Crossover between strong and weak measurement in interacting many-body systems
CCrossover between strong and weak measurement ininteracting many-body systems
Iliya Esin , Alessandro Romito , Ya. M. Blanter and YuvalGefen Department of Condensed Matter Physics, The Weizmann Institute of Science,Rehovot 76100, Israel Dahlem Center for Complex Quantum Systems and Fachbereich Physik, FreieUniversit¨at Berlin, 14195 Berlin, Germany Kavli Institute of Nanoscience Delft University of Technology Lorentzweg 1, 2628CJ Delft, The Netherlands14 October 2018
Abstract.
Measurements with variable system-detector interaction strength, rangingfrom weak to strong, have been recently reported in a number of electronicnanosystems. In several such instances many-body effects play a significant role. Herewe 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. Inthe context of a conditional which-path protocol, we define a generalized conditionalvalue (GCV), and determine its full crossover between the regimes of weak and strong(projective) measurement. We find that the GCV has an oscillatory dependence onthe system-detector interaction strength. These oscillations are a genuine many-bodyeffect, and can be experimentally observed through the voltage dependence of crosscurrent correlations.
Keywords : Weak measurement, Strong measurement, Weak value, Mach–Zehnderinterferometer
PACS numbers: 73.23.-b, 03.65.Ta, 07.60.Ly, 73.43.-f
Submitted to:
New J. Phys.
1. Introduction
Measurement in quantum mechanics is inseparable from the dynamics of the systeminvolved. The formal framework to describe quantum measurement, introduced byvon 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 withthe evasive notion of quantum collapse. In the other limit, that of weak (continuous) a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n measurement of an observable (reflecting weak coupling between S and D [2]), the systemis disturbed in a minimal way, and only partial information on the state of the latteris provided [3]. We note that this hindrance can be overcome, by resorting to a largenumber of repeated measurement (or a large ensemble of replica on which the sameweak measurement is carried out).Weak measurements, due to their vanishing back-action, can be exploited forquantum feedback schemes [4, 5] and conditional measurements. The latter is especiallyinteresting for a two-step measurement protocol (whose outcome is called weak value (WV) [6]), which consists of a weak measurement (of the observable ˆ A ), followed bya strong one (of ˆ B ), [ ˆ A, ˆ B ] (cid:54) = 0. The outcome of the first is conditional on theresult 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], andnon-collapsing observation of virtual states [27]. The particular features of WVs rely onweak measurement, and are washed out in projective measurements. Understanding therelation and the crossover between these two tenets of quantum mechanics is thereforean 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 betweenthe two. Indeed, within the two-step measurement protocol, it is possible to control thestrength of the first measurement. This allows to define a generalized conditional value (GCV), interpolating between WV and SV ( strong value ). The latter, in similitude toWV, refers to a 2-step measurement protocol. Unlike WV, in a SV protocol both stepsconsist of a strong measurement. The mathematical expression for GCV is depictedbelow in equation (1). It amounts to the average of the first measurement’s reading(whatever its strength is), conditional on the outcome of the second measurement. Thishas been studied in the context of single-degree-of-freedom systems [28–31], where theWV-to-SV crossover is quite straightforward and is a smooth function of the interactionstrength. We note that in experiments with electron nanostructures, interactionsbetween electrons play a crucial role. A many-body theory of variable strength quantummeasurement is called for. In many cases, the interaction strength can be controlledexperimentally [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 functionof the interaction strength. Conversely, these oscillations serve as a smoking gunmanifestation of the many-body nature of the system at hand, and present guidelines forobserving them as function of experimentally more accessible variables (e.g. the voltagebias). Our analysis sheds light on the relation between two seemingly very differentdescriptions of quantum measurement, with emphasis on the context of many-bodyphysics.Motivated by the two step WV protocol, we define the generalized conditional value Φ S Φ D S1S2S3S4 D1D2D3D4 V V L L
SQPC1 SQPC2DQPC1 DQPC2 α L α L Figure 1.
Two MZIs, the “system” and the “detector”, coupled through anelectrostatic interaction (wiggly lines). The sources S1 and S4 are biased by voltage Vand the sources S2 and S3 are grounded. Φ S and Φ D are the magnetic fluxes throughthe respective MZIs. The lengths of the arms 1 and 2 between SQPC1 and SQPC2are αL and L respectively, and similarly for the detector’s arms 3 and 4, as is shownin the figure. In the present analysis α = 1. (GCV) of the operator ˆ A as an average shift of the detector, δ ˆ q = ˆ q − (cid:104) ˆ q (cid:105) (cid:12)(cid:12)(cid:12) g =0 , during themeasurement process, projected onto a postselected subspace by the projection operator,Π f , and normalized by the bare S-D interaction strength, g . The GCV is given by (cid:68) ˆ A (cid:69) GCV = Tr (cid:110) δ ˆ q ˆ U † ρ ˆ U Π f (cid:111) g Tr (cid:110) ˆ U † ρ ˆ U Π f (cid:111) , (1)where ρ is the total density matrix which describes the initial state of S and D, and thetime ordered operator ˆ U = T e − i (cid:126) (cid:82) ∞−∞ H SD dt describes the evolution in time of the wholesetup during the measurement. Here, the system–detector coupling, H SD = − gw ( t )ˆ p ˆ A ,with w ( t ) – the time window of the measurement; ˆ q and ˆ p are the “position” and“momentum” operators of the detector ([ˆ q, ˆ p ] = i (cid:126) ). We note that equation (1) providesthe correct WV [6] and SV [33] in the respective limits ( g (cid:28) g (cid:29) S and Φ D independently [32].
