WEAK-VALUES AMPLIFICATION: WHEN PRECISION MEASUREMENTS BECOME EASY

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1 WEAK-VALUES AMPLIFICATION: WHEN PRECISION MEASUREMENTS BECOME EASY Julián Martínez-Rincón Department of Physics & Astronomy University of Rochester 03/05/14

2 OUTLINE A quick review on QM: Strong/projective measurements Time-symmetric (?) - Postselection Weak-values amplification = Weak Measurement + Postselection Experimental Applications. Noise advantages (Optical set-ups). Controversies! Future application?!

3 STRONG/PROJECTIVE MEASUREMENTS IN QM Let s consider one single quantum measurement: With probability Nevertheless, after N events (photons):

4 So, let s extract the measured value of g out from our N events: and Assuming an initial Gaussian distribution for q, and a judicious choice of A and the preparation of the initial state such that, Which is the well-known Standard Quantum Limit (SQL), or Shot Noise, or simply Central Limit Theorem! This result tells us the smallest uncertainty in the average measurement of g (interaction parameter) after N independent events.

5 TIME SYMMETRIC QM (?): TWO-VECTOR STATE AND POSTSELECTION Classical Mechanics: The final state is (in principle) determined by the initial one and the dynamics. Standard QM: New genuine information after a measurement. Collapse of the wave function. Non-symmetric!! However, the final state can be freely chosen, exactly like the initial one: Measurement Output?

6 ONE-particle system. What about if we want to measure the spin for an intermediate time? What about measuring them simultaneously? Is this possible? S x and S z are non-commuting operators.

7 Also, the formers are STRONG measurements: The results of the measurements depends of the timeorder! It might make impossible the post-selection. Then, a measurement must be done, but the post-selection still possible. Weak Measurement!

8 The trick: N-particle ensemble. An imperfect measurement is performed on each particle. Game with errors. The measurement is weak, not change the state of the particle at all. Uncertainty is large. N measurements reduce the error by a factor of 1/ N.

9 WEAK-VALUES AMPLIFICATION = WEAK MEASUREMENT + POSTSELECTION Let s put together what we just learned: Interaction parameter g is small =? Throwing away final events in System and Meter

10 Weak interaction/measurement approximation: The System s wavefunction is slightly perturbed and not relevant information can be extracted from one single event, where Aw is the weak value of the operator A, Then, the final state of the Meter is given by,

11 So, what is exactly the weak value? Strong/projective Measurement: Weak-Values Measurement: Recalling our previous result: So, is the probability of success in the postselection of The expectation value is the average of all possible weak values over all possible postselections!

12 So, measurements of q in the standard technique and weak-values: If almost orthogonal pre- and post-selection are chosen, the weak value is very large! Amplified Deflection. There is a price: The number of detected events decreases as. Aw is in general a complex number!

13 Is that all? Well, no Aw is imaginary! So, a shift is induced in the momentum of the Meter: The shift in the momentum offers an interesting advantage. The shift can be done very large choosing a very small variance in the preparation of the meter s state!

14 However, the final state can be freely chosen, exactly like the initial one Yakir Aharonov Lev Vaidman Jeff Tollaksen Sandu Popescu

15 EXPERIMENTAL APPLICATIONS First demonstration of the effect using Electromagnetic waves: An optical setup to tell apart the deflection in two orthogonal polarizations using a birefringent material: Importantly, the effect was demonstrated for a wave-type theory or system. Weak-values for quantum particles (as electrons) has not been experimentally demonstrated yet! So, we have a technique that amplifies a shift in a measurement, charging us with throwing away events (blah blah )

16 First use of the weak-values effect as a precision metrological technique: Sensitivity of ~ 1A Amplified up to 10 4

17 The next big step was given by people here in Rochester: Using imaginary weak-values, a small transverse momentum kick (k) was measured as a displacement!

18 There has been other applications: Measuring small phase shifts, frequency shifts, small velocities, and even temperature shifts. To amplify optical nonlinearities at the single-photon level. Decreasing the effective life-time of atomic states. Chiroptical spectroscopic (signals using perpendicularpolarization detection) method to detect chiral molecules as a weak-values technique. Measuring wave function tomography

19 NOISE ADVANTAGES - CONTROVERSIES So, we know now that weak-values can amplify a displacement, giving a high accuracy/sensitivity, But most of the events (photos) are thrown away, so can we still perform precise measurements? In other words, can we reach the SQL? The answer is YES!

20 Such a result is quite counterintuitive: the stronger the postselection, the better the sensitivity of the measurement. Even, after all the surprising experimental results

Nevertheless, these negative results are based on assuming small N (first paper for example), and are focused on real weak-value set-ups with not very realistic noise models!

21 Nevertheless, these negative results are based on assuming small N (first paper for example), and are focused on real weak-value set-ups with not very realistic noise models! We believe have responded to most of these papers making a deep analysis using imaginary weak-values and four different types of noise. We have shown that in a noiseless experiment, all cases (standard technique, real weak-values and imaginary weak-values) offer the same sensitivity. The win is for imaginary weak-values when some noise sources are included imaginary weak-values can mitigate technical noise!

22 How do we do that? We use the Fisher Information. For example, for a noise-free system:

23 We then consider two specific type of noise. Transverse detector jitter: The detector oscillates randomly with a variance J 2. vs Angular jitter noise: An external mirror produces a random tilt q with variance Q 2. vs Experimental results coming out!!!!!

24 CONCLUSIONS Time symmetric QM took us to Postselection and Weak Measurements. Some interesting phenomena (I did not talk about): Hardy s paradox, three box paradox, We have a new technique to measure small parameters: weak-values amplification. An amplification is obtained paying the price with undetected events. It offers amplification that helps mitigates technical noise, without compromising the precision. Importantly, every time a weak-value technique is to be used is important to verify the gain respect to the standard technique if there is one. Recycled photos!!

25 FUTURE APPLICATIONS?

Weak measurements: subensembles from tunneling to Let s Make a Quantum Deal to Hardy s Paradox

Weak measurements: subensembles from tunneling to Let s Make a Quantum Deal to Hardy s Paradox First: some more on tunneling times, by way of motivation... How does one discuss subensembles in quantum