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Abstract
We propose a scheme allowing us to observe the evolution of a quantum system in the semiclassical regime along the paths generated by the propagator. The scheme relies on performing consecutive weak measurements of the position. We show how "weak trajectories" can be extracted from the pointers of a series of devices having weakly interacted with the system. The properties of these weak trajectories are investigated and illustrated in the case of a time-dependent model system.
... Formulation of the seminal idea of weak measurement (abbreviated, WM) in quantum mechanics by Aharanov, Albert and Vaidmann (AAV) [1] and its subsequent clarifications as well as elaborations [2][3][4][5][6][7][8][9][10][11] over the years have given rise to a plethora of investigations, theoretical as well as experimental [for a recent comprehensive review see, for example, J.Dressel et al. [12]. This ranges from the use of WM in the analyses of intriguing quantum effects such as Hardy's paradox [13], the three box paradox [14], the quantum Cheshire Cat [15], to the application of WM in the context of quantum entanglement [16], for verifying the "error-disturbance uncertainty relations" [17], for observing the evolution of a quantum system in the semiclassical regime [18], for avoiding loopholes in showing quantum violation of hybrid Bell-Leggett-Garg inequalities [19], for experimentally verifying Bell's inequality in time [20], for shedding light on quantum discord [21], for demonstrating quantum contextuality [22] and for studying tunneling [23]arrival time [24], as well as for revealing interesting effects in the physics of telecommunication fibers [25]. Further, WM has been studied using neutron interferometry [26] and has been * somkanji@jcbose.ac.in † girish.muralidhara@students.iiserpune.ac.in ‡ dhome@jcbose.ac.in invoked for high precision measurements concerning quantum metrology [27] such as for identifying a tiny spin Hall effect [28], for detecting very small transverse beam deflections [29,30], and tiny temporal delay [31]. ...
... Here it is relevant to emphasize that, given any covariance matrix, it is possible to generate a multivariate distribution embodying the correlations given by the covariance matrix (Cholesky decomposition). Now, note that while the effects of multivariate pointer state distribution without correlations have already been discussed in the context of weak measurement [18,[35][36][37][38], curiously, the possible effects of multidimensionality embodying correlations among different pointer degrees of freedom have remained largely unexplored. This holds apart from a couple of works probing the effects of multidimensionality of pointer states in the special case of two dimensional Hermite-Gaussian and Laguerre-Gaussian optical modes as pointer states [39]. ...
... using Eqs. (17) and (18), and where the correlation term corr(q 1 , q 2 ) i is given by Eq. (5) for l, m = 1, 2. ...
In the weak measurement (WM) scenario involving weak interaction and postselection by projective measurement, the empirical significance of weak values is manifested in terms of shifts in the measurement pointer's mean position and mean momentum. In this context, a general quantitative treatment is presented in this paper by taking into account the hitherto unexplored effect of correlations among the pointer degrees of freedom which pertain to an arbitrary multidimensional preselected pointer state. This leads to an extension of the earlier results, showing that, for complex weak values, the correlations among different pointer degrees of freedom can crucially affect the way the imaginary parts of the weak values are related to the observed shifts of the mean pointer position and momentum. The particular relevance of this analysis is discussed in the case of sequential weak interactions, followed by a postselection (called sequential WM) which, in the special case, reduces to the usual WM scheme involving a single weak interaction and postseletion, modified by the effect of pointer state correlations.
... However there have been recent proposals to ascertain the paths taken by a quantum particle. In particular Vaidman examined the path of a photon in nested interferometers [3], while one of us investigated the dynamical paths compatible with a given final state when a quantum system evolution is generated by a semiclassical Feynman propagator [4]. ...
... The state of the weak pointer has picked up a shift (relative to its initial state) proportional to a quantity known as the weak value of A. When a weak value vanishes, the state of the quantum pointer remains unchanged, and Refs [3,4] interpreted this fact by asserting that the system property coupled to the pointer was not there (otherwise the pointer state would have changed). ...
... Unsurprisingly, any proposal to infer the past of a quantum system from the weak values is going to face criticism disputing the relevance of weak measurements concerning the properties that can be ascribed to a system during its evolution. In particular, Vaidman [3,4] noted that the weak values of the spatial projector were non-zero inside a Mach-Zehnder interferometer (MZI) inserted on one of the arms of another larger Mach-Zehnder, but the weak values along that arm did vanish before and beyond the nested MZI. The same feature was also remarked [11] in a 3 path interferometer: when 2 of the 3 branches are joined, the spatial projector weak value (that did not vanish on either of these 2 arms) vanishes once these 2 branches merge. ...
Non-destructive weak measurements (WM) made on a quantum particle allow to extract information as the particle evolves from a prepared state to a finally detected state. The physical meaning of this information has been open to debate, particularly in view of the apparent discontinuous trajectories of the particle recorded by WM. In this work we investigate the properties of vanishing weak values for projection operators as well as general observables. We then analyze the implications when inferring the past of a quantum particle. We provide a novel (non-optical) example for which apparent discontinuous trajectories are obtained by WM. Our approach is compared to previous results.
... From Eq. (31) (taken for q → ∞), we know that ψ(x, T /2; q) is a freely evolved Gaussian. We can thus repeat the same steps leading to (22), but starting from the time evolved Gaussian ...
... where τ (t) ≡ t 0 L −2 (t ′ )dt ′ . Contrary to the standard approaches for solving Gaussian problems in TDLOs, that involve nonlinear equations calling for numerical integration [22,23] ...
... still hold for the confined TDLO with moving walls, we can again write the time-evolved solution ψ(x, T ), here after a period T in terms of a Theta function. Formally ψ(x, t) is again given by Eq.(22), the only difference relative to the infinite potential well of Sec. III being that L(t) is a periodic function and not linear in t. ...
We investigate the issue of single particle nonlocality in a quantum system subjected to time-dependent boundary conditions. We discuss earlier claims according to which the quantum state of a particle remaining localized at the center of an infinite well with moving walls would be specifically modified by the change in boundary conditions due to the wall's motion. We first prove that the evolution of an initially localized Gaussian state is not affected nonlocally by a linearly moving wall: as long as the quantum state has negligible amplitude near the wall, the boundary motion has no effect. This result is further extended to related confined time-dependent oscillators in which the boundary's motion is known to give rise to geometric phases: for a Gaussian state remaining localized far from the boundaries, the effect of the geometric phases is washed out and the particle dynamics shows no traces of a nonlocal influence that would be induced by the moving boundaries.
... Our study encompasses two distinct scenarios: the first examines photons as measurement devices via their projection operators, and the second explores mirrors as probes. To that end, we extend Matzkin's path integral [20,21] approach for weak values to a sequence of weak measurements and study the probe shifts across the different branches of the interferometer. We also further consider the discontinuous photon trajectories as reported by Danan et al [22], a result which has sparked curious discussions. ...
... (15) The quantum state of the wavefunction after the interaction with the last mirror in this sequence is given by (see [21] and the supplementary material for derivation), ...
... Fig (S1) outlines the approach when considering mirrors, building on the approach in [21]. The initial state is propagated forward to mirror M E , where the photon, treated as a Gaussian centered at x 0 , receives a small shift. ...
We consider weak values in the Feynman propagator framework, to gain a broader understanding of their interpretation in terms of path integrals. In particular, we examine the phenomenon of seemingly discontinuous paths that particles take in nested Mach-Zender interferometer experiments. We extend on existing path integral approaches for weak values by deriving expressions to model a sequence of weak measurements, and study the probe shifts across the different branches of a weak value interferometer. We apply this to scrutinise two scenarios of interest, one which treats photons as measurement apparatus via their spatial projection operators, and the second treating mirrors as probes.
... WhenÔ is given by the position operatorX, or by the projection to a particular spatial position Π x ≡ |x x| , the weak value captures information about the position of the system as it evolves between the pre-and post-selected states. By inserting in different spatial regions a series of weakly coupled pointers that can be turned on and off at the desired time, it is possible to define "weak trajectories" [3]. Such trajectories are in principle observable signatures of the space-time evolution of the system. ...
