Transition Rates for a Rydberg Atom Surrounded by a Plasma

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by  Chengliang Lin

Transition Rates for a Rydberg Atom Surrounded by a Plasma
Chengliang Lin, Christian Gocke and Gerd R¨opke
Universita¨t Rostock, Institut fu¨r Physik, 18051 Rostock, Germany
Heidi Reinholz
Universita¨t Rostock, Institut fu¨r Physik, 18051 Rostock, Germany and
University of Western Australia School of Physics, WA 6009 Crawley, Australia
(Dated: January 19, 2016)
Wederiveaquantummasterequationforanatomcoupledtoaheatbathrepresentedbyacharged
particle many-body environment. In Born-Markov approximation, the influence of the plasma en-
6 vironment on the reduced system is described by the dynamical structure factor. Expressions for
1 the profiles of spectral lines are obtained. Wave packets are introduced as robust states allowing
0 for a quasi-classical description of Rydberg electrons. Transition rates for highly excited Rydberg
2 levelsareinvestigated. Acircular-orbitwavepacketapproachhasbeenapplied,inordertodescribe
the localization of electrons within Rydberg states. The calculated transition rates are in a good
n
a agreement with experimental data.
J

5
PACS number(s): 03.65.Yz, 32.70.Jz, 32.80.Ee, 52.25.Tx
1

]
h I. INTRODUCTION
p
-
m Open quantum systems have been a fascinating area of research because of its ability to describe the transition
from the microscopic to the macroscopic world. The appearance of the classicality in a quantum system, i.e. the loss
o
of quantum informations of a quantum system can be described by decoherence resulting from the interaction of an
t
a open quantum system with its surroundings [1, 2].
.
s An interesting example for an open quantum system interacting with a plasma environment are highly excited
c
atoms, so-called Rydberg states, characterized by a large main quantum number. Rydberg states play an important
i
s roleinastrophysicstostudystellaratmospheres[3,4]. Particularly,ionizationprocessesofRydbergstatesofhydrogen
y
and helium and their recombination processes are significant for hydrogen and helium plasmas in a very low-density
h
environment which exists in stellar atmospheres with weakly ionized layers [3, 5, 6]. Because the interaction with
p
[ the plasma, characterized by the plasma frequency, is no longer small compared to the energy differences of quantum
eigenstates, the surrounding plasma cannot be considered as a weak perturbation of the excited atom. The time
1
evolution, in particular transition rates, is modified as shown in this work. An essential problem is the construction
v
of optimum, robust states.
6
Note that Rydberg states are energetically near to the continuum of scattering states. The screening of a given
8
0 ion by the free electrons and neighboring ions in a plasma results in the reduction of the ionization potential and
4 line broadening of eigenenergy levels of the given atom. For the Rydberg states near the continuum edge, it may
0 be quite difficult to rigorously distinguish the borderline between the real continuum edge and bound states. For
1. example, it is known that in solar astrophysics spectral lines are visible up to main quantum numbers of about 17
0 [7]. The correct treatment of the Rydberg states which are near the continuum edge is a long-standing problem in
6 plasma spectroscopy, see Refs. [8, 9]. Thus a many-body approach to Rydberg states in a plasma is also of interest
1 for spectroscopy.
:
v Becauseoftheirmacroscopiccharactersandlonglifetimes,nowadaysRydbergstatesbecomeafundamentalconcept
i of open quantum systems in different fields of physics, such as quantum information research [10–12] and ultracold
X
plasmas [13–15]. Recently the existence of Rydberg excitons in the copper oxide Cu O [16] is demonstrated which
2
r enablevisiblemeasurementsofcoherentquantumeffects[17]. Actually,usingalocalizedsemi-classicalrepresentation
a
ofboundstatestostudytheconnectionbetweenclassicalmechanicsandthelarge-quantumnumberlimitofquantum
mechanics has been a topic of interest since the development of quantum mechanics [18, 19]. As a mesoscopic
object, the Rydberg atom may be regarded as an outstanding example demonstrating both macroscopic classical and
microscopic quantum behavior. In a series of papers of Stroud et al. [20–27], the dynamics of a hydrogenic Rydberg
atom has been discussed in detail. It has been shown that the behavior of a wave packet constructed by energy
eigenstates of the hydrogen atom is different for the short and the long-term evolution. This difference is essential for
the investigation of the connection between the quantum and classical description of nature and gives a possibility to
explain the emergence of classicality in a quantum system.
Motivated by these exciting perspectives, we study the properties of hydrogenic Rydberg atoms, in particular,
the transition rates of highly excited Rydberg states. Different environments are of interest: the interaction with
2
the radiation field, the interaction with phonons (Rydberg excitons), the interaction with charged particles. We
focus on the special case where the environment is described by a plasma background, see Refs. [28–30]. A similar
derivation for a test particle interacting through collisions with a low-density background gas by using the quantum
master equation approach is reported in Refs. [31, 32]. The influence of the plasma on the dynamics of the atom is
determinedbythedynamicalstructurefactorofthesurroundingplasma. Robuststatesarerepresentedbyoptimized
Gaussian wave packets. As an example, transition rates are calculated and compared to other theoretical approaches
and experimental data.
Another example which will be considered are the profiles of spectral lines. They are essentially determined by
the interaction of the bound states with the radiation field and the charge carriers of the plasma. Both of them can
be treated as thermal bath for the bound states, which are regarded as the reduced system in the theory of open
quantum systems. Various approaches can be used to calculate the spectral line profiles in a plasma environment, for
instance, unified theory [33], quantum mechanical scattering theory [34] and the Green’s function methods [35, 36],
which are based on the assumption that the plasma is in equilibrium. Quantum kinetic theory, as a nonequilibrium
approach, can also be applied to investigate the line profiles of the plasma which will be presented in this work.
Thispaperisorganizedasfollows: insectionIIAweoutlinethederivationofthegeneralquantummasterequation
inBorn-Markovapproximation. Thenwediscussthespecialcaseofplasmaasamany-bodyenvironmentinSec.IIB.
