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Quantum Limits to Precision Measurement

Quantum Limits of a Frequency Comb

The Steve Cundiff group began working in 2007 with theorist Curtis Menyuk of the University of Maryland, Baltimore County to explore the quantum limits of a frequency comb. Their original goal was to discover the fundamental limit on the width of a comb line (due to quantum mechanics in the gain medium) from a modelocked Ti:sapphire laser. This limit is similar to the Schawlow-Townes limit, which defines the quantum-limited linewidth of a continuous-wave laser. However, the nonlinear dynamics in modelocked lasers complicate the story. The theoretical calculation to answer this question was relatively straightforward, but it required as inputs parameters, such as the gain and nonlinear optical properties of the titanium sapphire crystal, that are hard to measure directly.

The Cundiff group designed a set of simple experiments to determine the needed laser parameters. As part of its efforts, the group developed a new technique for measuring timing and phase dynamics. The technique used frequency combs from two lasers, the first as a reference laser and the second for the study. The lasers were loosely locked, and a beat was generated by interfering the two combs. The researchers perturbed the second laser, and by measuring the beat at both ends of the comb, they were able to determine how much of the change they observed was due to timing and how much to phase dynamics.

By combining their experimental observations with Menyuk’s calculations, the researchers were able to determine that the frequency comb limit due to quantum noise is extremely small—a few mHz in the center of the laser spectrum. In other words, the phase of a comb line with respect to a perfectly stable source takes minutes to drift noticeably. This value is too small to be a limiting factor for current modelocked laser technology, though it could become more significant in the future as technology improves.

The Cundiff group also worked on a project to explore the quantum measurement limits of an optical atomic clock employing a frequency comb. The group used the results of this work to explore the possibility of predicting the intrinsic uncertainty of an optical atomic clock that uses a frequency comb to produce a radio-frequency (rf) output.

The group extended these methods to similar measurements of modelocked fiber lasers. The new work is challenging because the laser physics relevant to fiber lasers is very different than the laser physics of Ti:S modelocked lasers. For instance, the relevant time scale of the erbium atoms in a fiber laser is a thousand times slower than the titanium atoms in a Ti:S laser. As a result, the group and its collaborators have had to develop an entirely new method for observing the dynamics relevant to determining the quantum limits in a fiber laser.

Fiber lasers are also harder to investigate. For instance, optical fibers are slightly birefringent, which means that their index of refraction differs a little bit when light travels through them in different crystal directions. The way researchers coil the fiber on the laboratory bench may influence how the laser works. This sensitivity makes it harder to compare results from different laboratories. And, results observed in one experiment can disappear a few days later in the same lab if the fiber has been inadvertently disturbed.

Despite these difficulties, the group has found evidence of two stable states in which a fiber laser can operate. The researchers then worked on determining whether a laser can spontaneously switch between these two states and, if so, under what conditions.

Recently, the group performed an experimental study of the pulse behavior of a modelocked erbium fiber laser. Researchers injected a continuous-wave laser into the fiber laser as a way of modulating the fiber laser's gain. They then measured the response of the laser's pulse energy, central frequency, central pulse time, and phase to the gain modulation. These measurements allowed them to describe the coupling of these four laser characteristics and determine the comb linewidth and frequency uncertainty of the laser. The experiment also generated new ideas for optimizing fiber lasers.

Quantum Limits to Measuring Position

Because the Cindy Regal group is adept at using laser light to track the exact position of a tiny vibrating drum, it was able in 2012 to observe a limit imposed by the laws of quantum mechanics. This limit, imposed by the Heisenberg Uncertainty Principle, dictates that the closer an investigator comes to measuring the exact position of an object, the less that can be known about how fast the object is moving. It is a limit expected to come into play with the advanced Laser Interferometer Gravitational-Wave Observatory (LIGO), now undergoing testing and evaluation.

Advanced LIGO is expected to encounter a conundrum of how to balance the conflicting goals of precisely measuring position and velocity. This conundrum facing a major space U. S. observatory is one factor motivating the Regal group’s basic research into the quantum limits to measuring position.

