Mathematical Models of Dissipative Systems in Quantum Engineering

Abstract

The paper shows the results of theoretical research concerning the modeling and characterization of the dissipative structures generally, the dissipation being an essential property of the system with self-organization which include the laser-type systems also. The most important results presented are new formulae which relate the coupling parameters 𝑎_𝑖𝑛 from Lindblad equation with environment operators Γ_𝑖; microscopic quantitative expressions for the dissipative coefficients of the master equations; explicit expressions which describe the changes of the environment density operator during the system evolution for fermion systems coupled with free electromagnetic field; the generalized Bloch-Feynman equations for 𝑁-level systems with microscopic coefficients in agreement with generally accepted physical interpretations. Based on Maxwell-Bloch equations with consideration of the interactions between nearing atomic dipoles, for the dense optical media we have shown that in the presence of the short optical pulses, the population inversion oscillates between two extreme values, depending on the strength of the interaction and the optical pulse energy.

1. Introduction

An essential problem of the quantum information systems is the controllability and observability of the quantum systems. In this context, Fermi systems are essential for several important physical effects in quantum engineering as the dynamics of semiconductor nanostructures and high temperature superconductivity, nuclear resonances, fusion-fission reactions, and analysis of optical quantum systems. These effects are essentially determined by the dissipative coupling of the system.

Dissipation in quantum systems is a complex phenomenon which raises important theoretical investigations. A dissipative system is a system of interest, coupled with another system usually considered as being of much larger-environment. Fundamental and difficult problem of dissipative quantum theory is to design the total system (system of interest + environment) on the space system of interest. In this way obtain a quantum master equation describing the evolution of the system using two terms: (1) a hamiltonian term for processes with energy conservation and (2) a nonhamiltonian term with coefficients that depend on the dissipative coupling. A master equation is based on approximations that consist in mediating rapid oscillations of reduced density matrix describing the interaction.

Such an approximation is the assumption that the evolution operators of a dissipative system forms a semigroup, not a group like for isolated systems. In this framework was derived a quantum master equation with dissipative terms which is consistent with all principles of quantum mechanics. Considering two operators, coordinate 𝑞 and momentum 𝑝, master equation was used to describe the harmonic oscillator. In this theoretical framework, dissipation is described by the friction and diffusion coefficients that satisfy certain conditions called basic restrictions and Heisenberg’s uncertainty relations are observed during the whole evolution of the system.

A rigorous method for deducting the master equation with microscopic expressions of the dissipative coefficients is developed in the literature.

For a weak dissipative coupling one obtains a master equation of Lindblad form [1], but with the microscopic expressions of the dissipative coefficients.

In the development of quantum theory of dissipative systems an important step was the connection between Lindblad’s generator and the previous phenomenological descriptions, realized by Săndulescu and Scutaru [2]. Besides, we must mention Isar et al.’s contributions [3]. This school developed by the above-mentioned researchers in the field are well recognized in the scientific world [4–8].

Firstly, in the paper general expressions which relate the coupling parameters 𝑎_𝑖𝑛 in Lindblad equation with environment operatorsΓ_𝑖 have been established [9–11]. In this way, became possible deeper causality understanding of processes of friction and diffusion and of related quantum effects: broadening and shift of spectral lines, tunneling rates, bifurcations and instability [12, 13].

Secondly, for a system of fermions, coupled with a dissipative environment quantitative microscopic expressions for the coefficients of the dissipative master equation depending on the potential matrix elements, the densities of states of the environment and the occupation probabilities of these states are presented [14–19].

The study continue with the systems of fermions coupled by electric dipole interactions of free electromagnetic field for which has established general explicit expressions which describe the changes of the environment density operator during the system evolution. This description is not restricted to the Born approximation, taking into account the environment time evolution as a function of the system evolution. The results of the dissipative dynamics of the system of fermions in the presence of laser field are applicable to the dissipative structures [14, 20–29].

