Quantum coherence in the dynamical excitation, ionization, and decaying of neon gas induced by X-ray laser

1.5 MB·English

by  Yongqiang Li

Quantum coherence in the dynamical excitation, ionization, and decaying of neon gas
induced by X-ray laser
Yongqiang Li1, Cheng Gao1, Wenpu Dong1, Jiaolong Zeng1 and Jianmin Yuan1,2
1Department of Physics, National University of Defense Technology, Changsha 410073, P. R. China
2IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, P. R. China∗
(Dated: January 13, 2015)
We develop a large scale quantum master equation approach to describe dynamical processes of
practical open quantum systems driven by both coherent and stochastic interactions by including
5
more than one thousand truestates of the systems, motivated bythe development of highly bright
1
and fully coherent lasers in the X-ray wavelength regime. The method combines the processes of
0
coherentdynamicsinducedbytheX-raylaser andincoherent relaxations duetospontaneousemis-
2
sions,Augerdecays,andelectroniccollisions. Asexamples,theoreticalinvestigationofrealcoherent

n dynamicsofinner-shellelectronsofaneongas,irradiatedbyahigh-intensityX-raylaserwithafull
a temporal coherence, is carried out with the approach. In contrast to the rate equation treatment,
J
we findthat coherence can suppressthemultiphoton absorptions of a neon gas in the ultra-intense

2 X-ray pulse, due to coherence-induced Rabi oscillations and power broadening effects. We study
1 theinfluenceofcoherenceonionization processesofneon,anddirectlyprovethatsequentialsingle-
photonprocesses for both outer- andinner-shell electrons dominatetheionizations for therecently
] typical experiments with a laser intensity of ≈ 1018 W/cm2. We discuss possible experimental
h
implementationssuchassignaturesforcoherentevolution ofinner-shellelectronsviaresonance flu-
p
- orescence processes. The approach can also be applied to many different practical open quantum
m systems in atomic, quantum optical, and cold matter systems, which are treated qualitatively by a
few-level master equation model before.
o
t
a PACSnumbers: SubjectAreas: AtomicandMolecularPhysics,Optics,Computational Physics
.
s
c
i I. INTRODUCTION Tostudythesecoherence-inducedquantumfeatures,dra-
s
y matic success has also been achieved in preparing and
h controlling the coherent dominant systems with sup-
p Master equation approach is a standard technique for presseddissipations,suchaselectromagneticallyinduced
[ open quantum systems [1] and a successful theory in de- transparencyinquantumoptics[15]andsuperfluid-Mott

1 scriptions of light-matter interactions, such as in con- phase transition in condensed matter physics [16]. Re-
densed matter physics [2], chemistry and biology [3],
v cently, exploring coherence effects of complex systems
0 quantum optics [4], and ultracold gases [5]. The mas- has been arguably one of the most important topics
6 ter equation is quite general and encompasses various in physics, inspired by generating X-ray laser with ex-
6 physical phenomena, as long as these phenomena share
tremelyhighbrightnessandfullytemporalcoherence[8],
2 commonphysicalmechanisms,i.e. the interplaybetween
such as the Linac Coherent Light Source (LCLS) [17],
0
coherence and dissipation. Traditionally, these kinds of
where one opened a new era of exploring the interaction
.
1 systems are treated within the framework of few-level of high-intensity X rays with complex systems on fem-
0 models [5–7]. With the development of highly bright
tosecond(fs)timescales. Unfortunately,atypicalfeature
5 lasers [8], however, the few-level models lose the possi- ofX-ray-matterinteractionsistherapiddecayprocesses,
1
bility for describing complex open systems, since energy
duetothevastrelaxchannels. Asaresult,itisinevitable
:
v here is deposited in a broad range and relaxes in a vast to investigate the interplay between coherence-induced
i number of decay channels. In this case, new challenges
X effects and dissipations in X-ray-matter systems.
appear for real dynamics of these complex systems, and
r Here, the X-ray free electron laser, with an ultrashort
a a large-scale simulation is inevitable. Here, one open is- pulse duration and a high peak brilliance, provides the
sue, related to coherence effects in dynamical processes
possibility for the study of physical, chemical and bio-
of complex systems, is still unknown.
logical properties which are never accessed experimen-
Coherenceplaysanimportantroleindescribingcorre- tally before [18]. For example, a series of pioneer experi-
lation properties of quantum matters and understand- ments[19–28]focusonunderstandingofX-ray-matterin-
ing quantum phenomena including lasing [9], Fano teractions,including nonlinear photoionizations,hole re-
shape [10], superconductivity, superfluidity and Bose- laxations, Auger electron distributions, and X-ray emis-
Einstein condensate [11, 12], and novel phenomena aris- sion spectrums in atoms, molecules and solid materials.
ingfromquantumoptics[13]andattosecondphysics[14]. To simulate these experiments, a semiclassical descrip-
tion of radiation-field coupled systems, Einstein’s rate
equation approach [29–31], has recently been extended
to the regime of X-ray-matter interactions for gaining
∗Electronicaddress: [email protected] fundamentalinsightintothefastdecayedsystems,byin-
2
cluding photoexcitation and ionization, electron impact effects in the ultrafast decayed systems.