... Such trajectories are in principle observable signatures of the space-time evolution of the system. In simple cases, e. g. with narrow coherent states [3,4] or with point-like pre and post-selected states [5] and free propagation, the system is seen to follow a classical trajectory, or a superposition of such trajectories. This is reminiscent of Feynman's path integral approach [6], in which the evolution operator can be written (for free propagation, or when the potential is linear or quadratic in x) ...
... However, we may pick p 2 f along a different direction, as portrayed in Fig. 4. In this case, we know from the propagator [Eq. (3)] that χ * ...
The interference pattern produced by a quantum particle in Young's double-slit setup is attributed to the particle's wavefunction having gone through both slits. In the path integral formulation, this interference involves a superposition of paths, going through either slit, linking the source to the detection point. We show how these paths superpositions can in principle be observed by implementing a series of minimally-perturbing weak measurements between the slits and the detection plane. We further propose a simplified protocol in order to observe these "weak trajectories" with single photons.
... En particulier, une trajectoire unique relie deux points par la densité de courant, alors que les trajectoires de l'intégrale de Feynman sont en principe non-dénombrables et interfèrent. [90] qui montre que dans le régime semi-classique, on peut en principe observer la superposition des trajectoires de Feynman données par l'Eq (159) (c'est à dire que seuls les pointeurs situés sur ces trajectoires s'allument) ; ceci est le cas pourvu que la fonction d'onde sur chacune de ces trajectoires reste suffisamment bien localisée (c'est le cas par exemple pour une superposition d'états cohérents évoluant sur des trajectoires classiques). Le choix de la post-sélection est néanmoins important, puisque des pointeurs isolés non-situés sur ces trajectoires peuvent s'allumer [90]. ...
... [90] qui montre que dans le régime semi-classique, on peut en principe observer la superposition des trajectoires de Feynman données par l'Eq (159) (c'est à dire que seuls les pointeurs situés sur ces trajectoires s'allument) ; ceci est le cas pourvu que la fonction d'onde sur chacune de ces trajectoires reste suffisamment bien localisée (c'est le cas par exemple pour une superposition d'états cohérents évoluant sur des trajectoires classiques). Le choix de la post-sélection est néanmoins important, puisque des pointeurs isolés non-situés sur ces trajectoires peuvent s'allumer [90]. On peut par ailleurs choisir [91] une post-sélection bi-locale (en faisant interférer des détecteurs situés en deux points différents de l'espace), ce qui change radicalement les trajectoires faibles observées. ...
... L'existence de trajectoires discontinues lorsqu'on définit une trajectoire à partir d'une succession de mesures faibles de la position est en fait un résultat attendu, indépendant du schéma spécifique utilisé par Vaidman (voir [117], [90]). De plus, le schéma de Vaidman souffre d'un défaut important mentionné plus haut, à savoir le fait que la fonction d'onde du photon interfère destructivement sur la voie de sortie. ...
La théorie de la mesure, basée sur la mesure projective, constitue un aspect fondamental de la mécanique quantique. La mesure faible diffère de la mesure projective traditionnelle sur laquelle les axiomes élémentaires de la physique quantique sont bâtis. Bien que définies dans le cadre de la théorie quantique standard, les mesures faibles sont encore mal comprises. Le travail de thèse s'inscrit dans une large démarche qui vise à comprendre les implications conceptuelles et pratiques d'une telle mesure et à la comprendre dans le cadre de la théorie quantique.Le chapitre 1 est une introduction détaillée à la mesure faible et à la valeur faible. Ensuite, nous étudierons les "trajectoires faibles" dans un interféromètre à fentes d'Young. Nous abordons au chapitre 3 les implications de l'annulation d'une valeur faible au regard de l'apparition de trajectoires faibles dans des interféromètres de Mach-Zender imbriqués. Enfin, le chapitre 4 traite des critiques théoriques et expérimentales présentes dans la littérature concernant l'effet du Chat du Cheshire quantique qui est défini dans le cadre de mesures faibles.
... From equation (31) (taken for q → ∞), we know that ψ(x, T/2; q) is a freely evolved Gaussian. We can thus repeat the same steps leading to (22), but starting from the time evolved Gaussian ...
... Actually, since equations (14) and (17) still hold for the confined TDLO with moving walls, we can again write the time-evolved solution ψ(x, T), here after a period T in terms of a Theta function. Formally ψ(x, t) is again given by equation (22), the only difference relative to the infinite potential well of section 3 being that L(t) is a periodic function and not linear in t. To assess the relevance of geometric phases, we rescale the walls motion while leaving the Hamiltonian invariant as explained above by putting L (t) = kL(t). ...
... Writing ψ (x, T)/ψ(x, T) in terms of ϑ 4 functions as per equation (22), and noting that z = z/k, κ = κ/k 2 and L(T) = L 0 , we apply the Jacobi transformation (25) to find given by equation (18). From equation (23) we see that z = z/k and κ = κ/k 2 so that by using equation (22) and the Jacobi transformation (25) we are led tō ...
We investigate the issue of single particle nonlocality in a quantum system subjected to time-dependent boundary conditions. We prove that contrary to earlier claims, the quantum state of a particle localized at the center of an infinite well with moving walls is not modified by the wall's motion. This result is further extended to related confined systems in which the boundary's motion is known to give rise to geometric phases: the effect of the geometric phases is washed out and the particle dynamics shows no traces of a nonlocal influence that would be induced by the moving boundaries.
... However there have been recent proposals to ascertain the paths taken by a quantum particle. In particular Vaidman examined the path of a photon in nested interferometers [3], while one of us investigated the dynamical paths compatible with a given final state when a quantum system evolution is generated by a semiclassical Feynman propagator [4]. ...
... The state of the weak pointer has picked up a shift (relative to its initial state) proportional to a quantity known as the weak value of A. When a weak value vanishes, the state of the quantum pointer remains unchanged, and Refs [3,4] interpreted this fact by asserting that the system property coupled to the pointer was not there (otherwise the pointer state would have changed). ...
... Unsurprisingly, any proposal to infer the past of a quantum system from the weak values is going to face criticism disputing the relevance of weak measurements concerning the properties that can be ascribed to a system during its evolution. In particular, Vaidman [3,4] noted that the weak values of the spatial projector were non-zero inside a Mach-Zehnder interferometer (MZI) inserted on one of the arms of another larger Mach-Zehnder, but the weak values along that arm did vanish before and beyond the nested MZI. The same feature was also remarked [11] in a 3 path interferometer: when 2 of the 3 branches are joined, the spatial projector weak value (that did not vanish on either of these 2 arms) vanishes once these 2 branches merge. ...
Non-destructive weak measurements (WM) made on a quantum particle allow to extract information as the particle evolves from a prepared state to a finally detected state. The physical meaning of this information has been open to debate, particularly in view of the apparent discontinuous trajectories of the particle recorded by WM. In this work we investigate the properties of vanishing weak values for projection operators as well as general observables. We then analyze the implications when inferring the past of a quantum particle. We provide a novel (non-optical) example for which apparent discontinuous trajectories are obtained by WM.\ Our approach is compared to previous results.
... This is an enlightening result illuminating the physical meaning of the weak value, but in view of the indirect nature of the systematic treatment of the data, one may wonder if there are some other ways to define the trajectory based on the weak value. Indeed, an alternative and more direct trajectory can be delineated from the time-dependent weak value [7] with an iterative procedure of measurements [8]. This can be regarded as the dynamical version of the weak value that emerges naturally in the context of the time symmetric formulation of quantum mechanics [9]. ...
... When the given process consists of the pre-and post-selections given either by position eigenstates or by well-localized (Gaussian) states, the weak trajectory forms a curve with the two ends specified by the selections. The interest in this case is then to see how the trajectory deviates from the classical one, which has been analyzed earlier in [7,8,10]. In contrast, in our study we are interested in situations where the selections are performed not by those (semi)classical states but by genuinely quantum states, that is, by superposed states for which no par-ticular position is assignable to the particle at one of the ends at least. ...
