Quantum optical signatures in strong-field laser physics: Infrared photon counting in high-order-harmonic generation

0.52 MB·

by  I. A. Gonoskov

Quantum optical signatures in strong-field laser physics: Infrared photon counting in
high-order-harmonic generation
I. A. Gonoskov,1,2, N. Tsatrafyllis,1,3 I. K. Kominis,3 and P. Tzallas1,
∗ †
1Foundation for Research and Technology-Hellas, Institute of Electronic Structure & Laser,
P.O. Box 1527, GR-71110 Heraklion (Crete), Greece
2Max Planck Institute of Microstructure Physics, Weinberg 2, D-06120 Halle, Germany
3Department of Physics, University of Crete, 71103 Heraklion, Greece
6 Weanalyticallydescribethestrong-fieldlight-electroninteractionusingaquantizedcoherentlaser
1 state with arbitrary photon number. We obtain a light-electron wave function which is a closed-
0 form solution of the time-dependent Schr¨odinger equation (TDSE). This wave function provides
2 information about the quantum optical features of the interaction not accessible by semi-classical
n theories. With this approach we can reveal the quantum optical properties of high harmonic gen-
u eration (HHG)process ingases bymeasuring thephotonstatistics ofthetransmitted infrared (IR)
J laserradiation. Thisworkcanleadtonovelexperimentsinhigh-resolutionspectroscopyinextreme-
ultraviolet (XUV)andattosecond science without theneed tomeasuretheXUVlight,while it can
6
pavetheway for thedevelopment of intensenon-classical light sources.
1

]
h Strong-field physics and attosecond science [1–4] have the HHG process. Differently than previous approaches
p been largely founded on the electron recollision process [5–8],wedescribetheXUVemissionasfar-fielddipolera-
-
m described by semi-classical approaches [4] treating the diation by using an initially coherent laser state and the
electron quantum-mechanically and the electromagnetic obtainedclosed-formelectron-laserwavefunction,named
o
field classically. This is because the high photon number ”quantum-optical Volkov wave function”. Our approach
t
a limit pertinent to experiments with intense laser pulses consistently extends the well-known semi-classical theo-
.
s appears to be adequately accounted for by a classically- ries [4], since from the obtained quantum-optical wave
c
described electromagnetic wave,which is not affected by function we can retrieve the semi-classical Volkov wave
i
s the interaction. functions by averaging over the light states. This is of
y
h In the semi-classical approaches (known as three-step advantage, since all the results of the semi-classical the-
p models)usedfor the discriptionofthe HHG process,the ory (like harmonic spectrum, electron paths, ionization
[ electron tunnels through the strong-laser-field-distorted times, recombination times, etc.) can be retrieved from-

atomic potential, it accelerates in the continuum under and utilized in our quantized-field approach.
2
v the influence of the laser field and emits XUV radiation Goingbeyondthereachofthesemi-classicalapproach,
4 upon its recombination with the parent ion. Thus, the we find that the quantum-optical properties of the HHG
6 motion of the electron in an electromagnetic field is at process are imprinted in measurable photon statistics
7
the core of the recollision process. In the strong-field of the transmitted IR laser field, thus accessing HHG
2
regime, this motion is well described by non-relativistic dynamics does not require measuring the XUV radia-
0
. semi-classicalVolkov wavefunctions,obtained by solving tion. This is a unique advantage of our work since our
1
the TDSE for a free electron in a classically-described proposed measurements, dealing with high-resolution
0
6 electromagnetic field. spectroscopy in XUV and attosecond science, can be
1 Extending the semi-classical Volkov wave functions performed in open air without the need for specialized
v: into the quantum-optical region is non-trivial and, to optics/diagnostics required for the characterization of
i our knowledge, a closed form solution of the quantized the XUV radiation. Additionally, it has been found
X
TDSE with a coherent-state light field has never been that the interaction of strong laser fields with gas phase
r obtained before. Although an accurate calculation of media leads to the production of non-classical high
a
the properties of the XUV radiation emitted from a gas photon number light states.
phase medium requires the consideration of the driving
IR laser bandwidth and the propagation effects in the Full-quantum theoretical description of light-
atomicmedium, the fundamentalpropertiesofthe inter- electron interaction.
