Disorder-induced heating (DIH) prevents ultracold neutral plasma into the electron strong coupling regime. Here, we propose a scheme to suppress electronic DIH via optical lattice. We simulate the evolution dynamics of ultracold neutral plasma constrained by three-dimensional optical lattice using the classical molecular dynamics method. The results show that for experimentally achievable conditions, electronic DIH is suppressed by a factor of 1.3, and the Coulomb coupling strength can reach 0.8, which is approaching the strong coupling regime. Suppressing electronic DIH via optical lattice may pave a way for the research of electronic strongly coupled plasma.

Ultracold neutral plasma (UNP) provides an excellent platform for investigating strongly coupled and high-energy-density plasma (HEDP).1–3 Experimentally, UNPs can be produced by direct photoionization of laser-cooled atoms,4–7 a Bose–Einstein condensate,8 or from evolution of atomic,9 molecular Rydberg gases.10 The extremely low temperature of UNPs enables them to be in or near the strongly coupled regime, and dilute character implies long evolution timescale that ensures them to be precisely diagnosed.5,6,11–13 The unique nature of UNP has also inspired recent experimental investigations on magnetized plasmas,14–17 relaxation processes,18–21 as well as collective modes,6,22–24 while a series of theoretical works25–31 explored the physical processes in UNPs intensively. Moreover, UNPs are potentially advantageous as a source of high-brightness ion beams for nanoscale measurement and nanofabrication32–36 and as a source of high-coherence electron beams for coherent diffractive imaging and microscopy.37–39 From the plasma physics perspective, one of the main motivations for studying UNPs is the fact that they can be strongly coupled and have the possibility to create Wigner crystallization.40 

However, the Coulomb coupling strength is limited mainly by disorder-induced heating (DIH), since the plasma is created in a completely uncorrelated state, the Coulomb interaction potential energy is rapidly converted into kinetic energy and hence, both the electron and the ion components are heated.25,40 The Coulomb coupling strength Γ α = e 2 / ( 4 π ε 0 a α k B T α ), where e represents the elementary charge, a α = ( 3 / 4 π n α ) 1 / 3 is the Wigner–Seitz radius, ε 0 is the vacuum permittivity, kB is the Boltzmann constant, and T α denotes the corresponding ion or electron temperature, while n α is the number density of ions or electrons. Therefore, suppressing DIH and improving electronic and ionic Coulomb coupling strength are of great significance for research in UNPs and HEDPs. On the other hand, the coherent length of the UNP electron beam is also limited by the electron temperature in UNP. An electron beam with a longer coherence length will potentially permit the single-shot diffraction imaging of macromolecules, such as protein.

For the ionic component of UNPs, strong coupling is easier to be achieved.2,41 Numerous schemes have been proposed to suppress the DIH of ions based on pre-correlating the system before ionization by using a degenerate Fermi gas,25 Rydberg blockade,42,43 an optical lattice,44 or Penning ionization of a molecular Rydberg gas.45 Among these schemes, Rydberg blockade43 and Penning ionization of a molecular Rydberg gas45 have shown promising results. Moreover, it has been recognized that the coupling of ion plasma can be promoted by laser cooling the ions immediately after plasma formation,40,46 and recent experiment of Langin et al. demonstrated laser cooling of ions and achieved a value of Γ i as high as 11.47 

Compared to the strong coupling of ions, there are limited works on pursuing strong coupling of electrons. Several mechanisms contribute to the electron heating, such as DIH, three-body recombination (TBR), and Rydberg-electron collisions (REC).1 Vanhaecke et al.48 suggested that adding additional Rydberg atoms to UNP could increase or decrease the electron temperature, and Pohl et al.49 investigated this idea systematically by simulations and concluded that Γe could be increased to 0.5 by using more sophisticated processes of adding Rydberg atoms. More recent experiment and simulations by Crockett et al.50 showed that embedding Rydberg atoms into UNP has limited ability to push a plasma into electron strongly coupled regime. Moreover, Tiwari et al.51 suggested using strong external magnetic field to reduce DIH and TBR of electrons, the basic idea of which is constraining electron motion to reduce electron kinetic energy. This scheme needs a strong magnetic field and can only suppress the heating of electrons in the perpendicular direction to the magnetic field. At the density n e 10 8 cm 3, a strong magnetic field of one-tenth of a Tesla or higher is needed.

