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 Hînt (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 )AÛˆ (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

Hînt = g(t )AˆPfˆ w, (3) where Pˆ is the probe momentum, g(t ) is a function nonvanishing 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 + Hînt 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 Uînt, 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. For conve- (X1,x1 ) nience, we perform the propagation from the initial state t2 i | = | | ψ φ × exp L(Q, Q˙, q, q˙, t )dt , (7) i dXidxi Xi xi (xi ) (Xi ) up to the interaction time ∗ h¯ t |= | 1 tw. The postselected state b f dxbf (x f ) x f is in- where stead propagated backward to tw. The uncoupled evolution b f (tw )|ψi(tw )φi(X, t f ) involves the direct paths from each L = Ls + Lp − g(t )A(q) f (q, Qw )MQ˙ (8) xi where ψ is nonvanishing to each x f lying in the support of is the classical interacting Lagrangian [22]; f (q, Q )setsthe w b f (x). Factorizing this term in the full evolution, we get [22] range of the interaction (it becomes a Dirac δ function in the φ , = |ψ limit of a pointlike interaction). Qw will be taken here as a (X t f ) b f (tw ) i(tw ) parameter specifying the position of the probe, which makes X i t f [ Lpdt −gAw MQ˙] sense for a probe with a negligible kinetic term. Note that A(q) × dX1 D[Q(t )]e h¯ tw φi(X1, tw ), gives the configuration space value of the classical dynamical X1 variable A(q). (10)

032048-2 WEAK VALUES FROM PATH INTEGRALS PHYSICAL REVIEW RESEARCH 2, 032048(R) (2020) where Aw is the weak value of Aˆ given by ∗ A(q) f (q, Qw )Ks(x f , t f ; q, tw )Ks(q, tw; xi, ti )b (x f )ψ(xi )dqdxf dxi Aw = f . , , ∗ ψ (11) dxf dxiKs(x f t f ; xi ti )b f (x f ) (xi )

w Several remarks are in order. First note that for large M |x f , we have A = A(Qw )Ks(x f , Qw )ψ(Qw, tw )/ψ(x f , t f ), Eq. (10) implies a shift of the initial probe state, since the so if the wave functions are known, e.g., through a pre- Lagrangian maps each point X1 to X1 − gAw. Second, while vious weak-measurement-based procedure [23], the propa- the denominator in Eq. (11) involves all the paths connecting gator Ks(x f , Qw ) can be obtained from the measurement w the initial region to the final region, the numerator contains of the weak value A .Bothx f and Qw can be varied the sole paths connecting the initial and final regions passing in order to measure the propagator over the region of through a point q of the interaction region. Each such path interest. is weighed by the classical value of the configuration space Our weak value expression (11) is useful in cases in- function A(q) at that point. Note also that the weak value volving free propagation or in the semiclassical approxima- expressions (2) and (11) are both the results of asymptotic tion: For large actions, stationary phase integration trans- expansions, but the expansions are not exactly equivalent. forms the sum over all arbitrary paths to a sum containing For a contact interaction f (q, Qw ) = δ(q − Qw ), it is easy only the classical paths linking the initial and final points, to see that Eqs. (2) and (11) become identical under the so that both propagators (5) and (6) take the form Ksc = = | ˆ| = w A S / − πμ / A S condition A(q) q A q for q Qw. In this case, A is k k exp i[ k h¯ k 2], where k and k are the ampli- simply given by A(Qw ) × Tw/T , where Tw is the transition tude and classical action for the classical path k connecting the amplitude involving the paths connecting xi to x f by going initial and endpoints in time t f − ti [20,24]; the phase index μk through Qw at tw, while T connects xi to x f through any counts the number of conjugate points along each trajectory intermediate point) [see Fig. 1 and Eq. (12) below]. We and will be absorbed into the action to simplify the notation. then see the following: (i) The ratio Tw/T can take any The semiclassical regime, exact for free propagation, remains w complex value; hence if A(Qw ) is bounded, Re(A ) can lie quantum since the different classical paths still interfere, and beyond these bounds and Aw is said to be “anomalous.”1 Aw can be anomalous. Assuming again a pointlike interaction (ii) If the postselection state is chosen to be the space point at q = Qw, Eq. (11) becomes

∗ S / S / dx dx χ (x )ψ(x )[ A (x ; Q )ei W (x f ;Qw ) h¯ A (Q ; x )ei W (Qw ;xi ) h¯ ] w = f i f f i W W f w W w i . A A(Qw ) ∗ S / (12) χ ψ , A i J (x f ;xi ) h¯ dxf dxi f (x f ) (xi ti ) J J (x f ; xi )e

AW and SW label the amplitude and action of a path going (presumably observed) weak value and of a single Feynman through Qw, W runs over all the classical paths connecting xi path, the po: to Qw and Qw to x f , while J runs over all the classical paths − . w connecting directly xi to x f in time t f ti G(0)|G(t f )=A [Apo(x0, xp)A(xp)Apo(xp, x0 ) The form of the weak value given by Eq. (12) is par- / − × iSpo h¯ | , |2 1. ticularly well suited to the experimental investigation of the e G(x0 0) ] (13) contribution of individual orbits in semiclassical systems, such as the detection of quantum scars in wave functions Instead, the application of the standard weak value defini- that have recently received renewed interest [25]. The stan- tion (2) to infer the autocorrelation function would require the dard approach [26] to periodic orbit localization involves knowledge of the full Schrödinger propagator. the autocorrelation function G(0)|G(t ) where G(x, 0) is Our approach is also relevant to understand current con- a Gaussian with a maximum x0 placed on a periodic or- troversies involving WV such as the apparent observation bit (po) that quickly spreads over all the available phase of discontinuous trajectories [17–19] or the quantumness of space. If the system is prepared and postselected in state anomalous weak values [4–6], as we briefly detail. The first |G(0), the weak value of any observable Aˆ measured on of these issues involves a three-path interferometer depicted w a point xp of the periodic orbit is given by A =G(0)| in Fig. 2. The weakly measured observable is the projector ˆ , ˆ ˆ , | / | j ≡|x jx j|, indicating whether the particle is at point x j. Us(t f tw ) xp A xpUs(tw ti ) G(0) G(0) G(t f ) , where xp ≡| | By suitably choosing the preselected and postselected states, xxp xxp is the projector on xp.ByusingEq.(12), the autocorrelation function is obtained from the knowledge of the the following weak values are obtained [17]:

