Weak Measurements (WM): Concepts

The Measurement Problem in Quantum Physics

What was the state of the paticle before it was measured? Any answer to this question led by intuition built upon the philosophy of classical mechanics has led to paradoxes. Consider the famous Double Slit Interference for example. When light wave is made incident on a double slit, we observe an interference pattern with alternate dark and bright fringes on the screen.

Double Slit Interference Pattern obtained on screen when coherent light wave (LASER light) is incident on a double slit

If we send photons one by one, the particle arrive on the screen in chunks. But if we keep accumulating the photons we see the same interference pattern build up.

Interference Pattern build up with single photon source

How can photons - 'the particles', go one by one and create an interference pattern ? classical mechanics compels us to believe that each photon must have either gone through the left or right slit. But then knowing which slit each photon went through makes the interference pattern dissapear. Somehow, we are not able to assign a unique state to the particle before the measurement. Do particles have unique property before the measurement ?

Time Symmetry and Quantum Mechanics

In quantum mechanics, even if we know the wave function $\psi$ at time $t_i$ and the Hamiltonian for all time, we cannot in general predict the outcome of a measurement performed at a later time $t (> t_i)$ but we can only predict probabilities. However, given an outcome of a measurement at time $t$, we cannot retrodict the probability distribution at the earlier time $t_i$. Classical dynamics obey time symmetry but Quantum Mechanics do not seem to respect that. Dynamics in Quantum Mechanics consists of the Schrodinger's Equation and the state reduction upon measurement. The Schrodinger's equation is time symmetric.

Intially two Gaussian wave packets approach the barrier. The video is played in reverse and the backward evolution still satisfies the time-reversed Schrodinger's equation.

1964, Aharonov, Peter Bergmann, and Joel Lebowitz (ABL) realized that quantum mechanics can be formulated in a time symmetric manner where we not only ask to predict the probability distribution at time $t$ given $\psi(t_i)$ but also to retrodict given a result of subsequent measurement $\psi(t_f)$ where $(t_f>t)$. Satisfying these two constraints, if a property of the particle is both predicted and retrodicted with certainty we can call that property at time $t$ to be an element of physical reality.

Pre and Post Selection

Almost two decades after ABL paper, Aharonov along with Albert and D'Amato discovered that an assumption made by Gleason and subsequently by Kochen and Specker may not always hold true and that may open the possibility of simultaneous assignment of two non-commuting observables within the intervals between two measurements. The first measurement at time $t_1$ determines the pre-selected state $\psi(t_f)$ and the choice of considering a perticular set of outcomes of the the final measurement establishes the post-selected state $\psi(t_f)$. For pre and post-selected ensembles a novel property was discovered by Aharonov, Albert, Vaidman and Casher owing to the realization that the disturbance of measurement on the non-commutating properties can be reduced if the accuracy of measurement is traded off. After detailing out the procedure to measure all the Pauli observervables of a pre and post-selected ensemble simultaneously, it was discovered that the procedure sometimes yielded values outside the eigenspectrum .

For a free particle with spin, if we predict the $\sigma_z$ at time $t$, we expect it to be 1 since it was 1 at $t_i$. But since $\sigma_x$ does not commute with $\sigma_z$, we may not predict the $\sigma_x$ component at time $t$. But given additional contraint that the particle was measured to have $\sigma_x$ value as 1 at time $t_f$, we can retrodict that that the $\sigma_x$ component at time $t$ must be 1 as well. How can these be simultaneously satisfied? What about spin component in another direction say at an angle $\pi/4$. Since the outcome for the spin at the angle $\pi/4$ must be such that the $\sigma_z$ component in the past must have yielded 1 and its $\sigma_x$ component must yield 1 in the future. Thus, $\sigma_{\pi/4}$ can only be $\sqrt{2}$ : beyond the eigenvalue spectrum.

Weak Interaction

When do we say that a measurement disturbs a system ? What do we mean by measurement after all? We do measurement to infer something that is not observed directly either due to the inherent nature of the observable or due to the scale. To infer these properties we couple the system states with pointer states in a <\Von Neumann Measurement Scheme. By directly observing the pointer shifts, we infer the state of a particle. Say, for instance we need to measure the z-component of the spin of a spin half particle. For this, we need to couple the spin of the system to a pointer say the displacement along the transverse axis. This coupling is mediated by the Interaction Hamiltonian. If the displacement caused by the interaction is larger than the uncertainty in transverse position, then we can clearly infer the state of the particle. Interaction of such strength would lead to a strong measurement. On the other hand, if the displacement caused is much much smaller than the uncertainty in transverse position, for a single event, we cannot infer the state of the particle and the interaction is said to be weak.

When the width of the Gaussian is much smaller compared to the separation of the two Gaussians, if one particle gets detected we can distinguish to which Gaussian the particle belongs to hence know the property of the system. But when the Gaussians overlap consideriably, in a single event we cannot know the property of the particle. Thus, the particle remains effectively undisturbed during its evolution and hence the the interaction is said to be weak.

In 1988, Aharonov, Albert and Vaidman (AAV) evolved the system subjected to both pre-selection and post-selection with weak interaction Hamiltonian and and post-selection and derived how the result of measurement on a spin half particle can be 100 (or as high as possible). Sudarshan et.al and later Knight and Vaidman extended the scheme for weak measurement of an observable in polarization degree of freedom of light and soon the weak measuremnt scheme experimentally realized by Ritchie et. al. Today weak measurements have found numerous applications in quantum foundations and precision metrology.

Weak Value

The property which is inferred from the pointer shift using weak measurement of a system subjected to pre and post-selection is known as the weak value. In the limit of no interaction, the pointer shift indicates the property of the system and not the measuring apparatus. The weak value of a given observable $A$ between two states $|\psi(t_i)\rangle$ and $|\psi(t_f)\rangle$ is given by $$\langle A\rangle ^{(w)}=\frac{\langle \psi(t_f) |A| \psi(t_i) \rangle}{\langle \psi(t_f) | \psi(t_i) \rangle} $$

The interference between the two Gaussian (blue and green) with different phases yields the resultant Gaussian, the centroid of which can go beyond the bounds.

If we vary the post-selected state $|\psi(t_f)\rangle$, then the weak value also changes and by appropriate post-selection, the weak value can be amplified without any limits. The weak value is in general complex and the imaginary component physically manifests itself in the displacement of the momentum conjugate to the pointer variable.