Entangled light beams from four-wave mixing carry spatial information

Quantum entanglement is often associated with spin, polarization, or other discrete-valued measurements. But in the 1935 paper in which Albert Einstein, Boris Podolsky, and Nathan Rosen proposed what has come to be known as the EPR paradox, the system they considered involved the entanglement of two continuous-valued quantities, position and momentum. ¹ In 1992 researchers from Caltech presented an experimental realization of the EPR paradox for continuous variables. ² They used beams of light, and the entanglement was displayed in the fluctuations of the optical electric fields.

Now, Vincent Boyer, Alberto Marino, Raphael Pooser, and Paul Lett of the Joint Quantum Institute (JQI; a partnership between the University of Maryland, College Park, and NIST in Gaithersburg, Maryland) have created what they call entangled images, pairs of entangled beams whose cross sections can be represented as assemblages of many independently entangled pixels. ³ And the researchers’ method for producing the beams is no more involved than the production of entangled beams that lack such spatial detail.

Entangled light

Two quantum mechanical particles can have perfectly correlated positions and perfectly correlated momenta, so that from a measurement of the first particle’s position one can infer the second particle’s position exactly, and likewise for the momenta. Einstein, Podolsky, and Rosen argued that either the measurement of the first particle affects the quantum state of the second particle (the phenomenon Einstein famously termed “spooky action at a distance”) or the second particle was in a state of exact position and exact momentum all along. In the latter case, since quantum mechanics doesn’t allow a particle to simultaneously have an exact position and an exact momentum, there would have to be more to the physical reality of the system than quantum mechanics can describe.

Like the position and momentum of a particle, the amplitude and phase of a sinusoidal wave are subject to a minimum uncertainty constraint in quantum mechanics. But constructing a quantum mechanical phase operator is difficult, and researchers in quantum optics instead describe a sinusoidal wave of arbitrary phase as X times a cosine wave plus Y times a sine wave, where X and Y are the quadratures of the wave. When the quadratures are written as quantum mechanical operators, their commutator is a nonzero constant, so the Heisenberg uncertainty principle places a nonzero lower bound on the product of their uncertainties.

It’s possible for two light beams born of the same process to have highly correlated quadratures. In that case a measurement of X ₁ allows one to infer X ₂ with a very small uncertainty, and likewise for Y ₁ and Y ₂. If the product of those two uncertainties is less than the bound given by the uncertainty principle, then the waves demonstrate what’s known as EPR entanglement: They constitute evidence that either the measurement of wave 1 affects the quantum state of wave 2, or wave 2 was always in the state of reduced uncertainty and the uncertainty principle is wrong.

Thanks to Bell’s theorem and subsequent experiments, we have good reason to believe that the uncertainty principle isn’t wrong. That confidence has led researchers to develop a less stringent criterion for entanglement, known as inseparability. Two beams are inseparable if no product of two spatially localized quantum states—each of which is consistent with the uncertainty principle—can reproduce their degree of correlation. It’s a weaker criterion than EPR entanglement because it takes the uncertainty principle as a given, which the EPR argument does not.

Entangling light

Entangled light beams are produced by nonlinear optical media, materials that respond nonlinearly to light’s electric field. Up until now, the technique of choice has been parametric down-conversion, the process that occurs in an optical parametric oscillator. In an OPO, a crystal of nonlinear material converts photons from an input beam into pairs of lower-frequency photons, which emerge in two output beams. Because the photons are necessarily produced in pairs, the quantum fluctuations of the output beams are correlated: An anomalous burst of intensity in one beam is accompanied by a similar burst in the other. Parametric down-conversion occurs only weakly except at high optical powers, so an OPO houses the nonlinear crystal within an optical cavity, which improves the conversion efficiency.

The JQI researchers produced their beams using a different technique called four-wave mixing. In the JQI variant, an input beam, or pump, is passed through a cell of rubidium vapor. Two photons from the pump beam are converted into two output photons, the probe and the conjugate, via the scheme shown in figure 1. The output photons are again produced in pairs, and the resulting beams are entangled.

