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

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.

Introduction. Measurement in quantum mechanics is inseparable from the dynamics of the system involved. There are two disparate view points of the process of measurement. One relies on von Neumann's projection postulate, associated with the evasive notion of quantum collapse [1], and corresponds to strong measurement. Alternatively, one may consider the limit of weak (continuous) measurement of an observable (reflecting weak coupling between the system (S) and the detector (D)) [2]. Here one employs dynamics which involves the Hamiltonians of the system (H S ), the detector (H D ), and their coupling (H SD ), H = H S + H D + H SD . Weak measurement disturbs the system in a minimal way, and provides only partial information on the state of the latter [3].
Weak measurements, due to their vanishing backaction, can be exploited for quantum feedback schemes [4] and conditional measurements. The latter is especially interesting for a two-step measurement protocol (called weak value (WV) [5]), which consists of a weak measurement (of the observableÂ), followed by a strong one (ofB), [Â,B] = 0. The outcome of the first is conditional on the result of the second (postselection). WVs have been observed in experiments [6][7][8]. Their unusual expectation values [5,9] may be utilized for various purposes, including weak signal amplification [10][11][12][13][14][15][16], quantum state discrimination [17], and non-collapsing observation of virtual states [18]. 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 it is possible to modify the WV protocol, rendering the first measurement strong (strong value, SV). In this limit the protocol amounts to first collapsing the system onto an eigenstate ofÂ, which defines the initial state for the measurement ofB. It turns out that as far as single-degree-of-freedom systems are concerned, the WV-to-SV crossover is quite straightforward and is a smooth function of the interaction strength [19]. By contrast, in experiments with electron nanostructures, interactions between electrons play a crucial role. In many cases, the interaction strength can be controlled experimentally [8,20].
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 generalized conditional value (GCV) of the operatorÂ as an average shift of the detector, δq =q − q g=0 , during the measurement 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 where ρ 0 is the total density matrix which describes the initial state of S and D, andÛ is an operator which describes the evolution in time of the whole setup during the measurement. Here FIG. 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. ΦS and ΦD are the magnetic fluxes through the respective MZIs. The lengths of the arms 1 and 2 between SQPC1 and SQPC2 are αL and L respectively, and similarly for the detector's arms 3 and 4, as is shown in the figure. In the present analysis α = 1.
−gw(t)pÂ, with w(t) -the time window of the measurement;q andp are the "position" and "momentum" operators of the detector ([q,p] = i ). We note that Eq. (1) provides the correct WV [5] and SV [21] in the respective limits (g 1, g 1). Our specific setup is depicted in Fig. 1. It consists of two Mach-Zehnder interferometers (MZIs), the "system" and the "detector" respectively, that are electrostatically coupled [20,22]. It is possible to tune the respective Aharonov-Bohm fluxes, Φ S and Φ D independently [20].
A single-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 single-particle dynamics. According to this (over)simplified picture, particles going simultaneously through the interacting arms 2 and 3 (cf. Fig. 1), gain an extra phase e iγ [23,24], where γ takes values in the range [0, π]. First, we consider the intra-MZI operators, defined in a two-state single particle space, {|m }, 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 Q m = |m m|. The transition through the p-th QPC is described by the scattering 25]. The entries r p and t p encompass information about the respective Aharonov-Bohm flux and for p = 2 s , 2 d , about the orbital phase gained between the two QPCs. The dimensionless current operators at the source (S1, S2) and the drain (D1, D2) terminals of the system-MZI are given by I Sm = S 1s Q m S † 1s and I Dm = S † 2s Q m S 2s respectively, with m = 1, 2, and similarly for the detector with m = 3, 4 and employing the matrices S 1 d and S 2 d .
In view of Eq. (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 matrix ρ 0 = I S1 ⊗ I S4 operating in the two-particle product space, |m ⊗ |n (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 operatorÛ = e iγQ2⊗Q3 . A positive reading of the projective measurement consists of the detection of a particle at D2, and is described by the projection operator Π f = I D2 ⊗ 1. The detector reads the current at D3 (δq of Eq. (1) corresponds to 1 ⊗ δI D3 ). Plugging these quantities into Eq. (1) yields an expression for the single particle GCV, (2) The averages are calculated with respect to the total density matrix after the measurement, Ô = Tr ÔÛ † ρ 0Û . We have defined δI D3 I D3 − I D3 γ=0 , and I D2 I D3 I D2 I D3 − I D2 I D3 is the irreducible current-current correlator. A straightforward calculation yields where Ô 0 Tr Ô ρ 0 is an average with respect to the non-interacting setup, δI D3 Q 3 I D3 Q 3 0 − I D3 0 Q 3 0 and δQ 3 I D3 Q 3 3 0 . This result shows a trivial crossover between the weak (γ → 0) and strong (γ → π) limits.