In Sec. IIC, the general quantum master equation is investigated in detail by introducing the basis of the energy
eigenstates of the hydrogen atom. The Pauli equation and the spectral line profiles are derived in this section. The
wave packet description for the bound Rydberg electron is introduced in Sec. III. The robustness and validity of the
wave packet description are discussed in Sec. IIIA. The transition rates for the hydrogenic Rydberg atom derived
with the use of the circular-orbit wave packet and their comparisons with classical Monte-Carlo simulations and
experimental data are presented in IIIB. Conclusions are drawn in Sec. IV.
II. QUANTUM MASTER EQUATION FOR RYDBERG ATOMS IN A PLASMA
A. General quantum master equation
We are investigating the reduced system of a Rydberg atom (A) embedded in a bath (B) consisting of charged
particlesc,electrons(c=e)and(singly)chargedions(c=i),chargee ,massm ,particledensityn andtemperature
c c c
T. The microscopic model under consideration is a hydrogen atom coupled to a surrounding charge-neutral plasma,
(cid:80)
e n =0. In the bath, in general, the formation of bound states such as atoms is also possible. Furthermore, the
c c c
interaction of the atom is mediated by the Maxwell field which contains, besides the Coulomb interaction with the
charged particles, also single-particle states, the photons. The total system is then described by the Hamiltonian
Hˆ =Hˆ +Hˆ +Hˆ . (1)
A B int
In a plasma environment the Hamiltonian Hˆ includes both the kinetic energy and the Coulomb interactions of
B
chargedparticlesHˆ (seeEq. (15)below)aswellasthedegreesoffreedomofthephotonicfieldHˆ⊥ describing
Coul photon
the transversal Maxwell field of the plasma environment, i.e. Hˆ =Hˆ +Hˆ⊥ .
B Coul photon
The atomic Hamiltonian reads in the non-relativistic case
Pˆ2 pˆ2 e2
Hˆ = + − , (2)
A 2M 2m 4πε |ˆr|
0
where the center-of-mass (c.o.m.) motion is described by the total mass M = m +m and the variables Rˆ,Pˆ, the
e i
relativemotionbythereducedmassmandtherelativevariablesˆr,pˆ. Theeigenstates|Ψ (cid:105)oftheisolatedhydrogen
n,P
atomarethesolutionsoftheSchr¨odingerequationHˆ |Ψ (cid:105)=E |Ψ (cid:105)withtheeigenenergyE =P2/(2M)+
A n,P n,P n,P n,P
E . Thequantumnumbern={n¯,l,m,m }describestheinternalstateforboundstates E <0andn={p,m }for
n s n s
scattering states E =p2/(2m)>0. For the bound states, the wave function Ψ(R,r)=(cid:104)R,r|Ψ (cid:105)=Ψ (R)ψ (r)
p n,P P n
containstheeigenstatesψ (r)ofthehydrogenatom. Thec.o.m. motionΨ (R)isgivenbyaplanewave. Inthiswork
n P
we concentrate on the internal degrees of freedom of the bound states. The c.o.m motion, which, e.g., determines the
Doppler broadening of the spectral line profile, will not be discussed here in detail. In most cases it will be dropped
considering the adiabatic limit.
The interaction between the atomic electron and the plasma environment is given by the coupling of the atomic
current operator to the electromagnetic field of the bath
(cid:90)
Hˆ (t)= d3rˆjµ(x)Aˆ (x) (3)
int A µ,B
3
with xµ = {ct,r}. Introducing the creation (ψˆ†(x)) and anihilation (ψˆ(x)) operator for the atomic electron, the
current operator of the atomic subsystem ˆjµ(x)={c(cid:37)ˆ (x),ˆj (x)} can be explicitly written as (cid:37)ˆ (x)=−eψˆ†(x)ψˆ(x)
A A A A
(cid:104) (cid:16) (cid:17) (cid:105)
fortheelectronprobabilitydensityandˆj (x)= ie(cid:126) ψˆ†(x) ∂ ψˆ(x)− ∂ ψˆ†(x) ψˆ(x) fortheelectriccurrentdensity
A 2me ∂r ∂r
oftheelectron(non-relativisticlimit). Withoutfurtherexplanation,theoperatorsinthisworkaregiveninHeisenberg
picture
Oˆ(t)=eiHˆt/(cid:126)Oˆe−iHˆt/(cid:126). (4)
The source of the electromagnetic field of the bath Aˆµ(x) = (Uˆ (x),Aˆ (x)) is the current density ˆjµ(x) of all
B B B B
chargecarriersintheplasma. InthepresentworktheCoulombgauge∇×Aˆ (x))=0isused. TheFouriertransform
B
(cid:90) (cid:90) ∞
ˆj (ω)= d3r dteiωt−iq·rˆj (t,r) (5)
q,B B
Ω0 −∞
of the electrical current in the surrounding plasma can be decomposed into a transverse component (cid:80) ˆj⊥,c(ω)
c q,B
coupled only to the vector potential Aˆ (ω) and a longitudinal one (cid:80) ˆj||,c(ω)q/q which is related only to the
q,B c q,B
Coulomb potential. Because of the continuity equation, the relation q·ˆj (ω) = qˆj|| (ω) = ω(cid:37)ˆ (ω) holds, where
q,B q,B q,B
(cid:37)ˆ (ω) is the Fourier transform of the corresponding charge density operator (cid:37)ˆ(x).
q
The general form of the interaction (3) includes the Coulomb interaction via the longitudinal component of the
currents, andthecouplingofthetransversecomponentofthecurrentswiththeradiationfield. Wedonotinvestigate
the radiation interaction connected with the transverse component. The radiative field of the plasma determines the
natural broadening which has already been extensively discussed in [38, 39] by using the quantum master equation
approach. HoweverwefocusontheCoulombinteractionofthehydrogenatomwithitssurroundingchargedparticles
in this work. In this case, the distribution and the motion of the charge carriers in the plasma produce a scalar
potential which is given in terms of the longitudinal current [37]:
Uˆ (ω)=(cid:88)(cid:37)ˆcq,B(ω) =(cid:88)ˆjq||,,Bc(ω). (6)
q,B (cid:15) q2 (cid:15) ωq
0 0
c c
This results in the pressure broadening of the spectral lines as shown in Sec. IIC2.