The researchers didn’t set out to study the spatial properties of their beams. Rather, they were interested in producing entangled beams with the right frequencies to interact with atomic systems. By using an atomic gas instead of a crystal to produce the beams, they could achieve the desired frequencies more easily.

But then they noticed something. When they stimulated the process by inputting a probe beam with a complicated spatial profile (such as the one shown in figure 2), both the output probe beam and the conjugate beam took on that profile. That was a clue that something interesting was going on: If the shapes of the output beams retained spatial information from the mixing process, then maybe their fluctuations did also.

Detection

Producing entangled beams by four-wave mixing is straightforward. The challenge is in the detection: proving that the beams are entangled and studying the spatial character of the entanglement. For that, the JQI researchers used a technique called homodyne detection, as shown in figure 3. They generated bright beams called local oscillators at the same frequencies as the probe and conjugate beams. Then they mixed each beam with the corresponding local oscillator using a beamsplitter. Photodiodes, detecting the resulting beams, produced an electrical signal proportional to one quadrature of the probe or conjugate beam, averaged over the local oscillator’s cross section. The quadrature that was measured could be selected by using a piezoelectric crystal to tune the phase of the local oscillator. From the correlations in the quadrature fluctuations, the team found EPR entanglement between the probe and conjugate beams.

The spatial properties of the beams’ quadrature fluctuations were probed with local oscillators of different cross-sectional shapes. By using T-shaped local oscillators instead of round ones, the JQI researchers measured the fluctuations of the T-shaped subsets of the probe and conjugate beams. Like the full beams, the subsets were EPR entangled, with correlations nearly as large as those of the full beams.

There’s nothing special about the T shape. The researchers created it by passing a laser beam through a T-shaped hole in a piece of metal; other shapes they tried gave similar results. They concluded that any subset of the probe beam would be entangled with the corresponding subset of the conjugate beam—that the beams were, in fact, entangled point by point. The spatial resolution of the entanglement depends on the length of the four-wave-mixing cell: The shorter the cell, the finer the resolution. The group’s Rb cell produces probe and conjugate beams that can be broken down into about 100 independently entangled pixels.

Such multichannel entanglement is difficult to create with on OPO because of the limitations imposed by the OPO’s optical cavity. The input beam, after a full trip around the cavity, must match up with itself exactly. That criterion restricts not only the frequency of the light but also its spatial cross section, so T-shaped subsets of the beams from an OPO would display greatly reduced entanglement or none at all.

Applications

Entangled light beams offer the ability to beat the quantum noise limit, set by the uncertainty principle, in measurements of intensity. The fluctuations of one beam can be subtracted from the measured intensity of the other, thereby eliminating most of the quantum component of the noise. Similarly, in measurements of the intensity of some spatial subset of a beam, it is likely that point-by-point entangled beams will be useful.

For example, in one commonly used technique for measuring the position of a laser beam, the beam is directed at a two-part photodiode, and each part measures the intensity of half the beam. When the measured intensities are equal, the center of the beam must lie on the dividing line between the two parts. Quantum intensity fluctuations of each half of the beam limit the precision of each photodiode measurement and thus the precision of the beam position. In 2003 an Australian–French collaboration presented a way of beating the quantum beam-positioning limit with a so-called “quantum laser pointer.” ⁴ But those researchers created their beam by mixing two different spatial modes from two optical parametric amplifiers, close cousins of the OPO. Four-wave mixing could achieve the same end more easily.

Other applications may be found in the field of quantum information—that is, the use of quantum states for computation and communication. Quantum information protocols using discrete-variable systems may be more familiar (see, for example, the article by Andrew M. Steane and Wim van Dam, Physics Today, February 2000, page 35 ), but continuous-variable protocols have been widely considered as well. ⁵ And for that, the JQI group’s technique may prove advantageous. It produces, in effect, 100 independent sets of entangled quantum fluctuations as easily as an OPO can produce just one.

But the problem, once again, is in the detection. So far, the JQI researchers have looked at the entangled modes one at a time, but a quantum information application would require that different modes be manipulated and detected in parallel. The technology exists to separate them, but implementing it is a challenge.