- Email: [email protected]
Sequential versus direct multiphoton absorption
CHEMICAL
Volume 61, number 3
SEQUENTIAL Steven LJWWRE
PHYSICS
VERSUS DIRECT MULTIPHOTON and Robert
1 March 1979
LETTERS
ABSORPTION
*
E. WYATT
Department of Chemistry. The Umremt_v of Texas. Austin. Texas 78712. (is.4 Received 13 November 1978 Revised manuscript received I December 1978
We present results of eksct calcuhtions on a three-level Uorse osctiaror modelmg HI‘ 1%hich suggest that multiphoton absorption proceeds by sequential single-lrvd tmnsltions, transitions arwng from coupling bet\reen non-adJncent states bring dynamically ne&gble_ The time-dependent Schriidingx equation is integxed in the Floquet formaIism_
I_ Introduction The importance of anharmonicity in multiphoton absorption has recently been demonstrated [ 1 J _Two factors are generally associated with anharmonicity: a decrease in levei spacing with increasing quantum number, and the presence of non-zero coupling matriv eIements beyond the first off-diagonal_ The former is associated with the “anharmonicity bottleneck” 121, which causes all multiphoton resonance frequencies to be shifted below that for the highest frequency onephoton transition. The Iatter property of molecules gives rise to one contribution to multiphoton absorption, the “direct” process in which two or more photons are absorbed independent of the presence of intermediate molecular levels. Experimentally [3], this effect is observed when no combination of the photons involved can lead to resonance, and theoretically, the concept of virtual states is invoked. Another contribution comes from the “sequentraY absorption pathway, in which probability flows in oneIevel transitions between “physical” states. This is the exclusive mechanism for harmonic oscillator photon absorption. Of course the two contrrbutions cannot be distinguished in an “exact” calculation, but approximate models [4] generally assume sequential absorption.
In this paper we use exact, nonpenurbative dynamical calculations with model coupling matrices to assess: (a) the relative importance of the direct and sequential mechanisms, and (b) the relative importance of the two manifestations of anharmonicrty (level congestion and nonsequential coupling) in multiphoton absorption. Our method is to compare calculations for 3 threeIevel system chosen to modei HF with the full couphng matrh and with matrices where certain elements have been suppressed. The sequential mechanism calculations have no coupling between the first (u=O) and third (u=2) states, while those for the direct mechamsm have no coupling between states adjacent in energy.
2. Floquet
theory
Into the time-dependent ifi Cl*@, t)/at = [cl,+ with v(r, t) = A,r
Schrodinger
equation
V(r. f)] *(I-, t) ,
cos ot,
(1)
subsritute the e.upansion
2%’
where 3s usual the unperturbed basis is defined by (9 - En)xII = 0. -The usual projection of eq. (1) then yields coupled equations for the a,,(t),
* Supported in part by research grants from the Robert A. WeIch Foundation
and from the National Science Foundation.
