Abstract
We discover and characterise strong quantum scars, or quantum eigenstates resembling classical periodic orbits, in twodimensional quantum wells perturbed by local impurities. These scars are not explained by ordinary scar theory, which would require the existence of short, moderately unstable periodic orbits in the perturbed system. Instead, they are supported by classical resonances in the unperturbed system and the resulting quantum neardegeneracy. Even in the case of a large number of randomly scattered impurities, the scars prefer distinct orientations that extremise the overlap with the impurities. We demonstrate that these preferred orientations can be used for highly efficient transport of quantum wave packets across the perturbed potential landscape. Assisted by the scars, wavepacket recurrences are significantly stronger than in the unperturbed system. Together with the controllability of the preferred orientations, this property may be very useful for quantum transport applications.
Introduction
Quantum scars^{1} are enhancements of probability density in the eigenstates of a quantum chaotic system that occur around short unstable periodic orbits (POs) of the corresponding classical system. Scars have been observed experimentally in, e.g., microwave cavities^{2,3}, optical cavities^{4,5}, and quantum wells^{6,7}, and computationally in, e.g., simulations of graphene flakes^{8} and ultracold atomic gases^{9}.
Before the existence of scars was reported by Heller^{10}, eigenstates of a classically chaotic system were conjectured to fill the available phase space evenly, up to random fluctuations and energy conservation. If highenergy eigenstates of nonregular (i.e., generic) systems were indeed featureless and random, controlled applications in that regime would be difficult. Scars are therefore both a striking visual example of classicalquantum correspondence away from the usual classical limit, and a useful example of a quantum suppression of chaos.
In this work we describe quantum scars present in otherwise separable systems disturbed by local perturbations such as impurity atoms. In this case, scars are formed around POs of the corresponding unperturbed system. The scars are strikingly strong and common, and they have properties that cannot be explained by ordinary scar theory. Instead, we show that the symmetry of the unperturbed system and the local nature of the perturbations give rise to a new scarring mechanism.
In the following, values and equations are given in natural units where the quantum Hamiltonian is simply , where V is the unperturbed potential and V_{imp} represents the perturbation. The eigenstates of H are solved with imaginary time propagation in real space^{11}.
Model system
The scarring mechanism, explained later in this article, is very general; it requires only that the unperturbed system is separable, and that the perturbing impurities are sufficiently local. In the following we focus, for simplicity, on a few prototypical examples of a circularly symmetric, twodimensional potential well V(r) perturbed by randomly scattered Gaussian bumps.
The classical POs of any circularly symmetric V(r) can be enumerated directly (see Supplementary Section S1 and ref. 12). Each PO is associated with a resonance, where the oscillation frequencies of the radial and angular motion are commensurable. The PO structure is especially simple if V(r) is a homogeneous function (i.e., V(r) ∝ r^{a}), since then POs with different total energies differ only by a scaling, i.e., the shape of POs does not depend on the energy.
In the following we use a = 5. Its shortest nontrivial PO, i.e., besides the circular orbit and the case with zero angular momentum, is a fivepointed star, which is both easily distinguishable and short. This PO corresponds to a 2 : 5 classical resonance, where the orbit circles the origin twice in the time of five radial oscillations. The case of a = 2, the harmonic oscillator, is special and illsuited for drawing general conclusions, while for a = 1, 3, 4 the shortest nontrivial POs are longer.
For easier comparison between the results, we focus in the following on a single example shown in Fig. 1(a), where Gaussian bumps with amplitude M = 24 are distributed in an unperturbed potential . The average density of the bumps is two per unit square, and thus for a typical energy of 500, approximately a hundred bumps exist in the classically allowed region. The full width at half maximum (FWHM) of the Gaussian bumps is 0.235, which is similar to the typical local wavelength of the eigenstates we consider.