A full many-body analysis. We note that the system's Hamiltonian consists of . Here Γ p is the tunneling amplitude at QPC p and x p m 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 arms m = 3, 4. We next assume that the lengths of the interacting arms are equal, where the normal ordering with respect to the equilibrium (no voltage bias) state is defined as We are now at the position to construct the GCV for the actual many-body setup. We employ Eq. (2) to define the many-body GCV of Q 2 , where the current operator is given by, I(x, t) = ev F : Ψ † (x, t)Ψ(x, t) :. We average over time τ L v F . 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 λ, employing the Keldysh formalism. In this limit expectation values are taken with respect to tunneling decoupled edge states. The current is, and the irreducible current-current correlator [26] 1 τ Here the summation is over p, q = (1 s , 2 s ), r, s = (1 d , 2 d ) and repeating indices; γ cl = 1 0 0 −1 and γ q = 1 0 0 1 are the Keldysh γ matrices. G m is the fermionic propagator on the m-th arm (cf. Eq. (7)), and , whereM is the collision matrix of two electrons in the interacting arms 2 and 3, discussed below (Eq. (8)).
The expressions for the expectation values of Eqs. (5) and (6) can be represented diagrammatically in terms of the contributing processes. In these Feynman-Keldysh diagrams, each line corresponds to a propagator G (cf. Eq. (7)), and the vertices represent tunneling. The diagrams (to leading order in tunneling matrix elements) are depicted in Fig. 2. There are 16 diagrams contributing to the irreducible current-current correlator [26]. The leading diagrams ( Fig. 2 (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). [27] Explicit evaluation of GCV requires the calculation of the single electron G m and the collision matrixM [28]. We first compute the propagators on arms 2 (G 2 ) and 3 (G 3 ), where both the inter-and the intra-channel inter-action is present. This yields where α, β = ±1 are the Keldysh indices (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 The propagators in channels 1 (G 1 ) and 4 (G 4 ) are obtained by substituting g ⊥ = 0 in Eq. (7). This result recovers the simple non-interacting Green function with a renormalized velocity u = v F + 2g π due to intra-channel interaction. The maximal interaction between channel 2 and 3 is at g ⊥ = π 2 u (instability point). Similarly to the single particle analysis, here too the SV limit is reached at a finite value of the inter-channel interaction.
We finally find the collision matrix for two simultane- The relevant Feynman-Keldysh diagrams for the quantities in Eqs. (5) and (6) 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 (cf. Eqs. (7) and (8)). (a) The average current (Eq. (5)), O(Γ 2 ).
Only the system part of the setup (cf. Fig. 1 ously propagating particles through arms 2 and 3 [26]: where we have used the short notation x ij = x i − x j ; G is the single particle propagator given by Eq. (7), andζ represents two types of bosonic propagators (of the dressed photons that carry the interaction), and is given bỹ sinh[πA(1−s)] . Eq. (8) describes a general collision process between 2 interacting particles (also cf. Fig. S1 in [26] for more details).
The result is depicted in Fig. 3. We identify a high temperature regime, τ F L k B T (τ F L is the time of flight through the interacting arm of MZI, τ F L = L u ), where the GCV is exponentially suppressed by the factor e − τ F L k B T due to averaging over an energy window ∼ T . In the opposite, low temperature limit, the phase diagram shows novel oscillatory behaviour. We plot the phase diagram of GCV in a parameter space spanned by the applied voltage normalized by the temperature (eV /k B T ) and the renormalized interaction strength (λ) (cf. Fig. 3). In the low voltage limit (eV k B T ) 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 single particle result (cf. Eq. (3)). For eV > k B T , multiple particle interaction effects become important, and three different regimes are obtained as function of λ.
Here, as function of increasing λ, oscillatory behaviour (∼ J 0 λeV τ F L , where J 0 is the 0-th 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 in Fig. 3 (a), along with the dependence of the GCV on the interaction strength ( Fig. 3 (b-d)) and voltage bias (Fig. 3 (e)).
Discussion. It is interesting to note the similarity between the oscillations found here and the physics of "visibility lobes" that was found experimentally [29] and studied theoretically [30] in the context of coherent transport through a MZI. Both are related to interaction effects in an interferometry setup. Indeed the diagrams presented in Ref. [30] carry resemblance of the diagrams analyzed here, leading to an oscillatory behavior with voltage and interaction strength. Measurements on setups consisting of two electrostatically coupled MZI have been reported [20], albeit not in the context of the present work. By means of external gates one may control the magnitude of the coupling λ. More accessible experimentally would be to fix the distance between the MZIs and observe oscillations with V at moderately low values of λ.