625
Waker [I ] - Tfie matrix ekments of r (in ao) to five figures are 2-7072 C-2) tzz= ---x.2432 (-3) [ --f_z352
(-2)
-12432
f---1) -1_3_963_ (-a)
83213
(-2)
--I m71
f-1)
-
f,Tm
f-l)
f,423 7 (-1)
-
I
The relative magnitudes af the caupling rr;atrix elements determine whether a two-km9 transition will be favored over a sequence of twu on&eveI transitions, We solved the three-state problem over the first aptlti cycle by simple lWlmeri& integration af eq_ (3) for ertch of the three independent field-free canditions. Treating real and irnagimry parts separately BYE six coupled eqrratians; a Range-Kutta-Cti fourth order integration using IS0 steps gave good accuracy3*2 J!kc!x?rS~n=~Mn?S~n~~z~e f~eqtterrciies The FIrrquet formukttion aids &seatIy in determining resonance frequencies. In a plot &the Fioquet eigeav&es versus field frequency bruadening effects have in effect. ken rotated fsum the horizontal to the vertica1 axis 1[6]_Shifts fram the field-free remnant frequencies
can be: of two sorts; the peak in absorption cross section versus direquexy can actually be shifted by the t;tser field, QKthe anharmonici~y bottlene& r=an act in a multi-kvef event to shift the nz~onance to the frequency mast effective for tfoe overall wmsitisn, Table L sAaws some resunance frequencies and shifts of interest for the f&t 3 X 3 coupfing matrix, It shautd be noted that two-leve1 models, whik often invoked [8], da nat show the shift of resonance Freqtiencjr,
Volume 6 1. number 3
CHEMICAL PHYSICS LfTTCRS
3.3. Two-photon absorption Fig_ 1 shows the excited state probabilities as a function of time for the three types of coupling, starting from the field-free ground state, for the resonant transition HF(u = 0) -I-‘Biw + HF(u = 2). It is apparent from fig. I c that the direct coupling mechanism propagates
in a qualitatively different way from both the sequential and exact mechanisms, and that the latter are in close agreement_ The direct mechanism of coupling reduces the system to a two-level problem, which in the rotating wave approximation has the analytic solution [9]
qtt)
T!k.“E IN CYCLES
0
40 TIME
80 IN CYCLES
4 x CCJ
1 hfarch 1919
=
$ “6
7 sin+
(&qz)”
t/2) -
The Rabi frequency ,u corresponds to the high frequency oscillation appearing in fig_ 1 c. The lower frequency oscillation also appears to have the form of eq. (6) and indeed a scan in w shows the frequency of this oscillatron to be proportionai to w with a value at resonance of about 5.3 X 10e6 ~tu-’ or 3400 optical cycles. Note that no high-frequency oscill&trons are visible in the full marris and sequentA coupling matrix populations, figs. 1 a and 1 b. This can be explained qualitatively as follows. If the residence time in the u = 1 level is longer rhan the Rabi oscillation period, the oscillation ~vrll not induce a similar oscrllation in the u =2 population, but will be averaged by Interference effects Relatively strong coupling allows effective probability flow throughout the 9 stem as a whole. leadmg to further Interference and preventmg a description m terms of .I pair of weakly-coupled two-level systems. Floquet analysis is particularly useful for investigating the long-time behavior of systems. At the frequency for the two-photon transition considered_ a 500 ps laser pulse lasts for about 5SOOO optical cycles. This time is also about the interval of approximately constant field strength for a 100 ns pulse. A direct integration that long would require almost nine million steps, yet using the Floquet approach, probabrlity is still conserved to 0.3%. The agreement between sequential and full msttix calculations is still striking near 58000 optrcal cycles. One must shift the sequen-
0.8
TIME
IN CYCLES
Fis_ 1. State probJbiiities versus time in optical cycles. near the resonant frequency for HF(u = 0) + 2fiw - HF (v= 2). Solid Linesshow results for the full coupling mntrb and dotted lines shon the sequential coupiing results m parts (a) aad (b)- AU fiiures presented are sampled at two pomts per optxal cycle. The Rabi period at this frequency is 1.6 optical c> c&es-Part (c) shoas Pt(r) for the direct coupling model.
627
VoIume 6 1. number 3
CHEhfiCAL PHYSICS LElTERS
tial coupling data forward in time by 2.5 optical cycles to obtain this 3greement, however; this is the most signiticmt difference between the two coupling modes observed in this study3-C Three-photon
absorption
Fig_ 2 shows the vibrational energy expectation vahre versus time for the three coupling schemes near the resommt frequency for the process, HF(u = 0) + 3%~ * HF(u= 2)_ The behavior of Pt and P2 (not shown) indicates that the three-photon non-resonant transition is of less importance than off-resonant single level transitions; P, is always iess than P, by a frictor of 25. These seq&ntial si@e Ievei transitions are so far off resonamx
from the u= I Ievel that the small transition
0921
0
80
40
TIUE
1 hfuch 1979
probabilities observed are quantum mechanical in origin, analogous to tunneling in time-independent quanturn mechanics_ The presence of high frequency Rabi oscillations in fig_ 2a indicates that the system can be treated approximately as a pair of weakly-coupled twolevel systems. Because of the low probability of sequential transitions which are high!y detuned from the intermediate states, the direct coupling mechanism is slightly more important than in the previous case. At 39000 optic31 cycles, equivalent to 500 ps, the sequential coupling data must be shifted forward in time by 7.5 optical cycles_ Agree.ment is then quite close.