action can be adequately explored with the single-color The non-relativistic TDSE of an electron interact-
single-atominteraction,as has been done in the pioneer- ing with a single-mode long-wavelegth lineary-polarized
ing work of Lewenstein et al.[4]. In this work we de- quantized light field of frequency ω reads (in atomic
velop a quantized-field approach for an ionized electron units)
interacting with light field in a coherent state. We ob-
tain a closed-form solution of the TDSE, which contains i∂Ψ = 1 p Aˆ 2Ψ , (1)
complete information about the laser-electron quantum ∂t 2h − i
dynamics during the interaction, and use it to describe where p and Aˆ=−√β2 aˆe−iωt+aˆ+eiωt are the electron
(cid:0) (cid:1)
2
momentum and vector potential scalar operators along photon number limit demonstrates that the behavior of
thepolarizationdirection. Thecreationandannihilation the electron in a strong laser field can be accurately de-
operators are aˆ+ = 1 q ∂ and aˆ = 1 q+ ∂ , scribedbysemiclassicaltheorieswithnegligiblequantum
√2(cid:16) − ∂q(cid:17) √2(cid:16) ∂q(cid:17) corrections. However,ourfullquantum-opticalapproach
respectively, q is the in-phase quadrature of the field
can provide information about the IR laser field states
[9, 10], and β = c 2π/ωV is a constant determined by
during the interaction, inaccessible by the semi-classical
thequantizationvpolumeV,frequencyω,andlightveloc-
theories. This information can be experimentally
ity c. The detailed derivation of the analytical solution
extracted using balanced homodyne detection [11–13]
ofEq. (1),termedquantum-opticalVolkovwavefunction,
of the IR laser field transmitted from the harmonic
will be given elsewhere. Here we provide the result, the
generation medium.
validity of which can be checked by direct substitution
into Eq. (1). Based on this, we then analyze its fun-
Quantum-opticaldescriptionoftheHHGprocess.
damental features and their consequences for the HHG
Using the quantum-optical Volkov wave functions Ψ,
process. The closed-form solution of Eq. (1) reads: ′
the time evolution of the HHG process is described by
the following wave function
Ψ(p,q,t)=C0pM(t)ψ0(p)ea(t)q2+b(t)p2+d(t)pq+f(t)p+g(t)q+c0+c(t),
(2) Ψ˜ =agΨg+ biΨ′(p,q,t ti), (4)
where the functions a(t),b(t),...,g(t),M(t) are given in −
Xi
terms of the parameters of Eq.(1) in the Methods Sec- where Ψ = ψ ψ is the initial state of the system,
g g c
tion. The solution includes an arbitrary initial elec- ψ is the ground state of the electron, ψ is the ini-
g c
tron distribution ψ0(p) and an arbitrary initial photon tial coherent light state and Ψ′(p,q,t ti) are the con-
−
number N0 and the field phase θ. The wavefunction tinuum laser-electron states having different ionization
Ψ(p,q,t)providesthefullquantum-opticaldescriptionof times t . The complex amplitudes a and b satisfy the
i g i
the electron-lightinteraction. The termd(t)pq inthe ex- normalization condition Ψ˜ Ψ˜ = 1. In this case, the
ponentrenderstheelectronicandlightdegreesoffreedom time dependent dipole mhom|enit is r(t) = Ψ˜ rˆ Ψ˜ =
non-separable. In the high photon number limit where ∞ ∞ Ψ˜rˆΨ˜∗dqdp. As in the semi-classiahl th|eo|ryi[4],
AN00 →is ∞th,eβa→mp0li(tVud→e o∞f )t,haendcoωcrrβe√sp2oNn0di→ngωccAla0ss(iwcahlelyre- Rcwo−en∞tniRne−gul∞uemct grΨou′nrˆd-Ψg′rountrdanhsΨitgio|nrˆs|,Ψgaindancdoncsoindteirnuounmly-
described vector potential A = A0cos(ωt+θ)), Eqs.(2) ground-to-cohntin|u|umi(and continuum-to-ground) tran-
is simplified (see Methods) to Ψ′(p,q,t) maintaining all sitions given by the matrix elements Ψ′ rˆ Ψg (and
the quantum-optical properties of Eqs.(1). Ψ rˆ Ψ ). h | | i
g ′
h | | i
Acrucialpropertyofthequantum-opticalVolkovwave Integrating Ψ˜ rˆΨ˜ over q, using Eq. (4) and inte-
functionisthatthematrixelementsofanyq-independent h | | i
grating over p, we retrieve that r(t) ψ rˆ ψ which
operator Rˆ coincide with the matrix elements obtained ∝ V g
coincides with the expression given by(cid:10)the(cid:12)sem(cid:12) i-c(cid:11)lassical
fromusing the well knownsemi-classicalelectronVolkov (cid:12) (cid:12)
theories. Thus,allthesemi-classicalresults[4,14–16],in
wave functions ψ i.e.