... The point is that the (renormalized) weak trajectory yields a definite function of time such that it can be regarded as some trajectory, and the question whether it coincides with the classical one or not is secondary. However, once the trajectory is established, then it should be interesting to investigate the characteristic feature of the weak trajectory compared to the classical one, as has been done in [7,8] for the case of selections where semiclassical approximation is valid. ...
The notion of the trajectory of an individual particle is strictly inhibited in quantum mechanics because of the uncertainty
principle. Nonetheless, the weak value, which has been proposed as a novel and measurable quantity definable to any quantum
observable, can offer a possible description of trajectory on account of its statistical nature. In this paper, we explore
the physical significance provided by this “weak trajectory” by considering various situations where interference takes place
simultaneously with the observation of particles, that is, in prototypical quantum situations for which no classical treatment
is available. These include the double slit experiment and Lloyd's mirror, where in the former case it is argued that the
real part of the weak trajectory describes an average over the possible classical trajectories involved in the process, and
that the imaginary part is related to the variation of interference. It is shown that this average interpretation of the weak
trajectory holds universally under the complex probability defined from the given transition process. These features remain
essentially unaltered in the case of Lloyd's mirror where interference occurs with a single slit.
... This scheme was experimentally implemented in a two-slit interferometer [16] allowing to reconstruct Bohmian trajectories from the observed data. A method that can in principle allow to observe the Feynman paths with weak measurements of the position (including the coherent paths superposition) was also suggested recently [17] (see also [18]). In a first view it is therefore tempting to conclude that the type of trajectory that one sees depends eventually on what is being measured, which in turn calls for a definite experimental setup. ...
... Note that t ( ) α determines the time-dependent width of the evolving wavefunction; the initial state (17) ...
... See the supplemental material of[17].4 Since the spatial WMA wavefunction picks up an R dependent phase term proportional to the weak value, the WMA pointer is monitored in momentum space. ...
Weak values, obtained from weak measurements, attempt to describe the
properties of a quantum system as it evolves from an initial to a final state,
without practically altering this evolution. Trajectories can be defined from
weak measurements of the position, or inferred from weak values of the momentum
operator. The former can be connected to the asymptotic form of the Feynman
propagator and the latter to Bohmian trajectories. Employing a time-dependent
oscillator as a model, this work analyzes to what extent weak measurements can
shed light on the underlying dynamics of a quantum system expressed in terms of
trajectories, in particular by comparing the two approaches.
... This interesting relationship resonates with past works which studied nondemolition and continuous quantum measurements [1-4], while connecting them with path integration [5]. Recently, the above relationship was further analyzed and strengthened by different researchers [6][7][8][9][10][11], but here we focus on the notion of sequential weak values as a pivotal issue, which has not been mentioned before in the above literature. In particular, we show that sequential weak values are able to probe directly the quantum probability amplitudes along individual virtual Feynman histories thereby possibly supporting their physical meaningfulness. ...
... This interesting relationship resonates with past works which studied nondemolition and continuous quantum measurements [1][2][3][4], while connecting them with path integration [5]. Recently, the above relationship was further analyzed and strengthened by different researchers [6][7][8][9][10][11], but here we focus on the notion of sequential weak values as a pivotal issue, which has not been mentioned before in the above literature. In particular, we show that sequential weak values are able to probe directly the quantum probability amplitudes along individual virtual Feynman histories thereby possibly supporting their physical meaningfulness. ...
Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multi-time projection operators for a complete set of orthogonal quantum histories is unity, complete sets of weak values could be interpreted in agreement with the standard quantum mechanical picture. We also elucidate the relationship between sequential weak values of quantum histories with different coarse-graining in time and establish the incompatibility of weak values for non-orthogonal quantum histories in history Hilbert space. Bridging theory and experiment, the presented results may enhance our understanding of both weak values and quantum histories.
... While the meaning of weak values has been debated since their inception [2][3][4][5], several experimental implementations of the weak measurement protocol have been carried out: weak values have thus been measured for different observables in many quantum systems [6][7][8][9][10][11][12][13]. Concurrently, theoretical schemes based on weak measurements have been proposed with practical [14][15][16] or foundational [17][18][19][20][21] aims. Among the latter, Aharonov et al [22] introduced an interferometric based scheme baptized the Quantum Cheshire Cat (QCC). ...
... In Eq. (3) the interaction appears to take place precisely at t w ; this "midpoint rule" holds provided τ is small relative to the system evolution timescale (see Appdx A in the Supp. Mat. of Ref.[17]). ...
The Quantum Cheshire Cat (QCC) is an effect introduced recently within the Weak Measurements framework. The main feature of the QCC effect is that a property of a quantum particle appears to be spatially separated from its position. The status of this effect has however remained unclear, as claims of experimental observation of the QCC have been disputed by strong criticism of the experimental as well as the theoretical aspects of the effect. In this paper we clarify in what precise sense the QCC can be regarded as an unambiguous consequence of the standard quantum mechanical formalism applied to describe quantum pointers weakly coupled to a system. In light of this clarification, the raised criticisms of the QCC effect are rebutted. We further point out that the limitations of the experiments performed to date imply that a loophole-free experimental demonstration of the QCC has not yet been achieved.
... The wave function or the state function in the de Broglie-Bohm theory [17] also does not incorporate the particle property (the mass). As for the Feynman non-classical paths, regardless of whether there is a protocol-toobserve [18] or not, it has never been observed or confirmed. The existence of this protocol does not imply the observation has been made and/or confirmed. ...
Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics (due to Newtonian mechanics, special relativity, gravity and entropy). Here, we first prove the limitation of the Heisenberg uncertainty principle by invoking the wave-particle duality and apply it to revise and/or extend the Copenhagen postulates. In particular, even though the said uncertainty exists due to non-commuting operators, but these operators do not commute simply because one of the operators is well-defined in classical physics (particle-like), while the other is wave-like (well-defined in quantum mechanics), or the eigenvalue is not at all defined in classical physics. We also construct a new quantum mechanical postulate (Postulate 9) to deduce why the forbidden gap between discrete energy levels should exist in atoms, molecules and condensed matter phases. In fact, we tackle all the problems arising from the Copenhagen interpretation without violating established experiments and without proposing ideas that violate physical reality.
... The wave function or the state function in the de Broglie-Bohm theory [17] also does not incorporate the particle property (the mass). As for the Feynman non-classical paths, regardless of whether there is a protocol-toobserve [18] or not, it has never been observed or confirmed. The existence of this protocol does not imply the observation has been made and/or confirmed. ...
Interpreting quantum mechanics is a hard problem basically because it means explaining why and how the mathematics exploited to formulate wave-particle duality are related to observations or reality in classical physics. Consequently, interpretation of quantum mechanics and its formalism should involve proper physical mathematics, physical logic and classical physics, which is not the case from the Copenhagen interpretation. Here, we shall revisit all the postulates of quantum mechanics with proper physics and physical logic and reconstruct them to establish the less-complex interpretation of quantum mechanics with additional new postulates from classical physics. A new quantum mechanical postulate (Postulate 9) is also introduced to understand the meaning of the forbidden gap between discrete energy levels and/or electron orbitals in atoms that is valid for molecules and condensed matter based on quantum mechanics that is technically well-defined and experimentally observable. Here, we tackle all the problems arise from the Copenhagen interpretation systematically by revising and/or extending them with proper classical and quantum physics without violating established experiments and without proposing ideas that violate physical reality.
... As complex numbers, weak values give a direct access to the complex components of the quantum state in quantum tomography [31][32][33]. Weak values also provide essential insights into issues related to quantum foundations [34][35][36], such as paradoxes [37][38][39][40] or non-perturbative sensing of quantum particles along trajectories [41][42][43]. Weak values have also the potential to benefit quantum computing and quantum information processing [44][45][46][47][48][49]. ...
Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N -dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in function of three real vectors on the unit sphere in N ² − 1 dimensions, S N ² −2. These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of N − 1 dimensions, CP N −1 , which has a non-trivial representation as a 2 N − 2 dimensional submanifold of S N ² −2 (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N -level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in CP N −1 . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU( N ) given by the generalized Gell-Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates.
... Indeed, as complex numbers, they give a direct access to the complex components of the quantum state [31,32,33]. Resulting from weakly perturbing measurements, weak values also provide essential insights into issues related to quantum foundations [34,35,36], such as paradoxes [37,38,39,40] or sensing quantum particles along trajectories [41,42,43]. Weak values have also the potential to benefit quantum computing and quantum information processing [44,45,46,47,48,49]. ...
Observations in quantum weak measurements are determined by complex numbers called weak values. We present a geometrical interpretation of the argument of weak values of general Hermitian observables in N-dimensional quantum systems in terms of geometric phases. We formulate an arbitrary weak value in function of three real vectors on the unit sphere in dimensions, . These vectors are linked to the initial and final states, and to the weakly measured observable, respectively. We express pure states in the complex projective space of dimensions, , which has a non-trivial representation as a dimensional submanifold of (a generalization of the Bloch sphere for qudits). The argument of the weak value of a projector on a pure state of an N-level quantum system describes a geometric phase associated to the symplectic area of the geodesic triangle spanned by the vectors representing the pre-selected state, the projector and the post-selected state in . We then proceed to show that the argument of the weak value of a general observable is equivalent to the argument of an effective Bargmann invariant. Hence, we extend the geometrical interpretation of projector weak values to weak values of general observables. In particular, we consider the generators of SU(N) given by the generalized Gell-Mann matrices. Finally, we study in detail the case of the argument of weak values of general observables in two-level systems and we illustrate weak measurements in larger dimensional systems by considering projectors on degenerate subspaces, as well as Hermitian quantum gates.
... Some experimental and theoretical examples of other modal theories (i.e., with ontological descriptions different from the Bohmian theory, while still yielding experimental predictions identical to those of the orthodox quantum theory) can be found in Refs. [85][86][87][88]. Our paper also emphasizes that looking for a meaning of weak values without linking them to a given ontology (or mixing ontologies in a type of bipolarity that wants a reality independent of the measurement, but at the same time rejects it) is not the correct path to understand weak values. ...
The so-called eigenvalue-eigenstate link states that no property can be associated to a quantum system unless it is in an eigenstate of the corresponding operator. This precludes the assignation of properties to unmeasured quantum systems in general. This arbitrary limitation of orthodox quantum mechanics generates many puzzling situations such as for example the impossibility to uniquely define a work distribution, an essential building block of quantum thermodynamics. Alternatively, modal theories (e.g., Bohmian mechanics) provide an ontology that always allows one to define intrinsic properties, i.e., properties of quantum systems that are detached from any possible measuring context. We prove here that Aharonov, Albert, and Vaidman's notion of a weak value can always be identified with an intrinsic dynamical property of a quantum system defined in a certain modal theory. Furthermore, the fact that weak values are experimentally accessible (as an ensemble average of weak measurements which are postselected by a strong measurement) strengthens the idea that understanding the intrinsic (unperturbed) dynamics of quantum systems is possible and useful in a given modal theory. As examples of the physical soundness of these intrinsic properties, we discuss three intrinsic Bohmian properties, viz., the dwell time, the work distribution, and the quantum noise at high frequencies.
... Surprisingly, very few works have employed weak values in a path integral context. Even then, the interest was restricted to the weak measurement of Feynman paths in semiclassical systems [10][11][12], to WV of specific operators [13][14][15], or as a way to probe virtual histories [16]. ...
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).
... The wave function or the state function in the de Broglie-Bohm theory [17] also does not incorporate the particle property (the mass). As for the Feynman non-classical paths, regardless of whether there is a protocol-toobserve [18] or not, it has never been observed or confirmed. The existence of this protocol does not imply the observation has been made and/or confirmed. ...
In standard quantum mechanics using the ladder operator method, both the or-bital angular momentum eigenvalues, ls and the spin angular momentum eigenvalues, ss are always identical such that they are both integral and half-integral values. This is true despite the fact that l has got to be an integer from the solution derived using the Schrödinger equation. Here, we shall provide the proper technical reason as to why the orbital angular momentum eigenvalues are indeed integer values, while only the spin angular momentum eigenvalues can acquire both integral and half-integral values. We shall also furnish the unambiguous physical mechanism responsible for the said difference. 03.65.Ca; 03.65.Ta.
... Thus, although the latter is not a probabilistic approach itself, appropriate to describe single photons in Fock states, in the large-number limit the correspondence with classical electromagnetism shows there is a proportionality between probability distributions and energy densities, and therefore the electromagnetic-flow trajectories will accurately describe the average paths along which photons travel. In the last years, the laboratory implementation of weak measurements, used as an alternative to quantum tomography to determine the photon wave function [23,24], has also inspired a series of theoretical works concerning the interpretation [9], reconstruction [25], or new observations [26,27] of Bohmian trajectories. ...
Usual Gaussian beams are particular scalar solutions to the paraxial Helmholtz equation, which neglect the vector nature of light. In order to overcome this inconvenience, Simon et al. (J. Opt. Soc. Am. A 1986, 3, 536–540) found a paraxial solution to Maxwell’s equation in vacuum, which includes polarization in a natural way, though still preserving the spatial Gaussianity of the beams. In this regard, it seems that these solutions, known as Gauss-Maxwell beams, are particularly appropriate and a natural tool in optical problems dealing with Gaussian beams acted or manipulated by polarizers. In this work, inspired in the Bohmian picture of quantum mechanics, a hydrodynamic-type extension of such a formulation is provided and discussed, complementing the notion of electromagnetic field with that of (electromagnetic) flow or streamline. In this regard, the method proposed has the advantage that the rays obtained from it render a bona fide description of the spatial distribution of electromagnetic energy, since they are in compliance with the local space changes undergone by the time-averaged Poynting vector. This feature confers the approach a potential interest in the analysis and description of single-photon experiments, because of the direct connection between these rays and the average flow exhibited by swarms of identical photons (regardless of the particular motion, if any, that these entities might have), at least in the case of Gaussian input beams. In order to illustrate the approach, here it is applied to two common scenarios, namely the diffraction undergone by a single Gauss-Maxwell beam and the interference produced by a coherent superposition of two of such beams.
... Thus, although the latter is not a probabilistic approach itself, appropriate to describe single photons in Fock states, in the large-number limit the correspondence with classical electromagnetism shows there is a proportionality between probability distributions and energy densities, and therefore the electromagnetic-flow trajectories will accurately describe the average paths along which photons travel. In the last years, the laboratory implementation of weak measurements, used as an alternative to quantum tomography to determine the photon wave function [23,24], has also inspired a series of theoretical works concerning the interpretation [9], reconstruction [25], or new observations [26,27] of Bohmian trajectories. ...
Usual Gaussian beams are particular scalar solutions to the paraxial Helmholtz equation, which neglect the vector nature of light. In order to overcome this inconvenience, Simon et al. [J. Opt. Soc. Am. A 3, 536 (1986)] found a paraxial solution to Maxwell's equation in vacuum, which includes polarization in a natural way, though still preserving the spatial Gaussianity of the beams. In this regard, it seems that these solutions, known as Gauss-Maxwell beams, are particularly appropriate and a natural tool in optical problems dealing with Gaussian beams acted or manipulated by polarizers. In this work, inspired in the Bohmian picture of quantum mechanics, a hydrodynamic-type extension of such a formulation is provided and discussed, complementing the notion of electromagnetic field with that of (electromagnetic) flow or streamline. In this regard, the method proposed has the advantage that the rays obtained from it render a bona fide description of the spatial distribution of electromagnetic energy, since they are in compliance with the local space changes undergone by the time-averaged Poynting vector. This feature confers the approach a potential interest in the analysis and description of single-photon experiments, because of the direct connection between these rays and the average flow exhibited by swarms of identical photons (regardless of the particular motion, if any, that these entities might have), at least in the case of Gaussian input beams. In order to illustrate the approach, here it is applied to two common scenarios, namely the diffraction undergone by a single Gauss-Maxwell beam and the interference produced by a coherent superposition of two of such beams.
