s. kulin et al- plasma oscillations and expansion of an ultracold neutral plasma

8/3/2019 S. Kulin et al- Plasma Oscillations and Expansion of an Ultracold Neutral Plasma

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arXiv:physics/0006

027v1

[physics.plasm-ph]9Jun2000

Plasma Oscillations and Expansion of an Ultracold Neutral Plasma

S. Kulin, T. C. Killian, S. D. Bergeson, and S. L. RolstonNational Institute of Standards and Technology, Gaithersburg, MD 20899-8424

(Accepted Phys. Rev. Lett.)

We report the observation of plasma oscillations in an ul-tracold neutral plasma. With this collective mode we probethe electron density distribution and study the expansion ofthe plasma as a function of time. For classical plasma con-ditions, i.e. weak Coulomb coupling, the expansion is domi-nated by the pressure of the electron gas and is described by ahydrodynamic model. Discrepancies between the model andobservations at low temperature and high density may be dueto strong coupling of the electrons.

One of the most interesting features of neutral plasmas

is the rich assortment of collective modes that they sup-port. The most common of these is the plasma oscillation[1], in which electrons oscillate around their equilibriumpositions and ions are essentially stationary. This modeis a valuable probe of ionized gases because the oscillationfrequency depends solely on the electron density.

In an ultracold neutral plasma as reported in [2], thedensity is nonuniform and changing in time. A diag-nostic of the density is thus necessary for a variety ofexperiments, such as determination of the three-body re-combination rate at ultralow temperature [3], and obser-vation of the effects of strong Coulomb coupling [4] in atwo-component system. A density probe would also aid

in the study of the evolution of a dense gas of cold Ry-dberg atoms to a plasma [5], which may be an analog ofthe Mott insulator-conductor phase transition [6].

In this work we excite plasma oscillations in an ultra-cold neutral plasma by applying a radio frequency (rf)electric field. The oscillations are used to map the plasmadensity distribution and reveal the particle dynamics andenergy flow during the expansion of the ionized gas.

The creation of an ultracold plasma has been describedin [2]. A few million metastable xenon atoms are lasercooled to approximately 10K. The peak density is about2 1010 cm3 and the spatial distribution of the cloud isGaussian with an rms radius 220m. These parame-

ters are determined with resonant laser absorption imag-ing [7]. To produce the plasma, up to 25% of the atomsare photoionized in a two-photon excitation. Light forthis process is provided by a Ti:sapphire laser at 882 nmand a pulsed dye laser at 514 nm (10 ns pulse length).Because of the small electron-ion mass ratio, the result-ing electrons have an initial kinetic energy (Ee) approxi-mately equal to the difference between the photon energyand the ionization potential. In this study we vary Ee/kB

between 1 and 1000 K. The initial kinetic energy of the

ions varies between 10K and 4mK.For detection of charged particles, a small DC field

(about 1 mV/cm) directs electrons to a single channelelectron multiplier and ions to a multichannel plate de-tector. The amplitude of the rf field that excites plasmaoscillations, F, varies between 0.2 20 mV/cm rms. Allelectric fields are applied to the plasma with grids locatedabove and below the laser-atom interaction region.

In the absence of a magnetic field, the frequency ofplasma oscillations is given by fe = (1/2)

e2ne/0me

[1]. Here, e is the elementary charge, ne is the electrondensity, 0 is the permittivity of vacuum, and me is theelectron mass. This relation is most often derived for an

infinite homogeneous plasma, but it is also valid in ourinhomogeneous system for modes which are localized inregions of near resonant density. Corrections to fe dueto finite temperature [8] depend on the wavelength ofthe collective oscillation, which is difficult to accuratelyestimate. Such corrections are not expected to be largeand will be neglected. We observe plasma oscillationswith frequencies from 1 to 250 MHz. This correspondsto resonant electron densities, nr, between 1 104 cm3

and 8 108 cm3. The oscillation frequency is sensitiveonly to ne, but, as explained in [2], the core of the plasmais neutral. This implies that plasma oscillations measureelectron and ion densities in this region (ne = ni n).

Figure 1a shows electron signals from an ultracold neu-tral plasma created by photoionization at time t = 0.Some electrons leave the sample and arrive at the de-tector at about 1s, producing the first peak in the sig-nal. The resulting excess positive charge in the plasmacreates a Coulomb potential well that traps the remain-ing electrons [2]. In the work reported here, typically90 99% of the electrons are trapped. Debye shieldingmaintains local neutrality inside a radius re beyond whichthe electron density drops to zero on a length scale equalto D. The value of re depends on the fraction of elec-trons that has escaped, and D is the Debye screening

length, D =0kBTe/e

2

ne, where Te is the electrontemperature. For our conditions re > 2, and D .As the plasma expands, the depth of the Coulomb welldecreases, allowing the remaining electrons to leave thetrap. This produces the broad peak at 25s.

