Observation of the roton-like dispersion relation in a nonlocal acoustic metamaterial
The three-dimensional (3D) experimental sample of the acoustic metamaterial with the roton-like dispersion relation is fabricated with stereolithography 3D printing technology, as shown in Fig. 1a. It consists of 50 unit cells along z direction. Each unit cell has a cube resonator with side length d1 (see the red region of Fig. 1b). For the convenience of experimental measurement, each cube resonator has a small hole for detecting the acoustic waves in the resonator. These holes are sealed when not in use and have a negligible impact on the dispersion relation of the acoustic metamaterial. The vertical cylindrical tubes with a radius of r1 connecting two nearest-neighbour resonators serve as the nearest-neighbour interactions. The oblique cylindrical tubes with a radius of r2 connecting two third-nearest-neighbour resonators serve as the beyond-nearest-neighbour interactions with an order of N=3. The hollow frame with height d2 serves as an auxiliary structure to mediate the third-nearest-neighbour interactions. The oblique cylindrical tubes connect to the auxiliary hollow frame, from which another set of oblique cylindrical tubes are connected to the third-nearest-neighbour of the starting cube resonator to trigger the long-range interaction. The cube resonators, cylindrical oblique tubes, and the auxiliary frames are filled with air and made of hard plastics to ensure that the inner surfaces meet the hard boundary condition. Besides, the acoustic metamaterial structure is achiral, which exhibits mirror symmetry and inversion symmetry.
In the experiments, the sound wave generated by a broadband speaker is guided into the waveguide through a small hole at the leftmost cube resonator. A microphone probe is inserted into the other holes at each cube resonator to measure the acoustic pressure signal. This measurement is repeated for all cube resonators of the sample one by one (see Methods). Note that all cylindrical holes are plugged except for the one that opens to measure the acoustic signal. After performing a 1D Fourier transform to the measured acoustic field distributions, we obtain the dispersion relations of the nonlocal acoustic metamaterials (see the colour plotted in Fig. 1d). One can clearly observe the pronounced roton-like dispersion relation--the dispersion band starts from zero frequency and zero momentum with linear growth at low wavevectors, then evolves continuously to reach a maximum, and the dispersion slope (i.e., group velocity vg = dω/dk) reverses from positive to negative (see Fig. 1e), displaying a pronounced “roton” minimum (white dot in Fig. 1d) before increasing again. The dispersion relation closely resembles the dispersion curve of the elementary excitations of a Bose superfluid predicted by Landau1. The experimental results agree with the simulated counterpart (see the colour curve in Fig. 1d) very well.
The roton-like dispersion relation has several intriguing features, which may find potential applications to manipulate acoustic waves in uncommon ways. First, the “maxon” maximum and “roton” minimum with zero group velocity produce singularities with an extremely high density of states that can be utilized to significantly enhance the wave-matter interactions. Second, the negative slope of roton-like dispersion relation gives rise to broadband backward waves that could lead to acoustic negative refraction and superlensing14-18 based on a new mechanism of nonlocal effect. Third, at a given frequency, the dispersion curve can support multiple coexisting Bloch modes with different wavevectors, phase velocities, and group velocities.
To understand the origin of the roton-like behaviour in acoustic metamaterials, we plot the simulated mean energy flux along the z-direction in a unit cell for three different Bloch modes at the same frequency of 1.35 kHz in Fig. 1f, including two forward modes (A and C) and one backward mode (B) with different wavevectors and group velocities. For two eigenmodes with positive group velocity (A and C), the mean energy flux in both the vertical (nearest-neighbour interaction) and oblique connecting tubes (third-nearest-neighbour interaction) is positive. In sharp contrast, for eigenmode with negative group velocity (B), the mean energy flux is positive in the vertical connecting tube but is negative in the oblique connecting tubes, which support a backward propagating wave. It is clear that the sum of the two opposite energy fluxes is negative, consistent with the negative group velocity of the dispersion relations shown in Fig. 1d-1e. The two energy fluxes move forward and backward can give rise to a vortex-like behaviour of the energy flux, very similar to Feynman’s intuitive appealing picture of rotons which correspond to some sort of stirring motions due to a “return flow”2.
To further elaborate the experimental results, the practical acoustic metamaterial structure can be translated to a one-dimensional (1D) toy spring-mass model with the acoustic resonator and tubes playing roles of mass and spring (see Fig. 1c), respectively. Here, the black straight lines (green arc lines) represent the nearest-neighbour interactions K1 (the third-nearest-neighbour interactions K3). Newton’s equation of motion in this model can be written as: See formulas 1 and 2 in the supplementary files.