V particularthe short(S) andlong (L)electronpaths with
electronicVolkovwavefunctionsψS andψL respectively,
Ψx Rˆ Ψ′ ψx Rˆ ψV , (3) can be consistently used in the prVesent aVpproach. In a
h | | i−→h | | i
similar way, the same results can be obtained using the
where Ψ is an arbitrarily chosen electron-light wave IR wave functions ψ (see Method section).
x l
function and ψx is the corresponding state of the elec- A scheme which can describe the HHG process in the
tronincaseofclassically-describedelectromagneticfield. contextofthepresentmodelisshowninFig.1. Although
Thus, while Ψ′(p,q,t) goes beyond the semi-classicalap- the quantizationof the harmonicfield is notrequiredfor
proachtocompletelydescribethequantizedelectronand this work and thus was not considered in the previous
light interaction, it naturally reproduces the classical formalism, harmonic photons are included in Fig. 1
Volkov states after integrating over q. This has pro- for a complete understanding of the process. Fig.1a
found consequences for the description of HHG, since shows the electron states in case of integrating over q
the well known results of the semi-classical models [4] and Fig.1b shows the field states in case of integrating
can be retrieved, and more than that, utilized in our over p. The horizontal black lines in Fig.1a and 1b are
quantized-field approach. In the recollision process, Ψx the initial states of the electron ψg and IR-laser field
is the groundstate of the system (ψgψc), Rˆ is the dipole ψc with energy IP < 0 and W| ligiht(0), respectively.
| i
moment(rˆ), ψx isthegroundstateoftheatom(ψg)and At t > 0 the system is excited (small red arrows) in
ψc istheinitialcoherentlightstate. Detaileddescription an infinite number of entangled laser-electron Ψ′i states
ofaboveconsiderationscanbefoundintheMethodsSec- (gray area), resulting in a reduction of the average
tion. laser energy (small downwards red arrows in Fig. 1b)
The calculation of the dipole moment in the high and the enhancement of the average electron energy
3
Counting IR photons in HHG.
The probability for measuring n photons in a non-
interacting coherent light state is given by P = K 2,
n n
| |
where K is a probability amplitude appearing in
n
the expansion Ψ = ∞ K n , in terms of photon-
n
| i
nP=0
number (Fock) states. In this expression, K is time-
n
independent, since, as well known[18], the photon prob-
abilitydistributionin acoherentstate isconstantwithin
the cycle of the light field (Fig. 2a). When the coher-
ent light state is interacting with a single atom towards
the generation of XUV radiation, the probability dis-
tribution becomes time dependent, since Ψ(p,q,t t )
′ i
−
is changing at each moment of time within the cycle of
the laser field due the interaction with the ionized elec-
tron. Inthiscase,theprobabilitydistributionisgivenby
2
P (t) = K(kΩi,Ωi)(p(Ωi)(t),t,t(Ωi),t(Ωi)) , where
n (cid:12) n i i r (cid:12)
(cid:12)(cid:12)PΩi kPΩi (cid:12)(cid:12)
t(Ωi) and(cid:12)t(Ωi) aretheionizationandrecombinat(cid:12)iontimes
i (cid:12)r (cid:12)
of the corresponding electron paths k =S ,L with
Ωi Ωi Ωi
momentum p(Ωi)(t) which lead to the emission of XUV
i
radiation with frequency Ω (see Methods Section). The
i
parameterst(Ωi),t(Ωi) andp(Ωi)(t)areobtainedusingthe
i r i
3-step semi-classicalmodel [4].
