Path integral approach to the problem of rotational excitation of molecules by an ultrashort laser pulses sequence
aa r X i v : . [ qu a n t - ph ] J u l Path integral approach to the problem of rotational excitation of moleculesby an ultrashort laser pulses sequence
Alexander Biryukov ∗ and Mark Shleenkov † Samara State University, Pavlov Street 1, Samara 443011, Russia (Dated: July 3, 2018)The amplitude and probability of quantum transitions are represented as a path integrals in en-ergy state space of the investigated multi-level quantum system. Using this approach we considerrotational dynamics of nitrogen molecules N and N which interact with a sequence of ultra-short laser pulses. Our computer simulations indicate the complex dependency of the high rotationstates excitation probability upon ultrashort laser pulses sequence periods. We observe pronouncedresonances, which correspond to the results of some experiments. PACS numbers:
I. INTRODUCTION
The modern development of laser radiation technolo-gies induces theoretical and experimental investigationsof the dynamics of quantum objects (such as atoms ormolecules) under the action of intense electromagneticfield of different forms.This dynamics is principally non-linear, because theprobability is high of multiphoton processes (absorptionand emission more the one photon) and nonresonant pro-cesses (electromagnetic field frequency is far from quan-tum transitions frequency). We note the recent studiesof different rare gases multiphoton ionization [1–3], ofmultiphoton photoemission of the Au(111) surface statewith 800-nm laser pulses [4], of multiphoton transitions inGaSb/GaAs quantum-dot intermediate-band solar cells[5], of three-photon electromagnetically induced absorp-tion in a ladder-type atomic system [6].There are certain difficulties for theoretical studies ofthese processes and for simulations of quantum objectsdynamics that interact with laser field. Thus, differentapproximations are used. For example, there are two-or three-level quantum system models [7] and rotatingwave approximation [8]. For high-intensity laser field theperturbation theory runs into problems. It is necessaryto calculate the large number of terms. High-order per-turbation theory for miltilevel quantum system dynam-ics was considered in [9]. For theoretical researches ofthis processes the numerical solution of time-dependentSchr¨odinger equation is used [10]. For this reason differ-ent schemes of space-time discretization is realized. Thediscretization parameter should be small enough for sim-ulations of a minute error.The perspective approach to theoretical studies ofthis quantum processes is path integral (functional inte-gral) formalism, which are formulated by R.P. Feynman[11, 12] and based on P.A.M. Dirac ideas [13, 14]. Atpresent this formalism is an abundantly used approach ∗ [email protected] † [email protected] in many fields of physics: lattice theories in QCD simu-lations [15] and those in graphene [16], semiclassical ap-proachs in atom optics [17, 18], black-swan events prob-lem [19], influence functional approach in quantum the-ory [20, 21] and many others.In this paper we present a new theoretical approachfor describing the dynamics of a quantum system, in-teracting with laser radiation by path integration in en-ergy states space. We obtain formulas for calculating thequantum transition amplitude and probability as path in-tegrals in energy states space (the space of discrete non-negative variables i.e. quantum numbers).Recent experimental [22] and theoretical [23, 24] inves-tigations point at possibilities of selective excitations ofnitrogen isotopes by a sequence of ultrashort laser pulses(a pulse train). We have developed and are applying thetheoretical approach to quantum resonances problem inmolecule rotational excitation by ultrashort laser pulses. II. QUANTUM TRANSITION PROBABILITYAS PATH INTEGRAL IN ENERGY STATESSPACE
