Weak-field multiphoton femtosecond coherent control in the single-cycle regime

Weak-field multiphoton femtosecond coherent control in the single-cycle regime

Lev Chuntonov,1 Avner Fleischer2 and Zohar Amitay* The Shirlee Jacobs Femtosecond Laser Research Laboratory, Schulich Faculty of Chemistry, Technion – Israel

Institute of Technology, Haifa 32000, Israel 1Currently with Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel

2Currently with Faculty of Physics, Technion – Israel Institute of Technology, Haifa 32000, Israel *[email protected]

Abstract: Weak-field coherent phase control of atomic non-resonant multiphoton excitation induced by shaped femtosecond pulses is studied theoretically in the single-cycle regime. The carrier-envelope phase (CEP) of the pulse, which in the multi-cycle regime does not play any control role, is shown here to be a new effective control parameter that its effect is highly sensitive to the spectral position of the ultrabroad spectrum. Rationally chosen position of the ultrabroadband spectrum coherently induces several groups of multiphoton transitions from the ground state to the excited state of the system: transitions involving only absorbed photons as well as Raman transitions involving both absorbed and emitted photons. The intra-group interference is controlled by the relative spectral phase of the different frequency components of the pulse, while the inter-group interference is controlled jointly by the CEP and the relative spectral phase. Specifically, non-resonant two- and three-photon excitation is studied in a simple model system within the perturbative frequency-domain framework. The developed intuition is then applied to weak-field multiphoton excitation of atomic cesium (Cs), where the simplified model is verified by non-perturbative numerical solution of the time-dependent Schrödinger equation. We expect this work to serve as a basis for a new line of femtosecond coherent control experiments. ©2011 Optical Society of America OCIS codes: (020.4180) Multiphoton processes; (020.1670) Coherent optical effects.

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#137409 - $15.00 USD Received 18 Feb 2011; revised 25 Feb 2011; accepted 28 Feb 2011; published 25 Mar 2011(C) 2011 OSA 28 March 2011 / Vol. 19, No. 7 / OPTICS EXPRESS 6865

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1. Introduction

Femtosecond laser pulses are unique tools to control multiphoton excitation processes in atoms and molecules [1–31]. The control over the corresponding initial-to-final state-to-state transition probabilities is achieved by shaping the pulse [32–34] to manipulate the interferences among the coherent manifold of state-to-state multiphoton quantum transitions photo-induced by the broadband ultrashort pulse. The pulse-shaping control knobs are generally the relative phase, amplitude, and polarization of different spectral components of the pulse. Here we focus on rational coherent control approach, where the femtosecond pulse shaping is based on the initial identification of the interfering transitions and their interference mechanism. Such an approach has been shown to be very powerful when the photoexcitation picture is available in the frequency domain [1–31], which is possible when a corresponding perturbative description of finite order is valid. The weak-field regime corresponds to a valid description by perturbation theory of the lowest non-vanishing order. For two-photon processes it is second-order perturbation theory, while for three-photon processes it is third-order perturbation theory.

Over the last decade weak-field multiphoton femtosecond coherent control has been demonstrated very successfully in many studies that have been conducted in the multi-cycle femtosecond regime [1–31], where the pulse duration of the (unshaped) transform-limited pulse is of many optical cycles. A natural way to further enhance the possible control is to enrich the variety of multiphoton transitions by a significant broadening of the exciting pulse spectrum up to an octave-spanning corresponding to the single-cycle regime [35–41]. There, the transform-limited pulse duration is of about one optical cycle. The shaping of such ultrabroadband pulses has recently been reported [40, 41]. As opposed to the multi-cycle regime, where the electric field is effectively determined only by the instantaneous temporal frequency and temporal envelope profile, in the single-cycle regime it is also strongly affected by the global phase between them. This phase is referred to as the carrier-envelope phase (CEP) and is also equal to the global phase of the spectral field at the different pulse frequencies. During the process of coherent spectral broadening and creation of single-cycle pulses, the CEP is experimentally stabilized to preset values [35–39]. The CEP has shown to play a significant role in multiphoton processes such as photoionization, high harmonics generation, and others [42–49].

Here we study for the first time weak-field multiphoton coherent control using shaped ultrabroadband femtosecond pulses having a bandwidth of the single-cycle regime. We


investigate phase control of atomic bound-bound non-resonant multiphoton transitions. The CEP of the pulse, which in the multi-cycle regime does not play any control role, is shown here to be a new effective control parameter that its effect is highly sensitive to the spectral position of the ultrabroad spectrum. Rationally-positioned ultrabroad spectrum generally induces coherently several groups of multiphoton transitions from the ground to the excited state of the system: transitions involving only absorbed photons as well as Raman transitions involving both absorbed and emitted photons. For example, for a two-photon excitation process, one group is of transitions involving two absorbed photons and the other group is of Raman transitions involving one absorbed photon and one emitted photon. The intra-group interference is controlled by the relative spectral phases of the different frequency components of the pulse, while the inter-group interference is controlled jointly by the CEP and the relative spectral phases.

The corresponding effect and control principle are as follows. For N-photon excitation process, the global phase associated with all the transitions of N absorbed photons is N× CEP, while the global phase associated with all the transitions of M absorbed photons and N−M emitted photons is (2M−N) CEP. Thus, overall, the CEP contributes a value of 2(M−N)

CEP to the relative phase between the excitation amplitudes photo-induced by these different groups. This value is added then to the relative phase between the excitation amplitudes as determined by the intra-group interferences taking place separately within each of the transition groups. The higher is the order of the multiphoton excitation (N), the smaller is the minimal ultrabroad bandwidth allowing such a control. In the multi-cycle regime, the spectrum is not broad enough to photo-induce several types of N-photon transitions; it can induce only a single type. Thus, no CEP effect is possible in the multi-cycle regime.