2. A two-particle analysis
As a prelude to our analysis of a truly interacting many-body system, we briefly presentan analysis of the same system on the level of a single particle in the system, interactingwith a single particle in the detector. According to this (over)simplified picture,particles going simultaneously through the interacting arms 2 and 3 (cf. figure 1),gain an extra phase e iγ [35, 36], where γ takes values in the range [0 , π ]. First, weconsider the intra-MZI operators, defined in a two-state single particle space, {| m (cid:105)} ,with m=1,2 for the “system” (an electron propagating in arm 1 or 2) and similarlym=3,4 for the “detector”. The dimensionless charge operator (measuring the chargebetween the corresponding quantum point contacts (QPCs)), in this basis has a form Q m = | m (cid:105) (cid:104) m | . The transition through the p-th QPC is described by the scatteringmatrix S p = (cid:32) r p t p − t ∗ p r p (cid:33) , p = 1 s , s , d , d [37]. The entries r p and t p encompassinformation about the respective Aharonov-Bohm flux and for p = 2 s , d , about theorbital phase gained between the two QPCs. The dimensionless current operators atthe source ( S , S
2) and the drain ( D , D
2) terminals of the system-MZI are given by I Sm = S s Q m S † s and I Dm = S † s Q m S s respectively, with m = 1 ,
2, and similarly for thedetector with m = 3 , S d and S d .In view of equation (1), the initial state of the setup, which is described by theinjection of two particles into terminals S1 and S4 respectively, can be written as thedensity matrix ρ = I S ⊗ I S operating in the two-particle product space, | m (cid:105) ⊗ | n (cid:105) ( m = 1 , n = 3 , U = e iγQ ⊗ Q . A positive reading of the projective measurementconsists of the detection of a particle at D2, and is described by the projection operatorΠ f = I D ⊗ . The detector reads the current at D3 ( δq of equation (1) correspondsto ⊗ δI D ). Plugging these quantities into equation (1) yields an expression for thetwo-particle GCV (cf. Appendix A), (cid:104) Q (cid:105) T PGCV = (cid:104) I D δI D (cid:105) γ (cid:104) I D (cid:105) = 1 γ (cid:18) (cid:104) δI D (cid:105) + (cid:104)(cid:104) I D I D (cid:105)(cid:105)(cid:104) I D (cid:105) (cid:19) . (2)The averages are calculated with respect to the total density matrix after themeasurement, (cid:104) ˆ O (cid:105) = Tr (cid:110) ˆ O ˆ U † ρ ˆ U (cid:111) . We have defined δI D (cid:44) I D − (cid:104) I D (cid:105) (cid:12)(cid:12)(cid:12) γ =0 , and (cid:104)(cid:104) I D I D (cid:105)(cid:105) (cid:44) (cid:104) I D I D (cid:105) − (cid:104) I D (cid:105) (cid:104) I D (cid:105) is the irreducible current-current correlator. Astraightforward calculation (cf. Appendix B) yields (cid:104) Q (cid:105) T PGCV = 4 sin (cid:0) γ (cid:1) γ Re (cid:110) ie iγ (cid:104) I D Q (cid:105) (cid:104) δI D Q (cid:105) (cid:111) + sin (cid:0) γ (cid:1) (cid:104) Q I D Q (cid:105) (cid:104) δQ I D Q (cid:105)(cid:104) I D (cid:105) + 4 sin (cid:0) γ (cid:1) Re (cid:110) ie iγ (cid:104) I D Q (cid:105) (cid:104) Q (cid:105) (cid:111) + 4 sin (cid:0) γ (cid:1) (cid:104) Q I D Q (cid:105) (cid:104) Q (cid:105) (3)where (cid:68) ˆ O (cid:69) (cid:44) Tr (cid:110) ˆ Oρ (cid:111) is an average with respect to the non-interacting setup, (cid:104) δI D Q (cid:105) (cid:44) (cid:104) I D Q (cid:105) − (cid:104) I D (cid:105) (cid:104) Q (cid:105) and (cid:104) δQ I D Q (cid:105) (cid:44) (cid:104) Q I D Q (cid:105) − (cid:104) I D (cid:105) (cid:104) Q (cid:105) . Thisresult shows a smooth and trivial crossover between the weak ( γ →
0) and strong( γ → π ) limits. The specific form depends on the parameters of S and D (the magnitudeof the inter-edge tunneling; the value of the Aharonov-Bohm flux). For some range ofvalues (e.g., t s = t s = t d = t d = 0 .
1, Φ S / Φ = 0 . π , Φ D = 0) the function isnon-monotonic (but non-oscillatory), while for other values it is monotonic.
3. A full many-body analysis
The Hamiltonian H = H S + H D + H SD describes the system, the detector, and theirinteraction. The system’s Hamiltonian consists of H S = H S + H ST + H Sint , with H S = − iv F (cid:88) m =1 (cid:90) dx m : Ψ † m ( x m ) ∂ x m Ψ m ( x m ) : (4a) H ST = Γ s Ψ † ( x s )Ψ ( x s ) + Γ s Ψ † ( x s )Ψ ( x s ) + h.c. (4b) H Sint = (cid:88) m =1 g (cid:107) (cid:90) dx m : (cid:0) Ψ † m ( x m )Ψ m ( x m ) (cid:1) : . (4c)Here Γ p is the tunneling amplitude at QPC p and x pm is the coordinate at QPC p onarm m. A similar expression holds for the “detector” MZI, S ⇔ D , with a summationover the chiral arms m = 3 ,
4. We next assume that the lengths of the interacting armsare equal, x s − x s = x d − x d . The S-D interaction Hamiltonian is H SD = g ⊥ (cid:90) dx (cid:90) dx δ ( x − x ) : Ψ † ( x )Ψ ( x ) :: Ψ † ( x )Ψ ( x ) : , (5)where the normal ordering with respect to the equilibrium (no voltage bias) state isdefined as : Ψ † Ψ : (cid:44) Ψ † Ψ − (cid:10) (cid:12)(cid:12) Ψ † Ψ (cid:12)(cid:12) (cid:11) .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 Q , (cid:104) Q (cid:105) MBGCV = v F g ⊥ (cid:32) (cid:104) δI D (cid:105) + 1 τ (cid:90) τ/ − τ/ dt (cid:104)(cid:104) I D ( t ) I D (0) (cid:105)(cid:105)(cid:104) I D (cid:105) (cid:33) , (6)where the current operator is given by, I ( x, t ) = ev F : Ψ † ( x, t )Ψ( x, t ) :. We averageover time τ (cid:29) Lv F . The problem is now reduced to the calculation of average currentsand a current-current correlator. This is done perturbatively in the tunneling strength,but at arbitrary interaction parameter, employing the Keldysh formalism. In this limitexpectation values are taken with respect to tunneling decoupled edge states. Thecurrent is, (cid:104) I D ( x ) (cid:105) = − iev F (cid:88) p,q = { s , s } Γ p Γ q (cid:90) dω π G ,αβ ( x − x p , ω )ˆ γ clβγ G ,γδ ( x p − x q , ω )ˆ γ clδ(cid:15) G ,(cid:15)ζ ( x q − x, ω )ˆ γ qζα , (7)and the irreducible current-current correlator (cf. Appendix C)1 τ (cid:90) τ/ − τ/ dt (cid:104)(cid:104) I D ( x (cid:48) , t ) I D ( x, (cid:105)(cid:105) = (cid:40) (cid:88) pqrs Γ p Γ q Γ r Γ s e v F τ (cid:90) dω dω (2 π ) ×× G ,αβ (cid:16) x (cid:48) − x p , ω − ¯ ω (cid:17) ˆ γ clβγ G ,ηθ (cid:16) x − x r , ω + ¯ ω (cid:17) ˆ γ clθι ˜ M (cid:48) ικγδ ( x p , x q , x r , x s ; ω , ω , ¯ ω ) ×× ˆ γ clκλ G ,λµ (cid:16) x s − x (cid:48) , ω − ¯ ω (cid:17) ˆ γ clδ(cid:15) G ,(cid:15)ζ (cid:16) x q − x, ω + ¯ ω (cid:17) ˆ γ qζα ˆ γ qµη (cid:41)(cid:12)(cid:12)(cid:12) ¯ ω → √ πτ + (cid:40) ... (cid:41)(cid:12)(cid:12)(cid:12) ¯ ω →− √ πτ . (8) Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D V D V S Φ S Φ D S1S2S3S4 D1D2D3D4 Φ S S1S2 D1D2 V S
