Abstract
Quantum dynamics of light waves traveling through a timevarying turbulent plasma is investigated via the SU(1,1) Lie algebraic approach. Plasma oscillations that accompany timedependence of electromagnetic parameters of the plasma are considered. In particular, we assume that the conductivity of plasma involves a sinusoidally varying term in addition to a constant one. Regarding the time behavior of electromagnetic parameters in media, the light fields are modeled as a modified CK (CaldirolaKanai) oscillator that is more complex than the standard CK oscillator. Diverse quantum properties of the system are analyzed under the consideration of timedependent characteristics of electromagnetic parameters. Quantum energy of the light waves is derived and compared with the counterpart classical energy. Gaussian wave packet of the field whose probability density oscillates with time like that of classical states is constructed through a choice of suitable initial condition and its quantum behavior is investigated in detail. Our development presented here provides a useful way for analyzing time behavior of quantized light in complex plasma.
Introduction
The problem of quantum light waves propagating through complex plasma has attracted considerable attention for several decades thanks to its rich applications in various physical systems. Some interesting applications include the diagnosis and analysis of plasma^{1,2} and frequency transformer producing terahertz radiation (Tray)^{3,4,5}. The research for the propagation of light waves in general timevarying media began in the late 1960s by Auld et al.^{6} and some other research groups^{7,8}. Afterwards, wave phenomena in such media were actively investigated in the literature, especially focusing on the light waves traveling through timevarying plasma^{9,10,11,12,13,14,15,16}. Collective modes of selfsustained oscillations of turbulent dusty plasmas around their equilibrium positions were observed in some situations and their physical properties have been studied^{2,17,18,19,20,21,22}. The mechanism for realizing frequency upconversion of light waves via their interaction with a suddenly created plasma slab was also reported^{23,24,25}.
If we know analytical solutions of a quantum light wave, we can easily estimate quantum properties of the system and are able to predict its subsequent behaviors. However, most of the researches in this field have been performed numerically on the basis of the well known finitedifference timedomain (FDTD) method^{4,9,26} or some other methods, due to the difficulty of mathematical procedure for analytical description of waves in complicated plasma. Inspired by this situation, we investigate how to derive analytical solutions of the Schrödinger equation for light waves in a timevarying plasma medium. Further, quantum properties of the system will be studied in detail by taking advantage of their exact quantum wave functions. For this purpose, we consider light waves propagating through a selfoscillating turbulent plasma. Plasma, which is a quasineutral gas, is composed of charged particles and neutral particles. Due to chargedness of their elements, plasma allow currents and, on account of electrical force within a pair of plus and minus ions in plasma, there is a tendency for the plasma to become neutral as a whole. If we consider that the electrical force acts like a restoring force, this property is, in essence, the origin of plasma oscillation. Indeed, in macroscopic point of view, the plasma oscillation takes place by a restoring force when neutral plasma is slightly deviated from an equilibrium position.
When plasma oscillate, the resulting electromagnetic parameters in media, such as conductivity, permeability, and permittivity, may vary according to their specifically given oscillations. In this situation, the system is described by a timedependent Hamiltonian, i.e., it belongs to a timedependent harmonic oscillator. We will show that a periodical variation of the conductivity enables us to describe the light waves in media with a modified CaldirolaKanai (CK) oscillator. Our scheme for modifying the standard CK oscillator as a model for describing timevarying plasma in this situation is replacing the constant damping factor of the standard CK oscillator^{27,28} with a factor that involves a sinusoidally varying term in addition to a constant one. If we consider that conductivity acts as a damping factor for light wave propagation, this corresponds to the exact response of wave motion when sinusoidally varying conductivity is involved as a specific character of the plasma. Recent increasing interest in plasma oscillation is not only due to its great assortment of collective modes for which it sustains, but also due to its value in probing the density of ionized particles on the basis of the sole dependence of oscillation frequency on charge density of plasma^{22}. For other types of modified CK oscillator models, refer to Refs. 29,30,31.
