Structures far below sub-Planck scale in quantum phase-space through superoscillations

In 2001, Zurek derived the generic minimum scale $a_{Z}$ for the area of structures of Wigner's quantum phase distribution. Here we show by construction, using superoscillatory functions, that the Wigner distribution can locally show regular spotty structures on scales much below Zurek's scale $a_{Z}$. The price to pay for the presence of such structures is their exponential smallness. For the case we construct there is no increased interferometric sensitivity from the presence of patches with superoscillatory structure in phase-space.


I. INTRODUCTION
Based on the concept of interferences in phasespace [1], Zurek established that the minimum scale a Z for the area of structures of quantum phase distributions can, for one-dimensional quantum systems, be as small as a Z ≈ 2 /A, where A is the action representing the area of support of a system's Wigner distribution [2]. This was surprising [3], since Heisenberg's uncertainty principle was interpreted to limit the area of spots in phase-space to approximately /2.
When a bandwidth-limited signal contains segments which oscillate faster than what its spectrum suggests, it "superoscillates" [4].
Superoscillations have first been noticed in physics in the framework of weak measurement by Aharonov et al. [5], and then studied in detail by Aharonov et al. [6], Berry et al. [4,7], and others [8]. They have since been used experimentally, for example in super-resolution microscopy [9].
Translated to quantum phase-space: Wigner distributions can show regular spotty structures much below Zurek's scale a Z ; but such states cannot obviously be exploited for higher resolution in measurements.

II. ZUREK'S FUNDAMENTAL PHASE-SPACE TILES
The Wigner distribution of a "Schrödinger's cat" state of squeezed states G(x, p) = (π ) −1 e −x 2 /ξ 2 − p 2 ξ 2 2 , with squeezing parameter ξ, is Zurek's compass state is a coherent sum of two Schödinger's cat states rotated by π/2 with respect to each other. Starting from such compass states, Fig. 1 (a), Zurek showed that "Wigner functions can, and generally will, develop phase-space structures on scales as small as, but not generally smaller than" [3] Here, P and L are the phase-space distances between the squeezed states along the momentum and position axes respectively.

The
Zurek scale a Z is, e.g., the phase-space area of one fundamental tile [3] ("Zurek tile") associated with a compass state, highlighted in Fig. 1

Inspired by
Zurek's compass state, we construct a "cross-state" featuring superoscillations in quantum phase-space. This state features small patches with regular structures on scales much smaller than a Z , Fig. 2.
To map f (x) of Eq. (4) into phase-space, we use a superposition of suitably pairwise-displaced squeezed states where Φ j (x) = K j S(x − j∆x), K j = |D j | / N/2 l=0 D l and Here, N is even and Ψ contains N + 1 spikes, see One could use a coherent sum to form a cross, but this would lead to greater complexity in Fig. 2, which is unnecessary to illustrate our construction.

IV. SUBSTRUCTURES WITHIN ZUREK TILES
The area of sub-Planck structures of nonsuperoscillating states is limited by a Z [3]. Now we show that regular structures on scales much smaller than a Z can exist. Therefore, the superoscillating Wigner distribution W Ψ contains interference terms, equivalent to expression (1b), proportional to cos( 2p N ∆x 2 α) = cos(p L α). Thus for W + , analogously to Zurek's scale a Z in the compass state Fig. 1 (a), superoscillatory structures with αfold reduced length scales, Fig. 1 (d), yielding areas on the scale of arise.
For these superoscillatory structures to show, the 'overspill' from the two adjacent squeezed states Φ −1 and Φ +1 has to be so small that their Wigner distributions obey V. CONCLUSION We remind the reader of the fact that a quantum wave function cannot be strictly "bandwidth-limited", simultaneously in position and momentum. Our Wigner distributions are confined by a finite area A in phase-space, yet they feature regular sub-Zurek scale structures, in this sense they are superoscillating.
Zurek's compass states provide interferometric sensitivity at the Heisenberg limit. As Fig. 1 (c) illustrates, our superoscillating states change, under tiny displacements in x, p and t, in essentially the same way as regular compass states; they therefore do not perform better at the detection of small shifts or rotations.
At this stage, the formation of structures below the Zurek scale using superoscillations in phase-space is primarily a surprising curiosity. Superoscillating regions are known to be tiny in extent and amplitude [6][7][8]. This is why our superoscillating states cannot show sensitivity below the Heisenberg limit. We have shown, to paraphrase Zurek's statement cited above, that phase-space structures are frequently as small as, but not generally smaller than a Z . Yet, superoscillating states can generate localized small patches with regular structures on very much smaller scales.
An interesting open question raised by the existence of sub-Planck and sub-Zurek scale phase-space structures concerns their potential effects on simulations. Possibly, grids finer than commonly assumed for numerical calculations [10] have to be used.