Next, generalized Bloch-Feynman equations for 𝑁-level systems with microscopic coefficients in agreement with generally accepted physical interpretations are presented.

In the last part, we study the dynamics of dense media under the action of ultrafast optical pulses using Maxwell-Bloch formalism to include interaction between close atomic dipoles [30–34]. It is shown that, in a system initially without inversion, in the presence of optical pulses, the final population has two extreme values, results which contribute to understanding the specific mechanisms of switching for applications, with specific examples concerning the coherent radiation generation and amplification [35–43]. A computational specific software, to verify the experimental and numerical existing models and in the same time to discover new important situations for operative systems design and implementation, was developed [44–48].

2. Relationship between Coupling Coefficients in Lindblad Master Equation and Environment Observables

Research on dissipative processes has led to evidence for the first time concerning the relationship between coupling coefficients 𝑎_𝑖𝑛 in the Lindblad equation: 𝑖𝜌̇≡−ħ+1𝐻,𝜌2ħ^𝑛𝑋_𝑛𝜌,𝑋⁺_𝑛+𝑋_𝑛,𝜌𝑋⁺_𝑛 (2.1)

depending on the system Hamiltonian 𝐻 and the operators of opening 𝑋_𝑛: 𝑋_𝑛≡_𝑖𝑎_𝑖𝑛𝑠_𝑖, (2.2)

where 𝑠_𝑖 are system operators, 𝑎_𝑖𝑛 are complex coupling coefficients or amplitudes and Γ_𝑖 operators of environment defined using the interaction Hamiltonian as𝐻^𝑆𝐸=ħ_𝑖𝑠_𝑖Γ_𝑖. (2.3)

These relationships have been established under the form [10]_𝑛𝑎_𝑖𝑛𝑎^∗_𝑗𝑛=2ħΓ_𝑖Γ_𝑗, (2.4)

and allow an understanding of the physical causes of quantum processes of friction and diffusion, with their known effects: broadening and shift of spectral lines [11], increased rates of tunneling, nonlinear characteristics, leading to bifurcation, instability, and chaos.

3. Microscopic Quantitative Expressions of the Dissipative Coefficients in Master Equations

A general quantum master equation for a many-level many-particle system, with microscopic coefficients, that preserves the quantum-mechanical properties of the density matrix was obtained [12]:d𝜌(𝑡)𝑖d𝑡=−ħ𝐻,𝜌(𝑡)+_𝑖,𝑗𝜆_𝑖𝑗𝑐⁺_𝑖𝑐_𝑗𝜌(𝑡),𝑐⁺_𝑗𝑐_𝑖+𝑐⁺_𝑖𝑐_𝑗,𝜌(𝑡)𝑐⁺_𝑗𝑐_𝑖 (3.1)

with dissipative coefficients:𝜆_𝑖𝑗=𝜆^𝐹_𝑖𝑗+𝜆^𝐵_𝑖𝑗 (3.2)

including a component 𝜆^𝐹_𝑖𝑗 for a dissipative environment of fermions and a component 𝜆^𝐵_𝑖𝑗 for a dissipative environment of bosons.

Equation (3.1) is of Lindblad’s form, with dissipative operators depending on the transition/population operators 𝑐⁺_𝑖𝑐_𝑗. For a system with 𝑁 levels, the total number of these operators is 𝑁²−1 the number of the independent operators defined.