××

Specifically, the present study includes the investigation of non-resonant two- and three-photon excitation in a simple model system within the perturbative frequency-domain framework. Then, the developed intuition is applied to weak-field multiphoton excitation in atomic cesium (Cs), where the simplified model is verified by non-perturbative numerical solution of the time-dependent Schrödinger equation.

2. Two-photon theoretical perturbative description

Consider the simple model system shown in Fig. 1(a), with a weak-field non-resonant two-photon transition from the ground state g of energy to the excited state gE f of

energy fE , with the corresponding transition frequency of ( ) /Ω = −fg f g . The E E g and

f states are coupled via a manifold of intermediate states n (not shown in Fig. 1(a)) of the proper symmetry that are not accessed resonantly by the excitation. For a weak excitation pulse, the final amplitude (2)

fA of f afdependent perturbation theory as [9, 10]

ter the pulse is over is given by second-order time-

∝ Ω∫f fg( )(2) 2 ( )exp ,∞

A−∞

t i t dtε

with the temporal electric field of the pulse ( )tε given by

( )01( ) ( )exp . .,2

= +⎡ ⎤⎣ ⎦CEt E t i t c cε ω φ +

i

where ( )E t s the (Gaussian) temporal complex amplitude of the pulse, 0 ω is the carrier frequency, CEφ is the carrier-envelope phase (CEP), and c.c. stands for the co lex conjugate term

The spectral field, corre the

mp.

lated to temporal field via Fourier Transform, is given by ( )( ) ( )exp . .= +CEE iε ω ω φ c c , with ( )E ω being the spectral complex envelope. It is given by


[ ]rel( ) ( ) exp ( )= ΦE E iω ω ω , where ( )E ω and rel ( )Φ ω being, respectively, the spectral amplitude and relative phase of frequencyω . For the (unshaped) transform-limited (TL) pulse, rel ( )Φ =ω 0 for anyω .

Hence, (2)fA is given in the frequency domain as

(1) (2) (2) (2) ,= +abs RamfA A A

with

( )(2) 2

0

(2) 2 2,1 ,2

2exp ( ) ( )

( ) ( ) ( ) ( ) ,

∞

∞ ∞∗ ∗

Ω Ω

= Ω −

= −Ω + −Ω

∫

∫ ∫fg fg

abs CE abs fg

Ram Ram fg Ram fg

A i E E d

A E E d E E d

φ μ ω ω ω

μ ω ω ω μ ω ω ω

where the amplitude abs is coherently contributed by all the (2)A g − f two-photon absorption

transitions provided by the pulse spectrum, while the amplitude (2)RamA is coherently contributed

by all the possible g − f two-photon Raman transitions provided by the pulse spectrum. These two groups of transitions are schematically illustrated in Fig. 1(a).

Two-photon absorption transitions involve two absorbed photons of frequencies ω and ' = Ω −fgω ω (satisfying , ' < Ω fgω ω ), while two-photon Raman transitions involve an

absorbed of frequency photon ω (satisfying > Ω fgω ) and an emitted p oton of frequency −Ω

h

fgω . Absorption of a photon contributes a global phase of CEφ to its corresponding tr ion amplitude, while a photon emission contributes a correspon global phase of − CE

ansit ingdφ . H , as seen in Eq. (1), the two-photon absorpti mplitude (2)

absA has a global phase of 2 CE

ence on aφ , while the two-photon Raman amplitude (2)

RamA has zero global phase. So, the CEP value strongly affects the corresponding inter-group interferences. The 2

absμ , 2

,1Ramμ and 2,2Ramμ are, respectively, the effective non-resonant couplings of the two-photon

absorption transitions and different two-photon Raman transitions [9, 10]. These couplings of the different groups of transitions generally have different values since each two-photon transition group accesses a different range of intermediate excitation energies (see the dashed lines in Fig. 1(a)) and thus involves different off-resonance detunings from the various intermediate states n .

The possibility to simultaneously induce different types of two-photon transitions is a key feature of the ultrabroadband pulses of the single-cycle regime considered here as to the broadband pulses of the multi-cycle regime. For pulses with stabilized CEP CE

comparedφ , the

amplitudes associated with the different transition groups have relative phase of 2= CEΔφ φ interfe

control objective othat dictates the nature of the rence between them. The observable considethe f the CEP and relative spectral phase is the final population

red here as fP of state

f that is given by2(2)=f fP A .


Fig. 1. (a) Two-photon excitation scheme in the femtosecond single-cycle regime. Indicated are the two-photon absorption transitions and two-photon Raman transitions. (b) Ultrabroad spectrum of single-cycle pulse centered at fgΩ that provides two absorbed photons of

frequencies around to the two-photon absorption transitions as well as two photons,

one absorbed photon of frequency around 3 and one emitted photon of frequency

around , to Raman transitions. (c) Temporal electric field of the TL single-cycle pulse

with 0

/ 2fgΩ

/ 2fgΩ

/ 2fgΩ

ω =12500 cm−1 (800 nm) and different CEφ . (d) Relative population of the excited state

f induced by the TL single-cycle pulse with different values of CEφ and different carrier

frequencies 0ω (normalized to the case of =CEφ 0).