00 000 Φ S S1S2 D1D2 V S Φ S S1S2 D1D2 V S Φ S S1S2 D1D2 V S (b)(a) 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 theparticles are marked by solid lines (red) and dashed lines (blue), corresponding toforward and backward propagation in time (cf. equation (10)). (a) The average current(equation (7)), O (Γ ). Only the system part of the setup (cf. figure 1), while all degreesof freedom of the detector part have been integrated out. (b) The reducible current-current correlator (equation (8)), O (Γ ). Only the 2 most contributing diagrams outof 16 are shown (4 were included in calculations). Here { ... } reproduces the first part of the r.h.s, with ¯ ω → √ πτ replaced by ¯ ω → − √ πτ , thesummation is over p, q = (1 s , s ), r, s = (1 d , d ) and repeating indices; ˆ γ cl = (cid:32) − (cid:33) and ˆ γ q = (cid:32) (cid:33) are the Keldysh ˆ γ matrices. G m is the fermionic propagator on them-th arm (cf. equation (10)), and˜ M (cid:48) ( ω , ω ) (cid:44) ˜ M ( ω , ω ) − G ( ω ) G ( ω ) . (9)Here ˜ M δγβα ( r , r , r , r ) (cid:44) − (cid:10) T Ψ ,δ ( r ) ¯Ψ ,γ ( r )Ψ ,β ( r ) ¯Ψ ,α ( r ) (cid:11) is the collision matrixof two electrons in arms 2 and 3 (cf. Appendix D).The expressions for the expectation values of equations (7) and (8) can berepresented diagrammatically in terms of the contributing processes. In these Feynman-Keldysh diagrams, each line corresponds to a propagator G (cf. equation (10)), andthe vertices represent tunneling. The diagrams (to leading order in tunneling matrixelements) are depicted in figure G1. There are 16 diagrams contributing to theirreducible current-current correlator. The leading diagrams (figure G1 (b)) correspondto an electron in the system (going through arm 2) that maximally interacts with anelectron in the detector (going through arm 3). † † For these diagrams the time of the two particles being inside the interaction region is maximal; theother 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 G m andthe collision matrix ˜ M ‡ . We first compute the propagators on arms 2 ( G ) and 3 ( G ),where both the inter- and the intra-channel interaction is present. This yields G m,βα ( x, ω ) = − i v F [ F ( ω ) + α Θ( x ) − β Θ( − x )] ×× e iω xu ξ ( λ ) (cid:90) − ς (cid:18) T | x | u λ, s (cid:19) e isω | x | u λ ds , (10)where α, β = ± x and ω are thedistance traveled by and the energy of the particle, and T is the temperature. We definethe renormalized interaction λ = (cid:104) u g ⊥ π − (cid:0) u g ⊥ π (cid:1) − (cid:105) − , F ( ω ) = tanh( ω T ), Θ( x ) is theHeaviside function, ξ ( λ ) = λ √ λ +1 − , and ς ( A, s ) = A √ sinh[ πA (1 − s )] sin[ πA (1+ s )] .The propagators in channels 1 ( G ) and 4 ( G ) are obtained by substituting g ⊥ = 0in equation (10). This result recovers the simple non-interacting Green function witha renormalized velocity u = v F + g (cid:107) π due to intra-channel interaction. The maximalinteraction between channel 2 and 3 is at g ⊥ = π u (instability point). Similarly to thetwo-particle analysis, here too the SV limit is reached at a finite value of the inter-channel interaction.
4. Results
Plugging equations (10) and equation (S37) to equations (7) and (8), we obtain thefinal expression for the GCV in equation (6). The result is depicted in figure 3. Weidentify a high temperature regime, τ F L k B T (cid:29) (cid:126) ( τ F L is the time of flight throughthe interacting arm of MZI, τ F L = Lu ), where the GCV is exponentially suppressedby the factor e − τFLkBT (cid:126) due to averaging over an energy window ∼ T . In the opposite,low temperature limit, the phase diagram shows novel oscillatory behaviour. We plotthe phase diagram of GCV in a parameter space spanned by the applied voltagenormalized by the temperature ( eV /k B T ) and the renormalized interaction strength( λ ) (cf. figure 3). In the low voltage limit ( eV (cid:28) k B T ) the size of the injectedwave function is large compared with L . In this limit interaction effects should beless significant. The weak-to-strong crossover is smooth in similitude to the two particleresult (cf. equation (3)). For eV > k B T , multiple particle interaction effects becomeimportant, and three different regimes are obtained as function of λ . Here, as functionof increasing λ , oscillatory behaviour ( ∼ J (cid:0) λeV τ FL (cid:126) (cid:1) , where J is the 0-th order Besselfunction) of the crossover from WV to SV is predicted. The behavior of the GCV inthe different regimes is summarized in a phase diagram in figure 3 (a), along with thedependence of the GCV on the interaction strength (figure 3 (b-d)) and voltage bias(figure 3 (e)). ‡ As each channel is only slightly perturbed out of equilibrium, methods of equilibrium bosonizationmay be employed. λ λ = λ = λ Strong valueWeak value Oscillations � = ℏ� �� �� ℏ� �� � � � �� / � � � �� / � � � - - - - - - - - - - kB T τ FL ℏ = kB T τ FL ℏ = kB T τ FL ℏ = - - λ - -
10 100 1000
DC B A (a)(c) (b)(e) (d)(f) (C)(D)
Figure 3. (a) The phase diagram in the low temperature regime, τ F L k B T (cid:28) (cid:126) .Regions with different qualitative behavior are depicted by different colors. Thetransition between weak and strong values in the high-voltage regime goes through anintermediate phase where the GCV displays oscillations as a function of the couplingconstant. The latter feature is not present in the two-particle treatment of GCV(cf. equation (3)). (b) and (c). The normalized GCV, (cid:104) Q (cid:105) MBGCV e V/h , along the cutsA ( eV /k B T = 100), B ( eV /k B T = 0 . eV τ F L / (cid:126) = 1. (e) The normalized GCV along the cuts C ( λ =10) and D( λ = 1000) of (a) with a zoom on the relevant oscillatory regime. All the plots are forΦ S / Φ = 0 . π , Φ D = 0 at the low temperature phase, k B T τ
F L / (cid:126) = 0 .
01 except of(d) where the temperatures are specified explicitly.
