Weak Value Amplification

via Sagnac Interferometer

Justin Dressel

Institute for Quantum Studies

Schmid College of Science and Technology

Sagnac Interferometer

Prototype experiment: Howell lab, Rochester

PRL 102, 173601 (2009)

Ultra-sensitive to beam deflection: ~560 femto-radians of tilt detected

\phi\;\text{(phase offset)}
ϕ(phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k(tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x(signal)x\;\text{(signal)}
\sigma\;\text{(beam waist)}
σ(beam waist)\sigma\;\text{(beam waist)}

How is this possible?

  • Split-detector placed at the dark port of the Sagnac interferometer
     
  • Clockwise and counter-clockwise beams have phase fronts that tilt in opposite directions due to the piezo
     
  • The mismatch in phase fronts yields a spatially modulated signal from imperfect cancellation at the dark port
     
  • Slightly misaligning the phase cancellation (using an SBC) amplifies the spatial translation of the output signal, while remaining within the original beam waist 
\phi\;\text{(phase offset)}
ϕ(phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k(tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x(signal)x\;\text{(signal)}
x \propto k/\phi \quad (k\sigma \ll \phi \ll 1)
xk/ϕ(kσϕ1)x \propto k/\phi \quad (k\sigma \ll \phi \ll 1)
\sigma\;\text{(beam waist)}
σ(beam waist)\sigma\;\text{(beam waist)}

(Small tilt is amplified in the signal by the small phase offset from a perfect dark port)

I \propto \phi^2
Iϕ2I \propto \phi^2

Penalty: output intensity strongly attenuated

Weak Value Analysis

|i\rangle \propto i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle
ii+|i\rangle \propto i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle
|f\rangle \propto e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle
feiϕ/2ieiϕ/2|f\rangle \propto e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle
|\psi, i\rangle \to \langle p, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle \approx \langle f | i \rangle \langle p - k W_w | \psi \rangle
ψ,ip,fexp(ikx^W^)ψ,ifipkWwψ|\psi, i\rangle \to \langle p, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle \approx \langle f | i \rangle \langle p - k W_w | \psi \rangle
\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft | - |\!\circlearrowright\rangle\langle\!\circlearrowright |
W^=\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft | - |\!\circlearrowright\rangle\langle\!\circlearrowright |
W_w = \frac{\langle f |\hat{W}|i\rangle}{\langle f | i \rangle} = i\cot\frac{\phi}{2} \approx \frac{2i}{\phi}
Ww=fW^ifi=icotϕ22iϕW_w = \frac{\langle f |\hat{W}|i\rangle}{\langle f | i \rangle} = i\cot\frac{\phi}{2} \approx \frac{2i}{\phi}
\phi\;\text{(phase offset)}
ϕ(phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k(tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x(signal)x\;\text{(signal)}
\sigma\;\text{(beam waist)}
σ(beam waist)\sigma\;\text{(beam waist)}

Angular tilt (transverse momentum) amplified by large weak value.

Collimated Analysis

|i\rangle = \frac{i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle}{\sqrt{2}}
i=i+2|i\rangle = \frac{i|\!\circlearrowleft\rangle + |\!\circlearrowright\rangle}{\sqrt{2}}
|f\rangle = \frac{e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle}{\sqrt{2}}
f=eiϕ/2ieiϕ/22|f\rangle = \frac{e^{i\phi/2}|\!\circlearrowleft\rangle - ie^{-i\phi/2}|\!\circlearrowright\rangle}{\sqrt{2}}
|\psi, i\rangle \to \langle x, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle = \langle x|\hat{M}_f\,|\psi\rangle = i\sin(\phi/2 - kx)\langle x | \psi\rangle
ψ,ix,fexp(ikx^W^)ψ,i=xM^fψ=isin(ϕ/2kx)xψ|\psi, i\rangle \to \langle x, f | \exp(-i k \hat{x}\hat{W}) | \psi, i \rangle = \langle x|\hat{M}_f\,|\psi\rangle = i\sin(\phi/2 - kx)\langle x | \psi\rangle
\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft\! | - |\!\circlearrowright\rangle\langle\!\circlearrowright\! |
W^=\hat{W} = |\!\circlearrowleft\rangle\langle\!\circlearrowleft\! | - |\!\circlearrowright\rangle\langle\!\circlearrowright\! |
\phi\;\text{(phase offset)}
ϕ(phase offset)\phi\;\text{(phase offset)}
k\;\text{(tilt)}
k(tilt)k\;\text{(tilt)}
x\;\text{(signal)}
x(signal)x\;\text{(signal)}
\sigma\;\text{(beam waist)}
σ(beam waist)\sigma\;\text{(beam waist)}
\hat{M}_f = \langle f | \exp(-i k \hat{x}\hat{W}) | i \rangle = i \sin(\phi/2 - k\hat{x})
M^f=fexp(ikx^W^)i=isin(ϕ/2kx^)\hat{M}_f = \langle f | \exp(-i k \hat{x}\hat{W}) | i \rangle = i \sin(\phi/2 - k\hat{x})
P(x) = |\langle x,f| \psi,i\rangle|^2 = \sin^2(\phi/2 - kx)\,P_0(x) = \frac{\sin^2(\phi/2 - kx)}{\sqrt{2\pi\sigma^2}} e^{-(x/\sigma)^2/2}
P(x)=x,fψ,i2=sin2(ϕ/2kx)P0(x)=sin2(ϕ/2kx)2πσ2e(x/σ)2/2P(x) = |\langle x,f| \psi,i\rangle|^2 = \sin^2(\phi/2 - kx)\,P_0(x) = \frac{\sin^2(\phi/2 - kx)}{\sqrt{2\pi\sigma^2}} e^{-(x/\sigma)^2/2}

Gaussian profile of beam becomes modulated.