... Let us assume that the interaction takes place in a time window [t w − τ/2, t w + τ/2], i.e. t w is the average interaction time and τ the duration. If τ is small relative to the system evolution timescale, the interaction can be simply taken to take place precisely at t w (for a proof of this "midpoint rule", see Ref. [31]). As in von Neumann's impulsive measurement scheme (see below Eq. (4), g ≡ t w +τ/2 t w −τ/2 g(t)dt appears as the effective coupling constant, but we now require g to be very small. ...
We investigate in this work the meaning of weak values through the prism of property ascription in quantum systems. Indeed, the weak measurements framework contains only ingredients of the standard quantum formalism, and as such weak measurements are from a technical point of view uncontroversial. However attempting to describe properties of quantum systems through weak values—the output of weak measurements—goes beyond the usual interpretation of quantum mechanics, that relies on eigenvalues. We first recall the usual form of property ascription, based on the eigenstate-eigenvalue link and the existence of “elements of reality”. We then describe against this backdrop the different meanings that have been given to weak values. We finally argue that weak values can be related to a specific form of property ascription, weaker than the eigenvalues case but still relevant to a partial description of a quantum system.
... Although a subject of some controversy [2,3] and vivid scientific debate [4][5][6][7][8][9][10][11][12][13][14], weak values allow acquiring information on a quantum state without causing its collapse. With the usage of weak interaction, we are able to measure qubit without destroying it [15], directly measure the quantum wave function [16], observe trajectory in quantum systems in the semiclassical regime [17], study three-box paradox [18], determine the past of photons passing through an interferometer [14] and even amplify the nonlinear effect of the single photon [19]. Typically, weak value measurement requires an infinitesimally weak interaction between the measured system and a pointer state [20][21][22][23]. ...
Weak values are traditionally obtained using a weak interaction between the measured system and a pointer state. In this paper, we show that weak values can also be measured using strong interaction accompanied by either a suitably prepared pointer state or quantum erasure. Presented theoretical derivations prove analytical equivalence of these approaches. Moreover, we have performed an experimental verification of our model on a linear-optical controlled phase gate. Our results open new ways of performing non-invasive quantum measurements without collapsing the measured system.
... Some experimental and theoretical examples of other Modal theories (i.e. with an ontological descriptions different from the Bohmian theory, while still yielding experimental predictions identical to those of the Orthodox quantum theory) can be found in Refs. [88][89][90][91]. Our paper also emphasizes that looking for a meaning of weak values without linking them to a given ontology (or mixing ontologies in a type of bipolarity that wants a reality independent of the measurement, but at the same time rejects it) is not the correct path to understand weak values. ...
In contrast to observables and measured beables, which are synonyms referred to in this paper as measured properties, we define the concept of unmeasured property as the value of a beable belonging to a quantum system that is not interacting with the measuring apparatus. We show that, by construction, measured and unmeasured single-time expectation values are identical. Contrarily, multiple-time measured and unmeasured expectation values, defined, e.g., through time-correlations functions, are generally different. Unmeasured two-times correlation functions have, in general, no analogous observable. Even when the later is obtained through a two-time ideally weak measurement, with an almost negligible perturbation on the system state, measured and unmeasured dynamical properties are different. This work clarifies common misunderstandings when comparing measured and unmeasured properties of quantum systems, a common strategy to discredit hidden-variable theories. Furthermore, it highlights the usefulness of unmeasured properties to provide valuable, intermediate, information for calculating observables of large (computationally inaccessible) systems. While it is a standard practice in classical mechanics to compute the dynamics of certain subsystems and use this information to elaborate on the properties of a larger containing system, the extension of such procedure in quantum mechanics hides a conceptual difficulty. In this work we explore the possibility of extending the standard classical procedure by using Bohmian mechanics, which does not only reproduce any observable by construction, but allows a clear-cut definition of unmeasured properties of quantum systems. The tunneling time, the work distribution or the high-frequency electrical current are shown to be paradigmatic examples of the practical utility of unmeasured properties.
... It led to the development of new phenomena such as quantum random walks [8] and superoscillations [9,10]. It has influenced recent theoretical [11][12][13][14][15][16] and experimental [17][18][19][20] studies of quantum foundations. The weak value has been an important tool in the development of precision measurements [21][22][23][24][25], as well as state [26,27] and process [28,29] tomography. ...
Weak values have been shown to be helpful especially when considering them as the outcomes of weak measurements. In this paper we show that, in principle, the real and imaginary parts of the weak value of any operator may be elucidated from expectation values of suitably defined density, flux, and Hermitian commutator operators. Expectation values are the outcomes of strong (projective) measurements, implying that weak values are general properties of operators in association with pre- and postselection and they need not be preferentially associated with weak measurements. They should be considered as an important measurable property which provides added information compared with the “standard” diagonal expectation value of an operator. As the first specific example we consider the determination of the real and imaginary parts of the weak value of the momentum operator employing projective time-of-flight experiments. Then the results are analyzed from the point of view of Bohmian mechanics. Finally, we consider recent neutron interferometry experiments used to determine the weak values of the neutron spin.
... Weak values seek to represent observables of intermediate states, as the system evolves from an initial to a final or post-selection state. Judiciously choosing the postselection state onto which to project the ancilla (and system), have been shown to produce anomalous phenomena that have sparked debate over the interpretation of weak values [2][3][4][5][6][7][8][9]. Amongst the anomalous phenomena is the quantum Cheshire cat (QCC) effect [10], where the position of a photon exists in one arm of an interferometer, whilst its polarisation exists in the other arm; the whimsical name alludes to Lewis Carroll's Cheshire cat, whose grin (polarisation) could exist without its body (photon). ...
We show that under the weak measurement scheme, the double-slit experiment can produce an interference pattern even when one of the slits is completely blocked. The initial and final states are corpuscular, whilst the intermediate states are wave-like, in that it exhibits an interference pattern. Remarkably, the interference pattern is measured to be vertically polarised, whilst simultaneously the individual photons are measured to be horizontally polarised. We call this the \textit{phantom slit} effect. The phantom slit is the dual of the quantum Cheshire cat.
... Sequential weak values reflect the unique charac- ter of temporal correlations, as was also shown by Avella et al. [29]. Consider as another example, an experimenter changing the number of beam splitters from B n = 4 to B n = 22 on the history through x 1 , while measuring de- vices record the weak values at x 3 or x 4 . The presence of 18 extra beamsplitters on arm x 1 is felt by the weak mea- suring devices at arm x 3 or x 4 as they measure the very large weak values (x 3 ) w = −1024 and (x 4 ) w = +1024. ...
Feynman's sum-over-histories formulation of quantum mechanics has been considered a useful calculational tool in which virtual Feynman histories entering into a coherent quantum superposition cannot be individually measured. Here we show that sequential weak values, inferred by consecutive weak measurements of projectors, allow direct experimental probing of individual virtual Feynman histories, thereby revealing the exact nature of quantum interference of coherently superposed histories. Because the total sum of sequential weak values of multitime projection operators for a complete set of orthogonal quantum histories is unity, complete sets of weak values could be interpreted in agreement with the standard quantum mechanical picture. We also elucidate the relationship between sequential weak values of quantum histories with different coarse graining in time and establish the incompatibility of weak values for nonorthogonal quantum histories in history Hilbert space. Bridging theory and experiment, the presented results may enhance our understanding of both weak values and quantum histories.