FIG. 1. Excitation scheme for the quantum-optical descrip- In reality, an intense Ti:S femtoseond (fs) laser pulse
tion of the HHG process. (a) A schematic representation with 1017 photons/pulse (which corresponds to N
0
of the electron states in case of integrating over laser-state ∼ ∼
1012 photons/mode for a laser system based on a 100
parameter q. (b) A schematic representation the laser field
MHzoscillatorwhichdeliverspulsesof 30fsduration),
states in case of integrating overelectron momentum p. ∼
interactswithgas-phasemediumtowardstheemissionof
XUV radiation. In this case, where n(Qi) atoms coher-
a
ently emit XUV radiation with frequencies proportional
(small upwards red arrows in Fig. 1a). The Ω-frequency
(Q = Ω /ω) to the frequency of the IR laser, the in-
emission is taking place by constructive interference i i
teraction is imprinted in the photon number N¯ of the
of ψ states and recombination to the ground state
(dowVnwards red arrows in Fig.1a). In this case the final IR field as N¯ = N0 na(Qi)Qi, reflecting energy conser-
−
laser energy remains shifted by ~Ω compared to the vation. Since the signal of interest, na(Qi)Qi, is super-
initial energy Wlight(0). When the ψV states interfere imposed on a large background (≈ N0), a balanced in-
destructively the probability of Ω-frequency emission terferometer [11–13] is required in order to subtract the
is reduced and the average laser energy returns to the initial IR photon number N0 from N¯ and thus measure
initial value (black dashed arrows in Fig.1b). Among ∆N = N0 N¯ = na(Qi)Qi = C(Qi)nintQi. The num-
−
the infinite Ψ states, two are the dominant surviving ber of atoms interacting with the laser field is n and
′i int
the superposition,correspondingto the S and L electron C(Qi) is the conversionefficiency of a single-XUV-mode,
paths, described by ψS and ψL , respectively. which depends on the gas density in the interaction re-
| V,Ωi | V,Ωi
Correlated to them are the IR-laser states ψS and gion. Taking into account that for gas densities 1018
ψL (where ψS and ψL are the IR wave| flu,Ωncitions atoms/cm3 the conversion efficiency is 10−4 (f∼or Ar-
| l,Ωi l l gon, Krypton,Xenon in the 25-eVphoto∼n energy range)
correspond to the S and L electron paths, respectively)
, as well as the Ω-frequency states ψS and ψL , [19–23], it can be estimated that ∆N ranges from 0
| Ωi | Ωi (for zero gas density) up to 109 photons/mode ∼(for
respectively. This is consistent with the interpretation
∼
gas density 1018 atoms/cm3). Although the study is
of recent experimental data [17]. It is thus evident that
∼
validfor allnoble gases,in the following we will describe
by measuring quantum optical properties of the IR-light
the HHG process considering Xenon atoms interacting
we can access the full quantum dynamics of the HHG
process. Such properties, in particular photon statistics with a coherent IR laser field in case of low (na(Qi) =1),
to be discussed next, are not accessible by semi-classical intermediate (n(Qi) = 100,500) and high (n(Qi) = 108)
a a
models [1–4]. number of emitting atoms. ˆe For a single recollision, the
dependence ofP (t) ontime during the processis shown
n
4
FIG. 2. Probability distribution of the IR photons during the HHG process. (a) Probability distribution of a non-interacting
coherent IR laser state for N = 800. This is shown only for reasons of comparison with the interacting coherent laser
0
states. (b) Time dependence of the IR probability distribution during the recollision process calculated for na(Qi) = 100 and
I =1014W/cm2. In the calculation, the electron momentum, the ionization and the recombination times have been obtained
l
by the 3-step semi-classical model. (c) Expanded plot of Fig. 2b in the time interval 0.5TL < t < TL (d) IR photon number
absorbed by Xenon atoms duringthe recollision process. This has been obtained by then=n position of the peak of the
peak
distribution at each moment of time. (e) Overall IR photon number absorbed by the atoms during the recollision (red solid
line). This has been obtained after integration over the cycle of the IR field. The XUV spectrum shown in blue dashed line,
obtained using thesemi-classical 3-step model.
in Figs. 2b,c for n(Qi) = 100. It is seen that in the time classical three-step model (blue dashed line), including
a
interval 0 < t < t 300 asec where the ionization is the plateau and cut-off regions. Thus, we demonstrated
i
≈
taking place, the peak of the probability distribution is that all known features of the semiclassical three-step
located at n = n 800. Since the ionization of one model are imprinted in IR photon statistics.