3. Two-photon perturbative calculations for a model system

3.1 The model system

The model quantum system we consider here has two-photon transition frequency of Ω fg =12500 cm−1. When the ultrabroadband spectrum of th radiating pulse is centered in the region of the two-photon transition frequency, i.e., 0

e ir≈ Ω fgω , it effectively provides photon

pairs to both types of the two-photon transition groups: (i) Ea t absorption transition involves two abso photons wi ies 1

ch wo-photon rbed th frequenc ω , 2 2/≈ Ω fgω that are

symmetrically located around 2/Ω fg , such that 1 2+ = Ω fgω ω ; (iolves one of frequ

i) Each two-photon Raman transition inv absorbed pho ency 1 3 2/ton ≈ Ω fgω and one emitted photon of frequency 2 2/≈ Ω fgω , such that 1 2− ≈ Ω fgω ω . Hence, w sider in our study pulses having their Gaussian spectrum centered in the region of

e conΩ =fg 12500 cm−1. Their spectral


intensity bandwidth (FWHM) is Δω =5 5 cm−1 corresponding to TL pulse duration of 2.7 fs, which is equal to one optical cycle of 0

42ω =12500 cm−1 (80 m). The spectrum of the 800-nm

pulse and its TL tempo0 n

ral elect th differentric field wi CEφ values are schematically shown in

ribe e, thd

Figs. 1(b) and 1(c). As formulated and desc d abov e exact amplitudes contributed by the different two-

photon transition groups ( (2)absA an (2) Ram e ined, on one hand, by the values of the

different two ton couplings ( 2abs

A ) ar determ-pho μ , 2

,1Ramμ , 2,2Ramμ ) and, on the other hand, by the field of the

pulse ( )ε ω and its d we assume for simplicity

h d

ifferent characteristics. Here, that 2 2 2

,1 ,2= =abs Ram Ramμ μ μ .

3.2 Results for two-photon CEP and relative-phase control

First, we study in this section t e depen ence of the two-photon excitation probability on the value of the CEP for the unshaped TL pulses of different carrier frequencies. The corresponding dependence of , f TL on CEP φ , calculated using Eq. (1), is shown in Fig. 1(d) for different carrier frequencies 0ω , i.e., different positions of the pulse spectrum h trace is normalized by the corres nding value of ,

. Eacpo f TLP for zero CEP, i.e., 0)=CEP , (f TL φ . The

observed dependence on CEφ is explained very simply as follows. The field ( )E ω of the TL pulse is real since rel ( )Φ =ω 0. So intra-group interferences within (2)

,abs TLA and (2),, the Ram TLA are

onstru tive and maximize agnitu each of them, while the inter-group relative phase between them is 2 CE

fully c c the m de of φ with a corresponding periodicity of π . In th 0=CEe case of φ

and π , the amplitudes (2),abs TLA and (2)

,Ram TLA h ro r tive pg

ave ze ela hase [s . (1)] and, thus, their inter- roup interference is fully constructive and maximizes , ( )

ee Eq

f TL CEP φ for the given 0ω .

On the other hand 2/=, for CEφ π and 3 2/π , (2),abs TLA and (2)

,Ram TLA have a relative phase of π and enc e , thus, their inter-group interfer e is fully destructive and leads to th minimal value of

, ( )f TL C for the given 0ω . depth

EP φ

TLM obtained for a givenThe modulation 0ω , defined as

( ) ( ) ( ) ( ), , ,⎣ ⎦ ⎣f TL f TL CETL ,max min max min( ) ( ) ( ) ( )= − +⎡ ⎤ ⎡ ⎤⎦CE f TL CE f TL CEM P P P Pφ φ φ , equivalently, φ or

, , , ,( 0) ( / 2) ( 0) ( / 2)= == +⎡ ⎤ ⎡M P Pφ φ φ s determined by the

am e ratio

− = = ⎤⎣ ⎦ ⎣ ⎦f TL CE f TL CE f TL CE f TL CETL P Pπ πφ , i

plitud (2),=TL abs TLr A A elation (2)

,Ram TL via the r 22 1( )= +TL TL TLM r r . The ximal

value of TL

ma

M =1 tained w is ob hen (2) (2), ,=bs TL Ram TLA , while th inimal value of TLaA e m M =0 is

obtained when (2),abs TLA =0 or (2)

,Ram TLA =0. Othe es of r valu TLM , in between 0 and 1, are

obtained for non-equal non-zero magnitudes of (2),abs TLA and (2)

,RamA Here, as seen in Fig.

1(d), the modulation dept ncreases from close-to-zero value of TL

TL .

h i M ≈0.05 at 0ω = 14500 ink line) to TLcm−1 (p M ≈0.25 at 0ω =13500 cm−1 (gree ne) to TLn li M ≈0.80 at

0 Ω= fgω =12500 cm−1 (red line) and s maximal value of TLto it M =1 at 0ω =11 cm−1 (black line), which is red shifted

870 from Ω fg M. Then, it decreases a valagain to ue of TL =0.45

at 0ω =10300 cm−1 (blue line). This observed behavior can be understood as follows. Analyzing Eq. ( given Ω1) for a fg ,

it can be seen that in the case of a Gaussian TL p the m ude of ulse agnit (2),Ram TLA (involving


frequency d erences) depends on the couplings 2,1iff Ramμ and 2

,2Ramμ and on the pulse spectral bandwidth Δω , however ctually independent of the carrier frequency 0it is a ω . On the other

, the magnitudehand of (2),abs TLA (involving frequency sums) is determined by the coupling

abs

2μ as well as by ωΔ and 0ω . Thus, a given set of two-photon lings and

spectral bandwidth

for coup a given

Δω , a change in 0ω leads to change only in (2),abs TLA , while (2)

,Ram TLA

remains unchanged, and thus to a change in the ratio (2) (2), ,=TL abs TL Ram TLr A A . Specifically, as

the pulse spectrum is shifted to lower frequencies (i.e., shifted to the red), (2),abs TLA and TL

increase. In our model, due to the assumption of

r2 2

,1 ,2= = 2Ram Ram absμ μ μ , one obtains that

(2) (2), ,Ram TLA<abs TLA < and r 1 for TL 0 Ω= fgω and, thus, for any 0 Ω> fgω . Hence, for

any 0

1<TLM

≥ω 12500 cm−1 and the TLM value increases as 0ω decreases. As 0ω further decreases below Ω fg and becomes more and more red-shifted from it, the values of and TLr TLM

further increase (see above) up to the point of = 11870 cm−1, where (2),abs TLA and (2)

