We prove a new asymptotic formula for the Wigner d-functions, as a consequence of the discovery of superoscillatory behaviour in SO(3).
Superoscillations were introduced by Aharonov et al. (1988, 1990) and Berry (1994). A superoscillatory function is ‘band-limited’. That is, its Fourier representation has a maximum wavenumber (or spatial frequency), say , but it can nevertheless oscillate, over a substantial region, at a much faster rate, say by a factor a, arbitrarily larger than 1 and therefore well outside the ‘allowed’ spectrum: 1.1
A related phenomenon is super-resolution. Naively, one would think that waves of the form given by the left-hand side of equation (1.1) cannot distinguish features smaller than its smallest wavelength which is . Nevertheless, a much better resolution of Δx=π/a can be achieved within the superoscillating region.
Of great interest is the recent rapid application of superoscillations as a tool to obtain resolution far better than allowed by the diffraction limit, even in a purely classical optical setting. For example, a recent report on superoscillatory experiments (Rogers et al. 2012) states: ‘super-oscillation-based imaging has unbeatable advantages over other technologies’. In the past, super-resolution of very fine features was achieved with a very different type of phenomena, namely with evanescent waves. Superoscillations have many advantages over evanescent waves, such as the ability to penetrate much deeper into the media.
In a broader context, superoscillations are examples of unusual weak values (Aharonov & Vaidman 1990; Aharonov & Rohrlich 2005; Aharonov et al. 2010) that can be obtained for pre- and post-selected quantum systems. Using this intuition, it has been shown that regions of superoscillations are surprisingly typical in random fields (Dennis et al. 2008). The related issue of having weak values outside the spectrum of the operators involved has been discussed at length in the past and has most comprehensively been investigated for spin- systems (Berry 1994; Aharonov & Botero 2005; Berry & Popescu 2006; Berry & Dennis 2009; Berry et al. 2011, 2012; Botero in press). In addition, Berry et al. (2012) looked at superweak statistics for much more general situations, and the authors proved that if the Hilbert space is sufficiently high in dimensions, and if the pre- or post-selection are, in a sense, ‘generic’, then the existence of superweak values becomes common or typical. In general, the regions of superoscillations are created at the expense of having the function grow exponentially in other regions. It would therefore seem natural to expect that the superoscillations would be quickly ‘over-taken’ by tails coming from the exponential regions and would thus be short-lived. However, it has been shown that superoscillations are remarkably robust (Berry & Popescu 2006) and can last for a surprisingly long time (see also Aharonov et al. in press b). Furthermore, from the perspective of communication theory, it has been shown that this relationship is also related to a trade-off between signal-to-noise ratio and bandwidth (Ferreira & Kempf 2004), making it easier to engineer superoscillatory signals (Ferreira & Kempf 2006). Finally, the mathematical foundations for superoscillations have been clarified in a recent series of papers (Aharonov et al. 2011, in press a,b).
In this study, we show that superoscillations also occur in the framework of group representation theory where the analogue of the Fourier series are decompositions in terms of the special functions representing Lie groups. Although we analyse the case of the group SO(3), our results should extend to SO(n) for any integer n; we will come back to this in a forthcoming paper. The main by-product of superoscillations in SO(3) (and the key result of this study) is a new, unexpected asymptotic relationship involving the well-known Wigner d-functions (see Sakurai 1995; Dennis 2004). We also discuss the physical implications.
Before concluding §1, we want to offer a heuristic argument that suggests these superoscillations in SO(3). It is motivated by the same general reasoning that is applied when the weak value of operators is computed in the time-symmetric formulation of quantum mechanics (Aharonov et al. 2010). Aharonov et al. (1988) introduced the notion of the weak value of an operator . When a system is initially pre-selected in a state |Ψin〉 and subsequently post-selected in a final state |Ψfin〉, then the result of any measurement of an operator , which is performed in a gentle way during the intermediate time, is the weak value: 1.2As many experiments have confirmed, Aw can lie arbitrarily far outside the spectrum of a bounded operator 's eigenvalues. Unlike the expectation value, it can even be complex.
