–15]

may be utilized for various purposes, including weak signal amplification [16–23], quantum state discrimination

[24–26], and non-collapsing observation of virtual states [27]. The particular features of WVs rely on weak

measurement, and are washed out in projective measurements. Understanding the relation and the crossover between these two tenets of quantum mechanics is therefore an important issue on the conceptual level.

The WV protocol perfectly highlights the difference between weak and strong(projective) measurements, thus providing a platform to study the crossover between the two. Indeed, within the two-step measurement protocol, it is possible to control the strength of thefirst measurement. This allows to define a generalized conditional value(GCV), interpolating between WV and (strong value) (SV). The latter, in similitude to WV, refers to a two-step measurement protocol. Unlike WV, in a SV protocol both steps consist of a strong measurement. The mathematical expression for GCV is depicted below in equation(1). It amounts to the OPEN ACCESS

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REVISED

21 November 2015

ACCEPTED FOR PUBLICATION

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PUBLISHED

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average of thefirst measurement’s reading (whatever its strength is), conditional on the outcome of the second measurement. This has been studied in the context of single-degree-of-freedom systems[28–31], where the

WV-to-SV crossover is quite straightforward and is a smooth function of the interaction strength. We note that in experiments with electron nanostructures, interactions between electrons play a crucial role. A many-body theory of variable strength quantum measurement is called for. In many cases, the interaction strength can be controlled experimentally[10,32].

In this letter, we demonstrate theoretically that interactions can modify this weak-to-strong crossover in a qualitative way, in particular, making it an oscillating function of the interaction strength. Conversely, these oscillations serve as a smoking gun manifestation of the many-body nature of the system at hand, and present guidelines for observing them as function of experimentally more accessible variables(e.g. the voltage bias). Our analysis sheds light on the relation between two seemingly very different descriptions of quantum measurement, with emphasis on the context of many-body physics.

Motivated by the two step WV protocol, we define the (GCV) of the operatorAˆas an average shift of the detector, qd =ˆ qˆ- á ñqˆ ∣g 0= ,during the measurement process, projected onto a postselected subspace by the projection operator,P and normalized by the bare S–D interaction strength, g. The GCV is given byf,

A qU U g U U Tr Tr , 1 f f GCV 0 0 d r r á ñ = P P ˆ { ˆ ˆ ˆ } { ˆ ˆ } ( ) † †

whereρ0is the total density matrix which describes the initial state of S and D, and the time ordered operator

U =e-i

ò

_-¥SDdt ¥

ˆ _{describes the evolution in time of the whole setup during the measurement. Here, the} system–detector coupling,_SD_{= -}gw t pA,_{( ) ˆ ˆ with w(t)—the time window of the measurement; qˆ andpˆare} the‘position’ and ‘momentum’ operators of the detector ([ ˆ ˆ]q p, =i). We note that equation (1) provides the

correct WV[6] and SV [33] in the respective limits (g1,g1). Our approach here is in full agreement with earlier analyses of quantum measurement in the context of single particle systems[28–31].

Our specific setup is depicted in figure1. It consists of two Mach–Zehnder interferometers (MZIs), the ‘system’ and the ‘detector’ respectively, that are electrostatically coupled [32,34]. It is possible to tune the

respective Aharonov–Bohm fluxes,F andS FDindependently[32].

2. A two-particle analysis

As a prelude to our analysis of a truly interacting many-body system, we briefly present an analysis of the same system on the level of a single particle in the system, interacting with a single particle in the detector. According to this(over)simplified picture, particles going simultaneously through the interacting arms 2 and 3 (see figure1), gain an extra phaseeig_{[35,36], where γ takes values in the range 0,[ p].First, we consider the}

intra-MZI operators, defined in a two-state single particle space,{∣mñ},with m = 1, 2 for the ‘system’ (an electron propagating in arm 1 or 2) and similarly m = 3, 4 for the ‘detector’. The dimensionless charge operator (measuring the charge between the corresponding quantum point contacts (QPCs)), in this basis has a form Qm=∣m m .ñá ∣ The transition through the pth QPC is described by the scattering matrix

r t t r , p p p p p ⎛ ⎝ ⎜⎜ * ⎞_⎠⎟⎟  =

-Figure 1. Two MZIs, the‘system’ and the ‘detector’, coupled through an electrostatic interaction (wiggly lines). The sources S1 and S4 are biased by voltage V and the sources S2 and S3 are grounded.FSandFDare the magneticfluxes through the respective MZIs. The lengths of the arms 1 and 2 between SQPC1 and SQPC2 areaLand L, respectively, and similarly for the detector’s arms 3 and 4, as is shown in the_{figure. In the present analysis}a =1.

p=1 , 2 , 1 , 2s s d d[37]. The entries rpand tpencompass information about the respective Aharonov–Bohm flux

and for p=2 , 2 ,s d about the orbital phase gained between the two QPCs. The dimensionless current operators at the source(S1, 2S )and the drain D( 1,D2)terminals of the system-MZI are given by ISm=1sQm1s

†

and IDm=2sQm2s

† _{respectively, with m}

1, 2,

= and similarly for the detector with m= 3, 4 and employing the matrices1dand2d.

In view of equation(1), the initial state of the setup, which is described by the injection of two particles into

terminals S1 and S4 respectively, can be written as the density matrixr =₀ IS1ÄIS4operating in the two-particle

product space, m∣ ñ Ä ∣nñ(m=1, 2,n= 3, 4). The corresponding dynamics is that of two particles propagating simultaneously through arms m and n. The interaction between the particles is described by the operator Uˆ =eig ÄQ2 Q3._{A positive reading of the projective measurement consists of the detection of a particle at D2, and}

is described by the projection operatorP =f ID2Ä  The detector reads the current at D3. ( qd of equation(1) corresponds toÄdID3). Plugging these quantities into equation (1) yields an expression for the two-particle

GCV(see appendixA) Q I I I I 1 . 2 I I I 2 GCVTP D2 D3 D2 D3 D2 D3 D2 d g g d á ñ = á ñ á ñ = á ñ + áá ññ á ñ

(

)

( )

The averages are calculated with respect to the total density matrix after the measurement, O Tr OU r₀U .

