The Holometer: A Measurement of Planck-Scale Quantum Geometry
Stephan Meyer
November 3, 2014
1 The problem with geometry
Classical geometry is made of definite points and is based on “locality.”
Relativity is consistent with this point of view but makes geometry “dynamic” - reacts to masses.
Quantum physics holds that nothing happens at a definite time or place. All measurements are quantum and all known measurements follow quantum physics.
Anything that is “real” must be measurable. How can space-time be the answer?
Stephan Meyer SPS - November 3, 2014 2 Whats the problem?
Start with - Black Holes
What is the idea for black holes? - for a massive object there is a surface where the escape velocity is the speed of light.
Since nothing can travel faster than the speed of light, things inside this radius are lost. We can use Phys131 to figure this out
Stephan Meyer SPS - November 3, 2014 3 If r1 is ∞, then
To get the escape velocity, we should set the initial kinetic energy equal to the potential energy
and set the velocity equal to the speed of light. Solving for the radius, we get
Stephan Meyer SPS - November 3, 2014 4 having an object closer than r to a mass m, means it is lost to the world.
This is the definition of the Schwarzshild radius of a black hole.
So for stuff we can do physics with we need:
Stephan Meyer SPS - November 3, 2014 5 A second thing:
Heisenberg uncertainty principle: an object cannot have its position and momentum uncertainty be arbitrarily small
This can be manipulated, using the definition of p and E to be
What we mean is that to squeeze something to a size λ, we need to put in at least energy E.
Stephan Meyer SPS - November 3, 2014 6 Setting the these two physics limits equal, λ= r gives
and we can turn this into a time:
Stephan Meyer SPS - November 3, 2014 7 The “Planck Scale”
Max Planck was the first person to put these numbers together in this way.
the Planck time is
the Planck length is
This is much too small to measure directly.
Stephan Meyer SPS - November 3, 2014 8 What does it mean?
Quantum Particle Black Hole log (size) log
log (mass-energy) Planck length 10-35 m
Stephan Meyer SPS - November 3, 2014 9 Dynamical geometry must be indeterminate at the Planck scale
Quantum Particle Black Hole
Forbidden by Quantum Mechanics (size) log
Forbidden by Gravity
log (mass-energy) Planck length 10-35 m Forbidden by both QM and Gravity
Stephan Meyer SPS - November 3, 2014 10 Classical Physics Quantum Field Theory log (size) log
General Relativity
log (mass-energy) Planck length 10-35 m String Theory
Stephan Meyer SPS - November 3, 2014 11 So how do we deal with this?
Classical Geometry - “space-time” Dynamical but not quantum Responds to particles and fields Field Theory Quantum particles and fields play on the stage of classical geometry
This approximation explains all experiments with particles.
But cannot be the whole story.
Stephan Meyer SPS - November 3, 2014 12 But problems remain in Quantum Field Theory
Quantum states do not obey locality Proven by EPR-type experiments Nothing happens a definite time or place Yet locality is the basis of relativity, assumed by field theory
Classical space-time is emergent At the Planck scale, dynamical space-time is indeterminate Field theory cannot predict energy of the vacuum Quantum properties of macroscopic geometry are assumed away
Gravity is thermodynamical Black hole theory suggests that gravity is a statistical behavior Metric does not describe fundamental degrees of freedom
Actual physical states are holographic Information encoded with Planck density on 2D bounding surfaces States must have new forms of spatially nonlocal entanglement
Somehow, physics needs to go beyond the standard approximations of quantum field theory
Stephan Meyer SPS - November 3, 2014 13 Classical geometry is an approximation to a quantum system
Classical Physics Quantum Field Theory log (size) log
General Relativity
log (mass-energy) Planck length 10-35 m String Theory
Is there quantum behavior of nearly classical macroscopic geometry?
Stephan Meyer SPS - November 3, 2014 14 possibly...
In the `70s, Bekenstein and others were thinking about black holes and entropy. It would be nice to not have relativity “break” thermodynamics.
Inspired by thought experiments, it was realized there was a minimum entropy a black hole could have and that was proportional to its surface area.
More surprising, it was later shown that the constant of proportionality was the square of the Planck length.
Stephan Meyer SPS - November 3, 2014 15 Consequently, information is encoded with Planck density on 2D bounding surfaces
“This is what we found out about Nature’s book keeping system: the data can be written onto a surface, and the pen with which the data are written has a finite size.” - Gerard ‘t Hooft 1 bit of information This is the holographic principle. on every 0.724 x 10-65 cm2
on top of that comes the bound of information There are many fewer “The shocking thing is not the degrees of freedom holography, but rather the bandwidth limit.” - Raphael Bousso in physical systems than it appears.
Stephan Meyer SPS - November 3, 2014 16 How can we measure something?
Can we imagine an observable of the reduced number of degrees of freedom?
Hogan suggested that we could use diffraction to amplify the Planck-scale fuzziness.
Information is conveyed by null-spheres expanding and contracting at the speed of light.
Stephan Meyer SPS - November 3, 2014 17 Amplified Planck-scale fuzziness
Suppose the transverse space-time coordinate information is encoded at a particular plane on a grid with spacing lP .
The resulting diffraction over the later propigation, amplifies Planck-scale effects. The position uncertainty grows as
Stephan Meyer SPS - November 3, 2014 18 The Holometer Group • Fermilab A. Chou (co-PI), H. Glass, C. Hogan (Project Scientist),C. Stoughton, R. Tomlin • U Chicago B. Lanza, L. McCuller, S. Meyer (co-PI), J. Richardson • Vanderbilt B. Kamai • MIT M. Evans, S. Waldman, R. Weiss • U. Michigan D. Gustafson • Northwestern J. Steffen
And a number of undergraduate researchers from Chicago and REU programs.
