1. QUANTUM TRANSITIONS WITHIN THE FUNCTIONAL INTEGRATION REAL FUNCTIONAL
XXIII International Workshop
On High Energy Physics and Quantum Field Theory
QFTHEP’2017
Yaroslavl, Russia, June 26 – July 3, 2017
QUANTUM TRANSITIONS WITHIN THE FUNCTIONAL INTEGRATION REAL FUNCTIONAL
Alexander Biryukov, Yana Degtyareva, Mark Shleenkov
Samara University, Russia

2. INTRODUCTION
At present, nonlinear processes of interaction of microsystems with laser radiation of various configurations and varying degrees of intensity (excitation and dissociation of molecules, ionization of atoms under the action of laser radiation, etc.) are actively being studied.
It seems relevant to investigate the probabilities of transitions in multilevel quantum systems outside the framework of perturbation theory and approximations, which impose limitations on the structure of laser radiation, both in intensity and in the form of pulses.
In this r, it is proposed to construct a theory of the probabilities of quantum transitions in quantum structures, relying on the methods of functional integration in quantum theory.

3. MODEL OF THE QUANTUM SYSTEM
We investigate a model that is described by the Hamiltonian:
(1)
where
(2)
is the operator of interaction of a system with an external field, in the case of
a dipole Interaction it has the form:
(3)

4. GOALS AND OBJECTIVES OF THE STUDY
We will determine the probabilities of a quantum transition of the system from the ground state at time t = 0 into the state at

5. DESCRIPTION OF THE EVOLUTION OF THE QUANTUM SYSTEM
The state of a quantum system is described by a statistical operator
The evolution equation for the operator has the form:
(4)
where the evolution operator is represented by the expression:
(5)
(6)
- the operator of interaction of a quantum system with an electromagnetic field.

6. ENERGY REPRESENTATION FOR STATISTICAL OPERATOR AND DENSITY MATRIX
In the energy representation, the statistical operator of the density matrix is defined by the expression: where
(7)
- evolution operator in the Dirac representation

7. REPRESENTATION OF THE EVOLUTION OPERATOR BY THE FUNCTIONAL INTEGRAL IN THE SPACE OF QUANTUM STATES
Taking into account the completeness properties of the ground state vectors and the group properties of the evolution operator where и evolutionary kernel can be represented as and
(8)
(9)
(10)

8. REPRESENTATION OF THE EVOLUTION OPERATOR BY THE FUNCTIONAL INTEGRAL IN THE SPACE OF QUANTUM STATES
It was shown earlier that for small time intervals the evolution operator nucleus is representable in the form: where is the dimensionless (in units ) action in the energy representation For a period of time
(11)
(12)

9. TRANSITION PROBABILITY OF A QUANTUM SYSTEM
It was shown that the transition probability between levels
and
is determined by the formula :
(13)
where
The multiplicity of the integral is 2(К+1),
А is determined from condition :

10. TRANSITION PROBABILITY OF A QUANTUM SYSTEM (2)
where the action has the form :
(14)
- external field frequency,
- frequency of transition between energy levels and
- Rabi frequency

11. NUMERICAL SIMULATION
The proposed formula makes it possible to find the probabilities of quantum transitions for laser fields of any intensity and structure. However, in the process of calculating transition probabilities for specific models, it was found that calculation by this formula becomes extremely resource intensive. Computers with a finite number of nuclei allow us to find the transition probabilities for relatively small time intervals (the fraction of the Rabi oscillation period) and for quantum systems of small dimension. The analysis showed that calculation by this formula requires large computer resources, which seems to be limited in the practice of specific calculations. We propose to develop a method for calculating the reduced multidimensional integrals using recurrent formulas.

12. NUMERICAL SIMULATION USING RECURRENT RELATIONS
We introduce a function for calculating the transition probability in the form
(15):
where
then the transition probability will be determined by formula
(16)
where
(17)

13. NUMERICAL SIMULATION
The function (15) can be written:
(18)

14. NUMERICAL SIMULATION
In the expression (18) we introduce the notation: and Then
(19)
The formula (19) is a recurrence relation, allowing on each K-th step to find the transition
probabilities, using for calculations the values of the functions, calculated at the (K-1)-th step.

15. NUMERICAL SIMULATION
To calculate (19), we need to know the values of the function Taking into account the recurrence relation for the sine of the sum, this function can be calculated from the formula:
(20)

16. NUMERICAL SIMULATION
Thus, we obtain a chain of equations: The transition probability will be determined by the formula:
(21)

17. NUMERICAL SIMULATION
To calculate the transition probabilities according to the proposed scheme, it is necessary to know the state of the system at the initial instant of time. At the initial time the system is in the state , which can be interpreted as the probability of a transition from state to state for a time , therefore Therefore, for K = 1, the probability functions have the form :
(22)
(23)

18. RESULTS FOR THE TWO-LEVEL MODEL
The graphs show the probability curves for the transitions of a two-level system in case of resonance. The results obtained coincide exactly with the analytical solution.

19. RESULTS FOR THE TWO-LEVEL MODEL
In the case when the frequency of the external radiation does not coincide with the transition
frequency, the amplitude fluctuations in the probability of change. This fact was also verified for
the studied system.

20. THREE-LEVEL MODEL
In the experiments of Gentile and co-authors, two-photon transitions in calcium between Rydberg (highly excited) states of 52p and 51p were investigated. In the experimental setup, the valence electron in calcium atoms from its ground state is excited to the Rydberg state 52p, which we denote by | i>. From this state, an atom can go to the state 51d by single-photon radiation, which we denote by | a>. From the level | a> an electron with a single photon emission can go to the level | f>. Gentile, T. R. Experimental study of one- and two-photon Rabi oscillations / T. R. Gentile, B. J. Hughey, D. Kleppner [et al.] // Phys.Rev. A V. 40, no 9 - P

21. THREE-LEVEL MODEL
Let us consider a three-level system in the framework of the approximation of a rotating wave. The system interacts with an electromagnetic field whose amplitude is E = 12 V / m, and the frequency Ω / 2π = GHz. Under the action of the field, the electron initially in the state | i> passes with the emission of one photon into the state | a>. From the level | a> the electron can go to the level | f> with the emission of one photon. One- photon transitions between the levels | i> and | f> are forbidden. However, two-photon transitions between these levels are possible, which was shown experimentally.

22. THREE-LEVEL MODEL
In this work the numerical simulation of the evolution of a three-level system by the method of functional integration was carried out using recurrence relations. The frequencies of quantum transitions between levels of atoms are determined by the expressions: Parameters of Rabi oscillations are given by the following expressions:

23. RESULTS FOR THE THREE-LEVEL MODEL

24. RESULTS FOR THE THREE-LEVEL MODEL
It can be seen from the graphs that the period of the Rabi oscillations is 224 ns, which
corresponds to experimentally observed value.

25. CONCLUSION
The obtained results showed that the chosen method of numerical integration yields good results that satisfy the requirements of the required accuracy. The main advantage of the proposed method is the possibility of its implementation on modern personal computers and calculations take about several minutes depending on the complexity of the problem.

26. THANK YOU FOR ATTENTION!