... Quantum mechanics does not provide a clear answer to the question: What was the past of a photon which went through an interferometer [1]? Various welcher weg measurements [2], delayed-choice which-path experiments [3][4][5] and weak-measurements of photons in interferometers [6][7][8] presented the past of a photon as a trajectory or a set of trajectories. We have carried out experimental weak measurements of the paths of photons going through a nested Mach-Zehnder interferometer, discussed earlier in another context [9,10], which show a different picture: the past of a photon is not a set of continuous trajectories. ...
We present surprising experimental evidence regarding the past of photons passing through an interferometer. The information about the positions through which the photons pass in the interferometer is retrieved from modulations of the detected signal at the vibration frequencies of mirrors the photons bounce off. From the analysis we conclude that the past of the photons is not represented by continuous trajectories, although a "common sense" analysis adopted in various welcher weg measurements, delayed-choice which-path experiments, and counterfactual communication demonstrations yields a single trajectory. The experimental results have a simple explanation in the framework of the two-state vector formalism of quantum theory.
... One can minimize the back-action of the measurement on the system using weak measurements. Such measurements were initially developed by Aharonov, Albert and Vaidman (AAV) [1] more than two decades ago and they are receiving increasing attention [2][3][4][5][6][7][8][9][10] nowadays. As a relevant example, the spatial distribution of velocities of relativistic photons in a double slit scenario has been measured, and the associated quantum trajectories reconstructed [6]. ...
We present a protocol for measuring Bohmian, or the mathematically equivalent hydrodynamic, velocities based on an ensemble of two position measurements, defined from a positive operator-valued measure, separated by a finite time interval. The protocol is very accurate and robust as long as the first measurement uncertainty divided by the finite time interval between measurements is much larger than the Bohmian velocity, and the system evolves under flat potential between measurements. The difference between the Bohmian velocity of the unperturbed state and the measured one is predicted to be much smaller than 1% in a large range of parameters. Counterintuitively, the measured velocity is that at the final time and not a time-averaged value between measurements.
... The crucial difference between WMs and projective measurements is that the latter suppresses the entangled linear superposition of system-apparatus states (as if a collapse to a single term in the pointer basis had taken place) while the former retains the full wave aspect of the quantum system. For example in dynamical systems, where the system wavefunction is characterized by a "sum over paths" as prescribed by Feynman's propagator, an array of apparati weakly interacting with the system should allow in principle to detect the wavefunction simultaneously propagating along the available paths [20] provided the paths are sufficiently isolated from one another. The "strength" of weak measurements is to capture this wave phenomenon -too often thought of as being a computational artifact -with apparati weakly coupled to the system. ...
We propose in this work a definite theoretical implementation of the
three-box paradox - a scheme in which a single quantum particle appears to be
present with certainty in two separate boxes - with spin-1 atoms. We further
show how our setup can give rise to a "Cheshire cat grin" type of situation, in
which an atom can apparently be found with certainty in one of the boxes while
one of its properties (the angular momentum projection along a specifically
chosen axis) appears to be in a different box. The significance of our findings
are discussed relative to the status of the properties of a system obtained
from weak measurements.
Heisenberg-limited and weak measurements are the two intriguing notions, used in recent times for enhancing the sensitivity of measurements in quantum metrology. Using a quantum cat state, endowed with sub-Planck structure, we connect these two novel concepts. It is demonstrated that these two phenomena manifest in complementary regimes, depending upon the degree of overlap between the mesoscopic states constituting the cat state under consideration. In particular, we find that when sub-Planck structure manifests, the imaginary weak value is obscured and vice-versa.
Understanding how the interference pattern produced by a quantum particle in Young’s double-slit setup builds up—the “only mystery” of quantum mechanics according to Feynman—is still a matter of discussion and speculation. Recent works have revisited the possibility of acquiring which-way information based on weak measurements. Weak measurements preserve the interference pattern due to their minimally perturbing character while still leading to a final position detection. Here, we investigate a simplified double-slit setup by including weakly coupled pointers. We examine how the information provided by the weak pointers can be interpreted to infer the dynamics within a local picture through “weak trajectories”. We contrast our approach with non-local dynamical accounts, such as the modular momentum approach to weak values and the trajectories defined by the de Broglie–Bohm picture.
The interference pattern produced by a quantum particle in Young's double-slit setup is attributed to the particle's wave function having gone through both slits. In the path integral formulation, this interference involves a superposition of paths, going through either slit, linking the source to the detection point. We show how these paths superpositions can in principle be observed by implementing a series of minimally perturbing weak measurements between the slits and the detection plane. We further propose a simplified protocol in order to observe these “weak trajectories” with single photons.
It is shown, that the Aharonov-Albert-Vaidman concept of weak values appears to be a consequence of a more general quantum phenomenon of weak quantum evolution. Here the concept of weak quantum evolution is introduced and discussed for the first time. In particular, it is shown on the level of quantum evolution that there exist restrictions on the applicability of weak quantum evolution- and, hence, weak values approach. These restrictions connect the size of a given quantum ensemble with the parameters of pre- and post-selected quantum states. It is shown, that the latter requirement can be fulfilled for the model system, where the concept of weak values was initially introduced by Aharonov, Albert and Vaidman. Moreover, the deep connection between weak quantum evolution and conventional probability of quantum transition between two non-orthogonal quantum states is established for the first time. It is found that weak quantum evolution of quantum system between its two non-orthogonal quantum states is inherently present in the measurement-determined definition of quantum transition probability between these two quantum states.
We introduce a new semirelativistic quantum operator for the length of the worldline a particle traces out as it moves. In this article the operator is constructed in a heuristic way and some of its elementary properties are explored. The operator ends up depending in a very complicated way on the potential of the system it is to act on so as a proof of concept we use it to analyze the expected distance traveled by a free Gaussian wave packet with some initial momentum. It is shown in this case that the distance such a particle travels becomes light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This preliminary result has minor implications for the Weak Equivalence Principle (WEP) in quantum mechanics. In particular it shows that the logical relationship between two formulations of the WEP in classical mechanics extends to quantum mechanics. That our result is qualitatively consistent with the work of others emboldens us to start the task of evaluating the new operator in nonzero potentials. However, we readily acknowledge that the looseness in the definition of our operator means that all of our so-called results are highly speculative. Plans for future work with the new operator are discussed in the last section.
Weak values inferred from weak measurements have been proposed as a tool to investigate trajec-tories of pre-and post-selected quantum systems. Are the inferences drawn from the weak values about the past of a quantum particle fully true? Can the two-state vector formalism predict everything that the standard formalism of quantum mechanics can? To investigate these questions we present a "which-path" gedanken experiment in which the information revealed by a pre-and post-selected quantum system is surprisingly different from what one would expect from the weak values computed using the two-state vector formalism. In our gedanken experiment, a particle reveals its presence in locations where the weak value of the projection operator onto those locations was vanishingly small. Therefore our predictions turn out to be in contradistinction to those made based on the nonvanishing weak values as the presence indicators of the quantum particle. We propose a six port photon-based interferometer setup as a possible physical realization of our gedanken experiment.
Weak values inferred from weak measurements have been proposed as a tool to investigate trajectories of pre- and post-selected quantum systems. Are the inferences drawn from the weak values about the past of a quantum particle fully true? Can the two-state vector formalism predict everything that the standard formalism of quantum mechanics can? To investigate these questions we present a "which-path" gedanken experiment in which the information revealed by a pre- and post-selected quantum system is surprisingly different from what one would expect from the weak values computed using the two-state vector formalism. In our gedanken experiment, a particle reveals its presence in locations where the weak value of the projection operator onto those locations was vanishingly small. Therefore our predictions turn out to be in contradistinction to those made based on the nonvanishing weak values as the presence indicators of the quantum particle. We propose a six port photon-based interferometer setup as a possible physical realization of our gedanken experiment.