In this paper, we propose a scheme to suppress electronic DIH via optical lattice. The evolution of the UNPs are simulated by using the classical molecular dynamics method with open boundary conditions. The different initial lattice types and plasma densities are performed, and the influences of them on the DIH and electronic Coulomb coupling parameters are investigated. The paper is organized as follows. In Sec. II, the proposed scheme is elaborated and the numerical model is briefly outlined used in the simulations. Section III presents the results of our simulations, and a brief conclusion is given in Sec. IV.

Figure 1 illustrates the schematics of the scheme, taking rubidium-87 ( 87 R b) as a prototype. The creation of 87 R b UNP starts from laser cooled and trapped neutral atoms in a magneto-optical trap using 5 2 S 1 / 2 5 2 P 3 / 2 transition at 780 nm. Up to 109 atoms can be cooled to as low as microkelvin temperatures and confined at densities approaching 1011 cm−3. After the atoms are cooled, the cooling lasers are shut down and the cooled atoms are excited to 5 2 P 3 / 2 by a 780 nm laser pulse. Meanwhile, the excited atoms are photoionized near the threshold by a ∼480 nm laser pulse. UNPs are then formed having electron temperatures in the 1–1000 kelvin range and ion temperatures from tens of millikelvin to a few kelvin. The typical spherical Gaussian density distribution of atomic cloud and the threshold photoionization lead to a disordered configuration of ions and electrons. The subsequent rearrangement of ions and electrons establishes interparticle correlations and thus decreases the potential energy of plasma. The conservation of energy leads to a rapid heating of ions and electrons, i.e., DIH. This DIH prevents UNP into the electron strong coupling regime. In order to suppress the DIH, pre-ordering ions and electrons is a natural consideration. As shown in Fig. 1(a), the cooled atoms are first loaded into a three-dimensional optical lattice to establish the pre-ordering of atoms. For 87 R b atom, a 1064 nm high power laser is used to produce an optical lattice. The ionic component of the subsequently created UNP will inherit the ordering of atoms, and the initial ion–ion and electron-ion correlations are established by the optical lattice. The ordered ions form a periodic Coulomb potential field, which constrains the motion of electrons. As a result, the DIH of electrons will be suppressed.

FIG. 1.

Schematics of the scheme to suppress DIH. (a) Loading atomic cloud into a three-dimensional fractional filling optical lattice, where blue spheres represent cold atoms and the gray baseboard represents optical trap. (b) Two-color photoionization of 87 R b.

FIG. 1.

Schematics of the scheme to suppress DIH. (a) Loading atomic cloud into a three-dimensional fractional filling optical lattice, where blue spheres represent cold atoms and the gray baseboard represents optical trap. (b) Two-color photoionization of 87 R b.

Close modal

A classical molecular dynamics (MD)52 simulation is performed to simulate the evolution dynamics of the optical lattice constrained UNPs. The simulation model involved a cubic box with open boundary condition.53 The present work focuses on the DIH of electrons, which mainly occurs at early phase of timescale of 1–2 τe, where τe is equal to the inverse electron plasma frequency, i.e., τ e = ω p e 1 = m e ε 0 / ( n e e 2 ). Based on this character, the simulation runs for t max ω pe = 10 scaled time units. On the other hand, the characteristic plasma expansion time is determined by τ exp = m i σ ( 0 ) 2 / k B [ T e ( 0 ) + T i ( 0 ) ], typically 1μs, where σ ( 0 ) is the initial size of the plasma cloud and T e ( 0 ) , T i ( 0 ) are the initial electron and ion temperatures, respectively.54 Under the present simulation timespan, the plasma hardly expands. The radius of the simulation spherical volume is set at 0.5 mm, which is large enough to model the dynamics of UNPs and avoid the sample size effect. The UNP is put at the center of the spherical volume. In addition, the time step is set at a few femtoseconds that is sufficiently short to allow us revealing the “real” evolution of UNPs.