,Ram TLA

become equal and, thus, TL =1 and TLr M =1. Further red-shifting of 0ω leads to a further increase of TLr beyond the value of one ( TLr > ) and thus to a corresponding decrease of TL 1 M

a value of one ( <TLM deed is the case for 0below ), as in1 ω =10 0 cm−1. 30

Quantitatively, based on Eq. (1), the carrier frequency 0ω at which (2) (2)=abs TL RamA , ,TLA is

given by 2

20, 1

18

2ln= =

Δ⎛ ⎞⎢ Ω − ⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦TLM fg couplingRω

ω8 2ln

⎡ ⎤⎥Ω +fg , where

2 2,1 ,2

2

+= Ram Ram

couplinab

gs

μ μμ

R is

the ratio between the sum of the two-photon Raman couplings and the two-photon absorption coupling. As seen, the value of 0, 1=TLMω and its shift from Ω fg generally depends on Ω fg , Δω

coupliR hen couplinR one obtains 0,and . W =1, ng g 1= = ΩTLM fg , i.e., tω he maximal modulation occurs

when there is no shift of 0ω from Ω fg ch a case is, for example, the case of . Su

2 2,1 ,2

12

= = 2Ram Ram absμ μ μ . When R <1 or >1, one obtains, respectively, coupling couplingR

0, 1= > ΩTLM fgω or 0, 1= < Ω

TLM fg ,ω i.e., the maximal modulation occurs when there is,

respectively, a blue or red shift of 0ω from Ω fg . T2

he latter case corresponds to our model that

includes the assumption of 2 2,1 ,2= =Ram Ram absμ μ μ corre =2. Isponding to couplingR ndeed, we have

obtained the maximal modulation for 0, 1=TLMω =1 cm−1, which is red-shifted from Ω1870 fg .


Fig. 2. Coherent control of two-photon excitation in the model system: Excited state population induced by ultrabroadband pulses shaped with a spectral π phase step at different positions for several cases of carrier frequency 0ω and CEP CEφ . Each trace is normalized by the

population excited by the corresponding TL pulse having non-stabilized CEφ .

Next, as a benchmark case for the full phase control of the two-photon excitation, we consider shaping the pulses of different carrier frequencies and different CEPs with a relative spectral phase step of π, i.e., rel ( )Φ =ω 0 for < stepω and rel ( )Φ =ω π for ≥ stepω ω ω [10]. The position of the phase step stepω is scanned across the ultrabroadband pulse spectrum. The corresponding results for different values of 0ω in the range of moderate red-detuning to moderate blue-detuning from Ω fg are shown in the different panels of Fig. 2. For each value of 0ω , the corresponding dependence of fP on stepω is shown for the cases of CEφ =0 (blue line) and CEφ =π/2 (red line) as well as for the case of non-stabilized CEP (black line). The latter results from an average over all the different traces of fP , each corresponding to a different value of CEφ within the range of 0 to 2π. All the traces presented in Fig. 2 for a given

0ω are normalized by the value of fP excited by the corresp ng TL puls ng non-sta

to n

ondi e havibilized CEP. A feature ote of all the π-traces is that the asymptotic values of each trace, i.e., the fP

values when stepω is located far outside the spectrum either to the red or blue, are equal one to the other. It results from the fact that these two asymptotes correspond to a constant spectral phase of, spectively, π or 0 that is applied across the whole spectrum and thus is globally added to CE

reφ . So, effectively they are equivalent to two unshaped TL pulses with a difference

of π in their CEP values ( CEφ an CEd φ +π) that, as discussed above and shown in Fig. 1(d), both yield the same value of fP . It also worth noting that the di rence between the asymptotic values of the π-traces with different

ffeCEP values at a given 0ω also appears in

CEthe

P-dependence results presented in Fig. 1(d). A noticeable characteristic of the π-traces corresponding to 0ω =10300 cm−1, which are

Fig. 2(a), is the approximate symmetric double-well structure around presented in


/ 2= Ωstep fgω =62 cm−1. This feature can be explained ollows, considering the case of a spectrum having 0

50 as fω that is strongly red-detuned from Ω fg . S red-d

case discussed above, also here such a case one obtains that

imilar to the etuned TL

in (2) (2)>abs RamA A and 2(2)≈f absP A ,

i.e., the main contribution to fP originates from the two-photon absorption transitions. As previously observ two-photon absorption in the multi-cycle regime [10 corresponding to

ed for ], the π-trace2(2)

absA as a double-well structure that is symmetric around / 2= Ωstep fg h ω

and its two minima of fP =0 result from complete destructive interferences among the two-photon absorption transitions. This π-trace has n endence on the CEP value. The deviations from such a perfect symme y around / 2

o tr

depΩ fg as well as the C dependence,

which are observed in the π-traces of CE

EPφ =0 and π/2 that are presented for 0ω = 0 cm−1,

result from the still-existing non-zero small values of (2)

1030

RamA that interferes with (2absA hese

deviations from the perfect symmetry are of opposite direction (along the axis of

) . T

stepω ) in these two CEP cases, hence they disappear in the corresponding π-trace of the non-stabilized CEP case.

