## Abstract

Position and momentum were the first pair of conjugate observables explicitly used to illustrate the intricacy of quantum mechanics. We have extended position and momentum entanglement to bright optical beams. Applications in optical metrology and interferometry require the continuous measurement of laser beams, with the accuracy fundamentally limited by the uncertainty principle. Techniques based on spatial entanglement of the beams could overcome this limit, and high-quality entanglement is required. We report a value of 0.51 for inseparability and 0.62 for the Einstein-Podolsky-Rosen criterion, both normalized to a classical limit of 1. These results are a conclusive optical demonstration of macroscopic position and momentum quantum entanglement and also confirm that the resources for spatial multimode protocols are available.

Position and momentum are a fundamental example of conjugate quantum observables. Optical measurements of this conjugate pair are ubiquitous throughout many fields of research and across a broad range of scales. Applications in optics span from biology to astronomy, from the nanometer regime with atomic force microscopy (*1*) and optical tweezing (*2*) to the kilometer regime with free-space optical communication (*3*) and interferometry (*4*, *5*). These applications require continuous sampling of the data, although their accuracy is fundamentally limited by the quantum noise of these beams. Special squeezed beams, with the noise suppressed in one quadrature, have been used to improve the properties of many optical instruments (*6*), improving the signal-to-noise ratio for one selected observable.

Quantum entanglement, where two systems are quantum-correlated, can allow a near-perfect prediction from one system to the other and can enable new techniques for using information from one system to act on the other, thereby overcoming the limits set by quantum noise. In optics the systems are beams of light, with a single mode of electromagnetic radiation describing all their properties. Each beam can be represented by a single quantum operator. Such single-mode continuous wave (CW) optical beams have already been used to demonstrate strong entanglement between the amplitude and phase quadrature of pairs of beams (*7*–*9*) or the polarization of the beams (*10*–*13*). In combination with feed-forward control, these beams can be used for applications such as entanglement distillation (*14*) and teleportation (*15*).

However, laser beams have additional spatial properties that are described by higher-order spatial modes. A multimode description contains more information and allows the coding of more complex, multidimensional quantum information (*16*). The experimental techniques are direct and reliable because of the link between spatial properties and the well-known basis of Hermite-Gaussian transversely excited laser modes (TEM_{ij}, where *i* and *j* are the order numbers in the *x* and *y* directions). For a CW beam with the energy in a pure TEM_{00} mode (the reference beam) and detectable power of microwatts to milliwatts, the simplest spatial modulation is a displacement of the entire mode in the transverse directions *x* and *y*. For a periodic displacement at a frequency Ω, where the size of the displacement is much smaller than the diameter ω_{0} of the beam, all of the information about the displacement *X*(Ω) is contained in the real part (or amplitude quadrature) of the mode TEM_{10}, whereas the orthogonal displacement *Y*(Ω) appears in the real part of TEM_{01}. The accuracy of the measurement of the position of the beam is limited by quantum noise in the modes TEM_{10} and TEM_{01}. Previously, we have generated squeezed light in these higher-order modes (*17*). This technique of synthesizing is different from the idea of entangling images and complements the generation of intensity using four-wave mixing in an atomic vapor where a large number of modes can be entangled (*18*).

Concentrating on one transverse direction, the displacement *X* of the beam forms a conjugate pair of observables with the direction θ of the beam. The information for the direction is contained in the imaginary part (or phase quadrature) of the corresponding higher-order mode. This can be seen by the following expression for the electric field distribution *E*(*X*, θ) of a TEM_{00} mode that is both displaced by *X* and tilted by θ with the spatial information in the TEM_{10} mode (*19*) (1) where ω_{0} is the waist diameter, *i* is the imaginary unit, and λ is the wavelength of the beam. The direction of the laser beam θ corresponds to the transverse momentum of the photons in the beam. An improvement in the spatial accuracy beyond the quantum noise limit (QNL) for either position or direction measurement of a TEM_{00} beam has already been demonstrated (*6*).

We then investigated the spatial entanglement of a pair of beams (*20*). The existence of entanglement can be determined by measuring correlations between the displacements and directions of the two beams, respectively, to a level below the QNL. For beams that are coherent states, the spatial fluctuations of the two beams A and B are independent, and there are no correlations or anticorrelations. Thus, variance measurements for the sum *V*(*X*_{A} + *X*_{B}) and difference *V*(*X*_{A} – *X*_{B}) of the positions of the beams are both at the QNL. Similarly, the sum *V*(θ_{A} + θ_{B}) and difference *V*(θ_{A} – θ_{B}) of the directions of the beams are at the QNL. This is illustrated in Fig. 1, where the outer box shows the QNL for such independent beams. Inside the QNL box in this figure, the results are also shown for the entangled case. As well as measuring correlations between the spatial properties of the beams directly, we can infer the properties of B from a measurement of A, or vice versa, with an improved accuracy below the QNL. A measurement of these inferences allows us to demonstrate the Gedanken experiment discussed by Schrödinger and Einstein *et al*. (*21*), who proposed the link between the observables in two systems, and to quantify the extent of the entanglement.