peak
≈
Xenon atom requires the absorption of n 8 IR pho- We will now explore the new phenomena and poten-
≈
tons, this value corresponds to the energy absorbed by tialmetrologicalapplicationsonecanaddressutilizingIR
100 Xenon atoms. For t > t the variation of the IR photon statistics. To this end, we first elaborate on the
i
photonnumber n with time reflects the energy exchange atom-numberdependenceofP . WhilethenumberofIR
n
between the IR laser field and the free electron. The photonsabsorbedbythesystemisproportionalton(Qi),
a
peak of the probability distribution during the recolli- the width w (t) of the probability distribution is deter-
sion is located at n= n =Ω /ω, t=t(Ωi) (Fig. 2d). na
peak i r minedbyGaussianstatistics,w (t) n(Qi) (Figs3a).
This is due to the energyabsorbedby Xenonatoms dur- na ∝q a
However,thedistributionisdepartingfromtheGaussian
ing the recollision process towards the emission of XUV
statistics during the recollision process. This is clearly
radiation with frequency Ω at the moment of recombi-
i showninFig. 3bwhichdepictsincontourplotofthenor-
nation t(Ωi). Importantly, in Fig. 2d we demonstrate
r malizedprobabilitydistributionofFig3a. Forreasonsof
that the absorbed IR photon number reveals the funda-
comparison,a Gaussiandistribution is shownin Fig. 3c.
mental properties of the three-step semi-classical model:
The distortion of the probability distribution in Fig. 3b,
S and L paths lead to the emission of the same XUV
more pronounced in the time interval 0.5T <t<T , is
L L
frequency and degenerate to a single path in the cut-off
associated with energy/phase dispersion of the interfer-
region. Furthermore, as shown in Fig. 2e, the overallIR
ing electron wavepackets in the continuum, alluding the
photon number distribution (red solid line) reproduces
possibilities of producing non-classicallight-states.
the well-known XUV spectrum resulting from the semi-
Formulti-cyclelaserfield,theprocessisrepeatedevery
5
cantadvantagesforhighresolutionspectroscopyinXUV
and attosecond science. In Fig. 4c (left panel) we show
thedependenceoftheP ontheintensityofthelaserfield
n
(I =ε E 2/2 N ) and n for n(Q˜) =108 (for simplic-
l 0 0 0 a
ityweco|nsi|dero∝nlythecaseofQ˜ =11, 13, 15). Indeed,
theharmonicspectrumcanbeobtainedfromthemaxima
of P centered at ∆N = Q˜ n(Q˜). The spacing between
n a
the maxima δ(∆N) = n(Q˜)·δQ˜ with δQ˜=2 for consecu-
a
tive harmonics, and the width w = √2∆N = 2n(Q˜)Q˜
a
q
depend on C(Q˜) and n . The resolving spectral power
int
P = ∆N/w = ∆N/2 = λ /δλ increases with n
R Ω Ω int
and for values ofp∆N 109 photons, P can reach the
R
∼
valuesof 104 105inthespectralrangeof25eV,which
∼ −
competeswithstate-of-the-artXUVspectrometers. This
is shown in Fig. 4c (right panel) where the probability
distribution around Q˜ = 15 has been calculated in case
of recordingthe 799.95nm,800.00nm and 800.05nm IR
modes of a Ti:S laser pulse. This measurement can be
performed by collecting the photons of the IR modes of
the spectrally resolved multi-color IR pulse. This can
be done by means of an IR diffraction grating placed af-
ter the harmonic generation medium. This figure also
depicts the broadening effects introduced in a measured
distribution by the bandwith of the driving IR pulse in
caseofcollectingmorethanonemodesofthemulti-mode
laser pulse.