The Coulomb interaction potential of all charged particle pairs is adopted in the simulations. In order to avoid singularities, the Coulomb potential is given by the following form:27,31
± e 2 4 π ε 0 r 2 + ( η a α ) 2 ,
(1)
where r is the separation between two charged particles, and the sign + ( ) represents Coulomb repulsion (attraction) force, respectively. The choice of parameter η is critical to the dynamics of UNP. Tiwari et al.51 showed that the electron temperature evolution behavior does not appreciably change for η < 0.01, and thus, η is chosen to be 0.01 in all simulations. Next, we use Dormand–Prince 5 Runge–Kutta method to move the particles forward by one time step. The time step should be small enough to keep the total energy conserved to an accuracy of 0.1 % in the simulation. In addition, TBR is not included in the simulations because compared with DIH, TBR is a weaker heating mechanism on the timescale of our simulation27,51 and we mainly concern the suppression of DIH rather the longer-term slower TBR. We take a kinematic definition of temperature as the average kinetic energy per particle of type α, defined as
T α = 2 3 N k B i = 1 N ( 1 2 m α v i 2 ) ,
(2)
where N is the total number of electrons or ions. At the early phase of plasma evolution, electrons are not in thermal equilibrium state and the distribution of electron velocity is not Maxwellian. Niffenegger et al.31 showed that the electron temperature calculated by fitting the electron velocity distribution to a Maxwell–Boltzmann distribution and using the equipartition theorem are in good agreement. The electron temperature is not thermodynamic, but rather interpreted in terms of the kinetic temperature of Eq. (2).

As the typical plasma density is 108 10 11 cm 3, we perform the simulations at different average densities of 108 cm−3, 109 cm−3, 1010 cm−3, and 1011 cm−3. The plasma density is changed by varying the plasma volume. The ions are initialized at optical lattice sites for different lattice types such as simple-cubic (sc), body-centered-cubic (bcc), and face-centered-cubic (fcc), which can experimentally be realized with suitable laser arrangements.55–57 For saving the computational cost, we choose ∼1000 ions in the simulations. This number may differ slightly for different lattice types. Combined with average density and ion number, the initial plasma volume can be determined. Experimentally, lattice spacing is determined by the laser wavelength and arrangement. In the present simulations, the lattice spacing is determined by the plasma density and the filling fraction. Due to the neutral character of UNP, the number of electrons equals that of ions. The initial electron positions are randomly set. In order to give maximum correlations to develop, the initial ion and electron temperatures (or more precisely, kinetic energies) are set to zero, corresponding to Γ e ( t = 0 ) = and Γ i ( t = 0 ) = .27 For 87 R b UNP, m i / m e is equal to 1.65 × 10 5.

Furthermore, we evaluate the influence of optical lattice laser on electron temperature. The optical dipole potential arises from the interaction of the induced atomic dipole moment with the intensity gradient of the laser field.58 For a Gaussian laser beam, the typical trap depth can be up to millikelvin.55 For 87 R b +, the induced ionic dipole moment with the dipole laser is tremendously small so the interaction potential of induced ionic dipole moment with plasma electrons is relatively weak, compared to the coulomb potential between plasma ions and electrons. Therefore, the influence of optical lattice laser is negligible.

First, we simulate the evolution of normal UNPs where ions are randomly distributed. For the normal UNPs, the DIH of electrons occurs on a timescale of 1–2  τ e  1 ns. Then, the electrons undergo further heating by TBR and REC on a slower timescale, which is due to the formation of highly excited Rydberg atoms and the transfer of Rydberg atoms to more deeply bound levels. In addition, adiabatic cooling accomplished by plasma expansion will decrease electron temperature on a timescale of about 10 μs. Our simulation time spans the timescale of 10 τe and enables us to explore the main behavior of electronic DIH. Compared to the electronic DIH, the ionic DIH happens on a rather slower timescale of about 1 μs due to large ion mass.

