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1

Evolution of Ultracold, Neutral Plasmas

S. Mazevet, L. A. Collins, J. D. KressTheoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545

(submitted to PRL)

We present the first large-scale simulations of an ultracold,neutral plasma, produced by photoionization of laser-cooledxenon atoms, from creation to initial expansion, using classi-cal molecular dynamics methods with open boundary condi-tions. We reproduce many of the experimental findings suchas the trapping efficiency of electrons with increased ion num-ber, a minimum electron temperature achieved on approachto the photoionization threshold, and recombination into Ry-dberg states of anomalously-low principal quantum number.In addition, many of these effects establish themselves veryearly in the plasma evolution (∼ ns) before present experi-mental observations begin.

That a common characteristic connects such diverseenvironments as the surface of a neutron star, the initialcompression stage of an inertial confinement fusion cap-sule, the interaction region of a high-intensity laser withatomic clusters, and a very cold, dilute, and partially-ionized gas within an atomic trap, seems at first ratherremarkable. Yet all these cases embrace a regime knownas a strongly-coupled plasma. For such a plasma, the in-teractions among the various constituents dominate thethermal motions. The plasma coupling constant Γα, theratio of the average electrostatic potential energy Zα/aα [Zα, the charge; aα, the ion-sphere radius = 3/(4πnα

1/3);and nα, the number density for a given component α]to the kinetic energy kBTα, provides an intrinsic mea-sure of this effect [1]. When Γα exceeds unity, vari-ous strong-coupling effects commence such as collectivemodes and phase transitions. For multi-component plas-mas, the coupling constants need not be equal or evencomparable, leading to a medium that may contain bothstrongly- and weakly- coupled constituents.

Since temperatures usually start around a few hundredKelvin, most plasmas found in nature or engineered inthe laboratory attain strongly-coupled status from highdensities, as in the case of a planetary interior or a shock-compressed fluid [2,3], or from highly-charged states, asin colloidal or “dusty” plasmas [4]. In both situations,the particle density usually rises well above 1018/cm3.On the other hand, ion trapping and cooling methodshave produced dilute, strongly-coupled plasmas by rad-ically lowering the temperature. At first, these effortswere limited to nonneutral plasmas confined by a mag-netic field [5]. However, recently, new techniques [6–9]have generated neutral, ultracold plasmas, free of any

external fields at densities of the order of 108-1010/cm3

and temperatures at microkelvins (µK).Two methods, one direct and one indirect, but both

employing laser excitation of a highly-cooled sample ofneutral atoms, have successfully created such neutralplasmas. The direct approach employs the photoioniza-tion of laser-cooled xenon atoms [6–8], while the indirectgenerates a cold Rydberg gas in which a few collisionswith room-temperature atoms produces the ionization[9,10]. In both cases, the electron and ion temperaturesstart very far apart with the former from 1K to 1000Kand the latter remaining at the initial atom temperatureof a few µK. This implies that the coupling constant forthe electrons (Γe) ranges between 1 and 10, while theions begin with Γi ∼ 1000. Based on the results pre-sented below, the following picture emerges. As the sys-tem evolves, the electrons reach a quasi-equilibrium whileat the same time beginning to move the ions throughlong-range Coulomb interactions. Eventually, the ionsalso heat, and the whole cloud begins to expand withthe ions and electrons approaching comparable temper-atures. The former processes occur on the order of pico-to nano- seconds, while the full expansion becomes no-ticeable on a microsecond scale. Therefore, by followingthe progress of the system, we can study not only thebasic properties of a strongly-coupled plasma but also itsevolution through a variety of stages. The path to equi-libration still remains a poorly understood process forstrongly-coupled plasmas in general.