Let , , be the quantum operators associated with the components of angular momentum. These can be interpreted as the three generators of SO(3). Let |Ψin〉 and |Ψfin〉 be the states defined by 1.3and 1.4where is connected with the total angular momentum via (and/or ). Observing that we may compute the weak value of (Lz)w: 1.5 1.6In the first term of equation (1.6), the left-pointing arrow means that we apply to the left on 〈Ψfin| and, using equation (1.3), replace the resultant with 〈Ψfin|ℓ. Similarly, in the second term, the right-pointing arrow means that we apply to the right on |Ψin〉 and, using equation (1.4), replace the resultant with ℓ|Ψin〉.
We can now compute the weak value of with the same pre- and post-selection, assuming φ small, as follows:1 Therefore, for small φ, the last bracket can be re-exponentiated which suggests the existence of superoscillations in φ with a frequency much larger than ℓ (the maximal eigenvalue of ) that occurs when is very small.
To show this, using equation (1.1), we write 1.7where, again, and . The Baker–Campbell–Hausdorff identity for any two operators and is 1.8If and all the repeated commutators involving s and s can be neglected, then we may rewrite equation (1.7) as 1.9where, as mentioned earlier, we operate with to the right and with to the left. In the following, we restrict ourselves to large ℓ and small variations of φ so that φ=φ′/ℓ, with φ′ bounded. The first commutator that we need to consider is then where we used the fact that in general is bounded by ℓ. In the particular case of interest, is almost orthogonal to both angular momenta and defining the initial and final state so that its value is O(1) and the commutator is in fact approximately 1/ℓ2. Furthermore, k-fold repeated commutators in the BakerCampbellHausdorff (BCH) formula will always give us just one angular momentum operator with some pre-factor depending on θ which is bounded by one and with 1/ℓk suppressions. The strongly convergent nature of the series in the BCH formula allows us then to neglect all the commutators in the large ℓ limit. This then recoups equation (1.9) that exhibits superoscillations, namely extremely strong sensitivity to small rotations.
For the reader's convenience, we recall some notations and properties of quantum operators for the case of spin-. The three well-known matrices , , represent the spin- operators: It is immediate to verify that , , satisfy the commutation relations and We also have the raising and lowering operators: 2.1which satisfy 2.2
If we now consider a system of N non-interacting spin- particles with associated commuting operators ( operates only on particle j), then the total spin operator for all N particles is defined by the operator It is immediate to verify that the commutation relations between the individual spin- operators, , translates into the same commutation relations for the N particle system, :
In the product wave function |Ψ[N]〉, with each individual particle, j, in the state 2.3we have 2.4and 2.5
Equation (2.4) then follows directly from the definition and hence . Equation (2.5) then follows from 2.6To verify this last statement, we expand and in terms of the individual raising and lowering operators and (equation (2.1)). We may write the mixed terms in equation (2.6) as (for example) which clearly vanish owing to the presence of or that operate on and , respectively, thereby yielding 0 by equation (2.2). This leaves only the diagonal terms so that because and and we have N such terms.
From now on, we will consider systems with an even number of particles and we set N/2=ℓ.
The calculations we have carried out in this section have assumed that we are looking at the system of 2ℓ particles,2 all of which have spin- and this has allowed us to calculate the eigenvalues for , and . One could have, however, performed the same calculations for a system with ℓ+m particles in the state and ℓ−m particles . Clearly, the case we looked at is the case m=ℓ, but we expect that the superoscillatory behaviour we present in §3 to occur in the other cases as well. We will return to this situation in our next paper.
3. Weak values and superoscillations
In the following, we will again use the N spin- particles, but now we use an initial state Analogously, we use a final state |Ψ[N]in〉 and |Ψ[N]fin〉 can be realized in two distinct ways. Consider first |Ψ[N]in〉 that is the eigenfunction of the angular momentum operator , i.e. the result of rotating of the original spin- particles (equation (2.3)) by an angle θ (close to π/2) with respect to the y-axis that is realized by the operation . In the spin- product basis, this is obtained by . This amounts to rotating each of the individual spin- states in the same manner, yielding for each of the N spin- particles. Likewise, |Ψ[N]fin〉, the corresponding eigenstate of with maximum eigenvalue m=ℓ, is obtained by rotating |Ψ[N]in〉 by ϕ=π around the z-axis. In the z-basis of the original N spin- particles, the rotation of each individual spin- by ϕ modifies the above |ψin〉 to Thus, |ψfin〉 can be written as (apart from an overall phase of ei(π/2)): We now compute the weak value of the operator for a small angle δφ. To begin with, we note that the scalar product of the initial and final state for a single spin- particle is so that the scalar product of the initial and final state for the system of the N independent spin- particles is We now set δφ′=ℓδφ and compute the weak value 3.1Because , it suffices to calculate one of the products and take the result to the Nth power. Equation (3.1) can therefore be rewritten 3.2Because the eigenvalues of in the state [1 0]T and [0 1]T (where the superscript T denotes the transpose) are respectively and we find that for |ψin〉: Computing the scalar product with the post-selected state, we have 3.3When we take N=2ℓ spins, we finally obtain the weak value 3.4A key observation is that this has the same form as the canonical x-space superoscillating function (Aharonov et al. 2011; Berry et al. 2012): 3.5where .