A completely different characteristic is observed for the π-traces of 0ω =13500 cm−1 that are presented in Fig. 2(d). It is the symmetric wide double-well structure that is located around 0ω . This feature can be explained as follows, in the same spirit of the above explanation for the red-detu case but considering h at is strongly blue-detuned from Ω

ned ere the ca thse of a spectrum fg . Here, one obtains that (2) (2)<abs RamA A and

2(2)≈f RamP A , i.e.,

the main contribution to fP originates from the two-photon Raman transitions. The race

corresponding to

π-t2

RamA has a wide doub at is sym tric around 0(2) le-well structure th me ω and its

two a of minim fP =0 are located at 0 / 2)(±= Ωstep fgω ω (h 0ere, ±ω 6250 cm−1), i.e., they are Ω fg apart one from the other along the axis of stepω . They result from complete destructive

rferences among the two-photon Raman transitions. Also here, this π-trace has no nd

intedepe ence on e CEP e. The depen observe h rent ces of th valu CEP dence d for t e diffe π-tra

0ω =1 −13500 cm is due to the still-existing non-zero small values of absA that interferes with (2)

(2)

RamA . As seen in Fig. 2 for 0ω =11870 cm−1 [Fig. 2(b)] and 12500 cm−1 [Fig. 2(c)], the signatures

of these large-detuning π-trace features also show up when the pulse spectrum is located in between these two extreme spectral detuning cases. However, the exact shape of the corresponding π-trace re from the overall detailed intra- and inter-group interferences of the

sults phase-shaped (2)

absA and (2)RamA , where both of them are generally of significant magnitude.

Another eneral important feat of the single-cycle regime analyzed in e present work, i.e., when 0

g ure thω is in the region of Ω fg , is the fact that the final population fP excited by the

unshaped TL pulse can sometime be exceeded by a proper shaped pulse. This is as opposed to the multi-cycle regime, where the (CEP-independent)

ly fP value induced by the unshaped

TL pulse can never be exceeded by any shaped pu e; At most, it can be matched. This single-cycle feature is illustrated in all the π-traces of CE

lsφ =π/2 (red lines) own in Fig. 2,

where, for example, the population excited by the shaped pulse of 0

that are sh=stepω ω is higher than the

population excited by the unshaped TL pul . As discussed above, the latter corresponds to the asymptotic value of the π-trace. For CE

seφ =0 (blue lines), on the other hand, the relative


pop

e ca he T

ulation excited by the TL pulse is still the maximal one. Below we analyze in detail these cases.

In th se of t L pulse of CEφ , as also discussed above, the in ra-grou terferences

within (2)absA and (2)

t p in

RamA are fully constructive and thus maximize (2)absA and (2)

RamA , while the

relative phase between (2)absA and (2)

RamA is CEP dependent. For CEφ =π/2, this relative phase is equal to π and thus the inter-grodes

up interference between th amplitudes is completely e twotructive. For CEφ =0, the int relative ase is zero and the corresponding

interference is fully constructive. In the single-cycle case of

er-group ph

0=stepω ω , when 0ω is in the region of Ω fg , all the effectively contributing two-photon absorption transiti involve only pairs of photons that their frequencies are located near 2/Ω fg , for which t e π phase step contributes zero phase. Thus,

ntra-group interferences wi (2absA and thus maximize

oh

the i are fully constructive

ns

thin ) (2)absA ,

while the total phase of (2)absA is (2) 2 2)( 0= + =Φ abs CE CEA φ φ [see Eq. (1)]. On the o her hand, all

the effectively contributing two-photon Raman transitions involve pairs of photons that their frequencies are located in different spectral regions with respect to

t

stepω and thus the π phase step contributes a phase of 0 to one of the nd a phase of π to the other. Hence, als e intra-group interferences wi (2)

m a o thhin t RamA are fu uctive and maximize lly constr (2)

RamA , how er

the total phase of (2)

ev

RamA is actually (2)( ) =Φ RamA π [see Eq. (1)]. Hence, here, for CEφ = π/2, the relative phase between (2)

absA and (2)RamA is zero and thus the inter-group interference between

the two amplitudes is completely constructive. For CEφ =0, the inter-group relative phase is π .

/ 2)

and the corresponding interference is fully destructiveSo, overall, as seen in the results of Fig. 2, one indeed obtains

== < =CE CEstepf TL f shapedP P ω ω0, , ,( / 2) (φ π φ (red lines) and π

0, , ,( 0) ( 0)== > =CE CEstepf TL f shapedP P ω ωφ φ (blue lines).

4. Three-photon theoreticsystem

al perturbative description and calculations for a model

y in detail the case of the non-resonant three-photon excitation. In the weak-field regime the final amplitude

4.1 Theoretical description

In this section we stud(3)fA of the excited state f is given by the third-order

erturbation theory as

3( )exp .−∞

Ω g

p

( ))(3∞

∝ ∫f fA t i t dtε

In the frequency domain it is given by

(2)

ith

−

(3) (3) (3) ,= +abs RamfA A A

w

( )(3)1 2 1 2 1

0

32

0

3exp ( ) ( ) ( )∞ ∞

= Ω −∫ ∫abs CE abs fgA i E E E d dφ μ ω ω ω ω ω ω


3

exp ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ,

∞ ∞∗

∞ ∞ ∞ ∞∗ ∗

⎡= + −Ω +⎢

⎣

+ −⎤⎥−Ω + + Ω

∫ ∫

∫ ∫ ∫ ∫

Ram CE Ram fgE E E d d

E E E d d E E E d d

φ μ ω ω ω ω ω ω

μ ω ω ω ω ω ω μ ω ω ω ω ω ω

( )(3) 3,1 1 2 1 2 1 2

0 0

3,2 1 2 1 2 1 2 ,3 1 1 2 2 1 2

0 0 00 ⎦Ram fg Ram fg

A i

where the amplitude (3)absA is coherently contributed by all the g − f three-photon

absorption transitions provided b th pulse spectrum, while the amplitude (3)y e RamA is coherently contributed by all the possible g − f three-photon Raman transitions provided by the pul