Figure 2 presents the temporal evolution of average kinetic temperature of electrons (black line) and ions (red line). To make sure that the electron temperature is converged, we also perform simulations with 2000 electrons and ions and with 8000 electrons and ions. The results with more electrons and ions agree perfectly with those presented in Fig. 2. One can find that the electron temperature rapidly increases and approaches a plateau within nanosecond showing the obvious behavior of the DIH of electrons. The result agrees well with previous experiments5,13 and simulations.51,59 During the process of DIH, electrons gain kinetic energy largely via the ballistic motion associated with Coulomb attraction to the ions.51 The typical ion equilibration timescale τi is about 1 μs, which is equal to the inverse ion plasma frequency ω p i 1 = m i ε 0 / ( n i e 2 ). In the present simulation timespan, as expected, ions are far from equilibration.

FIG. 2.

(a) Electron and ion temperature evolution on timescale of 0–10 τe. Simulations were carried out with initial temperature T e = T i = 0 and a uniform random distribution of electrons and ions with the average density n e = n i = 10 8 cm 3. (b) Electron temperature evolution for different ion and electron numbers with the density of 10 10 cm 3: N i = N e = 1000 (black), N i = N e = 2000 (red), and N i = N e = 8000 (blue).

FIG. 2.

(a) Electron and ion temperature evolution on timescale of 0–10 τe. Simulations were carried out with initial temperature T e = T i = 0 and a uniform random distribution of electrons and ions with the average density n e = n i = 10 8 cm 3. (b) Electron temperature evolution for different ion and electron numbers with the density of 10 10 cm 3: N i = N e = 1000 (black), N i = N e = 2000 (red), and N i = N e = 8000 (blue).

Close modal

The influence of plasma density on electronic DIH and Coulomb coupling parameter is also investigated. The results are shown in Figs. 3(a) and 3(b). As can be seen, with the increase in plasma density, the plateau temperature TDIH increases. This can be easily understood because the electron kinematic energy gains from the Coulomb interaction k B T DIH e 2 / ( 4 π ε 0 a e ) n e 1 / 3. It is interesting to note that, although plateau temperature TDIH increases with the plasma density, the Coulomb coupling strength keeps unchanged, as shown in Fig. 3(b). When the electrons reach thermal equilibrium, energy conservation leads to k B T DIH e 2 / ( 4 π ε 0 a e ). Therefore, according to the definition of Coulomb coupling strength Γ e = e 2 / ( 4 π ε 0 a e k B T e ), which is the ratio of Coulomb interaction energy to electron kinematic energy, Γ e approaches a constant value regardless of the initial plasma density.

FIG. 3.

Electron temperature (a) and Coulomb coupling strength (b) evolution for random distribution of the ions at different initial average densities: 10 8 cm 3 (black), 10 9 cm 3 (red), 10 10 cm 3 (blue), and 10 11 cm 3 (green).

FIG. 3.

Electron temperature (a) and Coulomb coupling strength (b) evolution for random distribution of the ions at different initial average densities: 10 8 cm 3 (black), 10 9 cm 3 (red), 10 10 cm 3 (blue), and 10 11 cm 3 (green).

Close modal

Gericke et al.44 suggested confining atoms into an optical lattice to suppress the DIH of ions. Afterward, by using classical MD methods, several theoretical works53,59–61 demonstrated that the DIH of ions can indeed be suppressed. In the present work, we focus on possibility of suppression of the DIH of electrons by loading atoms into a three-dimensional optical lattice. The pre-ordering is obtained by setting ions into lattice sites. Figure 4 presents electron temperature and Coulomb coupling strength evolution for different lattice types and plasma densities under the condition of fully filled lattice. It is quite straightforward that, in all cases, the electronic DIH is significantly suppressed. The plateau temperature is reduced by a factor of 1.3. Meanwhile, the electronic Coulomb coupling strength increases to 0.8, approaching strong coupling regime. One can also find that electron plateau temperature increases with the increasing density, while the electronic Coulomb coupling strength keeps unchanged, which is similar to Fig. 3. For different types of the lattice, there is minor variation dependent upon sc, bcc, or fcc. The bcc and fcc lattice results are similar, owing to their nearly equal Madelung energy of the lattice, while the sc lattice has higher Γe due to its lower Madelung energy of the lattice. Generally, the plateau temperature and the electronic Coulomb coupling strength are almost independent of the lattice types.

FIG. 4.

Temperature and Coulomb coupling strength evolution of electrons at different cubic lattice geometries and densities for a fully filled optical lattice: (a) 10 8 cm 3, (b) 10 9 cm 3, (c) 10 10 cm 3, and (d) 10 11 cm 3. The legend represents different types of optical lattice.