4_ ConchLsiorls This application of Fioquet theory in time-dependent quantum mechanics has ihustrated two main advantages over direct integration of the quantum equations of motion First, the shifted resonance frequencies can be determined accurately without computing an actual absorption spectrum; transitions which are unresolved or too weak in transition strength to be easily observed in an absorption spectrum GUI be located. Second, the very long time behavior of the system can be followed with a great reduction in computational effort, if indeed a practical machine word size would allow a meaningful direct integr3tion at all_ The direct multilevel transitions studied here do not appear to be important, except for 3 slight over311 time shift at very long times_ The exact time dependent behavior in multiphoton absorption is highly dominated by the sequential excitation mechanism_ Since anharmonicity is important, if coupling matrix elements beyond the first offdiagonal nn be neglected, then the change in IeveI spacing with quantum number is the influential anharmonic property. Similar conclusions are expected for moIecuIes less anharmonic than HF, as the non-tridiagonal coupling matrix elements are directly related to anharmonicity_
IN CYCLES
Fz_ 7. Jkpcctation value of mokcuku ener=g~versus time in optiwl cycIes at the field frequency I-18 X IO-’ atu-‘. near the resonant frequency for HF(u=O) f- 3%~ -, HF(u=Z). 4) Full coupling matrix rendts; sequential marrix results show 110visibk difference on this sczle_ (b) Direcr meckuxism enew absoorption_
628
References [I]
R.B. Walker and R-K_ Preston, J. Cbem_ Phys. 67 (1977) 2017.
Volume 61. number 3
CHEMICAL
PHYSICS LETTERS
[2 ] N. Bloembergen. Opt. Commun. I5 (1975) 416; S. hfukamel and J. Jortner, J. Chem. Phys. 65 (1976) 5204; J. Jortner, Proc. Sot. Photo-Opt. Instr. Eng. 113 (1977) S8. [3] J-H_ Eberly and P_ Lambropoulos, eds., Multiphoton Processes, Proceedings of the Internatiomd Conference at the University of Rochester (WiIey, New York, 1978); especially thearticlesby JJ_\Vyme,J.A.Amstron~and P.Esherick and by G. Mainfray. [4] J.H. Eberly, B-W. Shore, Z. BiaIynicka-Birula and 1. BiaIynicka-Birula, Phys. Rev. A I6 (1977) 2038; M. Tamir and R-D_ Levine, Chem. Phys. Letters 46 (1977) 208; E-R- Grant, P-A- Schulz, AS. Sudbo, Y.R. Shen and Y-T. Lee. Phys. Rev. Letters 40 (1978) 115; M. Sargent III and P. Horo\\itz, Phys. Rev. 13 A (1976) 1962; B-W_ Shore and J_ AckerhaIt, Phys_ Rev. 15 A (1977) 1640.