The amplitude of the bumps is small compared to the total energy, making each individual bump a small perturbation. Nevertheless, together the impurities are sufficient to destroy classical longtime stability; any stable structures present in the otherwise chaotic Poincaré surface of section are tiny compared to ħ = 1.
Scar observations
Figure 1(b) shows an example of a scar found in the eigenstates of the example potential described previously. In this case approximately 80% of the probability density resides on the star path. Figure 1(c–e) show examples of scars in a homogeneous potential with exponent a = 8, and in a nonhomogeneous potential V(r) ∝ cosh(r) − 1. For more discussion about scars in other potentials please see Supplementary Sections S3 and S4. In all cases the scars follow POs in the corresponding unperturbed system. We emphasise that the strong scars are not a rare occurrence; for the potential in Fig. 1(a) over 10% of all eigenstates are clearly scarred by the fivepointed star orbit.
In ordinary scar theory^{10}, each scar corresponds to a moderately unstable PO in the classical system. In this case such orbits do not exist. For example, the shortest and least unstable PO near the scar shown in Fig. 1(f) for M = 16 closes on itself after two rounds around the scar, and has a oneperiod stability exponent^{13} χ ≈ 5. In linear scar theory^{1,14} this is too unstable to cause observable scarring, since scarring strength falls of exponentially with χ, although nonlinear contributions from homoclinic orbits (HOs)^{15,16,17} could conceivably increase the scarring strength in some cases assuming lucky constructive interference between many HOs.
Ordinary scar theory is more firmly excluded by the behaviour of the scars as a function of the bump amplitude M. If M is increased while keeping V_{imp} otherwise unchanged, the scars grow stronger and then fade away without changing their orientation, as shown in Fig. 1(f). With or without nonlinear contributions from HOs, a scar caused by ordinary scar theory would become rapidly weaker, since the stability exponent of a PO increases with M.
Comparing scars at different energies E also reveals that they occur in only a few distinct orientations, and these orientations change quite slowly with E. Later in this manuscript this stability is quantified by wavepacket methods. For the example V_{imp} used here there are three preferred orientations. For other impurity realisations the number and location of the preferred orientations vary, but the existence of preferred scar orientations is a generic feature (see Supplementary Section S5). This feature is also not explained by ordinary scar theory, as the existence and stability of POs should be very sensitive to changes in E.
We also note that the scars are not explained by dynamical localization^{18,19}. Although present in systems discussed here, dynamical localization corresponds to localization in angular momentum space, whereas the scars are localized in position space. Dynamical localization also does not explain the preferred orientations.
Wavepacket analysis
Gaussian wave packets are a standard tool for studying eigenstate scarring^{10}. If a Gaussian wave packet ϕ〉 is centred on a scarring PO, its autocorrelation function A(t) = 〈ϕ(0)ϕ(t)〉 shows clear shorttime recurrences with a period T matching the period of the PO. In the linear theory of ordinary scarring the strength of these recurrences A(nT)^{2} dies out as 1/cosh(χn), where χ is the stability exponent of the PO^{14}. In addition, 〈ϕ(0)ψ〉 is large for an eigenstate ψ〉 scarred by the particular PO, making wave packets useful for picking out eigenstates scarred by a particular PO (see Supplementary Section S2).
In our case we use the PO of the unperturbed system to initialise the wave packet. The energy and the orientation of the PO are matched to the scarred eigenstate. The width parameters of the Gaussian are matched approximately to the geometry of the scar (see the inset in Fig. 2).
The wave packets selected in this way have a considerable energy uncertainty, so that many scars, sharing the same approximate orientation, contribute to the recurrences. A typical FWHM of the wave packet is approximately 50 energy units, or 400 eigenstates.
Figure 2 shows the recurrence strength A(t)^{2} for a wave packet travelling on the scar shown in Fig. 1(b). Clear periodic recurrences are visible, with a period that matches the period of the unperturbed PO.