The timedependent harmonic oscillator model is very useful when investigating a system which has one or more timedependent parameters and has attracted great interest from the early days of modern physics^{32,33,34,35,36,37}. Quantum features of the modified CK oscillator model for light waves in plasma that have timevarying electromagnetic parameters will be described using an SU(1,1) Lie algebraic approach. Lots of novel quantum phenomena such as phase coherence, quantum correlation, and quadrature squeezing can be explained through the use of the SU(1,1) Lie algebra, since it provides a useful formulae for describing the majority of quantum states^{29,37,38}.
Results
Dynamics of light waves in timevarying plasma
Consider a light wave propagating in timevarying plasma which has no net charge density. Suppose that the magnetic permeability and the electric conductivity vary as time goes by, whereas the electric permittivity is constant for simplicity. Then, the speed of a light wave in the medium varies with time and is expressed as , where is the electric permittivity and μ(t) is the timedependent magnetic permeability, whereas the wave number k is constant. The basic relations between fields in such timevarying plasma are given by and B = μ(t)H. Meanwhile, the relation between current density and electric field is J = σ(t)E, where σ(t) is the timedependent electric conductivity.
Let us take the Coulomb gauge for the sake of simplicity. Then, the scalar potential vanishes in (net) charge free space and, hence, we only need to consider the expansion of the vector potential in order to develop electromagnetic theory of the system. By representing the vector potential in the form it is able to separate the position and the time functions. The solutions of the system are entirely determined by Maxwell's equations. The position function is determined by the geometry of boundaries of plasma while the time function follows the equation^{36} where ω(t) = kc(t). For σ(t) = σ(0) and μ(t) = μ(0), our model reduces to that described by the familiar standard CK oscillator. As is well known, the standard CK oscillator is an elementary system for describing a damped harmonic oscillator characterized by a constant damping factor. The modification of the system from the standard CK oscillator in our model is mainly determined by the time behavior of σ(t).
Each particle in the plasma is arranged in such a way that the net force exerted to a particle from all other particles becomes zero, leading to a homogeneous neutral state of charge. If an electron moves from such equilibrium position, a positive charge is generated in the equilibrium position and there arises electric forces between this new positive charge and surrounding electrons. This leads the electron to oscillate. Since the interaction between electrons is very large, each electron oscillates with a characteristic frequency determined depending on properties of the plasma. Under such plasma oscillations^{3}, we assume that there is a sinusoidally varying term in the conductivity of media, as well as a constant one, such that where a and b are arbitrary positive constants and ϖ is a constant frequency of conductivity oscillation. In this model, σ is always positive for the case that a > b, whereas it is both positive and negative, in turn, whenever a < b. The properties of plasma with the negative value of σ have been studied in Refs. 3 and 39. However, in most cases, b is very small compared to a. Another example of conductivity oscillation is the Shubnikovde Haas effect^{40,41,42}. This oscillation is a macroscopically observed intrinsic quantum nature of matter which appears when the material is subjected to very intense magnetic fields with the upkeep of low temperature.
The quantum Hamiltonian that corresponds to the classical equation of motion, Eq. (2), can be represented as where . The time function Λ(t) is given by where is a constant. From Eq. (4), we see that the Hamiltonian of the system is timedependent. Note that the time dependency of Λ(t) does not disappear even when b = 0. Therefore, the standard CK oscillator is also a family of timedependent harmonic oscillator.
Formulation of wave function from SU(1,1) Lie algebra
A useful way for investigating complex dynamical systems is to introduce SU(1,1) Lie Algebraic operators associated with a given system. A broad range of oscillatorlike problems in physics is well described in terms of SU(1,1) Lie algebra^{29,37,38,43,44}. Accordingly, many authors have used this algebraic method to unfold both classical and quantum theories of optical systems. The phase properties of a quantized light which exhibits nonclassical properties have been theoretically investigated by Gerry by means of SU(1,1) Lie algebras^{38}. This algebra can also be successfully used for the purposes of studying coherence and squeezing properties of photons, analyzing quantum mechanical interferometers and beam splitters, investigating dissipative processes in quantum optical systems, and so on^{44}.