The dilute character of the ultracold plasmas pro-vides a unique opportunity to explore the intricate natureof atomic-scale, strongly-coupled systems, both throughreal-time imaging of the sample to study wave phenom-ena, collective modes, and even phase transitions, andthrough laser probes to examine internal processes. Theinitial experiments have already discovered unexpectedphenomena such as more rapid than expected ion ex-pansion [7] and Rydberg populations that strongly de-viate from those predicted from standard electron-ionrecombination rates [8,11–13]. So far, the experimentshave examined the later expansion stage of the wholeplasma cloud, which occurs on the scale of microseconds.However, the nature of the plasma at this stage dependsstrongly on its evolution from its creation. While ultra-fast laser techniques offer the eventual prospect of prob-ing these early times, for now, only simulations can pro-vide an understanding of the full evolution of these sys-tems. We focus on the first such generated plasma [6]from the photoionization of cold Xenon atoms; however,

the findings also provide insight into other ultracold sys-tems.

Since the electron temperature greatly exceeds theFermi temperature for this system, we may effectivelyemploy a two-component classical molecular dynamics(MD) approach in which the electrons and ions aretreated as distinguishable particles, interacting through amodified Coulomb potential. We have employed severaldifferent forms from an ab-initio Xe+ + e− pseudopoten-tial [14] to a simple effective cut-off potential [15] basedon the de Broglie thermal wavelengths; we found littlesensitivity of the basic properties to this choice. Thisfinding stems from the dominance of the interactions bythe very long-range Coulomb tails since the average par-ticle separations are of the order of 104A(1 to 2 times thede Broglie wavelength for electrons at a temperature of0.1K). The short-range modification of the Coulomb po-tential basically prevents particles with opposite chargesfrom collapsing into a singularity. Due to the open andnonperiodic boundary conditions, sample size effects canplay a critical role in the simulations [16]. Therefore, wemust employ particle numbers of the basic order of theexperiments, requiring efficient procedures for handlingthe long-range forces. In addition, the extended simula-tion times needed to model the entire plasma evolution[fs to µs] demand effective temporal treatments. To thisend, we have employed multipole-tree procedures [17] inconjunction with reversible reference system propagatoralgorithms [18] (r-RESPA) through the parallel 3D MDprogram NAMD [19], generally used in biophysical ap-plications. These procedures allow us to treat samples ofbetween 102 and 104 particles [N = Ni/2 = Ne/2]; mostsimulations were performed with N = 103.

To reproduce the initial experimental conditions [6], wedistribute the ions randomly according to a 3D Gaussian,whose rms radius, σ, matches the desired ion numberdensity ni [ni = Ni/(2πσ2)3/2] and impose a Maxwell-Boltzmann velocity distribution at Ti = 1µK. To eachion, we associate an electron placed randomly in an orbitof radius ro= 50A. The electron velocities point in ran-dom directions but with fixed magnitude, determined bythe photoionization condition [Ki

e ≡1

2mev

2e = hω−1/ro,

with ω, the laser frequency and v2e as a sum over all three

spatial components]. This idealized prescription modelsthe photoionization process by allowing each electron toescape the Coulomb well with a final prescribed energy.By varying the laser frequency, we can control the finaleffective electron temperature, just as in the experiments.We tested this particular ionization model against simu-lations where both the initial electron-ion radius ro andthe form of the ion distribution are varied. Overall, wefind little sensitivity to these variations on the basic ini-tial conditions. Finally, we take a kinematic definitionof temperature as the average kinetic energy per parti-cle Tα = 1

3NαkB

∑

i mαv2α,i, where the sum runs over all

particles i of type α.