When ∆N is reduced, the probability distribution is
getting broader (Fig. 4d, left panel), while at the point
where the probability distribution between the consec-
utive harmonics overlaps, an interference pattern asso-
ciated with the relative phase between the consecutive
harmonics appears in Fig. 4d (right panel). Addition-
FIG.3. Generation oflight states with non-Gaussian photon
ally, the modulation of P with the intensity of the laser
n
distribution. (a) Time dependence of the IR probability dis-
field (clearly shown in the left panels of Figs. 4c,d) re-
tribution during the recollision process calculated na(Qi) = 1 flects the effect of the S and L path interferences in the
andI =1014W/cm2. Inthecalculation,theelectronmomen-
l contextofFig. 1,i.e. the maxima(minima)ofP versus
tum, the ionization and the recombination times have been n
N correspondtothoseIR-laserintensities N ,forwhich
obtainedbythe3-stepsemi-classicalmodel. (b)Contourplot 0 0
of the normalized probability distribution of Fig. 3a. (c) ΨV interferes destructively (constructively). These ob-
Contourplotofthenormalizedprobabilitydistributionwhich servationscanbeusedforattosecondscienceandmetrol-
follows the Gaussian photon statistics. This has been calcu- ogy,to be exploredindetail elsewhere. Since the photon
lated using a single electron path which contributes to the statisticsmeasurementsaresensitivetoshot-to-shotfluc-
emission of a monochromatic XUV radiation with frequency
tuationsoftheIRintensity,stablelasersystemsorIRen-
Qi=ω/11. Itisevident,thatincaseofreducingthenumber ergy tagging approaches are required in order to be able
of emitting atoms form na(Qi) =100 (Fig. 2b,c) to na(Qi) =1 to record an ”IR photon statistics spectrum”. Addition-
thewidth of the probability distribution is increasing.
ally, in order to avoid the influence of the laser intensity
variationalongthepropagationaxisintheharmonicgen-
erationmedium,agasmediumwithlengthmuchsmaller
half-cycleofthelaserperiod. Inthiscasetheprobability
comparedto the confocalparameter of the laser beam is
distribution consists a series of well confined peaks (Fig.
required. Any influence of the intensity variation along
4a,b) appearing at positions n = Q˜ = Ω/ω and reflects
the beam profile at the focus can be minimized (in case
the formationofwellconfinedhighorderharmonics(Q˜).
that is needed) using spatial filtering approaches where
Additionally, the atom-number dependence of the IR the IR photons of the specific area on the focal spot di-
photon distribution in the HHG process provides signifi- ameter can be collected.
6
FIG. 4. High resolution spectroscopy in XUV and attosecond science using IR photon statistics. (a) Probability distribution
for multi-cycle laser interaction calculated for na(Qi) = 500 and Il = 1014W/cm2. For this graph three laser cycles have been
considered. (b) ”IR photon statistics spectrum” obtained by time integrating the Fig. 4a. (c) (left panel) Dependence of Pn
on the laser intensity I (∝ N ) and on photon number n for Q˜ = 11, 13, 15 and n(Qi) = 108. The right panel shows the
l 0 a
probabilitydistributionaroundQ˜ =15incaseofrecordingthe799.95nm,800.00nmand800.05nmIRmodesofthelaserpulse
afterpassingthroughthegasmedium. (d)DependenceofPn onthelaserintensityandonphotonnumbernforQ˜ =11, 13, 15
and for na(Qi) = 100 (left panel) and na(Qi) = 10 (right panel). The dashed vertical lines depict the cut-off positions of the
harmonics. In these plots na(Qi) were taken independentof Il.
CONCLUSIONS XUV and attosecond science, can be performed without
the need for specialized XUV equipment ( gratings,mir-
rors, high vacuum conditions e.t.c.). Additionally, we
Concluding, we have developed a quantized-field ap-
have found that the HHG process in gases can lead to
proach which describes the strong-field light-electron in-
non-classicalIR lightstates. In general,this work estab-
teractionsusingaquantizedcoherentlaserstatewithar-
lishesapromisingconnectionofstrong-fieldphysicswith
bitrary photon number. The description is based on the
quantum optics.
