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Weak Values from Path Integrals PHYSICAL REVIEW RESEARCH 2, 032048(R) (2020) Rapid Communications Weak values from path integrals A. Matzkin Laboratoire de Physique Théorique et Modélisation, CNRS Unité 8089, CY Cergy Paris Université, 95302 Cergy-Pontoise cedex, France (Received 6 February 2020; accepted 7 August 2020; published 24 August 2020) We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in quantum-chaos-related studies (an illustration involving quantum scars is given), and also in solving current weak-value-related controversies (we discuss the existence of discontinuous trajectories in interferometers and the issue of anomalous weak values in the classical limit). DOI: 10.1103/PhysRevResearch.2.032048 There has been a growing interest in weak measurements— the path integral expression accounts for the discontinuous a specific form of quantum nondemolition measurements— trajectories observed in interferometers and currently wildly over the past decade. Weak measurements were indeed found debated [17–19]. We will also see that in the classical limit to be useful in fundamental or technical investigations, involv- the weak value is washed out by coarse graining, implying ing both experimental and theoretical works. Nevertheless, that anomalous pointer values are a specific quantum feature ever since their inception [1], weak values—the outcomes with no classical equivalent [4–6]. A nice feature arising from of weak measurements—have remained controversial. Since the present approach is the possibility to measure propagators weak values are entirely derived from within the standard through weak measurements. This could be a useful tool in quantum formalism, the controversies have never concerned quantum-chaos-related studies as a method to observe the the validity of weak values (WV), but their understanding semiclassical amplitudes; we give an illustration in which and their properties. For instance, are WV values similar to the autocorrelation function employed to investigate quantum eigenvalues, or are they akin to expectation values [2,3]? Do scars is inferred from weak values. The present approach anomalous WV represent a specific signature of quantum could also be the starting point to apply weak values in phenomena or can they be reproduced by classical conditional domains, such as quantum cosmology, for which the standard, probabilities [4–7]? Are WV related to the measured system nonrelativistic weak measurements framework, relying on a properties or do they represent arbitrary numbers characteriz- von Neumann interaction, cannot be applied. ing the perturbation of the weakly coupled pointer [8,9]? A weak measurement of a system observable Aˆ is charac- The path integral formulation of quantum mechanics is terized by four steps: (1) The system of interest is prepared in strictly equivalent to the standard formalism based on the the chosen initial state |ψi, a step known as preselection. A Schrödinger equation. Though it is often technically more quantum probe is prepared in a state |φi, and the initial state involved, path integrals give a conceptually clearer picture is thus though the natural, built-in connection with quantities de- | =|ψ |φ . fined from classical mechanics (Lagrangian, action, paths). (ti ) i i (1) Surprisingly, very few works have employed weak values in (2) The system and the probe are weakly coupled through a path integral context. Even then, the interest was restricted an interaction Hamiltonian Hˆint (a so-called von Neumann to the weak measurement of Feynman paths in semiclassical interaction). (3) After the interaction, the system evolves until systems [10–12], to WV of specific operators [13–15], or as a another system observable, say Bˆ, is measured through a stan- way to probe virtual histories [16]. dard measurement process. Of all the possible eigenstates |bk In this work, we connect the weak measurements frame- that can be obtained, a filter selects only the cases for which work to the path integral formulation. We will see that path |bk=|b f , where |b f is known as the postselected state. (4) integrals parse the quantum and classical aspects of weak val- When postselection is successful, the probe is measured. The ues, thereby clarifying many of the current controversies in- final state of the probe |φ f has changed relative to |φi by a volving weak measurements. In particular, we will show how shift depending on the weak value of Aˆ. Let Uˆs(t2, t1) denote the system evolution