If we denote by 𝑉^𝐹 and 𝑉^𝐵 the interaction dissipative potentials of the environment containing 𝑌^𝐹 fermions and 𝑌^𝐵 bosons, respectively, it is possible to write the expressions of the coefficients 𝜆^𝐹_𝑖𝑗 şi 𝜆^𝐵_𝑖𝑗 for the resonant transition |𝑗⟩→|𝑖⟩of the system coupled with |𝛽⟩→|𝛼⟩ environmental transition, with fermionic state having densitiesaly 𝑔^𝐹_𝛼,𝑔^𝐹_𝛽 and populations 𝑓^𝐹_𝛼(𝜀_𝛼), 𝑓^𝐹_𝛽(𝜀_𝛽), and bosonic states with densities 𝑔^𝐵_𝛼, 𝑔^𝐵_𝛽 and population 𝑓^𝐵_𝛼(𝜀_𝛼) şi 𝑓^𝐵_𝛽(𝜀_𝛽). The probability that the final state |𝛼⟩ of the environment to be free is 1−𝑓(𝜀_𝛼)while the probability the initial state of the environment to be occupied is 𝑓(𝜀_𝛽).

General expressions of the dissipative coefficients are written for this type of interaction in the form:𝜆^𝐹_𝑖𝑗=𝜋ħ𝑌^𝐹𝛼_𝑖𝑉^𝐹𝛽_𝑗²1−𝑓^𝐹_𝛼𝜀_𝛼𝑓^𝐹_𝛽𝜀_𝛽𝑔^𝐹_𝛼𝜀_𝛼𝑔^𝐹_𝛽𝜀_𝛽d𝜀_𝛽,𝜀_𝛼−𝜀_𝛽=𝜀_𝑗−𝜀_𝑖,𝜆^𝐵_𝑖𝑗=𝜋ħ𝑌^𝐵∫𝛼_𝑖𝑉^𝐵𝛽_𝑗²1+𝑓^𝐵_𝛼𝜀_𝛼𝑓^𝐵_𝛽𝜀_𝛽𝑔^𝐵_𝛼𝜀_𝛼𝑔^𝐵_𝛽𝜀_𝛽d𝜀_𝛽,𝜀_𝛼−𝜀_𝛽=𝜀_𝑗−𝜀_𝑖. (3.3)

4. The Environment Dynamics Correlated with that of a Fermion Systems Coupled with Free Electromagnetic Field

We consider a system of 𝑍 charged fermions with the coordinates 𝑟_𝑛 and momenta 𝑝_𝑛(𝑛=1,2,…,𝑍) in a single-particle potential 𝑈⁽¹⁾(𝑟_𝑛), while 𝑈⁽²⁾(𝑟_𝑛,𝑟_𝑚) represents the two-particle residual potential. This system is coupled to the modes 𝜈 of the free electromagnetic field. In order to describe the dynamics of this system, for simplicity, we neglect the particle spin and its dimensions with respect to the electromagnetic field wavelength (the electric dipole approximation). In this case, the total hamiltonian is of the form [14] 𝐻^𝑇=^𝑍_𝑛=1𝑝_𝑛−𝑒𝐴^𝐵22𝑚+^𝑍_𝑛=1𝑈⁽¹⁾𝑟_𝑛+12^{𝑍𝑛,𝑚=1}𝑈⁽²⁾𝑟_𝑛,𝑟_𝑚+𝐻^𝐵. (4.1)

In the total hamiltonian (4.1),𝑒𝑉=−𝑚^𝑍𝑛=1𝑝_𝑛𝐴^𝐵 (4.2)

is the system-field interaction potential, while𝐻^𝑆=^𝑍_𝑛=1𝑝²_𝑛2𝑚+^𝑍_𝑛=1𝑈⁽¹⁾𝑟_𝑛+12^{𝑍𝑛,𝑚=1}𝑈⁽²⁾𝑟_𝑛,𝑟_𝑚 (4.3)

is the fermion system hamiltonian, and𝐻^𝐵=_𝜈𝐻^(𝜈) (4.4)

is the field hamiltonian, where𝐻^(𝜈)=ħ𝜔_𝜈𝑎⁺_𝜈𝑎_𝜈+12 (4.5)

is the field mode 𝜈 hamiltonian.

Let us take the density operator 𝜒(𝑡) of the total system with hamitonian (4.1) and the reduced density matrix𝜌(𝑡)=Tr_𝐵𝜒(𝑡) (4.6)

over the environment states.

