by Tsukanov Roman

Quantum Optics by Tsukanov Roman Quantum-Mechanical Approach to the Nonlinear Optical Susceptibility, Density Matrix

Outline • Introduction and definitions • Perturbation solution for 1st 2nd and 3rd order susceptibilities • Density matrix formalism • Density matrix calculation for 1st and 2nd order susceptibility • Expansion terms representation by Feynman diagrams

Introduction • Nonlinear optics – field which studies the phenomena which occurs as a consequence of the modification of the optical properties of a material by the presence of light • Only laser light is sufficiently intense to modify the opt. properties of material system We look at the dependence of dipole moment per unit volume P of a material system upon the strength E of the applied field in case of linear optics: Where (1) is the linear susceptibility. In non linear optics we express P as a power series of E: Where (2) and (3) are the 2nd and 3rd order nonlinear optical susceptibilities, we will see that they depend on the frequencies of applied field

Formal Definition of the Nonlinear Susceptibility We want to consider the general case of a material with dispersion and loss. Then the susceptibility becomes a complex quantity relating the complex amplitudes of the electric field and polarization prime – summation over positive frequenves only We represent E as the sum of its positive and neg. frequency parts by requiring complex conjugate we get the physical field E to be real

Formal Definition of the Nonlinear Susceptibility definition: An-slowly varying field amplitude new notation: using new notations we can write the field in the more compact form where the summation runs over all frequencies

Formal Definition of the Nonlinear Susceptibility According to the new definitions, the field is given by where In similar way we define the nonlinear polarization where the summation runs over the positive and negative frequencies And finally we define the components of the second order susceptibility as const. of proportionality relating the amplitude to the product of field amplitude

Motivation Our goal is to calculate explicit expressions for the Nonlinear optical susceptibility. • These expressions display the functional form of the nonlinear susceptibility and hence show how the susceptibility depends on material parameters such as dipole transition moment and atomic energy levels • These expressions show the internal symmetries of the susceptibility • These expressions can be used to obtain numerical values of the non-linear susceptibilities

Schematic representation of the interaction processes We may consider the interaction in terms of the exchange of photons between The various frequency components of the field In the single quantum process three photons of frequency  are destroyed And a single photon of frequency 3 is simultaneously created If one of the real atomic levels is close to one of the virtual levels, the coupling Between the radiation and the atom is particularly strong and the nonlinear Susceptibility becomes large. 3rd harmonic generation in terms of virtual levels (a) and with real atomic levels indicated (b) Solid lines represents the eigenlevels of the atom and dashed lines –virtual levels. They represent the combined energy of one of the energy eigenstates of the atom and of one or more photons of the radiation field.

Schematic representation of the interaction processes Example – enhancing the efficiency of the 3rd harmonic generation (a) – one photon transition Is nearly resonant, (b) – two photon transition is Nearly resonant, (c) – the three photon transition is nearly resonant The method shown in part (b) is the preferred way, because: In (a) the incident field experiences linear absorption and can be rapidly attenuated As it propagates through the medium In (c) the generated field experiences linear absorption. In (b) the two photon absorption occurs with much lower efficiency than one-photon process

We assume that all of the properties of the atomic system can be described in terms of the atomic wave function (r,t), which is the solution to the time-dependent Schrödinger equation Schrödinger Equation Calculation of the Nonlinear Optical Susceptibility The sum runs over the positive and negative frequency components

Energy Eigenstates The solution: After substituting to the Shrodinger equation we see that un(r) must satisfy the eigenvalue equation (time independent S. equation) The solution is chosen in a such manner that they form a complete, orthonormal set:

Perturbation solution to Shrodinger Equation For the general case in which the atom is exposed to an electromagnetic field, Schrödinger's equation cannot be solved exactly. So we will use the perturbation theory in order to solve it where is a continuously varying parameter ranging from zero to unity that characterizes the strength of the interaction; The value =1 corresponds to the actual physical situation. Then the desired solution may be written in the form

Perturbation solution to Shrodinger Equation After substitution we require that the Schrödinger eq. will be fulfilled for Each order separately. We obtain the set of equations: under assumption that initially the atom is in the ground state, we get The Nth order contribution to the wave function may be represented as (remember the Schrödinger picture)

Perturbation solution to Shrodinger Equation Then We multiply each side of the equation by u* and integrate over all space, The result is As we see – once al(N-1) are determined, by integration we can get the next order

Perturbation solution to Schrödinger Equation We are interested to determine the linear, second order and third order optical Susceptibilities. For this we need to determine the probability amplitudes

Linear Susceptibility Now we want to use the results to determine linear optical properties of a material system. The expectation value of the electric dipole moment is given by Where  is given by perturbation expansion with =1.The linear contribution to <p> is given by We may formally replace p by -p in the second term, then

Linear Susceptibility Linear polarization Linear susceptibility defined through the relation then The first and the second terms are the resonant and anti resonant Contributions to the susceptibility

Second Order Susceptibility Then Now we may replace q by -q in the second term, q by -q and p by -p in the third term

Second Order Susceptibility We perform similar steps like we did while deriving 1st order: After introducing into the standard definition of the 2nd order susceptibility We obtain the following result PI-intrinsic permutation symmetry (averages the expression over Both permutations of the frequencies and q of the applied fields)

Second Order Susceptibility We may look at the energy level diagram in order to show where the levels m and n have to be located in order for each term to become resonant For the case of highly non resonant excitation ng and mg can be taken To be real, and the expression can by simplified further Where we used the full permutation operator, defined as the expression summed Over all permutations of the frequencies p , q and - The statement that can be made is: the non linear susceptibility of a lossless Medium possesses full permutation symmetry

