Structures far below sub-Planck scale in quantum phase-space through superoscillations
SStructures far below sub-Planck scale in quantum phase-space throughsuperoscillations
Maxime Oliva, Ole Steuernagel
School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK (Dated: April 27, 2017)In 2001, Zurek derived the generic minimum scale a Z for the area of structures of Wigner’squantum phase distribution. Here we show by construction, using superoscillatory functions, thatthe Wigner distribution can locally show regular spotty structures on scales much below Zurek’sscale a Z . The price to pay for the presence of such structures is their exponential smallness. Forthe case we construct there is no increased interferometric sensitivity from the presence of patcheswith superoscillatory structure in phase-space. I. INTRODUCTION
Based on the concept of interferences in phase-space [1], Zurek established that the minimum scale a Z for the area of structures of quantum phase distributionscan, for one-dimensional quantum systems, be as smallas a Z ≈ (cid:126) /A , where A is the action representing thearea of support of a system’s Wigner distribution [2].This was surprising [3], since Heisenberg’s uncertaintyprinciple was interpreted to limit the area of spots inphase-space to approximately (cid:126) / a Z ; but such states cannot obviously beexploited for higher resolution in measurements. II. ZUREK’S FUNDAMENTAL PHASE-SPACETILES
The Wigner distribution of a “Schr¨odinger’s cat” stateof squeezed states G ( x, p ) = ( π (cid:126) ) − e − x /ξ − p ξ (cid:126) , withsqueezing parameter ξ , is W ( x, p ) = G ( x − ∆ x, p ) + G ( x + ∆ x, p )2 (1a)+ G ( x, p ) cos (cid:18) p (cid:126) ∆ x (cid:19) . (1b)Zurek’s compass state is a coherent sum of twoSch¨odinger’s cat states rotated by π/ (a) ,Zurek showed that “Wigner functions can, and generallywill, develop phase-space structures on scales as small as,but not generally smaller than” [3] FIG. 1. (Color online) (a) shows a compass state [3] for L = P = 6 and ξ = 1 (atomic units [a.u.], (cid:126) = 1, are used in allfigures), the green frame borders its Zurek tile with area a Z . (b) , (c) and (d) are obtained using N = 8, α = 10, ξ = ,∆ x = 3 and (cid:126) = 1. (c) shows the squared wave functionΨ( x ) and its sign changes due to the complex coefficients inΨ( x ) [Eq. (5)]. (b) shows W Ψ (0 , p ), and (d) the logarithmof | W Ψ (0 , p ) | near the origin, in a panel of width h/L ≈ . h/Lα = h/L [see Eq. (8)]. a Z = hP × hL = h A . (2)Here, P and L are the phase-space distances betweenthe squeezed states along the momentum and positionaxes respectively.The Zurek scale a Z is, e.g., the phase-space area ofone fundamental tile [3] (“Zurek tile”) associated with acompass state, highlighted in Fig. 1 (a) . a r X i v : . [ qu a n t - ph ] A p r III. THE SUPEROSCILLATING CROSS-STATE
Inspired by Zurek’s compass state, we constructa “cross-state” featuring superoscillations in quantumphase-space. This state features small patches with regu-lar structures on scales much smaller than a Z , Fig. 2.We use the superoscillating function [4, 6, 7] f ( x ) = (cos( x ) + iα sin( x )) N , α > , N ∈ N . (3)For α = 1, f ( x ) = e iNx is a regular plane wave. For α > N (cid:29) f ( x ) becomes superoscillatory [seeFig. 1 (d) ] f ( x ) = N (cid:88) j =0 C j ( N, α ) e i ( N − j ) x , (4)where C j ( N, α ) = ( − j (cid:0) Nj (cid:1) ( α + 1) N − j ( α − j / N arethe Fourier coefficients [6].To map f ( x ) of Eq. (4) into phase-space, we use a su-perposition of suitably pairwise-displaced squeezed states S ( x ) = ( πξ ) − / e − x / (2 ξ ) [see Eq. (1)], to formΨ( x ) = Φ ( x ) + 1 √ N/ (cid:88) j = − N/ j (cid:54) =0 ( − i ) j Φ j ( x ) , (5)where Φ j ( x ) = K j S ( x − j ∆ x ), K j = (cid:113) | D j | / (cid:80) N/ l =0 D l and D j = (cid:40) C N/ if j = 0 ,C N/ j + C N/ − j if j (cid:54) = 0 . (6)Here, N is even and Ψ contains N + 1 spikes, seeFig. 1 (c) . The associated Wigner distribution W Ψ contains a suitable combination of plane wave terms[Eq. (1b)] to emulate f ( x ) of Eq. (4), see Fig. 1 (d) .An incoherent sum of two such Wigner distribu-tions, rotated by π/ W + ( x, p ) = [ W Ψ ( x, p ) + W Ψ ( − p, x )] /
2. This balanced mixed state features su-peroscillatory structures within Zurek tiles, on sub-Zurekscales [Fig. 2 (a) -inset, (b) and (c) ].One could use a coherent sum to form a cross, butthis would lead to greater complexity in Fig. 2, which isunnecessary to illustrate our construction.
