## Quantum Measurements

Presently, our knowledge of the Universe comes from observing the light that is radiated by stars; from the spectacular light shows that accompany the explosive births and deaths of stars; and ancient light from the Big Bang itself. But is not be the only messenger to carry the secrets of the distant Universe to us. Many of these spectacular events also emit gravitational waves. Gravitational waves are ripples in the fabric of space and time that were predicted to exist by Einstein in 1916. They are very faint, however, and have never been directly observed. My work involves making instruments, based on very sensitive interferometry using lasers to sense the position of test masses that move in response to gravitational waves (see LIGO).

When a pristine laser beam is stripped of all technical noise (such as fluctuations or drifts in the power or the frequency), it still has quantum mechanical noise on it. The easiest way to think of noise on the laser light is in terms of the Heisenberg Uncertainty Principle: amplitude and phase are complementary variables, so uncertainty in amplitude (DA) and in phase (Df) must satisfy (DA).(Df) > 1. For quantum-limited laser light with minimum uncertainty, DA = Df and the light is in a coherent state. In the amplitude-phase plane, the coherent state is best represented as a vector (stick) whose length and phase angle correspond to the classical amplitude and phase, respectively, while the ball at the end of the stick corresponds to the uncertainty.

The consequences of this uncertainty in amplitude and phase have a profound effect on our ability to use light to measure the position of a particle (mirrors of an interferometer, e.g.). Two effects are important.

1. Radiation-pressure noise: We know that light can impart momentum when it reflects of a mirror, and that momentum is proportional to the intensity of the light beam. Since the quantum uncertainty leads to fluctuations in the amplitude of the light, the momentum – and hence the force — acting on the mirror will also fluctuate.  This limits our ability to measure the position of the mirror, since our measurement itself disturbs the mirror position – an effect known as back-action.
2. Shot noise: The mirror position is inferred from phase shifts of the laser light traveling in two arms of the interferometer. The uncertainty in the phase of the light due to the quantum fluctuations then directly limits our ability to measure the mirror position.

So even quantum-limited laser light limits our ability to measure the position of the mirror: in the first case, the amplitude fluctuations kick the mirror around; in the second case, phase fluctuations make the light an inaccurate meter.

There’s an even more interesting effect hidden in all this: even if no light were impinging on the mirror, it would still be buffeted around due to vacuum fluctuations. Only in this case, we wouldn’t care, since without laser light we couldn’t measure the position of the mirror anyway.

One way to get around this problem is to create more exotic states of light, sometimes call a “squeezed” state. The Heisenberg Uncertainty Principle requires that area of the ball be unity, but it doesn’t say the ball can’t be turned into an ellipse of unit area. This leads to the concept of squeezing:  we can reduce the fluctuations in one quadrature at the expense of increasing them in the orthogonal quadrature.

Why is this useful? Well, say we squeeze the noise in the phase quadrature, then our ability to measure the phase shift of the light gets better (same “signal” but less “noise”). But wait, doesn’t that make the amplitude fluctuations larger and then doesn’t the back-action noise get worse? Yes, that’s true. So this trick only works if the mirrors are very heavy and aren’t easily kicked by the light, or if the light power is not too large, so the radiation pressure force is not too big.

This leads to a quantity called the standard quantum limit. Suppose we try to measure the position of a particle of mass M with power I0 circulating in our interferometer. If the radiation pressure noise (which is proportional to I0/(M W2)) is not correlated with the shot noise (which is proportional to1/I0), then we reach the standard quantum limit (SQL) when the radiation pressure noise is equal to the shot noise. In general we have a few knobs to turn in making a measurement at the SQL: The mass of the particle, the laser light power, and the frequency at which we make the measurement.

If we wish to make a more precise measurement than that allowed by the SQL (or the minimum uncertainty state), we must correlate the amplitude and phase quadratures.

There are a number of ways to create these correlations. We are two very different methods for generating squeezed states:

1. Squeezing using non-linear optical crystals
2. Squeezing using radiation pressure effects in interferometers

The radiation pressure force can also be used to alter the dynamics of movable mirrors, using the optical spring effect. When an optical cavity is detuned from resonance, the radiation pressure force becomes linearly dependent on the power in the cavity, resulting in a spring-like restoring force; we call this an optical spring.  Since the light bounces multiple times between the cavity mirrors before escaping the cavity, the light in an optical cavity does not respond instantaneously to mirror motion. This time delay gives rise to a velocity-dependent viscous damping force; we call this optical damping. These forces can be used to optically cool and trap mirrors, analogous to cooling and trapping of atoms. As in the case of atoms, this technique could be used to remove enough energy from a mechanical oscillator that its behavior would have to be governed by the laws of quantum mechanics. Mind-boggling, that a human-scale object like a mirror could be made to behave like a quantum particle. Our group also works on this, using mirrors of all sizes, from 250 nanograms to 1 gram to 10 kilograms.