- See supplementary materials.
- B. A. McGuire, Census of interstellar, circumstellar,
extragalactic, protoplanetary disk, and exoplanetary molecules.
arXiv:1809.09132v1 [astro-ph.GA] (2018). - C. Puzzarini, J. F. Stanton, J. Gauss,Int. Rev. Phys. Chem. 29 ,
273 – 367 (2010). - F. Wolfet al.,Nature 530 , 457–460 (2016).
- C. W. Chouet al.,Nature 545 , 203–207 (2017).
- M. Sinhal, Z. Meir, K. Najafian, G. Hegi, S. Willitsch,Science
367 , 1213–1218 (2020). - P. O. Schmidtet al.,Science 309 , 749–752 (2005).
- Ch. Rooset al.,Phys.Rev.Lett. 83 , 4713– 4716
(1999). - R. Rungangoet al.,New J. Phys. 17 , 035009 (2015).
- R. Blatt, D. Wineland,Nature 453 , 1008–1015 (2008).
23.K.Raghavachari,G.W.Trucks,J.A.Pople,M.Head-Gordon,
Chem. Phys. Lett. 157 , 479–483 (1989). - S. A. Kucharski, R. J. Bartlett,Theor. Chim. Acta 80 , 387– 405
(1991). - N. Oliphant, L. Adamowicz,J. Chem. Phys. 95 , 6645– 6651
(1991). - A. Bartels, C. W. Oates, L. Hollberg, S. A. Diddams,Opt. Lett.
29 , 1081–1083 (2004).
27. S. Schiller, V. Korobov,Phys. Rev. A 71 , 032505 (2005).
28. V. V. Flambaum, M. G. Kozlov,Phys. Rev. Lett. 99 , 150801
(2007).
29. M. Quack, J. Stohner,Phys. Rev. Lett. 84 , 3807– 3810
(2000).
30. R. Berger, M. Quack, J. Stohner,Angew. Chem. Int. Ed. 40 ,
1667 – 1670 (2001).
31. C. W. Chouet al., Data for“Frequency-comb spectroscopy on
pure quantum states of a single molecular ion,”NIST Public
Data Repository (2020).
ACKNOWLEDGMENTS
We thank F. C. Cruz and A. Kazakov for careful reading of this
manuscript.Funding:This work was supported by the U.S. Army
Research Office (grant no. W911NF-19-1-0172). A.L.C. is supported
by a National Research Council postdoctoral fellowship. Y.L.
acknowledges support from the National Key R&D Program of
China (grant no. 2018YFA0306600), the National Natural Science
Foundation of China (grant no. 11974330), and the Anhui Initiative
in Quantum Information Technologies (grant no. AHY050000).
C.K. acknowledges support from the Alexander von Humboldt
foundation. P.N.P. acknowledges support by the state of Baden-
Wurttemberg through bwHPC (bwUnicluster and JUSTUS, RVbw17D01).Author contributions:C.W.C., A.L.C., C.K., T.F., S.D.,
D.L., and D.R.L. developed components of the experimental
apparatus. C.W.C., A.L.C., and C.K. collected and analyzed data.
C.W.C., Y.L., D.L., and D.R.L. designed the experimental methods
and pulse sequences. M.E.H. and P.N.P. computed rotational
transition frequencies and constants. All authors provided
experimental suggestions, discussed the results, and contributed
to writing the manuscript. Competing interests: The authors
declare no competing interests. Data and materials availability:
The data from the main text and supplementary materials are
available from the NIST Public Data Repository ( 31 ). This is a
contribution of the National Institute of Standards and Technology,
not subject to U.S. copyright.SUPPLEMENTARY MATERIALS
science. /content/367/6485/1458/suppl/DC1 Supplementary Text
Fig. S1
Tables S1 to S5
References ( 32 – 45 )
27 November 2019; accepted 4 March 2020
10.1126/science.aba3628QUANTUM GASES
Observation of dynamical fermionization
Joshua M.Wilson, Neel Malvania, Yuan Le, Yicheng Zhang, Marcos Rigol, David S. Weiss*
The wave function of a Tonks-Girardeau (T-G) gas of strongly interacting bosons in one dimension maps
onto the absolute value of the wave function of a noninteracting Fermi gas. Although this fermionization
makes many aspects of the two gases identical, their equilibrium momentum distributions are quite
different. We observed dynamical fermionization, where the momentum distribution of a T-G gas evolves
from bosonic to fermionic after its axial confinement is removed. The asymptotic momentum distribution
after expansion in one dimension is the distribution of rapidities, which are the conserved quantities
associated with many-body integrable systems. Our measurements agree well with T-G gas theory. We
also studied momentum evolution after the trap depth is suddenly changed to a new nonzero value, and
we observed the theoretically predicted bosonic-fermionic oscillations.