Multiple rotons in acoustic metamaterials with higher beyond-nearest-neighbour interaction orders
Having established the exotic features of the roton-like dispersion in nonlocal acoustic metamaterials with long-range interactions, we discuss the dispersion engineering of the nonlocal acoustic metamaterials by tuning the long-range interaction orders. It has been theoretically predicted27 that a roton-like minimum occurs if and only if the interactions order is sufficiently high (i.e., N ≥ 3) and the strength of the beyond-nearest-neighbour interactions is sufficiently large (i.e., KN/K1 >1/N). More remarkably, the number of roton-minimum and the negative slope region of the roton-like dispersion can be controlled by tailoring the long-range interaction order. To verify the influence of the order of beyond-nearest-neighbour interactions on the dispersion band, we design and fabricate three acoustic metamaterials with second- (N=2), fourth- (N=4), and fifth-nearest-neighbour (N=5) interactions, respectively. Their unit cells are schematically shown in Figs. 2a, 2e and 2i, respectively, where the vertical cylindrical tubes (blue region) provide the nearest-neighbour interaction (N=1) and the oblique cylindrical connecting tubes with the auxiliary hollow frames (beige regions) provide the second- (N=2), fourth- (N=4) or fifth-nearest-neighbour (N=5) interactions independently.
For the case of N=2, the 1D toy model with second-nearest-neighbour interactions K2 (red arc line) is shown in Fig. 2b. The measured (red colour) and calculated (colour line) band dispersions only exhibit a maxon-maximum, and its group velocity switches sign for only once, as shown in Fig. 2c and 2d, respectively. In this situation, there are only two coexisting eigenmodes with different wavevectors and group velocities at the same frequency. This dispersion relation has no roton minimum and is similar to traditional metamaterial dispersion with a single negative slope region. We then study the case of N=4, whose 1D toy model with the fourth-nearest-neighbour interactions K4 (blue arc line) is shown in Fig. 2f. For the case of N=4, the measured (red colour) and simulated (colour line) dispersion relations are more complex with the sign of group velocity switches three times and support four different coexisting acoustic eigenmodes at one single frequency, displaying two maxon-maximums and one roton-minimum (white dot), as shown in Fig. 2g and 2h, respectively. Now we continue to increase the long-range interaction order and explore another case with N=5, whose 1D toy model with the fifth-nearest-neighbour interactions K5 (purple arc line) is shown in Fig. 2j. The measured (red colour) and simulated (colour line) dispersion relations reverse four times and support five different coexisting acoustic eigenmodes at one single frequency, displaying two maxon-maximums and two roton-minimums (white dots), as shown in Fig. 2k and 2l, respectively. Note that the bright regions above the roton-like dispersion in Figs. 2g and 2k come from a higher band. The mean energy fluxes of different coexisting eigenmodes at the same frequency of 1.35 kHz for N = 2, 4 and 5 are shown in Fig. S2. Both the measured and simulated results reveal, as expected, roton behaviour occurs only for beyond-nearest-neighbour interaction order N ≥ 3 and more slope inversions, and roton minimums can be brought into the first Brillouin zone by increasing the long-range interaction order. Note that multiple classical rotons on a single dispersion curve achieved when N ≥ 5 have never been experimentally demonstrated in both the quantum and classical-wave systems before.
Roton-like dispersion relation in a chiral acoustic metamaterial
Recently, it has been theoretically reported that the roton-like dispersion relations for transverse acoustical elastic waves can be realized in noncentrosymmetric micropolar crystal based on chiral micropolar elasticity theory, where chirality is a necessary condition for roton-like behaviours30. Hence, it is natural to ask whether chirality plays an important role in the roton-like behaviours in our airborne acoustic system. Here, we introduce chirality into our acoustic metamaterials with N=3 by twisting and doubling the oblique connecting tubes to construct an acoustic metamaterial with four-fold rotational symmetry around the z-axis, whose unit cell is schematically shown in Fig. 3a. As both the inversion and mirror symmetries are absent, the newly designed acoustic metamaterial is structurally chiral. Interestingly, both the measured (red colour) and simulated (colour line) dispersion relations of the chiral acoustic metamaterial (Fig. 3b) display a roton-like behaviour which is very similar to that of the achiral acoustic metamaterials in Fig. 1d. The energy flux of the three coexisting modes shown in Fig. 3c further confirmed the similarity between the chiral and achiral acoustic metamaterials with beyond-nearest-neighbour interactions. Therefore, we have experimentally verified that both the chiral and achiral acoustic metamaterials with beyond-nearest-neighbour interactions can support roton-like dispersion relations with similar properties.