Loosely bound van der Waals clusters
Doerte Blume
Center for Quantum Research and Technology (CQRT)
Department of Physics and Astronomy
The University of Oklahoma, Norman.

Collaborators:
Qingze Guan, AJ Yates, Maksim Kunitski,
Reinhard Doerner and group.
Supported by the NSF.

Superfluid Bulk Helium-4• 4He is the only substance that remains liquid under normal pressure atzero temperature (superfluid with condensate fraction of around 8%).
• Normal to superfluid transition at 2.17K.
Helium named after
the sun (greek “helios”).
Discovered in 1868.
Bulk liquid helium-4:
Binding energy per
particle E/N = −7K(1 K = 8.6 x 10−5 eV).
From Wikipedia

Helium Droplets =Quantum Liquid
Large N: E/N ~ −7K
Well known
literature results:

Quantum liquid:
No shell structure!

Small N (N<10):
E/N ~ # N.
(E/N changes by fourorders of magnitude.)N>20 energies are well described by liquid drop model withvolume and surface terms (no Coulomb, asymmetry, or pairing terms).
Rich interplay between many-body nuclear physics and quantum droplet
community [e.g., Pandharipande et al., PRL 50, 1676 (1983); Stringari etal., JCP 87, 5021 (1987); Sindzingre et al., PRL 63, 1601 (1989)].

Helium Dimer and Trimers
1 K = 8.6 x 10−5 eV• Dimer:• 4He-4He bound state energy Edimer = −1.625mK. No J > 0bound states. 4He-3He does not support bound state.
• Two-body s-wave scattering length as = 170.86a0.
• Two-body effective range reff = 15.2a0(alternatively, two-body van der Waals length rvdW = 5.1a0).
• Trimer:• Two J = 0 bound states with Etrimer = −131.8mK and−2.65mK.
• No J > 0 bound states.
4( He3)*
Discussed laterin this talk.

Helium Dimer and Trimers
1 K = 8.6 x 10−5 eV• Dimer:• 4He-4He bound state energy Edimer = −1.625mK. No J > 0bound states. 4He-3He does not support bound state.
• Two-body s-wave scattering length as = 170.86a0.
• Two-body effective range reff = 15.2a0(alternatively, two-body van der Waals length rvdW = 5.1a0).
• Trimer:
Close to hard wall at• Two J = 0 bound states with Etrimer = −131.8mK
andsmall internuclear
distances is a−2.65mK.
challenge for some• No J > 0 bound states.

( He )*• Nuclear physics: Deuteron and triton.

numerical 4
approaches: “Less 3soft” than “typical”
nuclear potentials.
Discussed laterin this talk.

Comparison With Other
Neutral Rare Gas Clusters• 4He, 10Ne, 20Ar: composite bosons (energy scales are such that theseatoms can be considered as point particles; consider only nuclear
degrees of freedom).
• Dimer (potential minimum at 5−10a0):§ 4He-4He binding energy: Edimer = −1.3mK.
§ 10Ne-10Ne binding energy: Edimer = −20.1K.
§ 20Ar-20Ar binding energy: Edimer = −101K.

4He

340Ar

340Ar

3

Finite s-wave Scattering Length:
Universally Linked Statesstronger attraction

V(r)rthreeatom
thresholdtrimern+1
trimern

Spectrum is determinedby 𝒂𝒔 and one three-bodyparameter.

atomdimer
thresholdV(r)rdimerReview by
Braatenand Hammer,
Physics
Reports 428,259 (2006).

Helium Trimer Excited Stateis an Efimov State
βVHe-He(r12) + βVHe-He(r23) + Three-body parameter is chosen suchthat ZR energy agrees with energy
βVHe-He(r31).
−|𝑬𝒕𝒓𝒊𝒎𝒆𝒓 /𝑬𝒉𝒂𝒓𝒕𝒓𝒆𝒆 |𝟏/𝟒

of scaled helium trimer excited state.
True helium(ground andexcited states)

Symbols:
Scaled helium

For the excitedstate, symbols
agree with line!
Molecular systemfollows predicted
Efimov behavior.

Line:
Universal
ZR theory

𝒔𝒈𝒏 𝒂𝒔 |𝒂𝒃𝒐𝒉𝒓 /𝒂𝒔 |𝟏/𝟐

D. Blume, FewBody Systems 56, 859 (2015)

β<1β>1

Probing Helium Trimer ExcitedEfimov State
very short and intense femtosecondsourceProbing Helium Trimer ExcitedEfimov State
Kunitski, Zeller, …, Blume, Doerner, Science 348, 551 (2015)

quantum+ +classi- +classicalcalHe3 signal contains ground state trimer *and* excited state trimer.
Laser beam ionizes trimer: Coulomb explosion of 4He3 (3 ions).

counts (arb. units)

Kinetic Energy Release+4classi- +( He )*Measurement: Observing
3mixture (exp.)

difference
spectrum (exp.)

cal

added by hand
(as a guide to the eye)groundstate only (experiment)exc. state(theory)kinetic energy release (KER) in eV (log scale)
The ionization is instantaneous and the He-ions are
distributed according to the quantum mechanical eigen
states of the ground and excited helium trimers.
Large r12, r23 and r31 correspond to small KER=1/r12+1/r23+1/r31.