á ñ =ˆ _{{ ˆ ˆ}† _{ˆ }}

We have defined IdD3 ID3- á ñID3 ∣g=0,andááI ID2 D3ññ  áI ID2 D3ñ - áID2ñá ñID3 is the

irreducible current–current correlator. A straightforward calculation (see appendixB) yields

Q I Q I Q Q I Q Q I Q I I Q Q Q I Q Q 4 sin ie sin 4 sin ie 4 sin , 3 TP 2 GCV 2 D2 2 0 D3 3 ₂ 2 D2 2 0 3 D3 3 D2 0 ₂ D2 2 0 3 0 2 ₂ 2 D2 2 0 3 0 i 2 i 2 Re Re g d d á ñ = á ñ á ñ + á ñ á ñ á ñ + á ñ á ñ + á ñ á ñ g g g g g g

{

}

{

}

( )

( )

( )

( )

( )

where Oá ñ ˆ 0 Tr{ ˆOr0}is an average with respect to the non-interacting setup,ádI QD3 3ñ

I QD3 3 0 ID3 0 Q3 0

á ñ - á ñ á ñ

 andádQ I Q3 D3 3ñ áQ I Q3 D3 3 0ñ - áID3 0ñ áQ32ñ0.This result shows a smooth and trivial crossover between the weak(g 0) and strong (gp) limits. The specific form depends on the parameters of S and D(the magnitude of the inter-edge tunneling; the value of the Aharonov–Bohm flux). For some range of values(e.g., t1s =t2s =t1d=t2d=0.1,F F =S 0 0.99 ,p F =D 0) the function is non-monotonic (but non-oscillatory), while for other values it is monotonic.

Figure 2. The relevant Feynman–Keldysh diagrams for the quantities in equations (7) and (8) to leading order in tunneling matrix

elements.‘Semi-classical’ paths of the particles are marked by solid lines (red) and dashed lines (blue), corresponding to forward and backward propagation in time(see equation (10)). (a) The average current (equation (7)),O_(G2₎._{Only the system part of the setup(see}

figure1), while all degrees of freedom of the detector part have been integrated out. (b) The reducible current–current correlator

3. A full many-body analysis

The Hamiltonian_=S_+D_+SD_{describes the system, the detector, and their interaction. The system’s}

Hamiltonian consists of S _T , 0 S S int S  = + + with v x x x a i d : , 4 m m m m x m m 0 S F 1 2 m  = -

å

ò

Y ¶ Y = ( ) ( ) ( ) † x x x x h. c., 4b T S 1s 1 11 2 21 2 1 12 2 22 s s s s s  = G Y†( )Y ( )+ G Y†( )Y( )+ ( ) g dx : x x : . 4c m m m m m m int S 1 2 2  =

å

ò

Y Y =  ( †( ) ( )) ( )

HereΓpis the tunneling amplitude at QPC p and xmp is the coordinate at QPC p on arm m. A similar expression

holds for the‘detector’ MZI,SD,with a summation over the chiral armsm=3, 4.We next assume that the lengths of the interacting arms are equal, x₂2s -x₂1s =x₃2d-x .₃1d _{The S}–D interaction Hamiltonian is

g dx dx x x : x x :: x x : , 5

SD

2 3 2 3 2 2 2 2 3 3 3 3

 = _^

ò ò

d( - ) Y†( )Y( ) Y†( )Y( ) ( )

where the normal ordering with respect to the equilibrium(no voltage bias) state is defined as :Y Y† :Y Y - ᆠ0∣Y Y ñ† ∣0_.

We are now at the position to construct the GCV for the actual many-body setup. We employ equation(2) to

define the many-body GCV of Q2

Q v g I t I t I I 1 d 0 , 6 2 GCVMB F D3 2 2 D2 D3 D2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

ò

d t á ñ = á ñ + áá ññ á ñ t t ^ -( ) ( ) ( )

where the current operator is given by, I x t( , )=e :vF Y†(x t, ) (Y x t, ): .We average over time _vL.

F

t  The problem is now reduced to the calculation of average currents and a current–current correlator. This is done perturbatively in the tunneling strength, but at arbitrary interaction parameter, employing the Keldysh

formalism. In this limit expectation values are taken with respect to tunneling decoupled edge states. The current is, I x iev G x x G x x G x x 2 d 2 , , , , 7 p q p q p cl p q cl q q D2 F , 1 ,2 1, 1 2, 2 2 1, 1 s s  

ò

å

w_p w g w g w g á ñ = - G G ab - bg gd - d z - za = ( ) ( ) ˆ ( ) ˆ ( ) ˆ ( ) { }

and the irreducible current–current correlator (see appendixC)

t I x t I x e v G x x G x x M x x x x G x x G x x 1 d , , 0 2 d d 2 , 2 , 2 , , , ; , , , 2 , 2 ... . 8 pqrs p q r s p cl r cl p q r s cl s cl q q q 2 2 D2 D3 2 F2 2 3 2 1, 1 2 4, 4 3 2 2 3 3 3 2 4, 4 3 1, 1 2 2 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎪ ⎪ ⎧ ⎨ ⎩ ⎛ ⎝ ⎞⎠ ⎛⎝ ⎞⎠ ⎛ ⎝ ⎞⎠ ⎛⎝ ⎞⎠ ⎫ ⎬ ⎭  

ò

å

ò

t t w w p w w g w w g w w w g w w g w w g g áá ¢ ññ = G G G G ´ ¢ - - - + ¢ ´ - ¢ - - + + t t ab bg hq qi ikgd kl lm d z za mh _w p t w tp - -( ) ( ) ( ) ¯ ˆ ¯ ˆ ˜ ( ¯ ) ˆ ¯ ˆ ¯ ˆ ˆ ∣ { }∣ ( ) ¯ ¯

Here{ }... reproduces thefirst part of the rhs, withw ¯ _t2p replaced byw  -¯ 2_tp,the summation is over p q, = (1 , 2 ,s s) r s, = (1 , 2d d)and repeating indices; 1 0

0 1

cl

g =ˆ

(

_-

)

and 1 0 0 1 q

g =ˆ

( )

are the Keldyshgˆ matrices. Gmis the fermionic propagator on the mth arm(see equation (10)), and

M˜ (¢ w w2, 3) M˜ (w w2, 3)-G2(w2)G3(w3). ( )9 Here M˜dgba(r r r r4, , ,3 2 1) - á Y 3,d( ) ¯r4 Y3,g( )r3 Y2,b( ) ¯r2 Y2,a( )r1 ñis the collision matrix of two electrons in

arms 2 and 3(see appendixD).