The plasma evolves through several stages. In the firstor photoionization stage, which lasts on the order of fs,the electrons climb out of the Coulomb well and becomebasically freely-moving particles. In the next stage, theelectrons reach a quasi-equilibrium at Te due to theirfast intra-particle collisions. Following this stage, theelectron-ion collisions begin the slow process of heatingthe ions, which can require up to µs. This process canalso be viewed in terms of an electron pressure term in ahydrodynamical formulation [7]. Then, the whole cloudof ions and electrons begins a systematic expansion. Thisprogression clearly evinces itself in Fig. 1, which showsthe temporal behavior of electron and ion average kineticenergies, using an effective mass for the ion of mi = 0.01amu in order to accelerate the evolutionary process. Theaveraged kinetic energy of the electrons and ion becomecomparable and the cloud perceptibly expands in about

20 ns. Scaling this value by√

mXe

miyields an estimate for

this time in a Xe plasma of about 1µs, in line with theexperiments [6]. Also consistent with the experimentalmeasurements [7], these preliminary calculations indicatethat a large fraction of the kinetic energy transferred bythe electrons results in outward translational motion forthe ions. This implies that the kinetic temperature de-fined above coincides with the usual themodynamic tem-perature (random thermal motion) for the electrons only.In all other simulations discussed, the ions carry the massof Xe.

0.0 2.0 4.0 6.0 8.0 10.0

time (ns)

10-2

10-1

100

101

Ave

rage

kin

etic

ene

rgy

(K)

FIG. 1. Schematic variation of the electron (open square)and ion (filled square) average kinetic energy per particle asa function of time for a model plasma with mi=0.01 amu.

Since we shall use the simulations to understand in-termediate stages in the development of the ultracoldplasma, we need to establish the validity of the procedureby comparing to experiments. Two initial observationshave particular significance: 1) the number of trappedelectrons rises with the number of ions created for a fixedfinal temperature, and 2) the electrons attain a certainminimum temperature no matter how small hω becomes.Figure 2a shows that the number of trapped electronsas a function of the number of ions and initial electron

temperature basically follows the general experimentaltrends [6]. Some of the electrons, freed by the photoion-ization process from ions with sufficient energy, escapethe atomic cloud and never return, leaving an overallpositively charged system. This residual charge then ef-fectively traps the remaining electrons so that the cen-ter of the distribution resembles a neutral plasma. Thelarger the number of ions produced, the more effectivethe confinement of the electrons.

In Fig.2b, we present for ni=4.32x109 ions/cm3, theelectron temperature Tf

e after the initial equilibration inthe simulations as a function of the excess initial temper-ature Ti

e[=2

3Ki

e] given to the electrons during the pho-toionization process. The plateau at about 5K [Γe ∼ 1]for small Ti

e appears quite pronounced. This effect arisesdue to a mechanism usually designated as “continuumlowering” [1]. Despite the small energies and enormousdistances involved, at least by usual atomic standards,the interaction of an ion with its neighbors still has a no-ticeable effect due to the very long-range of the Coulombpotential. This shifts the appropriate zero of energy fromthe isolated atom to the whole system. Using a simplebinary interaction, we can estimate this energy differ-ence for an electron halfway between the ions [∼ 2/aion]at around 3K for this density, in qualitative agreementwith the simulations. Even for a photon energy near theatomic threshold, the electron within the plasma will stillgain this minimum energy. This same effect also explainsthe dependence of the final electron temperature on theion density. In this situation, an increase in the ion den-sity enhances the zero of energy shift and leads to anincrease in the final electron temperature.

100 1000

Number of ions

20

40

60

80

100

% e

lect

rons

trap

ped

Te=+10K

Te=+3K

0.1 1 10 100

initial temperature Ti

e(K)

0

10

20

30

40

fina

l tem

pera

ture

Tf e (

K)

(a) (b)

FIG. 2. (a) Percentage of electrons trapped as a function ofthe number of ions created for two electron temperatures (3Kand 10K). (b) Variation of the final electron temperature Tf

e

as a function of the initial excess kinetic energy representedan associated temperature Ti

e.