excitationandionization, Augerdecay, andtheir reverse These questions motivate our study in this paper to
processes. In Einstein’s rate equation model, all the ab- establish a general method for describing coherent dy-
sorption and emission processes are treated with transi- namics of the X-ray-matter systems in the framework of
tionprobability andthe coherencebetweendifferent lev- master equation approach. In principle, master equa-
elsisneglected,whicharegoodapproximationsforeither tion approach can deal with different kinds of dynam-
incoherentlightfieldsordominantdecayprocessesofthe ics of the intense X-ray-matter systems by including the
systemic coherence [32]. The agreement that one finds microscopic processes due to photons, electrons and en-
betweentheoreticalpredictions[31]andexperiments[20] vironments. In dilute atomic gases [20, 26], for exam-
atrelativelylowatomicnumberdensities,verifiestheca- ple,one shouldinclude photoexcitationsandionizations,
pability of the model to simulate X-ray-matter interac- Auger and spontaneous decay processes, and their re-
tions. The physical reason is that the free-electron laser verse processes. For dilute molecules [21], additional
beam has limited temporal coherence with each pulse processes, such as photon dissociations, should be taken
consisting of a random number of incoherent intensity intoaccount. Forsolid-densewarmandhotmatters[27],
spikes in a fs duration. electron impact excitations and ionizations play an im-
portant role, in addition to environmental screening ef-
One big challenge, however, which has not been over-
fects[60]. Therefore,wedevelopathousandstatemaster
comeyet,isthetemporalcoherenceofX-rayfree-electron
equation for describing real dynamics of these complex
lasergeneratedfromself-amplifiedspontaneousemissions
systems induced by an X-ray laser, and discuss possi-
from electron beams [33]. One goal of the X-ray free-
ble experimental implementations such as signatures for
electron laser is to enhance temporal coherence of the
coherent evolution of inner-shell electrons. Our studies
field,andthequestforimprovedX-raysources,basedon
willprovidethebasisforunderstandingthecoherenceef-
the free electronlaser,is being tackledwith severaltech-
fects in X-ray absorption mechanisms at a fundamental
niques [25, 34–41]. Infact, the new X-raypulse withim-
level. Actually,ifalltheoffdiagonalelementsoftheden-
provedtemporalcoherence[41]ismoresuitabletolocally
sitymatrixareignored,ourmethodreducestoEinstein’s
deposit energy and prepare electronic states, study dy-
rate equation approach, which is a semiclassical descrip-
namical properties via photon correlation spectroscopy,
tion for X-ray-matter interactions [29–31]. The present
and image biological specimens and long-rang orders in
approachcanbeappliedtomanydifferentpracticalopen
liquid and condensed matters [18]. Recently, one gener-
quantumsystems,whicharewidely treatedqualitatively
atesthefirstsuccessfulcoherentfree-electronlaserradia-
beforebyafew-levelmasterequationapproach[49],such
tionpulsesinthesoftX-rayregime,basedontheseeding
as those in atomic physics [50], quantum optics [51] and
experiment [42], which experimentally provides the pos-
cold matter physics [5, 52].
sibility for investigating the interplay between coherence
and dissipation in X-ray-matter systems. Then the cru-
cial issue, related to an X-ray laser with an improved
temporal coherence, is how to model the ultrafast dy-
namics of the X-ray-matter systems and understand the
underlyingphysics. Toobtainamoreprecisedescription
of dynamical mechanics, in general, we need a quantum
mechanicaltoolforsimulationsofitstimeevolution,such
as time-dependent Schr¨odinger equation. In contrast to
the case of low-Z species in dilute gases [43–48], time-
dependent Schr¨odinger equation for complex systems is
difficult to tackle directly,due to electron-electroncorre-
lations, collision processes, and spontaneous and Auger
FIG.1: Sketchofmultiphotonabsorptionsinneoninducedby
decay processes. Approximations for the X-ray-matter ultra-intense X-ray pulses. Excitations and ionizations from
systemsareinevitable,suchasmasterequationapproach. the 1s shell dominate the photoabsorptions, followed by the
Inthemasterequationapproach,onecanincludetheim- consequentfurtherouter-shellionizations(a)andAugerdecay
portant microscopic processes as much as possible and processes(b),endingupwithahighlyionizedstage. Coherent
investigate the interplay between coherence and dissipa- RabioscillationsinterplaywithdissipationsintheX-ray-atom
tions in the X-ray-mattersystems. Insteadof a few-level systems, such as Augerand spontaneous decay processes.
simulations [5–7, 49–53], one need to simulate real dy-
namics of the complex X-ray-matter systems based on In the recently typical experiment [20], however,
large scale simulations, due to the vast decay channels atomicgasesarediluteandtheinfluenceofelectroncolli-
induced by the intense X-ray laser. As far as we know, sionsandphotonscatteringcanbeneglected[30]. Inthis
thisproblemisneverexploredinthe frameworkofquan- paper, we take dilute atomic gases as examples for dis-
tummasterequationapproachbefore,coherentdynamics cussing coherentdynamics of the rapidly decayedX-ray-
of the complex systems is still unclear, and it is still an matter systems (sketch in Fig. 1), where there are still
openissuewhethernewphenomenaarisefromcoherence few studies related to real dynamics of complex atoms
3
based on multilevel master equation approach [7]. Here, With the semi-classical treatment for the laser field,
comparisons between master equation and rate equation the quantizationofthe lightis ignored. Insteadthe light
approachwill be made to investigate the influence of co- isconsideredasaelectricfield,E(t),whichinteractswith
herence on the dynamical mechanics, as related to the thei-thdipoled forthetransitionbetweenstates|kiand
i
ongoing experiments with different temporal coherence. |k′i to give
In parallel, we also perform an approximate calculation
based on a degenerate atomic master equation approach ~Ω (t)
which is computationally more affordable. HˆI =− 2i (Dˆie−iωLt+H.c.), (4)
The paper is organized as follows: in section II we Xi
give a detailed description of the approach. Section III
wherethe transitionoperatorDˆ =|kihk′|,andRabifre-
coversourresults forcoherentdynamics ofneoninduced i
by anX-raylaser,anddiscusses Augerprocessesof neon quency ~Ωi(t) = ediE(t) at time t with di = hk′|dˆi|ki.
based on thousand state master equation approach. We Weremarkherethatwehaveusedthedipoleapproxima-
summarize with a discussion in Section IV. tionforthecouplinginEq.(4),andcanalsotraceoutthe
externalfielddegreesoffreedomwithinthesemi-classical
treatment indicating the Eq. (3) can be ignored in the
II. THEORETICAL MODEL simulations. This semiclassical approximation forms the
basis of most investigations on the many-body systems
A. Hamiltonian bothcoherentlyandincoherentlycoupledbyastrongex-
ternal field.