We consider interaction of multilevel quantum system(such as an atom or a molecule) with electromagneticfield. The Hamiltonian ˆ H full describing our model isgiven as ˆ H full = ˆ H syst + ˆ V , (1)where ˆ H syst is Hamiltonian of the investigated quantumsystem. We define stationary eigenstates | l i with energies E l having the following properties:ˆ H syst = N − X l =0 E l | l ih l | , (2) N − X l =0 | l ih l | = 1 , h l ′ | l i = δ l ′ l ; (3)ˆ V – the interaction operator.Our main goal is to define the probability P ( l f , t | l in , | l in i at the moment t = 0 to the one | l f i at themoment t > ρ ( t ). The evolution equation of ˆ ρ ( t ) in Dirac (in-teraction) picture [14] is as follows:ˆ ρ ( t ) = ˆ U D ( t )ˆ ρ (0) ˆ U + D ( t ) , (4)where ˆ U D ( t ) = T exp[ − ı ~ t Z ˆ V D ( τ ) dτ ] (5)– the evolution operator in Dirac picture,ˆ V D ( τ ) = exp[ ı ~ ˆ H syst τ ] ˆ V ( τ ) exp[ − ı ~ ˆ H syst τ ] (6)– the operator of quantum system and electromagneticfield interaction in Dirac picture.Eq. (4) in energy representation is ρ l f m f ( t ) = X l in ,m in h l f | ˆ U D ( t ) | l in i ρ l in m in h m in | ˆ U + D ( t ) | m f i , (7) where ρ l f m f ( t ) = h l f | ˆ ρ ( t ) | m f i , ρ l in ,m in = h l in | ˆ ρ (0) | m ′ in i . By the use of evolution operator ˆ U group propertiesand completeness condition Eq. (3) of eigenvectors | l k i basis the kernel of evolution operator h l f | ˆ U D ( t ) | l in i canbe expressed as h l f | ˆ U D ( t ) | l in i = N − X l ,..,l K =0 K +1 Y k =1 h l k | ˆ U D ( t k , t k − ) | l k − i , (8)as long as t k > t k − and whereˆ U D ( t k , t k − ) = exp[ − ı ~ t k Z t k − ˆ V D ( τ ) dτ ] , (9)here we introduce the notations t K +1 = t , l K +1 = l f , t = 0, l = l in , K +1 P k =1 ( t k − t k − ) = t .In Appendix A we show that for small time inter-val ( t k − t k − ) → h l k | ˆ U D ( t k , t k − ) | l k − i can be expressed as h l k | ˆ U D ( t k , t k − ) | l k − i = Z exp[ ıS [ l k , l k − ; ξ k − ]] dξ k − , (10)where S [ l k , l k − ξ k − ] – dimensionless (in ~ units) actionin energy representation during time interval ( t k − t k − ) S [ l k , l k − ; ξ k − ] = 2 π ( l k − l k − ) ξ k − − t k Z t k − V l k l k − ( τ ) ~ π ( l k − l k − ) ξ k − − ω l k l k − τ ] dτ, (11)where V l k l k − ( τ ) = h l k | ˆ V ( τ ) | l k − i – interaction operatormatrix element.The probability P ( l f , t | l in ,
0) of transition from purequantum state ˆ ρ (0) = | l in ih l in | ( ρ l in m in (0) = δ l in m in ) atthe initial moment t = 0 to the quantum state ˆ ρ ( t ) = | l f ih l f | ( ρ l f m f ( t ) = P l f ( t ) = δ l f m f ) at the final moment t can be defined by using Eq. (7) P ( l f , t | l in ,
0) = U ∗ D ( l f , t | l in , U D ( l f , t | l in , , (12)where U D ( l f , t | l in ,