The molecular dynamics simulations yield additionalproperties of the ultracold plasma. We paid particu-lar attention to the electron plasma frequency fe since

experiments [7] have used this quantity as an indirectmeans of deducing the density of the ultracold plasma.Given the nature of the system with open boundary con-ditions and quasi-equilibration among the electrons, thequestion arises as to whether fe has a precise definition.Neglecting temperature dependence, the electron plasmafrequency is given by

fe =1

2π

√

e2ne

me, (1)

where e is the elementary charge, ne the electron den-sity, and me the electron mass. From the MD simula-tions, the electron plasma frequency is obtained from theFourier transform of the electron velocity autocorrela-tion function [20]. We find, reassuringly, that a distinctthough broadened peak arises near the fe predicted byEq.1 in the regime in which the electrons reach a quasi-equilibration. For example, at a density of 4.32x1010

ions/cm3, an electron temperature of 3K, and simula-tion time extending up to 18ns, MD gives plasma fre-quencies of 1.2GHz (1.4 GHz) while Eq.1 yields 1.57GHz(1.87GHz) for trapped electrons at 70% (100%) of the iondensity. When the ion density is decreased to a value of4.32x109 ions/cm3 at the same electron temperature, fe

from the MD simulations becomes 0.5Ghz, also in accor-dance with Eq.1 . In general, we find that the number ofparticles used in the simulation cell has little effect on thedetermination of the electron plasma frequency beyondproviding a better statistical sample.

100 1000 100000

0.2

0.4

0.6

0.8

Gie

(r)

100 1000 100000

0.1

0.2

0.3

0.4

Gie

(r)

100 1000 100000

0.2

0.4

0.6

0.8

Gie

(r)

100 1000 100000

0.1

0.2

0.3

0.4G

ie(r

)

100 1000 10000

Ri-e

(A)

0

0.2

0.4

0.6

0.8

Gie

(r)

100 1000 10000

Ri-e

(A)

0

0.1

0.2

0.3

0.4

Gie

(r)

2.5ns

8.5ns

16ns

4.32.1010

ion/cm3 T

e=+10K 4.32.10

9ion/cm

3 T

e=+3K

(d) 7.4ns

(e) 16.2ns

(f) 20.2ns

(a)

(b)

(c)

FIG. 3. Time evolution of the modified ion-electron paircorrelation function Gie as a function of density and electrontemperature: (a)-(c) 4.32x1010 ions/cm3 and 10K, and (d)-(f)4.32x109 ions/cm3 and 3K.

We now turn to the nature of the constituents as theplasma evolves and pay particular attention to the for-mation of Rydberg atoms as noted in the experiments

[8–10]. To this end, we examine over space and timethe ion-electron pair correlation function gie(r), whichgives the probability of finding an electron a distance rfrom a reference ion [15]. To enhance the examination ofthe closer encounters, we have employed a modified formGie(r) by subtracting a uniform distribution and aver-aging gie(r) over a small time interval of 1.5 ns. Figure3 displays Gie(r) for two representative cases. The openboundary conditions, which force the distribution to zeroat some large distance, give Gie(r) a distinctly differentbehavior than seen in a fluid or periodic system.

Figures 3(a)-(c) display the development of a plasmafor ni=4.32x1010 ions/cm3 and Te ∼ 10K in the regime inwhich the electrons have reached quasi-equilibration andthe ions have started to heat. After 2.5ns, the electrondistribution departs from its initial form, especially be-yond a radius of 10000A. By 8.5ns, we begin to observe anincrease in Gie just below 1000A. This signifies the forma-tion of Rydberg states in an average principal quantumnumber n ∼ 40, a conclusion supported by an examina-tion of movies of the simulation. These movies also showstable Rydberg atom configurations over many orbitalperiods of the electrons. A simple model [13] that bal-ances various collisional processes, using strictly atomiccross sections, predicts a much larger value (n ≥ 100).For conditions closer to the experimental measurements(lower density), as depicted in Figs. 3(d)-(f), we find asignificant Rydberg population after only 20ns (Fig.3f).The electron distribution shifts to around 2000A, corre-sponding to a principal quantum number n ∼ 60. In bothcases, the Rydberg population is estimated to be between5% and 10 % of the total number of electrons. Thesefindings closely resemble those of the experiments [8],though for later times (∼ µs) and indicate that collectiveor density effects may play an important role by changingthe accessible Rydberg-level distribution or cross sectionsthrough long-range interactions. By examining the elec-tron mean square displacement over time, we can identifytwo stages in the evolution of the cold ionized gas. First,after the ionization of the atoms, the electrons diffusethroughout the system for several nanoseconds with nosignificant Rydberg atom formation. Second, the elec-trons reach the edge of the cloud and begin systematicmultiple traverses of the system. The Rydberg atom pop-ulation becomes noticeable only several nanoseconds intothis second stage. The feature that attracts the most in-terest is the relatively short time scale required to estab-lish the neutral plasma and produce a noticeable Rydbergpopulation.