FIG. 4.

Temperature and Coulomb coupling strength evolution of electrons at different cubic lattice geometries and densities for a fully filled optical lattice: (a) 10 8 cm 3, (b) 10 9 cm 3, (c) 10 10 cm 3, and (d) 10 11 cm 3. The legend represents different types of optical lattice.

Close modal

The above-mentioned behavior suggests that, for a fully filled optical lattice, cooling of the electrons is always possible and independent of the plasma density and the type of optical lattice. Pohl et al.61 and Murphy et al.60 found that the filling fraction of lattice is an important factor for the suppression of the ionic DIH. Moreover, they presented an analytical model for calculating the ionic Coulomb coupling parameter following DIH of lattice-correlated ions as a function of the filling fraction. Here, we also explore the effect of the filling fraction on the suppression of the electronic DIH for different plasma densities and different types of optical lattice. In the simulations, for a fixed density and optical lattice type, in order to keep the density constant, the filling fraction is changed by varying the spacing distance of optical lattice. The Coulomb coupling parameter of electron Γ e is evaluated from the simulations by taking the average of the coupling parameters at each time step between plasma evolution time 2 τ e and 10 τ e. As shown in Fig. 5, the higher the filling fraction, the better cooling effect of the electrons, independent of the plasma density and optical lattice type. However, unlike ions (refer to Fig. 2 of Ref. 60), the filling factor does not affect the suppression of electronic DIH by orders of magnitude.

FIG. 5.

Simulated Coulomb coupling parameters of the electrons for (a) simple-cubic (sc), (b) body-centered-cubic (bcc), and (c) face-centered-cubic (fcc) lattices at different filling fractions. Random distribution (disorder) is equivalent to f = 0. Γ e is evaluated in simulations over plasma evolution time between 2 τ e and 10 τ e. The legend indicates the different plasma densities.

FIG. 5.

Simulated Coulomb coupling parameters of the electrons for (a) simple-cubic (sc), (b) body-centered-cubic (bcc), and (c) face-centered-cubic (fcc) lattices at different filling fractions. Random distribution (disorder) is equivalent to f = 0. Γ e is evaluated in simulations over plasma evolution time between 2 τ e and 10 τ e. The legend indicates the different plasma densities.

Close modal

As we mentioned, Tiwari et al.51 suggested using one-dimensional strong external magnetic field to constrain electron motion to reduce electron kinetic energy and thus to cool electrons. This scheme needs strong magnetic field and can only suppress the heating of electrons in the direction perpendicular to the magnetic field. Furthermore, due to large ion mass, the motions of ions can hardly be constrained by the external magnetic field, and thus, the DIH of ions could not be extenuated. Previous theoretical works53,59–61 have demonstrated that optical lattices can push the ionic components into a deeper coupling regime. Therefore, based on our simulations, both the DIH of electrons and ions can be suppressed via optical lattice.

We propose a scheme to suppress the DIH of electrons, which is based on pre-ordering UNPs by loading atoms into three-dimensional optical lattices. By performing the classical MD simulations with open boundary conditions, we demonstrate that the DIH of electrons could be suppressed by a factor of 1.3, compared to the conventional UNPs. We also show that the degree of electron temperature reduced is independent of plasma density and lattice types. Moreover, the effect of the filling fraction on the suppression of electronic DIH has been explored, and we find that the higher the filling fraction, the greater the cooling effect of the electrons. Both electronic and ionic Coulomb coupling strength can be pushed into a deeper coupling regime, which will allow for exploring novel and intriguing phenomena. Furthermore, decreasing electronic temperature in UNP will increase the coherent length of the UNP-based electron beam, which will potentially permit the single-shot diffraction imaging of macromolecules, such as protein.

This work is supported by the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0303300), the National Key Research and Development Program of China (Grant No. 2022YFA1602502), and the National Natural Science Foundation of China (Grant No. 12127804).

The authors have no conflicts to disclose.