The state of the total system is described by the statistical operator ρˆ(t). We assume that the observables Aˆof the
subsystem A commute with the observables Bˆ of the bath B. If only the properties of the subsystem A are relevant,
we can consider the corresponding statistical operator
ρˆ (t)≡Tr ρˆ(t) (7)
A B
performing the trace over all bath variables. Then, the average value of any observable Aˆ of the subsystem A is
calculated as (cid:104)Aˆ(cid:105)t ≡Tr{Aˆ ρˆ(t)}=Tr {Aˆ ρˆ (t)}.
A A
The equation of motion for the total statistical operator ρˆ(t) [39] reads
∂ 1
ρˆ(t)− [Hˆ,ρˆ(t)]=−ε[ρˆ(t)−ρˆ (t)] (8)
∂t i(cid:126) rel
with the relevant statistical operator ρˆ (t) = ρˆ (t)ρˆ which implies that the quantum systems A and B are uncor-
rel A B
related. The equilibrium state ρˆ of the bath B is assumed as the grand canonical distribution
B
1 (cid:34) Hˆ −(cid:80) µ Nˆ (cid:35) (cid:34) Hˆ −(cid:80) µ Nˆ (cid:35)
ρˆ = exp − B c c c , Z =Tr exp − B c c c (9)
B Z k T B B k T
B B B
with the chemical potentials µ of the species c. The limit ε → 0+ has to be performed after the thermodynamic
c
limit.
A closed equation of motion can be derived for the reduced statistical operator ρˆ (t) of the subsystem A by
A
performingtheaveragewithrespecttothebathin(8). Ifthebathisassumedtohaveshortmemoryinthesensethat
the correlation in the bath decays very quickly in comparison to the time evolution of the reduced system (Markov
approximation),andthedynamicsofthereducedsystemisconsideredonlyinsecondorderwithrespecttoHˆ (Born
int
approximation), we obtain [39]
∂ 1
ρˆ (t)− [Hˆ ,ρˆ (t)]=D[ρˆ (t)] (10)
∂t A i(cid:126) A A A
4
with the influence term
1 (cid:90) 0 (cid:104) (cid:104) (cid:105)(cid:105)
D[ρˆ (t)]=− dτeετTr Hˆ , Hˆ (τ),ρˆ (t)ρˆ . (11)
A (cid:126)2 B int int A B
−∞
Thisisthequantummasterequation(QME)inBorn-Markovapproximation. TogobeyondtheBornapproximation,
a more general solution has been given in [41].
Born approximation indicates that higher orders of the interaction Hamiltonian in the time evolution of the op-
erator (4) can be dropped. Consequently, the time dependence in Born approximation is given by the interaction
picture
OˆI(t,t )=ei(HˆA+HˆB)(t−t0)/(cid:126)Oˆe−i(HˆA+HˆB)(t−t0)/(cid:126). (12)
0
At t = t , the interaction picture coincides with the Schr¨odinger picture. Note that the time of reference t is often
0 0
taken as zero. In interaction picture, the QME in Born-Markov approximation reads
∂
ρˆI (t,t )=DI(t,t ), (13)
∂t A 0 0
i.e., only the perturbation determines the time evolution of ρˆI (t,t ) (note that Hˆ commutes with ρˆ (t)). The
A 0 B A
influence term in interaction representation follows as
1 (cid:90) 0 (cid:104) (cid:104) (cid:105)(cid:105)
DI(t,t )=− dτeετTr HˆI (t,t ), HˆI (t+τ,t ),ρˆI (t,t )ρˆ . (14)
0 (cid:126)2 B int 0 int 0 A 0 B
−∞
In zeroth order with respect to the perturbation, ρˆI (t,t ) is constant, no changing with time t.
A 0
B. The Influence Term for a Charged Particle System
Inthissectionthemasterequationforthereducedstatisticaloperator(13)shallbeappliedtoatomicboundstates
in a many-particle plasma environment. However, most of the discussion is valid for a much more general case.
For the plasma, surrounding the radiating atom, the Hamiltonian is described by
Hˆ =(cid:88)(cid:126)2p2cˆ†cˆ + 1 (cid:88) eced δ δ δ cˆ† dˆ† dˆ cˆ (15)
Coul c,p 2mc p p 2c,d,p1p2,p(cid:48)1p(cid:48)2 (cid:15)0Ω0|p(cid:48)1−p1|2 p1+p2,p(cid:48)1+p(cid:48)2 σ1,σ1(cid:48) σ2,σ2(cid:48) p1 p2 p(cid:48)2 p(cid:48)1
where we used second quantization cˆ ,cˆ† for free particle states |p(cid:105)=|p,σ(cid:105) (wave vector and spin) of charge c . The
p p
grand canonical equilibrium (9) contains also the particle number operator Nˆ = (cid:80) cˆ†cˆ . The macroscopic state
c p p p
of the bath is fixed by the Lagrange multipliers µ and T. Ω is the volume of the total system. Because of charge
c 0
neutrality(cid:80) e Nˆ ≡0bothµ ,µ arerelated. ThephotonicfieldHˆ⊥ isnotrelevantinourpresentconsideration
c c c e i photon
which is focussed on the Coulomb interaction with the charged particles of the bath.
