# Simultaneously measuring position and momentum of optical fields

Efficient quantum measurement is critical for enabling and scaling quantum technologies (e.g., precision metrology, quantum cryptography, and quantum computing). The peculiar and non-intuitive laws of quantum mechanics, however, can also make these measurement techniques challenging. A particular quantum feature is the measurement disturbance that is represented by the uncertainty principle, i.e., a precise measurement of a quantum particle's position causes a disturbance that makes the subsequent measurement of its momentum unpredictable. These so-called projective measurements are commonly thought to ‘collapse’ the quantum wave function.

In recent years, researchers have begun to investigate more gentle strategies for interrogating quantum systems, which leave them less disturbed. For example, weak measurement schemes can be used to dramatically reduce the interaction between a system and a detector.^{1} Such schemes can thus allow an effective peak inside ‘Schrödinger's box.’ For these and other more general measurements, the uncertainty principle can be recast in terms of information, i.e., the more information a measurement provides about position, the less information it can provide about momentum.^{2} Critically, however, this principle does not specify how the system is disturbed. It should therefore be possible to extract the most important information about a quantum ensemble, while limiting the disturbance to a tolerable type and level.

We have designed an experimental setup to measure both position and momentum distributions. The system of interest in our work is the transverse optical field at an object mask, which we illuminate with a collimated laser (see Figure 1). The position distribution is measured by determining the probability of a photon arriving from a certain location on the mask. In a classical sense, this would involve obtaining an image of the mask. In contrast, the momentum distribution is measured by determining the probability of a photon traveling in a specific direction away from the mask. In a classical sense, this would be represented by the mask's diffraction pattern. With our setup, we randomly filter the position of our system and we then make a traditional projective measurement of the system's momentum.^{3}

We use a 4F (i.e., four focal lengths long) imaging system to reproduce an image of the mask, at which point we filter the light with a sequence of random binary filters. Each pixel in a filter either fully transmits or blocks the light. As such, the filters have minimal impact on the momentum distribution of the photons and only a partial—rather than a full—collapse of the quantum wave function is created. Our random filtering technique encodes a small amount of information about position onto the total intensity. A small amount of noise is also introduced into the momentum distribution through this process. The effect of our filtering technique on a triple slit object mask is shown in Figure 2. The diffraction patterns of the filtered and unfiltered objects differ only by a small noise floor. The filtered momentum distribution, however, remains sharp to an arbitrary resolution. We can therefore directly image the object (at point *Z*_{k} in Figure 1) by placing a camera in the focal plane of a Fourier-transform lens.

In our system, we note that the amount of light passing through each filter is a measure of the correlation between the object and the filter. We can therefore take a weighted average over many filters to recover the true position distribution. Furthermore, we can substantially reduce the required number of filters with the use of a compressive sensing^{4} optimization technique. Compressive sensing approaches—in which a signal is compressed during measurement—rely on prior information about the possible structures of the signal.

We have formulated a linear model for our measurements where a vector (with *m* dimensions) of the measurements is equal to the product of a sensing matrix (the rows correspond to our filters) and the object of interest (with *n* dimensions). We use far fewer random filters than pixels (i.e., *m* is much less than *n*) and there are many possible signals (*X*) that are consistent with our measurements. According to the compressive sensing approach, the value of *X* that is most compressible by an appropriate scheme is the correct signal. To find *X*, we therefore must solve an optimization problem. The results of our experiments for several object masks are shown in Figure 3. In all these cases we were able to recover high-fidelity position and momentum distributions at 256×256 pixel resolution. For the double slit, triple slit, and ℏ masks, we used only 6533 filters, which is much fewer than the 65,536 position measurements that would otherwise have been required.

We have designed and tested an experimental setup that can be used to simultaneously measure the position and momentum distributions of an optical field. With our approach, we use standard quantum mechanics and do not violate any uncertainty relations. In our technique (based on compressive sensing), we simply economize the use of information. As these concepts become more commonplace in quantum measurement, we expect many new innovations and discoveries to be made. We are currently researching extensions to our work. We hope to measure the position and momentum correlations in pairs of optical fields simultaneously. This will allow us to witness quantum entanglement more efficiently.

*We gratefully acknowledge support from the Air Force Office of Scientific Research (grant FA9550-13-1-0019) and the Defense Advanced Research Projects Agency (Defense Sciences Office Information in a Photon program grant W911NF-10-1-0404).*

Gregory Howland received his PhD from the University of Rochester in 2014 and is currently a postdoctoral fellow at the Air Force Research Laboratory. He specializes in continuous variable entanglement and compressive sensing at the single-photon level.

James Schneeloch is a PhD candidate in the Howell Research Group at the University of Rochester. His research interests include quantum optics and quantum information associated with optical experiments.

*Rev. Mod. Phys.*86, p. 307, 2014.

*Phys. Rev. Lett.*74, p. 3307-3311, 1995.

*Phys. Rev. Lett.*112, p. 253602, 2014. doi:10.1103/PhysRevLett.112.253602

*IEEE Signal Process. Mag.*24, p. 118-121, 2007. doi:10.1109/MSP.2007.4286571