To account for purely classical effects, we compare the results with the recurrence of the corresponding classical density. This is given by the Wigner transform G of the wave packet. Its recurrence strength was calculated by sampling 80 000 classical initial states {(r_{i}, p_{i})}_{i} from the distribution G, propagating each in time, and computing . Classical time integration was performed with the sixthorder symplectic integrator of Blanes & Moan^{20}. Once normalised so that I(0) = 1, I(t) corresponds to the quantum recurrence strength. The classical recurrences are significantly weaker than the quantum recurrences even at t = T, and this difference grows rapidly at later times, illustrating the quantum nature of the phenomenon.
The existence of stable preferred scar orientations can be demonstrated by systematically detecting scarred states with wave packets. Figure 3 shows how the overlap of the initial wave packet with the target eigenstate depends on the energy of the eigenstate and the orientation of the PO used to initialise the wave packet. The orientation coordinate α is such that at α = 0 the wave packet starts on the positive yaxis and heads to the right.
In an angular window of 2π/5 (after which the PO is the same), three branches of high overlaps are visible, corresponding to the preferred orientations. Note that the rightmost branch is roughly vertical for E = 400 …700, corresponding to an increase of the average radius of the PO by roughly 0.3 units, which is the length scale of the individual bumps.
Figure 4 shows how the amplitude of the recurrence peaks in Fig. 2 depend of the orientation angle α. Both quantum and classical wavepackets show stronger shorttime recurrences at the preferred orientations, indicating that the preferred orientations can also be explained classically. However, especially at the preferred orientations, quantum latetime recurrences are much stronger than in the classical case.
Note that the quantum recurrences are stronger than the classical ones even on average, suggesting that there is also an effect that strengthens quantum recurrences at all orientations.
To highlight how strong the quantum recurrences are, a comparison to recurrences in the unperturbed system is also shown in Fig. 4. For the preferred orientations, the quantum latetime recurrences greatly exceed the strength of both the quantum and the classical recurrences in the unperturbed system. Via the creation of strong scars with stable preferred orientations, the randomly scattered impurities enhance the coherent propagation of quantum wave packets in the potential well!
Perturbation Theory
Both the existence of scars and the preferred orientations can be explained by perturbation theory. This explanation is based on two ingredients. Firstly, special nearlydegenerate subspaces exist in the basis of unperturbed eigenstates. Secondly, the local perturbations select scarred eigenstates from these subspaces.
The eigenstates of the unperturbed, circularly symmetric system are labelled by two quantum numbers (r, m), corresponding to radial and angular motion, respectively. States (r, ±m) are exactly degenerate, but in addition there are neardegeneracies that correspond to classical POs.
By the Bohr–Sommerfeld quantisation condition (a good approximation at high quantum numbers) the energy difference from increasing r or m by one is proportional to the classical oscillation frequency of the corresponding action. Thus, if a state (r, m) is nearby in action to a classical PO with a ratio a : b between the oscillation frequencies, the state (r + a, m − b) will be nearby in energy. The smaller a and b are the closer the neardegeneracy is. This creates “resonant sets” of unperturbed basis states, and a part of a resonant set can be almost degenerate. Expanding in the unperturbed basis reveals that the scarred eigenstates are localised to such neardegenerate subspaces.
A superposition of two resonant states will exhibit beating in both the radial and angular directions. Because the ratio of the beat frequencies is also a:b, the interference pattern will trace out the shape of the classical PO. Adding more resonant states with appropriate phases will narrow the region of constructive interference and sharpen the scar, and even a few basis states can create a distinctly classicallooking linear combination.
Similar reconstruction of classicallike states from (nearly) degenerate basis states has also been studied previously^{21,22,23,24}. However, to create a lot of strongly scarred eigenstates with preferred orientations a mechanism that favours these scarred linear combinations is required.
Within the impurity strength regime that results in the strongest scarring, the perturbation V_{imp} couples predominantly only states that are nearly degenerate. One can therefore approximate the perturbed eigenstates by degenerate perturbation theory (DPT), i.e., by diagonalising V_{imp} within the neardegenerate subspace of resonant states.
