Administrative details:

(1)

Administrative details:

• Anything from your side? • Paper presentations

(2)

Where do we stand?

• Optomechanics

• Interferometric position measurement • Photon shot noise

• Measurement imprecision

• Force sensing with a harmonic oscillator

• Thermal noise: the fluctuation dissipation theorem • Thermal limit of force sensing

• Measurement backaction

(3)

Power spectral density of photon shot noise

• Photon shot noise is white, so PSD is constant

oscilloscope

For perfectly uncorrelated detection events,

(4)

Measurement imprecision

Measurement imprecision:

• In limit of strong pump power: pump power drops out!

(5)

Measurement imprecision

Compare to localization problem in microscopy:

• In imaging measurement, each photon gave us position information with precision of ca. wavelength

detector

Source plane Imaging system

(6)

Measurement imprecision

Compare to localization problem in microscopy:

• In imaging measurement, each photon gave us position information with precision of ca. wavelength

(7)

Where do we stand?

• Optomechanics

• The imprecision backaction product and the Heisenberg limit • The standard quantum limit

(8)

Force sensing

• Why would you want to measure the position of a mirror? • To sense a force!

• If I can measure x(t), I can (in post-processing, taking the derivatives) get F(t) • In Fourier-space:

(9)

Force sensing

• If I can measure x(t), I can (in post-processing, taking the derivatives) get F(t) • In Fourier-space:

(10)

Force sensing

• Measuring x(t), we can (in post-processing, taking derivatives) get F(t) • In Fourier-space:

(11)

Force sensing

So far things look great!

It seems like we can sense any force, now matter how small, if we just probe the system strongly enough, such that

(12)

Where do we stand?

• Optomechanics

(13)

The fluctuation dissipation theorem

• Consider a harmonic oscillator in the absence of an external force

• What is the energy of the oscillator you expect after waiting long enough?

(14)

The fluctuation dissipation theorem

• What is the energy of the oscillator you expect after waiting long enough? • There must be a force at work to heat the oscillator to kT

Assume force are random “kicks” (fully uncorrelated):

(15)

The fluctuation dissipation theorem

• There must be an external force for the oscillator to carry an average energy kT

(16)

The fluctuation dissipation theorem

• Damping rate along gives rise to cooling

• Damping rate times temperature gives rise to heating • Differential equation for energy of system:

(17)

(18)

Thermal limit of force sensing

• Any oscillator coupled to a thermal bath shows thermal fluctuations (Brownian motion)

• Any signal force to be measured has to exceed the thermal force • What helps?

(19)

Thermal limit of force sensing

(20)

Example: Charge on levitated particle

(21)

Where do we stand?

• Optomechanics

(22)

Measurement backaction

• Each reflected photon transfers momentum to the mirror • DC effect: radiation pressure

• But what about the fluctuations?

• The discreteness of the photons gives rise to a fluctuating force • Poissonian statistics: white force power spectral density

(23)

Measurement backaction

• Each reflected photon transfers momentum to the mirror • DC effect: radiation pressure

• But what about the fluctuations?

• The discreteness of the photons gives rise to a fluctuating force • Poissonian statistics: white force power spectral density

(24)

The imprecision backaction product

• In the best case (we collect and optimally detect all light, system only interacts with measurement system):

(25)

The imprecision backaction product

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The imprecision backaction product

(27)

Radiation cooling

• See Homework 3

• Interaction with measurement heats oscillator

• Assume we had no intrinsic damping, would the oscillator heat to infinity?

(28)

Where do we stand?

• Optomechanics

(29)

The standard quantum limit

• For small measurement strength, measurement imprecision dominates (we simply don’t know much about the oscillator)

• For large measurement strength, measurement backaction dominates (the probe dominates the oscillator’s motion)

(30)

The standard quantum limit

• For small measurement strength, measurement imprecision dominates (we simply don’t know much about the oscillator)

• For large measurement strength, measurement backaction dominates (the probe dominates the oscillator’s motion)

(31)