5. Discussion
The oscillations found here and the physics of visibility lobes that was foundexperimentally [38] and studied theoretically [39–41] in the context of coherent transportthrough a MZI, are both related to interaction effects in an interferometry setup. Tounderstand this similarity we employ a caricature semi-classical picture: a single particlewave-packet, whose energy components are in the interval [0 , eV ], is injected into thesystem MZI (arm 1 of figure 1). During its propagation through the interacting arm, itsdynamics is affected by Coulomb interaction with the entire out-of-equilibrium Fermisea of electrons inside the interaction region of the detector MZI (arm 3 of figure 1),producing a phase shift of the systems wave-packet. When this single particle wave-packet interacts with a single electron in the detector (cf. the discussion precedingequation(2)), its phase shift is 0 ≤ γ ≤ π . If the detector’s arm consists of N electrons,a phase shift of N γ is produced, giving rise to oscillations as function of the interactionstrength or N . More qualitatively: the number of background non-equilibrium electronsinside the detector MZI, (cid:104) N (cid:105) = LeV πu [39–41], splits into n and (cid:104) N (cid:105) − n in arms 3 and 4respectively, with probability P ( n ) = T n R (cid:104) N (cid:105)− n (cid:0) (cid:104) N (cid:105) n (cid:1) , R = | r d | , T = | t d | .Neglecting, for the sake of this caricature, time dependent quantum fluctuationsin the number of particles (we have treated those in full), the incremental additionto 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 ⊥ L (cid:82) t + τ FL t n ( t ) dt . Here t ∈ [0 , eV ]is the injection time of an electron wave packet. The added phase to the singleparticle wave-function is: ψ → ψe i ∆ S . It follows that the current at D2 per aspecific n is I D ( n, t ) = e Vh (cid:16) R + T + 2 RT · Re (cid:110) e πi Φ S Φ0 e i ∆ S ( n,t ) (cid:111)(cid:17) . The meancurrent is a weighted average over all { n } and t , leading to a lobe structure. Forexample, when (cid:104) N (cid:105) (cid:28)
1, then ∆ S ( n ) = g ⊥ τ FL nL , and the total current is I = e Vh (cid:16) R + T + 2 RT · D Re (cid:110) e πi Φ S Φ0 + iη D (cid:111)(cid:17) , where De iη D = R + T e ig ⊥ τFL (cid:104) N (cid:105) L , which isperiodic in g ⊥ with a period of (2 π ) uτ FL eV . We can repeat the same argument for the detectorMZI and obtain the same lobe structure dependence there.Measurements on setups consisting of two electrostatically coupled MZI have beenreported [32], albeit not in the context of the present work. By means of external gatesone may control the magnitude of the coupling λ . More accessible experimentally wouldbe to fix the distance between the MZIs and observe oscillations with V at moderatelylow values of λ .The present analysis interpolates between two conceptually distinct views ofmeasurement in quantum mechanics: the von Neumann projection postulate, and thecontinuous time evolution in the weak system-detector coupling limit. Admittedly thesetwo views could be obtained as limiting cases of the same formalism. The analysispresented 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 todemonstrate this SV-to-WV crossover consists of two coupled MZIs (the “system” andthe “detector”). Measurements on such a setup have been reported in the literature (seee.g., Ref. [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, ItamarSivan, Lev Vaidman and Emil Weisz. YG acknowledges the hospitality of the Dahlem0Center for Complex Quantum Systems. This work is supported by the GIF, ISF andDFG (Deutsche Forschungsgemeinschaft) grant RO 2247/8-1.
Appendix A. Derivation of the formula for two-particle GCV in terms ofthe irreducible correlation function
Here we present an extended derivation of equation (2). The two-particle GCV of Q isdefined by, (cid:104) Q (cid:105) T PGCV = (cid:104) I D δI D (cid:105) γ (cid:104) I D (cid:105) = (cid:104) I D ( I D − (cid:104) I D (cid:105) ) (cid:105) γ (cid:104) I D (cid:105) (A.1)This can be rewritten as, (cid:104) I D (cid:105) (cid:104) I D (cid:105) − (cid:104) I D (cid:105) (cid:104) I D (cid:105) + (cid:104) I D I D (cid:105) − (cid:104) I D (cid:105) (cid:104) I D (cid:105) γ (cid:104) I D (cid:105) (A.2)which yields equation (2), (cid:104) Q (cid:105) T PGCV = 1 γ (cid:18) (cid:104) δI D (cid:105) + (cid:104)(cid:104) I D I D (cid:105)(cid:105)(cid:104) I D (cid:105) (cid:19) . (A.3) 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)). Inaccordance with equation (A.1) we compute the current-current correlator (cid:104) I D I D (cid:105) andthe average current (cid:104) I D (cid:105) , defined with respect to the density matrix ρ = e iγQ Q I S ⊗ I S e − iγQ Q , (cid:104) I D I D (cid:105) =Tr (cid:8) I D I D e iγQ Q I S I S e − iγQ Q (cid:9) ==Tr (cid:8) I D I D (cid:0) e iγ − Q Q (cid:1) I S I S (cid:0) e − iγ − Q Q (cid:1)(cid:9) (B.1)where in the last step we employed e γQ Q = 1 + ( e iγ − Q Q because the eigenvaluesof Q i are only 0 or 1. Then, (cid:104) I D I D (cid:105) == (cid:104) I D (cid:105) (cid:104) I D (cid:105) (cid:0) γ (cid:1) Re (cid:110) ie iγ (cid:104) I D Q (cid:105) (cid:104) I D Q (cid:105) (cid:111) + 4 sin (cid:0) γ (cid:1) (cid:104) Q I D Q (cid:105) (cid:104) Q I D Q (cid:105) (cid:104) I D (cid:105) (cid:104) I D (cid:105) (B.2)where (cid:104)(cid:105) denotes average with respect to the noninteracting setup ( γ → (cid:104) I D (cid:105) yields (cid:104) I D (cid:105) = (cid:104) I D (cid:105) (cid:104) Q (cid:105) (cid:0) γ (cid:1) Re (cid:110) ie iγ (cid:104) I D Q (cid:105) (cid:111) + 4 sin (cid:0) γ (cid:1) (cid:104) Q I D Q (cid:105) (cid:104) I D (cid:105) . (B.3)1Plugging equations (B.2) and (B.3) in equation (A.1) yields an expression for a twoparticle GCV, (cid:104) Q (cid:105) T PGCV = 4 sin (cid:0) γ (cid:1) γ Re (cid:110) ie iγ (cid:104) I D Q (cid:105) (cid:104) δI D Q (cid:105) (cid:111) + sin (cid:0) γ (cid:1) (cid:104) Q I D Q (cid:105) (cid:104) δQ I D Q (cid:105)(cid:104) I D (cid:105) + 4 sin (cid:0) γ (cid:1) Re (cid:110) ie iγ (cid:104) I D Q (cid:105) (cid:104) Q (cid:105) (cid:111) + 4 sin (cid:0) γ (cid:1) (cid:104) Q I D Q (cid:105) (cid:104) Q (cid:105) . (B.4)In the weak limit ( γ →
0) this expression simplifies tolim γ → (cid:104) Q (cid:105) T PGCV = 2 Re (cid:26) i (cid:104) I D Q (cid:105) (cid:104) δI D Q (cid:105)(cid:104) I D (cid:105) (cid:27) , (B.5)and in the strong limit ( γ → π ),lim γ →∞ (cid:104) Q (cid:105) T PGCV = 4 π (cid:104) Q I D Q (cid:105) (cid:104) δQ I D Q (cid:105) − Re {(cid:104) I D Q (cid:105) (cid:104) δI D Q (cid:105)}(cid:104) I D (cid:105) − Re {(cid:104) I D Q (cid:105) (cid:104) Q (cid:105) } + 4 (cid:104) Q I D Q (cid:105) (cid:104) Q (cid:105) . (B.6) Appendix C. Perturbative calculation of expectation values
In this section we derive the expression for expectation values of the current and thecurrent-current correlator. Employing a path integral formalism, a general formula forthe expectation value of an operator ˆ O [Ψ † , Ψ] is, (cid:68) ˆ O [Ψ † , Ψ] (cid:69) = (cid:82) D [ ¯Ψ , Ψ] ˆ O [ ¯Ψ , Ψ] e iS [ ¯Ψ , Ψ] (cid:82) D [ ¯Ψ , Ψ] e iS [ ¯Ψ , Ψ] , (C.1)where S = S + S int + S T is the full action over the Schwinger-Keldysh contour with S [ ¯Ψ , Ψ] = (cid:88) m =1 (cid:90) drdr (cid:48) ¯Ψ m,α ( r ) ˘ G − m,αβ ( r − r (cid:48) )Ψ m,β ( r (cid:48) ) , (C.2) S int [ ¯Ψ , Ψ] = (cid:88) m,n =1 (cid:90) drρ m,α ( r ) g mn ˆ η clαβ ρ n,β ( r ) (C.3)and S T [ ¯Ψ , Ψ] = (cid:88) m,n =1 (cid:90) drdr (cid:48) ¯Ψ m,α ( r )Γ mn ( r, r (cid:48) )ˆ γ clαβ Ψ n,β ( r (cid:48) ) . (C.4)where α , β are the Keldysh indices in forward/backward basis, m,n are the wire indices,r denotes the spacial 2-vector (r=(x,t)), ρ m,α ( r ) = ¯Ψ m,α ( r )Ψ m,α ( r ) is the density of theparticles, ˆ η clαβ is the Keldysh matrix (cf. Table C2), g mn = g (cid:107) g (cid:107) g ⊥ g ⊥ g (cid:107)
00 0 0 g (cid:107) , (C.5)2 (cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72) χ α, β (+ / − ) (cl/q)+ ˆ γ + αβ = (cid:32) (cid:33) ˆ γ + αβ = (cid:32) (cid:33) − ˆ γ − αβ (cid:32) − (cid:33) ˆ γ − αβ = (cid:32) − − (cid:33) cl ˆ γ clαβ = (cid:32) − (cid:33) ˆ γ clαβ = (cid:32) (cid:33) q ˆ γ qαβ = (cid:32) (cid:33) ˆ γ qαβ = (cid:32) (cid:33) Table C1.
A list of Keldysh ˆ γ χαβ matrices (for fermions) in different bases of bosonic( χ ) indices and fermionic indices ( α, β ). Γ mn ( r, r (cid:48) ) = s ( x, x (cid:48) ) 0 0Γ ∗ s ( x, x (cid:48) ) 0 0 00 0 0 Γ ∗ d ( x, x (cid:48) )0 0 Γ d ( x, x (cid:48) ) 0 δ ( t − t (cid:48) ) (C.6)and Γ s ( x, x (cid:48) ) = Γ s δ ( x − x s ) δ ( x (cid:48) − x s ) + Γ s δ ( x − x s ) δ ( x (cid:48) − x s ) and Γ d ( x, x (cid:48) ) =Γ d δ ( x − x d ) δ ( x (cid:48) − x d ) + Γ d δ ( x − x d ) δ ( x (cid:48) − x d ). ˘ G − m,αβ ( k, ω ) is the inverse of thefermionic Green function for particles whose dynamics is described by H S + H D , whichin ( k, ω ) representation is given by [42]˘ G m,βα ( k, ω ) = 12 (cid:20) F ( ω ) + αω − v F k + i(cid:15) − F ( ω ) − βω − v F k − i(cid:15) (cid:21) . (C.7)Here we assume the setup was in thermal equilibrium with a temperature T (describedby the fermionic population function F ( ω ) = tanh (cid:0) ω T (cid:1) at the time t → −∞ , whenthe tunneling Γ, and the interaction g were adiabatically turned on. By assumingsmall tunneling the action can be expanded in power series to desired order in Γ, thenequation (C.1) gets a form, (cid:68) ˆ O [Ψ † , Ψ] (cid:69) = (cid:80) n n ! (cid:68) ˆ O [ ¯Ψ , Ψ]( iS T [ ¯Ψ , Ψ]) n (cid:69) Ω (cid:80) n n ! (cid:10) ( iS T [ ¯Ψ , Ψ]) n (cid:11) Ω (C.8)where (cid:104)(cid:105) Ω denotes averaging with respect to the action S + S int .The current in a chiral system with linear dispersion is linearly proportional to thedensity ( (cid:104) I (cid:105) = ev F (cid:104) ρ (cid:105) ). The expectation value of the density is obtained by weaklyperturbing the system by a quantum potential probe V q , which should be taken to zeroat the end to restore causality [42]. Therefore, we obtain an expression for the currentmeasured at Dm ( m = 1 , , ,
4) (cf. figure 1), (cid:104) I Dm ( x, t ) (cid:105) = − iev F (cid:110) ˜ G m ( x, t ; x, t )ˆ γ q (cid:111) , (cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72)(cid:72) χ α, β (+ / − ) (cl/q)+ ˆ η + αβ = (cid:32) (cid:33) ˆ η + αβ = (cid:32) (cid:33) − ˆ η − αβ (cid:32) − (cid:33) ˆ η − αβ = (cid:32) − − (cid:33) cl ˆ η clαβ = (cid:32) − (cid:33) ˆ η clαβ = (cid:32) (cid:33) q ˆ η qαβ = (cid:32) (cid:33) ˆ η qαβ = (cid:32) (cid:33) Table C2.