0) is transition amplitude Eq. (8).If at the initial moment t = 0 the state of the quantumsystem under investigation is expressed as distribution ˆ ρ (0) = N − P l in =0 P l in (0) | l in ih l in | ( ρ l in m in (0) = P l in (0) δ l in m in )over the pure eigenstates | l in i , the probability of quan-tum system observation in eigenstate l f at moment t hasthe following form: P ( l f , t | ρ (0)) = N − X l in =0 P ( l f , t | l in , P l in (0) . (13)We note that using Eq. (8), Eq. (10), Eq. (11) quan-tum transition amplitude U D ( l f , t | l in ,
0) for any t can beexpressed as path integral in energy eigenstates space h l f | ˆ U D ( t ) | l in i = U D ( l f , t | l in ,
0) = lim K →∞ N − X l ,..,l K =0 1 Z .. Z exp[ ıS [ l f , l K , ξ K ; .. ; l k , l k − , ξ k − ; .. ; l , l in , ξ ]] dξ ..dξ K , (14)where S [ l f , l K , ξ K ; .. ; l k , l k − , ξ k − ; .. ; l , l in , ξ ] == K +1 X k =1 S [ l k , l k − , ξ k − ] (15)– dimensionless action. It is a functional, which is de- fined on a path set in discrete variables l k space of size N (quantum system levels number) and continuous c-number variables ξ k space [0 , ρ l f ,m f ( t ) = lim K →∞ N − X l in ,..,l K =0 N − X m in ,..,m K =0 1 Z .. Z dξ ..dξ K dζ ..dζ K exp[ ı ( S [ l f , l K , ξ K ; .. ; l k , l k − , ξ k − ; .. ; l , l in , ξ ] −− S [ m f , m K , ζ K ; .. ; m k , m k − , ζ k − ; .. ; m , m in , ζ ])] ρ l in ,m in (0) , (16) P ( l f , t | l in ,
0) = lim K →∞ N − X l ,..,l K =0 N − X m ,..,m K =0 1 Z .. Z dξ ..dξ K dζ ..dζ K exp[ ı ( S [ l f , l K , ξ K ; .. ; l k , l k − , ξ k − ; .. ; l , l in , ξ ] −− S [ l f , m K , ζ K ; .. ; m k , m k − , ζ k − ; .. ; m , l in , ζ ])] . (17)Thus, Eq. (16), Eq. (17) with Eq. (15) and Eq. (11)are the closed equations system for describing of transi-tions of miltilevel quantum system interacting with elec-tromagnetic field by interaction operator ˆ V . III. ROTATIONAL DYNAMICS OF N AND N INTERACTING WITH LASER PULSESSEQUENCES
Recent results of experimental observation of N and N high rotational states excitation were published in[22]. Detailed discussions of the results were in [23, 24].In the experiments the groups of N and N molecules were investigated. At the initial moment thedistribution of rotational population is thermal and cor-responds to T = 6 . . . I = 5 ∗ W/cm . The relative populations were measured of therotational levels of N and N and the functional de-pendence of the populations on the pulse train periodwas obtained.The results of these experiments show that there arequantum nonlinear resonances i.e. the nonlinear increaseof rotational excitation efficiency under specific valuesof the pulse train period. The most efficient populationtransfer up the rotational ladder occurs around 8 . N and 9 ps for N .We analyse these experiments using the method devel-oped by us which is based on path integral formulationin energy states space.The initial distribution of rotational population is ther-mal and corresponds to T = 6 . P l in = 1 Z exp[ − E l in k B T ] , (18)where Z = N − X l in =0 exp[ − E l in k B T ] (19)– particle function, k – Boltzmann factor, T – absolutetemperature, N – rotational states number in the theo-retical model.We calculate the energy E l of investigated moleculesrotational levels for quantum rigid rotor model [25] − ~ I θ ∂∂θ (sin θ ∂∂θ ) Y l ( θ ) = E l Y l ( θ ) , (20)where I = µR – moment of inertia, µ – molecule re-duced mass, R – atom distances, Y l ( θ ) = Y l ( θ, φ ), where Y ml ( θ, φ ) – spherical harmonics.Eq. (20) defines the rotational energy spectrum of adiatomic molecule E l = ~ I l ( l + 1) , where l – azimuthal quantum number.It is known, that nonpolar molecule dipole moment isequal to zero. However, strong laser fields induce themolecular dipole by exerting an angle-dependent torque.The interaction is described by the potential [26, 27] V ( τ ) = −