According to the SU(1,1) Lie algebraic theory^{29,37,38}, we introduce SU(1,1) generators that are suitable for our system such that where Ω is a real positive constant and ρ(t) is a timefunction that satisfies the differential equation of the form Equation (9) is a modified form of the Milne equation^{45}. Although we focus on analytical investigation of light waves, the Milne equation is also useful for studying numerical analysis of mechanical systems^{46}. If where ω_{0} is a constant, Eqs. (6)–(8) recover to those of the simple harmonic oscillator. The commutation relations between generators are , , and . At this stage, it can be easily checked that the direct differentiation of Eq. (6) results in zero: This means that is a timeconstant.
Now we define the raising and the lowering operators to be These also yield the usual commutation relations which are and . Considering that the eigenvalue of the Casimir operator, , is given by k(k − 1) = −3/16, we can confirm that the Bargmann index is k = 1/4 or k = 3/4. The basis of the unitary space is a set of an even boson number for k = 1/4 and an odd boson number for k = 3/4.
If we use the properties of SU(1,1) operators, it is possible to derive the wave functions of the light wave propagating through the plasma medium. From the procedure represented in the latter section, Methods, we can confirm that the wave functions in the Fock state are given by where are the eigenstates of : and θ_{n}(t) are timedependent phases of the form These wave functions are very useful when investigating quantum characteristics of the system and agree well with those obtained from other methods irrelevant to Lie algebraic scheme^{47}. For instance, the quantum fluctuation of canonical variables and the expectation value of quantum energy can be identified using the wave functions. Notice that the exact knowledge of ρ(t) that can be obtained from Eq. (9) is crucial for analyzing the time behavior of the wave functions.
Analysis of quantum behaviors
Our research in this work can be applied to diverse particular cases by choosing the timedependent angular frequency ω(t) given in Eq. (2). Let us consider one of the solvable cases in which ω(t) is given by where ω_{0} is a positive constant and Let us first see the classical features of this system. The classical energy of the light wave can be expressed as where . The classical solution of Eq. (2) in this case is given by where q_{0} and φ are arbitrary constants while, from Hamilton's equation, the canonical momentum becomes . If we insert these quantities, q(t) and p(t), in Eq. (17), we have The envelope of this energy decays with time due to the exponential factor e^{−Λ(t)}.
Now we turn our attention to quantum features of the system. From previous sections, we have seen that the quantum solutions are expressed in terms of the classical solution ρ(t). Hence, to investigate the quantum systems completely for particular cases, it is necessary to solve Eq. (9). In this case, we can easily verify that the solution of Eq. (9) is represented in the form Using this, Eq (12) can be rewritten as The probability densities which are the square of this equation, , are illustrated in Fig. 1, where we have chosen n = 5 as an example. Six peaks, that each corresponds to a highly concentrated local probability density, are found in this figure along the qaxis at any given time. This proves to be the same result as in the case of the well known simple harmonic oscillator. In general there are n + 1 regions of locally high probability density for nth quantum state. As you can see, the probability density converges to the origin where q = 0 as a whole. This means that the amplitude of the wave packet dissipates with time due to the existence of conductivity.
We can investigate various quantum properties of the system using the wave functions given in Eq. (21). As an example, we consider the expectation value of quantized energy of the system. The energy operator of the system is easily obtained by replacing q and p in Eq. (17) with operators and : The expectation value of this operator can be obtained with the use of Eq. (21), such that where These quantum energies dissipate over time according to the decrease of wave amplitude caused by the nonzero conductivity in plasma. The comparison of the quantum energy and the classical energy represented in Eq. (19) is given in Fig. 2. The time behavior of the quantum energy is very similar to that of the classical energy in most cases [see Fig. 2(a)], but sometimes they are somewhat different from each other [see Fig. 2(b) and 2(c)]. One of the reasons for this discrepancy of quantum energy with the classical one is the existence of the zeropoint energy in quantum energy as a pure quantum effect.
If we take the time average of Eq. (19) only for the oscillation relevant to the frequency represented in terms of ω_{0}, we have Thus, we can confirm that the time behavior of the quantum energy Eq. (23) is very similar to this timeaveraged classical energy.