The expressions for the expectation values of equations(7) and (8) can be represented diagrammatically in

terms of the contributing processes. In these Feynman–Keldysh diagrams, each line corresponds to a propagator G(see equation (10)), and the vertices represent tunneling. The diagrams (to leading order in tunneling matrix

elements) are depicted in figure2. There are 16 diagrams contributing to the irreducible current–current

correlator. The leading diagrams(figure2(b)) correspond to an electron in the system (going through arm 2) that

maximally interacts with an electron in the detector(going through arm 3)5.

5

For these diagrams the time of the two particles being inside the interaction region is maximal; the other diagrams are almost reducible(i.e., decoupled from each other), and are thus neglected.

Explicit evaluation of GCV requires the calculation of the single electron Gmand the collision matrix6M˜.

Wefirst compute the propagators on arms 2 (G2) and 3 (G3), where both the inter- and the intra-channel

interaction is present. This yields

G x v F x x T x u s s , i 2 e , e d , 10 m s , F i 1 1 i x u ⎜⎛ ⎟ xu ⎝ ⎞⎠

ò

w w a b V l = - + Q Q -´ ab w x l w l -( ) [ ( ) ( ) ( )] ∣ ∣ ( ) ( ) ∣ ∣

where ,a b =  are the Keldysh indices1 (in forward/backward basis), x and ω are the distance traveled by and the energy of the particle, and T is the temperature. We define the renormalized interaction

, u g u g 1 2 1 2 1 1 ⎡ ⎣⎢ ⎤⎦⎥ l = _p^ - _p^

-( )

F tanh , T 2 w =

_{( )}

w

( ) Q( )x is the Heaviside function, 2 ,

4 1 1 2 2 x l = l l + -( ) and A s, A . A s A s sinh 1 sinh 1 V = p - p + ( ) [ ( )] [ ( )]

The propagators in channels 1(G1) and 4 (G4) are obtained by substituting g 0^= in equation(10). This result

recovers the simple non-interacting Green function with a renormalized velocity u vF g

2

= + _p

due to intra-channel interaction. The maximal interaction between intra-channel 2 and 3 is at g u

2

=p

^ (instability point). Similarly

to the two-particle analysis, here too the SV limit is reached at afinite value of the inter-channel interaction.

4. Results

Plugging equations(10) and equation (D.14) to equations (7) and (8), we obtain the final expression for the GCV

in equation(6). The result is depicted in figure3. We identify a high temperature regime,tFL Bk T(tFLis the

time offlight through the interacting arm of MZI, L u

FL

t = ), where the GCV is exponentially suppressed by the factor e-FL Bk T

t

due to averaging over an energy window~T.In the opposite, low temperature limit, the phase diagram shows novel oscillatory behavior. We plot the phase diagram of GCV in a parameter space spanned by the applied voltage normalized by the temperature eV k T( B )and the renormalized interaction strength(λ) (see

figure3). In the low voltage limit (eVk TB ) the size of the injected wave function is large compared with L. In

this limit interaction effects should be less significant. The weak-to-strong crossover is smooth in similitude to the two particle result(see equation (3)). For eV>k T,B multiple particle interaction effects become important,

and three different regimes are obtained as function ofλ. Here, as function of increasingl,oscillatory behavior ( J~0

(

l teV_FL

)

,where J0is the 0th order Bessel function) of the crossover from WV to SV is predicted. The

behavior of the GCV in the different regimes is summarized in a phase diagram infigure3(a), along with the

dependence of the GCV on the interaction strength(figures3(b)–(d)) and voltage bias (figure3(e)).

5. Discussion

The oscillations found here and the physics of visibility lobes that was found experimentally[38] and studied

theoretically[39–41] in the context of coherent transport through a MZI, are both related to interaction effects

in an interferometry setup. To understand this similarity we employ a caricature semi-classical picture: a single particle wave-packet, whose energy components are in the interval[0,eV],is injected into the system MZI(arm 1 offigure1). During its propagation through the interacting arm, its dynamics is affected by Coulomb

interaction with the entire out-of-equilibrium Fermi sea of electrons inside the interaction region of the detector MZI(arm 3 of figure1), producing a phase shift of the systems packet. When this single particle

wave-packet interacts with a single electron in the detector(see the discussion preceding equation (2)), its phase shift is

0 g p If the detector. ’s arm consists of N electrons, a phase shift of Ng is produced, giving rise to oscillations as function of the interaction strength or N. More qualitatively: the number of background non-equilibrium electrons inside the detector MZI, N LeV

u

2

á ñ = _p [39–41], splits into n and Ná ñ - in arms 3 and 4n respectively, with probability P n T R N

n , n N n⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = á ñ- á ñ ( ) R= ∣ ∣ Tr1d2, t1 . 2 d = ∣ ∣

Neglecting, for the sake of this caricature, time dependent quantumfluctuations in the number of particles (we have treated those in full), the incremental addition to the (system) wave packet action due to an electron in arm 2 interacting with n background electrons in arm 3 is S n t, g n t d .t

L _t t 0 0 0 FL

ò

D _{( )}= ^ +t _{( ) Here t 0,eV} 0Î [ ]is

the injection time of an electron wave packet. The added phase to the single particle wave-function is:

6

ei S.

yy D _{It follows that the current at D2 per a specific n is}

I n t, e V R T 2RT e e . h S n t D2 0 2 2 2 i i , 2 S 0 0 Re =

(

+ +

{

pF_F D

}

)

( ) · ( ) _{The mean current is a weighted average over all}

n

{ }and t0, leading to a lobe structure. For example, whená ñ N 1,then S n ,

g n L

FL

D = ^t

( ) and the total current

is I e V R T 2RT D e , h 2 2 2 i i 2 S 0 D Re =

(

+ + _·

_{

p_FF+h

_}

)

_{where De}iD _{R Te ,} g N L i FL = + h ^t á ñ which is periodic in g_^ with a period of u. eV 2 2 FL p t ( )

We can repeat the same argument for the detector MZI and obtain the same lobe structure dependence there.

