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.

1.
M. V.
Berry
and
N. L.
Balazs
, “
Nonspreading wave packets
,”
Am. J. Phys.
47
(
3
),
264
267
(
1979
).
2.
D. M.
Greenberger
, “
Comment on ‘Nonspreading wave packets’
,”
Am. J. Phys.
48
(
3
),
256
(
1980
).
3.
K.
Unnikrishnan
and
A. R. P.
Rau
, “
Uniqueness of the Airy packet in quantum mechanics
,”
Am. J. Phys.
64
(
8
),
1034
1035
(
1996
).
4.
J. W.
Goodman
,
Introduction to Fourier optics
(
Roberts and Company Publishers
,
Colorado
,
2005
).
5.
G. A.
Siviloglou
and
D. N.
Christodoulides
, “
Accelerating finite energy Airy beams
,”
Opt. Lett.
32
(
8
),
979
981
(
2007
).
6.
G. A.
Siviloglou
,
J.
Broky
,
A.
Dogariu
, and
D. N.
Christodoulides
, “
Observation of accelerating Airy beams
,”
Phys. Rev. Lett.
99
(
21
),
213901
213901
(
2007
).
7.
K.
Dholakia
, “
Optics: Against the spread of the light
,”
Nature
451
(
7177
),
413
413
(
2008
).
8.
J.
Broky
,
G. A.
Siviloglou
,
A.
Dogariu
, and
D. N.
Christodoulides
, “
Self-healing properties of optical Airy beams
,”
Opt. Express
16
(
17
),
12880
12891
(
2008
).
9.
H. I.
Sztul
and
R. R.
Alfano
, “
The Poynting vector and angular momentum of Airy beams
,”
Opt. Express
16
(
13
),
9411
9416
(
2008
).
10.
S.
Nomoto
,
A.
Aadhi
,
S.
Prabhakar
,
R. P.
Singh
,
R.
Vyas
, and
S.
Singh
, “
Polarization properties of the Airy beam
,”
Opt. Lett.
40
(
19
),
4516
4519
(
2015
).
11.
P.
Polynkin
,
M.
Kolesik
,
J. V.
Moloney
,
G. A.
Siviloglou
, and
D. N.
Christodoulides
, “
Curved plasma channel generation using ultraintense Airy beams
,”
Science
324
(
5924
),
229
232
(
2009
).
12.
J.
Baumgartl
,
M.
Mazilu
, and
K.
Dholakia
, “
Optically mediated particle clearing using Airy wavepackets
,”
Nat. Photon.
2
(
11
),
675
678
(
2008
).
13.
J. R.
Klauder
and
E. C. G.
Sudarshan
,
Fundamentals of Quantum Optics
(
Dover
,
New York
,
2006
).
14.
J.
Gazeau
,
Coherent States in Quantum Physics
(
Wiley
,
Weinheim
,
2009
).
15.
K.
Gottfried
and
T.
Yan
,
Quantum Mechanics: Fundamentals
(
Springer
,
New York
,
2013
).
16.
A.
Perelomov
,
Generalized Coherent States and their Applications
(
Springer
,
Berlin
,
2012
).
17.
R. R.
Puri
,
Mathematical Methods of Quantum Optics
(
Springer
,
Berlin
,
2001
).
18.
A familiar example of these coherent states is given by angular momentum coherent states, obtained by the action of SO(3) operators on the L̂z eigenstate exp(iθxL̂x+iθyL̂y)|m, where L̂z|m=m|m.
19.
What is presented here is the essence of the coherent-state construction due to Perelomov. For a more mathematically rigorous and nuanced treatment, the reader is referred to Refs. 14 and 16.
20.
One can consider any other fiducial state, say |x=x0;t=t0, without affecting subsequent conclusions about Perelomov coherent states. The state |x=0;t=0 is chosen only for simplicity.
21.
M. V.
Berry
, “
Stable and unstable Airy-related caustics and beams
,”
J. Opt.
19
(
5
),
055601
(
2017
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.