The recent single-photon double-slit experiment of Steinberg et al., based on a weak measurement method proposed by Wiseman, showed that, by encoding the photon's transverse momentum behind the slits into its polarization state, the momentum profile can subsequently be measured on average, from a difference of the separated fringe intensities for the two circular polarization components. They then integrated the measured average velocity field, to obtain the average trajectories of the photons enroute to the detector array. In this paper, we propose a modification of their experiment, to demonstrate that the average particle velocities and trajectories change when the mode of detection changes. The proposed experiment replaces a single detector by a pair of detectors with a given spacing between them. The pair of detectors is configured so that it is impossible to distinguish which detector received the particle. The pair of detectors is then analogous to the simple pair of slits, in that it is impossible to distinguish which slit the particle passed through.To establish the paradoxical outcome of the modified experiment, the theory and explicit three-dimensional formulas are developed for the bilocal probability and current densities, and for the average velocity field and trajectories as the particle wavefunction propagates in the volume of space behind the Gaussian slits. Examples of these predicted results are plotted. Implementation details of the proposed experiment are discussed.
Weak-measurement-based experiments [Kocsis et al., Science 332 (2011) 1170] have shown that, at least for pure states, the average evolution of independent photons in Young’s two-slit experiment is in compliance with the trajectories prescribed by the Bohmian formulation of quantum mechanics. But, what happens if the same experiment is repeated assuming that the wave function associated with each particle is different, i.e., in the case of mixed (incoherent) states? This question is investigated here by means of two alternative numerical simulations of Young’s experiment, purposely devised to be easily implemented and tested in the laboratory. Contrary to what could be expected a priori, it is found that even for conditions of maximal mixedness or incoherence (total lack of interference fringes), experimental data will render a puzzling and challenging outcome: the average particle trajectories will still display features analogous to those for pure states, i.e., independently of how mixedness arises the associated dynamics is influenced by both slits at the same time. Physically this simply means that weak measurements are not able to discriminate how mixedness arises in the experiment, since they only provide information about the averaged system dynamics.
The weak value, introduced by Aharonov et al. to extend the conventional
scope of physical observables in quantum mechanics, is an intriguing concept
which sheds new light on quantum foundations and is also useful for precision
measurement, but it poses serious questions on its physical meaning due to the
unconventional features including the complexity of its value. In this paper we
point out that the weak value has a direct connection with the wave-particle
duality, in the sense that the wave nature manifests itself in the imaginary
part while the particle nature in the real part. This is illustrated by the
double slit experiment, where we argue, with no conflict with complementarity,
that the trajectory of the particle can be inferred based on the weak value
without destroying the interference.
Heisenberg-limited and weak measurements are the two intriguing notions, used
in recent times for enhancing the sensitivity of measurements in quantum
metrology. Using a quantum cat state, endowed with sub-Planck structure, we
connect these two novel concepts. It is demonstrated that these two phenomena
manifest in complementary regimes, depending upon the degree of overlap between
the mesoscopic states constituting the cat state under consideration. In
particular, we find that when sub-Planck structure manifests, the imaginary
weak value is obscured and vice-versa.
That quantum mechanics is a probabilistic theory was, by 1964, an old but still troubling story. The fact that identical measurements of identically prepared systems can yield different outcomes seems to challenge a basic tenet of science and philosophy. Frustration with the indeterminacy intrinsic to quantum mechanics was famously expressed in Albert Einstein's assertion that "God doesn't play dice."
With the exception of the harmonic oscillator, quantum wave packets usually spread as time evolves. This is due to the non-linear character of the classical equations of motion which makes the various components of the wave packet evolve at various frequencies. We show here that, using the non-linear resonance between an internal frequency of a system and an external periodic driving, it is possible to overcome this spreading and build non-dispersive (or non-spreading) wave packets which are well localized and follow a classical periodic orbit without spreading. From the quantum mechanical point of view, the non-dispersive wave packets are time periodic eigenstates of the Floquet Hamiltonian, localized in the non-linear resonance island.We discuss the general mechanism which produces the non-dispersive wave packets, with emphasis on simple realization in the electronic motion of a Rydberg electron driven by a microwave field. We show the robustness of such wave packets for a model one-dimensional as well as for realistic three-dimensional atoms. We consider their essential properties such as the stability versus ionization, the characteristic energy spectrum and long lifetimes. The requirements for experiments aimed at observing such non-dispersive wave packets are also considered.The analysis is extended to situations in which the driving frequency is a multiple of the internal atomic frequency. Such a case allows us to discuss non-dispersive states composed of several, macroscopically separated wave packets communicating among themselves by tunneling. Similarly we briefly discuss other closely related phenomena in atomic and molecular physics as well as possible further extensions of the theory.
Bohmian mechanics (BM) is a popular interpretation of quantum mechanics (QM) in which particles have real positions. The velocity of a point x in configuration space is defined as the standard probability current j(x) divided by the probability density P(x). However, this 'standard' j is in fact only one of infinitely many that transform correctly and satisfy . In this paper, I show that a particular j is singled out if one requires that j be determined experimentally as a weak value, using a technique that would make sense to a physicist with no knowledge of QM. This 'naively observable' j seems the most natural way to define j operationally. Moreover, I show that this operationally defined j equals the standard j, so, assuming , one obtains the dynamics of BM. It follows that the possible Bohmian paths are naively observable from a large enough ensemble. Furthermore, this justification for the Bohmian law of motion singles out x as the hidden variable, because (for example) the analogously defined momentum current is in general incompatible with the evolution of the momentum distribution. Finally I discuss how, in this setting, the usual quantum probabilities can be motivated from a Bayesian standpoint, via the principle of indifference.
Unlike the real part of the generalized weak value of an observable, which
can in a restricted sense be operationally interpreted as an idealized
conditioned average of that observable in the limit of zero measurement
disturbance, the imaginary part of the generalized weak value does not provide
information pertaining to the observable being measured. What it does provide
is direct information about how the initial state would be unitarily disturbed
by the observable operator. Specifically, we provide an operational
interpretation for the imaginary part of the generalized weak value as the
logarithmic directional derivative of the post-selection probability along the
unitary flow generated by the action of the observable operator. To obtain this
interpretation, we revisit the standard von Neumann measurement protocol for
obtaining the real and imaginary parts of the weak value and solve it exactly
for arbitrary initial states and post-selections using the quantum operations
formalism, which allows us to understand in detail how each part of the
generalized weak value arises in the linear response regime. We also provide
exact treatments of qubit measurements and Gaussian detectors as illustrative
special cases, and show that the measurement disturbance from a Gaussian
detector is purely decohering in the Lindblad sense, which allows the shifts
for a Gaussian detector to be completely understood for any coupling strength
in terms of a single complex weak value that involves the decohered initial
state.
A consequence of the quantum mechanical uncertainty principle is that one may not discuss the path or “trajectory” that a
quantum particle takes, because any measurement of position irrevocably disturbs the momentum, and vice versa. Using weak
measurements, however, it is possible to operationally define a set of trajectories for an ensemble of quantum particles.
We sent single photons emitted by a quantum dot through a double-slit interferometer and reconstructed these trajectories
by performing a weak measurement of the photon momentum, postselected according to the result of a strong measurement of photon
position in a series of planes. The results provide an observationally grounded description of the propagation of subensembles
of quantum particles in a two-slit interferometer.
The concept of a modular value of an observable of a pre- and postselected quantum system is introduced. It is similar in form and in some cases has a close connection to the weak value of an observable, but instead of describing an effective interaction when the coupling is weak, it describes a coupling of any strength but only to qubit meters. The generalization of the concept for a coupling of a composite system to a multiqubit meter provides an explanation of some current experiments.
The essential feature of weak measurements on quantum systems is the
reduction of measurement back-action to negligible levels. To observe the
non-classical features of weak measurements, it is therefore more important to
avoid additional back-action errors than it is to avoid errors in the actual
measurement outcome. In this paper, it is shown how an optical weak measurement
of diagonal (PM) polarization can be realized by path interference between the
horizontal (H) and vertical (V) polarization components of the input beam. The
measurement strength can then be controlled by rotating the H and V
polarizations towards each other. This well-controlled operation effectively
generates the back-action without additional decoherence, while the visibility
of the interference between the two beams only limits the measurement
resolution. As the experimental results confirm, we can obtain extremely high
weak values, even at rather low visibilities. Our method therefore provides a
realization of weak measurements that is extremely robust against experimental
imperfections.