Because can be arbitrarily small, depending on the pre- and post-selection, the weak value of (which is ) can be arbitrarily large, well outside the spectrum of which is [−ℓ,ℓ]. If we perform a binomial expansion of f(x), then we see that the smallest wavelength in the expansion is one. However, around , f(x) can be approximated as f(x)≈ei2πax, that is, with a wavelength much shorter than one, seemingly a violation of the Fourier theorem. This surprising phenomenon is very general and holds for a wide range of functions and coefficients. In our particular case, when ℓ becomes large, this expression behaves like which, recalling that δφ′=ℓδφ and δφ is small, is just .
This SO(3) superoscillatory phenomenon depends on the choice of |Ψ[N]fin〉. In the example mentioned earlier, we generated |Ψ[N]fin〉 by rotating |Ψ[N]in〉 by an angle φo:=φ=π. And around the resultant, we performed a further small rotation δφ. In general, this phenomenon depends on the choice of φo.
To better understand the conditions under which superoscillations occur, we consider the case , corresponding to φ=0. In this case, we can repeat the argument given earlier, where |ψfin〉 is now given by the vector so that 〈Ψ[N]fin|Ψ[N]in〉=1, and therefore the weak value is given by because , superoscillations do not occur.
We now look at the case in which we consider a small rotation δφ centred around φ0 and for simplicity we will still consider |Ψin〉=|Ψfin〉. The same computations we have carried out before show that the expression is given by and because δφ′ is small, we can approximate the above expression as By taking the 2ℓ-th power, we obtain The right-hand side for ℓ large converges to ei(a+ib)(δφ′/ℓ), where a corresponds to the imaginary part of that equals We now note that if φ0 approximates 0, then and thus there are no superoscillations, while if φ0 is near π then and the superoscillation phenomenon occurs. This is fully consistent with the previous analysis.
To conclude this section, we note that if we apply the same argument to the case in which Ψfin is associated with the vector , we obtain that the frequency is, up to the factor δφ/ℓ, given by which reproduces the superoscillations when
4. Asymptotic expansion for the Wigner functions
We next use the above discussion to show a new interesting relationship for Wigner's matrices. These matrices are the representation of the rotation group in the standard |ℓ,m〉 basis defined by 4.1The are orthogonal functions on the sphere, and for any given θ, form a (2ℓ+1)(2ℓ+1) unitary matrix. In general, are polynomials in and . This polynomial simplifies into just one term for the special case that interests us here m=ℓ (Sakurai 1995): Thus, is the (j−m′)-th term in the binomial expansion of As a consequence, we have as expected . We can now compute the scalar product 〈Ψfin|Ψin〉 in two different ways that gives us the following (probably very well-known) result.
With the notations mentioned earlier, for any angle θ, we have
We consider the initial and final state |Ψin〉, |Ψfin〉 introduced earlier. We first note that and, owing to the π rotation around the z-axis (leading to the (−1)m′ factor): and so the scalar product of the initial and final state , which we found above (equation (1.3)), is given by We can also compute the same scalar product directly and the statement follows.
More interesting and novel, however, is the following asymptotic formula.
With the notations mentioned earlier, for any angle θ and any small value of δφ we have, for large ℓ: where which, again, is of the canonical superoscillating form, equation (3.5).
The weak value of is given by equation (1.3). On the other hand from which the statement follows.
We can also directly compute yielding the same result as shown earlier.
The phenomenon of superoscillations is well-known and well-studied for the case of spatial oscillations. Our main purpose here was to extend the phenomena of super-oscillations to angular superoscillations and to express these in terms of new relations for the rotation matrices .