For magnetic sublevels, we rewrite the Hamiltonian of
We consider a many-body system, such as dilute
the matter-field interaction in a more explicit form
atomic and molecular gases [20, 21, 26], and solid-state
materials [27], coupled with incoherent sources such as
~Ω (t)
vacuum,irradiatedbyahighintensityX-raylaser. Inner- HI = i (DJJ′σe−iωt+H.c.), (5)
shell electrons of these atoms and molecules will be ex- XJ,J′ 2
cited, forming a far-off-equilibrium system and typically
relaxing in a fs timescale via spontaneous, Coulombic where ~Ω (t)=eE(t)hJ|dˆ |J′i and
i i
and Auger decay processes. These processes compete
with other mechanics,such as processes including coher-
J 1 J′
ethnet ptohtoatloHexacmitialttoionniasnanodf tihoenizXa-triaoyn-sm. aCttoerrressypstoenmdisncgalyn, DJJ′σ =(−1)J−MJ (cid:18)−MJ σ MJ′ (cid:19)|J,mJihJ′,mJ′|.
be written as
Hereσ denotesthepolarizationoftheexternallaser,and
Hˆ =HˆA+HˆF +HˆI +Hˆinc, (1) the dipole operator dˆi describes a J →J′ transition be-
tweenstates|J,M iand|J′,M′iwithZeemansubstruc-
J J
where the total Hamiltonian is the sum of the Hamilto- tures.
nian HˆA of the many-body system in vacuum, the ex- Actually, we can remove the explicit time dependence
ternal field Hamiltonian HˆF including the coherent and from the Hamiltonian, if Rabi frequency Ω is time inde-
incoherentexternalfield,thelaser-matterinteractionHˆI, pendent and Ω ≪ ǫi,ω, transforming the Hamiltonian
and the incoherent-field-matter interaction Hˆinc. of the system to an arbitrarily specified rotating frame.
The Hamiltonian The interaction Hamiltonian in a frame rotating at the
laser frequency ω reads
N
Hˆ = ǫ Aˆ , (2)
A k kk ~Ω
kX=1 HI = (DJJ′σ +H.c.). (6)
2
governsthetimeevolutionofthesysteminvacuum,such XJ,J′
asdiluteatomicandmoleculargases,andcondensedma-
terials, with Aˆ = |kihk| and N being the total energy Thesystemcanalsobepumpedbyanincoherentfield,
kk
levels included. Here, |ki and ǫ denote the eigenstate such as the black-body radiation field in a hot plasma
k
andeigenvalueofthe system,respectively. TheHamilto- environment, and it reads
nian of the external field is given by
Hˆ =~ (d e ·e ˆb Dˆ +H.c.), (7)
inc i i n n i
HˆF = ~ωiaaˆ†iaˆi+ ~ωibˆb†iˆbi, (3) Xi Xn
Xi Xi
where e and e denote the directions of the dipole mo-
i n
where ~ denotes the reduced Planck constant, aˆi (aˆ†i) ment di, the polarization of the incoherent field, re-
denotes the annihilation (creation) operator that corre- spectively. Actually, the contributions of the incoherent
sponds to the i-th mode of the laser with frequency ωia, field can directly be included in the master equation ap-
andˆb and ωb denote those for the incoherent field. proach [6, 54], as discussed in Sec. IIB.
i i
4
B. Method the low intensity X-ray laser, we consider ionization as
coherent processes for multiphoton dominant processes,
Now,thenextquestionishowtotheoreticallysimulate whichfeaturesanalogiestobound-statetransitionsofthe
the time evolutionof complex systems in the presenceof system. Formultiphotondominantprocesses,theionized
the X-raylaser,describedbyEq.(1). Incontrasttolow- state composing of the residual system and ionized elec-
Z atomic and molecular gases [43–48], the X-ray-matter trons reads
system is dominated by ultrafast decayed mechanics in
|ki=|j ,κ;J,ǫ ,Pi, (9)
the experimental timescales, such as spontaneous decay core k
processes,whereonelosesthepossibilitytokeeptrackof
where j , κ, J, ǫ , P denote the angular momentum
core k
the couplings between the system and the environment
of the residual system, relativistic angular momentum
with infinite degrees of freedom. In solid-state materi-
of free electrons, total angular momentum, total energy
als, electron-ion, electron-electron and ion-ion collisions
and parity of the system, respectively. In the physical
occur rapidly and randomly,where screeningand broad-
systems, these states can be populated by multiphoton
eningeffectsduetothesolid-densityenvironmentshould
excitations. Considering the large amount of continuous
be taken into account in dynamical simulations. Hence,
states, selection rules should be used to solely include
a statistical description for the X-ray-matter system is
dominantstates, suchas degenerateinitial, intermediate
needed,andherewestudy the time evolutionbasedona
and finial continuous states connected by multiphoton
generalizedthousandstatemasterequationapproachfor
energies, and at the same time neglect states detuned
the reduced density matrix of the system, where the de-
from resonance excitations since the finite time dura-
greesoffreedomofboththeenvironmentandX-raylaser
tion of the X-ray laser implies a finite Fourier width for
have been traced out in a perturbative treatment. The
the bound-free or free-free transitions. These selections
generalizedthousand-levelmasterequationforacomplex
are a good approximation for hydrogen in the presence
system,coupledtovacuummodesoftheelectromagnetic
of strong laser beams, and therefore we anticipate that
field and irradiatedby an X-ray laser and incoherent ra-
the dominant states for ionizations of a complex system
diation field, reads
should be similar.
dρˆ(t) i Complementary to the multilevel master equation, in
dt =−~[Hˆ,ρˆ(t)]+Lρˆ(t), (8) this work we employ a degenerate master equation ap-
proachtoexplorethephysicsofEq.(1),whichiscompu-
where ρˆ= kpk|kihk| denotes the reduced density ma- tationally more affordable. Here, the Rabi frequency is
trix operatPor of the multilevel system, and Lρˆ(t) = (2MJ +1)(2MJ′ +1)Ωi and the decay rate is defined
iΓi/2[2Dˆiρˆ(t)Dˆi†−Dˆi†Dˆiρˆ(t)−ρˆ(t)Dˆi†Dˆi] with Γi de- pas a total transition probability from one upper state
nPotingtransitionrateforthei-thdipoleduetotheback- |J,MJitoallthedegeneratelowerstatesofthelevel|J′i.