The longitudinal part of the interaction Hamiltonian can be extracted from the general form Hˆ (3) by using the
int
expression (6) and performing the Fourier transform with respect to the time for the atomic charge density operator
(cid:90) ∞ dω
(cid:37)ˆI (t,t )= e−iω(t−t0)(cid:37)ˆI (ω) (16)
q,A 0 2π q,A
−∞
so that
HˆI,||(t,t )=(cid:88) 1 (cid:90) dω e−iω(t−t0)(cid:37)ˆI (ω)(cid:37)ˆI (t,t ) (17)
int 0 (cid:15) q2Ω 2π q,A −q,B 0
0 0
q
with (cid:37)ˆ = (cid:80) (cid:37)ˆc and (cid:37)ˆc = (cid:80) e cˆ† cˆ . In this work only the contribution of the electrons in the
q,B c q,B q,B p c p−q/2,σ p+q/2,σ
plasma is considered. The ionic contribution should be treated in another way, see Sec. IV. Coming back to the
influence term (14), the factorization of the interaction Hamiltonian allows us to perform the average over the bath
degrees of freedom separately
DI(t,t )=− 1 (cid:90) 0 dτeετ(cid:88) 1 (cid:90) dω (cid:90) dω(cid:48) e−i(ω+ω(cid:48))(t−t0)−iω(cid:48)τ
0 (cid:126)2 (cid:15)2q2q(cid:48)2Ω2 2π 2π
−∞ q,q(cid:48) 0 0
×(cid:8)(cid:2)(cid:37)ˆI (ω)(cid:37)ˆI (ω(cid:48))ρˆI (t,t )−(cid:37)ˆI (ω(cid:48))ρˆI (t,t )(cid:37)ˆI (ω)(cid:3)(cid:104)(cid:37)ˆI (t,t )(cid:37)ˆI (t+τ,t )(cid:105)
q,A q(cid:48),A A 0 q(cid:48),A A 0 q,A −q,B 0 −q(cid:48),B 0 B
−(cid:2)(cid:37)ˆI (ω)ρˆI (t,t )(cid:37)ˆI (ω(cid:48))−ρˆI (t,t )(cid:37)ˆI (ω(cid:48))(cid:37)ˆI (ω)(cid:3)(cid:104)(cid:37)ˆI (t+τ,t )(cid:37)ˆI (t,t )(cid:105) (cid:9) (18)
q,A A 0 q(cid:48),A A 0 q(cid:48),A q,A −q(cid:48),B 0 −q,B 0 B
5
with (cid:104) ··· (cid:105) =Tr {···ρˆ }. The charge density autocorrelation function (cid:104)(cid:37)ˆI (t,t )(cid:37)ˆI (t+τ,t )(cid:105) is calculated
B B B −q,B 0 −q(cid:48),B 0 B
in thermodynamic equilibrium. Because of homogeneity in space and time it is ∝ δ and not depending on the
q(cid:48),−q
time t as well as t . We introduce the Laplace transform of the bath auto-correlation functions which can be also
0
defined as the response function
1 (cid:90) 0
Γ (q,ω)= dτeετe−iωτ(cid:104)(cid:37)ˆI (t ,t )(cid:37)ˆI (t +τ,t )(cid:105) . (19)
r (cid:126)2 −q,B 0 0 q,B 0 0 B
−∞
The response function Γ (q,ω) is a complex physical quantity which is related to the dynamical structure factor of
r
the plasma or the dielectric function, as shown in the App. A. It can be decomposed into real and imaginary parts,
1
Γ (q,ω)= γ (q,ω)+iS (q,ω), (20)
r 2 r r
where γ (q,ω) and S (q,ω) are both real functions. They fulfill the Kramers-Kronig relation and are related to the
r r
damping and the spectral line shift, respectively (see Eqs. (B9) and (B10) in App. B).
With the response function (19), we find that the influence term (14) can be rewritten as
DI(t,t )=−(cid:88) 1 (cid:90) dω (cid:90) dω(cid:48)ei(ω(cid:48)−ω)(t−t0)Γ (q,−ω(cid:48))(cid:2)(cid:37)ˆI (ω),(cid:37)ˆI (−ω(cid:48))ρˆI (t,t )(cid:3)+h.c. (21)
0 (cid:15)2q4Ω2 2π 2π r q,A −q,A A 0
q 0 0
The second contribution of the r.h.s. of Eq. (21) is the hermitean conjugate of the first contribution so that DI(t,t )
0
is a real quantity. Approximations for the response function Γ (q,ω) are obtained from the approximations for the
r
dielectric function such as the random-phase approximation and improvements accounting for collisions.
C. Atomic Quantum Master Equation
In a next step we introduce the orthonormal basis of the hydrogen bound states in the Hilbert space of the atomic
subsystem to obtain the Pauli equation for population numbers and the spectral line profiles.