A list of Keldysh ˆ η χαβ matrices (for bosons) in different bases of bosonic χ, α and β indices. where ˜ G m,βα ( x, t ; x, t ) = − i (cid:10) T Ψ m,β ( x, t ) ¯Ψ m,α ( x, t ) (cid:11) is the fermionic Green functionof the system (averaged with respect to the full action, S ) at point (x,t) of the m-th arm. The trace is over the Keldysh indices, where ˆ γ q is the Keldysh matrix (cf.Table C1). For the sake of simplicity we compute first (cid:104) I D ( x, t ) (cid:105) by expanding it tosecond (leading) order in Γ. We then employ the current conservation to find (cid:104) I D (cid:105) , (cid:104) I D ( x, t ) (cid:105) = I − (cid:104) I D ( x, t ) (cid:105) , where I = e h V . To this order, particle tunnels twice. Weemploy equation (C.8) to expand ˜ G in S Γ . This yields (cid:104) I D ( x, t ) (cid:105) = iev F (cid:90) dt dt (cid:88) p,q = { s , s } (cid:110) Γ ∗ p Γ q G ,αβ ( x − x p , t − t )ˆ γ clβγ G ,γδ ( x p − x q , t − t )ˆ γ clδ(cid:15) G ,(cid:15)ζ ( x − x, t − t )ˆ γ qζα (cid:111) . (C.9)Here G m ( x, t ) βα = − i (cid:10) T Ψ m,β ( x, t ) ¯Ψ m,α (0 , (cid:11) (C.10)is the fermionic Green function averaged with respect to the interacting action, S + S int .We perform Fourier transform over the time variable to obtain, (cid:104) I D ( x, (cid:105) = iev F (cid:90) dω π (cid:88) p,q = { s , s } Γ ∗ p Γ q G ,αβ ( x − x p , ω )ˆ γ clβγ G ,γδ ( x p − x q , ω )ˆ γ clδ(cid:15) G ,(cid:15)ζ ( x − x, ω )ˆ γ qζα . (C.11)To find the current-current correlator, we generalize the last procedure, employing (cid:104)(cid:104) I D I D (cid:105)(cid:105) = (cid:104)(cid:104) I D I D (cid:105)(cid:105) , to obtain,41 τ (cid:90) τ/ − τ/ dt (cid:104)(cid:104) I D ( x (cid:48) , t ) I D ( x, (cid:105)(cid:105) = − e v F τ (cid:90) τ/ − τ/ dt (cid:88) pqrs Γ ∗ p Γ q Γ ∗ r Γ s (cid:90) dt dt dt dt ×× G ,αβ ( x (cid:48) − x p , t (cid:48) − t ) ˆ γ clβγ G ,ηθ ( x − x r , − t ) ˆ γ clθι ˜ M (cid:48) ικγδ ( x s , x r , x q , x p ; t , t , t , t ) ×× ˆ γ clκλ G ,λµ ( x s − x (cid:48) , t −
0) ˆ γ clδ(cid:15) G ,(cid:15)ζ ( x q − x, t − t ) ˆ γ qζα ˆ γ qµη (C.12)where ˜ M (cid:48) ( r , r , r , r ) (cid:44) ˜ M ( r , r , r , r ) − G ( r − r ) G ( r − r ). And˜ M δγβα ( r , r , r , r ) (cid:44) − (cid:10) T Ψ ,δ ( r ) ¯Ψ ,γ ( r )Ψ ,β ( r ) ¯Ψ ,α ( r ) (cid:11) (C.13)is the collision matrix. We perform Fourier transform over the time differences, suchthat ω corresponds to t − t , ω to t − t and ¯ ω to ( t + t ) − ( t + t ). Finally, ityields1 τ (cid:90) τ/ − τ/ dt (cid:104)(cid:104) I D ( x (cid:48) , t ) I D ( x, (cid:105)(cid:105) = − e v F τ (cid:90) τ/ − τ/ dt (cid:88) pqrs Γ ∗ p Γ q Γ ∗ r Γ s (cid:90) d ¯ ωdω dω (2 π ) e i ¯ ωt ×× G ,αβ (cid:16) x (cid:48) − x p , ω − ¯ ω (cid:17) ˆ γ clβγ × × G ,ηθ (cid:16) x − x r , ω + ¯ ω (cid:17) ˆ γ clθι ˜ M (cid:48) ικγδ ( x s , x r , x q , x p ; ω , ω , ¯ ω ) ×× ˆ γ clκλ G ,λµ (cid:16) x s − x (cid:48) , ω − ¯ ω (cid:17) ˆ γ clδ(cid:15) G ,(cid:15)ζ (cid:16) x q − x, ω + ¯ ω (cid:17) ˆ γ qζα ˆ γ qµη . In order to find a simpler expression for the time integral over τ , we denotethe current-current correlator by F ( t ): F ( t ) = (cid:104)(cid:104) I D ( x (cid:48) , t ) I D ( x, (cid:105)(cid:105) , and its Fouriertransform F (¯ ω ). equation (C.14) can be written in these terms as¯ F (cid:44) τ (cid:90) τ/ − τ/ dtF ( t ) = 1 τ (cid:90) τ/ − τ/ dt (cid:90) d ¯ ω π e i ¯ ωt F (¯ ω ) . (C.14)It is easy to find an expression for F (¯ ω ) by comparing equations (C.14) and (C.14).First, we write τ [ F (¯ ω ) + F ( − ¯ ω )] = τ (cid:82) ∞−∞ [ F ( t ) + F ( − t )] ( e i ¯ ωt + e − i ¯ ωt ) dt. From theother hand we approximate the average by,¯ F ≈ τ (cid:90) ∞−∞ [ F ( t ) + F ( − t )] e − π ( t/τ ) dt where we have assumed that F ( t ) grows much slower than e π ( t/τ ) , and the antisymmetricpart of F ( t ) is cancelled by the averaging. By comparing the exponentials in the twoequations we obtain ¯ ω = √ πτ . Then ¯ F = τ (cid:104) F ( √ πτ ) + F ( − √ πτ ) (cid:105) . Appendix D. Calculation of the fermionic correlators
Here we derive the expressions for the fermionic propagator (cf. equation (C.10)) andthe collision matrix (cf. equation (C.13)) averaged with respect to the action S + S int ,5within an interacting arms (2,3) of MZI (the propagator in arms 1 and 4 can be found bytaking g ⊥ → S + S int as [45], S + S int [ ¯Ψ , Ψ; Φ] = ¯Ψ G − Ψ + 14 Φ g − Φ , (D.1)with the notation,¯Ψ G − Ψ = (cid:88) m =2 , (cid:90) drdr (cid:48) ¯Ψ m,α ( r ) G − m,αβ ( r − r (cid:48) )Ψ m,β ( r (cid:48) )where G − m,αβ ( r − r (cid:48) ) = ˘ G − m,αβ ( r − r (cid:48) ) − ˆ γ χαβ Φ m,χ ( r ) δ ( r − r (cid:48) )and Φ g − Φ = (cid:88) m,n =2 , (cid:90) dr Φ m,α ( r ) g − mn ˆ η clαβ Φ n,β ( r ) . where we implicitely sum over the Keldysh indices α, β, χ = ± g − mn is the inverse of the m, n = 2 , g mn (cf. equation (C.5)).Following the functional bosonization procedure [45], we obtain a general expression foran n-fermion correlator, (cid:42) T n (cid:89) i Ψ a i ( r i ) ¯Ψ b i ( q i ) (cid:43) = (cid:42) T n (cid:89) i Ψ a i ( r i ) ¯Ψ b i ( q i ) (cid:43) e − (cid:68) T ( (cid:80) ni θ ai ( r i ) − θ bi ( q i ) ) (cid:69) Φ , (D.2)where a, b = ( α, m ) denote the Keldysh and the wire indices, r, q = ( x, t ), (cid:104)(cid:105) is thefermionic correlator with respect to the free action S [ ¯Ψ , Ψ] = ¯Ψ ˘ G − Ψ , (D.3)and (cid:104)(cid:105) Φ is the Φ-field correlator with respect to the action S Φ [Φ] = 14 Φ g − Φ + Φ ˆΠΦ (D.4)respectively. Here Φ ˆΠΦ = (cid:88) m =2 , (cid:90) drdr (cid:48) Φ m,α ( r ) ˆΠ m,αβ ( r, r (cid:48) )Φ m,β ( r (cid:48) )with the polarization matrix,ˆΠ m,αβ ( r − r (cid:48) ) = i (cid:110) ˆ γ α ˘ G m ( r − r (cid:48) )ˆ γ β ˘ G m ( r (cid:48) − r ) (cid:111) , (D.5)where the trace is taken over the Keldysh fermionic indices [42]. The θ field is definedby θ m,α ( r ) = − i (cid:88) βγ = ± (cid:90) dr (cid:48) G Bm,αβ ( r − r (cid:48) )ˆ η clβγ Φ m,γ ( r (cid:48) ) , (D.6)6where G B is the bosonic free Green function with linearized spectrum, G Bm,βα ( k, ω ) = 12 (cid:20) B ( ω ) + αω − v F k + i(cid:15) − B ( ω ) − βω − v F k − i(cid:15) (cid:21) . (D.7)The action for the Φ field (cf. equation (D.4)) is quadratic due to Larkin-Dzyaloshinskii[46] theorem, therefore an exact expression for the Φ-field correlator is i ˘ Q mn,αβ ( r − r (cid:48) ) (cid:44) (cid:104)T Φ m,α ( r )Φ n,β ( r (cid:48) ) (cid:105) Φ = i (cid:16) g − mn ˆ η clαβ δ ( r − r (cid:48) ) + δ mn ˆΠ m,αβ ( r − r (cid:48) ) (cid:17) − . We reduce the problem of finding an inverse of an infinite-dimensions matrix, invertingit to the finite (4) dimensions by Fourier-transforming it to a diagonal ( k, ω ) basis.Employing equation (D.6) we obtain the θ -field correlator, i ˘ K mn,αβ ( r − r (cid:48) ) (cid:44) (cid:104)T θ m,α ( r ) θ n,β ( r (cid:48) ) (cid:105) Φ == − i (cid:90) dqdq (cid:48) (cid:104) G B ( r − q )ˆ η cl ˘ Q ( q − q (cid:48) )ˆ η cl G B ( q (cid:48) − r (cid:48) ) (cid:105) mn,αβ , (D.8)where we implicitly sum over the Keldysh and the wire indices. This yields,˘ K mn = δ mn (cid:32) B (cid:104) ˘ K R (cid:107) − ˘ K A (cid:107) (cid:105) ˘ K R (cid:107) ˘ K A (cid:107) (cid:33) + σ xmn (cid:32) B (cid:104) ˘ K R ⊥ − ˘ K A ⊥ (cid:105) ˘ K R ⊥ ˘ K A ⊥ (cid:33) (D.9)where ˘ K R/A (cid:107) ( k, ω ) = πk (cid:20) ω − v ρ k ± i(cid:15) + 1 ω − v σ k ± i(cid:15) − ω − v F k ± i(cid:15) (cid:21) , (D.10)and ˘ K R/A ⊥ ( k, ω ) = πk (cid:20) ω − v ρ k ± i(cid:15) − ω − v σ k ± i(cid:15) (cid:21) . (D.11)Here, v ρ = u + g ⊥ π , v σ = u − g ⊥ π , with u = v F + g (cid:107) π . We plug this result inequation (D.2) to compute the Green function (equation (C.10)) and the collisionmatrix of the particles in arms 2 and 3 (equation (C.13)). The calculation requirestransformation of equations (D.10) and (D.11) to real (x,t) space. Here we present thefinal result, G m,βα ( x, t ) = − T v F (cid:114) sinh (cid:104) πT (cid:16) t − xv ρ + i Λ [ α Θ( t ) − β Θ( − t )] (cid:17)(cid:105) ×× (cid:114) sinh (cid:104) πT (cid:16) t − xv σ + i Λ [ α Θ( t ) − β Θ( − t )] (cid:17)(cid:105) . (D.12)Fourier-transforming the time coordinate yields, G m,βα ( x, ω ) = − i v F [ F ( ω ) + α Θ( x ) − β Θ( − x )] e iω xu ξ ( λ ) (cid:90) − ς (cid:18) T | x | u λ, s (cid:19) e isω | x | u λ ds (D.13)7with the definitions λ = (cid:104) u g ⊥ π − (cid:0) u g ⊥ π (cid:1) − (cid:105) − , ξ ( λ ) = λ √ λ +1 − , and ς ( A, s ) = A √ sinh[ πA (1 − s )] sinh[ πA (1+ s )] . For the sake of consistency check, lim g ⊥ → ,g (cid:107) → G = ˘ G . Andthe collision matrix reads,˜ M δγβα ( x , x , x , x ) = G ,δγ ( x ) G ,βα ( x ) ˜ ζ (1) γα ( x ) ˜ ζ (1) δβ ( x ) ˜ ζ (2) γβ ( x ) ˜ ζ (2) δα ( x )where, ˜ ζ (1) βα ( x, t ) = (cid:114) sinh (cid:104) πT (cid:16) t − xvρ + i Λ [ α Θ( t ) − β Θ( − t )] (cid:17)(cid:105)(cid:113) sinh [ πT ( t − xvσ + i Λ [ α Θ( t ) − β Θ( − t )] )] and ˜ ζ (2) βα ( x, t ) = (cid:16) ˜ ζ (1) βα ( x, t ) (cid:17) − . Fourier-transforming the time coordinates yields,˜ M δγβα ( x , x , x , x , ω , ω , ω ) = 12 (cid:90) dω (cid:48) dω (cid:48) dω (cid:48) (2 π ) G ,δγ ( x , ω − ω (cid:48) ) G ,βα ( x , ω − ω (cid:48) ) ×× ˜ ζ (1) γα ( x , ω − ω (cid:48) + ω (cid:48) − ω (cid:48) ζ (1) δβ ( x , ω − ω (cid:48) − ω (cid:48) + ω (cid:48) ζ (2) γβ ( x , ω (cid:48) − ω (cid:48) − ω (cid:48) ζ (2) δα ( x , ω (cid:48) + ω (cid:48) + ω (cid:48) , (D.14)where we have used the short notation x ij = x i − x j ; G is the single particle propagatorgiven by equation (D.13), and ˜ ζ (1 / βα ( x, ω ) = 2 πδ ( ω ) cosh (cid:0) πT xλu (cid:1) − xλ ˜ Z (1 / βα ( x, ω ), where˜ Z (1 / βα is given by,˜ Z (1 / βα ( x, ω ) = − i u [ B ( ω ) + α Θ( x ) − β Θ( − x )] e iω xu ξ ( λ ) (cid:90) − κ ( T | x | u λ, s ) e ± isω xu λ ds, (D.15)where B ( ω ) = coth( ω T ) is the Bose function and κ ( A, s ) = (cid:113) sinh[ πA (1+ s )]sinh[ πA (1 − s )] .equation (D.14) has a pictorial interpretation, presented in figure D1, according towhich, the ˜ Z particles are the dressed bosons that carry the interaction between theelectrons. Appendix E. Passage of the electron through the MZI: a semiclassicalpicture
Here we present the propagation of a localized wave packet (according to a semiclassicalpicture) through an interacting arm of MZI, and derive the condition to be in thesemiclassical regime. We assume semiclassically a propagating rectangular shaped wavepacket with a width ∼ (cid:126) eV in time domain (cf. figure E1). The propagation of the singleparticle wave function can be derived by convolving the initial state with the retardedGreen function, Ψ( x, t ) = i (cid:90) G R ( x − x (cid:48) , t ) ∗ Ψ( x (cid:48) , dx (cid:48) . (E.1)An expression for the zero temperature retarded Green function is (this is simply derivedfrom equation (10)). G R ( x, t ) = iuπλxv F Π (cid:16) uxλ ( t − xξ ( λ ) u ) (cid:17)(cid:114) − (cid:16) uxλ ( t − xξ ( λ ) u ) (cid:17) (E.2)8 x x x x ω - ω ’ ω - ω ’ ½ ( ω - ω ’ + ω ’ - ω ’ ) ½ ( ω ’ + ω ’ + ω ’ ) ½ ( ω ’ - ω ’ - ω ’ ) ½ ( ω - ω ’ - ω ’ + ω ’ ) ω +½ ω ω -½ ωω -½ ω ω +½ ωα β γ δ Figure D1.