The general wave function is the collection of all possible wave functions with order n in the Fock space: where c_{n} are amplitudes of nth Fock state wave function, which yield ∑_{n}c_{n}^{2} = 1. As a further illustration, let us consider an initial wave function of the form where β_{1} and β_{2} are arbitrary constants which are complex. Without loss of generality, we can put where ϑ_{1} is an arbitrary constant. β_{2} can also be represented in the same way. If we denote that β_{j} = Re(β_{j}) + iIm(β_{j}) (j = 1, 2) where Re(β_{j}) and Im(β_{j}) are real and imaginary parts of β_{j}, from the normalization condition of this wave function, we have the following relation that should be imposed here The amplitude of the nth order wave function is easily computed from Using Eqs. (21) and (27), c_{n} becomes By inserting the above equation and Eq. (21) into Eq. (26), we have This is the central result of our research. For t = 0, this equation recovers to Eq. (27) as expected. If we consider that no approximation is used up to now, the time behavior of the wave function, Eq. (31), is exact. Many authors rely on numerical methods^{4,9,26,48,49,50} in order to analyze complicated phenomena of light propagation in timevarying plasma. However, we no longer need numerical simulation in case it is possible to know exact analytical solutions. The analytical solution given in Eq. (31) is very convenient when estimating the time behavior of the light waves thanks to its flexible applicability in any situation. Moreover, its derivation requires no iteration procedure which is necessary when we find a new solution at different times or with other choices of parameters using numerical simulation.
Now, one can easily verify that the probability density can be represented as a simple form which is a Gaussian wave packet: where
In the derivation of Eq. (32), we used the relation given in Eq. (28). Because we see from Eq. (33) that the width ν(t) of the Gaussian wave packet is proportional to e^{−Λ(t)/2} and it is known from Eq. (5) that Λ(t) increases with time, the leading behavior of the width of the packet is that it becomes exponentially narrower with time while the height increases. This implies the appearance of dissipative characteristics of the Gaussian wave packet in response to the presence of conductivity σ(t). If we consider that the value in the parenthesis of Eq. (34) can be rewritten as , the center of the Gaussian wave packet oscillates sinusoidally with time like a classical state. We can also confirm these facts from Fig. 3. Figure 3(a) exhibits a highfrequency oscillation of the wave packet with relative high ω_{0}, whereas Fig. 3(b) a lowfrequency oscillation. Except for such dynamical oscillations of the packet, the timevarying behavior of the amplitude of packets given in Fig. 3 is largely similar to that of the Fock state represented in Fig. 1.
The probability that represents the contribution of nth Fock state wave to the total wave packet is given by Using this and Eq. (23), the mean value of energy for the Gaussian wave is evaluated to be where is the mean value of quantum numbers contributed to the packet, that is given by . In addition, the dispersion of energy is obtained from A straightforward calculation gives This also decreases with time and is proportional to .
We have shown up until now that the quantum wave packets in timevarying plasma converge to the origin, where q = 0, with time. Along this convergence, the quantum energy also dissipates. In the case that σ(t) → 0, the overall dissipation discussed in this section vanishes. We confirmed that the Gaussian wave packet, Eq. (32), oscillates like that of the classical states. The analysis of quantum light waves given here may provide a useful method for treating quantum behavior of light waves in complex plasma.
Discussion
The modified CK oscillator model is considered in order to investigate the analytic properties of light waves in timevarying plasma whose electromagnetic parameters such as magnetic permeability, and electric conductivity vary with time. The main factor that characterizes our model is the time behavior of the conductivity given in Eq. (3). One of prospect applications of the modified CK oscillator is the quantization problem of radiation fields in timevarying media like dynamical plasma. The conductivity of a plasma medium is responsible for the damping of a light wave propagating through it. Hence, if the conductivity vanishes and other electromagnetic parameters are constant, the radiation field does not dissipate with time and it merely corresponds to that of the case described by the simple harmonic oscillator.
Through different choices of timedependent parameters, our research can be applied to analyzing quantum behaviors of light waves in plasma under various situations of plasma process. When b → 0 and μ(t) → μ_{0} (constant), our model recovers to that of the standard CK oscillator, whereas, in the case that a → 0, it becomes a somewhat different model of a modified CK oscillator given in Ref. 51.