1. Pauli Equation for Occupation Numbers
We use the basis of hydrogen-like states |ψ (cid:105) of the Hamiltonian Hˆ . For the charge density operator
n A
(cid:90) (cid:90)
(cid:37)ˆ = d3r¯eiq·¯r(cid:37)ˆ (¯r)= d3r¯eiq·¯r[e δ(ˆr −¯r)+e δ(ˆr −¯r)]=e eiq·ˆre +e eiq·ˆri, (22)
q,A A e e i i e i
the time dependence in the interaction picture can be written in matrix representation as (e =−e )
e i
(cid:37)ˆIq,A(t,t0)=e(cid:126)iHˆA(t−t0)(cid:37)ˆq,Ae−(cid:126)iHˆA(t−t0) =(cid:88)eeTˆn(cid:48)nFn(cid:48)n(q)e−iωnn(cid:48)(t−t0) (23)
nn(cid:48)
with
Tˆ =|ψ (cid:105)(cid:104)ψ |, (24)
n(cid:48)n n(cid:48) n
E −E
ω = n n(cid:48), (25)
nn(cid:48) (cid:126)
(cid:90)
F (q)= d3rψ∗ (r)ψ (r)(1−e−iq·r), (26)
n(cid:48)n n(cid:48) n
in adiabatic approximation m (cid:28) m . Furthermore, the atom is assumed to be localized at R = 0. Performing the
e i
Fourier transformation with respect to t we obtain the atomic charge density in Fourier-space
(cid:37)ˆI (ω)=(cid:88)e Tˆ F (q)2πδ(ω−ω ). (27)
q,A e n(cid:48)n n(cid:48)n nn(cid:48)
nn(cid:48)
With Eq. (27) the influence function (21) can be represented as
DI(t,t0)=− (cid:88) e−i(ωnn(cid:48)+ωmm(cid:48))(t−t0)Kmm(cid:48);n(cid:48)n(q,ωmm(cid:48))(cid:110)Tˆn(cid:48)nTˆm(cid:48)mρˆIA(t,t0)−Tˆm(cid:48)mρˆIA(t,t0)Tˆn(cid:48)n(cid:111)+h.c. (28)
nn(cid:48),mm(cid:48),q
6
with
e2
K (q,ω)= e F∗ (q)F (q)Γ (q,ω) (29)
mm(cid:48);n(cid:48)n (cid:15)2q4Ω2 mm(cid:48) n(cid:48)n r
0 0
containing informations about the atomic system, the plasma bath and the interaction between them. In matrix
representation the atomic QME (13) can be represented as (|ψ (cid:105) - initial state, |ψ (cid:105) - final state)
i f
∂
ρI (t,t )=(cid:104)ψ |DI(t,t )|ψ (cid:105) (30)
∂t A,if 0 i 0 f
with the influence function
(cid:88) (cid:110)
(cid:104)ψi|DI(t,t0)|ψf(cid:105) =− eiωim(t−t0)Kmn;in(q,ωmn)ρIA,mf(t,t0)+eiωmf(t−t0)Km∗n;nf(q,ωnf)ρIA,im(t,t0)
mn,q
−ei(ωim+ωnf)(t−t0)(cid:2)Kmi;fn(q,ωmi)+Kn∗f;mi(q,ωnf)(cid:3)ρIA,mn(t,t0)(cid:111) (31)
with the density matrix ρI (t,t ) = (cid:104)ψ |ρˆI (t,t )|ψ (cid:105). The corresponding atomic QME in Schr¨odinger picture is
A,mn 0 m A 0 n
obtained with ρIA,mn(t,t0)=eiωmn(t−t0)ρA,mn(t), see Appendix B.
Weinvestigatethediagonalelementsofthedensitymatrixbysettingi=f intheaboveexpression(30). Thisleads
to an equation for the population number P (t)=ρI (t,t )=ρ (t)
i A,ii 0 A,ii
∂Pi(t) =(cid:88)(cid:104)k (q,ω )P (t)−k (q,ω )P (t)(cid:105)− (cid:88) 2Re(cid:104)eiωim(t−t0)K (q,ω )(cid:105)ρI (t,t )
∂t ni ni n in in i mn;in mn A,mi 0
n,q n,m(cid:54)=i,k
(cid:88) (cid:26) (cid:104) (cid:105)(cid:27)
+ 2Re eiωnm(t−t0) K∗ (q,ω )+K (q,ω ) ρI (t,t ) (32)
ni;mi ni mi;in mi A,mn 0
m>n,q
withk (q,ω )=2Re K (q,ω )=e2|F (q)|2γ (q,ω )/((cid:15)2q4Ω2),whereexpression(20)isusedandtheindices
ab ab ab;ab ab e ab r ab 0 0
m and n are interchanged in the derivation. The interaction picture shows a slow time dependence in ρI (t,t )
A,nm 0
owingtotheinfluenceofthebath,Eq. (13),andaquicktimevariationduetothefactoreiωnm(t−t0)withωnm (cid:54)=0. The
secondandthirdtermoscillatewiththecharacteristictransitionfrequenciesω andω ,respectively. Subsequently
nm im
their contributions vanish when averaging over relative larger time interval in comparison with the inverse of the
characteristic transition frequencies, because the population numbers are approximately constant. This is the so-
called Rotating Wave Approximation (RWA). For the long-term evolution of the reduced system the nondiagonal
elements in Eq. (32) can be neglected and consequently we obtain a closed rate equation for the population number
– the Pauli equation:
∂Pi(t) =(cid:88)(cid:104)k (q,ω )P (t)−k (q,ω )P (t)(cid:105). (33)
∂t ni ni n in in i
n,q
Comparing with the standard form of the Pauli equation ∂ P (t) = (cid:80) [w P (t)−w P (t)], we have for the
∂t i n n→i n i→n i
transition rates
w =(cid:88)k (q,ω )=(cid:88)e2e|Fni(q)|2γr(q,ωni), w =(cid:88)k (q,ω )=(cid:88)e2e|Fin(q)|2γr(q,ωin). (34)
n→i ni ni (cid:15)2q4Ω2 i→n in in (cid:15)2q4Ω2
q q 0 0 q q 0 0
To derive the Pauli equation we used the RWA which neglects quickly oscillating terms. Also the dependence on the
time t where the interaction picture coincides with the Schr¨odinger picture disappears. The validity of the RWA
0
in the theory of open quantum systems is under discussion. The dynamics is modified if contributions of the right
hand side of Eq. (32) are dropped. In our investigation we found that if the RWA is carried out prematurely, it will
be inappropriate to describe the dissipative properties of the relevant atomic system (Rydberg states) and result in
erroneous transition rates. More details on that can be found in Appendix B. The nondiagonal elements of Eq. (30)
are also discussed there.
2. Quantum Kinetic Approach to Spectral Line Profile
In open quantum system theory one separates a reduced subsystem out from the total quantum system, which
includes all relevant observables that one is interested in. The remaining degrees of freedom are treated as irrelevant
7
for the dynamical behavior and are denoted as the observables of the bath. However, the selection of the relevant
observables that are appropriate to describe the dynamics of the system depends sensitively on the physical problems
that we tackle.
For instance, the degrees of freedom of the emitted photons are irrelevant for the dynamics of the population
numbers of the atomic energy eigenstates and therefore can be considered as part of the bath in the derivation of
the Pauli equation. This consideration is also applied in the derivation of the natural line width of the spectral line
profile[38,39]. Incontrast,thesedegreesoffreedomaremostimportantforthedescriptionofthespectrallineprofile
in a plasma environment where we obtain the spectral line shapes by measuring the energy of the emitted photons.
