Temperature estimates for quantum systems after an ionization induced rapid switch of the spin statistics
ABSTRACT It has recently become possible to create ultracold plasmas in the microkelvin temperature regime. Unfortunately, these plasmas are created in a disordered state and the build up of Coulomb correlations leads to rapid electron and ion heating that can be understood in terms of energy conservation. We explore this disorder–induced heating for the ionization of degenerate gases. In particular, we consider the ionization of fermionic gases, which are partially ordered by the Pauli exclusion principle, and the subsequent heating of the bosonic ions. The introduction of fermionic correlations mitigates the heating of the ions. However, we find from energy conservation that the final state for typical experimental situations is a classical plasma where the bosonic character of the particles is negligible.
INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS A: MATHEMATICAL AND GENERAL
J. Phys. A: Math. Gen. 36 (2003) 6095–6101 PII: S0305-4470(03)55337-9
Temperature estimates for quantum systems after an
ionization induced rapid switch of the spin statistics
D O Gericke1, M S Murillo1, M Bonitz2and D Semkat2
1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2FB Physik, Universit¨ at Rostock, Universit¨ atsplatz 3, 18051 Rostock, Germany
Received 28 October 2002
Published 22 May 2003
Online at stacks.iop.org/JPhysA/36/6095
It has recently become possible to create ultracold plasmas in the microkelvin
temperature regime. Unfortunately, these plasmas are created in a disordered
state and the build up of Coulomb correlations leads to rapid electron and ion
disorder–induced heating for the ionization of degenerate gases. In particular,
we consider the ionization of fermionic gases, which are partially ordered
by the Pauli exclusion principle, and the subsequent heating of the bosonic
ions. The introduction of fermionic correlations mitigates the heating of the
ions. However,we find from energyconservationthat the final state for typical
experimentalsituations is a classical plasma wherethe bosoniccharacterofthe
particles is negligible.
PACS numbers: 52.27.Gr, 05.30.Fk, 05.30.Jp
In the recent years impressive progress has been made on the experimental techniques to
create and diagnose dilute, ultracold many-body systems. For very low temperatures, clear
signatures of quantum degeneracy effects have been observed. For instance, Bose–Einstein
condensates (BEC) [1–3] and degenerate Fermi systems [4, 5] have been created in systems
of trapped atoms. Due to the weak atom–atominteractions and the large interatomic distance,
such systems behave almost as ideal gases.
Furthermore, ultracold plasmas, i.e. systems of charged particles have been created by
laser ionization of ultracold gases [6, 7] and spontaneous ionization of Rydberg atoms . In
to a build up of correlation energy shortly after the ionization process, which drives a rapid
0305-4470/03/226095+07$30.00 © 2003 IOP Publishing LtdPrinted in the UK6095
6096D O Gericke et al
Moreover,theionizationprocessestablishes asuddenswitchofthestatistics theparticlesobey
since the fermionic atoms become bosonic ions plus electrons. Therefore, the question arises
if the switch of statistics further reduces the ion heating. If heating could be avoided or the
final temperaturecouldbe at least held below a few microkelvins,novelsystems rangingfrom
BEC consisting of charged particles to Coulomb crystals could be created.
We introduce here an approach that estimates the temperature achieved after a change of
the statistics. We emphasize again that this is related to a switch of the interactions from a
weak atom–atom to a screened Coulomb potential. Our calculations are only based on the
conservationof total energywhich requiresestimates forthe correlation,exchangeandkinetic
energy of a degenerate system. First we consider ideal quantum and correlated classical
systems as limiting cases. The results indicate that an ideal treatment is only possible for
weakly interacting atomic systems. In particular, we demonstrate that the correlations have to
be included for ionic systems since, otherwise, energy conservation cannot be achieved while
switching from Fermi to Bose statistics. Finally, we give the results for the final temperature
of a plasma created by ionizing a degenerate Fermi gas.
2. Connection between final temperature and initial conditions
We consider an ultracold, degenerate atomic gas in a trap. These atoms are ionized by a
short-pulse laser at t = 0 . Due to the large mass ratio of ions and electrons, the latter pick up
almost the entire laser energy. For the same reason, the time for energy equilibration between
electronsand ions is much longerthan the relaxationof the ion system. We, therefore,neglect
energy transfer between the species and treat the electrons adiabatically. We also consider
situations where the ionization process is much faster than any relaxation process in the ion
subsystem. Therefore, we can describe the ionization as a sudden switch of interactions and
statistics. Due to the sudden change, the momentum and the pair distributions of the ions
directly after the ionization, i.e. at t = 0+, are still the ones of the atoms. However, the
interactionsare now givenby the muchstrongerscreened Coulombpotential. To calculatethe
final temperature, we employ energy conservation
Etotal(0+) ≡ Ekin(0+) + Eex(0+) + Ecorr(0+)
= Ekin(∞) + Eex(∞) + Ecorr(∞)
where E(∞) denotes the quasi-equilibrium energies achieved after the relaxation of the ion
subsystem, but for shorter times than the equilibration between electron and ions. The initial
kinetic energy Ekin(t = 0+) is given by the atoms; the exchange Eex(t = 0+) and correlation
Ecorr(t = 0+) energies have to be calculated with the pair distribution of the atoms and the
new screened Coulomb potential. The final energies depend on the final temperature, which
has to be chosen in a way that ensures energy conservation (1).