Measurements on setups consisting of two electrostatically coupled MZI have been reported[32], albeit not

in the context of the present work. By means of external gates one may control the magnitude of the coupling .l More accessible experimentally would be tofix the distance between the MZIs and observe oscillations with V at moderately low values of .l

The present analysis interpolates between two conceptually distinct views of measurement in quantum mechanics: the von Neumann projection postulate, and the continuous time evolution in the weak system– detector coupling limit. Admittedly these two views could be obtained as limiting cases of the same formalism. The analysis presented here demonstrates that the interpolation between the two is non-trivial. Oscillatory crossover is a unique feature of our many-body analysis. The setup chosen to demonstrate this SV-to-WV crossover consists of two coupled MZIs(the ‘system’ and the ‘detector’). Measurements on such a setup have

Figure 3.(a) The phase diagram in the low temperature regime,tFL Bk T.Regions with different qualitative behavior are depicted by different colors. The transition between weak and strong values in the high-voltage regime goes through an intermediate phase where the GCV displays oscillations as a function of the coupling constant. The latter feature is not present in the two-particle treatment of GCV(see equation (3)). (b) and (c) The normalized GCV, Q ,

e V h

MB 2 GCV 2

á ñ

along the cuts A(eV k TB =100), B (eV k TB =0.001) in (a). The zoom in (c) highlights the oscillatory behavior. (d) The oscillatory regime along A for various temperatures keeping eVtFL =1.(e) The normalized GCV along the cuts C (λ=10) and D (l =1000) of (a) with a zoom on the relevant oscillatory regime. All the plots are forF F =S 0 0.99 ,p F =D 0at the low temperature phase, k TB tFL =0.01except of (d) where the temperatures are specified explicitly.

been reported in the literature(see e.g., [32]), with a considerable latitude of controlling the system–detector

coupling. We conclude that our predictions are, then, within the realm of experimental verification.

Acknowledgments

We gratefully acknowledge discussion with Yakir Aharonov, Moty Heiblum, Itamar Sivan, Lev Vaidman and Emil Weisz. YG acknowledges the hospitality of the Dahlem Center for Complex Quantum Systems. This work is supported by the GIF, ISF and DFG(Deutsche Forschungsgemeinschaft) grant RO 2247/8-1.

Appendix A. Derivation of the formula for two-particle GCV in terms of the irreducible

correlation function

Here we present an extended derivation of equation(2). The two-particle GCV of Q2is defined by,

Q I I I I I I I . A.1 TP 2 GCV D2 D3 D2 D2 D3 D3 0 D2 d g g á ñ = á ñ á ñ = á - á ñ ñ á ñ ( ) ( )

This can be rewritten as,

I I I I I I I I I A.2 D2 D3 D2 D3 0 D2 D3 D2 D3 D2 g á ñá ñ - á ñá ñ + á ñ - á ñá ñ á ñ ( )

which yields equation(2)

Q I I I I 1 . A.3 TP 2 GCV D3 D2 D3 D2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ g d á ñ = á ñ + áá ññ á ñ ( )

Appendix B. Strong-to-weak crossover of GCV for two particle system

Here we present the derivation of GCV for two particle system(i.e. equation (3)). In accordance with

equation(A.1) we compute the current–current correlator I IáD2 D3ñand the average current IáD2ñ defined with,

respect to the density matrix e Q QI I e ,

S S Q Q i 1 4 i 2 3 2 3 r = g Ä -g I I I I I I I I Q Q I I Q Q Tr e e Tr 1 e 1 1 e 1 , B.1 Q Q S S Q Q S S D2 D3 D2 D3 i 1 4 i D2 D3 i 2 3 1 4 i 2 3 2 3 2 3 á ñ = = + - + -g g g g -{ } { ( ( ) ) ( ( ) )} ( )

where in the last step we employedeQ Q 1 ei 1 Q Q

2 3

2 3= +

-g ₍ g _{) because the eigenvalues of Q}

iare only 0 or 1. Then I I I I I Q I Q Q I Q Q I Q I I 1 4 sin ie 4 sin , B.2 D2 D3 D2 0 D3 0 2 D2 2 0 D3 3 0 2 2 2 D2 2 0 3 D3 3 0 D2 0 D3 0 i 2 Re ⎛ ⎝ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟⎟ á ñ = á ñ á ñ ´ + á ñ á ñ + á ñ á ñ á ñ á ñ g g g

{

}

( )

( )

( )

whereáñ denotes average with respect to the noninteracting setup0 (g 0). Similar calculation foráID2ñyields

I I Q I Q Q I Q I 1 4 sin ie 4 sin . B.3 D2 D2 0 3 0 2 D2 2 0 2 ₂ 2 D2 2 0 D2 0 i 2 Re ⎛ ⎝ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟⎟ á ñ = á ñ + á ñ á ñ + á ñ á ñ g g g

{

}

( )

( )

( )

Plugging equations(B.2) and (B.3) in equation (A.1) yields an expression for a two particle GCV

Q I Q I Q Q I Q Q I Q I I Q Q Q I Q Q 4 sin ie sin 4 sin ie 4 sin . B.4 TP 2 GCV 2 D2 2 0 D3 3 ₂ 2 D2 2 0 3 D3 3 D2 0 ₂ D2 2 0 3 0 2 ₂ 2 D2 2 0 3 0 i 2 i 2 Re Re g d d á ñ = á ñ á ñ + á ñ á ñ á ñ + á ñ á ñ + á ñ á ñ g g g g g g

{

}

{

}

( )

( )

( )

( )

( )

In the weak limit(g 0) this expression simplifies to

Q I Q I Q I lim TP 2 i B.5 0 2 GCV D2 2 0 D3 3 D2 0 Re⎧⎨ ⎩ ⎫ ⎬ ⎭ d á ñ = á ñ á ñ á ñ g ( )

and in the strong limit(gp) Q Q I Q Q I Q I Q I Q I I Q Q Q I Q Q lim 4 4 4 . B.6 TP 2 GCV 2 D2 2 0 3 D3 3 D2 2 0 D3 3 D2 0 D2 2 0 3 0 2 D2 2 0 3 0 Re Re p d d á ñ = á ñ á ñ - á ñ á ñ á ñ - á ñ á ñ + á ñ á ñ g¥ { } { } ( )

