Calculation of the probability of quantum transitions in the system by the method of functional integration

Describing the interaction between a quantum system and an intense electromagnetic field is quite difficult when using means of the theory of perturbation [1, 2]. Developing nonperturbative ways of describing the evolution of a quantum system exposed to an electromagnetic field is therefore important. One non-perturbative approach is the formalism of functional integration (integrating over trajectories) [3, 4]. In this work, the probability of transitions over any interval of time between the states of a quantum system interacting with an electromagnetic field are presented in the form of an integral over trajectories in the space of the energy states of a system. Since it is difficult to obtain an analytical solution for these integrals, we propose an algorithm for their numerical integration that is based on recurrent relationships.


Introduction
Describing the interaction between a quantum system and an intense electromagnetic field is quite difficult when using means of the theory of perturbation [1,2]. Developing nonperturbative ways of describing the evolution of a quantum system exposed to an elec-tromagnetic field is therefore important. One non-perturbative approach is the formalism of functional integration (integrating over trajectories) [3,4].
In this work, the probability of transitions over any interval of time between the states of a quantum system interacting with an electromagnetic field are presented in the form of an integral over trajectories in the space of the energy states of a system. Since it is difficult to obtain an analytical solution for these integrals, we propose an algorithm for their numerical integration that is based on recurrent relationships.

Probabilities of transitions in the formalism of functional integrals in the space of energy states
Let us consider a multilevel quantum system that interacts with an electromagnetic field. It is described by HamiltonianĤ, written asĤ whereĤ syst is the Hamiltonian of the quantem system that determines its stationary states |n with energies E n : H syst |n = E n |n , n |n = δ n n N n=1 |n n| = 1, n = 1, 2, . . .
whereV is the operator of interaction between the system and the electromagnetic field, and N is the number of quantum states of the system. We shall describe a system at moment t by state vector |t , whose equation of evolution in our concept of interaction has the form whereÛ D (t),V D (τ) are the operators of evolution and interaction in Dirac's concept. We shall assume that in our model, the operator of interaction between an electron of the system and a plane electromagnetic wave is determined by the expression wherex, e are the coordinate and the electron charge, respectively; E 0 x (τ) is the projection of the wave amplitude onto axis x, wich can change over time; and Ω is the field's frequency of vibration. In the basis presented by vectors (2), we write operator (4) in the form where Ω R n n (τ) = e| n |x|n |E 0 x (τ)/ is the Rabi frequency and ω n n = (E n − E n ) / is the frequency of a transition between levels.
Our problem is to determine probabilities P(n f , t|n in , 0) of transitioning between quantum states |n in and |n f when t = 0 and t > 0 for a system under the influence of an electromagnetic field.
We write Eq. (3) in the energy representation, using the basis presenter by vectors (2), where n f |Û D (t)|n in is the amplitude of a quantum transition from state |n in at moment t = 0 to state |n f at moment t > 0. Probability P(n f , t|n in , 0) of this quantum transition is correspondingly expressed as To calculate probability of transition (8), we must specify the explicit form of transition amplitude n f |Û D (t)|n in for finite interval of time t, using the group properties of operator U D (t) and the entirety of state vectors |n , which we write in the form where t k = t, n K = n f , t 0 = 0, n in = n 0 and |n k are the vectors of state at moment t k ; is the amplitude of the quantum transition between states |n k−1 and |n k for the brief interval of time t k − t k−1 . Once amplitude n k |Û D (t)|n k−1 of transition is plotted in explicit form according to formula (9), we find explicit amplitude of transition for any range of times t. We now show for proposed model of interaction (6) that amplitude (10) of quantum transition for the fairly brief interval of time t k − t k−1 = ∆t k has the form [5] where operation S [n k , t k ; n k−1 , t k−1 ; ξ k−1 ] in units of is defined by the expression where t k = 1 2 (t k + t k−1 ) and ω n k n k−1 = 1 E n k − E n k−1 is frequency of thatsition between levels. The amplitude of a system transitioning between quantum states for any interval of time t is determined by (9) with allowance for formulas (11) and (12): where S n f , n K−1 , ξ K−1 ; . . . ; n 1 , n i n, k and operatorsR k have the form It is clear that the conjugate amplitude of transition has exactly the same structure with a change in the sign before imaginary unit i. Substituting formula (13) and conjugate amplitude into expression (8), we obtain the probability of a quantum transition from state |n in in to state |n f for finite interval of time t.

Calculation of transitions probabilities for hydrogen atom
The proposed formalism is applicable to the description of quantum transitions in a hydrogen atom under the action of a monochromatic polarized electromagnetic wave field. The probabilities of transition between quantum states 2s, 3s, 3p, 4s, 4p, 4d with energies respectively E 2 , E 3 , E 4 for the time interval t will be calculated by the formula (8). The probability amplitudes corresponding to these transitions will be determined by (11) where the action S is concretized by the model.
For the numerical calculation of the amplitude, we present it in the form of a recurrent formula where S is defined by (12). We calculate the probabilities of hydrogen atom staying at the levels 3s, 3p, 4s, 4p, 4d, initially prepared in the state 2s. The electic field E is directed along the z axis. Non-zero Rabi frequencies between quantum states for E = 0.5 · 10 6 V/m are Ω R 2s,3p /2π = 1.80 GHz, Ω R 2s,4p /2π = 0.75 GHz, Ω R 3s,4p /2π = 3.22 GHz, Ω R 3p,4s /2π = 2.87 GHz, Ω R 3p,4d /2π = 3.98 GHz.  The results of the numerical calculation are presented in figures 1-4 (there are the results for the amplitude of the electric field E = 0.5 · 10 6 V/m (left) and E = 10 6 V/m (right).
The figure 1 illustrate the classical Rabi oscillations. The figure 2 shows the probability of detection of the system in the 4p state. Here we illustrate the process of multiphoton transition from the 2s state. Figures 3 and 4 shows the probability of system detection in the 4s and 4d states from the 3p state.
Thus, the proposed the method successfully describes both single-photon and multiphoton transitions and transitions through virtual levels. The presence of multiphoton processes is confirmed by the probability of transitions increasing with an increase in the electric field strength. The presence of multiphoton processes is confirmed by the dependence of the probability of transitions on the electric field.

Conclusion
The proposed method successfully describes quantum transitions in the system under the action of an electromagnetic wave. This formalism describes both single-photon processes and multi-photon processes. The analysis shows that it is suitable for describing a wide class of quantum systems.