3. Energy contributions for weakly correlated systems
Let us first consider the energy of an ideal quantum system consisting of one componentwith
mass m. In this case, the equilibrium momentum distribution is known to be a Fermi/Bose
?p2/2m − µ
Temperature estimates for quantum systems after an ionization induced rapid switch of the spin statistics6097
where the chemical potential µ depends on density n, temperature T and the spin of the
particles σ. The upper and lower signs refer to the Bose and fermion cases, respectively. The
average kinetic energy of the system is then given by
which is smaller than classical result3
The potential energy (which is in this case the exchange energy) of the system can be
easily calculated from the pair distribution g(r,σ), namely
where V(r) is the pair interaction potential. For an ideal quantum system in equilibrium, the
pairdistributioncan becalculatedfromthe one-particlemomentumdistribution. Forisotropic
distribution, one obtains 
If the particles occupy a restricted number of spin states Ns(e.g., for spin polarized systems),
the spin factor 2σ + 1 has to be replaced by the number of occupied states Ns. It is worth
mentioning that the exchange energy (4) with the pair distribution (5) is identical with the
Fock energy defined by
EFock= ±(2σ + 1)
From the last formula, one can easily see that the exchange energy of an ideal Fermi system
is negative whereas that for an ideal Bose system is positive.
In figure 1, results for binary distribution assuming ideal behaviour are shown for a
potassium gas. Clearly, there is more structure in the system for lower temperatures and for
of the exchange for bosons and fermions, respectively.
2kBT in the case of bosons and larger for fermions.
drV(r)[gideal(r,σ) − 1] (4)
gideal(r,σ) = 1 ±
(2σ + 1)
(2π¯ h)3V(p1− p2)f(p1,σ1)f(p2,σ2).
3.1. Exchange and total energy for weakly interacting Fermi and Bose systems
Let us now consider a weakly interacting atomic gas. Here, the interactions can be modelled
by the Morse-type potential
V(r) = D0[(1 − e−α(r−R0))2− 1].
This empirical formula involves the binding energy of molecules D0, the binding length R0
and a parameter α which is connected with the vibrational frequency of the molecule. At low
density this is a very weakly interacting system; therefore the pair distribution is dominated
by exchange effects at very low temperatures.
We again consider a potassium gas to demonstrate quantum degeneracy effects on the
energies (3) and (4). We estimate the exchange energy (4) using the ideal binary distribution
(5) and the Morse-type potential (7). For potassium, the parameters of the Morse potential
are D0 = 0.594 eV, R0 = 3.91 ˚ A and α = 0.715 ˚ A−1. Figure 2 shows results for total,
kinetic and exchange energies calculated with the Morse potential. For temperatures down
to T = 5 × 10−8K, no signs of degeneracy effects can be found. Interestingly, degeneracy
effects are first present in the exchange energy contribution, which slightly increases (lowers)
the total energy for bosons (fermions) around T = 10−8K. The exchange contributions to
6098D O Gericke et al
0 50 100150
binary distribution g(r, )
0 50100 150 200 250
Fermions - two spin states
Fermions - spin polarized
Bosons - three spin states
Bosons - two spin states
Bosons - spin polarized
Figure 1. Binary distribution of an ideal quantum system of40K atoms (fermion case) and39K
atoms (boson case) occupying a different number of spin states. The system temperature and
density are T = 10−9K and n = 109cm−3, respectively. In addition, results for T = 0 (dashed
lines) are shown for fermions.
atom temperature [K]
Etotal, Ekin, and Eex [Ekin
Figure 2. Total (solid), kinetic (dashed), and exchange (dash–dotted) energies of a potassium
gas with n = 109cm−3normalized by the kinetic energy of the corresponding classical system.
Interactions are modelled by the Morse potential (7). The blue and red lines correspond to bosonic
(39K) and fermionic (40K) atoms, respectively.
the kinetic energy are dominant for lower temperatures, which gives rise to an increase of
total and kinetic energy of a fermionic gas and a decrease for bosons. The kinetic energy of
a fermionic system for such low temperatures is, of course, given by the Fermi energy that is
much larger than the classical prediction.
It should be mentioned that the total energy is monotonically increasing with the system
temperaturefor atomic systems. The small dips in the curves in figure 2 are overcompensated
by the energy unit, i.e.3
2kBT, which is linearly increasing with temperature.