Haibo Wang: Investigation (lead); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Zonglin Yao: Investigation (equal); Methodology (equal); Software (equal). Haoyu Huang: Investigation (supporting); Methodology (supporting). Jianing Sun: Investigation (supporting); Methodology (supporting). Fuyang Zhou: Investigation (supporting); Methodology (supporting). Yong Wu: Funding acquisition (equal); Methodology (equal); Supervision (equal). Jianguo Wang: Funding acquisition (equal); Supervision (equal). Xiangjun Chen: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
T.
Killian
,
T.
Pattard
,
T.
Pohl
, and
J.
Rost
,
Phys. Rep.
449
,
77
(
2007
).
2.
M.
Lyon
and
S. L.
Rolston
,
Rep. Prog. Phys.
80
,
017001
(
2016
).
3.
S. D.
Bergeson
,
S. D.
Baalrud
,
C. L.
Ellison
,
E.
Grant
,
F. R.
Graziani
,
T. C.
Killian
,
M. S.
Murillo
,
J. L.
Roberts
, and
L. G.
Stanton
,
Phys. Plasmas
26
,
100501
(
2019
).
4.
T. C.
Killian
,
S.
Kulin
,
S. D.
Bergeson
,
L. A.
Orozco
,
C.
Orzel
, and
S. L.
Rolston
,
Phys. Rev. Lett.
83
,
4776
(
1999
).
5.
C. E.
Simien
,
Y. C.
Chen
,
P.
Gupta
,
S.
Laha
,
Y. N.
Martinez
,
P. G.
Mickelson
,
S. B.
Nagel
, and
T. C.
Killian
,
Phys. Rev. Lett.
92
,
143001
(
2004
).
6.
E. A.
Cummings
,
J. E.
Daily
,
D. S.
Durfee
, and
S. D.
Bergeson
,
Phys. Rev. Lett.
95
,
235001
(
2005
).
7.
D.
Feldbaum
,
N. V.
Morrow
,
S. K.
Dutta
, and
G.
Raithel
,
Phys. Rev. Lett.
89
,
173004
(
2002
).
8.
T.
Kroker
,
M.
Großmann
,
K.
Sengstock
,
M.
Drescher
,
P.
Wessels-Staarmann
, and
J.
Simonet
,
Nat. Commun.
12
,
596
(
2021
).
9.
M. P.
Robinson
,
B. L.
Tolra
,
M. W.
Noel
,
T. F.
Gallagher
, and
P.
Pillet
,
Phys. Rev. Lett.
85
,
4466
(
2000
).
10.
J. P.
Morrison
,
C. J.
Rennick
,
J. S.
Keller
, and
E. R.
Grant
,
Phys. Rev. Lett.
101
,
205005
(
2008
).
11.
J. L.
Roberts
,
C. D.
Fertig
,
M. J.
Lim
, and
S. L.
Rolston
,
Phys. Rev. Lett.
92
,
253003
(
2004
).
12.
P.
Gupta
,
S.
Laha
,
C. E.
Simien
,
H.
Gao
,
J.
Castro
,
T. C.
Killian
, and
T.
Pohl
,
Phys. Rev. Lett.
99
,
075005
(
2007
).
13.
R. S.
Fletcher
,
X. L.
Zhang
, and
S. L.
Rolston
,
Phys. Rev. Lett.
99
,
145001
(
2007
).
14.
X. L.
Zhang
,
R. S.
Fletcher
, and
S. L.
Rolston
,
Phys. Rev. Lett.
101
,
195002
(
2008
).
15.
J.-H.
Choi
,
B.
Knuffman
,
X. H.
Zhang
,
A. P.
Povilus
, and
G.
Raithel
,
Phys. Rev. Lett.
100
,
175002
(
2008
).
16.
G. M.
Gorman
,
M. K.
Warrens
,
S. J.
Bradshaw
, and
T. C.
Killian
,
Phys. Rev. Lett.
126
,
085002
(