Appendix C. Perturbative calculation of expectation values

In this section we derive the expression for expectation values of the current and the current–current correlator. Employing a path integral formalism, a general formula for the expectation value of an operatorOˆ [Y Y†, ]_is

O , , O , e , e , C.1 S S i , i , D D

ò

ò

á Y Y ñ = Y Y Y Y Y Y Y Y Y Y ˆ [ ] [ ¯ ] ˆ [ ¯ ] [ ¯ ] ( ) † [ ¯ ] [ ¯ ]

where S=S0+Sint+STis the full action over the Schwinger–Keldysh contour with

S , d dr r r G r r r , C.2 m m m m 0 1 4 , , 1 ,

ò

å

Y Y = ¢Y _{a ab} - ¢ Y _b ¢ = - [ ¯ ] ¯ ( ) ( ) ( ) ( ) S , dr r g r C.3 m n m mn cl n int , 1 4 , ,

ò

å

r h r Y Y = _{a ab b} = [ ¯ ] ( ) ˆ ( ) ( ) and ST , d dr r r r r, r , C.4 m n m mn cl n , 1 4 , ,

ò

å

g Y Y = ¢Y _a G ¢ _abY _b ¢ = [ ¯ ] ¯ ( ) ( ) ˆ ( ) ( )

wherea,b are the Keldysh indices in forward/backward basis, m, n are the wire indices, r denotes the spacial two-vector(r = (x, t)),r_m,a( )r = Y¯m,a( )r Ym,a( )r is the density of the particles,hˆ_abcl is the Keldysh matrix(see tableC2) g g g g g g g 0 0 0 0 0 0 0 0 0 0 , C.5 mn ⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟ = ^ ^     ( ) r r x x x x x x x x t t , 0 , 0 0 , 0 0 0 0 0 0 , 0 0 , 0 C.6 mn s s d d ⎛ ⎝ ⎜ ⎜ ⎜ ⎜⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟⎟ * * d G ¢ = G ¢ G ¢ G ¢ G ¢ - ¢ ( ) ( ) ( ) ( ) ( ) ( ) ( ) and s x x, 1s x x11 x x21 2 x x12 x x22 s s s s s d d d d G( ¢ = G) ( - ) ( ¢ - )+ G ( - ) ( ¢ - )andGd(x x, ¢ = G) 1dd x-x31d d x¢ -x41d + G2dd ( ) ( ) ₍x-x₃2d₎d _{x x .} 42d ¢

-( ) G_m,-1_ab(k,w)is the inverse of the fermionic Green function for particles whose dynamics is described by0S+0D,which in(k, w)representation is given by[42]

G k F v k F v k , 1 2 i i . C.7 m, F F ⎡ ⎣⎢ ⎤ ⎦⎥   w w a w w b w = + - + -- -ba  _{( )} ( ) ( ) _{( )}

Here we assume the setup was in thermal equilibrium with a temperature T(described by the fermionic population function F tanh

T

2

w =

_{( )}

w

( ) at the time t  -¥, when the tunnelingG,and the interaction g were adiabatically turned on. By assuming small tunneling the action can be expanded in power series to desired order inG,then equation(C.1) gets a form

O O S S , , i , i , , C.8 n n n n n n 1 T 1 T

å

å

á Y Y ñ = á Y Y Y Y ñ á Y Y ñ W W ˆ [ ] ˆ [ ¯ ]( [ ¯ ]) ( [ ¯ ]) ( ) † ! !

where áñWdenotes averaging with respect to the action S0+S .int

The current in a chiral system with linear dispersion is linearly proportional to the density( Iá ñ =evFá ñr).

The expectation value of the density is obtained by weakly perturbing the system by a quantum potential probe Vq, which should be taken to zero at the end to restore causality[42]. Therefore, we obtain an expression for the

I x t, iev G x t x t

2 Tr , ; , ,

Dm F m gq

á ( )ñ = - { ˜ ( ) ˆ }

where G˜m,ba(x t x t, ; , )= - á Yi  m,b(x t, ) ¯Ym,a(x t, )ñis the fermionic Green function of the system(averaged with respect to the full action, S) at point (x,t) of the mth arm. The trace is over the Keldysh indices, where_gˆq_is the Keldysh matrix(see tableC1). For the sake of simplicity we compute first I x táD1( , )ñby expanding it to second(leading) order in .G We then employ the current conservation tofind I ,áD2ñ

ID2 x t, I0 ID1 x t, ,

á ( )ñ = - á ( )ñ where I e V .

h

0

2

= To this order, particle tunnels twice. We employ equation(C.8)

to expandG˜in S_Γ. This yields

I x t v t t G x x t t G x x t t G x x t t , ie 2 d d , , , . C.9 p q p q p cl p q cl q D2 F 1 2 , 1 ,2 1, 1 1 2, 1 2 1 2 1, 1 1 s s * _{ }

ò

å

g g g á ñ = ´ G G ab - - bg gd - - d z - - za = ( ) { ( ) ˆ ( ) ˆ ( ) ˆ } ( ) { } Here Gm(x t, )ba= - á Yi  m,b(x t, ) ¯Ym,a(0, 0)ñ (C.10) is the fermionic Green function averaged with respect to the interacting action, S0+S .int We perform Fourier

transform over the time variable to obtain

I x, 0 iev G x x G x x G x x 2 , , , . C.11 p q p q p cl p q cl q D2 F d₂ , 1 ,2 1, 1 2, 1 2 1, 1 s s * _{ }

ò

å

w g w g w g á ñ = w G G - - -p ab bg gd d z za = ( ) ( ) ˆ ( ) ˆ ( ) ˆ ( ) { }

Tofind the current–current correlator, we generalize the last procedure, employing I ID2 D3 I ID1 D4 ,

áá ññ = áá ññ to obtain

Table C1. A list of Keldyshgˆ_abc matrices(for fer-mions) in different bases of bosonic (χ) indices and fermionic indices( ,a b). , a b χ (+ -) (cl/q) + 1 0 0 0 g =_ˆab+

( )

1 1 1 1 1 2 g =_ˆab+

( )

− 0 0 0 1 g_ˆab-

(

_-

)

1 1 1 1 1 2 g = -ab-

(

)

ˆ cl 1 0 0 1 cl g =ˆ_ab

(

_-

)

1 0 0 1 cl g =ˆ_ab

( )

q 1 0 0 1 q g =ˆ_ab

( )

0 1 1 0 q g =ˆ_ab

( )

Table C2. A list of Keldyshhˆ_abc matrices(for bosons) in different bases of bosonic ,c a andbindices.