In the presence of an rf field an additional peak appearsin the electron signal (Fig. 1a). We understand the gen-eration of this peak as follows: The applied rf field excitesplasma oscillations only where the frequency is resonant.Energy is thus pumped into the plasma in the shell with

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the appropriate electron density (n = nr). The ampli-tude of the collective electron motion is much less than, but the acquired energy is collisionally redistributedamong all the electrons within 10 1000ns [9], raisingthe electron temperature. This increases the evaporationrate of electrons out of the Coulomb well, which producesthe plasma oscillation response on the electron signal.

The resonant response at a given time, S(t), is pro-

portional to the number of electrons in the region wherethe density equals nr. If we make a simple local den-sity approximation and neglect decoherence of the oscil-lations, S(t) F2

d3r n(r, t) [n(r, t) nr]. The width

in time of the observed signal (Fig. 1a) reflects the den-sity distribution of the sample [10]. At early times whenthe density is higher than nr almost everywhere, S(t) isnegligibly small. As the cloud expands and the densitydecreases, the response grows because the fraction of theplasma which is in resonance increases. The peak of theresponse appears approximately when the average den-sity, n, becomes resonant with the rf field. S(t) vanisheswhen the peak density is less than nr.

The resonant response arrives later for lower frequency(Fig. 1b) as expected because n decreases in time. As-suming that the plasma density profile remains Gaussianduring the expansion, S(t) can be evaluated and its am-plitude scales as F2/nr. In Fig. 1b the data have beennormalized by this factor and the resulting amplitudesare similar for all conditions.

By equating n to nr when the response peak arrives,we can plot the average plasma density as a function oftime (Fig. 2). The data are well described by a self similarexpansion of a Gaussian cloud, n = N/[4(2

0+v2

0t2)]3/2,

where 0 is the initial rms radius and v0 is the rms radial

velocity at long times. N is determined independently bycounting the number of neutral atoms with and withoutphotoionization. The extracted values of0 are equal tothe size of the initial atom cloud. In such an expansion,the average kinetic energy per particle is 3mv2

0/2.

Figure 3 shows the dependence of v0 on density andinitial electron energy. We first discuss data with Ee 70 K, for which the expansion velocities approximatelyfollow v0 =

Ee/mi, where mi is the ion mass and

= 1.7 is a fit parameter. For the plasma to expandat this rate, the ions must acquire, on average, a veloc-ity characteristic of the electron energy. This is muchgreater than the initial ion thermal velocity. Electron-

ion equipartition of energy would yield v0 =Ee/3mi,

close to the observed value. However, due to the largeelectron-ion mass difference, this thermalization requiresmilliseconds [9]. The observed expansion, in contrast, oc-curs on a time scale of tens of microseconds. One mightexpect the expansion to be dominated by the Coulombenergy arising from the slight charge imbalance of theplasma, but this energy is about an order of magnitudeless than the observed expansion energy. Also, by Gausslaw, it would only be important in the expansion of the

non-neutral outer shell of the plasma. The oscillationprobe provides information only on the neutral core be-cause it relies on the presence of electrons.

A hydrodynamic model [11], which describes theplasma on length scales larger than D, shows that theexpansion is driven by the pressure of the electron gas.The pressure is exerted on the ions by outward-movingelectrons that are stopped and accelerated inward in the

trap. For the hydrodynamic calculation, ions and elec-trons are treated as fluids with local densities na(r) andaverage velocities ua(r) = va(r). Here, a refers to ei-ther electrons or ions, and denotes a local ensembleaverage. Particle and momentum conservation lead tothe momentum balance equations

mana

uat

+ (ua )ua

= (nakBTa) + Rab.

Here nakBTa represents a scalar pressure [11]. The ionand electron equations are coupled by Rab, which is therate of momentum exchange between species a and b.

The exact form of this term is unimportant for this study,but Rab = Rba. Plasma hydrodynamic equations typi-cally have electric and magnetic field terms, but appliedand internally generated fields are negligible when de-scribing the expansion.

We can make a few simplifying approximations thatare valid before the system has significantly expanded.The directed motion is negligible, so we set ua 0 ev-erywhere. Because ne ni = n, ue/t ui/t. Dueto the small electron mass, the rate of increase of aver-age electron momentum is negligible compared to thatof the ions. The electron momentum balance equationthen yields (nkBTe) Rei, which describes a balance

between the pressure of the electron gas and collisionalinteractions. This is the hydrodynamic depiction of thetrapping of electrons by the ions.