The effect of nonresonant states that are nearly degenerate by chance is mitigated by the weak coupling of states with very different quantum numbers. The product of the wave functions of two such states oscillates rapidly even on the length scale of a single perturbation bump. As a result, the coupling term 〈ψ_{1}V_{imp}ψ_{2}〉 decreases rapidly as a function of the difference in quantum numbers between the states ψ_{1}〉 and ψ_{2}〉.
The DPTproduced eigenstates corresponding to the extremal eigenvalues of V_{imp} are, by the variational principle, the states ψ〉 that extremise the expectation value 〈V_{imp}〉: = 〈ψV_{imp}ψ〉. This quantity for a nonscarred, spatially delocalised state is essentially an average value of V_{imp} over the entire accessible region, and never an extremum. Compared to the nonscarred states, the scarred states in a neardegenerate subspace have their probability amplitude concentrated in a much smaller region of space. For a V_{imp} consisting of local impurities the state that maximises (minimises) 〈V_{imp}〉 will therefore generally be a scarred state oriented so as to coincide with an anomalously many (few) impurities.
By the previous argument the scar orientations are mostly selected by the positions of the impurities. Since the inner and outer radii of the POs change slowly with energy, the orientations that extremise 〈V_{imp}〉 will be determined largely by the same impurities for many different resonant sets. This is the origin of the stability of the preferred orientations seen in Fig. 3.
Though a simple DPT approximation does not reproduce the true eigenstates exactly, it both explains the scarring and predicts the orientations of the observed scars well, as illustrated in Fig. 5. Taking into account the imperfect degeneracy amounts to diagonalising the full Hamiltonian instead of only the perturbation^{25}. This means that in the matrix that is diagonalised the diagonal elements are shifted by the (small) spacings of the neardegenerate basis states. This “quasiDPT” (qDPT) approximation improves the agreement between the DPT approximation and the exact eigenstates, as shown in Fig. 5(b).
Summary and Outlook
To summarise, we have shown that a new type of quantum scarring is found in separable systems perturbed by local impurities. The scars are very common, and they tend to occur in discrete preferred orientations. This allows wave packets to propagate through the perturbed system with higher fidelity than in the unperturbed system.
The theoretical basis of the scarring is very general, requiring only classical resonances and local perturbations. Therefore the scarring should have consequences well beyond the simple models discussed here.
The implications of the enhanced wave packet recurrences for quantum transport will be an important area for future work. In an experiment, local perturbations similar to the ones used here could be generated by a conducting nanotip^{26,27,28}, selecting particular scar orientations and enhancing the local conductance in a controlled way.
Additional Information
How to cite this article: Luukko, P. J. J. et al. Strong quantum scarring by local impurities. Sci. Rep. 6, 37656; doi: 10.1038/srep37656 (2016).
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Acknowledgements
P.J.J.L. thanks the Finnish Cultural Foundation, the Magnus Ehrnrooth Foundation, and the Emil Aaltonen Foundation for financial support. L.K. acknowledges support from the U.S. NSF under Grant No. 1205788, and E.R. from the from the Academy of Finland under project no. 126205. A.K. acknowledges that this work was supported by the STC Center for Integrated Quantum Materials, NSF Grant No. DMR1231319. We are grateful to CSC – the Finnish IT Center for Science – for providing computational resources for numerical simulations. We wish to thank Li Ge and E. G. Vergini for useful discussions.
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The scars were initially observed by P.J.J.L. and E.R. The existence and preferred orientations of the scars were explained with perturbation theory by B.D., A.K., L.K. and E.J.H. All authors contributed to the theoretical understanding and further research of the phenomenon. The numerical calculations were performed by P.J.J.L.
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Luukko, P., Drury, B., Klales, A. et al. Strong quantum scarring by local impurities. Sci Rep 6, 37656 (2016). https://doi.org/10.1038/srep37656
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