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. Oscillatory crossover is a unique feature of our many-body analysis. The setup chosen to demonstrate this SV-to-WV crossover is amenable to experiment within present technology. In this supplemental material we present some details of the calculations leading to the main results in the manuscript. In particular we provide a full calculation of the correlation functions in Eq. (6), and the derivation of the expressions for the propagator and the collision matrix in Eqs. (7) and (8) of the manuscript.
Derivation of the formula for single particle GCV in terms of the irreducible correlation function Here we present an extended derivation of Eq. (2). The single particle GCV of Q 2 is defined by, This can be rewritten as, which yields Eq. (2), Strong-to-weak crossover of GCV for single particle system Here we present the derivation of GCV for single particle system (i.e. Eq. (3)). In accordance with Eq. (S1) we compute the current-current correlator I D2 I D3 and the average current I D2 , defined with respect to the density matrix ρ = e iγQ2Q3 I S1 ⊗ I S4 e −iγQ2Q3 , I D2 I D3 = Tr I D2 I D3 e iγQ2Q3 I S1 I S4 e −iγQ2Q3 = Tr I D2 I D3 1 + (e iγ − 1)Q 2 Q 3 I S1 I S4 1 + (e −iγ − 1)Q 2 Q 3 (S4) where in the last step we employed e γQ2Q3 = 1 + e iγ − 1 Q 2 Q 3 because the eigenvalues of Q i are only 0 or 1. Then, where 0 denotes average with respect to the noninteracting setup (γ → 0). Similar calculation for I D2 yields Plugging Eqs. (S5) and (S6) in Eq. (S1) yields an expression for a single particle GCV, In the weak limit (γ → 0) this expression simplifies to and in the strong limit (γ → π), 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 operatorÔ[Ψ † , Ψ] is, where S = S 0 + S int + S T is the full action over the Schwinger-Keldysh contour with drdr Ψ m,α (r)Γ mn (r, r )γ cl αβ Ψ n,β (r ). (S13) 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 the particles, η cl αβ is the Keldysh matrix (cf. Table S2), is the inverse of the fermionic Green function for particles whose dynamics is described by H S 0 + H D 0 , which in (k, ω) representation is given by [S1] G m,βα (k, ω) = 1 2 Here we assume the setup was in thermal equilibrium with a temperature T (described by the fermionic population function F (ω) = tanh ω 2T at the time t → −∞, when the tunneling Γ, and the interaction g were adiabatically turned on. By assuming small tunneling the action can be expanded in power series to desired order in Γ, then Eq. (S10) gets a form, where Ω denotes averaging with respect to the action S 0 + S int .
The current in a chiral system with linear dispersion is linearly proportional to the density ( I = ev F ρ ). The expectation value of the density is obtained by weakly perturbing the system by a quantum potential probe V q ,  which should be taken to zero at the end to restore causality [S1]. Therefore, we obtain an expression for the current measured at Dm (m = 1, 2, 3, 4) (cf. Fig. 1), is the fermionic Green function of 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 γ q is the Keldysh matrix (cf. Table S1). For the sake of simplicity we compute first I D1 (x, t) by expanding it to second (leading) order in Γ. We then employ the current conservation to find I D2 , I D2 (x, t) = I 0 − I D1 (x, t) , where I 0 = e 2 h V . To this order, particle tunnels twice. We employ Eq. (S17) to expandG in S Γ . This yields (S18) Here is the fermionic Green function averaged with respect to the interacting action, S 0 + S int . We perform Fourier transform over the time variable to obtain, To find the current-current correlator, we generalize the last procedure, employing I D2 I D3 = I D1 I D4 , to obtain, is the collision matrix. We perform Fourier transform over the time differences, such that ω 2 corresponds to t 2 − t 1 , ω 3 to t 4 − t 3 andω to 1 2 (t 3 + t 4 ) − 1 2 (t 1 + t 2 ). Finally, it yields 1 τ In order to find a simpler expression for the time integral over τ , we denote the current-current correlator by F (t): F (t) = I D2 (x , t)I D3 (x, 0) , and its Fourier transform F (ω). Eq. (S23) can be written in these terms as It is easy to find an expression for F (ω) by comparing Eqs. (S23) and (S24). First, we write 1 From the other hand we approximate the average by, where we have assumed that F (t) grows much slower than e π(t/τ ) 2 , and the antisymmetric part of F (t) is cancelled by the averaging. By comparing the exponentials in the two equations we obtainω = √ 2π

Calculation of the fermionic correlators