We report on the use of an interferometric weak value technique to amplify very small transverse deflections of an optical beam. By entangling the beam's transverse degrees of freedom with the which-path states of a Sagnac interferometer, it is possible to realize an optical amplifier for polarization independent deflections. The theory for the interferometric weak value amplification method is presented along with the experimental results, which are in good agreement. Of particular interest, we measured the angular deflection of a mirror down to 400+/-200 frad and the linear travel of a piezo actuator down to 14+/-7 fm.
It has been proposed that the ability to perform joint weak measurements on postselected systems would allow us to study quantum paradoxes. These measurements can investigate the history of those particles that contribute to the paradoxical outcome. Here we experimentally perform weak measurements of joint (i.e., nonlocal) observables. In an implementation of Hardy's paradox, we weakly measure the locations of two photons, the subject of the conflicting statements behind the paradox. Remarkably, the resulting weak probabilities verify all of these statements but, at the same time, resolve the paradox.
The question in the title may be answered by considering the outcome of a ``weak measurement'' in the sense of Aharonov et al. Various properties of the resulting time are discussed, including its close relation to the Larmor times. It is a universal description of a broad class of measurement interactions, and its physical implications are unambiguous.
We have detected a spin-dependent displacement perpendicular to the refractive index gradient for photons passing through
an air-glass interface. The effect is the photonic version of the spin Hall effect in electronic systems, indicating the universality
of the effect for particles of different nature. Treating the effect as a weak measurement of the spin projection of the photons,
we used a preselection and postselection technique on the spin state to enhance the original displacement by nearly four orders
of magnitude, attaining sensitivity to displacements of ∼1 angstrom. The spin Hall effect can be used for manipulating photonic
angular momentum states, and the measurement technique holds promise for precision metrology.
We re-examine the status of the weak value of a quantum mechanical observable as an objective physical concept, addressing its physical interpretation and general domain of applicability. We show that the weak value can be regarded as a \emph{definite} mechanical effect on a measuring probe specifically designed to minimize the back-reaction on the measured system. We then present a new framework for general measurement conditions (where the back-reaction on the system may not be negligible) in which the measurement outcomes can still be interpreted as \emph{quantum averages of weak values}. We show that in the classical limit, there is a direct correspondence between quantum averages of weak values and posterior expectation values of classical dynamical properties according to the classical inference framework.
Aharonov-Albert-Vaidman's weak values are investigated by a semiclassical method. Examples of the semiclassical calculation that reproduces "anomalous" weak values are shown. Furthermore, a complex extension of Ehrenfest's quantum-classical correspondence between quantum expectation values of the states with small quantum fluctuation, and classical dynamics, is shown.
We investigate the experimental spectra of excited NO molecules in the diamagnetic regime and develop a quantitative semiclassical framework to account for the results. We show the dynamics can be interpreted in terms of classical orbits provided that in addition to the geometric orbits, diffractive effects are appropriately taken into account. We also show how individual orbits can be extracted from the experimental signal and use this procedure to reveal the first experimental manifestation of inelastic diffractive orbits.
We describe how to reconstruct individual classical trajectories from spectroscopic data. The ac dipole moment of a trajectory can be found from the effect of an oscillating field on the spectrum. The inverse Fourier transform of such data yields the component of the electron trajectory along the direction of the oscillating field. We demonstrate the method by experimentally extracting zt for two electron trajectories that influence the Stark spectrum of Rydberg lithium. Within the experimental resolution, the reconstructed orbits agree well with classical predictions.
A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.
Using scaled-energy Stark spectroscopy, we report the observation of recurrences due to closed orbits, both geometric and diffractive, in the ν=0, R=1, nd Rydberg series of H2 (16<n<26) interacting with the ν=0, R=3 series (13<n<15). The data support the molecular closed-orbit theory prediction of diffractive trajectories due to inelastic scattering of the excited electron on the molecular core. We have made similar measurements in He, and a comparison between the recurrence properties of H2 and its united atom equivalent is given.
Emphasizes the many applications that have been found for path integrals in quantum mechanics, statistical physics, field theory, and optics. Covers both the Feynman integral of quantum mechanics and the Wiener integral of probability theory. Initial experience is gained with solvable integrals and some of the formal and general properties developes.
By weakly measuring the polarization of a photon between two strong polarization measurements, we experimentally investigate the correlation between the appearance of anomalous values in quantum weak measurements and the violation of realism and nonintrusiveness of measurements. A quantitative formulation of the latter concept is expressed in terms of a Leggett-Garg inequality for the outcomes of subsequent measurements of an individual quantum system. We experimentally violate the Leggett-Garg inequality for several measurement strengths. Furthermore, we experimentally demonstrate that there is a one-to-one correlation between achieving strange weak values and violating the Leggett-Garg inequality.
We show how a single trapped ion may be used to test a variety of important physical models realized as time-dependent harmonic oscillators. The ion itself functions as its own motional detector through laser-induced electronic transitions. Alsing et al., [Phys. Rev. Lett. 94, 220401 (2005)] proposed that an exponentially decaying trap frequency could be used to simulate (thermal) Gibbons-Hawking radiation in an expanding universe, but the Hamiltonian used was incorrect. We apply our general solution to this experimental proposal, correcting the result for a single ion and showing that while the actual spectrum is different from the Gibbons-Hawking case, it nevertheless shares an important experimental signature with this result.
The anisotropic diamagnetic Kepler problem (ADKP) is realized experimentally by the orbital electrons of a P donor in Si under magnetic fields. The interference of electron wave packets which leads to quasi-Landau resonances (QLR) were observed. Applying the closed-orbit theory to an anisotropic solid state environment, we have identified orbits responsible for the QLR manifesting the quantum chaotic behavior in Rydberg atoms. The excellent consistency between the measured spectra and theoretical calculation provides unambiguous evidence of quantum chaotic dynamics of electrons in the ADKP.
We have found that the usual measuring procedure for preselected and postselected ensembles of quantum systems gives unusual results. Under some natural conditions of weakness of the measurement, its result consistently defines a new kind of value for a quantum variable, which we call the weak value. A description of the measurement of the weak value of a component of a spin for an ensemble of preselected and postselected spin-(1/2 particles is presented.
A protocol for steering Rydberg electrons towards targeted final states is realized with the aid of a chirped train of half-cycle pulses (HCPs). Its novel capabilities are demonstrated experimentally by transporting potassium atoms excited to the lowest-lying quasi-one-dimensional states in the n(i)=350 Stark manifold to a narrow range of much higher-n states. We demonstrate that this coherent state transfer is, to a high degree, reversible. The protocol allows for remarkable selectivity and is highly efficient, with typically over 80% of the parent atoms surviving the HCP sequence.
The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterisation of the notion of counterfactual quantum computation.
Square billiards are quantum systems complying with the dynamical quantum-classical correspondence. Hence an initially localized wavefunction launched along a classical periodic orbit evolves along that orbit, the spreading of the quantum amplitude being controlled by the spread of the corresponding classical statistical distribution. We investigate wavepacket dynamics and compute the corresponding de Broglie-Bohm trajectories in the quantum square billiard. We also determine the trajectories and statistical distribution dynamics for the equivalent classical billiard. Individual Bohmian trajectories follow the streamlines of the probability flow and are generically non-classical. This can also hold even for short times, when the wavepacket is still localized along a classical trajectory. This generic feature of Bohmian trajectories is expected to hold in the classical limit. We further argue that in this context decoherence cannot constitute a viable solution in order to recover classicality. Comment: Figures downgraded to low resolution; To be published in Found. Phys. (2009).
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