The emitted photons are therefore relevant degrees of freedom. To correctly describe the spectral line shapes via the
open quantum system theory, we must extend the reduced system by including the set of the degrees of freedom of
the emitted photons. This means that the radiation field together with the atomic system should be considered as
the reduced system to be described by the QME, and the surrounding plasma is the bath coupled to the system by
Coulomb interaction.
Absorption as well as spontaneous and induced emission coefficients, related by the Einstein relation, are obtained
from QED where the transverse part of the Maxwell field
Hˆ⊥ =(cid:88)(cid:126)ω nˆ (35)
photon k,s k,s
k,s
isquantizedanddenotedbythephotonmodes|k,s(cid:105). Thefrequencyω =c|k|=2πc/λisthedispersionrelationforthe
k
frequencyasafunctionofthewavenumberλ. nˆ =ˆb† ˆb istheoccupationnumberwiththepolarizations=1,2.
k,s k,s k,s
As mentioned above, the photon field must be treated as part of the reduced system with the Hamiltonian Hˆ =
S
Hˆ +Hˆ⊥ , and the eigenstates will be denoted by the expression |n˜(cid:105) = |ψ ,N (k,s)(cid:105) containing corresponding
A photon n n
quantum numbers for the eigenenergy E˜ = E +(cid:80) N (k,s)(cid:126)ω with the occupation number N (k,s) of the
n n k,s n k,s n
mode |k,s(cid:105).
Emission and absorption are described by the interaction Hamiltonian, see Eq. (3), Hˆrad = (cid:82) d3rˆj⊥ · Aˆ =
A ph
(cid:82) d3rdˆ ·Eˆ after integration by parts with the atomic dipole operator dˆ . The decomposition of the electric field
A ph A
of the photon subsystem (two polarization vectors eˆ ) is
k,s
Eˆ =i(cid:88)(cid:114)(cid:126)ωk eˆ [ˆb −ˆb† ]. (36)
ph 2Ω k,s k,s k,s
0
k,s
For a given measured photon mode |k¯,s¯(cid:105) in the experiment, only the mode with k=k¯ and s=s¯in the Hamiltonian
Hˆrad contributes. This allows us to introduce a new operator describing emission and absorption
dˆS =dˆA⊗(ˆbk¯ −ˆb†k¯), (37)
where the polarization index is suppressed. The initial and final states in this case are given by |˜i(cid:105) = |ψ ,N (k)(cid:105)
i i
and |f˜(cid:105) = |ψf,Nf(k)(cid:105) with Nf(k) = Ni(k)+δk,k¯, respectively. This means that for the measured photon mode
k¯ the occupation number fulfills N (k¯) = N (k¯)+1, while for all other photon modes their occupation numbers
f i
remain unchanged. A shift of the eigenenergy levels is caused by the interaction with the plasma environment via
the momentum exchange. Subsequently, this leads to a deviation of the measured transition frequency ωk¯ from the
characteristic transition frequencies ω between the unperturbed atomic eigenstates |ψ (cid:105). We define the deviation
nn(cid:48) n
by using the eigenenergies E˜ via
n
∆ω =(E˜ −E˜ )/(cid:126). (38)
nn(cid:48) n n(cid:48)
We use the interaction picture with Hˆ0 =HˆS+HˆB so that the power spectrum P(ωk¯)=(cid:82)0∞e−(cid:15)te±iωk¯t(cid:104)dˆA(cid:105)tdt as
shown in [40] in the framework of the linear-response theory can be rewritten as
(cid:90) ∞
P(ωk¯)= e−(cid:15)t(cid:104)dˆS(cid:105)tdt=(cid:88)Li,f, (39)
0 if
wherethephotonfrequencyisabsorbedbythenewdipoleoperatordˆ ofthereducedsystem(includingphotons)and
S
(cid:104)dˆ (cid:105)t =Tr{ρˆ (t)dˆ }=(cid:88)ρI (t)dI (t) (40)
S S S S,fi S,if
if
8
with ρI (t) being the solution of the QME in interaction picture (see Eq. (44)), and the matrix elements dI (t)=
S,fi S,if
(cid:104)ψi|dA|ψf(cid:105)e−i∆ωift. Consequently, the spectral line shape Li,f in Eq. (44) can be written as
(cid:90) ∞
L = dte−(cid:15)tρI (t)dI (t). (41)
i,f S,fi S,if
0
In order to obtain the solution of the QME, a similar reduced charge density operator containing the photon
information as in Eq. (37) can be introduced for the extended reduced system
(cid:37)ˆq,S =(cid:37)ˆq,A⊗(ˆbk¯ −ˆb†k¯). (42)
Using the basis set |n˜(cid:105) of the unperturbed reduced system, we obtain the matrix elements of the reduced charge
density operator (cid:37)ˆI (t) at time t
q,S
(cid:104) (cid:105)
(cid:104)n˜(cid:48)|(cid:37)ˆIq,S(t)|n˜(cid:105)=eeFn(cid:48)n(q)ei∆ωn(cid:48)nt δNn(cid:48)(k¯),Nn(k¯)−1−δNn(cid:48)(k¯),Nn(k¯)+1 ,
where the Kronecker’s delta is connected to the atomic emission and absorption with the transition frequency ω .
n(cid:48)n
Performing the Fourier transform with respect to the time t, we obtain the reduced charge density operator in
Fourier-space
(cid:37)ˆ (ω)= (cid:88) e Tˆ− ·F (q)δ(ω−∆ω )− (cid:88) e Tˆ+ ·F (q)δ(ω+∆ω ) (43)
q,S e n(cid:48)n n(cid:48)n n(cid:48)n e n(cid:48)n n(cid:48)n nn(cid:48)
n(cid:48)>n n(cid:48)<n
with Tˆn−(cid:48)n =|n˜(cid:48)(cid:105)(cid:104)n˜|·δNn(cid:48)(k¯),Nn(k¯)−1 denoting the one photon absorption and Tˆn+(cid:48)n =|n˜(cid:48)(cid:105)(cid:104)n˜|·δNn(cid:48)(k¯),Nn(k¯)+1 - the one
photon emission.