The total density operator 𝜒(𝑡) satisfies the equation of motion:d𝜒𝑖d𝑡=−ħ𝜀𝑉^𝑅(𝑡)+𝜀𝑉(𝑡),𝜒(𝑡), (4.7)

where the sign above 𝜒 designs operators within the framework of interaction picture of the system and environment𝜒(𝑡)=𝑒^{(𝑖/ħ)(𝐻𝐵+𝐻𝑆0)𝑡}𝜒(𝑡)𝑒^{−(𝑖/ħ)(𝐻𝑆0+𝐻𝐵)𝑡}, (4.8)

while 𝜀 is an intensity parameter used to show the orders of the series expansion of this density. Considering the radiation field of the black body in the initial state 𝑅, the total density operator of the system can be taken under the form:𝜒(𝑡)=𝑅⊗𝜌(𝑡)+𝜀𝜒⁽¹⁾(𝑡)+𝜀²𝜒⁽²⁾(𝑡)+⋯, (4.9)

where 𝜒⁽¹⁾(𝑡), 𝜒⁽²⁾(𝑡) represent modifications of the field during the system evolution. The first term of this expression corresponds to the Born approximation when the environment state is a constant state 𝑅, while the higher-order terms, which satisfy the normalization relationsTr_𝐵𝜒⁽¹⁾(𝑡)=Tr_𝐵𝜒⁽²⁾(𝑡)=⋯=0, (4.10)

describe the environment dynamics that is correlated to the system dynamics. For an equation of motion of the formd𝜌𝐵d𝑡=𝜀⁽¹⁾𝜌(𝑡),𝑡+𝜀²𝐵⁽²⁾𝜌(𝑡),𝑡 (4.11)

From (4.7), (4.9), and (4.11) we get a system of coupled equations:𝑅⊗𝐵⁽¹⁾𝜌(𝑡),𝑡+d𝜒⁽¹⁾𝑖d𝑡=−ħ𝑉^𝑅(𝑡)+𝑉(𝑡)(𝑡,𝐵,𝑅⊗𝜌)𝑅⊗⁽²⁾𝜌(𝑡),𝑡+d𝜒⁽²⁾𝑖d𝑡=−ħ𝑉^𝑅(𝑡)+𝑉(𝑡),𝜒⁽¹⁾(𝑡). (4.12)

By calculating the partial traces over the environment states and using the normalization conditions (4.10), from these equations we get successively the terms of the equation of motion (4.11):𝐵⁽¹⁾(𝑡)𝑖𝜌,𝑡=−ħTr_𝐵𝑉^𝑅(𝑡)+𝑉(𝑡)(𝑡,𝐵,𝑅⊗𝜌)₍₂₎(𝑡)𝑖𝜌,𝑡=−ħTr_𝐵𝑉^𝑅(𝑡)+𝑉(𝑡),𝜒⁽¹⁾(𝑡), (4.13)

while, integrating by time, we get "excitation" terms of the total density operator (4.9):𝜒⁽¹⁾(𝑡)=^𝑡₀−𝑖ħ𝑉^𝑅𝑡′+𝑉𝑡′,𝑅⊗𝜌𝑡′−𝑅⊗𝐵⁽¹⁾𝜌𝑡′,𝑡′d𝑡′,𝜒⁽²⁾(𝑡)=^𝑡₀−𝑖ħ𝑉^𝑅𝑡′+𝑉𝑡′,𝜒⁽¹⁾𝑡′−𝑅⊗𝐵⁽²⁾𝜌𝑡′,𝑡′d𝑡′. (4.14)

The first-order equation (4.13) represents the system evolution when the environment is considered as being in a constant state 𝑅, while for the higher-order term (28), we take into consideration some changes of the environment matrix (4.14). Further on, we will show that the first-order terms (4.13) describe the hamiltonian dynamics of the system, while the second-order term (28) describes system one-particle transitions related to environment.