Reconstructing Real Space
Properties
ground state
only (experiment)

excitedstate(theory)

Pair distributionfunction of (4He3)*
excited state(experiment, “differencemeasurement”)
mixture (exp.)

The excited state is eight times larger than the ground state.
Assuming an “atom-dimer geometry”, the tail can be fit to extract thebinding energy of the excited helium trimer.
Fit to experimental data yields 2.6(2)mK.
Theory 2.65mK [Hiyama et al., PRA 85, 062505 (2012)].

Normalized Structural
4
Other
Structuralfix length 1 fix
Properties of Characteristics
He3plot

excited state:
theoryexcited state:experimentground state:
theoryDivide all three interparticle distances by largest rij and plot
kth atom (positive y): Corresponds to placing atoms i and jat (−1/2,0) and (1/2,0).

Ground state and excited states have distinct characteristics!!!
Message: Reconstruction of quantum mechanical trimer density.

Structural Properties: Short- andLong-Range Characteristics
Atomic systems:••

Van der Waals universality.
Highly repulsive short-range potential.

Which long-range behaviors “collapse”?
Which short-range behaviors “collapse”?
Strategy:Consider different two-body potentials (realistic and effectivemodels), both at the physical point and at unitarity.

Different (Helium-Helium)
Interaction Potentials
Strategy: Consider different two-body potentials [realistic (Model I) andeffective models (Model II)], both at the physical point and at unitarity.
Model I: Realistic potential with hard inner wall.

HFD-HE2.
CPKMJS.
𝑉!"! = $ $ 𝑉*+,-./!.0 (𝑟#( )TTY.
#$% ()#
Unitarity realized by applying overall scaling factor. LM2M2.
&'% &Model II: Effective low-energy potential model.
&'% &𝑉!"! = $ $ 𝑉1,3,4// 𝑟#(#$% ()#matched to+ $ $ $ 𝑉5,3,4// (𝑅#(- ) HFD-HE2 andscaled-HFD-HE2.
#$% ()# -)(&'1 &'% &See Kievsky et al., PRA 96, 040501 (2017); PRA 102, 063320 (2020):
2-body range and depth → s-wave scattering length and effective range.
3-body range and depth → three- and four-body energy.

4He : Nth Atom Relative toN

Center-of-Mass of N-1 Atoms
energy.

N=3
N=10

HFD-HE2 & Gauss
Solid: true.
Gauss:
dashed.

physical point

CPKMJS (log.)
CPKMJS (linear)
Overview: AJ’s work
Dashed lines:
Phrasing the questions
Based onbinding
Start with some “older” works

unitarity
Results obtained using forward walking: Reynolds et al., J. Stat. Phys. 43, 1017 (1986).

Pair Distribution Function(2)
P (r) for 𝑵 = 𝟐 − 𝟏𝟎“secondlayer”

N=3

N=2

HFD-HE2 & Gauss
Solid: true.
Gauss:
dashed.

physical pointCPKMJSCPKMJS
Overview: AJ’s work
Phrasing theN=10
questionsStart with some “older” works

unitarity

Yates, Blume, PRA 105, 022824 (2022)

(𝟐)
(2)
P (r), Two-Body Contact 𝑪 ,van der Waals Universality𝒂𝒔𝒓𝒗𝒅𝑾= 𝟒𝟔. 𝟗𝟓 for

𝑵GaussHFD-HE2.
𝒂𝒔

CPKMJS“CPKMJS”“HFD-HE2”

𝒓𝒗𝒅𝑾= 𝟑𝟑. 𝟔𝟑 for

CPKMJS.

HFD-HE2

See Naidon et al., PRL 112, 105301(2014); PRA 90, 022106 (2014) for vdW
universality of trimer at unitarity.

See Kim et al., Annu.
Rev. Nucl. Sci. 24, 96
Flambaum et al., PRA 59, 1998 (1999). (1974): “asymptoticGao, PRA 58, 4222 (1998).
normalization constant”

Three-Body (Sub-Cluster)
Correlations for 𝑵 = 𝟑 − 𝟏𝟎
HFD-HE2

N=3Gauss
physical point

N=10

CPKMJS

unitarity

Next: Dynamics of Dimer

Prepare universal
initial state (i.e.,state that is
dominated by swave scattering
length).