In the ion momentum balance equation, we eliminateRie using the electron equation, and we drop the pres-sure term because the ion thermal motion is negligible.Thus minui/t (nkBTe), which shows that thepressure of the electron gas drives the expansion [12].This result implies that the ions acquire a velocity of or-der

kBTe/mi, which is in qualitative agreement with

the high Ee data of Fig. 3. To calculate the expansionvelocity more quantitatively, one must consider that aselectrons move in the expanding trap, they perform workon the ions and cool adiabatically. The thermodynamicsof this process [13] is beyond the scope of this study.

The data in Fig. 3 indicate that about 90 % of the ini-tial kinetic energy of the electrons is transferred to theions kinetic energy, 3miv20/2 = 3Ee/2. This does notimply that the temperature of the ions becomes compara-ble to Ee/kB in this process. For the ions, ui increases,but mi|vi ui|

2, which measures random thermal mo-tion and thus temperature, is expected to remain small.

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This follows from slow ion-electron thermalization [9] andcorrelation between position and velocity during the ex-pansion [14].

We now turn our attention to systems with Ee < 70 K(Fig. 3). They expand faster than expected from an ex-trapolation of v0 =

Ee/mi, and thus do not even

qualitatively follow the hydrodynamic model. A relativemeasure of the deviation is (miv

2

0Ee/)/(Ee/). Fig-

ure 4 shows that the relative deviation increases with in-creasing electron Coulomb coupling parameter [4], e =(e2/40 a)/kBTe. Here, a = (4n/3)

1/3 is the Wigner-Seitz radius, n is the peak density at t = 0, and thetemperature is calculated by 3kBTe/2 = Ee.

The fact that the relative deviation depends only one, and that it becomes significant as e approaches 1,suggests that we are observing the effects of strong cou-pling of the electrons [15]. The hydrodynamic model ofthe plasma is only valid when e 1. When e > 1, elec-tron and ion spatial distributions show short range corre-lated fluctuations that are not accounted for in a smoothfluid description [16]. Correlations between the ion andelectron positions would provide the excess kinetic en-ergy observed in the expansion by lowering the potentialenergy of the plasma. This satisfies overall energy con-servation and it may also explain the systematically poorfits of the data for high e (See Fig. 2).

Strong coupling is also predicted to alter the relationfor the frequency fe [17], with which we extract theplasma density, size, and expansion velocity. The trendof this effect agrees qualitatively with the observed devi-ation, but knowledge of the wavelength of the collectiveoscillation is needed for a quantitative comparison.

Other possible explanations for the deviation are re-

lated to how the ultracold plasma is created. The10 ns duration of the photoionization pulse is long com-pared to the time required for electrons to move an in-terparticle spacing. Photoionization late in the pulsethus occurs in the presence of free charges, which willdepress the atomic ionization threshold by EIP 1

2kBTe{[(3e)3/2 + 1]2/3 1} [18]. This effect might in-

crease the electron kinetic energy by EIP above whathas been assumed. However, as shown in Fig. 4, the cal-culated EIP is about an order of magnitude smallerthan the observed effect. The random potential energyof charged particles when they are created may also yielda greater electron energy than Ee [19].

High e (high density and low temperature) conditionsare desirable for studying the three-body recombinationrate in an ultracold plasma. The theory [20] for this pro-cess was developed for high temperature, and is expectedto break down in the ultracold regime [3]. Measuring orsetting an upper limit for the recombination rate is notpossible until the dynamics of high e systems is un-derstood. We are currently studying this problem withmolecular dynamics calculations.

We have shown that plasma oscillations are a valuable

probe of the ion and electron density in an ultracold neu-tral plasma. This tool will facilitate future experimentalstudies of this novel system, such as the search for othercollective modes in the plasma and further investigationof the effects of correlations due to strong coupling.

We thank Lee Collins for helpful discussions andMichael Lim for assistance with data analysis. S. Kulinacknowledges funding from the Alexander-von-Humboldt

foundation. This work was funded by the ONR.

Present address: Department of Physics and Astronomy,Brigham Young University, Provo UT 84602-4640.

[1] L. Tonks and I. Langmuir, Phys. Rev. 33, 195 (1929).[2] T. C. Killian, S. Kulin, S. D. Bergeson, L. A. Orozco,

C. Orzel, and S. L. Rolston, Phys. Rev. Lett. 83, 4776(1999).

[3] Y. Hahn, Phys. Lett. A 231, 82 (1997).[4] S. Ichimaru, Rev. Mod. Phys. 54, 1017 (1982).[5] S. Kulin, T. C. Killian, S. D. Bergeson, L. A. Orozco, C.

Orzel, and S. L. Rolston, in Non-Neutral Plasma PhysicsIII, edited by J. J. Bollinger, R. L . Spencer, and R. C.Davidson, (AIP, New York, 1999), p. 367.

