Post-selected von Neumann measurement with Hermite–Gaussian and Laguerre–Gaussian pointer states

Abstract

Through the von Neumann interaction followed by post-selection, we can extract not only the eigenvalue of an observable of the measured system but also the weak value. In this post-selected von Neumann measurement, the initial pointer state of the measuring device is assumed to be a fundamental Gaussian wave function. By considering the optical implementation of the post-selected von Neumann measurement, higher-order Gaussian modes can be used. In this paper, we consider the Hermite-Gaussian (HG) and Laguerre-Gaussian (LG) modes as pointer states and calculate the average shift of the pointer states of the post-selected von Neumann measurement by assuming the system observable  with A Î2 = and A Â2 = for an arbitrary interaction strength, where Ir epresents the identity operator. Our results show that the HG and LG pointer states for a given coupling direction have advantages and disadvantages over the fundamental Gaussian mode in improving the signal-to-noise ratio. We expect that our general treatment of the weak values will be helpful for understanding the connection between weak-and strong-measurement regimes and may be used to propose new experimental setups with higher-order Gaussian beams to investigate further the applications of weak measurement in optical systems such as the optical vortex.

Blurred quantum Darwinism across quantum reference frames

“…In a typical measurement scenario one usually assumes that the evolution is dominated by the central interaction, and then the so-called generalized von Neumann measurement is performed [35,36]. It is reasonable to assume that this part is acting only for a limited period of time, as one expect the measurement to occur after a finite number of time units.…”

Generalized modular-value-based scheme and its generalized modular value

“…A straightforward calculation the modular value Eq. (11) gives (A) s m = (2ǫ − 1) 2 . To change the modular value, e.g., from 0 to 9, we vary the parameter ǫ from 0.5 to 2.0 as shown in the Insert Fig.…”

“…So far, weak values have been studied in the context of weak measurements, and in such weak measurements, not only classical pointer states (such as Gaussian states [6][7][8][9] or Hermite-Gaussian/Laguerre-Gaussian beams [10,11] but also nonclassical pointer states (squeezed states, and Schrödinger cat states) are used [12,13]. Nonclassical pointer states have an advantage in controlling the precision of weak measurements, which have been recently demonstrated [11][12][13].…”

Toward tangible quantum nature

“…The second part involves a non-unitary transformation that achieves the realization of a particular measurement outcome by the projection postulate. Note that we do not consider a detection process to obtain observational data corresponding to a macroscopic experimental result 5 . All the above processes are summarized in the book [18], and illustrated in Fig.…”

“…However, to build up our intuitive picture of the quantum-mechanical Nature, tangible quantum Nature, quantum measurement theory has been developed such as the non-observable measurement setup [1,2]. Furthermore, there have been developed several applications of quantum measurement theory, e.g., the visualization of the quantum state [3,4,5,6]. Also, by using the unique quantum phases [7,8], the quantum-mechanical tunneling dynamics can be indirectly observed [9].…”