Beyond The Standard Quantum Limit

B. W. BarrInstitute for Gravitational Research

University of Glasgow

2

Standard Quantum Limit?

Heisenberg Uncertainty principal for position-momentum

Applying this minimum relation to interferometer mirror masses gives an uncertainty in differential position between the two arms of:

(min)

- signal frequency

- mass of mirror

- measurement bandwidth

gMf

L - arm length

2

22

)()(

x

hp

hpx

fM

hx

gdiff

22 4)(

fML

hh

gSQL

22

4

3

Alternative Approach

If all other noise sources are absent (laser amplitude and frequency noise, thermoelastic noise, etc…) the quantum measurement process dominates.

The quantity measured at the detector output is the light power. Noise on this signal is due to interaction of the light with the mirror masses and shot noise at the detector.

Shot Noise -

Radiation Pressure Noise -

Poisson statistics of photon countingPoisson statistics of force on the mirror masses due to photons

4

Optical Quantum Noise

Applying these statistics to the output signal of an interferometer an equivalent mirror displacement can be calculated and from this derive a limit to the measurable strain sensitivity.

These derivations are a simplified arrangement purely for the quantum noise and ignoring interferometer response.

An obvious balance is observed between the two noise sources in choice of input power. By increasing the input power the SN is reduced but the RPN is increased.

Also note that increasing the mass of the mirrors decreases both the RPN and the SQL.

fP

hc

Lhshot

02

1

fc

hP

LMh

gRP

0

2

2

5

Resulting Overall Noise

P increasing

Assuming the sensitivity signals are uncorrelated and add in quadrature the following plot is obtained:

SQL

Quantum

noise

shotRP hh Minimum found when

In this picture shot noise and radiation pressure are statistical properties of the light and uncorrelated.

6

Interferometric Scenario

Incident beam

The incident laser light is coherent and any noise associated with the laser is treated as correlated by definition when the interferometer is on a dark fringe.

Quantum noise on the output light is treated as vacuum fluctuations entering the interferometer from the dark port and mixing with the outgoing signal on detection.

Re(E)

Im(E)

7

Vacuum Fluctuations?

Re(E)

Im(E)

Light noise comes in two flavours: amplitude and phase.

tiA NeE

ti

tiN

NeiNn

enNE

)/1(

)( )(

Re(E) is the amplitude noise quadrature and Im(E) is the phase noise quadrature.

Thus radiation pressure is associated with the Re(E) quadrature and shot noise with the Im(E) quadrature.

It should be noted that gravitational wave signals are due to anti-correlated light at the beamsplitter and thus are measured on the Im(E) axis.

8

Interferometer Output Signals

Re(E)

Im(E)

As long as there are no correlations between shot noise and radiation pressure noise the light firmly enforces the SQL.

Squeezing: involves ‘reshaping’ the optical quantum noise.

vacuum state

amplitude squeezed state

phase squeezed state

9

Signal Recycling Effect

Originally, signal recycling was intended to beat the shot noise limit. By enhancing the signal response of the interferometer at a given frequency the shot noise curve dips to give improved sensitivity.

BUT, treating the interferometer as a quantum mechanical system some interesting effects become apparent… The signal exiting the

detector is reflected back. This produces correlations between the radiation pressure and shot noise causing a squeezing of the output signal.

10

Optical Springs?

Recent work in the field has (through a full quantum mechanical analysis) shown a signal recycled interferometer to be an ‘optical spring’.

The mirrors in the arms of the interferometer are connected by radiation pressure. The result is an opto-mechanical resonance structure on the output signal.The structure comes from two dynamic noise correlations: the resonance of a SR interferometer with fixed cavity mirrors and the free mass resonance. As laser power is increased these resonances become more and more coupled with the resulting response being a mixture of the two.

11

How Can We Use This?

The GW signal is measured along the Im(E) axis. But looking at the noise ellipse the noise is clearly less if measured off axis.

Measuring off the axis will reduce the maximum detectable signal. There will be an optimal angle of measurement for signal to noise.

Off axis the GW signal is a projection of the rotated measurement onto the Im(E) axis.

12

Obtaining An Output Signal

How can one measure a different quadrature?

The standard method of measurement is to use RF sidebands and beat them with the carrier (heterodyne). The disadvantage of this is to couple in the noise from both good and bad quadratures.

-+

One option is single RF sideband (or unbalanced sideband) operation. Another option is homodyne detection.

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Is That All?

On a superficial level, yes.

This is a well explored field in terms of theory and gives us many variables to play with: Choice of detection method, laser input power, demodulation phase (be it heterodyne or homodyne), SR mirror position…Experimentally the field is just about ripe for the picking.

14

Not Quite. What Else Is There?

Another possibility is the use of squeezed states in a constructive manner.We now know that quantum noise in an interferometer is due to vacuum fluctuations entering from the dark port. If instead squeezed states can be injected into the dark port then we can reshape the input noise as required.

Squeezing has been demonstrated at high frequencies (> MHz) – some way to go before it can be used as a practical solution.

15

The End

The work and abstract futuristic ideas presented here are the result of a lot of hard work by many talented individuals. I’d just like to thank them all.