1 March 1979
1.51 F-R. hlouIton. Differential ea_uations (MacMrIIian, New York, 1930) ch. 17; D-R. Dion and J-0. Hirschfelder, Advan. Chem. Phys. 35 (1976) 265. [6] C. Besset, J. Horoaitz, A. Messiah and J. Winter, J. Phys. (Paris) 15 (1954) 251; J-H. Shirley, Phys. Rev. 138 B (1965) 979; S. Leasure and R.E. Wyatt, to be published. [7] DG. Truhlar, J. Comp. Phys. 10 (1972) 123. (81 S-H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703; A.M.F. lau, Phys. Rev_ 14A (1976) 279; J-V. Moloney and W-J- Meath. Mol. Phys. 35 (1978) 1163. [9] MS. Sargent III, MO. ScuIIy and W-E. Lamb Jr., Laser physics (Addison-Wesley, Reading, 1974) pp_ 25,26_
629
Volume 61, number 3
SEQUENTIAL Steven LJWWRE
PHYSICS
VERSUS DIRECT MULTIPHOTON and Robert
1 March 1979
LETTERS
ABSORPTION
*
E. WYATT
Department of Chemistry. The Umremt_v of Texas. Austin. Texas 78712. (is.4 Received 13 November 1978 Revised manuscript received I December 1978
We present results of eksct calcuhtions on a three-level Uorse osctiaror modelmg HI‘ 1%hich suggest that multiphoton absorption proceeds by sequential single-lrvd tmnsltions, transitions arwng from coupling bet\reen non-adJncent states bring dynamically ne&gble_ The time-dependent Schriidingx equation is integxed in the Floquet formaIism_
I_ Introduction The importance of anharmonicity in multiphoton absorption has recently been demonstrated [ 1 J _Two factors are generally associated with anharmonicity: a decrease in levei spacing with increasing quantum number, and the presence of non-zero coupling matriv eIements beyond the first off-diagonal_ The former is associated with the “anharmonicity bottleneck” 121, which causes all multiphoton resonance frequencies to be shifted below that for the highest frequency onephoton transition. The Iatter property of molecules gives rise to one contribution to multiphoton absorption, the “direct” process in which two or more photons are absorbed independent of the presence of intermediate molecular levels. Experimentally [3], this effect is observed when no combination of the photons involved can lead to resonance, and theoretically, the concept of virtual states is invoked. Another contribution comes from the “sequentraY absorption pathway, in which probability flows in oneIevel transitions between “physical” states. This is the exclusive mechanism for harmonic oscillator photon absorption. Of course the two contrrbutions cannot be distinguished in an “exact” calculation, but approximate models [4] generally assume sequential absorption.
In this paper we use exact, nonpenurbative dynamical calculations with model coupling matrices to assess: (a) the relative importance of the direct and sequential mechanisms, and (b) the relative importance of the two manifestations of anharmonicrty (level congestion and nonsequential coupling) in multiphoton absorption. Our method is to compare calculations for 3 threeIevel system chosen to modei HF with the full couphng matrh and with matrices where certain elements have been suppressed. The sequential mechanism calculations have no coupling between the first (u=O) and third (u=2) states, while those for the direct mechamsm have no coupling between states adjacent in energy.
2. Floquet
theory
Into the time-dependent ifi Cl*@, t)/at = [cl,+ with v(r, t) = A,r
Schrodinger
equation
V(r. f)] *(I-, t) ,
cos ot,
(1)
subsritute the e.upansion
2%’
where 3s usual the unperturbed basis is defined by (9 - En)xII = 0. -The usual projection of eq. (1) then yields coupled equations for the a,,(t),
* Supported in part by research grants from the Robert A. WeIch Foundation
and from the National Science Foundation.