-photon absorpti involve three se spectrum. These two groups of transitions are schematically illustrated in Fig. 3(a). Three on transitions absorbed photons of frequencies 1ω , 2ω

and 3 1 2= Ω − −fgω ω ω (satisfying 1 2 3, , < Ω fgω ω ω ), while three-pho ransitions involve two absorbed photons 1

ton Raman t of frequency ω , 2ω (satisfying 1 2+ > Ω fgω ω ) and an em

photo fr

itted

n of equency 1 2+ −Ω fgω ω . Hence, t phases of and he global the amplitudes (3)absA (3)

RamA

Fig. 3. (a) Three-photon excitation scheme in the femtosecond single-cycle regime. Indicated are the three-photon absorption transitions and three-photon Raman transitions. (b) Ultrabroad spectrum of the single-cycle pulse centered at Ω fg /2 provides three absorbed photons of

frequencies around Ω fg /3 to the three-photon absorption transitions as well as three photons,

bed photon of frequency around 2two absor Ω fg /3 and one emitted photo

around Ω

n of frequency

fg /3, to the Raman transitions. (c) Relative population of the excited state f

induced by the TL single-cycle pulse with differ values of ent CEφ and different carrier

frequencies 0ω (normalized to the case of =CEφ 0).


are, respectively, 3 CEφ and 2 − =CE CE CEφ φ φ , resulting in a relative phase between them of 2Δ = CEφ φ . The 3

absμ , 3,1Ramμ , 3

,2Ramμ and 3,3Ramμ are, respectively, the effective non-resonant

couplings of the three-photon absorption transitions and different three-photon Raman transitions.

The observable considered here as the control objective of the CEP and relative spectral phase is the final population fP of state f that is given by

2(3)=f fP A .

4.2 The model system

The model system we consider has three-photon transition frequency of Ω fg =25000 cm−1. In the three-photon excitation case, the rational choice of the excitation pulse spectrum is with a carrier frequency 0ω in the region of one-half of the three-photon transition frequency, i.e.,

0 / 2≈ Ω =fgω 12500 cm−1. As illustrated in Fig. 3(b), such spectral position effectively provides significant intensity at frequencies around / 3Ω fg as well as around .

Hence, as desired for CEP control, it generally yields significant magnitude to both abs and

2 /Ω fg 3

A(3)

(3)RamA . The three-photon absorption amplitude results from transitions involving three

photons of 1

(3)absA

ω , 2ω , 3 / 3≈ Ω fgω with 1 2 3+ + = Ω fgω ω ω , while the three-photon Raman

amplitude (3)Ram results from transitions involving three photons of 1A ω , 2 2 / 3Ω≈ fgω and

3 / 3≈ Ω fgω with 1 2 3+ − = Ω fgω ω ω . For consistency, we use here for the three-photon case the same pulse of the two-photon case, i.e., a p e of 5425-cm−1 bandwidth corresponding to a 2.7-fs TL pulse that is of a single cycle at 0

ulsω =12500 cm−1 (800 nm). Generally, it worth

noting that the higher is the order of the multiphoton excitation, the smaller is the minimal bandwidth of the ultrabroad spectrum that is required for observing a CEP control effect. This is important for some experimental configurations, where the pulse-shaping implementation is somewhat easier with a reduced bandwidth. Regarding the effective three-photon couplings, si n excitation case, we assume here for simplicity that milar to th

absμe two-photo

3 3 3 3,1 ,2 ,3= = =Ram Ram Ramμ μ μ .

In order to avoid here an additional source of interferences, the excitation amplitude induced via one-photon transition at frequency Ω fg , which is described theoretically by the first-order perturbative term, is not included in our model. This is justified since, as seen in Fig. 3(b), this spectral component falls far outside the main spectral region, with a

ng intensity that is about three orders of magnitude lower than e intensity at 0 / 2= Ω fg

correspondi thω . Thus, experimentally the spectral tail located in the region of Ω fg can easily be

the following we will only describe in sho

eliminated by proper amplitude shaping of the pulse [32–34].

4.3 Results for three-photon CEP and relative phase control

The results for the phase control of the three-photon excitation are shown in Figs. 3(c) and 4. The former presents the CEP control results for unshaped TL pulses, while the latter presents the results for the full phase control utilizing both the CEP and relative spectral phase. Conceptually and qualitatively, the three-photon results are very similar to the control results of the two-photon case [Figs. 1(d) and 2], hence in

rt some of the features of the three-photon results. The dependence of the three-photon excited population ,f TLP on the CEφ value for the

unshaped TL pulses is shown in Fig. 3(c) carrier frequencies 0for different ω . Each trace is ized by orresponding value of , 0( )normal the c =f TL CEP φ . With a TL pulse, the magnitudes of


(3),abs TLA and (3)

,Ram TLA are maximal due to f erences within each of them, and the inter-group relative phase between them is 2 CE

ully constructive intra-group interfφ having a corresponding

periodicity of π. With =CEφ 0 and π, (3),abs TLA and 3)

,(Ram TLA have zero relative phase [see Eq. (2)]

and, thus, their inter-group interference is constructive leading to the maximal , ( )f TL CEP φ for

the given 0ω . With =CEφ π/2 and 3π/2 ),abs TL and (3)

,, (3A Ram TLA have a relative phase of π and, thus, their in p interference is destructive and ter-grou , ( )f TL CEP φ is the minimal one for the given 0ω .