The QME in RWA in interaction picture can be written in terms of the matrix element ρI (t)=(cid:104)f˜|ρˆI(t)|˜i(cid:105):
S,fi S
∂ρI (t)
S∂,fti =−ΓBfiS(ωk¯)ρIS,fi(t)+ΓVfiρIS,fi(t), (44)
which is shown in detail in Appendix C. The influence function, the right side of the Eq. (44), characterizes the
spectral intensity of the emitted photons by a coefficient ΓBfiS(ωk¯) describing the shift of the eigenenergy levels and
the pressure broadening
ΓBfiS(ωk¯)=(cid:88)(cid:8)Knf;fn(q,∆ωnf)+Knf;fn(q,−∆ωfn)+Kn∗i;in(q,∆ωni)+Kn∗i;in(q,−∆ωin)(cid:9) (45)
n,q
and a coefficient ΓV describing the vertex correction
fi
ΓV =(cid:88)(cid:8)K (q,∆ω )+K (q,−∆ω )+K∗ (q,∆ω )+K∗ (q,−∆ω )(cid:9), (46)
fi ii;ff ff ii;ff ff ff;ii ii ff;ii ii
q
The vertex correction has no dependence on the photon frequency ωk¯ and contributes only beyond the dipol approx-
imation. Formally integrating the expression (44) yields
ρI (t)=ρI (0)·e−{ΓBfiS(ωk¯)−ΓVfi}t. (47)
S,fi S,fi
Inserting this formal solution into the Eq. (41), the line shape function can be expressed as
1
L(ωk¯)i,f ∝ ωk¯ −ωif +i(cid:15)−iΓBifS(ωk¯)+iΓVif. (48)
The expression (48) coincides with the result of the unified theory for spectral line profiles [42] if only the electron
contribution(impactapproximation)isconsidered. NotethattheunifiedtheorygivestheresultinBornapproximation
with respect to the interaction with the surrounding plasma, what corresponds to the Born-Markov approximation
for the coupling to the plasma considered as the bath. Strong coupling of the radiator to the perturbing environment
has been treated in the Green function approach using a T-matrix approximation, see [42]. The improvement of the
Born-Markov approximation for the QME considering strong interactions and the ionic contribution of the plasma
environment, given by the microfield distribution, will be discussed below in Sec. IV.
9
III. ROBUST CIRCULAR WAVE PACKET AND TRANSITION RATES
An advantage of the QME is the possibility to introduce optimal (robust) states which allows the transition from
a quantum description to a classical one. In the case of Rydberg atoms, one considers electrons in highly excited
hydrogen states. With increasing quantum number n¯, a pure hydrogen orbital can be formed only if the atom is well
isolatedfromexternalinfluences. Whentheinteractionwiththebathiscomparableorgreaterthanthedifferencesof
atomic energy eigenstates E for n¯ near a fixed value n , the wave packet description is more appropriate to describe
n 0
the evolution of the system, in particular transition rates. For a local interaction such as the Coulomb potential, the
position r of the atomic electron enters the interaction part of the Hamiltonian, and localization is favored because r
commutes with Hˆ and is a conserved quantity with respect to this part of the Hamiltonian.
int
In addition, the introduction of the wave packet description may allow us to investigate the boundary between the
quantum and classical descriptions of systems. In fact, since the introduction of quantum mechanics many physicists
attempted to establish the connections between these descriptions of nature by exhibiting the so-called coherent
wave packet. One of the famous examples is the well known coherent state of the linear harmonic oscillator [43]
which may be regarded as an excellent example to describe the macroscopic limit of a quantum mechanical system
according to the correspondence principle. For the Coulomb problem, e.g. the hydrogen atom, many attempts to
constructlocalizedsemi-classicalsolutionsofthecoherent-statetypehavebeenmade[44–48]. Notethatthehydrogen
atom is equivalent to the four-dimensional harmonic oscillator so that coherent wave packets can be introduced
accordingly[46]. Recently, MakowskiandPeplowskiconstructedverywell-localizedtwo-dimensionalwavepacketsfor
twodifferentpotentials[49,50]whereaverygoodquantum-classicalcorrespondenceisobserved. Inthepresentpaper
we use Brown’s circular-orbit wave packet [44, 51] as a quasiclassical representation to describe the highly excited
Rydberg states of the hydrogen atom.
A. Wave Packet for Circular Motion
Within the QME approach, the state of the relevant system ρˆ is of interest, and we can represent the statistical
S
operator by the density matrix ρ =(cid:104)m|ρˆ |n(cid:105) with respect to the states |n(cid:105) of the system. For the representation,
S,mn S
one can use the orthonormal basis set of energy eigenstates of the unperturbed bound system according to the
interactionpicture. Inthecaseconsideredherethesearethehydrogenorbitals. Foracompleteorthonormalbasisthe
scattering states must be also included. The hydrogen orbitals are long-living if the perturbation by the surrounding
plasma is small. If the broadening of the energy levels remains small compared to the distance between neighbored
energy eigenvalues, the transition rate due to collisions with the plasma is small.
At high excitation (Rydberg states), the interaction effects are no longer small compared to the level distance,
and the pure quantum state has only a short life time. Therefore one can look for more robust states that are
formed as superposition of energy eigenstates but are more stable in the time evolution. In particular, the Coulomb
interaction contains the position operator, and localized states are more robust with respect to the interaction with
the surrounding plasma. To find the robust states one has to optimize the quantum states. For simplicity, we restrict
ourselvestocircular-orbiteigenfunctionsofthehydrogenatom. Inthissection,weusethenotationnfortheprincipal
quantum number,
(cid:18) r (cid:19)n−1
ψ (r)=(cid:104)r|ψ (cid:105)=c e−r/(naB)sinn−1(θ)ei(n−1)φ, (49)
n n,n−1,n−1 n a
B
where c = (2/(na ))3/2[2n(2n+1)!]−1/2 denotes the normalization constant. Furthermore, in this section we use
n B
theabbreviationψ (r)forthecircularwavefunctionψ (r). ItcouldbeseenfromEq.(49)thatthehydrogen
n n,n−1.n−1
electron in this eigenstate is already excellently localized in the radial (r) and polar (θ)-direction. To achieve the
localization with respect to the azimuthal direction angle φ, the wave packet can be introduced.