, a b χ (+ -) (cl/q) + 1 0 0 0 h =_ab+

( )

ˆ 1 1 1 1 1 2 h =_ˆab+

( )

− 0 0 0 1 h_ˆab-

(

_-

)

1 1 1 1 1 2 h =_ab- -

-(

)

ˆ cl 1 0 0 1 cl h = -ab

(

)

ˆ 0 1 1 0 cl h =ˆ_ab

( )

q 1 0 0 1 q h =ˆ_ab

( )

1 0 0 1 q h =ˆ_ab

( )

t I x t I x v t t t t t G x x t t G x x t M x x x x t t t t G x x t G x x t t 1 d , , 0 e d d d d d , , 0 , , , ; , , , , 0 , , C.12 D D pqrs p q r s p cl r cl s r q p cl s cl q q q 2 2 1 4 2 F2 2 2 1 2 3 4 1, 1 1 4, 4 4 3 3 2 2 3 4 2 1 4, 4 3 1, 1 2 * *  

ò

ò

å

ò

t t g g g g g g áá ¢ ññ = - G G G G ´ ¢ - ¢ - - - ¢ ´ - ¢ - - -t t t t ab bg hq qi ikgd kl lm d z za mh -( ) ( ) ( ) ˆ ( ) ˆ ˜ ( ) ˆ ( ) ˆ ( ) ˆ ˆ ( ) where M r r r r˜ (¢ 4, , ,3 2 1) M r r r r˜ (4, , ,3 2 1)-G r2(2-r G r1) 3(4-r3).And M˜dgba(r r r r4, , ,3 2 1) -á Y 3,d( ) ¯r4 Y3,g( )r3 Y2,b( ) ¯r2 Y2,a( )r1ñ (C.13) is the collision matrix. We perform Fourier transform over the time differences, such thatω2corresponds to

t2-t ,1 ω3to t4- andt3 w¯to₂1(t3+t4)- 1₂(t1+t .2) Finally, it yields

t I x t I x e v t G x x G x x M x x x x G x x G x x 1 d , , 0 d d d d 2 e , 2 , 2 , , , ; , , , 2 , 2 . C.14 D D pqrs p q r s t p cl r cl s r q p cl s cl q q q 2 2 1 4 2 F2 2 2 2 3 3 i 1, 1 2 4, 4 3 3 3 2 2 3 2 4, 4 3 1, 1 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ ⎝ ⎞⎠ ⎛⎝ ⎞⎠ ⎛ ⎝ ⎞⎠ ⎛⎝ ⎞⎠ * *  

ò

ò

å

ò

t t w w w p w w g w w g w w w g w w g w w g g áá ¢ ññ = - G G G G ´ ¢ - - ´ ´ - + ¢ ´ - ¢ - - + t t t t w ab bg hq qi ikgd kl lm d z za mh - ( ) ( ) -¯ ( ) ¯ ˆ ¯ ˆ ˜ ( ¯ ) ˆ ¯ ˆ ¯ ˆ ˆ ( ) ¯

In order tofind a simpler expression for the time integral overt,we denote the current–current correlator byF t :( ) F t( )= ááID2(x t I¢, ) D3(x, 0)ññ,and its Fourier transformF( ¯ )w .Equation(C.14) can be written in these terms as F 1 dtF t 1 dt d F 2 e . C.15 t 2 2 2 2 i

ò

ò

ò

t t w p w = t t t t w - - ¯ _{( )} ¯ ¯ _{( ¯ ) ( )}

It is easy tofind an expression for F w( ¯ )by comparing equations(C.14) and (C.15). First, we write

F F F t F t e t e t d .t 1 2 1 4 i i

ò

w + -w = + - + t t _-¥ w w ¥

-[ ( ¯ ) ( ¯ )] [ ( ) ( )]( ¯ ¯ ) _{From the other hand we approximate the} average by, F 1 F t F t t 2 e d , t 2

ò

t » + - p t -¥ ¥ -¯ _{[ ( ) ( )]} ( )

where we have assumed that F(t) grows much slower than e_p(t _t)2,

and the antisymmetric part of F(t) is cancelled by the averaging. By comparing the exponentials in the two equations we obtainw =¯ 2_tp.

Then F 1 F F . 2 2 2 ⎡ ⎣ ⎤⎦ = _{t t}p + - _tp ¯ _{( ) ( )}

Appendix D. Calculation of the fermionic correlators

Here we derive the expressions for the fermionic propagator(see equation (C.10)) and the collision matrix (see

equation(C.13)) averaged with respect to the action S0+S ,int within an interacting arms(2, 3) of MZI (the

propagator in arms 1 and 4 can be found by taking g_^0). In this calculation we employ the functional bosonization approach for system out of equilibrium[43,44]. We apply the Hubbard–Stratonovich

transformation, and introduce the bosonic auxiliaryfieldF,writing an action S0+Sintas[45]

S S , ; G 1 g

4 , D.1

0+ int[ ¯Y Y F = Y] ¯ [ ]-F1Y + F -1F ( )

with the notation

G d dr r r G r r r , m m m m 1 2,3 , 1 , ,

ò

å

Y -_F Y = ¢Y _{a ab} - ¢ Y _b ¢ = F -¯ _{[ ]} ¯ _{( ) [ ] ( ) ( )} where G 1_m, r r G_m, r r m r r r 1 , g d - ¢ = - ¢ - F - ¢ ab ab abc c F - _{( ) }- _{( ) ˆ ( ) ( )} [ ] and g dr r g r , m n m _mn cl n 1 , 2,3 , 1 ,