A = 0, E = 0, B = 0, C = 0, F = 0. (14) 1The standard deﬁnition of anomalous—Re(Aw ) can lie above the largest or below the lowest eigenvalue—is more restrictive since by This means that weakly coupling a probe to the system leaves canonical quantization the eigenvalues range is typically contained a trace on paths A, B, and C, but none on the segments E and within the possible values of A(q) in conﬁguration space. F: The particle is seen inside the loop, but not on the entrance

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remarked in Ref. [19]. The genuinely interesting case is case (iii): The paths do not interfere destructively in the interaction region, but the fraction of the preselected wave function propagated by the paths passing through this region (here E or F) is incompatible with the postselected state. In other interaction regions (here B or C), the paths contributing to the propagator in Eq. (12) are not orthogonal to the postselected state anymore. The crucial point is that the paths contained in the propagator are continuous, but each probe’s motion results from the overlap at t = t f between the preselected state propagated by the paths hitting the probe region and the postselected state. The other current controversy mentioned above concerns the quantumness of anomalous weak values that has been recently questioned as resulting from a mere statistical effect [4–7]. A path integral approach is convenient to obtain heuristically the classical limit of the weakly coupled probe’s motion. This involves two well-known steps. The first step, deriving the semiclassical propagator Ksc, was recalled above. The second one involves retaining only the diagonal terms of the semiclassically propagated density matrix [27], due to FIG. 2. A quantum particle propagating in an interferometer and postselected at D displays discontinuous trajectories when observed coarse graining over a classical scale [28]. We then obtain [22] with weak measurements [the weakly coupled probes depicted in classical amplitudes obeying the classical transport equation, black remain unaffected by the interaction with the particle, while the and the weak probe’s dynamics is given by solving for each probes depicted in orange see their pointers shift after postselection; point of the initial distribution the equations of motion for see Eq. (14)]. The Feynman paths of the particle, represented as the Lagrangian given by Eq. (8). Postselection is specified in pencils of trajectories along the arms, are continuous and propagate terms of a domain B f at time t f defined such that the cho- the interactions with the probes up to D, where superposition with sen configuration space function b(q, t f ) ∈ B f . The average the postselected state yield zero WV at E and F even when the wave probe shift with postselection is then function on these segments does not vanish (see text for details). ρ , = , , s(q tw ) . Q g A(q tw ) f (q Qw ) dq ρs(q , tw )dq and exit paths. A point that has caused some confusion in Bw Bw the literature [18,19] is that in the initial proposal [17]the (15) wave function vanished (by destructive interference) on paths E and F, so that an interpretation in terms of vanishing weak values appeared to be flawed. It is possible, however, The integral is taken at tw over the configuration space domain to enforce Eq. (14) without having destructive interference Bw such that at t f the condition b(q, t f ) ∈ B f is obeyed. on E and F. The present path-integral approach naturally Q can never be anomalous. Classically, the only way to parses the aspects relevant to a vanishing classical value, have an anomalous probe shift would be to replace A(q, tw ) to a vanishing superposition, or to a partially propagated by a different configuration space function or to have the state being incompatible with postselection. Indeed, the WV numerator and denominator integrated over different domains, expression (12) vanishes as follows: say, B f and B f . In both cases, this would be the result (i) If A(Qw ) = 0, which for a projector implies that the of a perturbation, due to the detection process in the latter particle is not there; case. (ii) if the term between brackets vanishes, corresponding To sum up, we have obtained an expression of weak to destructive interference of the wave function at Qw by values from a path integral approach. We have shown this summing over the different continuous paths W ; expression to be useful when semiclassical propagators are (iii) if the integral vanishes, that is the ensemble of points involved. We have seen that the present approach gives a of the preselected state propagated by the sole paths going consistent account of two current controversies involving through Qw up to x f is orthogonal to the postselected state. weak values (the observation of discontinuous trajectories The case for asserting that the particle is not at Qw = E or and the quantumness of anomalous weak values). Other F when the WV vanishes is unproblematic in case (i): This recent perplexing results [29] involving pre- and postse- is independent of postselection and would also be the case lected ensembles can be treated similarly. We have also classically. Case (ii) corresponds to the initial proposal [17] suggested a method for measuring the Feynman propagator in which two paths with opposite phases interfere destruc- through weak values. The approach introduced in this work tively. This is a nonclassical effect [since we have A(Qw ) = will be fruitful to tackle extensions of the weak measure- 0] but it does not necessarily depend on postselection and ments framework to relativistic and cosmological settings, an interpretation in terms of weak values appears moot: If where quantities analogous to weak values are known to there is no wave function, there is nothing to measure, as emerge [30].

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