14 ∆ αE ( τ ) cos θ, (21)
80 82 84 86 88 90 92Train period, τ per , 10 -1 ps 0 1 2 3 4 5 6 7 Rotational number, l (a) N
80 82 84 86 88 90 92Train period, τ per , 10 -1 ps 0 0.2 0.4 0.6 0.8 1 (b) N FIG. 1. (color online) Nitrogen N (a) and N (b) molecules observation probabilities in rotation states l = 0 , , . . . l and the pulses train period τ . We normalize them by their maximum values for eachrotational state. where ∆ α describes the molecular polarizability, θ is theangle between the molecular axis and the field polariza-tion. Matrix elements of interaction operator are V l ′ l ( τ ) = −
14 ∆ αE ( τ ) h l ′ | cos θ | l i , (22)where h l ′ | cos θ | l i = 2 π π Z Y ∗ l ′ ( θ ) cos θY l ( θ ) sin θdθ. (23)Matrix elements h l ′ | cos θ | l i were numerically calcu-lated by Eq. (22) and Eq. (23).The investigated molecules parameters are [28]: ∆ α =1 . ∗ − C ∗ m /V, I = 1 . ∗ − kg ∗ m for N , I = 1 . ∗ − kg ∗ m for N .We consider a sequence of ultrashort laser pulses whichwas used in [22]. The electric field value is as follows E ( τ ) = X n = − J n ( A ) E exp[ − ( τ − nτ per ) τ pul ] , (24)were J n ( A ) is Bessel function of the first kind, A = 2 . E ≈ × V/m is electric field value, τ pul ≈
500 fs is each laserpulse duration, 7 .
98 ps ≤ τ per ≤ .
38 ps is pulse trainperiod.We are considering the model of N with N = 8 ro-tational levels ( l = 0 , , . . . , − . The results are givenin Fig. 1. In Fig. 1 (two-dimensional map) we present normalizedprobability of N and N molecules rotational stateobservation after they have interacted with 7 laser pulsesunder different pulses train periods. For the pulse trainperiod equal to 8 .
38 ps for N and 8 .
98 ps for N thepopulation is efficiently transferred from the initial (ther-mal distribution) states l = 0 , , l = 3 , , , , IV. CONCLUSION
In this paper we present new method of calculatingthe transition probability of a quantum system interact-ing with electromagnetic field by the path integral for-malism. We construct the amplitude and probabilityof quantum transition as path integrals in energy statesspace. The algorithm of path integral calculation was de-veloped. This approach enables us to perform computersimulations of molecule dynamics induced by a laser field.By the deduced formulas we describe quantum res-onances in dynamics of nitrogen molecules, that inter-act with a sequence of ultrashort laser pulses. The ob-tained results are in good agreement with the experimen-tal data [22] and the theoretical investigations [23, 24] bySchr¨odinger equation numerical solution.The approach developed is appliable to nonperturba-tive studies of different multiphoton and nonresonantprocesses.
ACKNOWLEDGMENTS
The work is supported by the Ministry of Educationand Science of Russian Federation (grant 2.870.2011).Numerical calculations were performed at Samara StateAerospace University by supercomputer ”Sergey Ko-rolev”.
Appendix A: Path integral formulationin energy representation