Here we derive the expressions for the fermionic propagator (cf. Eq. (S19)) and the collision matrix (cf. Eq. (S22)) averaged with respect to the action S 0 + 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 [S2]. We apply the Hubbard-Stratonovich transformation, and introduce the bosonic auxiliary field Φ, writing an action S 0 + S int as [S3], with the notation,Ψ where and where we implicitely sum over the Keldysh indices α, β, χ = ±1 (in forward/backward basis) and g −1 mn is the inverse of the m, n = 2, 3 submatrix of g mn (cf. Eq. (S14)). Following the functional bosonization procedure [S3], we obtain a general expression for an n-fermion correlator, where a, b = (α, m) denote the Keldysh and the wire indices, r, q = (x, t), 0 is the fermionic correlator with respect to the free action and Φ is the Φ-field correlator with respect to the action respectively. Here where the trace is taken over the Keldysh fermionic indices [S1]. The θ field is defined by where G B is the bosonic free Green function with linearized spectrum, The action for the Φ field (cf. Eq. (S28)) is quadratic due to Larkin-Dzyaloshinskii [S4] theorem, therefore an exact expression for the Φ-field correlator is We reduce the problem of finding an inverse of an infinite-dimensions matrix, inverting it to the finite (4) dimensions by Fourier-transforming it to a diagonal (k, ω) basis. Employing Eq. (S30) we obtain the θ-field correlator, where we implicitly sum over the Keldysh and the wire indices. This yields, We plug this result in Eq. (S26) to compute the Green function (Eq. (S19)) and the collision matrix of the particles in arms 2 and 3 (Eq. (S22)). The calculation requires transformation of Eqs. (S33) and (S34) to real (x,t) space. Here we present the final result, . (S35) Fourier-transforming the time coordinate yields, . For the sake of consistency check, lim g ⊥ →0,g →0 G =G. And the collision matrix reads, Fourier-transforming the time coordinates yields, where we have used the short notation r ij = r i − r j ; G is the single particle propagator given by Eq. (S36), and ζ 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 (cf. Fig. S2). The propagation of the single particle wave function can be derived by convolving the initial state with the retarded Green function, An expression for the zero temperature retarded Green function is (this is simply derived from Eq. (S36)).
FIG. S1. The collision matrixM (cf. Eq. (S37)). 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, cf. Eq. (S38)). 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 by α, β, γ, δ. Electrons enter the interacting region with energies ω2 + 1 2 ω and ω3 − 1 2 ω and exit with energies ω2 − 1 2 ω and ω3 + 1 2 ω respectively, exchanging energy ω via 4 possible different bosons. where Π(x) = 1 −1 < x < 1 0 o.w. is a rectangle function. The wave packet at 4 different points is shown in Fig. S2. 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 0 ) = eV + 2λx0 u . The center of mass of the wave packet then propagates with velocity v CM = u ξ(λ) . 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, ∆t(L) L/v CM . From this condition we deduce, eV u L and λ 1.

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 of O(N 2 ). The initial state of the system is a mixed state, which is represented by the density matrix ρ s = n,m R nm |α n α n |. The detector is initialized in the zeroth coherent state (we denote the α s coherent state by |α ) such that its density matrix is ρ 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 H I = w(t)gÂ(b † + b) with b, b † are the ladder operators of the detector,Â = n a n |α n α n | and w(t) is a window function around the time of the measurement. The post-selection is represented by the projection operator, Π f = n,m P nm |α n α n |. Plugging into Eq. (1) and considering, ρ tot = ρ s ⊗ ρ d and δq = b, yields Â GCV = n,m a n R nm P mn e − g 2 2 (an−am) 2 n,m R nm P mn e − g 2 2 (an−am) 2 . (S41) The numerator and the denominator consist of sums of Gaussian (in g) functions, with different coefficients and prefactors. Each Gaussian is a monotonic function (for g > 0), thus the maximal number of extremas in the weak-tostrong crossover (g ∈ [0, ∞)) is of the order of O(N 2 ), where N is the number of system's states.
A full list of diagrams A FULL LIST OF DIAGRAMS Fig. S3 depicts 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 (cf. Fig. S3(a)), diagrams which are dependent on either Φ S or Φ D (cf. Figs. S3(b) and S3(c)), and diagrams which are depend on both Φ S and Φ D , cf. Fig. S3(d). FIG. S3. The full list of irreducible diagrams to fourth (leading) order in tunneling which should be taken in account for the current-current correlator (cf. Eq. (6)).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. Eqs. (7) and (8)). The diagrams are divided to four groups by their Aharonov-Bohm flux dependence. The leading diagrams which were included in the calculation of the GCV, are in 3(d).