[6] Metal-Insulator Transitions Revisited, edited by P. P. Ed-wards and C. N. R. Rao, (Taylor & Francis Ltd., Lon-don, 1995); G. Vitrant, J. M. Raimond, M. Gross, andS. Haroche, J. Phys. B: At. Mol. Phys. 15, L49 (1982).

[7] M. Walhout, H. J. L. Megens, A. Witte, and S. L. Rol-ston, Phys. Rev. A 48, R879 (1993).

[8] D. Bohm and E. P. Gross, Phys. Rev. 75, 1851 (1949).[9] L. Spitzer, Jr., Physics of Fully Ionized Gases (John Wi-

ley & Sons, Inc., New York, 1962), chap. 5.[10] By applying an rf pulse, we found the response time

for excitation and detection of plasma oscillations to beabout 1s. This is short compared to the width of S(t)and is probably set by the electron thermalization rateand time-of-flight to the detector.

[11] R. J. Goldston and P. H. Rutherford, Introductionto Plasma Physics (Institute of Physics, Philadelphia,1995), chap. 6.

[12] This equation would also describe the ballistic expansionof a single-component cloud of noninteracting particlesat temperature Te. It also preserves a Gaussian densitydistribution, which supports our assumption of such adistribution in the data analysis.

[13] G. Manfredi, S. Mola, and M. R. Feix, Phys. Fluids B 5,388 (1993).

[14] C. Orzel, M. Walhout, U. Sterr, P. S. Julienne, and S. L.Rolston, Phys. Rev. A 59, 1926 (1999).

[15] We expect that the ions are also strongly coupled, al-though we have no direct evidence for this. The ion-ion thermalization time, equal to that of the electrons,is short, and the low initial ion kinetic energy yieldsi >> 1. Strong coupling of ions is not expected to affectthe experiments discussed here.

[16] S. Ichimaru, Statistical Plasma Physics, Volume I

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(Addison-Wesley Publishing Co., Reading, MA, 1992),chap. 7.

[17] G. Kalman, K. I. Golden, and M. Minella, in StronglyCoupled Plasma Physics, edited by H. M. Van Horn, andS. Ichimaru, (University of Rochester Press, Rochester,1993), p. 323.

[18] J. C. Stewart and K. D. Pyatt, Jr., Astrophys. J. 144,1203 (1966); M. Nantel, G. Ma, S. Gu, C. Y. Cote, J.Itatani, and D. Umstadter, Phys. Rev. Lett. 80, 4442(1998).

[19] E. Eyler and P. Gould, private communication.[20] P. Mansbach, and J. Keck, Phys. Rev. 181, 275 (1969).

FIG. 1. Electron signals from ultracold plasmas created byphotoionization at t = 0. (a) 3 104 atoms are photoionizedand Ee/kB = 540K. Signals with and without rf field areshown. The rf field is applied continuously. (b) 8104 atomsare photoionized and Ee/kB = 26 K. For each trace, the rffrequency in MHz is indicated, and the nonresonant responsehas been subtracted. The signals have b een offset for clarityand have been normalized by F2/nr. The resonant responsearrives later for lower frequency, reflecting expansion of the

plasma. For 40 MHz, nr = 2.0 107 cm3, and for 5MHz,

nr = 3.1 105 cm3.

FIG. 2. Expansion of the plasma for N = 5 105

photoionized atoms. The expansion is well described byn = N/[4(20 + v

2

0t2)]3/2. Horizontal error bars arise from

uncertainty in peak arrival times in data such as Fig. 1b. Un-certainty in N is negligible in this data set, but is significantfor smaller N. The fits are consistently poor at low Ee, as inthe 3.9 K data.

FIG. 3. Expansion velocities, v0, found from fits to datasuch as in Fig. 2. The initial average density, n0, varies from6 106 to 2.5 109 cm3. The solid line, v0 =

Ee/mi,

with = 1.7, is a fit to data with Ee/kB 70 K. The be-havior of low Ee data is discussed in the text. Uncertaintyin v0 is typically equal to the size of the symbols. There is a0.5 K uncertainty in Ee/kB reflecting uncertainty in the dyelaser wavelength. Note that for Ee/kB < 70K, v0 shows asystematic dependence on n0.

FIG. 4. Excess expansion energy, E= miv2

0 Ee/, rel-ative to Ee/, as a function of e, the Coulomb couplingparameter for the electrons at t = 0. The solid line resultsfrom equating E to the predicted suppression of the atomicionization potential in the plasma. Horizontal error bars arisefrom uncertainty in Ee. Vertical error bars reflect uncertaintyin both Ee and v0.

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s. kulin et al- plasma oscillations and expansion of an ultracold neutral plasma

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