ground radiation pump, spontaneous, Coulombic and a. population distribution— Free states |ki form a
Augerdecays,photodissociations,andcollisionprocesses. completebasisset,andwecanworkinthesetandobtain
Here, the first term in the rightside describes the coher- a number of algebraic equations for Eq. (8). Due to infi-
ent dynamics of the multilevel system coupled with the nitenumberofmatrixelements,however,truncationfora
laser field, and the second term denotes the incoherent finiteN-statebasisisrequired. AndthenthefullEq.(1)
processes, such as spontaneous and Auger decays, tran- is projected onto this subspace spanned by the N-basis
sitionsduetotheblack-bodyradiationfieldandcollision states, and it is expected this projection gives the best
processes. The contributionsofthe black-bodyradiation possible description of dynamics of the system induced
pump are only nontrivial in warm and hot matters, and by an X-ray laser, such as a neon gas by including thou-
collisionprocessescanbe neglectedfor the dilute atomic sands of energy levels. After solving the multilevel mas-
and molecular gases since it occurs in a fs timescale be- terequation,we obtainpopulationdistributions foreach
ingmuchshorterthantheaverageparticle-collisiontime. levelandcoherencebetweendifferenttransitionstatesas
Notethatthebroadeningcontributionsoftheincoherent- a function of time, by calculating ρkk′ =hk|ρˆ(t)|k′i, and
field-system interactions in Eq. (1) and plasma environ- then comparison can be made with experimental data
ments are included to the second term in Eq. (8), while by utilizing ion charge-state spectra recorded by a time-
the corresponding energy shifts are incorporated in the of-flight analyzer [20], even though the recently typical
energy levels of the system in vacuum. We remark here free-electron lasers have limited temporal coherence and
that the thousand state master equation has unique fea- new techniques are needed to improve it.
tures,whichneedslarge-scalesimulationsforpropagating MasterequationapproachcanreducetoEinstein’srate
intime amatrixof≈106×106 andisnottrivialtosolve equation approach, if the pump field is fully stochastic,
directly,suchas stability ofnumericallinearalgebraand suchasabroadbandisotropiclightfield. Inthiscase,co-
parallel procedure demanding. herence embedded in the off-diagonalterms is neglected,
There are two possible ways to take photoionization and the time evolution of the system is characterizedby
processes into account in our simulations. While we the population changes for each energy level.
treat the ionization as incoherent processes by adding b. resonance fluorescence— The spectrum of reso-
photoionization cross section in the incoherent terms for nancefluorescenceinhigh-intensityx-raypulsescanalso
5
becalculatedviamasterequationapproach[7,53,55–58]. bound and continuum states of different successive ion-
The time-dependent spectrum of the fluorescent light, ization stages in the single atom. For the bound states,
describedby autocorrelationfunctionofthe electricfield a fully relativistic approach based on the Dirac equa-
operator Eˆ(t), is given by [53, 59] tion is utilized, while for the continuum processes the
distorted wave approximation is employed. The bound
t t
states of the atomic system are calculated in the config-
S(t,ω)= e−iω(t1−t2)hEˆ−(t )Eˆ+(t )idt dt ,(10)
Z Z 1 2 1 2 uration interaction approximation. The radial orbitals
0 0
for the construction of basis states are derived from a
whereEˆ−(t)andEˆ+(t)denotenegativeandpositivefre- modified self-consistent Dirac-Fock-Slater iteration on a
quency parts of the electric field, respectively. fictitious mean configuration with fractional occupation
The change in time of the occupation number of pho- numbers, representing the average electron cloud of the
tons can be related to the dipole moment operator, and configurations included in the calculation. The detailed
it reads discussion can be found in Refs. [60–62].
c. Rabi frequency— In the dipole approximation,
Eˆ−(t)=C(r)Dˆ†(t), (11) the Rabifrequencyforthe bound-statetransitioncanbe
i
obtainedfromthedegenerateemissionoscillatorstrength
where C(r) is a proportionality factor at position r and of the electric dipole dˆ ,
i
canbeneglectedinahomogeneoussystem. Andthenwe
obtain gf = 2me∆E|hdˆ i|2
i 3~2 i
t t−t2
S(t,ω)=2Z dt2Z Re(cid:20)e−iωτhDˆi†(t2+τ)Dˆi(t2)i(cid:21)dτ(,12) = 8.749×1018×∆E|hdˆii|2, (14)
0 0
and then it yields.
where we only include the contributions in the region
t1 ≥t2 and introduce the time delay τ ≡t1−t2. Ω = eEhdˆ i
Thus we need the knowledge of two-time expecta- i ~ i
tion values of Aˆij(t1,t2) = Dˆj(t1)Dˆi(t2) for the time- = 1.409×109 I×gfi/∆E
dependent fluorescentspectrumofthe i-thdipole transi- p J 1 J′
tion. Applying the quantum regressiontheorem [55, 56], × (−1)J−Mj , (15)
we obtain the coupled equation for the i-th dipole tran- (cid:18)−MJ q MJ′ (cid:19)
sition, where laser intensity I and transition energy ∆E are in
unitsofW/cm2 andeV,respectively. Hereq denotesthe
dAˆ(dtt11,t2) =−~i[Hˆ(t1),Aˆ(t1,t2)]+LAˆ(t1,t2), (13) pdioplaorleizmatoiomneonfttfhoer tehxeteJrn→al lJa′setrr,adnˆisidtieonnotbeestwtheeenelsetcattreics
|J,M iand|J′,M′iwithZeemanstructure,m denotes
where t ≥ t , and the initial condition can be ob- J J e
1 2 the mass of atom, E denotes the electric field, and e de-
tained by the one-time evolution operator A (t ,t ) =
ij 2 2 notes the charge of electron.
hDˆ (t )Dˆ (t )i via Eq. (8).
j 2 i 2 For photoionization processes (bound-free transition),
the cross section is given by the differential oscillator
strength df,
C. Examples: dilute atomic gases dǫ
df
σ(ǫ) = 4π2αa2
Before proceed to construct the master equation, we 0dǫ
should first calculate the wavefunctions and energy lev- df
= 8.067×10−18× , (16)
els of the system, and obtain the required data, includ-
dǫ
ing oscillator strength, dipole moment, Rabi frequency,
where the energy ǫ is in unit of Ry, α denotes the fine-
spontaneousandAugerdecayrates,andphotoionization
structure constant and a denotes the Bohr radius. And
cross section. Actually, dilute atomic gases are widely 0
then Rabi oscillation for bound-free processes can be
used in the recently typical experiment [20]. From now
written as:
on,wetakedilutemultielectronatomicgasesasexamples
for discussing coherent dynamics of the rapidly decayed eE
Ω = hdˆ i (17)
X-ray-matter systems. Without loss of generality, these ~ i
discussions can be easily applied to other X-ray-matter
systems, such as molecular gases and solid-state mate- = 0.135×1018 g(ǫ) I ×σ(ǫ)/∆Edǫ,
Z
rials by calculating corresponding wavefunctions, energy p
levelsandmicroscopictransitionprocesses. Here,wecan where, I, σ and∆E areinunits ofW/cm2,cm2 andeV,
mapthediluteatomicsystemintoasingle-atomproblem respectively, and g(ǫ) is the lineshape of the bound-free
each of which can be written as in Eq. (1). The compu- transition,such as broadeningdue to the finite periodof
tations of various atomic radiative processes involve the time duration for the X-ray laser.