To study the quantum problem of light waves in timevarying plasma, the SU(1,1) Lie algebraic formulation of quantum theory is used. The SU(1,1) generators , , and are introduced as shown in Eqs. (6)–(8). The raising and the lowering operators associated with this algebra are also constructed and we took advantage of them in order to derive the exact wave functions of the system in the Fock state. Quantum solutions and further results for light waves are described in terms of the classical solution ρ(t) of a modified Milne equation^{45} given in Eq. (9). We can say that the obtainability of quantum solutions is determined by the solvability of Eq. (9). In fact, the solution of the Milne equation can be represented in terms of two linearly independent solutions of classical light wave, i.e., the solutions of Eq. (2)^{46}. For this reason, we are able to obtain ρ(t) provided that two linear independent solutions of Eq. (2) are known or derivable. This implies that the complete quantum solutions can be obtained according to our theory based on the SU(1,1) Lie algebraic approach, so long as the solutions of corresponding classical light waves are solved. Hence, we can analyze the overall quantum characteristics of light waves on the basis of such solutions. Indeed, this point is a great advantage when implementing our novel method in the light wave research. Notice that our description of quantum light waves requires no approximation or perturbation technique whatsoever.
As a particular case, we considered the light wave that has the natural frequency of Eq. (15). The wave functions in the Fock state are given by Eq. (21) and the corresponding probability densities are illustrated in Fig. 1. As time goes by, the wave packet converges to the origin where q = 0. It is shown that the time behavior of the quantum energy of light waves agrees well with that of the classical energy in most cases under the specific choice of the value of parameters, as shown in Fig. 2(a). However, in some cases, the quantum energy somewhat deviates from the classical one [see Figs. 2(b) and 2(c)]. One of the reasons for this discrepancy is the existence of the zeropoint energy in quantum cases. By choosing the initial wave packet as Eq. (27), we obtained the Gaussian wave packet that oscillates like a classical wave. The width of the Gaussian wave packet given in Fig. 3 was gradually narrowed as time went by due to the existence of conductivity in the media. This implies that the light waves dissipate with time. Indeed, conductivity acts like a damping factor for the traveling fields.
The current mainstream of the research for the properties of light propagation in complex timevarying media is to use numerical methods such as the FDTD method^{4,9,26}, the TMM (transfer matrix method)^{48}, the BPM (beam propagation methods)^{26,49}, and the FEM (finite element method)^{50}. However, in principle, numerical simulation is a second alternative chosen when we are unable to find analytical solutions. Due to the intrinsic limitation inherent in numerical methods, a solution obtained from a numerical simulation at a certain time t is inadequate to be used for further analysis at a later time. For this reason, analytical solutions for the time evolution of light waves, such as Eq. (31), are very important for identifying their overall time behavior in complicated timevarying media. Apparently, even if the time variation of the system is somewhat complex, we can flexibly use this analytical solution without necessity of paying attention to the change of the parameters in plasma.
As a final remark, our research can be applied to various fields in science and technology. Among them, the realtime state of plasma in a tokamak where nuclear fusion takes place can be analyzed and diagnosed by monitoring the radiation fields propagating through it, leading to keep the optimal condition for nuclear fusion^{4}. The physical conditions of plasma in a tokamak, such as temperature, pressure, and ionization, are closely related to the state of radiation fields in it, which we have studied here. The exact realtime check for the status of plasma in a tokamak is crucial for efficient and rigorous control of the whole procedure during the generation of fusion energy. Another useful application is biomedical THz imaging^{10} which is a very promising research field in biomedical science and engineering. The radiation fields within the frequency range of 0.1–10 THz are known as Trays. The Trays occupy a large part of the frequency band, involving both microwave and infrared, and relatively do not developed further when compared to other rays. Our theory developed here can be used to provide a theoretical background for frequency transformers which are necessary for producing Trays. Trays are obtained by transforming frequencies of beams that initially have other range of frequencies, because there is no known natural source of Trays with sufficient efficiency yet.
