We study the dynamics of quantum system interacting with electromagnetic field. We present density matrix and transition probability as a path integrals in energy state space without resonance and rotating wave approxinations. By the use of obrained equations we develop an algorithm for numerical simulations of the dynamics of quantum system interacting with electromagnetic field. Using this approach we consider rotational dynamics of nitrogen molecules 14N2 and 15N2 which interact with a sequence of ultrashort laser pulses. Our computer simulations indicate the complex dependency of the high rotation states excitation probability upon ultrashort laser pulses sequence periods. We observe pronounced resonances, which correspond to the results of some experiments.
2. Mathematical model of quantum system interacting with electromagnetic field N−1 X |lihl| = 1, l=0
Our main goal is to define the probability P(l f , t|lin, 0) of investigated quantum system transition from eigenstate |lini at the moment t = 0 to the one |l f i at the moment t > 0.
We describe the investigated system by statistical operator ρˆ(t). The evolution equation of ρˆ(t) in Dirac (interaction) picture [15] is as follows:
ρˆ(t) = Uˆ D(t)ρˆ(0)Uˆ D+(t), where ρˆ(0) — statistical operator at initial time moment t = 0, – the evolution operator in Dirac picture, – the operator of quantum system and electromagnetic field interaction in Dirac picture.
ı Z Uˆ D(t) = T exp[− ~
Vˆ D(τ)dτ] 0
t
Vˆ D(τ) = exp[ ~ı Hˆ systτ]Vˆ (τ) exp[− ~ı Hˆ systτ] ρlf mf (t) =
X hl f |Uˆ D(t)|liniρlinmin hmin|Uˆ D+(t)|m f i,
lin,min
ρlf mf (t) = hl f |ρˆ(t)|m f i,
ρlin,min = hlin|ρˆ(0)|mi′ni
(
The probability of a quantum state observation is to define as diagonal matrix element. At initial time moment t = 0 it is equal to ρlinlin (t = 0) = ρlin . At final time moment t it is equal to ρlf lf (t) = ρlf (t).
where lin, l f = 1, 2, . . ..
The quantum transition probability from state |lini or ρlin (0) = δlinnin ) at time moment t = 0 to state |l f i or ρlf (t) = ρlf mf (t)δlf mf at time moment t > 0 is to describe as follows
For numerical calculation ρlf mf (t), ρlf (t), P(l f , t|lin, 0) by the use of eq. (
For that reason we use evolution operator Uˆ group properties and express the evolution operator hl f |Uˆ D(t)|lini as ρlf (t) = Xhl f |Uˆ (t)|linihlin|Uˆ †|l f iρlin ,
lin P(l f , t|lin, t) = hl f |Uˆ D(t)|linihl f |Uˆ D(t)|lini ρlf (t) = X P l f , t|lin, 0 ρlin (0) lin
K+1 Uˆ D(t) = Y Uˆ D(tk, tk−1), k=1
ı Z Uˆ D(tk, tk−1) = exp[− ~
Vˆ D(τ)dτ], tk tk−1 hl f |Uˆ D(t)|lini =
N−1 K+1
X Yhlk|Uˆ D(tk, tk−1)|lk−1i, l1,...lK=0 k=1
as long as tk > tk−1 and where K+1 here we introduce the notations tK+1 = t, lK+1 = l f , t0 = 0, l0 = lin, P (tk − tk−1) = t.
k=1
By the use of eq. (
hlk|Vˆ D(τ)|lk−1i = Vlklk−1 (τ) exp[ıωlklk−1 τ]. where Vlklk−1 (τ) = hlk|Vˆ (τ)|lk−1i – interaction operator matrix element, ωl′l = (El′ − El)/~ – frequency of quantum transition between eigenstates with eigenvalues (energies) El′ and El.
It is possible to prove that for small time interval (tk − tk−1) → 0 the evolution operator kernel hlk|Uˆ D(tk, tk−1)|lk−1i eq. (
exp[ıS [lk, lk−1; ξk−1]]dξk−1, 0 where S [lk, lk−1ξk−1] – dimensionless (in ~ units) action in energy representation during time interval (tk − tk−1) S [lk, lk−1; ξk−1] = 2π(lk − lk−1)ξk−1 −
2 cos[2π(lk − lk−1)ξk−1 − ωlklk−1 τ]dτ, where Vlklk−1 (τ) = hlk|Vˆ (τ)|lk−1i – interaction operator matrix element.