6
For free-free processes, the Rabi frequency of the κ→ withenergydensityρ(ω)=~ω3n /π2c3. Weremarkhere
λ
κ′ transitionwith the residualionin the state |j i can thatthetransitionrateB =A n(ω),withn(ω)beingthe
core i i
be given by: meanphotonnumberoftheincoherentfieldatfrequency
ω [6, 54], such as the black-body radiation field.
eE
Ω= hj ,κ;J|dˆ |j ,κ′;J′i, (18) As for laser driven atomic system, one can use the
~ core i core relationbetweenlaserintensityI andphotonnumbernǫ
λ
with polarization ǫ and wavelength λ
whereκandJ denotetherelativisticangularmomentum
of free electrons and the total angular momentum of the ω2
system, respectively. I(Θ)/c=nλ(2πc)3~ω, (24)
d. Spontaneous decay— Even in the absence of an
applied field, spontaneous emission cannot be ignored, and then obtains:
since the excited state interacts with the vacuum fluctu-
dB e2
ationsoftheelectromagneticfield. Afterintegratingover i = |he|ˆr·ǫ|gi|2I(Θ)g(ω)dω
allpossible modesandsumming overthe twoorthogonal dΘ Z 2~2ǫ0c
polarizations possible for each wave vector, the sponta- e2
= |he|ˆr|gi|2I(Θ)g(ω)dω, (25)
neous|ei→|gitransitionrateisgivenbyaperturbation Z 6~2ǫ c
0
treatment,
whereone averagesoverallthe polarizationdirectionsof
the dipole moment. Normally, the laser injects from one
A = g(ω)|Ω |2dω
i Z i certain direction with central frequency ν (I(Θ)dΘ =
e2ω3 I(ν)δ(Θ)dΘ), and in this case the total transition rate
= 3πǫ ~0c3|hg|ˆr|ei|2, (19) can be written as:
0
πe2
where c is the speed of light, ǫ denotes the permit- B = |he|ˆr|gi|2 I(ν)g(ν)dν (26)
tivity, and ω denotes the transi0tion frequency. Here, ge 3~2ǫ0c Z
0
one assumes the lineshape function g(ω) is sharply where g(ν) is the line shape of the |gi→|ei transition.
peaked around ω0, so that ωg(ω)|hg|ˆr|ei|2ω3d3ω ≈ If one defines absorbtion (emission) cross section as:
|hg|ˆr|ei|2ω3 g(ω)d3ω =2πω3|Rhg|ˆr|ei|2.
0 ω 0
e. StimuRlated transition— If the |gi → |ei transi- σi(ν) = Bihν/I(ν)
tionis drivenby the externalradiationfieldwith photon 2π2e2ν 2
= he|ˆr|gi g(ν), (27)
polarizationǫ inthe differentialsolidangledΘ,the tran- 3ǫ0~c (cid:12) (cid:12)
sitionrateinthissolidangleΘ,basedonFermi’sGolden (cid:12) (cid:12)
rule, is written as: one obtains (cid:12) (cid:12)
I(ν)
e2 ~ω V B = σ (ν) dν (28)
dBi = Z ~2|he|ˆr·ǫ|gi|22ǫ V (cid:18)nǫλ(2πc)3ω2(cid:19)g(ω)dωdΘ i Zν i hν
0
I(ν) πe2
e2ω3 = f g(ν)dν
= Z 2~ǫ (2πc)3|he|ˆr·ǫ|gi|2nǫλg(ω)dωdΘ (20) Zν hν 2ǫ0mec i
0 = 109.761×10−18×
with |E|2 = 2ǫ~0ωV nǫλ(2πVc)3ω2. I(ω)fi ~Ai/2π dω,
As for black-body radiationfield, energy density of its Zω ~ω (~ω−~ωi)2+(~Ai/2)2
radiation varies slowly over the range of transition ener-
where the full-width-half-maximum ∆ν = A /2π, and
gies, i
laser intensity I and photon energy ~ω are in units of
W/cm2 and eV, respectively.
g(ω)ρ(ω)dω ≈ρ(ω) g(ω)dω =2πρ(ω). (21)
Z Z f. Autoionization— Complex atoms, irradiated by
ω ω
X-ray laser, can form hole atoms and then relax in a fs
Considering the isotropic and unpolarized properties of timescale via Auger decay processes, i.e. spontaneously
the radiation field, the total transition rate can be ob- emitting one of the electrons in along with refilling the
tained by integrating over all possible modes and sum- holebyouter-shellelectrons. Thecorrespondingautoion-
ming over the two polarizations of the field: ization rate is expressed as
e2ω3 2
and corresponBdiin=gly3πEǫin0~s0tce3innλB|hec|oˆre|gffii|c2i,ent (22) whAeraie=κ2iXsκth(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)e(cid:28)rjecolaret,ivκi;sJti,cMaJn(cid:12)(cid:12)(cid:12)(cid:12)gmXu<lanrrqm1unan(cid:12)(cid:12)(cid:12)(cid:12)Jtu′,mMJn′(cid:29)um(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) b,e(r29o)f
πe2 the ionized electron, and r denotes the relative dis-
B = |he|ˆr|gi|2, (23) mn
Ein 3ǫ ~2 placement of between electrons m and n. Here, J and
0
7
M denote total angular moment and magnetic quan-
J 1
tum number of the system consisting of the residual ion
Master
and ionized electrons, respectively.