We motivated the phenomenon with a heuristic argument, involving unusually large weak values generated by appropriate pre- and post-selections. Specifically, in the present case, we chose the angular wave functions to be the almost orthogonal eigenstates of (pre-selected) and (post-selected). With θ near , the weak value of is far outside its usual spectral range of [−ℓ,ℓ].
Originally, the study of the weak value started as a curious observation used to probe the foundations of quantum theory (Aharonov & Vaidman 1990; Aharonov et al. 2002; Aharonov & Rohrlich 2005). However, soon it was experimentally confirmed in many physical settings and found many applications. For example, large weak values have been used as a technique in quantum metrology to amplify small shifts in order to discover unknown aspects of a system's parameters. This technique also reduces technical noise, and, because it does not rely on entanglement (as do other quantum metrology approaches), it is much more robust to noise. The precision that has been reached in these experiments is unprecedented in both polarization (Hosten & Kwiat 2008) and interferometry (Dixon et al. 2009a,b) (resolving sub-picometre scales).
We strongly believe that this will also occur for the case of large weak values of angular momenta and attendant angular superoscillations presented in this study.
Thus, let us consider a system of many identical atoms prepared in very high Rydberg states. These states can have large angular moments (ℓ∼n) and can also live sufficiently long in order to perform the following experiment: using two consecutive Stern–Gerlach set-ups with magnetic fields equal in magnitude yet pointing in almost exactly opposite directions, we can pre- and post-select the states discussed earlier. We then split the pre-selected atoms into two paths. Along one path, we apply a weak magnetic field in the z-direction. We then re-combine the two beams, perform the post-selection, and then perform an interference experiment. The large intermediate weak value interacts with the applied weak magnetic field along one of the paths and this would then manifest as an ultra-sensitivity (via the shift in interference pattern)—within the pre- and post-selected sample—to weak magnetic fields in the region of space intermediate between the two Stern–Gerlach set-ups—weak fields that point along the z-axis bisecting the angle between the above two almost opposite directions and almost orthogonal to each.
In contrast to atomic interferometry, quantum optics has in the past provided a much better testing ground for a variety of quantum effects and weak measurements in particular. It may well be the case for the angular superoscillations as well. Thus, laser light beams could be used where each of the photons carries not only spin polarization of ±1, but also significantly higher orbital angular momenta. Such beams have been experimentally constructed (Yao & Padgett 2011). The large number of photons in such systems may allow us to overcome the barrier of having typically only a small fraction of all photons in the pre- and post-selected ensemble, and the relative ease of manipulating such beams may both help to find physical manifestations of the large weak values of angular momenta and attendant angular superoscillations. For example, there are several analogues of the Zeeman effect but with the use of orbital angular momentum of light. The rotational Doppler effect could be used to implement the weak measurement interaction with a rotating Dove prism (Bliokh 2006; Bliokh et al. 2009). This would induce a relative phase shift. Methods exist to implement a post-selection (Berkhout et al. 2010; Lavery et al. 2011). One could use this type of measurement in the same applications which measure small rotations with Sagnac interferometers (Dixon et al. 2009a,).3
We have shown that superoscillatory phenomena occur in group theory too, and in this study, we have specifically shown this behaviour for the case of SO(3). As a consequence, we have been able to discover a new asymptotic relation for the Wigner d-matrices. We believe that it will be possible to generalize this result to other groups of physical interest.
This work was supported in part by Israel Science Foundation (grant no. 1125/10), and The Wolfson Family Charitable Trust. F.C., and I.S., D.C.S. and J.T. are also grateful to Tel Aviv University for the kind hospitality during the period in which this paper was written. We thank Franco Nori and Konstantin Bliokh for discussions.
↵1 This is true under the assumption of sufficient regularity for the functions on which the operators act.
↵2 ℓ denotes the orbital angular momentum quantum number and is related to the angular momentum by (though we set throughout this study); m is the magnetic quantum number and is the z component of L.
↵3 Another physics-motivated study (Dennis 2004) that derives many other classical results concerning the rotation group (particularly from the spinor perspective) has been pointed out to us by one of the referees.
- Received March 1, 2012.
- Accepted June 1, 2012.
- This journal is © 2012 The Royal Society