Third Order Susceptibility Now we want to treat the 3rd Order. The dipole moment per atom, correct to third order in perturbation theory is Then

Third Order Susceptibility We can replace the values of p, q and r by their negatives in those Expressions where the complex conjugate of a field amplitude appears. Then

Third Order Susceptibility Like before we let The definition of the 3rd order susceptibility And the result for the 3rd order is

Third Order Susceptibility The illustration of the locations of the resonances we may see in the figure Case of highly non resonant excitation – permutation symmetry

Density Matrix Formalism Definition • A density matrix, or density operator, is used in quantum theory to describe the statistical state of a quantum system. • The formalism was introduced by John von Neumann in 1927. • The need for a statistical description via density matrices arises because it is not possible to describe a quantum mechanical system that undergoes general quantum operations such as measurement, using exclusively states represented by ket vectors. • In general a system is said to be in a mixed state, except in the case the state is not reducible to a combination of other statistical states. In that case it is said to be in a pure state.

Density Matrix Formalism Motivation: Using the D.M. Formalism we may treat effects, such as collisional broadening of the atomic resonances, that cannot be treated by the simpler theoretical formalism based on the atomic wave function

Density Matrix Formalism - Introduction if the system is known to be in a particular state s then s describes all the Physical properties of the system. Also this wave function obeys the S. E.: We assume that H can be represented as Where H0 - - the Hamiltonian for a free atom and V(t) – an interaction field-atom. We make explicit use of the fact that the energy eigenstates of the free atom Hamiltonian H0 form a complete set of basis functions.

We multiply by u*m and integrate over all space Density Matrix Formalism - Introduction The coefficient C(t) gives the probability evolution that the atom, Which is known to be in state s, is in energy eigenstate n at time t. To determine the time evolution we introduce the expansion into The Schrödinger equation to obtain The expectation value in terms of the wave function of the system (the 3rd postulate of quantum mechanics):

Density Matrix Formalism - Introduction In terms of probability amplitudes Cns(t) we obtain As long as the initial state of the system and the Hamiltonian operator H For the system are known, the formalism described above is capable of Providing a complete description of time evolution of the system and all of Its observable properties. BUT there are circumstances under which the state of system is not known In a precise manner. We define the elements of the density matrix of the system by

Diagonal elements = probabilities Off-diagonal elements = "coherences" (provide info. about relative phase) Density Matrix Formalism Interpretation of matrix elements

Density Matrix Formalism The Density matrix formalism is useful because it can be used to Calculate the expectation value of any observable quantity Integrating the notation we have used The last expression may be simplified as follows

Density Matrix Formalism In order to determine how any expectation value evolves in time, it is necessary Only to determine how the density matrix evolves in time. So we need to Differentiate the equation And we will get Now we assume that p(s) does not vary in time, so the first term vanishes and use the Schrödinger equation for the probability amplitudes for the second term evaluation

Density Matrix Formalism After substitution Using the density matrix notation we may write The last equation describes how the density matrix evolves in time as the Result of interactions that are included in H

Density Matrix Formalism • Till now we found how the DM evolves in time as a result of interactions • that are included in H. • But in addition there are interactions that change the state of the system • and cannot conveniently be included in H. • One of the ways to include such an effects in the formalism is to add • phenomenological damping terms to the equation of motion. Then And the meaning Is that nm relaxes to its equilibrium value nmeq with decay rate  The additional physical assumption is that This means that thermal excitation is incoherent process and cannot produce any coherent superpositions of atomic states

Density Matrix - Example Two-Level Atom: There are only two atomic states a and b interacting appreciably with the incident optical field. The wave function describing s state is given by

Density Matrix - Example DM for the atom is given by The dipole moment operator The expectation value of the dipole moment is given by As seen the expectation of the dipole moment depend upon the off-diagonal Elements of the density matrix

Perturbation Solution of the Density Matrix Equation of Motion The density matrix equation of motion with phenomenological inclusion of Damping is This equation cannot be solved exactly for physical systems of interest and We should use the perturbavite technique for solving it: We suppose that V is given by the electric dipole approximation

Perturbation Solution of the Density Matrix Equation of Motion We require that the expansion of  will be the solution of the original equation For any value of the parameter , so the coefficients of each power of  must Satisfy the equation separately (for derivation see Appendix 1). Then

Perturbation Solution of the Density Matrix Equation of Motion

Perturbation Solution of the Density Matrix Equation of Motion Then after substitution S into  All the higher-order corrections to the density matrix can be obtained By appropriate index change

Density Matrix of the Linear Susceptibility Easy to show

Density Matrix of the Linear Susceptibility

Density Matrix of the Linear Susceptibility By few operations and simplifying assumption that all of the population Is in one level (typically the ground state - a) As we may see the first term is resonant for positive frequencies p . The second one is antiresonant and can be dropped when p is Close to one of the resonance frequencies of the atom. Then to good Approximation the linear susceptibility is given by

Density Matrix Calculation of the Second Order Susceptibility The expectation value of the atomic dipole moment is

Density Matrix Calculation of the Second Order Susceptibility We will with to look at atomic dipole moment oscillating at frequency p+ q The nonlinear susceptibility is defined

Density Matrix Calculation of the Second Order Susceptibility By comparison of the equalities we obtain m, n and  are dummy indices then can be replaced, so the susceptibility May be recast in the form

Density Matrix Calculation of the Second Order Susceptibility

References • R. W. Boyd, Nonlinear Optics (1992) • A. Maitland and M.H. Dunn, Laser Physics (1969)

APPENDIX 1 Perturbation Solution of the Density Matrix Equation of Motion

by Tsukanov Roman