IV. SUBSTRUCTURES WITHIN ZUREK TILES
The area of sub-Planck structures of non-superoscillating states is limited by a Z [3]. Nowwe show that regular structures on scales much smallerthan a Z can exist. FIG. 2. (Color online) (a)
Cross-state W + ( x, p ) =[ W Ψ ( x, p ) + W Ψ ( − p, x )] / N = 4, α = 6, ξ = 1, and∆ x = 6 ( (cid:126) = 1, a.u.). In (a) ’s inset and panels (b) and (c) ,red (positive) and blue (negative) regions depict superoscilla-tory structures contained within Zurek tiles (green frames). (b) and (c) show ln | W + ( x, p ) | and demonstrate the scalingwith a SO , see Eq. (8). State parameters N = 12, ξ = ,∆ x = 3, with α = 10 in (b) , and α = 16 in (c) , respectively. The local expansion of Eq. (3) around the origin hasthe local superoscillatory plane wave form f ( x ) = e N ln[cos( x )+ iα sin( x )] ≈ e iNαx e Nα x / . (7)Therefore, the superoscillating Wigner distribution W Ψ contains interference terms, equivalent to expres-sion (1b), proportional to cos( p (cid:126) N ∆ x α ) = cos( p L (cid:126) α ).Thus for W + , analogously to Zurek’s scale a Z in the com-pass state Fig. 1 (a) , superoscillatory structures with α -fold reduced length scales, Fig. 1 (d) , yielding areas onthe scale of a SO ≈ h/Pα × h/Lα ≈ a Z α (8)arise.For these superoscillatory structures to show, the ‘over-spill’ from the two adjacent squeezed states Φ − and Φ +1 has to be so small that their Wigner distributions obey W Φ − (0 ,
0) + W Φ +1 (0 , (cid:28) | W Ψ (0 , | . (9) V. CONCLUSION
We remind the reader of the fact that a quantum wavefunction cannot be strictly “bandwidth-limited”, simul-taneously in position and momentum. Our Wigner dis-tributions are confined by a finite area A in phase-space,yet they feature regular sub-Zurek scale structures, inthis sense they are superoscillating.Zurek’s compass states provide interferometric sensi-tivity at the Heisenberg limit. As Fig. 1 (c) illustrates,our superoscillating states change, under tiny displace-ments in x , p and t , in essentially the same way as regularcompass states; they therefore do not perform better atthe detection of small shifts or rotations.At this stage, the formation of structures below the Zurek scale using superoscillations in phase-space is pri-marily a surprising curiosity. Superoscillating regions areknown to be tiny in extent and amplitude [6–8]. This iswhy our superoscillating states cannot show sensitivitybelow the Heisenberg limit.We have shown, to paraphrase Zurek’s statement citedabove, that phase-space structures are frequently as smallas, but not generally smaller than a Z . Yet, superoscil-lating states can generate localized small patches withregular structures on very much smaller scales.An interesting open question raised by the existence ofsub-Planck and sub-Zurek scale phase-space structuresconcerns their potential effects on simulations. Possibly,grids finer than commonly assumed for numerical calcu-lations [10] have to be used. [1] W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, 2001).[2] E. Wigner, Phys. Rev. , 749 (1932).[3] W. H. Zurek, Nature , 712 (2001), 0201118.[4] M. Berry, Quantum Coherence and Reality; in celebrationof the 60th Birthday of Yakir Aharonov , edited by J. S.Anandan and J. L. Safko (World Scientific, Singapore,1994) pp. 55–65.[5] Y. Aharonov, J. Anandan, S. Popescu, and L. Vaidman,Phys. Rev. Lett. , 2965 (1990). [6] Y. Aharonov, F. Colombo, I. Sabadini, D. Struppa, andJ. Tollaksen, J. Phys. A: Math. Theor. , 365304 (2011).[7] M. V. Berry and S. Popescu, J. Phys. A: Math. Theor. , 6965 (2006).[8] M. S. Calder and A. Kempf, J. Math. Phys. , 012101(2005).[9] E. T. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad,M. R. Dennis, and N. I. Zheludev, Nature Mat. , 432(2012).[10] R. Kosloff, J. Chem. Phys.92