2021
).
17.
R. T.
Sprenkle
,
S. D.
Bergeson
,
L. G.
Silvestri
, and
M. S.
Murillo
,
Phys. Rev. E
105
,
045201
(
2022
).
18.
G.
Bannasch
,
J.
Castro
,
P.
McQuillen
,
T.
Pohl
, and
T. C.
Killian
,
Phys. Rev. Lett.
109
,
185008
(
2012
).
19.
T. S.
Strickler
,
T. K.
Langin
,
P.
McQuillen
,
J.
Daligault
, and
T. C.
Killian
,
Phys. Rev. X
6
,
021021
(
2016
).
20.
M. A.
Viray
,
S. A.
Miller
, and
G.
Raithel
,
Phys. Rev. A
102
,
033303
(
2020
).
21.
R. T.
Sprenkle
,
L. G.
Silvestri
, and
S. D.
Bergeson
,
Nat. Commun.
13
,
15
(
2022
).
22.
K. A.
Twedt
and
S. L.
Rolston
,
Phys. Rev. Lett.
108
,
065003
(
2012
).
23.
J.
Castro
,
P.
McQuillen
, and
T. C.
Killian
,
Phys. Rev. Lett.
105
,
065004
(
2010
).
24.
S.
Kulin
,
T. C.
Killian
,
S. D.
Bergeson
, and
S. L.
Rolston
,
Phys. Rev. Lett.
85
,
318
(
2000
).
25.
26.
F.
Robicheaux
and
J. D.
Hanson
,
Phys. Rev. Lett.
88
,
055002
(
2002
).
27.
S. G.
Kuzmin
and
T. M.
O'Neil
,
Phys. Rev. Lett.
88
,
065003
(
2002
).
28.
S.
Mazevet
,
L. A.
Collins
, and
J. D.
Kress
,
Phys. Rev. Lett.
88
,
055001
(
2002
).
29.
T.
Pohl
,
T.
Pattard
, and
J. M.
Rost
,
Phys. Rev. Lett.
94
,
205003
(
2005
).
30.
T.
Pohl
,
D.
Vrinceanu
, and
H. R.
Sadeghpour
,
Phys. Rev. Lett.
100
,
223201
(
2008
).
31.
K.
Niffenegger
,
K. A.
Gilmore
, and
F.
Robicheaux
,
J. Phys. B: At., Mol. Opt. Phys.
44
,
145701
(
2011
).
32.
J. J.
McClelland
,
A. V.
Steele
,
B.
Knuffman
,
K. A.
Twedt
,
A.
Schwarzkopf
, and
T. M.
Wilson
,
Appl. Phys. Rev.
3
,
011302
(
2016
).
33.
M. P.
Reijnders
,
P. A.
van Kruisbergen
,
G.
Taban
,
S. B.
van der Geer
,
P. H. A.
Mutsaers
,
E. J. D.
Vredenbregt
, and
O. J.
Luiten
,
Phys. Rev. Lett.
102
,
034802
(
2009
).
34.
K. A.
Twedt
,
J.
Zou
,
M.
Davanco
,
K.
Srinivasan
,
J. J.
McClelland
, and
V. A.
Aksyuk
,
Nat. Photon.
10
,
35
(
2016
).
35.
D.
Murphy
,
R.
Speirs
,
D.
Sheludko
,
C.
Putkunz
,
A.
McCulloch
,
B.
Sparkes
, and
R.
Scholten
,
Nat. Commun.
5
,
4489
(
2014
).
36.
M.
Viteau
,
M.
Reveillard
,
L.
Kime
,
B.
Rasser
,
P.
Sudraud
,
Y.
Bruneau
,
G.
Khalili
,
P.
Pillet
,
D.
Comparat
,
I.
Guerri
,
A.
Fioretti
,
D.
Ciampini
,
M.
Allegrini
, and
F.
Fuso
,
Ultramicroscopy
164
,
70
(
2016
).
37.
B. J.
Claessens
,
S. B.
van der Geer
,
G.
Taban
,
E. J. D.
Vredenbregt
, and
O. J.
Luiten
,
Phys. Rev. Lett.
95
,
164801
(
2005
).
38.
W.
Engelen
,
M.
van der Heijden
,
D.
Bakker
,
E.
Vredenbregt
, and
O.
Luiten
,
Nat. Commun.
4
,
1693
(
2013
).
39.
A. J.
McCulloch
,
D. V.