Methods
Here we represent how to derive wave functions of quantum light wave via the SU(1,1) Lie algebraic approach. If we denote the nth eigenfunction of as ϕ_{n}〉, it can be easily verified that
These properties are necessary when we derive quantum states of the system. From the condition that we derive the ground state eigenfunction of K_{0} to be On the same ground, the first excited eigenfunction, 〈qϕ_{1}〉, is also obtained by solving . Further, the even order excited eigenfunctions 〈qϕ_{n}_{ = 2m}〉 () can be evaluated by operating m times in 〈qϕ_{0}〉 [Eq. (42)] and the odd order excited eigenfunctions 〈qϕ_{n}_{ = 2m+1}〉 can also be obtained by the same operation in 〈qϕ_{1}〉. Eventually, by merging the corresponding two results^{37}, we have the normalized full eigenfunctions, 〈qϕ_{n}〉, which are represented in Eq. (13).
Now, by solving the eigenvalue equation, , we get the eigenvalues of as . If we recall from Eq. (10) that is a timeconstant, this result which exhibits that λ_{n} is not a function of t is natural.
According to the LewisRiesenfeld theory for a quantization of the timedependent harmonic oscillator^{32}, the wave functions are represented in terms of the eigenfunctions 〈qϕ_{n}〉 as shown in Eq. (12). However, the phases θ_{n}(t) of the wave functions are undetermined yet. By inserting the equation , where 〈qϕ_{n}〉 are given by Eq. (13), together with Eq. (4) into the Schrödinger equation, we can derive the exact phases^{32,37,47}. Thus, through this procedure, the full wave functions of the system in the Fock state are derived as given in Eq. (12) with Eqs. (13) and (14).
References
 1.
Heald, M. A. & Wharton, C. B. Plasma Diagnostics with Microwaves (New York, Wiley, 1965).
 2.
Vyacheslavov, L. N. et al. Diagnostics of strong Langmuir turbulence. Plasma Phys. Rep. 24, 183–190 (1998).
 3.
Ryzhii, V., Ryzhii, M., Shur, M. S. & Mitin, V. Negative terahertz dynamic conductivity in electrically induced lateral pin junction in graphene. Physica E 42, 719–721 (2010).
 4.
Kalluri, D. K. Electromagnetics of Time Varying Complex Media 2nd ed. (Boca Raton, CRC Press, 2010).
 5.
Knap, W. et al. Terahertz emission by plasma waves in 60 nm gate high electron mobility transistors. Appl. Phys. Lett. 84, 2331–2333 (2004).
 6.
Auld, B. A., Collins, J. H. & Zapp, H. R. Signal processing in a nonperiodically timevarying magnetoelastic medium. Proc. IEEE 56, 258–272 (1968).
 7.
Felsen, L. & Whitman, G. Wave propagation in timevarying media. IEEE Trans. Antennas Propagat. AP18, 242–253 (1970).
 8.
Fante, R. L. Transmission of electromagnetic waves into timevarying media. IEEE Trans. Antennas Propagat. AP19, 417–424 (1971).
 9.
Lee, J. H. & Kalluri, D. K. Threedimensional FDTD simulation of electromagnetic wave transformation in a dynamic inhomogeneous magnetized plasma. IEEE Trans. Antennas Propagat. 47, 1146–1151 (1999).
 10.
Budko, N. V. Electromagnetic radiation in a timevarying background medium. Phys. Rev. A 80, 053817 (2009).
 11.
Monroe, R. L. Electromagnetic radiation in a timevarying plasma. J. Appl. Phys. 41, 560–562 (1970).
 12.
Yang, L.X., Shen, D.H. & Shi, W.D. Analyses of electromagnetic scattering characteristics for 3D timevarying plasma medium. Acta Phys. Sin. 62, 104101 (2013).
 13.
He, G. et al. Channel characterization and finitestate Markov channel modeling for timevarying plasma sheath surrounding hypersonic vehicles. Prog. Electromag. Res. 145, 299–308 (2014).
 14.