JD et al., PRA 88, 023801 (2013)

Example Dark Port Profiles

Left: Wavefront tilt mechanism producing spatial modulation

Right: Asymmetric dark port profiles in different regimes

Dashed envelope: input beam intensity

Solid curve:

dark port intensity

 

Top right:

weak value regime
 

 

Middle right:

double lobe regime
 

 

Bottom right:

misaligned regime

k\sigma \ll \phi \ll 1
kσϕ1k\sigma \ll \phi \ll 1
\phi\ll k\sigma \ll 1
ϕkσ1\phi\ll k\sigma \ll 1
\phi\ll 1 \ll k\sigma
ϕ1kσ\phi\ll 1 \ll k\sigma

Weak value regime

k\sigma \ll \phi \ll 1
kσϕ1k\sigma \ll \phi \ll 1

Dark port has single lobe that approximates displaced a Gaussian centered at:

\langle x/\sigma \rangle \propto k\sigma / \phi
x/σkσ/ϕ\langle x/\sigma \rangle \propto k\sigma / \phi

Tiny beam deflections can be distinguished, but with low output intensity.

I \propto \phi^2
Iϕ2I \propto \phi^2

Split-detected signal

Raw split-detected signal has linear response around                 .

Changing       just attenuates the observable signal at the meter.

(Parameter-invariant response curves)

Renormalizing by the collected power sharpens the linear response to a

                region bounded by             .

The slope of this response involves an anomalously large weak value.

\phi/2
ϕ/2\phi/2
\pm\phi/2
±ϕ/2\pm\phi/2
k\sigma
kσk\sigma
\phi
ϕ\phi
k\sigma=0
kσ=0k\sigma=0

Frequency Measurement

Mirror vertically misaligned to produce phase shift

\phi
ϕ\phi

Frequency-dependent horizontal tilt

Frequency-modulated and power-locked source laser

Prototype experiment: Howell lab, Rochester,  PRA 82, 063822 (2010)

Idea: Use prism to convert frequency modulation into beam tilt

Frequency Results

  • 780nm light (frequency                      ),  2mW CW power
  •                                         shift detected over 100ms
  • Frequency sensitivity of 
  • 10 Hz frequency modulation 35dB above noise floor
f = 340 \mathrm{THz}
f=340THzf = 340 \mathrm{THz}
\Delta f = 743 \mathrm{kHz} \approx 10^{-9} f
Δf=743kHz109f\Delta f = 743 \mathrm{kHz} \approx 10^{-9} f
129 \pm 7 \mathrm{kHz}/\sqrt{\mathrm{Hz}}
129±7kHz/Hz129 \pm 7 \mathrm{kHz}/\sqrt{\mathrm{Hz}}

Split-detection

maximum excursion

(weak value regime)

Spectral density of split detected signal

Confirming Noise Performance

WVA vs. Focused spot

Two injected noise sources:

  1. Modulated position jitter at detector (simulated vibration)
  2. Modulated mirror tilt outside interferometer setup

Prototype experiment: Howell lab, Rochester, PRA 92, 032127 (2015)

Goal: compare robustness against technical noise for two theoretically similar (and near optimal) beam deflection techniques

Injected Noise Robustness

  1. Modulation of signal mirror enhanced by 10dBV
  2. Modulation noise of detector position suppressed by 20dBV
  3. Modulation noise of external tilt suppressed by 30dBV
  4. Non-injected noise floor generally suppressed

Passive Laser Noise Results

  1. No modulated signal added - passive random background
  2. No spatial mode filtering of raw laser output:
    random mode fluctuations below 300 Hz expected.
  3. WVA suppresses passive laser noise by 20dBV

(Real noise in the wild)

Signal Information vs. Port

  1. Dark port contains almost all information about the signal
  2. Discarded bright port has almost no information
  3. WVA filters the relevant information and suppresses irrelevant noise

Shot Noise Robustness

Consider an input signal with ~12,000 photons per modulation cycle.  After post-selection, WV has only ~120 photons remaining per cycle.

JD (unpublished)

With such a small signal, shot noise becomes visible.

Nevertheless, the WV technique performs equally well.

Shot Noise Robustness

An input signal with ~120 photons per modulation cycle.  

After post-selection, WV keeps only ~1.2 photons per cycle.

JD (unpublished)

The shot noise sampling drops below the Nyquist limit.

Averaging over many integration periods still compensates.

Shot Noise Robustness

Even including incoherent background light incident on the detector,

the detection scales identically between a standard focused spot as more photons are added per cycle, in spite of the lower flux.

JD (unpublished)

Shot Noise Robustness

The incoherent background light incident degrades the visibility of the spectral information uniformly for both a focused spot and WV.

 

Averaging over multiple phase-locked integrations restores visibility.

JD (unpublished)

Take-aways

  • WV techniques enable sensitive precision measurement by amplifying the response of the meter per unit interaction.
     
  • Despite dramatic postselection loss, WV techniques retain nearly all relevant information while discarding irrelevant background.
     
  • Identical scaling of WV vs. standard methods is expected for temporally uncorrelated noise (like shot noise).
     
  • WV techniques may additionally suppress correlated noise originating outside the interferometer (like laser noise).

Thank You!

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