625
Waker [I ] - Tfie matrix ekments of r (in ao) to five figures are 2-7072 C-2) tzz= ---x.2432 (-3) [ --f_z352
(-2)
-12432
f---1) -1_3_963_ (-a)
83213
(-2)
--I m71
f-1)
-
f,Tm
f-l)
f,423 7 (-1)
-
I
The relative magnitudes af the caupling rr;atrix elements determine whether a two-km9 transition will be favored over a sequence of twu on&eveI transitions, We solved the three-state problem over the first aptlti cycle by simple lWlmeri& integration af eq_ (3) for ertch of the three independent field-free canditions. Treating real and irnagimry parts separately BYE six coupled eqrratians; a Range-Kutta-Cti fourth order integration using IS0 steps gave good accuracy3*2 J!kc!x?rS~n=~Mn?S~n~~z~e f~eqtterrciies The FIrrquet formukttion aids &seatIy in determining resonance frequencies. In a plot &the Fioquet eigeav&es versus field frequency bruadening effects have in effect. ken rotated fsum the horizontal to the vertica1 axis 1[6]_Shifts fram the field-free remnant frequencies
can be: of two sorts; the peak in absorption cross section versus direquexy can actually be shifted by the t;tser field, QKthe anharmonici~y bottlene& r=an act in a multi-kvef event to shift the nz~onance to the frequency mast effective for tfoe overall wmsitisn, Table L sAaws some resunance frequencies and shifts of interest for the f&t 3 X 3 coupfing matrix, It shautd be noted that two-leve1 models, whik often invoked [8], da nat show the shift of resonance Freqtiencjr,
Volume 6 1. number 3
CHEMICAL PHYSICS LfTTCRS
3.3. Two-photon absorption Fig_ 1 shows the excited state probabilities as a function of time for the three types of coupling, starting from the field-free ground state, for the resonant transition HF(u = 0) -I-‘Biw + HF(u = 2). It is apparent from fig. I c that the direct coupling mechanism propagates
in a qualitatively different way from both the sequential and exact mechanisms, and that the latter are in close agreement_ The direct mechanism of coupling reduces the system to a two-level problem, which in the rotating wave approximation has the analytic solution [9]
qtt)
T!k.“E IN CYCLES
0
40 TIME
80 IN CYCLES
4 x CCJ
1 hfarch 1919
=
$ “6
7 sin+
(&qz)”
t/2) -
The Rabi frequency ,u corresponds to the high frequency oscillation appearing in fig_ 1 c. The lower frequency oscillation also appears to have the form of eq. (6) and indeed a scan in w shows the frequency of this oscillatron to be proportionai to w with a value at resonance of about 5.3 X 10e6 ~tu-’ or 3400 optical cycles. Note that no high-frequency oscill&trons are visible in the full marris and sequentA coupling matrix populations, figs. 1 a and 1 b. This can be explained qualitatively as follows. If the residence time in the u = 1 level is longer rhan the Rabi oscillation period, the oscillation ~vrll not induce a similar oscrllation in the u =2 population, but will be averaged by Interference effects Relatively strong coupling allows effective probability flow throughout the 9 stem as a whole. leadmg to further Interference and preventmg a description m terms of .I pair of weakly-coupled two-level systems. Floquet analysis is particularly useful for investigating the long-time behavior of systems. At the frequency for the two-photon transition considered_ a 500 ps laser pulse lasts for about 5SOOO optical cycles. This time is also about the interval of approximately constant field strength for a 100 ns pulse. A direct integration that long would require almost nine million steps, yet using the Floquet approach, probabrlity is still conserved to 0.3%. The agreement between sequential and full msttix calculations is still striking near 58000 optrcal cycles. One must shift the sequen-
0.8
TIME
IN CYCLES
Fis_ 1. State probJbiiities versus time in optical cycles. near the resonant frequency for HF(u = 0) + 2fiw - HF (v= 2). Solid Linesshow results for the full coupling mntrb and dotted lines shon the sequential coupiing results m parts (a) aad (b)- AU fiiures presented are sampled at two pomts per optxal cycle. The Rabi period at this frequency is 1.6 optical c> c&es-Part (c) shoas Pt(r) for the direct coupling model.
627
VoIume 6 1. number 3
CHEhfiCAL PHYSICS LElTERS
tial coupling data forward in time by 2.5 optical cycles to obtain this 3greement, however; this is the most signiticmt difference between the two coupling modes observed in this study3-C Three-photon
absorption
Fig_ 2 shows the vibrational energy expectation vahre versus time for the three coupling schemes near the resommt frequency for the process, HF(u = 0) + 3%~ * HF(u= 2)_ The behavior of Pt and P2 (not shown) indicates that the three-photon non-resonant transition is of less importance than off-resonant single level transitions; P, is always iess than P, by a frictor of 25. These seq&ntial si@e Ievei transitions are so far off resonamx
from the u= I Ievel that the small transition
0921
0
80
40
TIUE
1 hfuch 1979
probabilities observed are quantum mechanical in origin, analogous to tunneling in time-independent quanturn mechanics_ The presence of high frequency Rabi oscillations in fig_ 2a indicates that the system can be treated approximately as a pair of weakly-coupled twolevel systems. Because of the low probability of sequential transitions which are high!y detuned from the intermediate states, the direct coupling mechanism is slightly more important than in the previous case. At 39000 optic31 cycles, equivalent to 500 ps, the sequential coupling data must be shifted forward in time by 7.5 optical cycles_ Agree.ment is then quite close.
