1. Sub-Planck Structure and Weak Measurement
International Conference on Quantum Information and Quantum Computing
Indian Institute of Science
Sub-Planck Structure and Weak Measurement
Prasanta K. Panigrahi
IISER Kolkata

2. INTRODUCTION
Measurement plays a central role in quantum mechanics. Precision of measurement is restricted not only by technology but also by the theory itself.
Heisenberg-limited metrology and weak measurements are the two notions that find use in enhancing sensitivity of measurements. The first can increase precision whereas the second can be used to amplify small signals.
Both of these essentially rely on quantum mechanical interference, but in two opposite regimes.

3. Sub-Planck and Weak Value
The cat states and their generalization owe this precision to sub-Planck structure in the phase space.
Weak value of a dichotomic observable can also be inferred from the cat state if the two coherent states comprising the cat state are identified with the post-measurement meter state in measuring that observable.
We link these two notions of sub-Planck structure and weak measurement in terms of the cat state.
Specifically, we show that the parameter regime relevant to Heisenberg limited measurement is opposite of what is required for weak measurement.

4. Sub-Planck Structure
Sub-Planck structure is a striking effect of interference in phase space. It explains oscillating photon number distribution of squeezed states in a phase plane.
The Wigner distribution ideally captures such non-classical interference phenomena, as it can be negative in a phase-space region.
Zurek first demonstrated that such oscillatory structures resulting from quantum interference, can lead to a fundamental area in the phase space, smaller than the Planck’s constant.
Heisenberg-limited sensitivity has been demonstrated for superposition of two coherent states and a classical wave optics analogue has been tested experimentally .

5. Wigner function for compass state showing sub-Planck
scale structures in the interference region

6. Heisenberg Limited Sensitivity
For quasiclassical states ( e.g., coherent states) the sensitivity of phase estimation is restricted by the standard quantum limit, also known as the shot-noise limit.
Sub-shot-noise sensitivities can be obtained using the sensitivity of the quantum state to displacements, which is related to the sub-Planck phase space structure extracted from its Wigner function.
A number of proposals have been advanced for generating single particle cat and generalized states, showing the above feature .

7. Weak Measurement
The path-breaking idea of ‘weak measurement’(WM) in QM, proposed by Aharonov, Albert and Vaidman (AAV) .
In this measurement scenario, the empirically measured value (coined as ‘weak value’) of an observable can go beyond the eigenvalue spectrum of the measured observable.
WM has several implications, one being that it provides insight into conceptual quantum paradoxes.
It has been used in identifying a tiny spin Hall effect, detecting very small transverse beam deflections and measuring average quantum trajectories for photons.

8. Ideal Quantum Measurement
System +
| 𝜓 𝑖 = 𝑐 𝑗 | 𝑎 𝑗 〉
Apparatus
𝑄 0 = 𝛷 𝑖
System + apparatus
𝐻 =𝑔′ 𝑡 𝐴 ⊗ 𝑃
𝛷 𝑓 = 𝑐 𝑗 𝑎 𝑗 | 𝑄 𝑗 〉 ⊗
System is coupled to apparatus, which is prepared in a known initial state
Through appropriate interaction, one-to-one correlation established between apparatus and system property that is being measured.
Superposition in system leads to entangled system+apparatus state.
Experimenter records final apparatus state – entangled state collapses stochastically; change in apparatus state proportional to measurement outcome (eigenvalue of A).

9. Weak Measurement and Weak Value
Ideal measurements require initial apparatus state to be sharply peaked – allow uncertainty.
ideal apparatus
𝑞 | 𝑄 0 =𝛿(𝑞− 𝑞 0 )
practical apparatus
𝑞 | 𝑄 0 =N exp − 𝑞 2 /4 𝜎 2
After measurement interaction, post-select a final state of the system, | 𝜓 𝑓 and then observe apparatus. Then, apparatus state is 𝑄 𝑓 = 𝜓 𝑓 𝑒 −𝑖 𝑑𝑡𝑔′ 𝑡 𝐴 𝑃 𝜓 𝑖 𝑄 0 = 𝜓 𝑓 | 𝑒 −𝑖𝑔𝐴𝑃 𝜓 𝑖 𝑄 0
The measurement interaction is weak, i.e., coupling g is small.