Temperature estimates for quantum systems after an ionization induced rapid switch of the spin statistics6099
ion temperature [K]
Etotal and Ekin [eV]
Figure 3. Total (solid), exchange (dashed–dotted), and kinetic (dotted) energies of potassium ions
in a plasma. The interactions are modelled by a screened Coulomb potential (8) with an inverse
screening length of κ = 10−4a−1
B. The blue (red) lines correspond to particles obeying Bose
3.2. Exchange and total energy for the initial plasma state
We will nowdemonstratethat the strongerinteractionsin a system ofchargedparticles change
the general behaviourof the total energyas a functionof temperature. Again, we calculate the
exchange energy using the binary distribution (5). We describe the interaction by a screened
where the inverse screening length κ is determined by the plasma electrons, and Z is the ion
charge. Figure 3 shows the energies (3) and (4) for a system of K+ions in a system with
a screening parameter of κ = 10−4a−1
the total energy is dominated by the exchange contribution for low temperatures (here below
T < 10−5K) if correlations are neglected.
Since the exchange energy of a fermion system becomes more negative with decreasing
temperature, the total energy is monotonically decreasing. For bosons the behaviour is quite
with temperature, but, when degeneracy becomes important, the total energy increases again
since the positive exchange energy is the dominant contribution. This behaviour results in an
unstable situation, because two system temperatures belong to one total energy. Furthermore,
it demonstrates that, in principle, the statistics switch cannot be described by the ideal
calculation. For example, the total energy of the system directly after the ionization, which is
still characterized by fermionic distributions, is negative for T < 10−5K; since the bosonic
total energy, where the system should relax to, is always positive, we cannot match the total
energies for t = 0+and t = ∞.
Of course, this failure is due to the ideal description of an interacting system. The
inclusion of correlations would include a negative energy contribution to the total energy,
which makes in turn the total energy a unique function of the temperature and, therefore,
allows for switching the statistics.
B(aBis the Bohr radius). It clearly demonstrates that
6100D O Gericke et al
4. Energy of an interacting system of charged particles
butnowg(r) includescorrelationsandnotexchangeeffects. We calculatethe pairdistribution
by means of a hypernetted chain scheme
g(r) = exp[−V(r)/kBT + h(r) − c(r)]
h(r) = c(r) + n
g(r) = 1 + h(r)
with a screened interaction potential. This scheme is known to show good agreement with
plasmas which should be considered here .
For systems as considered in figure 3, we should include correlations and degeneracy
effects. Of course, a rigorous solution of this problem is quite complicated, and one needs to
rely on techniques such as quantum Monte Carlo simulations.
Here, we first want to estimate how important both contributions are. The correlations,
which are due to repulsive forces, are proportional to the strength of the screened Coulomb
potential. To estimate the strength of exchange effects in a bosonic system, we go back to the
ideal case where they can be exactly considered as the exchange potential
where λ2= 2π¯ h2/kBT is the thermal wavelength . For bosons, this potential is attractive
and, therefore, competes with the Coulomb repulsion.
To estimate which contribution, attractive exchange or Coulomb repulsion is more
important, we compare the strength of both potentials.
the screening is typically large (in units of the interparticle distance), it turns out that the
screened Coulomb potential is in the important regime with r < κ−1always much larger than
the exchange potential (10). This means that the system is dominated by Coulomb repulsion.
We, therefore, will use a classical calculation to obtain the total energy of a charged particle
dr?h(|r − r?|)c(r?)
Vex(r) = −kBT ln 1 + exp
For an ultracold plasma, where
5. Temperature estimates for systems long after the ionization
Nowwe cancalculatethe finaltemperatureofthe systemafter theionization. Thetotal energy
at t = 0+is calculated with the distributions of the atomic system, but with the screened
Coulomb potential (8). The final state is described by an interacting, classical system. The
results will further justify such a treatment since degeneracyeffects are negligiblefor the final
temperatures. Since the total energyshould be a uniquefunction of the temperature,there can
be no other solution for matching the initial and the final energy in the quantum regime.
Figure 4 gives results for different initial situations: different initial temperatures and
differentnumbersofoccupiedspin states are considered. Forcomparison,results for a system
without any initial correlation are shown also. Clearly, the introduction of initial correlations
considerably reduces the amount of heating where lower final temperatures follow for more
degeneratesystems (colder and spin polarized). The results for Tinitial= 0 give the theoretical
limit of possible initial correlations. However, Tinital=1
4TFis the experimental limit so far,
Temperature estimates for quantum systems after an ionization induced rapid switch of the spin statistics6101
plasma density [cm-3]
final temperature [K]
two spin states
Figure 4. Final temperature of potassium ions in a plasma with an electron temperature of Te=
5 Kversus particle density. The screening length is calculated accordingly. The initial temperature
of the Fermionic atoms is Tinitial = 0 (dashed lines), Tinitial =
Tinitial= TF(dotted line).
4TF (dash–dotted lines), and
due to internal heating mechanisms in trapped gases . Interestingly, degeneracy has only
a small effect for plasma densities n < 108cm−3, even for Tinitial= 0.
Reviewing the results in figure 4, we conclude that the heating due to the build up of
correlation energy can be reduced by a factor of two to four for realistic situations. However,
iontemperaturestill increases by severalordersof magnitude;the final system is a moderately
coupled, classical ion system. Therefore, additional laser cooling of the ions (see, e.g., )
may be needed should a strongly coupled ion system be created.
The work was supported by the US Department of Energy (partially under the Los Alamos
LDRD program) and by the Deutsche Forschungsgemeinschaft(SFB 198).
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