5. The Generalized Bloch-Feynman Equations

An alternative description of dissipative system dynamics is given by Bloch-Feynman equations for systems of fermions obtained by defining the pseudo-spin operators [14].

In particular, for a system with two-level known Bloch-Feynman, equations are obtained, where, 𝑄₁₂ is the field operator, 𝑃₁₂ is the polarization operator, and 𝑁₂ is population operator:d𝑄d𝑡¹²=−𝛾_⟂𝑄₁₂+𝜔₂₁𝑃₁₂,d𝑃d𝑡¹²=−𝜔₂₁𝑄₁₂−𝛾_⟂𝑃₁₂,d𝑁d𝑡²=−𝛾_||𝑁₂−𝑁₂⁽⁰⁾, (5.1)

with microscopic coefficients 𝛾_⟂ şi 𝛾_|| expressed by dissipative coefficients 𝜆_𝑖𝑗 of the master equation:𝛾_⟂=𝜆₁₂+𝜆₂₁+𝜆₁₁+𝜆₂₂,𝛾_||=2𝜆₁₂+𝜆₂₁,𝑁₂⁽⁰⁾=𝜆₂₁𝜆₁₂+𝜆₂₁. (5.2)(5.3)(5.4)

The condition 2𝛾_⟂≥𝛾_|| is a confirmation of master equation (4.7) which led to the establishment of Bloch equations-Feynman, because this condition is verified experimentally.

6. Dynamics of Dense Media under the Action of Short Optical Pulses

Maxwell-Bloch equations of a two-level atomic medium generalized to include interactions between the dipoles approach [15, 16, 32] have been used to describe the system dynamics under the action of ultrafast optical pulses. These equations, for systems with homogeneous broadening of spectral lines in about semiclassical treating, were established using the density matrix formalism asd𝑤d𝑡=−𝛾_𝐿()+𝜇𝑤+1ħ𝐸^∗𝑅_𝑎𝑏+𝐸𝑅^∗_𝑎𝑏,d𝑅_𝑎𝑏𝛾d𝑡=−_𝑇+𝑖(Δ+𝜀𝑤)𝑅_𝑎𝑏−𝜇2ħ𝐸𝑤. (6.1)(6.2)

In the above equations, 𝑤 is the inversion of population, 𝑅_𝑎𝑏 nondiagonal elements of density matrix slow variable, indices 𝑎 and 𝑏 refer to lower and higher energy states, with the gap ħ𝜔₀, 𝐸_𝐿 is slowly varying local field Δ=𝜔₀−𝜔 is the frequency deviation in relation to the center frequency of the field resonance frequency, 𝜇 is the transition matrix element of the electric dipole, and𝛾_||, 𝛾_⟂ are longitudinal and transverse relaxation rates.

Contributions of the dipole-dipole interactions occur in (6.2) by term 𝑖𝜀𝑤𝑅_𝑎𝑏, where 𝜀=𝑛𝜇²/3ħ𝜀_𝑑≪𝜔₀ is the strength parameter of dipole-dipole interactions having a dimension of a frequency. Equations (6.1) and (6.2) for the atomic variables and for field variables realise the description Maxwell-Bloch of optically dense environment. These equations were generalized and used to study intrinsic optical bistability, propagation effects in nonlinear media, and so forth.

For numerical simulation, we considered the case resonant (Δ=0), a characteristic distance between dipoles much smaller than the wavelength of the central field (propagation effects are negligible) and ultrafast pulses (pulses much shorter than 𝛾⁻¹_||; this enables us to neglect dissipation processes). In these conditions, the matrix element 𝑅_𝑎𝑏 is decomposed into its real and imaginary parts 𝑅_𝑎𝑏=1/2(𝜈+𝑖𝑢), resulting in systemd𝑢d𝑡′=−𝜀𝜏_𝑝𝜈𝑤,d𝜈=d𝑡′𝜀𝜏_𝑝Ω𝑢+𝜀𝑤,d𝑤d𝑡′=−𝜀𝜏_𝑝Ω𝜀𝜈 (6.3)

whose outcome is possible only numerically.

