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High Energy Physics and Quantum Field Theory XXI International Workshop
Saint Petersburg Area, June 23­30, 2013

The Influence Functional approach to the quantum systems dynamics

Alexander Biryukov, Mark Shleenkov Samara State University


The paper structure
1. Model QS+EMF 2. Density matrix of QS+EMF as a paths integral 3. EMF influence functional

4. Quantum transition probability as a paths integral of real functional
5. Some applications


The studied model


The system evolution


The mixed representation


The density matrix

where


The kernel of the evolution operator as a paths integral

Fadeev L. D., Slavnov A. A. Gauge Fields. Introduction to Quantum theory // Addison-Wesley P. C., 1991. Feynman R. P., Hibbs A. R. Quantum mechanics and Path Integrals // N.Y.: McGraw-Hill, 1965


The density matrix as a paths integral


The density matrix for quantum system


The influence functional definition
The equation of this density matrix evolution can be written as


The influence functional definition


The Influence Functional calculation
In general, the influence functional explicit form depends on a) EMF initial and finite states b) Model of EMF and investigated system interaction
In 1963 R. Feynman introduced the influence functional and studied some of its properties. We have done influence functional calculations for different models in holomorphic space of EMF paths.

Feynman R. P., Vernon F. L. The Theory of a General Quantum System Interacting with a Linear Dissipative System // Annals of Physics, 1963, 24, Issue 1, p.118-173 Feynman R. P., Hibbs A. R. Quantum mechanics and Path Integrals // N.Y.: McGraw-Hill, 1965


Vacuum electromagnetic field influence functional


The coherent electromagnetic field influence functional
The initial state of EMF is pure coherent


The mixed coherent electromagnetic field influence functional


The multimode field influence functional
Influence functional multimode EMF can be obtained by the product of all independent modes influence functionals, following Feynman:

These rules are valid for all considered models.


The influence functional general form
On the basis of the calculated influence functionals its general form can be written so

These functionals lead to different effects of the EMF influence on the target system.


Having used this general form for influence functional the density matrix is to be represented so


Quantum transition probability


Quantum transition probability


Quantum transition probability
Using Euler 's formula we express transition probability in the following form


Quantum transition probability as paths integral of real functional

If this expressions are valid, then

Ryazanov G. V. Quantum-mechanical probability as a sum over path // JETP. 1958. V. 35. 1


Model for calculations
For this approach demonstration we are to deal with 1. A QS model is a particle in one-dimensional infinite potential well.
2. Initial state of EMF is a onemode pure coherent state. We are also considering such processes in which it is possible to neglect influence vacuum EMF. 3. Interaction between QS and EMF is chosen by us in the dipole approximation.


Quantum transition probability as a paths sum of real functional
The energy representation is more convenient for numerical calculations. Thus we write transition probability in the following form:

Energy representation Scully M. O., Zubairy M. S. Quantum Optics // Cambridge University Press. 1997


Here the action in energy representation is


Supercomputer "Sergey Korolev "

The program was made in C environment


Applications

one-photon Rabi oscillation without RWA

Two-level QS in the resonance case

E2

E1


Applications
P 1, t 1, 0
1 .0

one-photon Rabi oscillation without RWA

0 .8

0 .6

0 .4

0 .2

500

1000

1500

2000

t


Applications
P 1, t 1, 0
0 .6 0

one-photon Rabi oscillation without RWA

0 .5 5

0 .5 0

0 .4 5

250

260

270

280

290

300

310

320

t


Applications
Multi-photon processes

two-photon Rabi oscillation without RWA

Three-levels QS Two-photon resonance case
E3

E2
E1


Applications
1 .0

two-photon Rabi oscillation without RWA

0 .8

0 .6

0 .4

0 .2

0

500

1000

1500

2000

2500

3000


Applications
Coherent population trapping
Three-levels QS Two one-photons resonance cases E3

E2 E1


Applications
Coherent population trapping


Conclusion
· The Influence Functional approach allows us to describe the investigated quantum system dynamics by integrating of functional along path.
· This functional is sign variable and it is given by the product of exponent and cosine.

· The obtained formula can be used to describe multi-photon processes in many cases.


Thanks for your attention