The wavefunction is the complex distribution used to completely describe a quantum system, and is central to quantum theory. But despite its fundamental role, it is typically introduced as an abstract element of the theory with no explicit definition. Rather, physicists come to a working understanding of the wavefunction through its use to calculate measurement outcome probabilities by way of the Born rule. At present, the wavefunction is determined through tomographic methods, which estimate the wavefunction most consistent with a diverse collection of measurements. The indirectness of these methods compounds the problem of defining the wavefunction. Here we show that the wavefunction can be measured directly by the sequential measurement of two complementary variables of the system. The crux of our method is that the first measurement is performed in a gentle way through weak measurement, so as not to invalidate the second. The result is that the real and imaginary components of the wavefunction appear directly on our measurement apparatus. We give an experimental example by directly measuring the transverse spatial wavefunction of a single photon, a task not previously realized by any method. We show that the concept is universal, being applicable to other degrees of freedom of the photon, such as polarization or frequency, and to other quantum systems--for example, electron spins, SQUIDs (superconducting quantum interference devices) and trapped ions. Consequently, this method gives the wavefunction a straightforward and general definition in terms of a specific set of experimental operations. We expect it to expand the range of quantum systems that can be characterized and to initiate new avenues in fundamental quantum theory.

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Recent work [J.S. Lundeen et al. Nature, 474, 188 (2011)] directly measured the wavefunction by weakly measuring a variable followed by a normal (i.e. 'strong') measurement of the complementary variable. We generalize this method to mixed states by considering the weak measurement of various products of these observables, thereby providing the density matrix an operational definition in terms of a procedure for its direct measurement. The method only requires measurements in two bases and can be performed 'in situ', determining the quantum state without destroying it.PACS numbers: 03.65. Ta, 03.65.Wj, 42.50.Dv, The wavefunction Ψ is at the heart of quantum mechanics, yet its nature has been debated since its inception. It is typically relegated to being a calculational device for predicting measurement outcomes. Recently, Lundeen et al. proposed a simple and general operational definition of the wavefunction based on a method for its direct measurement: "it is the average result of a weak measurement of a variable followed by a strong measurement of the complementary variable [1,2]." By 'direct' it is meant that a value proportional to the wavefunction appears straight on the measurement apparatus itself without further complicated calculations or fitting. The 'wavefunction' referred to here is a special case of a general quantum state, known as a 'pure state.' The general case is represented by the density operator ρ, which can describe both pure and 'mixed' states. The latter incorporates both the effects of classical randomness (e.g., noise) and entanglement with other systems (e.g., decoherence). The density operator plays an important role in quantum statistics, quantum information, and the study of decoherence. Because of its generality and because it follows naturally from classical concepts of probability and measures, some consider ρ, rather than Ψ, to be the fundamental quantum state description. In this letter, we propose two methods to directly measure general quantum states, one of which directly gives the matrix elements of ρ.The standard method for experimentally determining the density operator is Quantum State Tomography [3]. In it, one makes a diverse set of measurements on an ensemble of identical systems and then determines the quantum state that is most compatible with the measurement results. An alternative is our direct measurement method, which may have advantages over tomography, such as simplicity, versatility, and directness. A quantitative comparison of measures such as the signal to noise ratio, resolution, and fidelity, has not been undertaken but some limitations of the direct method have been identified in [4]. As compared to tomography, which works with mixed states, the most significant limitation of the direct measurement of the wavefunction is that it has only been shown to work with pure states.Previous works have developed direct methods to measure quasi-probability distributions, such as the Wigner function [5], Husimi Q-function [6], and the GlauberSudarshan P-function [7]...

We r e p o r t on measurements of quantum electrodynamic processes in an intense electromagnetic wave, where nonlinear e ects (both multiphoton and vacuum polarization) are prominent. Nonlinear Compton scattering and electronpositron pair production have been observed in collisions of 46.6 GeV and 49.1 GeV electrons of the Final Focus Test Beam at SLAC with terawatt pulses of 1053 nm and 527 nm wavelengths from a Nd:glass laser. Peak laser intensities of 0:5 10 18 W/cm 2 have been achieved, corresponding to a value of 0:4 for the parameter = eErms=m!0c, and to a value of 0:25 for the parameter e = E ? rms =Ecrit = eE ? rms h=m 2 c 3 , where E ? rms is the rms electric eld strength of the laser in the electron rest frame. We p r e s e n t data on the scattered electron spectra arising from nonlinear Compton scattering with up to four photons absorbed from the eld. A convolved spectrum of the forward high energy photons is also given. The observed positron production rate depends on the fth power of the laser intensity, as expected for a process where ve photons are absorbed from the eld. The positrons are interpreted as arising from the collision of a high-energy Compton scattered photon with the laser beam. The results are found to be in agreement with theoretical predictions.