In terms of the modulati h TLon dept M , also here, a spectrum that is largely red or blue

shifted from 2/Ω fg =12500 cm−1 leads to a magnitude (3),abs TLA that is much larger or much

than smaller, respectively, (3),Ram TLA . Such detuned spectrum supports effectively only one

group out of th wo groups of three-photon transitions. The resulting ratio e t(3) (3)

, ,TL abs TL Ram TLr A A is, respectively, either much smaller or much larger than 1, and thus the

lation depth

=

modu TLM in both cases is small [see blue and green lines in Fig. 3(c) for 0ω =10500 and 13500 cm−1]. As seen in Fig. 3(c), the maximal modulation depth of

TLM =1 and the corresponding minimal TL population of zero, obtained when (3) (3)

, ,=abs TL Ram TLA A , occur here at 0ω =11750 cm−1 (black line), i.e., at a moderate red shift from

2/ . The cause or this maximal-modulation red-shif lar to the Ω fg f t effect is qualitatively simione of the two-photon excitation case. It is the combination of the following facts: (i) TLr <1 for 0 2/= Ω fgω due to our model assumption of 3 3 3 3

,1 ,2 ,3= = =abs Ram Ram Ramμ μ μ μ , and (ii) TLr increases as 0ω increases.

gure 4 presents the r lts for the full phase control of the ee-photon excitation using ed pulses of different carrier frequencies and d ferent CEPs applied with a relative

spectral phase step of π. The different panels of Fig. 4 present the results f different values of 0

Fi esu thrshap if

or ω . For each lue of 0 va ω , the corresponding dependence of fP on the phase step position

steω p is shown for the cases of CEφ =0 (blue line) and CEφ =π/2 (red line) as well as for the case of non-stabilized CEP (black line). All the π-traces presented for a given 0ω are normalized by the value of fP excited by the co esponding TL pulse having non-stabilized CEP.

As in the two-photon case, the asymptotic values of each π-trace are equal one to the other. It results from the fact that the two asymptotes correspond to a constant spectral phase of 0 or π that is globally added to CE

rr

φ and, thus, they correspond to two unshaped TL pulses wit ere wnh a diff nce of π in their CEP values. As sho in Fig. 3(c), such two TL pulses induce the same population fP . The difference between the ptotic values of the π-traces with different CEP values at a given 0

asymω also appears in the CEP-dependence results of Fig. 3(c).

When 0ω is far detuned to the red from Ω fg /2 [Fig. 4(a)], as in the TL case, the dominant

contribution to the total excited amp e flitud (3)A is from the absorption (3)absA . Then, the

resulting π-trace is similar to the one obtained for the three-photon absorption case in the multi-cycle regime [10], with three zero-value minima located around two peaks of small magnitude. On th

part

e other hand, when 0ω is uned to the blue from far det Ω fg /2 [Fig. 4(d)], the

dominant ribution to cont (3)fA comes from the Raman part (3)

RamA leading to a π-trace that is of similar structure to the red-detuned case but much wider. As the CEP value affects only the


Fig. 4. Coherent control of three-photon excitation in the model system: Excited state population induced by ultrabroadband pulses shaped with a spectral π phase step at different positions for several cases of carrier frequency 0ω and CEP CEφ . Each trace is normalized by

the population excited by the corresponding TL pulse having non-stabilized CEφ .

inter-group interferences, when one of the group amplitudes abs or (3)A (3)Ram dominates, the π-

trace is expected to have only weak dependence on the CEP value. Indeed, as seen in Figs. 4(a) and 4(d), in these two highly-detuned cases there is only a small difference between the π-traces obtained with

A

CEφ =0 and π/2. When 0 ≈ Ω fgω /2 [Figs. 4(b) and 4(c)], the exact shape of the corresponding π-trace is

determined by the overall detailed intra- and inter-group interferences of the phase-shaped and (3)

absA (3)RamA , where both of them are generally of significant magnitude. For every three-

photon transition that contributes to the absorption or Raman amplitudes, the position of the π-step determines the relative phases of the three involved photons and, hence, the way this three-photon transition interferes with the other transitions of his group. All such intra-group interferences determine the magnitudes of abs and (3)A (3)

Ram , and, together with the CEP, they also determine the phases of and

A(3)absA (3)

RamA . All in all, these resulting amplitudes yield the final outcome of fP

Last, it is important to note that also the three-photon results exhibit the general important feature of the single-cycle regime that, as opposed to the multi-cycle regime, the final population excited by the TL pulse can sometime be exceeded by a properly shaped pulse. This is illustrated in the π-traces of

.

CEφ =π/2 (red lines) presented in Figs. 4(a) and 4(b).

5. Non-perturbative calculations for cesium (Cs) atom

In this section, the control strategy presented above is applied to atomic Cs, demonstrating it for its two excited states 1 ≡f 5d and 3/2 2 ≡f 5d 5/2 that are populated by two-photon excitation from the ground state ≡g 6s. The corresponding two-photon couplings between the ground and excited states are provided by the manifold of p-states of Cs. As opposed to our model system analyzed above, in a real atom, depending on its specific structure, the effective two-photon couplings 2

absμ , 2,1Ramμ , and 2

,2Ramμ generally have different magnitude


and might also have different sign. The Cs excitation scheme is shown in Fig. 5, with examples of the two-photon absorption and Raman transitions. The results for the two-photon excitation of the Cs atom irradiated by weak unshaped (TL) and shaped ultraboradband pulses have been calculated by the numerical propagation of the time-dependent Schrödinger equation using the fourth-order Runge-Kutta method. The calculations have considered the atomic states 6s to 9s, 6p to 8p, and 5d to 7d, with all their fine-structure states. The data of the states and the corresponding transition dipole moments are based on Refs [50–53].