1 = Z 0
ı Z exp[2πı(lk − lk−1)ξk−1] exp[− ~
tk tk−1
hlk|Uˆ D(tk, tk−1)|lk−1i =
Vlklk−1 (τ)2 cos[2π(lk − lk−1)ξk−1 − ωlklk−1 τ]dτ]dξk−1 ı Z exp[− ~ where
2 cos[2π(lk − lk−1)ξk−1 − ωlklk−1 τ] = e−ı[2π(lk−lk−1)ξk−1−ωlklk−1 τ + e+ı[2π(lk−lk−1)ξk−1−ωlklk−1 τ.
Using eq. (
tk ı Z − ~
tk−1 tk ı Z − ~ tk−1
1 Z 0
1 Z 0 Vlklk−1 (τ)
(exp[4πı(lk − lk−1)ξk−1 − ıωlklk−1 τ] + exp[ıωlklk−1 τ])dξk−1dτ = Vlklk−1 (τ) exp[ıωlklk−1 τ] + exp[4πı(lk − lk−1)ξk−1] · exp[−ıωlklk−1 τ] dτdξk−1 1 = Z 0
hlk|Uˆ D(tk, tk−1)|lk−1i =
exp[2πı(lk − lk−1)ξk−1]dξk−1 −
exp[2πı(lk − lk−1)ξk−1]dξk−1 −
3rd International conference Information Technology and Nanotechnology, ITNT-2017
(
if n = 1, 2, . . . is integer.
exp[4πı(lk − lk−1)ξk−1]dξk−1 = δlklk−1 , ı Z hlk|Uˆ D(tk, tk−1)|lk−1i = δlklk−1 − ~
ı Z Vlklk−1 (τ) exp[ıωlklk−1 τ]dτ − ~
Vlklk−1 (τ) exp[−ıωlklk−1 τ]dτ
By the use of eq. (
We note that using Eq. (
The quantum transition amplitude eq. (
However analytical calculation eq. (
We consider algorithm for numerical calculation of quantum transition amplitude U(l f , t|lin, 0) and probability P(l f , t|lin, 0).
Using eq. (
U(lK , tK |lin, 0) = X U(lK , tK |lk−1, tk−1)U(lk−1, tk−1|lin, 0),
lk−1
U(lK , tK |lin, 0) = hlK |Uˆ D(tK )|lini, U(lK , tK |lk−1, tk−1) = hlK |Uˆ D(tK , tk−1)|lk−1i,
U(lk−1, tk−1|lin, 0) = hlk−1|Uˆ D(tk−1)|lini. where hl f |Uˆ D(t)|lini = UD(l f , t|lin, 0) = lim
K→∞
XN−1 Z1 .. Z1
N−1 ℜ[U(lk, tk|lin, 0)] = X
N−1 ℑ[U(lk, tk|lin, 0)] = X lk−1=0 lk−1=0 ℜ[U(lk, tk|lk−1, tk−1)]ℜ[U(lk−1, tk−1|lin, 0)] − ℑ[U(lk, tk|lk−1, tk−1)]ℑ[U(lk−1, tk−1|lin, 0)] ℑ[U(lk, tk|lk−1, tk−1)]ℜ[U(lk−1, tk−1|lin, 0)] + ℜ[U(lk, tk|lk−1, tk−1)]ℑ[U(lk−1, tk−1|lin, 0)]
ℜ[U˜ (lk, tk|lin, 0)] ! = ℑ[U˜ (lk, tk|lin, 0)]
Quantum transition probability P(lk, tk|lin, 0) of investigated system from the state |lini at moment t = 0 to the state |lki at moment tk can be expressed as
P(lk, tk|lin, 0) =
ℜ[U(lk, tk|lin, 0)] 2 + ℑ[U(lk, tk|lin, 0)] 2 , The transition probability P(lk, tk|lin, 0) must be normalized for each time moments tk For this we calculate ℜ[U(lk, tk|lin, 0)] and ℑ[U(lk, tk|lin, 0)] using eq. (34) and product them on normalizing factor A: N−1 X P(lk, tk|lin, 0) = 1.
lk=0 ℜ[U(lk, tk|lin, 0)] ! = A ℑ[U(lk, tk|lin, 0)] ℜ[U(lk, tk|lin, 0)] ! .
ℑ[U(lk, tk|lin, 0)] N−1 −1/2 A = X(ℜ[U˜ (lk, tk|lin, 0)]2 + ℑ[U˜ (lk, tk|lin, 0)]2) lk=0 .