quantized-Volkov light-electron wave function resulting
from the closed-form solution of TDSE. The obtained
wave function provides information about the quantum METHODS
optical features of the interaction, which are not accessi-
blebythesemi-classicalapproachesusedsofarinstrong-
On the closed-form solution of TDSE: In order to
field physics and attosecond science. The approach has
obtain a closed-form solution of Eq. (1) of the main
been used for the description of HHG in gases. We have
text of the manuscript, we consider as an initial state,
foundthatthe quantumopticalfeaturesofthe HHGcan
a state where the electron is decoupled from the light
be unraveled by measuring the photon statistics of the
i.e. Ψ = Ψ(p,q,t = 0) = ψ (q)ψ (p) to be a separable
0 c 0
IR laser beam transmitted from the gas medium with- product of a coherent state of light ψ (q) = eλq2+g0q+c0
out the need of measuring the XUV radiation. This is c
and an arbitrary field-independent electron state ψ (p)
a unique advantage of the work since our proposedmea- 0
surements, dealing with high-resolution spectroscopy in in momentum representation, where λ = 1 1+ β2 is
−2q ω
7
the parameter which introduces the light dispersion due of the manuscriptcan be rigorouslyproved,since the in-
to the presence of the electron [24, 25], g0 = √2N0e−iθ tegrationoverq, ∞ ∞ Ψ RˆΨ dpdq,leadstothedelta
x ′∗
carries the information about the phase of the light θ,
R R
N0 is the average photon number of the initial (t = 0) function δ q √−2∞N−0∞cosθ , and results in integral over
−
coherent light state, and c0 is a normalizing constant. p which is(cid:0)equal to ψ Rˆ(cid:1)Ψ .
x V
The parametersappearing inEq. (2) ofthe main textof h | | i
On the description of HHG using the IR wave
the manuscript are
functions: The results obtained by the semi-classical
theories regarding HHG can be also obtained by inte-
1 1
a(t)= + +λ Me2iωt, grating Ψ˜ rˆΨ˜ over p, using Eq. (3) and integrat-
−2 (cid:18)2 (cid:19) h | | i
ing over q. In this case the dipole moment r(t)
b(t)= it 1 β2 + β2 (1 e2iωt) ψlV ψc is expressed in terms of the corresponding t∝o
− (cid:18)2 − 8λ2ω(cid:19) 8λ3ω2 − (cid:10)the (cid:12)Vol(cid:11)kov-electron states IR wave functions ψV =
(cid:12) l
+ 32λβ32ω2(1−e4iωt)+ 2(1γ2m2λ), sRtaΨt′e(po,fq,tth−e teVle)c∇trponψg∗i(np)mdopm(ewnhteurme ψregp(rpe)seinstaatigorno,untVd
− are the ionization times Volkov electron paths con-
d(t)=γMeiωt,
tribute to the harmonic generation). The IR wave
g0γe2iλωt g0ωγ2 functions which correspond to the S and L electron
f(t)= m ,
(1−2λ) − 2β paths are ψlS = Ψ′(p,q,t−tS)∇pψg∗(p)dp and ψlL =
g(t)=g0Me2iλωteiωt, Ψ′(p,q,t−tL)∇Rpψg∗(p)dp,respectively,withtS andtL
Rbeing the ionization times of the short and long electron
1 g2 g2e4iλωt
c(t)=iωt +λ 0(1 e4iλωt)+ 0 m paths.
(cid:18)2 (cid:19)− 8λ − 2(1 2λ) On the calculations of the IR probability dis-
−
where γ = β (1 e2iλωt), M = tribution: The probability to measure n photons in
2λω −
(1 λ)+(1 +λ)e2iωt −1, m = 1 Me2iωt, a non-interacting light field state Ψ is Pn = Kn 2,
g(cid:2)02=− √2N02e−iθ, N0 (cid:3)= ψc aˆ+aˆ ψc and −C0 is nor- where Kn is a probability amplitude appearing i|n th|e
h | | i
malization constant. From the general solution (2) expansion Ψ = ∞ K n , in terms of photon-number
n
| i
we can recover energy conservation, i.e. the instan- nP=0
(Fock) states. In q-representation the Fock states are
taneous interaction energy of the electron is given by
exp( q2/2)