operator, and let ti, tw and t f represent the preparation, interaction, and postselection times respectively. The weak value [1]ofAˆ is Published by the American Physical Society under the terms of the then given by Creative Commons Attribution 4.0 International license. Further b (t )|Uˆ (t , t )AˆUˆ (t , t )|ψ(t ) distribution of this work must maintain attribution to the author(s) Aw = f f s f w s w i i . (2) and the published article’s title, journal citation, and DOI. b f (t f )|Uˆs(t f , ti )|ψ(ti ) 2643-1564/2020/2(3)/032048(5) 032048-1 Published by the American Physical Society A. MATZKIN PHYSICAL REVIEW RESEARCH 2, 032048(R) (2020) Typically, the probe state |φi is a Gaussian pointer initially centered at the position Qw and Hˆint is of the form Hˆint = g(t )AˆPfˆ w, (3) where Pˆ is the probe momentum, g(t ) is a function non- vanishing only in a small interval centered on tw, and fw reflects the short-range character of the interaction that is only nonvanishing in a small region near Qw. Under these −igAw Pˆ conditions, it is well known [1] that |φ f =e |φi, with FIG. 1. Schematic illustration of the paths involved in the weak g = dt g(t ) and the final probe state is the initial Gaussian value expression when the pre- and postselected states are well shifted by gRe(Aw ). localized, and the propagator is given as a sum over the classical In order to determine the evolution of the coupled system- trajectories [see Eq. (12)]. For the specific choice of postselection = δ − probe problem from the initial state |i(ti )=|ψi|φi,we b(x f ) (x x f ), the weak value resulting from the coupling at = , need the full Hamiltonian Hˆ = Hˆs + Hˆp + Hˆint where Hˆs q Qw becomes proportional to the propagator Ks(x f Qw ). The propagator can hence be observed from weak value measurements and Hˆp are the Hamiltonians for the uncoupled system and probe respectively. In terms of the corresponding evolution varying Qw and x f . operators, the system and the probe evolve first indepen- Nonseparable propagators are notoriously difficult to han- dently, Uˆ0(t, ti ) = Uˆs(t, ti )Uˆ p(t, ti ). Then, assuming for sim- plicity that the interaction takes place during the time interval dle except when they can be treated perturbatively [21], which is the case here. Standard path integral perturbation techniques [tw − τ/2, tw + τ/2] (τ is the duration of the interaction), the (see Chap. 6 of Ref. [21]) applied to the system degrees of total evolution operator from ti to the postselection time t f is given as freedom give Kint in terms of the uncoupled propagators and a first-order correction in which the system propagates as if ˆ , = ˆ , + τ/ ˆ + τ/ , − τ/ U (t f ti ) U0(t f tw 2)Uint (tw 2 tw 2) it were uncoupled except that each path q(t ) is weighed by the perturbative term g(t )A(q) f (q, Qw )MQ˙. If the duration ×Uˆ0(tw − τ/2, ti ). (4) τ of the interaction is small relative to the other timescales The propagators K ≡x2|Uˆ (t2, t1)|x1 for the uncoupled evo- (as is usually assumed in weak measurements), the time- lution of the system and probe are given respectively by dependent coupling can be integrated to an effective coupling tw +τ/2 x2 t2 constant g = g(t )dt and the uncoupled paths see an i tw−τ/2 Ks(x2, t2; x1, t1) = D[q(t )] exp Ls(q, q˙, t )dt , effective perturbation gA(q) f (q, Qw )MQ˙ at time tw. Equa- x1 h¯ t1 (5) tion (7) becomes (the derivation is given in the Supplemental X2 Material [22]) K (X , t ; X , t ) = D[Q(t )] p 2 2 1 1 X2 i t2 X1 h t Lpdt Kint = KpKs + D[Q(t )]e ¯ 1 dqKs(x2, t2; q, tw ) t2 i X1 × exp L (Q, Q˙, t )dt , (6) p ig h¯ t1 × A(q) f (q, Qw )Ks(q, tw; x1, t1) − MQ˙ . (9) where as usual [20,21] D[.] implies integration over all paths h¯ connecting the initial and final space-time points, and Ls = From the point of view of the system, the interpretation mq˙2 − = MQ˙ 2 2 V (q) and Lp 2 are the classical system and probe of Eq. (9) is straightforward (see Fig. 1): The transition Lagrangians respectively. amplitude from xi to x f involves a sum over the paths directly For the coupled evolution Uˆint, the propagator becomes joining these two points in time t f − ti as well as those that nonseparable, interact with the probe in the region determined by f (q, Qw ), hence going from xi to some intermediate point q within this Kint (X2, x2, t2; X1, x1, t1) region, and then from this point q to x . , f (X2 x2 ) We now take into account the pre- and postselected = D[Q(t )]D[q(t )], states and focus on the probe evolution.
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