(32) (33) (34) (35) (36) (37) (38) (39)
Using Eq. (30)–(37) we calculate the amplitude U(l f , t|lin, 0), the transition probability P(l f , t|lin, 0) and the probability of quantum state observation for any t. 4. Rotational dynamics of 14N2 and 15N2 interacting with laser pulses sequences
Recent results of experimental observation of 14 N2 and 15 N2 high rotational states excitation were published in [12]. Detailed discussions of the results were in [14, 13].
In the experiments the groups of 14 N2 and 15 N2 molecules were investigated. At the initial moment the distribution of rotational population is thermal and corresponds to T = 6.3 K. Molecules interact with a sequence of ultrashort laser pulses with period from 6.5 ps to 9.5 ps. Each laser pulse has duration equal 500 fs. Laser radiation intensity reaches the value I = 5 ∗ 1012 W/cm2. The relative populations were measured of the rotational levels of 14N2 and 15 N2 and the functional dependence of the populations on the pulse train period was obtained.
The results of these experiments show that there are quantum nonlinear resonances i.e. the nonlinear increase of rotational excitation efficiency under specific values of the pulse train period. The most efficient population transfer up the rotational ladder occurs around 8.4 ps for 14 N2 and 9 ps for 15 N2.
We analyse these experiments using the method developed by us which is based on path integral formulation in energy states space.
We calculate the energy El of investigated molecules rotational levels for quantum rigid rotor model [16] − ~2
1 ∂ 2I sin θ ∂θ ∂ ∂θ (sin θ )Yl(θ) = ElYl(θ), where I = μR2 – moment of inertia, μ – molecule reduced mass,
Yl(θ) = Yl0(θ, φ), where Ylm(θ, φ) – spherical harmonics.
2I
3rd International conference “Information Technology and Nanotechnology 2017” ~2 El = l(l + 1), where where were Jn(A) is Bessel function of the first kind, A = 2.5 is the spectral phase modulation amplitude,
τpul ≈ 500 fs is each laser pulse duration,
We are considering the model of N2 with N = 8 rotational levels (l = 0, 1, . . . , 7). This model is a good approximation, because in experiments [12] higher rotational states are practically not excited.
The initial distribution of rotational population is thermal and corresponds to T = 6.3 K: where l – azimuthal quantum number.
It is known, that nonpolar molecule dipole moment is equal to zero. However, strong laser fields induce the molecular dipole by exerting an angle-dependent torque.
where Δα describes the molecular polarizability, θ is the angle between the molecular axis and the field polarization.
Plin = Z exp[− Elin ], kBT Z = lin=0 N−1 X exp[− Elin ] kBT — particle function, k — Boltzmann factor,
By the use of Eq. (44)-(45) and numerical simulation algorithm we calculate the probability of excitation from the initial state (Boltzmann distribution) to different rotational states having interacted with a sequence of ultrashort laser pulses as a function of pulse train period. The absolute error of our probability calculation was not more than 10−3. The results of our numerical simulations are given in fig. 1, fig. 2 and agree well with experimental data as for 14N2 both for 15N2 molecules.
In fig. 1 we present the population of 14N2 molucels on diffent rotational quantum level l after interaction with pulse train. For the pulse train period equal to 2.79 ps, 5.58 ps and 8.38 ps for 14N2 the population is efficiently transferred from the initial (thermal distribution) states l = 0, 1, 2 to the higher states l = 3, 4, 5, 6, 7. The resonanse train period value τ = 8.38 ps was observed in experiment [12].
In fig. 2 we present normalized probability of 14N2 molecules rotational state observation after they have interacted with 7 laser pulses with period τ = 8.38 ps and different values of laser pulses maximum intensity 0.5I0, I0 and 2I0, where I0 = 5 ∗ 1012 W/cm2. We note the population of high rotational state is depend on intensity of laser pulses non-linearly.
In this paper we present new method of calculating the transition probability of a quantum system interacting with electromagnetic field by the path integral formalism. We construct the amplitude and probability of quantum transition as path integrals (40) (41) (42) (43) (44) (45) l ,P 0.1 y iilt b a b o r p n o it a v r e sb 0.01 O in energy states space. The algorithm of path integral calculation was developed. This approach enables us to perform computer simulations of molecule dynamics induced by a laser field.
By the deduced formulas we describe quantum resonances in dynamics of nitrogen molecules, that interact with a sequence of ultrashort laser pulses. The obtained results are in good agreement with the experimental data [12] and the theoretical investigations [14, 13] by Schro¨ dinger equation numerical solution.
The approach developed is appliable to nonperturbative studies of different multiphoton and nonresonant processes.