I
ntegrable many-body quantum systems
have been extensively studied theoret-
ically since 1931, when Bethe solved the
one-dimensional (1D) Heisenberg model
( 1 ). The theoretical appeal of these sys-
tems stems from the deep symmetries they
exhibit and the fact that it is possible to ex-
actly solve for their many-body wave func-
tions ( 2 ). Over the past 20 years, there have
been more than a dozen experimental imple-
mentations of very nearly integrable models.
Systemsofbosons( 3 ), spins ( 3 ), and fermions
( 4 ) have been realized, using a range of ultra-
coldatom,trappedion,andcondensedmatter
techniques. Each of these integrable many-
body systems has a set of conserved quantities,
the distribution of rapidities, which fully char-
acterizes the many-body state. The rapidities
embody what makes integrable systems spe-
cial, including the fact that integrable systems
do not reliably thermalize under unitary dy-
namics [see ( 5 ) for a recent set of reviews on
this topic]. The rapidities depend on com-
plicated interactions among many particles,
which makes it difficult if not impossible to
extract their distribution directly from equi-
librium measurements. However, when the
particles in an integrable system are allowed
to expand in one dimension, the interparticle
interactions vanish asymptotically and the
momentum distribution of the system ap-
proaches the distribution of rapidities ( 6 – 11 ).
Here, we report such an expansion measure-
ment with a Lieb-Liniger gas ( 12 ), an integrable
system of 1D bosons with contact interactions.
We operate in the Tonks-Girardeau (T-G) gas
limit ( 13 – 15 ), where the interactions are very
strong. The many-body wave function of the
T-G gas is the same as that of a noninteract-
ing Fermi (NIF) gas, to within an absolute
value ( 13 ). All properties that depend on the
square of the wave function, such as total
energy ( 14 ) and local pair correlations ( 16 ),
arethesameforthetwotypesofgases.Other
properties, such as momentum distributions,
are typically different. One central property
shared by the two gases is the rapidity dis-
tribution ( 3 ). In the absence of a confiningpotential, the momentum distribution of a
NIF gas is its rapidity distribution. There-
fore, the T-G gas rapidity distribution is the
same as the momentum distribution of a
NIF gas. Hence, as interactions become neg-
ligible upon expansion in a flat potential, the
T-G momentum distribution transforms into
a NIF gas momentum distribution ( 7 ). The
observation of this“dynamical fermionization”
constitutes a direct measurement of the dis-
tribution of rapidities in this many-body inter-
acting quantum system, thus bringing these
theoretical constructs into the realm of exper-
iment. Our experimental results for time-
of-flight (TOF) measurements are in almost
complete agreement with exact theoretical cal-
culations. We have also measured momentum
distribution dynamics after quenches to differ-
ent nonvanishing trap strengths ( 8 ).
The momentum distributions of equilibrium
1D Bose gases have been measured with TOF,
Bragg spectroscopy, observation of phase fluc-
tuations, and momentum focusing techniques
( 3 , 17 ). These measurements have all been ini-
tiated by shutting off both axial and transverse
trapping simultaneously, which precludes the
expansion in one dimension that is required
for a rapidity measurement. In our experi-
mental setup, we can remove the axial poten-
tial without affecting the transverse trapping
that makes the system 1D, thus allowing for
free expansion in one dimension. We initi-
ate the momentum measurement at control-
lable times,tev, during the 1D expansion by
suddenly shutting off the transverse trapping
(Fig. 1A). The wave functions rapidly expand
transversely, which markedly decreases their
interaction energy before the axial wave func-
tion appreciably changes. After a long TOF, the
spatial distribution approaches the momentum
distribution attev(Fig. 1B).
The experiment starts with a Bose-Einstein
condensate of 105 87Rb atoms in theF= 1,mF=27 MARCH 2020•VOL 367 ISSUE 6485 1461Department of Physics, Pennsylvania State University,
University Park, PA 16802, USA.
*Corresponding author. Email: [email protected]
SCIENCE
RESEARCH | REPORTS