1. Quantum Non-Demolition in Optics
Novice
Pedagogical Review by Nergis
Grangier et al, Nature 398 (1998)
Chiao et al, quant-ph (1995)
Buchler, Ph.D. thesis, ANU (2001)

2. Quantum mechanics basics
Non-commuting operators
Obey Heisenberg uncertainty principle
Operator terminology
standard deviation, dispersion, uncertainty
variance
expectation value

3. Measurement Back Action
Heisenberg Uncertainty Principle Measure A very precisely  DA arbitrarily small  DB very large
Large DB directly imposes no restriction on precision of A unless the large fluctuations in B couple back to A
Precision measurement introduces a perturbation or ‘back action’
How to evade this back action ‘noise’?

4. QND to the rescue
Devise a measurement where back action noise is kept entirely within unwanted observables and without being coupled back to quantity of interest
Types of QND
Mechanical oscillators (GW bars, NEMs)
Quantum Optics

5. Is it QND? Pedagogical example
Stern-Gerlach (SG) apparatus
SG filter
A good quantum state preparation (QSP)
Once –/2 is filtered out and there are no perturbations, only ever get +/2
+/2
-/2 e-
+/2
-/2 e-
SG filter

6. Is it QND? But what if +/2 is perturbed between two SG filters?
If the perturbation flips +/2 to –/2, nothing passes through the next filter and the +h/2 state is “demolished”
Filter: selects a quantum state with given eigenvalue
QND device: measures the quantum state and keeps the particle going regardless of the outcome of the measurement
 SG filter  QND device since +/2 state prepared by demolishing –/2

7. Good QND observables? Spin Generally, constants of motion
Without external perturbation spin doesn’t evolve between successive measurements
Generally, constants of motion
Position, e.g., is not ideal since Dp  Dx(t)

8. QND schemes need a readout
Even if a good QND variable (e.g. spin) exists still need to ‘know’ (measure) its eigenstate
Measuring eigenstate destroys it
READOUT to the rescue
Couple the spin of test particle to the spin of a meter or probe particle
The coupling is such that it does not affect signal particle’s eigenstate
Measure the spin of the meter particle (so what if meter particle’s eigenstate is destroyed)

9. EPR paradox  measurement of ym gives ys
Entangled States
EPR paradox  measurement of ym gives ys
+/2
-/2
meter
particles
signal
interaction
regions
Entangled state
y = ays + bym t0 t1
Entangled state  simultaneous eigenstate of deltaX and SigmaP. Non-factorizable superposition of product states.
EPR  measure X (or P) of particle 1, predict with certainty X (or P) of particle 2 even if separation is greater than light cone (space-like intervals)
Aspect expt two blue photons emitted from atomic transition with 0 net angular momentum  orthogonal polarzations. Coincidence count rate as function of relative angle between polarizers in the two beams is measure of the correlation between two well separated photons.
Once meter particle is detected, all other non-commuting observables of the signal particles are randomized but that’s okay since they are not back coupled onto original QND variable

10. QND in Optics
Squeezed light paved way for manipulation of quantum noise of light
QND used to control of the light quantum noise
Standard technique
Signal and meter beams coupled via a non-linear optical medium
Usual observables
Quadrature amplitudes of (quantized) EM field
Photon number and phase
signal beam
meter beam
non-linear medium

11. Quadrature Fields
Add a steady carrier field E0 to fluctuating electric field with quadratures E1 and E2 
ET(t) = E0 + E1(t) cos(w0t) + E2(t) sin(w0t)  E0 (1 + E1(t)/E0) cos[w0 (1 + E2(t)/E0) t]
E1(t) gives amplitude modulation
E2(t) gives phase modulation
  fluctuating part is small (linear approximation)

12. Quantization of quadrature fields
E-field operators using two-photon formalism (W = light frequency and j = 1,2)
Ej(t) = (aj e-iWt + aj+ eiWt) dW/(2p)
Commutation relations
[E1(t), E1(t’)] = 0 [E2(t), E2(t’)] = 0
[E1(t), E2(t’)] = i(t – t’)
E1(t), E2(t)  X(t), Y(t)  position, momentum operators of QHO

13. Standard Quantum Limit
Precision scale set by shot noise limit 
Dn = n DX
Df = (1/2 n) DY
Heisenberg 
Dn Df  1/2  DX DY  1
Standard Quantum Limit
DX DY = 1

14. Is it QND?  QSP criterion  Measurement efficiency criterion
After measurement is performed, system should be left in a well-defined eigenstate
Quantified using conditional uncertainty in DXs after measurement of m, DXs|m
For laser beam, limit set by shot noise  DXs|m < 1
 Measurement efficiency criterion
Noise of measurement device
Quantified by ‘input’ refered noise’ of meter, DXm
 Non-demolition criterion
How much is measured observable disturbed by the measurement?
Quantified by signal noise, DXm

15. QND quantified
, ,   efficient measurement minimizes DXs and DXm  DXs DXm < 1
T is the transfer function of signal-to-quantum-noise for s and m, respectively
Ts = 1/(1+DXs2) Tm = 1/(1+DXm2)
Then DXs DXm < 1  Ts + Tm > 1
General conditions for a QND measurement
Ts + Tm > 1
(DXs|m ) 2 < 1

16. QND in Optics Need non-linearity to couple signal and meter beams
Second-order non-linearities
Parametric amplification  amplifcation of signal beam at w using a pump beam at 2w
Produced with special crystal, e.g. KTP
Third-order non-linearities
Kerr effect  intensity-dependent refractive index
Cross-Kerr effect  n for one beam is modified by intensity of a second beam
Produced in fibers, resonant 3-level (cold, trapped) atoms

17. Second-order non-linearities
Parametric processes useful in QND
allows meter beam to carry amplified copy of signal beam
How? Noiseless amplification
Amplify one quadrature of signal while preserving SNQR (i.e. Ts  1)
Relative phase between signal and pump beam
(Conventional optical amplifiers (lasers) have Ts = 0.5 for large gains  3 dB loss in SQNR)
Noiseless amplification
Deamplification  shot noise limited light becomes squeezed

18. Third-order non-linearities
Cross-phase modulation  index of beam 1 modulated by intensity of beam 2
n(s) = n0(s) + n2(s) I(m)
n(m) = n0(m) + n2(m) I(s)
Gain  g2 = (Fs Fm) where Fs,m = ks,ml n2(s,m)I
Ts = 1  input beam is shot noise limited
Tm = g2/(1 + g2)
(DXs|m)2 = 1/(1 + g2)  perfect QND as g  
Sout
Mout

19. QND(3)
Input-output transformations for quantum fluctuations of the beams
XOUT(s) = XIN(s) YOUT(s) = YIN(s) - g XIN(m)
XOUT(m) = XIN(m) YOUT(m) = YIN(m) - g XIN(s)
Intensity noise of meter beam XIN(m) is fed back onto phase noise of signal beam YOUT(s)
Measurement back action  variance of signal phase noise (DXs|m)2 = 1 + g2  Heisenberg or worse as g > 0
Conditional variance of signal intensity noise  (DXs|m)2 = 1/(1 + g2)  Heisenberg or better

20. Present state of QND
(DXs|m)2