The collision matrix ˜ M (cf. equation (D.14)). A diagrammaticrepresentation of the renormalized inelastic collision between two chiral fermionsinside the interacting region. Straight lines correspond to fermionic Green functions(gray- outside the interacting region and black- inside). Wavy lines correspond tobosonic Green functions (red and blue for the two different types of bosons, cf.equation (D.15)). The vertices x , x ( x , x ) correspond to the two entry (exit) pointsof the interaction region on the edges. The Keldysh indices ( ± ) at these points areindicated by α, β, γ, δ . Electrons enter the interacting region with energies ω + ω and ω − ω and exit with energies ω − ω and ω + ω respectively, exchanging energy ω via 4 possible different bosons. where Π( x ) = (cid:40) − < x < o.w. is a rectangle function. The wave packet at 4 differentpoints is shown in figure E1. We observe, the wave packet has been broadened asa result of the interaction, its width in time at different space points is given by∆ t ( x ) = (cid:126) eV + λx u . The center of mass of the wave packet then propagates withvelocity v CM = uξ ( λ ) . Consistent with the semiclassical picture, we require the width ofthe wave packet to be much smaller compared with the propagation time through theMZI, ∆ t ( L ) (cid:28) L/v CM . From this condition we deduce, eV (cid:29) (cid:126) uL and λ (cid:28) Appendix F. General GCV for an N-state system
Here we present a derivation of GCV for a general system with N-states being measuredby a Gaussian detector. We show that the weak-to-strong crossover in such a casemay be oscillatory with a bounded number of periods of the order of O ( N ). Theinitial state of the system is a mixed state, which is represented by the density matrix ρ s = (cid:80) n,m R nm | α n (cid:105) (cid:104) α n | . The detector is initialized in the zeroth coherent state (wedenote the α (cid:48) s coherent state by | ˜ α (cid:105) ) such that its density matrix is ρ d = (cid:12)(cid:12) ˜0 (cid:11) (cid:10) ˜0 (cid:12)(cid:12) .We neglect the dynamics of the system and the detector assuming the measurementprocess was short in time compared to the typical timescales of the system and thedetector. The coupling Hamiltonian is H I = w ( t ) g ˆ A ( b † + b ) with b, b † are the ladderoperators of the detector, ˆ A = (cid:80) n a n | α n (cid:105) (cid:104) α n | and w ( t ) is a window function around thetime of the measurement. The post-selection is represented by the projection operator,9 � ℏ�� | Ψ ( � , � )| (a) (b) (c) (d) Figure E1.
A propagation of the wave packet through an interacting arm of theMZI, at zero temperature, for λ = 1 for different points (a) x = 0, (b) x = u (cid:126) eV , (c) x = 2 u (cid:126) eV , (d) x = 3 u (cid:126) eV . As can be derived from equation (E.2), the width of thewave packet is given by, ∆ t = (cid:126) eV + λx u . Π f = (cid:80) n,m P nm | α n (cid:105) (cid:104) α n | . Plugging into equation (1) and considering, ρ tot = ρ s ⊗ ρ d and δq = b , yields (cid:68) ˆ A (cid:69) GCV = (cid:80) n,m a n R nm P mn e − g ( a n − a m ) (cid:80) n,m R nm P mn e − g ( a n − a m ) . (F.1)The numerator and the denominator consist of sums of Gaussian (in g) functions, withdifferent coefficients and prefactors. Each Gaussian is a monotonic function (for g > g ∈ [0 , ∞ )) is ofthe order of O ( N ), where N is the number of system’s states. Appendix G. A full list of diagrams
Figure G1 depicts a full list of irreducible diagrams to fourth (leading) order in tunnelingwhich should be taken in account for the current-current correlator. It is divided todiagrams with no flux dependence (cf. figure G1(a)), diagrams which are dependent oneither Φ S or Φ D (cf. figures G1(b) and G1(c)), and diagrams which are depend on bothΦ S and Φ D , cf. figure G1(d).0 Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D (a) Diagrams independent of the Aharonov-Bohm flux. Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D (b) Diagrams with Φ S dependence. Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D (c) Diagrams with Φ D dependence. Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D V D V S Φ S Φ D S1S2S3S4 D1D2D3D4
00 00 Φ S Φ D S1S2S3S4 D1D2D3D4 V S V D V D V S Φ S Φ D S1S2S3S4 D1D2D3D4
00 00 (d) Diagrams depending on both Φ S and Φ D . Figure G1.
The full list of irreducible diagrams to fourth (leading) order in tunnelingwhich should be taken in account for the current-current correlator (cf. 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 (cf. equations (7)and (8)). The diagrams are divided to four groups by their Aharonov-Bohm fluxdependence. The leading diagrams which were included in the calculation of the GCV,are in 1(d).
References [1] von Neumann J 1955
Mathematical Foundations of Quantum Mechanics (Princeton UniversityPress)[2] Korotkov A N and Averin D V 2001
Phys. Rev. B Rev. Mod. Phys. Phys. Rev. B Nature
Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett. Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. Lett.
Nat. Commun. Phys. Rev. Lett.
Proc. Natl.Acad. Sci.
Phys. Rep.
Science
Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. Lett.
Rev. Mod. Phys. Phys. Rev. Lett.
Science
Nature
Phys. Rev. Lett.
Phys. Rev. B Phys. Rev. Lett.
Phys. Rev. A Phys. Rev. Lett.
Phys. Rev. B Science
Phys. Rev.
B1410–B1416[34] Chalker J, Gefen Y and Veillette M 2007
Phys. Rev. B Phys. Rev. B Phys. Rev. Lett. Phys. Rev. Lett.
Phys. Rev. Lett. Phys. Rev. Lett.
Phys. Rev. Lett.
Phys. Rev. B Field Theory of Non-Equilibrium Systems (Cambridge University Press)[43] Gutman D B, Gefen Y and Mirlin A D 2010
Phys. Rev. B Phys. Rev. B Bosonisation as the Hubbard-Stratonovich transformation (Springer)[46] Dzyaloshinskii I E and Larkin A I