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.

prospects of proposals such as quantum-interferometric optical lithography. The method can be adapted to generate entangled states of arbitrarily large photon number. Because prior entangle-ment is not required, the procedure would work well with single-photon-on-demand sources 29,30 , which promise to be more efficient and scalable than down-conversion sources. Scalability would also be enhanced by the use of photon-number-resolving detectors. The construction proceeds from spatially separated, unentangled pho-tons to a maximally entangled state in a single spatial mode, a state suitable for Heisenberg-limited phase measurements. A

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It has been proposed that the ability to perform joint weak measurements on post-selected 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 these statements but, at the same time, resolve the Paradox.Retrodiction is a controversial topic in quantum mechanics [1]. How much is one allowed to say about the history (e.g. particle trajectories) of a post-selected ensemble? Historically this has been deemed a question more suitable for philosophy (e.g. counterfactual logic) than physics; since the early days of quantum mechanics, the standard approach has been to restrict the basis of our physical interpretations to direct experimental observations. On the practical side of the question, postselection has recently grown in importance as a tool in fields such as quantum information: e.g. in linear optics quantum computation (LOQC) [2], where it drives the logic of quantum gates; and in continuous variable systems, for entanglement distillation [3]. Weak measurement is a relatively new experimental technique for tackling just this question. It is of particular interest to carry out weak measurements of multi-particle observables, such as those used in quantum information. Here, we present an experiment that uses weak measurement to examine the two-particle retrodiction paradox of Hardy [4,5], confirming the validity of certain retrodictions and identifying the source of the apparent contradiction.Hardy's Paradox is a contradiction between classical reasoning and the outcome of standard measurements on an electron E and positron P in a pair of MachZehnder interferometers (see Fig. 1). Each interferometer is first aligned so that the incoming particle always leaves through the same exit port, termed the "bright" port B (the other is the "dark" port D). The interferometers are then arranged so that one arm (the "Inner" arm I) from each interferometer overlaps at Y. It is assumed that if the electron and positron simultaneously enter this arm they will collide and annihilate with 100% probability. This makes the interferometers "InteractionFree Measurements" (IFM) [6]: that is, a click at the dark port indicates the interference was disturbed by an object located in one of the interferometer arms, without the interfering particle itself having traversed that arm. Therefore, in Hardy's Paradox a click at the dark port of the electron (positron) indicates that the positron (electron) was in the Inner arm. Consider if one were to detect both particles at the dark ports. As IFMs, these results would indicate the particles were simultaneously in the Inner arms and, therefore should have annihilated. But this is in contradi...

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We show that weak measurement can be used to "amplify" optical nonlinearities at the singlephoton level, such that the effect of one properly post-selected photon on a classical beam may be as large as that of many un-post-selected photons. We find that "weak-value amplification" offers a marked improvement in the signal-to-noise ratio in the presence of technical noise with long correlation times. Unlike previous weak-measurement experiments, our proposed scheme has no classical equivalent.An interaction between two independent photons could be used to serve as a "quantum logic gate," enabling the development of optical quantum computers [1][2][3], as well as opening up an essentially new field of quantum nonlinear optics [4]. Typical optical nonlinearities are many orders of magnitude too weak to create a π phase shift as required in initial proposals, but more recently it was realized that any phase shift large enough to be measured on a single shot could be leveraged into a quantum logic gate [5]. Much recent work has shown that atomic coherence effects [6][7][8][9] and nonlinearities in microstructured fiber [10,11] can generate greatly enhanced Kerr nonlinearities. While even a very small phase shift can be made larger than the quantum (shot) noise, by using a sufficiently intense probe, present experiments are limited by technical rather than quantum noise and difficult to carry out even with much averaging. For example, in Ref.[11], a phase shift of 10 −7 rad was measured by averaging over 3 × 10 9 classical pulses with singlephoton-level intensities. To date, no one has yet been able to observe the cross-Kerr effect induced by a single propagating photon on a second optical beam [12]. In this Letter, we show that using weak-value amplification (WVA) [13][14][15], a single photon can be made to "act like" many photons, and it is possible to amplify a cross-Kerr phase shift to an observable value, much larger than the intrinsic magnitude of the single-photon-level nonlinearity. In so doing, we also demonstrate quantitatively how WVA may improve the signal-to-noise ratio (SNR) in appropriate regimes, a result of broad general applicability to quantum metrology.Weak measurement is an exciting new approach to understanding quantum systems from a time-symmetric perspective, obtaining information from both their preparation and subsequent post-selection [16]. In the past several years, it has been widely studied to address foundational questions in quantum mechanics [17], as well as for its potential application to ultrasensitive measurements [14,15,18,19]. If a quantum system is coupled only weakly to a probe, then very little information may be obtained from a single measurement, and in compensation, this measurement disturbs the sys-