In a recent experiment, researchers were able to measure the motion of the drum by sending light back and forth through it many times. However, during the measurement, 100 million photons from the laser beam also struck the drum at random, making the drum vibrate. This extra vibration obscured the motion of the drum at exactly the level predicted by the uncertainty principle.

This particular result indicated that the experimenters had reached a limit on successive measurements. It was no longer possible to both precisely measure the position of the drum and how fast it was vibrating at the same instant. Of course, how fast the drum was vibrating had lots to do with its exact position in the future. The experimenters in the Regal lab, like the LIGO scientists, faced the same conundrum: Do we make the best position measurement now or obscure the motion later?

The easiest way to get the best precision, according to Regal, is to give up precise knowledge of an initial position to balance the combined uncertainty in position and velocity. However, there are also fancy measurement techniques that, in theory, would allow researchers to work around, or avoid, the limits to successive measurement imposed by the uncertainty principle.

The challenge of working around quantum limits is irresistible for several physicists at JILA. It’s especially enticing for Regal, whose experimental system is running up against quantum mechanics even though it is large enough to be visible with the naked eye. The square drum at the heart of the experiment measures about 0.5 mm on a side—about half the length of the world’s smallest ant.

Two aspects of the 2012 experiment made it possible to observe the tiny vibrations due to quantum mechanical effects in such a “large” system. First, the experiment was done at the very low temperature of 5 K (-451 ºF). The low temperature reduced the amount of vibration caused by heating of the experiment by its surrounding environment. Second, the researchers had designed drums that only very slowly lose vibrational energy to the environment.

Because of these factors, they were able to determine that quantum mechanical fluctuations of light were responsible for about half of the measured vibrations. The group is now looking forward to investigating creative ways to work around the Heisenberg Uncertainty Principle.

Working Around Quantum Limits to Precision Measurement

The James Thompson group knows the secret for reducing quantum noise in a precision measurement of spins in a collection of a million atoms: Premeasure the quantum noise, then subtract it out at the end of the precision measurement. In so doing, it is essential not to do anything that detects and measures the spins of individual atoms in the ensemble. If states of individual atoms are measured, then those atoms stop being in superposition with the ensemble, and any subsequent precision measurements are ruined.

To avoid ruining a precision measurement, the Thompson group uses a nondemolition measurement that does not alter the quantum states of specific atoms. The group also makes sure that its technique preserves coherence, i.e., the quantum mechanical phase of each atom before being probed. It’s challenging to make a nondemolition measurement that also preserves coherence. However, in 2011 the group succeeded in making a precision coherence-preserving quantum nondemolition measurement of a million cold rubidium (87Rb) atoms inside an optical cavity.

What the group measured were signals from the energy levels of 87Rb atoms that were well localized between two mirrors in the cavity. There, the atoms interact, or “talk,” to the cavity resonances. This quantum conversation splits the cavity resonance in two—a phenomenon called Rabi splitting. The size of the frequency difference directly depends on the number of atoms in a spin-up state!

The researchers were able to accurately and precisely count the total number of atoms in spin-up and spin-down states. But, they weren’t counting individual atoms, and they had no way of identifying which atoms were in a particular spin state. Coherence was preserved in the experiment because it was designed to have all the atoms rapidly talk in unison to the cavity resonance.

The group discovered it could reduce quantum noise in their experiment with a longer atom-cavity conversation to measure cavity resonances. However, a longer conversation led to a reduced signal. Taking this tradeoff into account, the researchers were able to surpass the standard quantum limit on quantum phase estimation by a factor of 2.

The researchers are now working on improving their measurement technique with a goal of surpassing the same quantum limit by a factor of 10. A key factor in meeting this goal will be to ensure that even if laser photons interact badly with atoms in the cavity, the quantum states of the atoms won’t change and destroy the quantum superposition.

The group anticipates that the technique will become an important tool for quantum metrology and spur the development of more precise atomic sensors, magnetometers, rotation and inertial sensors, gravity meters, and atomic clocks.

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