0.8 Master(incoherent)
For some cases, double Auger processes are nontrivial
and should be included in the simulations, and the code n Rate
o
for the atomic data of direct two-electron processes has ti 0.6
a
been developed recently [63]. In this work, we do not ul
include the contributions from the direct double Auger op 0.4
P
processes,yet its implementation in the master equation
is straightforward. 0.2 Ne9+ : 1s → 2p
0
III. RESULTS 0 20 40 60 80 100 120
t(fs)
In this section, we discuss coherent dynamics of the
1
rapidly decayed X-ray-matter systems. As examples, we Master
Master(T=1000eV)
investigate ultrafast dynamics of inner-shell electrons of
0.8 Master(incoherent)
complex atoms irradiated by an X-ray laser via a thou- n Rate
sand state master equation approach. The influence of o
ti 0.6
coherence on dynamical mechanics will be investigated a
ul
throughcomparisonsbetweenatomicsystemsinducedby p
o 0.4
X-ray pulses with different temporal coherence. For ion- P
izations,multiphotonejectionsofelectronsfromanatom
0.2 Ne8+ : 1s2 → 1s2p
subjectedtoastronglaserfieldwillbeinvestigatedinthe
framework of master equation approach, where the reli-
0
ability of our numerical results will be verified against
0 100 200 300 400
the time-dependent Schr¨odinger equation. For inner- t(fs)
shell dynamicalprocesses,Auger and spontaneousdecay
processes typically occur in a fs timescale and compete
FIG. 2: Comparison with rate equation. Upper: State pop-
withothermechanics,suchascoherentRabioscillations,
ulations of a two-level H-like neon (between |J = 0.5,MJ =
where we will discuss possible experimentalimplementa- 0.5i of 1s and |J = 1.5,MJ = 1.5i of 2p) as a function of
tions such as signatures for coherent evolution of inner- time. Lower: Degenerate state populations for the transi-
shell electrons. Finally, we will discuss a real coherent tion |J =0i→|J =1i (1s2-1s2p) of He-like neon. Here, the
dynamics of complex atoms induced by an X-ray laser, solid and dashed lines denote populations of the ground and
based on master equation approach by including thou- excitestates,respectively,coupledbyalaserwithanintensity
sands of atomic levels. I0=1012W/cm2. Fortheincoherentpump,masterequation
yields identical results with those from therate equation.
A. Coherent dynamics between bound states
ing Ω2/(Γ2+2Ω2). We observe that the stable states
In this section, we investigate coherent dynamics be- of mpaster equation at a longer time are identical with
tween bound states of an atomic system coupled by a rate equation approach, whose conclusion is also consis-
X-ray laser,and pay special attention to the influence of tent with analytical results.
coherence on dynamical evolution by comparising with Thenweinvestigateamultilevelatominteractingwith
Einstein’s rate equation, which is a reduction of master an isotropic and unpolarized radiation beam, as rele-
equation approach and a method for the system illumi- vant to Einstein’s rate equation. We focus on the dipole
nated by a broadband isotropic light field (incoherent transition |J = 0i → |J = 1i (1s2-1s2p) of He-like
light). neon, where there are four states in the reduced Hilbert
First we study the time evolution of a two-level H- spaceandtheupperthreelevelswithangularmomentum
like neon with atomic data being obtained via solving |J = 1i are triply degenerate due to magnetic splitting.
Dirac equation [60–62]. Due to the pump of the laser We assume the ground state |J = 0i are populated at
beam and coupling with the external environment, it is t=0, and then switch on the pump laser. We observe a
expected that the system exhibits a Rabi-oscillating fea- decayed Rabi-flopping structure in the master equation,
ture, and then decays to a steady state after a longer due to the coupling with the pump laser and the exter-
time. As shown in the upper panel of Fig. 2, we observe nal environment, while in the rate equation there is a
Rabi oscillations between |J = 0.5,M = 0.5i (1s) and monotonous decay for the ground state, as shown in the
J
|J = 1.5,M = 1.5i (2p), and, after a few fs, steady lowerpanelofFig.2. Wealsoobservethatbothmethods
J
states areachievedwith the excited-statepopulationbe- cannot yield identical stable states at a longer time. It
8
indicates that, for a multilevel atom coupled by a laser
1
field,coherenceinfluencesbothshort-timestructuresand Scho.
long-time stable states, and Einstein’s rate equation ap-
0.8
proach cannot give reasonable results since coherence is Rate
totally neglected in this method. The coherence effects n
can also be clearly seen in Sec. IIID. tio 0.6 Master
a
Another issue needed to be addressed is the influence ul
of different temporal coherence of the pump field on the Pop 0.4 H : 1s → eǫl
dynamics of complex atoms. To simplify our discussion,
an incoherent thermal radiation field is taken into ac- 0.2
count as incoherent pump sources, in addition to a co-
herent X-ray laser. It is expected that the interplay be- 0
tween coherent and incoherent pump influences dynam- 0 5 10 15 20
ical properties of the atomic system. For example, the t(fs)
incoherent thermal radiation field can be introduced via
RJ,J′ =ΓJ,J′n˜ =ΓJ,J′{exp[~(ωJ′ −ωJ)/kT]−1}, where 1
T isthe temperature ofthe incoherentthermalradiation Scho.
field, n˜ is average number of thermal photons per mode 0.8
ateachtransitionfrequency,andΓJ,J′ denotes the spon- n Rate
taneous decay induced by vacuum (the energy shift due tio 0.6 Master
tocouplingwithvacuumisincludedinHˆa). Inthelower ula
panelofFig.2,weshowresultsforanatomicsystemcou- p 0.4
o
pTle=d1w0it0h0aeVco,haenrdenfitnfideltdhaatndthaetshtearbmlealstraatdeisatsihoinftficellodseorf P H : 1s → eǫl
0.2
tothoseobtainedfromrateequation. Forafullyincoher-
entpumpintheframeworkofmasterequationapproach,
0
we find that the time evolution of the multilevel system 0 20 40 60 80 100
coincides with Einstein’s rate equation approach. t(fs)
B. Coherent dynamics for photoionizations FIG. 3: Comparison with time-dependent Schr¨odinger equa-
tion. Ground-state populations of hydrogen as a function
of time for a laser intensity I0 = 1014W/cm2 with photon
g. Comparison with time-dependent Schro¨dinger
energiesof14eV(upper)and30eV(lower),respectively,ob-
equation for hydrogen— Ionization is a basic process tainedviarateequation,masterequationandtime-dependent
of atoms subjected to an external electromagnetic field. Schr¨odingerequation.