Interrogate theinitial state: fastand intense pumplaser that takes thesystem out ofequilibrium.

Wait for a variable time(delay) and apply evenshorter and more
intense probe laser thatallows us to look attime-evolved system.

Next: Dynamics of Dimer
Sort of like…

From vectorstock.com

Prepare universal
initial state (i.e.,
state that isdominated by swave scatteringlength).

Interrogate theinitial state: fastand intense pumplaser that takes thesystem out ofequilibrium.

Wait for a variable time(delay) and apply evenshorter and more
intense probe laser thatallows us to look attime-evolved system.

Selected Works in This
Direction(ultra)cold atoms: fast intenselasersuniversal physics

Pump-Probe Spectroscopy ofIsolated Helium Dimers

Pump pulse: pulse length of 311 fs and intensity of 1.3 × 1014 W/cm2.
Probe pulse rips off two electrons (Coulomb explosion).
What do we expect to happen as a function of the delay time???

What Does The Pulse Do To
Helium Dimer?
𝜽moleculelaser polarization
Without the laser, themolecule is sphericallysymmetric (no alignment):
The helium dimer has
vanishing relative orbital
angular momentum.

(positive)alignment

antialignment

Will show: Helium dimer can be aligned.
However, since the 𝑱 > 𝟎 partial wave
components are not bound, they will“run away” (dissociative wave packet).
Heavier non-universal dimers behave
very differently.

4He-4He In Time-Dependent

Electric Field

In what follows, the initial state will be the 𝑱 = 𝟎
eigenstate of the zero-field Hamiltonian of 4He4He system.
Scenario 1 (non-adiabatic laser kick):𝜺 𝒕 = 𝜺𝟎 𝐞𝐱𝐩 −𝟐 𝐥𝐧𝟐
20

J=42
J=00-20

246R [A° ]

𝝉80

(a)
Veff,l(R) [K]

Veff,l(R) [K]40𝒕0𝒕𝒓𝒆𝒇 𝟐8; 𝝉 ≈ 𝟑𝟎𝟎fs.
(b)flux40

𝑱3𝟎,𝟐,…

0-40
2.5

Solve timedependent
Schroedinger
equation using
spherical
coordinates:𝜳 𝑹, 𝜽, 𝒕𝒖𝑱 𝑹, 𝒕?)= ;𝒀𝑱𝟎 (𝑹𝑹

3.5
4.5
R [A° ]

5.5

Scenario 2 (“slow”): Gaussian turn-on,
hold for several ps, Gaussian turn-off.

time in ps

Laser couples
different partialwaves.
When laser is off,the channels aredecoupled.

Scenario 1: Theory Result
Solve timedependent
Schroedinger
equation using
spherical
coordinates:𝜳 𝑹, 𝜽, 𝒕𝒖𝑱 𝑹, 𝒕?)= ;𝒀𝑱𝟎 (𝑹𝑹
𝑱3𝟎,𝟐,…

Interference between J=0 and J=2 partial
waves. J=2 portion “travels” on structurelessbackground.

Pulse couplesdifferent partialwaves.
When pulse is off,the channels aredecoupled.

ExperimentParameterfree theory(usingmeasuredpulselength,intensity,spatialimagingresolution)

Kunitski,Guan,…,Blume,Doerner,
Nature Physics(2021).

Kicking the 4He Dimer
For the first time: Intense laser used to probe dynamics at single-atomlevel using universal, scattering length dominated initial state.
“Rotationless” 4He dimer can be aligned! Note, it’s the continuum
portion of the wave packet…
Pattern due to interference between J=0 and J=2 channels:
Measurement of spatially and time dependent relative phase between
these two partial wave channels. State tomography!
Many outstanding challenges:
Resonances as in ultracold atoms? Need longer pulses…
Time-dependent modulation of interaction strength?
Dynamics of (Efimov) trimers? Need to populate it first…
Larger clusters.

Scenario 2: Longer Pulses
max. intensity
𝟐. 𝟓 % 𝟏𝟎𝟏𝟒 W/cm2

4ps hold time

12ps hold time

unbound portion

boundportion
Fingerprint of revivals in time-dependent response of system: Dimer oscillates
between electric field-induced deeply-bound state and weakly-bound state.

Summary•

4He

N droplets can be realized experimentally in size-selectivemanner.
•Access to structural properties (accessing real-space structures
beyond N=3 is non-trivial due to reconstruction algorithm).

•

4He : Ground state has Efimov characteristics and excited state is3

Efimov trimer (s-wave scattering length and three-bodyparameter).
•Some aspects of 4HeN clusters well described by effective lowenergy model.

•4He : Alignment of dissociate wave packet portion.
2

Thank You!