The circular-orbit wave packet of the hydrogen atom is a coherent state constructed from the superposition of
circular-orbit eigenfunctions of the hydrogen atom with a Gaussian weighting function around a large principal
quantum number n [51]:
0
|Gn0,φ0(cid:105)=(cid:88)(cid:112)gnN0,n ei(n−1)φ0|ψn(cid:105) (50)
n n0
with the Gaussian factor and the normalization factor respectively
g =exp(cid:26)−(n−n0)2(cid:27), N = (cid:88)∞ exp(cid:26)−(n−n0)2(cid:27), (51)
n0,n 4σ2 n0 2σ2
n0 n=1 n0
10
where σ is the standard deviation considered as fixed parameter for n . Without loss of generality we can put
n0 0
φ = 0 because it fixes, as a phase factor, only the initial position of the wave packet at the azimuthal angle φ. We
0
dropφ inthefollowing. DuetothesuperpositionwithaGaussianfactor,wehavealsogoodlocalizationwithrespect
0
to φ in the wave packet description (50). The actual Hilbert space H considered here is only a subspace of
n,n−1.n−1
the entire Hilbert space H of the hydrogen atom. The generalization to the full Hilbert space to include all bound
and scattering states could be done straightforwardly.
The time-dependent wave packet in the coordinate representation in terms of spherical coordinates is given by
(cid:104)r|Gn0(cid:105)t =(cid:88)(cid:112)gnN0,n eiEnt/(cid:126)ψn(r) (52)
n n0
with E = Ry/n2 and Ry = 13.6eV. For an appropriate Gaussian factor only the terms with principal quantum
n
number adjacent to n contribute. Therefore we can use the central quantum number n to approximate other states
0 0
in radial and θ-direction. In addition, for short-term time evolution the energy En in the factor eiEnt/(cid:126) in Eq. (52)
can be expanded around n up to the second order, which relates directly to the quantum revival, see below. The
0
probability distribution of the wave packet can be represented as
|(cid:104)r|Gn0(cid:105)t|2 =(cid:88)c2n0(cid:18)ar (cid:19)2n0−2e−2r/(n0aB)sin2n0−2(θ)·e−(a1−iωrevt)(n−n0)2−(a1+iωrevt)(m−n0)2+i(φ−ωclt)(n−m) (53)
B
m,n
with a = 1/(4σ2 ), ω = |E(cid:48) |/(cid:126) = 2Ry/((cid:126)n3) and ω = |E(cid:48)(cid:48) |/(2(cid:126)) = 3Ry/((cid:126)n4), where E(cid:48) and E(cid:48)(cid:48) are the
1 n0 cl n0 0 rev n0 0 n0 n0
first and second derivatives of E with respect to the main quantum number n at n , respectively. As pointed out
n 0
in [52], ω relates to the classical Kepler period T = 2πr /v for the Kepler trajectory with r = n2a and
cl cl cl cl cl 0 B
(cid:112)
v = Ry/(m n2), and the quantum revival period can be defined by ω .
cl e 0 rev (cid:80)
For highly excited states |x|(cid:28)n with x=n−n and |y|(cid:28)n with y =m−n , the sum can be replaced
0 0 0 0 m,n
(cid:82)∞ (cid:82)∞
bytheintegral dx dy. Integratingoverthevariablesr andθ andperformingtheintegraloverx, y yieldsthe
−∞ −∞
probability distribution of the wave packet
(cid:115)
π2 (cid:20) φ2(t) (cid:21)
|G (φ,t)|2 ∼ exp − cl (54)
n0 a2+(ω t)2 2[a2+(ω t)2]/a
1 rev 1 rev 1
with φ (t)=φ−ω t. From this probability distribution the time-dependent width of the wave packet for a Rydberg
cl cl
electron can be extracted
(cid:115)
(cid:113) 1 σ2 (E(cid:48)(cid:48) t)2
σφ (t)= [a2+(ω t)2]/a = + n0 n0 . (55)
n0 1 rev 1 4σ2 (cid:126)2
n0
For the initial time t = 0, we have σφ = 1/(2σ ). The expression (54) also shows that on such a short time scale
n0 n0
the central position of the probability distribution is exactly determined by the Kepler motion. The localized wave
packet for the hydrogen atom moves along the classical Keplerian trajectory of the electron and its width broadens.
With time evolution the localization of the wave packet is destroyed and interference fringes of different eigenstates
are displayed. On a much longer time scale T =2π/ω , the wave packet finally reverses itself, which is the above
rev rev
mentioned quantum revival as indicated in Eq. (54).
The dynamics of the wave packet shown above is purely due to quantum mechanical evolution without plasma
surroundings. Within a plasma environment, the hydrogen atom undergoes interactions with the plasma particles
whichresultsintheshiftoftheeigenenergylevels,thebroadeningofplasmaspectrallines,thescreeningoftheCoulomb
potential, thelocalizationofthehydrogenatom(protonandboundelectron), etc. Inthisworkweconcentrateonthe
localization of the bound electron immersed in a plasma environment.
The scattering of the bound electron by free plasma electrons results in the localization of the electron of the
hydrogen atom, i.e. the collisions with the plasma tend to localize the Rydberg electron and to narrow the wave
packet. As in the case of free particles in a surrounding environment [2], the spreading of the wave packet competes
withthelocalizationeffectinducedbytheplasmaenvironment. TheoptimumwidthofaGaussianwavepacketwhere
both effects, localization and quantum diffluence, nearly compensate, describes a state which is nearly stable in time
and is denoted as robust state.
Inthisworkweareinterestedintimescales,whichareevensmallerthantheclassicalKeplerianperiodicityT . We
cl
assume that on such a short time scale a Rydberg electron behaves like a free electron because of the weak coupling
between the electron and the proton. Comparing with the relaxation processes, which describe the inelastic coupling