Forweaklaserfields,perturbationtheorycanbe utilized
to capture these processes. For example, one considers
a system prepared in an initial state |ii and perturbed
the atomic master equation approach, and the validity
by a periodic harmonic potential V(t) = Veiωt which
of approach will be verified via comparisons with time-
is abruptly switched on at time t = 0. Based on time-
dependent Schr¨odingerequationwhere both the discreet
dependent perturbation theory,the first two basis coeffi-
and continuous states are treated in the same level.
cient c can be expanded as: c(1) = 2π|hf|V|ii|2δ(ω −
n n ~2 fi In the calculations here, remarkable agreements be-
ω) corresponding to Fermi’s golden rule, and c(2) =
n tween master equation and time dependent Schr¨odinger
2~π4 m|hf|ωVm|m−iωhim−|ωV|ii|2δ(ωfi−2ω)associatingwithtwo- equationhave been obtained for realdynamics of hydro-
phoPton processes, where |ii, |mi and |fi denote the ini- gen in the intermediate-wavelength regime. In Fig. 3,
tial, intermediate and final states, respectively. In the for example, we show the ground-state evolution of hy-
study of atomic dynamics, one can simply include pho- drogen induced by a strong laser field with an intensity
tonionizationsasdecaycoefficientsuptothecorrespond- of 1014 W/cm2, and photon energies of 14 eV and 30
ing order, just as used in rate equation approach, where eV, respectively, and find that the master equation ap-
thecoherenceinionizationprocessesistotallyneglected. proach can capture photoionization processes of hydro-
For intense laser fields, however, multiphoton ejections geninthe intermediate-wavelengthregime. Surprisingly,
of one electron typically occur, and manifest as domi- rate equation offers another possibility to address this
nant mechanics [20, 64–66]. In this case, the perturba- issue at these parameter regimes, even though the de-
tion theory lacks the possibility for understanding mul- tailed structures are different, which indicates that the
tiphoton absorptions, and a nonperturbative treatment leading order process dominates ionizations here, irre-
is inevitable to address this issue. Here, we describe spective of coherence or incoherence. We anticipate that
coherent multiphoton ionizations in the framework of photoionization mechanism for complex atoms in the X-
9
ray-wavelengthregimeshouldalsobecapturedbymaster
1
equation approach.
h. Coherence in photoionizations of neon— In this Ne : 2p6 → 2p5 + eǫl
0.8
section, we discuss ionizations of complex atoms in the
short-wavelengthregime. Inthisregime,time-dependent on
Schr¨odinger equation encounters difficulties for dealing ti 0.6
a
with electron-electron correlations and irreversible pro- ul
cesses, such as Auger and spontaneous decay processes, op 0.4
P
whilerateequationapproach,mainlybasedonphotoion-
ization cross sections, fully discards coherence effects in 0.2
thedynamicalprocesses. Theunderlyingphysicsforion-
izations of both outer- and inner-shell electrons of com- 0
plex atoms andits relevantdynamics induced by intense 0 10 20 30 40 50 60 70
X-ray lasers are still unclear. Here, we will address this t(fs)
issue related to ionizations of complex atoms driven by
a strongX-raylaser,basedonmasterequationapproach 1
whosevalidityhasbeenverifiedagainstionizationsofhy-
Master
drogen as a benchmark. 0.8
In contrast to simple species such as hydrogen, the n Rate
o
responses of neon subjected to X-ray laser beams for va- ti 0.6
lferenqcueeanncdy fionrnvera-lsehnecleleelleeccttrroonnssiasrceomdipffaerraenbtle, tsoinfcreeeR-farebei pula 0.4 Ne : 1s2 → 1s1 + eǫl
o
excitations, while the frequency for inner-shell electrons P
is much larger than free-free processes. We find that
0.2
inner-shell electrons are ionized much faster than outer-
shell electrons, as shown in Fig. 4 where ionizations of
0
valence (upper panel) and inner-shell electrons (lower
0 2 4 6 8 10
panel)ofneonaretriggeredbyanX-raylaserbeamwith t(fs)
a typicalexperimentalintensityof1018W/cm2 andpho-
ton energy of 875 eV. Comparisons between rate equa-
FIG.4: Ionizationofvalenceelectrons(upper)andinner-shell
tionandmasterequationaremadeandremarkableagree-
electrons(lower)ofaneongasinducedbyanX-raylaserfora
ments are obtained, even though tiny differences can be laser intensity of 1018 W/cm2 with photon energy of 875 eV,
found in the detailed structures, due to coherent oscil-
obtained by master equation (red) and Einstein’s rate equa-
lations for ionizations in master equation approach. As tionapproach(blue),respectively. Wefindthatsingle-photon
far as we know, this is the first direct proof for coher- processes for both outer- and inner-shell electrons dominate
ent ionizations of both outer- and inner-shell electrons ionizations for the recently typical experiments with a laser
of neon in the short-wavelength regime, where sequen- intensity of ≈1018 W/cm2.
tialsingle-photonprocessesdominatetheabsorptionsfor
the recently typical experimental conditions. This con-
clusion is consistent with the recent X-ray experiments, and can smear out coherence-induced signals embedded
where one did not observe clear signatures of multipho- in Rabi oscillations. In this section, we will investigate
tonsingle-electronionizations. Hence it is notsurprising the competitions between Rabi oscillations,photoioniza-
that comparisonsbetweentheories andexperiments sug- tions, Auger and spontaneous decays, since the magni-
gestthedominantabsorptionsofelectronsaresequential tudes of these processes are typically of the same or-
single-photon processes in Young’s experiment [20]. der. These processes, however, are normally beyond the
time revolution of the X-ray experiments. Fortunately,
resonant fluorescence provides another available tool for
C. Coherent dynamics in Auger decay processes studyingthesecompetitionsandunderlyingcoherenceef-
fects.