We consider the evolution operator kernel Eq. (9) as aseries and for the time interval ( t k − t k − ) → h l k | ˆ U D ( t k , t k − ) | l k − i = h l k | l k − i −− ı ~ t k Z t k − h l k | ˆ V D ( τ ) | l k − i dτ. (A1)By using Eq. (2) and Eq. (6), the quantum system andelectromagnetic field interaction operator is expressed asˆ V D ( τ ) = N X l ′ ,l =1 V l ′ l ( τ ) exp[ ıω l ′ l τ ] | l ′ ih l | , (A2)where V l ′ l ( τ ) – interaction operator matrix element, ω l ′ l = ( E l ′ − E l ) / ~ – frequency of quantum transitionbetween eigenstates with eigenvalues (energies) E l ′ and E l . Using Eq. (A2), interaction operator matrix elementin Dirac picture h l k | ˆ V D ( τ ) | l k − i is expressed h l k | ˆ V D ( τ ) | l k − i = V l k l k − ( τ ) exp[ ıω l k l k − τ ] . (A3)Thus, we conclude h l k | ˆ U D ( t k , t k − ) | l k − i = δ l k l k − −− ı ~ t k Z t k − V l k l k − ( τ ) exp[ ıω l k l k − τ ] dτ. (A4)Now we prove, that the kernel h l k | ˆ U ( t k , t k − ) | l k − i ofevolution operator can be expressed as h l k | ˆ U D ( t k , t k − ) | l k − i = Z exp[ ıS [ l k , l k − ; ξ k − ]] dξ k − , (A5)where dimensionless action S [ l k , l k − ; ξ k − ] is found inEq. (11).For this proof, by using Eq. (11) we transform Eq. (A5)into Eq. (A4). h l k | ˆ U D ( t k , t k − ) | l k − i = Z exp[2 πı ( l k − l k − ) ξ k − ] exp[ − ı ~ t k Z t k − V l k l k − ( τ )2 cos[2 π ( l k − l k − ) ξ k − − ω l k l k − τ ] dτ ] dξ k − == Z exp[2 πı ( l k − l k − ) ξ k − ](1 − ı ~ t k Z t k − V l k l k − ( τ )2 cos[2 π ( l k − l k − ) ξ k − − ω l k l k − τ ] dτ ) dξ k − == Z exp[2 πı ( l k − l k − ) ξ k − ] dξ k − − ı ~ t k Z t k − V l k l k − ( τ ) Z (exp[4 πı ( l k − l k − ) ξ k − − ıω l k l k − τ ] + exp[ ıω l k l k − τ ]) dξ k − dτ == δ l k l k − − ı ~ t k Z t k − V l k l k − ( τ ) exp[ ıω l k l k − τ ] dτ. For this we use the facts, that the diagonal matrix ele-ment V ll ( τ ) is equal to zero and integral representationsof Kronecker symbol has the form: δ l k l k − = Z exp[2 πın ( l k − l k − ) ξ k − ] dξ k − , (A6)where n – integer. So, we have proved the equivalence of Eq. (A5) andEq. (A4), which define the quantum transition amplitudefor time interval t k − t k − → Appendix B: Numerical simulation algorithm
In this appendix we consider algorithm for nu-merical calculation of quantum transition amplitude U ( l f , t | l in ,
0) and probability P ( l f , t | l in , (cid:18) ℜ [ ˜ U ( l k , t k | l in , ℑ [ ˜ U ( l k , t k | l in , (cid:19) = N − X l k − =0 1 Z (cid:18) cos[ S [ l k , l k − ; ξ k − ]] − sin[ S [ l k , l k − ; ξ k − )]]sin[ S [ l k , l k − ; ξ k − ]] cos[ S [ l k , l k − ; ξ k − )]] (cid:19) (cid:18) ℜ [ U ( l k − , t k − | l in , ℑ [ U ( l k − , t k − | l in , (cid:19) dξ k − , (B1)where ℜ [ . . . ], ℑ [ . . . ] – real and imaginary components;explicit form of S [ l k , l k − ; ξ k − ] is defined by Eq. (11).The initial condition for pure quantum state | l in i is asfollows (cid:18) ℜ [ U ( l , | l in , ℑ [ U ( l , | l in , (cid:19) = (cid:18) δ l l in (cid:19) . (B2)Quantum transition probability P ( l k , t k | l in ,
0) of inves-tigated system from the state | l in i at moment t = 0 tothe state | l k i at moment t k can be expressed as P ( l k , t k | l in ,
0) = ℜ [ U ( l k , t k | l in , + ℑ [ U ( l k , t k | l in , , (B3)where normalized real and imaginary components of thetransition amplitude are (cid:18) ℜ [ U ( l k , t k | l in , ℑ [ U ( l k , t k | l in , (cid:19) = A − (cid:18) ℜ [ ˜ U ( l k , t k | l in , ℑ [ ˜ U ( l k , t k | l in , (cid:19) . (B4)The normalizing factor A is calculated by the followingformula: A = N − X l k =0 ( ℜ [ ˜ U ( l k , t k | l in , + ℑ [ ˜ U ( l k , t k | l in , ) . (B5)Using Eq. (B1)–(B5) we calculate the amplitude U ( l f , t | l in ,
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