Liu, S., Liu, S. & Yuan, N. FDTD simulation of bistatic scattering by conductive cylinder covered with inhomogeneous timevarying plasma. Plasma Sci. Technol. 8, 190–194 (2006).
 15.
Liu, S., Mo, J. & Yuan, N. FDTD simulation of electromagnetic reflection of conductive plane covered with inhomogeneous timevarying plasma. Int. J. Infrared Milli. Waves 23, 1179–1191 (2002).
 16.
Lee, J. H., Kalluri, D. K. & Nigg, G. C. FDTD simulation of electromagnetic wave transformation in a dynamic magnetized plasma. Int. J. Infrared Milli. Waves 21, 1223–1253 (2000).
 17.
Cho, S. N. Mechanism behind selfsustained oscillations in direct current glow discharges and dusty plasmas. Phys. Plasmas 20, 043708 (2013).
 18.
Zubtsov, V. M., Sinkevich, O. A. & Chuklova, V. T. Origination of a selfoscillating mode (magnetic striations) in a nonequilibrium magnetized plasma. J. Appl. Mech. Tech. Phys. 19, 296–302 (1978).
 19.
Fleishman, G. D. & Toptygin, I. N. Diffusive radiation in onedimensional Langmuir turbulence. Phys. Rev. E 76, 017401 (2007).
 20.
Vyacheslavov, L. N. et al. Strong Langmuir turbulence with and without collapse: Experimental study. Plasma Phys. Control. Fusion 44 (12 B SPEC), B279–B291 (2002).
 21.
Ossipenko, M. V. & Tsaun, S. V. Description of turbulent convection in a plasma with the help of interacting Lorentz oscillators. Plasma Phys. Rep. 26, 465–476 (2000).
 22.
Kulin, S., Killian, T. C., Bergeson, S. D. & Rolston, S. L. Plasma oscillations and expansion of an ultracold neutral plasma. Phys. Rev. Lett. 82, 318–321 (2000).
 23.
Savage, R. L., Jr, Joshi, C. J. & Mori, W. B. Frequency upconversion of electromagnetic radiation upon transmission into an ionization front. Phys. Rev. Lett. 68, 946–949 (1992).
 24.
Wilks, S. C., Dawson, J. M. & Mori, W. B. Frequency upconversion of electromagnetic radiation with use of an overdense plasma. Phys. Rev. Lett. 61, 337–340 (1988).
 25.
Kuo, S. P. Frequency upconversion of microwave pulse in a rapidly growing plasma. Phys. Rev. Lett. 65, 1000–1003 (1990).
 26.
Vukovic, A., Bekker, E. V., Sewell, P. & Benson, T. M. Efficient time domain modeling of rib waveguide RF modulators. J. Lightwave Technol. 24, 5044–5053 (2006).
 27.
Caldirola, P. Porze non conservative nella meccanica quantistica. Nuovo Cimento 18, 393–400 (1941).
 28.
Kanai, E. On the quantization of dissipative systems. Prog. Theor. Phys. 3, 440–442 (1948).
 29.
Ozeren, S. F. The effect of nonextensivity on the time evolution of the SU(1,1) coherent states driven by a damped harmonic oscillator. Physica A 337, 81–88 (2004).
 30.
Abdalla, M. S. & Colegrave, R. K. Harmonic oscillator with strongly pulsating mass under the action of a driving force. Phys. Rev. A 32, 1958–1964 (1985).
 31.
Ikot, A. N., Akpabio, L. E. & Antia, A. D. Path integral of timedependent modified CaldirolaKanai oscillator. Arab. J. Sci. Eng. 37, 217–224 (2012).
 32.
Lewis, H. R., Jr & Riesenfeld, W. B. An exact quantum theory of the timedependent harmonic oscillator and of a charged particle in a timedependent electromagnetic field. J. Math. Phys. 10, 1458–1473 (1969).
 33.
Yeon, K. H., Kim, D. H., Um, C. I., George, T. F. & Pandey, L. N. Relations of canonical and unitary transformations for a general timedependent quadratic Hamiltonian system. Phys. Rev. A 55, 4023–4029 (1997).
 34.