In summary, we have followed the evolution of an ul-tracold, neutral plasma over a broad range of tempo-ral stages with classical molecular dynamics simulationtechniques. We find general agreement with experimen-tal observations of the number of trapped electrons, theminimum of electron temperature, and the production ofRydberg atoms in low-lying states. The latter two con-

ditions especially demonstrate the importance of strong-coupling or density effects on the basic atomic inter-actions. In addition, we have found that the electronplasma frequency appears as a valid tool to probe thestate of the system. Important to the understanding ofthe temporal development of these plasmas, we discov-ered that recombination and the formation of long-livedRydberg states occur rapidly on the order of nanosec-onds. Our studies continue in an effort to push into thefully expanded regime by using hydrodynamical methodstied to the molecular dynamics.

We wish to acknowledge useful discussions with Dr. S.Rolston (NIST), Prof. T. Killian (Rice), and Prof. P.Gould (U. of Connecticut). We thank Dr. N. Troullierfor providing the Xe pseudopotential. Work performedunder the auspices of the U.S. Department of Energy(contract W-7405-ENG-36) through the Theoretical Di-vision at the Los Alamos National Laboratory.

[1] S. Ichimaru, Statistical Plasma Physics, ed. Addison-Wesley, 1994.

[2] W.J. Nellis et. al., Science 269,1249 (1995).[3] G. W. Collins et. al., Science 281(5380) 1178 (1998) and

references therein.[4] M. Rosenberg, J. de Phys. IV, 10, 73 (2000) and refer-

ences therein.[5] D.H.E. Dubin and T.M. O’Neil, Rev. Mod. Phys. 71, 87

(1999).[6] T.C. Killian et. al., Phys. Rev. Lett. 83, 4776 (1999).[7] S. Kulin et al., Phys. Rev. Lett., 85, 318 (2000).[8] T.C. Killian et. al., Phys. Rev. Lett. 86, 3759 (2001).[9] M.P. Robinson et. al., Phys. Rev. Lett. 85, 4466, (2000).

[10] P. Gould and E. Eyler, Phys. World 14, 19-20 (2001).[11] D.R. Bates et. al. Proc. R. Soc. London A 267, 297

(1962).[12] Y. Hahn, Phys. Lett. A 231, 82 (1997); Phys. Lett. A

264, 465 (2000).[13] J. Stevefelt et. al. Phys. Rev. A 12, 1246 (1975).[14] N. Troullier and J.L. Martins, Phys. Rev. B 43, 1993

(1991).[15] J.P. Hansen and I.R. McDonald, Phys. Rev. A 23, 2041

(1981).[16] Far more restrictive calculations using periodic bound-

ary conditions have been reported by A. Tkackev et al.,Quant. Elect. 30, 1077 (2000) and, after submission ofthis article, by M. Morrillo Phys. Rev. Let. 87, 115003-1(2001).

[17] L. Greengard and V. Rokhlin, J. Comp. Phys. 73, 325(1987); J. Barnes and P. Hut, Nature 324, 446 (1986).

[18] F. Figueirido et. al., J. Chem. Phys. 106, 9835 (1997)and references therein.

[19] L. Kale et. al., J. Comp. Phys.151, 283-312 (1999).[20] J.P. Hansen, and I. McDonald, Theory of Simple liquid,

ed. Academic Press (1986).