The entangled beams are created by combining a spatially squeezed reference beam (TEM_{00} copropagating with squeezed TEM_{10}) with another squeezed beam (TEM_{10}) on a 50:50 beamsplitter. If both of the entangled beams were in a single spatial mode and the relative phase of the beams was locked to ϕ_{c} = π/2, we would have a conventional entangler (*22*). However, in our case the spatial information of the strong reference beam is now distributed into both output beams A and B, together with the quantum entanglement created by the two squeezed higher-order modes. Thus, the quantum entanglement can be measured by using two balanced homodyne detectors (HDs), each with carefully mode matched TEM_{10} modes (Fig. 2). By selecting the phases of the two local oscillator beams (ϕ_{LOA} and ϕ_{LOB}), we can choose to measure either the differential position or differential momentum of the two beams. We used optical parametric amplifiers (OPAs) as the generators of the squeezed light. In this experiment, the OPAs are linear resonators—each with one LiNbO_{3} nonlinear crystal—and operate with a pump wavelength of 532 nm and a seed of 1064 nm. The entire experiment is driven by one lownoise monolithic Nd-YAG (Nd–yttrium-aluminum-garnet) laser with a second harmonic generator, with all beams phase-locked to each other. The OPAs generate up to –3.8 dB of squeezing when tuned to TEM_{10} (–5 dB for TEM_{00}), and the technical noise level from the custom made photodetectors is –14 dB, compared to the QNL. To avoid technical noise from the laser at low frequencies and the various modulation frequencies that we need for the locking loops that maintain the correct beam conditions, we concentrated on measurements at Ω = 3 to 4 MHz.

Entanglement can be quantified in a variety of ways, each motivated by a specific physical property of the entangled beams. For entangled beams produced from beams with infinite squeezing and measured with perfect efficiency, all of these measures give a value of zero. For the practical case of imperfect entanglement due to the finite degree of squeezing available and the limited detection efficiency, the different measures give different values. A measure that is simple to implement is the degree of inseparability between the two beams. This can be quantified by the product of the normalized variances of the sum of the beam positions, *V*(*X*_{A} + *X*_{B}), and the difference of the beam directions, *V*(θ_{A} – θ_{B}). For a simple symmetric arrangement, the condition for inseparability is (2)

The central tower of Fig. 1 shows the actual experimental spectra for the two variances resulting in *I* = *V*(*X*_{A} + *X*_{B})*V*(θ_{A} – θ_{B}) for a scan from 3 to 4 MHz. It is directly apparent from Fig. 1 that both variances are below the QNL, so the directions are correlated and the positions are anti-correlated below the QNL. The two beams are spatially entangled, and we obtain a result of *I* = 0.51 ± 0.02 at 3.3 MHz. Not only is the product of the variances <1, but we also find that *V*(*X*_{A} + *X*_{B}) = 0.82 ± 0.03 and *V*(θ_{A} – θ_{B}) = 0.76 ± 0.01 independently, which is the requirement for a lowest-order cluster state of only two neighboring elements (*23*).

A more complex measure of entanglement is the evaluation of the Einstein-Podolsky-Rosen (EPR) criterion (*24*). This can be measured directly by evaluating the conditional variance of the measurements of *X*_{A} and θ_{A} from one beam and *X*_{B} and θ_{B} from the other beam, as was measured with the use of single photons in (*25*). The condition for entanglement is given by the conditional variance (3)

The direct detection of the conditional variance is not possible with an electronic spectrum analyzer, and the value of ϵ can be calculated with several different combinations of variances from measurements of the two beams (*20*). More details of this process are provided in the supporting online material (SOM). Figure 3 shows ϵ(Ω) plotted for a range of frequencies. We find strong EPR entanglement for a broad frequency interval, with a value of ϵ = 0.62 ± 0.03 at 3.3 MHz.

Practical systems suffer from asymmetries in the apparatus, where the losses on the two entangled beams are different, and in this case there are two different values for ϵ. For our experiment, the two values are given by *V*(*X*_{B}|*X*_{A})*V*(θ_{B}|θ_{A}) and *V*(*X*_{A}|*X*_{B})*V*(θ_{A}|θ_{B}). In the first case, we take a measurement of beam A, which has the fewest losses, and infer what we will measure on beam B. This is the result that has been quoted above, and more details are given in the SOM. The other case, where we take beam B with its higher losses, and attempt to infer beam A, gives us a value for ϵ of 0.94. The two different cases, with experimental data, are shown in Fig. 3. Finally, we must test the quality of the intensity mode of the entangled beams before we can claim to have spatial entanglement for our laser beams. The beam intensity must be dominantly in the TEM_{00} mode, as this is the mode for which we are measuring the position and momentum. Our entangled beam is close to being entirely TEM_{00} mode, with over 90% of the intensity being in the TEM_{00} mode.

We have entangled the position and momentum of laser beams by combining squeezed TEM_{10} with the original beam. By combining other higher-order squeezed modes TEM_{ij} with the reference beam TEM_{00}, we can entangle beams that carry other spatial information with reduced quantum fluctuations. Additionally, by using homodyne detection with a local oscillator in the mode TEM_{ij}, we can detect one specific spatial property of the reference TEM_{00} beams (*16*), or each mode independently. We have achieved a value for the EPR parameter that is a genuine proof of the entanglement of position and momentum of two laser beams.

This technology can be used as a resource for new quantum information applications, particularly those that require multimode entanglement. With this system of multiple modes we can create complex correlations within one beam. For example, we can start with several squeezed sources for TEM_{ij} and consider the creation of the equivalent of cluster states by correlating these modes using spatial techniques, in analogy to the creation of such states from several TEM_{00} beams (*26*). The spatial techniques may provide advantages in the simplicity of such systems. In addition, these complex quantum correlations can be sent in free-space communication in one beam or can potentially be stored as multimode spatial information (*27*).