In the above equations 𝑡^′=𝑡/𝜏_𝑝 is the normalized time, 𝜏_𝑝 is the measured width pulse, Ω(𝑡)=𝜇𝐸(𝑡)/ℎ is the instantaneous Rabi frequency, and 𝐸(𝑡) is the intensty of electrical pulse.

In Figure 1, we present the final population inversion function of maximum Rabi frequency for hyperbolic secant pulses 𝐸(𝑡)=𝐸₀sech(𝑡/𝜏_𝑝). As long as the Rabi frequency has a value so that Ω₀/𝜀<1, the final population inversion is 𝑤=−1. In the region Ω₀/𝜀>1, the final population inversion has an oscillatory behavior, almost rectangular wave. As the parameter 𝜀𝜏_𝑝 value is greater, the oscillation period decreases, the transitions become abrupt, and the first half cycle of the rectangular wave becomes more centered to Ω₀/𝜀=1.

Figure 1 

Final state of population inversion, depending on the hyperbolic secant pulse maximum value 𝐸(𝑡)=𝐸0sech(𝑡/𝜏𝑝)(solid line), 𝜀𝜏𝑝=20 (continuous line) and 𝜀𝜏𝑝=30.

In Figure 2, temporal evolution of the system is presented for a hyperbolic secant pulse with a peak higher than one (when 𝑡→∞, the population inversion performs a number of oscillations before reaching a value 1; under certain conditions when 𝑡→∞, after a number of oscillations, the system remains in the ground state).

Figure 2 

Temporal evolution of the system for a hyperbolic secant pulse, 𝐸(𝑡)=𝐸0sech(𝑡/𝜏𝑝).

7. Conclusions

General expressions which relate the coupling parameters 𝑎_𝑖𝑛 in Lindblad equation with environment operatorsΓ_𝑖 have been established. These expressions allow deeper understanding of causal processes of friction- and diffusion-related quantum effects: broadening and shift of spectral lines, tunneling rates, bifurcations, and instability.

For a system of fermions coupled with a dissipative environment quantitative microscopic expressions for the coefficients of the dissipative master equation are presented.

These coefficients depend on the potential matrix elements, the densities of states of the environment, and the occupation probabilities of these states.

Expressions of the dependence of the particle distributions on temperature are taken into account. It can be shown that a system of fermions located in a dissipative environment of bosons tends to a Bose-Einstein distribution.

Studying the systems of fermions coupled by electric dipole interactions of free electromagnetic field, has established general explicit expressions which describe the changes of the environment density operator during the system evolution for fermion systems coupled with free electromagnetic field. This description is not restricted to the Born approximation, taking into account the environment time evolution as a function of the system evolution. The study can be continued with the calculation of the higher-order term of the reduced matrix equation in order to describe the correlated transition of the system particles. The results of the dissipative dynamics of the system of fermions in the presence of laser field are applicable to the dissipative structures.

Generalized Bloch-Feynman equations for 𝑁-level systems with microscopic coefficients in agreement with generally accepted physical interpretations are presented. On this basis, the problem of a quantum system control is explicitly formulated in terms of microscopic quantities: matrix elements of the dissipative two-body potential, densities of the environment states, and occupation probabilities of these states.

Studying the dynamics of dense media under the action of ultrafast optical pulses using Maxwell-Bloch formalism to include interaction between close atomic dipoles showed that, in a system initially without inversion, in the presence of optical pulses, the final population has two extreme values, the ratio of Rabi frequency and the parameter that describes the interactions between close dipoles, which contribute to understanding the specific mechanisms of switching.