In addition to ionizations of inner-shell electrons, In our simulations, we study dynamical evolution of
Auger decay is another dominant process for complex ground-state neon subjected to X-ray pulses with dif-
atoms, which carries valuable information about inner- ferent temporal coherence and resonant photon energy
shell electronic structures and dynamical properties of relative to 1s22s22p6 → 1s12s22p63p1 transition. After
atoms, molecules, and solids. Specifically, inner-shell shining the X-ray laser on neon, inner-shell electrons are
electrons can be excited or ionized by the X-ray laser coherently excited and coupled between 1s and 3p or-
beam and a hole atom is formed. The atom is normally bitals. On the other hand, the hole state 1s12s22p63p1
unstableandrelaxesthroughAugerandspontaneousde- is unstable and decays in a fs timescale, mainly due to
cay processes, which typically occurs in the fs timescale Auger decay processes by refilling the inner 1s orbital
10
1111 11 11
((((aaaa)))) 1111ssss22222222ssss22222222pppp6666 ((bb)) ∆∆==00 ((cc)) 11ss2222ss2222pp66
0000....8888 111sss222222sss222222ppp555 00..88 −2eV 00..88 1s12s22p63p1
nnnn nn nn
PopulatioPopulatioPopulatioPopulatio 00000000........46464646 Sa.u.Sa.u.Sa.u.()()()484848000000000000000000000888666000 888666555 888777000 PopulatioPopulatio 0000....4646 Sa.u.Sa.u.Sa.u.Sa.u.Sa.u.()()()()()484848484800000000000000000000000000000000000 PopulatioPopulatio 0000....4646 Sa.u.Sa.u.Sa.u.()()()484848000000000000000000000
0000....2222 ωωω 00..22 888886666600000 888886666655555ωωωωω 888887777700000 888887777755555 00..22 888666000 888666555ωωω888777000 888777555
0000 00 00
0000 5555 11110000 11115555 22220000 22225555 00 55 1100 1155 2200 2255 00 1100 2200 3300 4400 5500 6600
tttt((((ffffssss)))) tt((ffss)) tt((ffss))
FIG. 5: Competition between Rabi oscillations, Auger and spontaneous decays, and photoionizations of a neon gas induced
by an X-ray laser beam with a peak intensity of 1018 W/cm2. (a) Populations of degenerate states 1s22s22p6 (solid) and
1s22s22p5 (dashed) coupled by a flat-topped X-ray laser beam with a resonant photon energy for the 1s → 3p transition,
obtained via master equation (red) and rate equation approach (blue), with the inset for the fluorescence spectrum of the
1s22s22p6 → 1s12s22p63p1 transition. (b) Ground-state populations of Ne for a resonant 1s → 3p (red) and a red detuned
pump ∆ = −2 eV (blue), where the inset shows the fluorescence spectrum of the 1s22s22p6 → 1s12s22p63p1 transition with
the black dashed line being the power-broadening guideline for the central peak. (c) Neon gases driven by a Gaussian X-ray
pulse of a FWHM duration 15 fs, with the inset for fluorescence spectrum of 1s22s22p6 → 1s12s22p63p1 transition (red) and
those driven bya flat-topped pulse with theidentical fluence(blue).
by the outer-shellelectrons, or spontaneous decays of 2p also observe a ≈ 5 eV broadening due to the extreme
and 3p electrons. The hole state can also be further ion- strong X-ray laser with an intensity of 1018 W/cm2, as
ized with double-core forming or via sequential valence shown in the guideline in the inset of Fig. 5(b), which
electrons. Toinvestigatethe interplaybetweenthese dif- is normally referred to as power broadening as will be
ferent processes driven by X-ray pulses, we include all discussed in Sec. IIID. We expect that our discussions
the dominant microscopic processes. We observe a de- provide valuable insight for investigating inner-shell co-
cayed Rabi-flopping structure between 1s and 3p states herent dynamics in the upcoming experiments.
coupled by anX-ray laser,based onmaster equationap-
proach, while it is absent in rate equation method with
amonotonouspopulationchanging. Itindicatesthatthe D. Real coherent dynamics of a neon gas irradiated
atomic coherence is distinctly embodied in Rabi oscilla- by an X-ray laser
tionsdrivenby the X-raylaser,inspite ofextremelyfast
Auger decays in the fs timescale. At present, however,
one lacks reliable tools for observing these fast Rabi os-
cillationsbyreal-timeX-rayimagesexperimentally. For- 00..55 LLYY22001100eexxpp..
tunately, the coherent dynamical information for inner-
shell processes can be extracted from fluorescence spec- 00..44 LLYY22001100tthh..
tra, based on master equation approach.
onon OOCC22001111tthh..
ii 00..33
As shown in the inset of Fig. 5(a), we observe a tt
cc
triple-peak structure for the resonance fluorescence of rara RRaattee
the 1s22s22p6 →1s12s22p63p1 transition,while onlyone FF 00..22
MMaasstteerr
central peak occurs for incoherent light pump. Con-
00..11
sidering the remarkable shift (≈ 2 eV) of the satel-
lite line from the central peak for a laser intensity of
1018W/cm2, the triple-peak spectra provide a valuable 00
11 22 33 44 55 66 77 88 99 1100
tool for studying Rabi floppings and are expected to be
IIoonnssttaaggee
detected via recording photons from spontaneous radia-
tive decays [25]. The next issue is related to the sta-
FIG. 6: Neon charge-state yields by a far off-resonant beam
bility of the triple-peak structure against different ex-
withphotonenergyof800eV.Goodagreementsbetweendif-
perimental conditions, such as laser detuning and pulse
ferent theories[20,30]andexperiments[20]indicatethatco-
shapes. We find that the triple-peak structure is stable.
herence, induced by a far off-resonant laser beam, plays a
Specifically, we observe that the side peaks demonstrate tiny role in the time evolution for the present experimental
asymmetricalongwith a redshift ofthe centralpeak for conditions.
a red-detuned pump, as shown in Fig. 5(b), whereas a
Gaussian X-ray pulse with a FWHM duration of 15 fs Informersections,coherenceeffects havebeeninvesti-
yeids a broadenedtriple-peak structure compared to the gated through ultrafast dynamics of model systems cou-
flat-topped pulse, as shown in Fig. 5(c). Note that we pled by X-ray pulses with different temporal coherence.