10. Weak Measurement and Weak Value
𝑄 𝑓 ≈ 𝜓 𝑓 𝐼−𝑖𝑔 𝐴 𝑃 +… 𝜓 𝑖 𝑄 0 ≈ 𝜓 𝑓 𝜓 𝑖 1−𝑖𝑔 𝐴 𝑤 𝑃 | 𝑄 0 𝑞 | 𝑄 𝑓 ≈ 𝜓 𝑓 𝜓 𝑖 𝑒 − 𝑞− 𝐴 𝑤 2 /4 𝜎 2
𝐴 𝑤 = 𝜓 𝑓 𝐴 𝜓 𝑖 𝜓 𝑓 𝜓 𝑖
Weak Value
Final pointer approximately a gaussian peaked at weak value
Weak value can be outside range of eigenvalues, depending on pre- and post-selection
The approximations used are valid in limited cases; we require 𝜎≪1/𝑔 𝐴 𝑤 and 𝜎≪ 𝑚𝑖𝑛 𝑛=2,3… 𝜓 𝑓 𝐴 𝜓 𝑖 𝑔 𝜓 𝑓 𝐴 𝑛 𝜓 𝑖 1/(𝑛−1)
Single gaussian arising from interference effects due to large variance in initial apparatus state.
Duck, Stevenson, Sudarshan, PRD (1989)

11. Weak Measurement and Weak Value
In contrast to strong measurement, the formalism of weak measurement allows to extract information of a quantum system in the limit of vanishingly small disturbance to its state.
In strong measurement, while measuring an observable, the pointer indicates the eigenvalues of the given observable.
In weak measurement the pointer may indicate a value beyond the eigenvalues range.
The weak value can be complex. Real part gives pointer shift in position space whereas imaginary part gives shift in momentum space.

12. Weak Measurement with Stern-Gerlach
𝐻=𝑔𝛿 𝑡−𝜏 𝑥 𝐴
𝜓 0 𝑥 = 2𝜋 𝜎 2 − 𝑒 − 𝑥 2 4 𝜎 2
Weak
Strong
Post-selected spin state: | 𝜒 𝑓 〉
Initial state: |Ψ〉=𝜓 0 𝑥 |𝜒 𝑖 〉
After first SG the state is | Ψ ′ 〉=𝑒 −𝑖𝑔 𝑥 𝐴 /ℏ 𝜓 0 𝑥 |𝜒 𝑖 〉
If the interaction is weak, 𝑒 −𝑖𝑔 𝑥 𝐴 /ℏ =1−𝑖𝑔 𝑥 𝐴 /ℏ + O(2)
If particles are post-selected in the spin state | 𝜒 𝑓 〉, the device state is
𝜓 𝑓 𝑥 = 𝜒 𝑓 𝛹 ′ = 〈 𝜒 𝑓 |𝜒 𝑖 〉[1−𝑖𝑔 𝑥 𝐴 𝑤 /ℏ] 𝜓 0 𝑥 = 〈 𝜒 𝑓 |𝜒 𝑖 〉 𝑒 −𝑖𝑔 𝑥 𝐴 𝑤 /ℏ 𝜓 0 𝑥
The final pointer state in momentum space: 𝜙 𝑓 𝑝 𝑥 = 〈 𝜒 𝑓 |𝜒 𝑖 〉 2 𝜎 2 𝜋 ℏ 𝑒 − 𝜎 2 (𝑝 𝑥 −𝑔 𝐴 𝑤 )^2 ℏ 2
Aharonov, Albert, and Vaidman, PRL, 60, 1351 (1988)