We,int(t) = We(t) − We(0) = Wlight(0) − Wlight(t), written as |ni = π1/42−n/2√n!Hn(q), where Hn(q) =
where We(0) is the initial kinetic energy of the electron, 2nexp[ 1 ∂2 ]qn areHermitepolynomials. Forcoherent
Wlight(0) = ω 1+ βω2 21 + N0 is the initial energy lightst−at4es∂q[92],thephotonstatisticsaredescribedbythe
oisf tthhee lifigehldt fieeqnlder,gyWliaghtht(ta)ny=miωoqme1n+t βωo2fh12tim+eN¯a(tn)di PbyoisasonGdaiusstsriiabnutioPnn Pn≈=√N2nπ10n!Ne0−Nex0,pw−el(ln-a−2pNNp00r)o2ximwahteend
N¯(t) = N = Ψ aˆ+aˆ Ψ . In the high photon N0 1. In case of HHG process(cid:2), the pro(cid:3)bability
number limh itithe qh-de|pend|enit part of the total wave distr≫ibution during the recollision process for a single
function becomes exponentially small everywhereexcept path i of ionization time ti and electron momentum
the region around q √2N0. Thus, Eq. (2) of the pi(t) which contributes to the production of XUV ra-
main text of the ma|n|us≈cript leads to diation with frequency the harmonic Ωi is given by
2
Ψ′(p,q,t)=C0′ψ0(p)ea′(t)q2+b′(t)p2+d′(t)pq+f′(t)p+g′(t)q, Pn = (cid:12)(cid:12)Kn(i)(t)(cid:12)(cid:12) , wheren Kni(t) ≈ ci(t)√1n!(cid:20)√N0e−iθi +
where now1the1pβa2rameters in the exponent are A2ω0√piN(t0)(cid:12) 1−eiω(cid:12)(t−tr) (cid:21) = ci(t)K˜ni(t) is determined
a′(t)= e2iωt+O(β4), (cid:0) (cid:1)
−2 − 4 ω through the expansion Ψ(p (t),q,t ) = ∞ Ki(t) n .
1 ′ i i n | i
b′(t)=−it2 +O(β2), ci(t)aren-independentcomplexnumberspnPr=o0portionalto
d(t)= β(eiωt 1)+O(β2) theq-independentpartoftheΨ′ andθi =ω(t−ti)isthe
′ −ω − phase of the laser field at the moment of ionization. In
f (t)= 1A e iθ(1 cosωt)+O(β), the high photon number limit, ci(t)K˜ni(t) → Ai(t)eiΦi
′ −ω 0 − − (where Ai(t) is real), and the probability distribution
g′(t)= 2N0e−i(cid:16)θ+β22t(cid:17)(cid:20)1− 41βω2(1−e2iωt)(cid:21)+O(β3) reads Pn =(cid:12)Ai(t)eiΦi(cid:12)2, with
and C0′ ips normalization constant. Φi ≈−(t−t(cid:12)r)[pi(2t)]2 +(cid:12) pi(tω)A0 sin(θi(1,2))[1−cos(ω(t−tr))]
On the validity of Eq. (3): Eq. (3) of the main text −pi(t)A0 n cos(θi)sin(ω(t−tr))
ω N
0
8
where An(t)2 exp (n N¯(t))2/2N¯(t) and N¯(t) = tween quantum Volkov states. Phys. Rev. A 61, 043812
| | ∝ − −
N0 ξ(t)istheaverage(cid:2)numberofphotonsd(cid:3)uringtherec- (2000).
olli−sion,withξ(t)=Ω(t)/ω andΩ(t)=((p2(t)/2) IP). [8] Hu, H., and Yuan, J., Time-dependent QED model
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When multiple paths contribute to the emission of mul-
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SΩi,LΩi denotes the e(cid:12)(cid:12)lectron paths cont(cid:12)(cid:12)ribute to the [10] Schleich, W. P., Quantum optics in phase space, (John
emissionoftheΩ frequency. Sincetheprobabilitydistri-
i Wiley & Sons, 2001).
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ACKNOWLEDGMENTS
testofcoherentstateinvariancebysingle-photonannihi-
We acknowledge support by the Greek funding pro- lation, New. J. Phys. 10, 123006 (2008).
gram NSRF and the European Union’s Seventh Frame- [14] Bellini, M., et al. Temporal Coherence of Ultrashort
work Program FP7-REGPOT-2012-2013-1 under grant High-Order Harmonic Pulses. Phys. Rev. Lett. 81, 297,
agreement 316165. (1998).
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theoretical calculations shown in the figures and con-
[17] Kominis, I. K., Kolliopoulos, G., Charalambidis D., and
tributed on the data analysis; I.K.K. contributed on the
Tzallas, P., Quantum-optical nature of the recollision
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