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.PACS numbers: 03.65. Bz,73.40.Gk The question posed in the title has remained controversial since the early days of quantum theory [1][2][3][4][5]. One commonly cited reason for this is the nonexistence of a quantum-mechanical time operator. However, it is quite possible to construct an operator Θ B which measures whether a particle is in the barrier region or not. Such a projection operator is Hermitian, and may correspond to a physical observable. It has eigenvalues 0 and 1, and its expectation value simply measures the integrated probability density over the region of interest-it is this expectation value divided by the incident flux which is referred to as the "dwell time" [6]. Thus the central problem is not the absence of an appropriate Hermitian operator, but rather the absence of well-defined histories (or trajectories) in standard quantum theory. For the dwell time measures a property of a wave function with both transmitted and reflected portions, and does not display a unique decomposition into portions corresponding to these individual scattering channels. Some workers point out that it can be fruitful to consider the expectation value for a particular outgoing channel rather than a particular incident state [7]. However, this approach answers the present title's question no better than does the usual dwell time; instead of discarding information about late times it discards information about early times. Other related approaches follow phase space trajectories [7], Bohm trajectories [4], or Feynman paths [8]. No consensus has been reached as to the validity and the relationship of these various approaches. Ideally, transmission and reflection times τ T and τ R would, when weighted by the transmission and reflection probabilities |t| 2 and |r| 2 , yield the dwell time τ d :this relation has served as one of the main criteria in a broad review of tunneling times [3], but has also been criticized [5]. In this paper I present an approach to tunneling times which adheres strictly to the standard formulation of quantum mechanics, defining the "dwell" time by explicitly considering a general von Neumann-style measurement interaction. The mean value of the measurement outcome may then be calculated for transmitted and reflected particles individually, and I show that it automatically satisfies the above equation. This time is in general complex, and is closely related to the Larmor times and to the complex time of Sokolovski, Baskin, and Connor [8]; the consideration of a measuring apparatus, however, makes interpretation of its real and imaginary parts straightforward, as will be demonstrated by examining a generalization of the Larmor time. Before presenti...

The three-box problem is a gedankenexperiment designed to elucidate some interesting features of quantum measurement and locality. A particle is prepared in a particular superposition of three boxes, and later found in a different (but nonorthogonal) superposition. It was predicted that appropriate "weak" measurements of particle position in the interval between preparation and post-selection would find the particle in two different places, each with certainty. We verify these predictions in an optical experiment and address the issues of locality and of negative probability.Weak measurements have been controversial ever since the concept was developed by Aharonov, Albert, and Vaidman (AAV) [1].In contrast to the usual, von Neumann, approach to measurement, weak measurement uses an apparatus whose pointer has a very large quantum mechanical uncertainty when compared with its typical shift. After the system-pointer interaction, the shift in the pointer position is much smaller than its initial uncertainty and almost no information is gained about the quantum system. Nevertheless, after a sufficiently large number of measurements on an ensemble of identically prepared quantum systems, the mean pointer position can be determined to any degree of precision. In such a measurement strategy, one sacrifices knowledge of the value of an observable on any given experimental run to avoid entanglement with the measurement device and the ensuing 'collapse' of the wavefunction. In particular, this makes it possible to contemplate the behavior of a system defined both by state preparation and by a later post-selection, without significant disturbance of the system in the intervening period.AAV [1] calculated the shift in the pointer of a measurement apparatus that weakly measured an observable A between two strong measurements. The initial strong measurement pre-selects (or prepares) the state, |ψ i , of the quantum system and the final strong measurement post-selects the quantum state, ψ f . In between, consider a von Neumann-style interaction Hamiltonian of the formwhere is the hermitian operator corresponding to an observable A of the quantum system, g is a (real) coupling constant, andP x is the momentum operator conjugate to the pointer positionX. In the absence of postselection, the effect of having this measurement interaction on for a time T (assumed short enough that A is constant during the measurement) shifts the pointer position by an amount ∆x = K  ≡ gT  ,such that one can infer a value for A by dividing the pointer shift by the interaction strength K. The main result of AAV's seminal work [1] is that for sufficiently weak coupling strength K and in the presence of postselection, the inferred value of