ò

å

h F -F = F a _abF b = -( ) ˆ ( )

where we implicitely sum over the Keldysh indices ,a b c = , 1(in forward/backward basis) andg_mn-1is the inverse of the m, n= 2, 3 submatrix of gmn(see equation (C.5)). Following the functional bosonization procedure

[45], we obtain a general expression for an n-fermion correlator r q r q e , D.2 i n a i b _i i n a i b _i r q 0 i i i i i n ai i bi i 1 2 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟   



Y ( ) ¯ ( )Y =



Y ( ) ¯ ( )Y - åq -q F ( ) ( ) ( )

where a b, = (a,m)denote the Keldysh and the wire indices, r q, = (x t, ),áñ is the fermionic correlator with0

respect to the free action

S0 , G , D.3

1

Y Y = Y- Y

[ ¯ ] ¯ ( )

and áñFis theΦ-field correlator with respect to the action

S 1 g 4 D.4 1 F = F F + FPF F[ ] - ˆ ( ) respectively. Here r r r r r r d d , m m m m 2,3 , , ,

ò

å

FPF = ¢F _a P _ab ¢ F _b ¢ = ˆ _{( ) ˆ ( ) ( )}

with the polarization matrix

r r i G r r G r r

2Tr , D.5

m, g m g m

Pˆ ab( - ¢ =) { ˆa ( - ¢) ˆb ( ¢ - )} ( ) where the trace is taken over the Keldysh fermionic indices[42]. The θ field is defined by

r i dr G r r r , D.6 m, m cl m 1 , B ,

ò

å

q _a = - ¢ - ¢ h F ¢ bg ab bg g = ( ) ( ) ˆ ( ) ( )

where GBis the bosonic free Green function with linearized spectrum,

G k B v k B v k , 1 2 i i . D.7 m,B F F ⎡ ⎣⎢ ⎤ ⎦⎥   w w a w w b w = + - + -- -ba( ) ( ) ( ) ( )

The action for theΦ field (see equation (D.4)) is quadratic due to Larkin–Dzyaloshinskii [46] theorem, therefore

an exact expression for theΦ-field correlator is

Q r r r r g r r r r

imn,ab( - ¢) á F m,a( )Fn,b( )¢ ñ =F i( _mn-1h dˆ_{a b}cl_, ( - ¢ +) dmnPˆm, ,a b( - ¢))-1. We reduce the problem offinding an inverse of an infinite-dimensions matrix, inverting it to the finite (4) dimensions by Fourier-transforming it to a diagonal(k, w)basis. Employing equation(D.6) we obtain the

θ-field correlator K r r r r q q G r q Q q q G q r i i d d , D.8 mn m n cl cl mn , , , B B , 

ò

q q h h - ¢ á ¢ ñ = - ¢ - - ¢ ¢ - ¢ ab a b ab F  _  ( ) ( ) ( ) [ ( ) ˆ ( ) ˆ ( )] ( )

where we implicitly sum over the Keldysh and the wire indices. This yields,

K B K K K K B K K K K 0 0 , D.9 mn mn R A R A mn x R A R A ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ d s = - + ^ - ^ ^ ^              [ ] [ ] ( ) where K k k , D.10 R A v k v k v k 1 i 1 i 2 i F ⎡ ⎣⎢   ⎤⎦⎥ w = p + -w- r  w- s  w-  _{ ( ) ( )} and K k k v k , 1 i . D.11 R A v k 1 i ⎡ ⎣ ⎢ ⎤ ⎦ ⎥   w p w = - r  -w ^ _{- s }  _{( ) ( )} Here, vr=u+ 2_pg^,vs=u- 2_pg^,with u v . g F 2 = + p 

We plug this result in equation(D.2) to compute the

Green function(equation (C.10)) and the collision matrix of the particles in arms 2 and 3 (equation (C.13)). The

calculation requires transformation of equations(D.10) and (D.11) to real (x , t) space. Here we present the final

G x t T v T t x v t t T t x v t t , 2 1 sinh i 1 sinh i . D.12 m, F ⎡ ⎣ ⎢ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦ ⎥ ⎥ ⎡ ⎣ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟⎤ ⎦ ⎥ p a p a b = -- + L Q Q -´ - + L Q Q -ba r s ( ) [ ( ) ( )] [ ( ) ( )] ( )

Fourier-transforming the time coordinate yields,

G x v F x x T x u s s , i 2 e , e d D.13 m, s F i 1 1 i x u _⎜⎛ _⎟ xu ⎝ ⎞ ⎠

ò

w = - w + Qa - Q - V l ba w x l w l -( ) [ ( ) ( ) ( )] ( ) ∣ ∣ ∣ ∣ ( )

with the definitions u g u g 1 2 1 2 , 1 1 ⎡ ⎣ ⎢ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎤ ⎦ ⎥ l p p = ^ - ^ - - 2 4 1 1 2 2 x l l l = + -( ) and A s A A s A s , sinh 1 sinh 1 . V p p = - + ( )

[ ( )] [ ( )] For the sake of consistency check,limg^0,g0G=G. 

 And

the collision matrix reads

M x4,x x x3, 2, 1 G3, x43 G2, x21 x x x x 1 31 1 42 2 32 2 41 z z z z = dgba dg ba _{ga db gb da} ˜ ( ) ( ) ( ) ˜( )( ) ˜( )( ) ˜( )( ) ˜( )( ) where, x t, T t t t T t t t 1 sinh sinh x v x v i i ⎜ ⎟ ⎡ ⎣⎢ ⎛⎝ ⎞⎠⎤⎦⎥ ⎡ ⎣⎢ ⎤⎦⎥ z_ba = p a b p a b - + Q Q -- + Q Q -r s L L

(

)

˜( )_{( )} [ ( ) ( )] [ ( ) ( )] and_z2 x t, _{= z} 1 x t, 1. ba ba -˜( )_{( ) ( ˜}( )_{( ))} Fourier-transforming the

time coordinates yields,

M x x x x G x G x x x x x , , , , , , 1 2 d d d 2 , , , 2 , 2 , 2 , 2 , D.14 1 2 3 4 3 2 2₃ 3 3, 43 3 3 2, 21 2 2 1 31 2 3 1 42 2 3 2 32 2 3 2 41 2 3

