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arXiv:1710.10302 · American Journal of Physics

Airy wavepackets are Perelomov coherent states

Vivek M. Vyas

Published Sep 19, 2018• 6 citations
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Abstract

Accelerating non-spreading wavepackets in a nonrelativistic free-particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs [Am. J. Phys. 47(3), 264–267 (1979)], and have been subsequently realised in several optics experiments. It is shown that these wavepackets are actually Perelomov coherent states. It is found that the Galilean invariance of the Schrödinger equation plays a key role in making these states unique and gives rise to their unusual propagation properties.

DOI
10.1119/1.5051181

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Airy wavepackets are coherent states

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Accelerating non-spreading wavepackets in nonrelativistic free particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs (Am. J. Phys., 47, 264 (1979)), and have been subsequently realised in several optical experiments. It is shown that these wavepackets are actually coherent states. It is found that the Galilean invariance of the Schrodinger equation plays a key role in making these states unique and giving rise to their unusual propagation properties.

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In 1979, Berry and Balazs [M V Berry and N L Balazs 1979 Am. J. Phys. 47 264] obtained a strict solution of the Schrödinger equation with Airy function used as the initial condition, and described the wave function represented by such solution as the Airy wave-packets. They discovered that infinite Airy wave-packet has unique properties such as non-spreading and free acceleration, proving that it is the only nontrivial non-spreading solution of the time-dependent Schrödinger equation in one dimension. However, the observing of the finite Airy beam seems to be more meaningful since wave-packets in reality is inevitably band limited. A certain form of finite Airy beam was investigated by Siviloglou et al. in 2007 [Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901; Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979]. They noted that the Airy wave packet still exhibits its most exotic feature, i.e., its trend toward free acceleration. While in the present paper we discuss the properties of Airy beam in a few steps further and propose several conclusions. On the one hand, a theoretic explanation is given to solve the matter of the centre of mass of infinite Airy beam. On the other hand, deeper research is conducted on the unique properties of finite Airy beam. Another form of finite Airy beam is discussed by reduction to absurdity, and its field distribution is put forward by numerical simulation. We find that the trajectory of the centroid holds its position, which means that the beam cannot accelerate freely as a whole. Ultimately, we have the conclusion that finite Airy beam can neither freely accelerate nor be non-diffractive.

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We present the solution of Schrödinger’s equation for time-dependent quadratic potentials in terms of an Airy packet. We demonstrate new features of the Airy packet in light of its apparent unfamiliar properties discovered by Berry and Balazs (who found a nonspreading and accelerating Airy packet to be a unique solution to the free-particle Schrödinger equation). Our solution, found using space-time scale transformations, not only overcomes the packet’s tendency to accelerate, but also make it possible to spread or contract in time.

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The general form of a nonspreading wave packet in one-dimensional free space is derived from first principles, employing a decomposition of the quantum mechanical evolution operator of the free particle. In agreement with the classical analysis of Berry and Balazs, the corresponding probability density is proportional to the square of an Airy function, and the packet propagates with constant acceleration.

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Motivated by the work of Berry and Balazs and Greenberger on the 1-D Schrödinger equation, we have investigated a class of nonspreading solutions to the 3-D Schrödinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy-type solutions to the 3-D Schrödinger equation has led us also to analogous solutions for the 3-D scalar wave equation.

American Journal of Physics•Jun 1, 1994•50 citations
Accelerating non-spreading wavepackets in a nonrelativistic free-particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs [Am. J. Phys. 47(3), 264–267 (1979)], and have been subsequently realised in several optics experiments. It is shown that these wavepackets are actually Perelomov coherent states. It is found that the Galilean invariance of the Schrödinger equation plays a key role in making these states unique and gives rise to their unusual propagation properties.
Accelerating non-spreading wavepackets in nonrelativistic free particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs (Am. J. Phys., 47, 264 (1979)), and have been subsequently realised in several optical experiments. It is shown that these wavepackets are actually coherent states. It is found that the Galilean invariance of the Schrodinger equation plays a key role in making these states unique and giving rise to their unusual propagation properties.
In 1979, Berry and Balazs [M V Berry and N L Balazs 1979 Am. J. Phys. 47 264] obtained a strict solution of the Schrödinger equation with Airy function used as the initial condition, and described the wave function represented by such solution as the Airy wave-packets. They discovered that infinite Airy wave-packet has unique properties such as non-spreading and free acceleration, proving that it is the only nontrivial non-spreading solution of the time-dependent Schrödinger equation in one dimension. However, the observing of the finite Airy beam seems to be more meaningful since wave-packets in reality is inevitably band limited. A certain form of finite Airy beam was investigated by Siviloglou et al. in 2007 [Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901; Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979]. They noted that the Airy wave packet still exhibits its most exotic feature, i.e., its trend toward free acceleration. While in the present paper we discuss the properties of Airy beam in a few steps further and propose several conclusions. On the one hand, a theoretic explanation is given to solve the matter of the centre of mass of infinite Airy beam. On the other hand, deeper research is conducted on the unique properties of finite Airy beam. Another form of finite Airy beam is discussed by reduction to absurdity, and its field distribution is put forward by numerical simulation. We find that the trajectory of the centroid holds its position, which means that the beam cannot accelerate freely as a whole. Ultimately, we have the conclusion that finite Airy beam can neither freely accelerate nor be non-diffractive.
We present the solution of Schrödinger’s equation for time-dependent quadratic potentials in terms of an Airy packet. We demonstrate new features of the Airy packet in light of its apparent unfamiliar properties discovered by Berry and Balazs (who found a nonspreading and accelerating Airy packet to be a unique solution to the free-particle Schrödinger equation). Our solution, found using space-time scale transformations, not only overcomes the packet’s tendency to accelerate, but also make it possible to spread or contract in time.
Motivated by the work of Berry and Balazs and Greenberger on the 1-D Schrödinger equation, we have investigated a class of nonspreading solutions to the 3-D Schrödinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy-type solutions to the 3-D Schrödinger equation has led us also to analogous solutions for the 3-D scalar wave equation.