Choi, J. R. & Gweon, B. H. Operator method for a nonconservative harmonic oscillator with and without singular perturbation. Int. J. Mod. Phys. B 16, 4733–4742 (2002).
 35.
Choi, J. R. & Choi, S. S. Investigation of the coherent wave packet for a timedependent damped harmonic oscillator. J. Appl. Math. Comput. 17, 495–508 (2005).
 36.
Choi, J. R. & Yeon, K. H. Quantum properties of light in linear media with timedependent parameters by LewisRiesenfeld invariant operator method. Int. J. Mod. Phys. B 19, 2213–2224 (2005).
 37.
Choi, J. R. & Nahm, I. H. SU(1,1) Lie algebra applied to the general timedependent quadratic Hamiltonian system. Int. J. Theor. Phys. 46, 1–15 (2007).
 38.
Gerry, C. C. Phase operators for SU(1,1): Application to the squeezed vacuum. Phys. Rev. A 38, 1734–1738 (1988).
 39.
Ryzhii, V., Ryzhii, M., Mitin, V., Satou, A. & Otsuji, T. Effect of heating and cooling of photogenerated electronhole plasma in optically pumped graphene on population inversion. Jpn. J. Appl. Phys. 50, 094001 (2011).
 40.
Wosnitza, J. et al. Shubnikovde Haas effect in the superconducting state of an organic superconductor. Phys. Rev. B 62, R11973–R11976 (2000).
 41.
Linke, H. et al. Application of microwave detection of the Shubnikovde Haas effect in twodimensional systems. J. Appi. Phys. 73, 7533–7542 (1993).
 42.
Balicas, L. et al. Shubnikovde Haas effect in the metallic state of Na_{0.3}CoO_{2}. Phys. Rev. Lett. 97, 126401 (2006).
 43.
Inomata, A., Kuratsuji, H. & Gerry, C. C. Path Integrals and Coherent States of SU(2) and SU(1,1) (Singapore, World Scientific, 1992).
 44.
Ban, M. SU(1,1) Lie algebraic approach to linear dissipative processes in quantum optics. J. Math. Phys. 33, 3213–3228 (1992).
 45.
Milne, W. E. The numerical determination of characteristic numbers. Phys. Rev. 35, 863–867 (1930).
 46.
Korsch, H. J. & Laurent, H. Milne's differential equation and numerical solutions of the Schrödinger equation I. Boundstate energies for single and doubleminimum potentials. J. Phys. B: At. Mol. Phys. 14, 4213–4230 (1981).
 47.
Choi, J. R. Nonclassical properties of superpositions of coherent and squeezed states for electromagnetic fields in timevarying media. Quantum Optics and Laser Experiments Lyagushyn, S. (ed.), 25–48, (Rijeka, Intech, 2012).
 48.
Carretero, L., PerezMolina, M., Acebal, P., Blaya, S. & Fimia, A. Matrix method for the study of wave propagation in onedimensional general media. Opt. Express 14, 11385–11391 (2006).
 49.
Fesenko, V. I., Sukhoivanov, I. A., Shulga, S. N. & Andrade Lucio, J. A. Propagation of electromagnetic waves in anisotropic photonic structures. Advances in Photonic Crystals Passaro, V. M. N. (ed.), 79–105 (Rijeka: Intech; 2013).
 50.
Ninan, M., Zhengyi, J. & Dongbin, W. Analysis of multilayer sandwich structures by finite element method. Adv. Sci. Lett. 4, 3243–3248 (2011).
 51.
Ikot, A. N., Akpabio, L. E., Akpan, I. O., Umo, M. I. & Ituen, E. E. Quantum damped mechanical oscillator. Int. J. Opt. 2010, 275910 (2010).
Acknowledgements
This research was supported by the Basic Science Research Program of the year 2014 through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No.: NRF2013R1A1A2062907).
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Department of Radiologic Technology, Daegu Health College, Bukgu, Daegu 702722, Republic of Korea
 Jeong Ryeol Choi
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J.R.C. wrote the paper and approved it.
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The author declares no competing financial interests.
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Correspondence to Jeong Ryeol Choi.
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