ò

w w w w w w p w w w w z w w w w z w w w w z w w w z w w w = ¢ ¢ ¢ - ¢ - ¢ ´ - ¢ + ¢ - ¢ - ¢ - ¢ + ¢ ´ ¢ - ¢ - ¢ ¢ + ¢ + ¢ dgba dg ba ga db gb da ˜ _{( )} ( ) ( ) ( ) ˜ _{( ) ˜ ( )} ˜ _{( ) ˜ ( ) ( )} ( ) ( ) ( ) ( )

where we have used the short notation xij=xi-x ;j G is the single particle propagator given by equation(D.13), and x, 2 cosh Tx 2x Z x, ,

u

1 2 1 2

z˜_ba( )₍ w₎= pd w_{( )}

_{( )}

p l - l˜_ba( )₍ w₎

where Z˜ba(1 2)is given by Figure D1. The collision matrixM˜ _{(see equation (}D.14)). A diagrammatic representation of the renormalized inelastic collision

between two chiral fermions inside the interacting region. Straight lines correspond to fermionic Green functions(gray- outside the interacting region and black- inside). Wavy lines correspond to bosonic Green functions (red and blue for the two different types of bosons, see equation(D.15)). The vertices x1, x3(x2, x4) correspond to the two entry (exit) points of the interaction region on the edges.

The Keldysh indices( ± ) at these points are indicated bya b g d, , , .Electrons enter the interacting region with energies 2 1 2 w + w and 3 1 2

w - wand exit with energies 2 1 2

w - wand 3 1 2

Z x u B x x T x u s s s , i 2 e , e i d , D.15 x u 1 2 _i 1 1 x u

ò

w = - w + Qa - Q -b k l w l ba w x l - ˜( )_{( ) [ ( ) ( ) ( )] ( ) (} ∣ ∣ _{) ( )} where B coth T 2 w =

_{( )}

w

( ) is the Bose function and A s A s

A s , sinh 1 sinh 1 . k p p = + -( ) [ ( )] [ ( )] Equation(D.14) has a pictorial interpretation, presented infigureD1, according to which, the Z˜ particles are the dressed bosons that carry the interaction between the electrons.

Appendix E. Passage of the electron through the MZI: a semiclassical picture

Here we present the propagation of a localized wave packet(according to a semiclassical picture) through an interacting arm of MZI, and derive the condition to be in the semiclassical regime. We assume semiclassically a propagating rectangular shaped wave packet with a width

eV



~ in time domain(see figureE1). The propagation

of the single particle wave function can be derived by convolving the initial state with the retarded Green function,

x t, i

ò

GR x x t, _* x, 0 d .x E.1

Y( )= ( - ¢ ) Y ¢( ) ¢ ( )

An expression for the zero temperature retarded Green function is(this is simply derived from equation (10))

G x t u xv t t , i 1 , E.2 R u x x u u x x u F 2 pl = P -- -l x l l x l

(

)

(

)

( ) ( ) ( ) ( ) ( ) ( ) where x x o w 1 1 1 0 . .

P( )=

{

- < < is a rectangle function. The wave packet at four different points is shown in figureE1. We observe, the wave packet has been broadened as a result of the interaction, its width in time at different space points is given by t x .

eV x u

0  2 0

D ( )= + l The center of mass of the wave packet then propagates with velocity vCM= _{x l( )}u .Consistent with the semiclassical picture, we require the width of the wave packet to be much smaller compared with the propagation time through the MZI,Dt L( ) L vCM.From this condition we deduce,eV u

L



 andl 1.

Appendix F. General GCV for an

N-state system

Here we present a derivation of GCV for a general system with N-states being measured by a Gaussian detector. We show that the weak-to-strong crossover in such a case may be oscillatory with a bounded number of periods of the order ofO N .₍ 2_{) The initial state of the system is a mixed state, which is represented by the density matrix}

R .

s

å

n m, nm n n

r = ∣a ñáa ∣ The detector is initialized in the zeroth coherent state(we denote the sa¢ coherent state by añ∣ ˜ ) such that its density matrix isr = ñá_d ∣ ˜0 0 .˜ ∣ We neglect the dynamics of the system and the detector assuming the measurement process was short in time compared to the typical timescales of the system and the detector. The coupling Hamiltonian is =I w t gA b( ) ˆ ( †+b)with b b, †are the ladder operators of the

Figure E1. A propagation of the wave packet through an interacting arm of the MZI, at zero temperature, forl =1for different points(a)x0=0,(b)x0 u_eV,  = (c) x 2u , eV 0  = (d)x 3u . eV 0 

= As can be derived from equation(E.2), the width of the wave packet is

given by, t . eV x u 2 0  D = + l

detector, Aˆ =

å

_nan ∣anñáan∣and w(t) is a window function around the time of the measurement. The post-selection is represented by the projection operator,P =f

å

_{n m nm,} P ∣anñáan∣.Plugging into equation(1) and considering,r_tot=r_sÄr_dand qd =b,yields

A a R P R P e e . F.1 n m n nm mn a a n m nm mn a a GCV , , g n m g n m 2 2 2 2 2 2

å

å

á ñ = - -- -ˆ _{( )} ( ) ( )

The numerator and the denominator consist of sums of Gaussian(in g) functions, with different coefficients and prefactors. Each Gaussian is a monotonic function(forg>0), thus the maximal number of extremas in the weak-to-strong crossover(gÎ[0,¥)) is of the order of O N ,( 2) where N is the number of system’s states.

Appendix G. A full list of diagrams

FigureG1depicts a full list of irreducible diagrams to fourth(leading) order in tunneling which should be taken in account for the current–current correlator. It is divided to diagrams with no flux dependence (see

figureG1(a)), diagrams which are dependent on eitherF orS FD(see figuresG1(b) and (c)), and diagrams which

are depend on bothF andS F seeD, figureG1(d).

References

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Figure G1. The full list of irreducible diagrams to fourth(leading) order in tunneling which should be taken in account for the current–current correlator (see equation (8)). ‘Semi-classical’ paths of the particles are marked by solid lines (red) and dashed lines

(blue), corresponding to forward and backward propagation in time (see equations (7) and (8)). The diagrams are divided to four

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