Stephan Meyer SPS - November 3, 2014 19 Interferometer end mirror Introduction
L1
L2 SP Laser end mirror
ASP
20 The power in the ASP The power is
where the angle brackets are time averages and the “delay” is
Stephan Meyer SPS - November 3, 2014 21 In operation, the interferometer is held a small distance from exactly null
then changes in delay make changes in the intensity on the photodiode depending only on the slope of the curve at the operating point.
In this way we measure the change of differential arm length
Stephan Meyer SPS - November 3, 2014 22 end mirror
L1
L2 SP Laser end mirror
beam splitter
ASP
23 end mirror
Power Recycling
L1
Power recycling mirror L2
Laser end mirror
beam splitter
In addition we put in a power recycling mirror on the SP.
It is possible to build up the power in the interferometer to larger than the laser power 24 Holographic signal in a Michelson Interferometer
The beam comes in from the left, samples the beamsplitter position and goes down the arms and back.
We can think of the upward arm as making a time measurement.
During that time the transverse uncertainty L of the up/down position of the beamsplitter grows. When the two beams return this uncertainty results in a variation of relative phase of the two return beams.
On return the position of the beam on the beamsplitter is uncertain by where for a 40 meter armlength, L, and λoptical is the wavelength of the laser light, 1.064 μm.
Stephan Meyer SPS - November 3, 2014 25 Conventional noise in an Interferometer
The conventional quantum noise in an interferometer is “shot noise,” the minimum noise from a “rain” of discrete photons,
also, the Heisenberg uncertainty principal for a coherent state in an interferometer arm is
so the phase spectral density from shot noise for a 1kW beam is
Stephan Meyer SPS - November 3, 2014 26 The need for correlation
The ratio of holometric signal to shot noise is
so its a tiny signal even with 40 m arms and 1 kW in the interferometer. This can be overcome by using two interferometers.
The holographic signal is correlated between the two instruments while the shot noise is uncorrelated. By sampling the noise many times, we can reduce the uncorrelated part:
Stephan Meyer SPS - November 3, 2014 27 We can find the required N, the number of observations, needed to make the correlated part equal to the uncorrelated part
which is a large number.
Fortunately, each measurement is independent of the next after one light -7 passing time, τsample = 3 x 10 sec. The total time needed to get to a signal-to-noise-ratio of 1 is
Stephan Meyer SPS - November 3, 2014 28 Optics block diagram
The optics are actively locked to the laser and the differential length is adjusted by moving the interferometer end mirrors to keep the system “locked.”
Stephan Meyer SPS - November 3, 2014 29 End Mirror The pair of interferometers End Mirror 5 mW (arms made 2 m long)
Power recycling mirror 1 kW
100 mW 1 W Vacuum tubing Laser 1 Photo Diode
Laser 2
Beam Splitters
Arm for null test
30 31 32 EndEnd station station optics optics and vibration vibration isolation isolation
PZT (1 of 3) end mirror T ~ 1 PPM transmission optics
23 kg mass
viton ball (1 of 3)
Stephan Meyer SPS - November 3, 2014 The Holometer is located at FermiLab in an old meson beamline
Stephan Meyer SPS - November 3, 2014 34 Center stations of the Holometer
35 36 37 38 The signal
The expected cross correlation signal will be
Ξ(τ)
τ
The zero lag signal is the total variation. When the lag is longer than the round- trip travel time in the interferometer, the correlation is zero.
Stephan Meyer SPS - November 3, 2014 39 The spectral density
uncorrelated noise after 100 hours of integration The power spectral density describes how much fluctuation power is at each frequency. It is the Fourier Transform of the correlation function shown on the previous page.
The spectral noise density shape is dictated by the transfer function of the instrument. Seeing this shape, including nulls at frequencies, f=c/2L, will be an important indicator that the signal is understood.
The spectrum from the instrument will be this curve plus the uncorrelated noise that is reduced with integration time.
Stephan Meyer SPS - November 3, 2014 40 How do we know if it is holographic noise?
• The holographic noise has a predicted spectral shape - The normalization of the spectrum scales as L2. - The interferometer response function has a known shape and cuts off at f=c/2L. • Convention RF backgrounds are usually frequency dependent - narrow lines, 1/f... - This gives us a way to discriminate against conventional backgrounds such as radio stations. • Experimental tests: - Move the two interferometers to operate in a null configuration.
Stephan Meyer SPS - November 3, 2014 41 Power on output photodiode
The antisymmetric (AS) port output vs differential phase in the arms (no power recycling)
Optical gain vs phase. Gain is high when no light is lost at AS port.
AS port power vs differential phase with power recycling on.
Stephan Meyer SPS - November 3, 2014 42 Holding the lock point
• The lock point is 3 nm from the null. A servo system send commands to the end mirror positions to move them if the light on the AS port photodiode changes from the setpoint.
• This keeps the differential arm lengths locked (at low frequencies). • At high frequencies (where the holographic signal is) the interferometer arm length is free to move. The photodiode signal is recorded and analyzed.
ΔΦ (radians) -1 0 1
Stephan Meyer SPS - November 3, 2014 43 Actual length spectrum of Holometer
• This is an early differential arm spectrum taken without mechanical isolation. Floor motion is strongly coupled through mount resonances.
• The control loop does not have enough gain at f < 100Hz to compensate for these motions,
• The large, low frequency motions prevent us from getting a stable lock.
Stephan Meyer SPS - November 3, 2014 44