Sheludko
,
S. D.
Saliba
,
S. C.
Bell
,
M.
Junker
,
K. A.
Nugent
, and
R. E.
Scholten
,
Nat. Phys.
7
,
785
(
2011
).
40.
T.
Pohl
,
T.
Pattard
, and
J. M.
Rost
,
Phys. Rev. Lett.
92
,
155003
(
2004
).
41.
M.
Lyon
,
S. D.
Bergeson
,
A.
Diaw
, and
M. S.
Murillo
,
Phys. Rev. E
91
,
033101
(
2015
).
42.
G.
Bannasch
,
T. C.
Killian
, and
T.
Pohl
,
Phys. Rev. Lett.
110
,
253003
(
2013
).
43.
M.
Robert-de Saint-Vincent
,
C. S.
Hofmann
,
H.
Schempp
,
G.
Günter
,
S.
Whitlock
, and
M.
Weidemüller
,
Phys. Rev. Lett.
110
,
045004
(
2013
).
44.
D.
Gericke
and
M.
Murillo
,
Contrib. Plasma Phys.
43
,
298
(
2003
).
45.
H.
Sadeghi
,
A.
Kruyen
,
J.
Hung
,
J. H.
Gurian
,
J. P.
Morrison
,
M.
Schulz-Weiling
,
N.
Saquet
,
C. J.
Rennick
, and
E. R.
Grant
,
Phys. Rev. Lett.
112
,
075001
(
2014
).
46.
S. G.
Kuzmin
and
T. M.
O'Neil
,
Phys. Plasmas
9
,
3743
(
2002
).
47.
T. K.
Langin
,
G. M.
Gorman
, and
T. C.
Killian
,
Science
363
,
61
(
2019
).
48.
N.
Vanhaecke
,
D.
Comparat
,
D. A.
Tate
, and
P.
Pillet
,
Phys. Rev. A
71
,
013416
(
2005
).
49.
T.
Pohl
,
D.
Comparat
,
N.
Zahzam
,
T.
Vogt
,
P.
Pillet
, and
T.
Pattard
,
Eur. Phys. J. D
40
,
45
(
2006
).
50.
E. V.
Crockett
,
R. C.
Newell
,
F.
Robicheaux
, and
D. A.
Tate
,
Phys. Rev. A
98
,
043431
(
2018
).
51.
S. K.
Tiwari
and
S. D.
Baalrud
,
Phys. Plasmas
25
,
013511
(
2018
).
52.
M. P.
Allen
and
D. J.
Tildesley
,
Computer Simulation of Liquids
(
Oxford University Press
,
2017
), p.
95
.
53.
J. W.
Gao
,
Y.
Wu
,
Z. P.
Zhong
, and
J. G.
Wang
,
Phys. Plasmas
23
,
123507
(
2016
).
54.
S.
Laha
,
P.
Gupta
,
C. E.
Simien
,
H.
Gao
,
J.
Castro
,
T.
Pohl
, and
T. C.
Killian
,
Phys. Rev. Lett.
99
,
155001
(
2007
).
55.
I.
Bloch
,
J.
Dalibard
, and
W.
Zwerger
,
Rev. Mod. Phys.
80
,
885
(
2008
).
56.
G.
Grynberg
,
B.
Lounis
,
P.
Verkerk
,
J.-Y.
Courtois
, and
C.
Salomon
,
Phys. Rev. Lett.
70
,
2249
(
1993
).
57.
C. S.
Adams
,
S. G.
Cox
,
E.
Riis
, and
A. S.
Arnold
,
J. Phys. B: At., Mol. Opt. Phys.
36
,
1933
(
2003
).
58.
R.
Grimm
,
M.
Weidemüller
, and
Y. B.
Ovchinnikov
, “Optical Dipole Traps for Neutral Atoms” in
Advances in Atomic, Molecular, and Optical Physics
(
Academic Press
,
Cambridge, MA
,
2000
), Vol. 42, pp.
95
170
.
59.
L.
Guo
,
R. H.
Lu
, and
S. S.
Han
,
Phys. Rev. E
81
,
046406
(
2010
).
60.
D.
Murphy
and
B. M.
Sparkes
,
Phys. Rev. E
94
,
021201(R)
(
2016
).
61.
T.
Pohl
,
T.
Pattard
, and
J. M.
Rost
,
J. Phys. B: At., Mol. Opt. Phys.
37
,
L183
(
2004
).