4_ ConchLsiorls This application of Fioquet theory in time-dependent quantum mechanics has ihustrated two main advantages over direct integration of the quantum equations of motion First, the shifted resonance frequencies can be determined accurately without computing an actual absorption spectrum; transitions which are unresolved or too weak in transition strength to be easily observed in an absorption spectrum GUI be located. Second, the very long time behavior of the system can be followed with a great reduction in computational effort, if indeed a practical machine word size would allow a meaningful direct integr3tion at all_ The direct multilevel transitions studied here do not appear to be important, except for 3 slight over311 time shift at very long times_ The exact time dependent behavior in multiphoton absorption is highly dominated by the sequential excitation mechanism_ Since anharmonicity is important, if coupling matrix elements beyond the first offdiagonal nn be neglected, then the change in IeveI spacing with quantum number is the influential anharmonic property. Similar conclusions are expected for moIecuIes less anharmonic than HF, as the non-tridiagonal coupling matrix elements are directly related to anharmonicity_
IN CYCLES
Fz_ 7. Jkpcctation value of mokcuku ener=g~versus time in optiwl cycIes at the field frequency I-18 X IO-’ atu-‘. near the resonant frequency for HF(u=O) f- 3%~ -, HF(u=Z). 4) Full coupling matrix rendts; sequential marrix results show 110visibk difference on this sczle_ (b) Direcr meckuxism enew absoorption_
628
References [I]
R.B. Walker and R-K_ Preston, J. Cbem_ Phys. 67 (1977) 2017.
Volume 61. number 3
CHEMICAL
PHYSICS LETTERS
[2 ] N. Bloembergen. Opt. Commun. I5 (1975) 416; S. hfukamel and J. Jortner, J. Chem. Phys. 65 (1976) 5204; J. Jortner, Proc. Sot. Photo-Opt. Instr. Eng. 113 (1977) S8. [3] J-H_ Eberly and P_ Lambropoulos, eds., Multiphoton Processes, Proceedings of the Internatiomd Conference at the University of Rochester (WiIey, New York, 1978); especially thearticlesby JJ_\Vyme,J.A.Amstron~and P.Esherick and by G. Mainfray. [4] J.H. Eberly, B-W. Shore, Z. BiaIynicka-Birula and 1. BiaIynicka-Birula, Phys. Rev. A I6 (1977) 2038; M. Tamir and R-D_ Levine, Chem. Phys. Letters 46 (1977) 208; E-R- Grant, P-A- Schulz, AS. Sudbo, Y.R. Shen and Y-T. Lee. Phys. Rev. Letters 40 (1978) 115; M. Sargent III and P. Horo\\itz, Phys. Rev. 13 A (1976) 1962; B-W_ Shore and J_ AckerhaIt, Phys_ Rev. 15 A (1977) 1640.
1 March 1979
1.51 F-R. hlouIton. Differential ea_uations (MacMrIIian, New York, 1930) ch. 17; D-R. Dion and J-0. Hirschfelder, Advan. Chem. Phys. 35 (1976) 265. [6] C. Besset, J. Horoaitz, A. Messiah and J. Winter, J. Phys. (Paris) 15 (1954) 251; J-H. Shirley, Phys. Rev. 138 B (1965) 979; S. Leasure and R.E. Wyatt, to be published. [7] DG. Truhlar, J. Comp. Phys. 10 (1972) 123. (81 S-H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703; A.M.F. lau, Phys. Rev_ 14A (1976) 279; J-V. Moloney and W-J- Meath. Mol. Phys. 35 (1978) 1163. [9] MS. Sargent III, MO. ScuIIy and W-E. Lamb Jr., Laser physics (Addison-Wesley, Reading, 1974) pp_ 25,26_
629