13. Weak Measurement with Stern-Gerlach
The final pointer state in momentum space :
𝜙 𝑓 𝑝 𝑥 = 〈 𝜒 𝑓 |𝜒 𝑖 〉 2 𝜎 2 𝜋 ℏ 𝑒 − 𝜎 2 (𝑝 𝑥 −𝑔𝜏 𝐴 𝑤 )^ ℏ 2
Pre-selected state : |𝜒 𝑖 〉= +𝜃 =𝐶𝑜𝑠 𝜃 2 +𝑧 +𝑆𝑖𝑛 𝜃 2 |−𝑧〉
Post-selected state: 𝜒 𝑓 =|+𝑧〉
Measure A= 𝜎 𝑥
Weak Value: (𝜎 𝑥 )_𝑤 =𝑇𝑎𝑛 𝜃 2
But, prob. of post-selection 〈 𝜒 𝑓 |𝜒 𝑖 ​ 2 =𝐶𝑜 𝑠 2 𝜃 2 →0

14. Experiments with weak measurement
Observation of the Spin Hall Effect
O. Hosten and P. Kwiat Science 319, 787(2008).
Ultrasensitive Beam Deflection
P.B. Dixon, et al., PRL,102, (2009).
Violation of Leggett-Garg inequality
N. S. Willams and A. N. Jordan, PRL, 100, (2008) .
J. Dressel et al. PRL, 106, (2011).
Probing Quantum paradoxes
J. S. Lundeen and A. M. Steinberg, PRL 102, (2009).
K. Yokota, et al. New J. Phys. 11, (2009).
Direct measurement of the quantum wavefunction
J. S. Lundeen et al. Nature, 474, 188(2011).

15. Cat State The cat state is a superposition of two coherent states
where are essentially shifted Gaussians.
The inner product is crucial:

16. Sub-Fourier Sensitivity of Cat State
Let the phase be shifted by 𝛿:
The overlap is given by
where
Given other parameters, the overlap function vanishes for multiple values of 𝛿, allowing Heisenberg-limited metrology.
𝜓 𝑐 ′ 𝜓 𝑐

17. Sub-Fourier Sensitivity of Cat State

18. Sub-Fourier Sensitivity of Cat State
Sub-Planck structure can also be inferred from the Wigner function

19. Sub-Fourier Sensitivity of Cat State

20. Weak measurement & cat-states
Weak value:
Suppose ,
For , weak value is
Weak value can be arbitrarily large as 𝑎 1 is made small.
We now generate the cat state as our final pointer state after a weak measurement using a Stern-Gerlach apparatus.
Then we will explore the connection with sub-Planck.

21. Generating cat state with Stern-Gerlach
Initial state
Interaction Hamiltonian

22. Generating cat state with Stern-Gerlach
After interaction , final state is
where the components satisfy the Pauli equation

23. Generating cat state with Stern-Gerlach
After post-selecting in the state final state is
where
and
The overlap between the two components
Weak interaction requires 𝐼≈1
But sub-Planck structures require 𝐼≈0

24. Probe state after weak interaction
Taking 𝛼= 𝑝′ 𝑥 𝜏/2𝑚 and 𝑝 0 = 𝑝 𝑥 we rewrite
𝐼≈1 ensured by small 𝛼 and 𝑝 0 .
Weak value is
Expanding to first order in 𝑝 0
Thus, after weak interaction, we get as probe state
whose momentum distribution is

25. Illustrations 𝐼≈0, 𝑎=𝑏=1/ 2
Oscillatory overlap function
Interference in phase space (zeros in Wigner function)
No weak value – strong measurement condition; two peaks in probability distribution

26. Illustrations 𝐼≈0, 𝑎= 1+𝑖 2 ,𝑏= 1−𝑖 2
Oscillatory overlap function
No interference in phase space
Weak value is imaginary, so position distribution peaks at weak value

27. Illustrations 𝐼≈1, 𝑎=0.866, 𝑏=-0.5
No oscillatory overlap
No interference in phase space
Real weak value, momentum distribution peaked at weak value

28. Conclusions
There is no sub-Planck structure in the limit where we obtain a weak value
Sub-Planck structures are obtained in the opposite limit that coincides with the requirement for strong measurement
Oscillatory overlap function does not guarantee sub-Planck structure

29. Dr. Alok K. Pan, Anirban Ch. N. Chowdhury
Thanks
Collaborators
Dr. Alok K. Pan, Anirban Ch. N. Chowdhury