Similar to the model results presented above, the Cs results also show high degree of CEP control using unshaped TL pulses, with a high sensitivity to the value of the pulse carrier frequency 0ω as compared to the value of the two-photon transition frequency. The transition frequencies corresponding to the Cs states 5d3/2 and 5d5/2 are, respectively,

1,Ω f g = 14499.3

cm−1 and 2 ,f gΩ = 14596.8 cm−1. We have found that the most effective CEP control over the

Cs two-photon excitation occurs with 0ω = 16667 cm−1 (600 nm), where the single-cycle TL pulse is of 2-fs duration and bandwidth of Δω =7350 cm−1. The corresponding results of the CEP control with such 600-nm single-cycle unshaped TL pulses are shown for the final populations of the Cs states 5d and 5d in, respectively, Figs. 5(a) and 5(b). As seen, there is a strong dependence of the final populations of both states on the CE

3/2 5/2

φ value, also here, with a periodicity of π. The maximal modulation depth of TLM =1 is obtained for the 5d5/2-state population [Fig. 5(b)], while a very high modulation depth of TLM =0.83 is obtained for the 5d3/2-state population [Fig. 5(a)]. As also seen in Figs. 5(a) and 5(b), the minimal CEP-dependent TL populations of both states 5d5/2 and 5d3/2 are obtained for CEφ =0.16 rad and

CEφ =0.16+π=3.3 rad, indicating that at these CEP values the inter-group interference between the corresponding overall TL two-photon absorption and Raman amplitudes is the most destructive one. These CEP values are different from the corresponding model-system values of CEφ =π/2 and 3π/2 due to the resonance-mediated nature [11, 20] of some of the two-photon transitions involved in the Cs excitation with the ultrabroadband pulses. Such resonance-mediated transitions do not exist in the non-resonant model excitation analyzed above.

The Cs results also exhibit effective full phase control using the CEP and relative spectral phase of the pulse. As an illustrative example, Figs. 5(c) and 5(d) show the results for, respectively, the 5d 3/2 d 5d 5/2 popu tions that are controlled with the above ultrabroadband pulses ( 0

an laω =16667 cm−1, Δω =7350 cm−1) plied with CEP of CEap φ =3.3 rad and sh ped with

a π phase step. One main feature is that, as seen, setting the π-step position in the central spectral region of the pulse (i.e., 0

a

≈stepω ω ) conv bove-mentioned destructive inter-group interference of the TL case into a constructive one and leads to an enhancement of the excited populations. Also this effect is indeed predicated by the above model-system results.

erts the a

Additional prominent features of the π-traces presented in Figs. 5(c) and 5(d) are the several enhancement peaks appearing there. They also originate from the above-mentioned resonance-mediated nature [11, 20] of some of the two-photon transitions induced in the ultrabroadband excitation of Cs. When stepω is set exactly at a spectral frequency that is equal to a transition frequency between the initial ground state (6s) and an intermediate state (np) or between an intermediate state (np) and the final state (5d or 5d ), the two-photon excitation probability is significantly enhanced. As previously explained [11], such an enhancement results from the interferences among two-photon transitions that are near-resonant with the intermediate state and their corresponding detunings are of different signs. These (intra-group) interferences are of destructive nature when induced by the TL pulse, while they become to be constructive when induced by a pulse shaped with a properly-

3/2 5/2


Fig. 5. Weak-field two-photon femtosecond coherent phase control of Cs in the single-cycle regime. Upper panel: Excitation scheme of Cs (fine structure splitting is not shown), with examples of two-photon absorption and Raman transitions. Panels (a) and (b): Relative population of the excited states 5d 3/2 anel (a)] and 5d 5/2 [ el (b)] induced by TL single-

cycle pulses of 0

[p pan

ω = 1 67 cm−1 (66 Δω = 7 0 cm−1) having different values of CE35 φ . Pa s

(c) and (d): Relative population of the excited states 5d 3/2 [pane )] and 5d 5/2 [panel ]

induced by such ultrabroadband pulses applied with CE

nel

l (c (d)

φ = 3.3 rad and shaped with a spectral π phase step at different positions. Indicated are the different state-to-state transitions associated with the different resonance-mediated enhancement peaks of the π-traces (see text).

positioned π phase step. The different state-to-state resonant transitions leading to the different resonance-mediated enhancement peaks in the π-traces of the 5d3/2 and 5d5/2 populations are indicated in Figs. 5(c) and 5(d). It worth noting that the 5d3/2 and 5d5/2 populations are not equally enhanced even when the same intermediate state is involved. For example, the resonance-mediated enhancement of the 5d3/2 population is either larger or smaller than the enhancement of the 5d5/2 population when the involved intermediate state is either 6p or 7p, respectively. The corresponding reason is the difference in the effective two-photon couplings associated with the different excitation pathways, which in the above example are 6s-6p-5d and 6s-7p-5d.

6. Conclusions

To summarize, we have shown theoretically the power of weak-field multiphoton femtosecond coherent control in the single-cycle regime. In this regime, for a rationally chosen pulse spectrum, the CEP of the pulse serves as a powerful control parameter in addition to the relative spectral phase. For weak-field N-photon excitation process, the ultrabroadband pulse induces several groups of initial-to-final N-photon transitions, where each group corresponds to a different combination of the number of absorbed photons (M) and the number of emitted photons (N-M). The intra-group interference is phase controlled by the relative spectral phase, while the inter-group interference is phase controlled jointly by the


CEP and relative spectral phase. Specifically, here, we have revealed and illustrated the corresponding control mechanism in non-resonant two- and three-photon excitation of a simple model system using frequency-domain perturbative analysis. Then, the phase control mechanism of the single-cycle regime has been confirmed also for weak-field two-photon excitation of atomic cesium (Cs) studied by non-perturbative numerical solution of the time-dependent Schrödinger equation. The same mechanism generally applies also to multiphoton excitations of higher order. We expect this work to serve as a basis for a new line of femtosecond coherent control experiments, most prominently for selective control in excitation scenarios that simultaneously involve several channels.

Acknowledgements

This research was supported by The Israel Science Foundation (Grant No. 1450/10) and by The James Franck Program in Laser Matter Interaction.


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Weak-field multiphoton femtosecond coherent control in the single-cycle regime

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