# Ion Plasma Responses to External Electromagnetic Fields

Authors: H. W. L. Naus
Journal: SRX Physics
 (2010)

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## Abstract

The response of ion plasmas to external radiation fields is investigated in a quantum
mechanical formalism. We focus on the total electric field within the plasma. For general bandpass signals three frequency regions can be distinguished in terms of the plasma frequency. For low frequencies, the external field is shielded. For high frequencies, the field is not modified. Resonant behavior of the plasma appears for frequencies near the plasma frequency: large internal electric fields and induced currents are present. These effects may be relevant for biological systems. The model is therefore extended to a two-species plasma and additional interactions are studied. The response is not essentially altered. To make the models more realistic, a so-called bath is included. In the weak coupling approximation the resonance frequency is shifted and some damping occurs. Finite temperature effects on the electric field are absent. The energy of the system, however, depends on temperature.


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## Body

## 1. Introduction

Ion plasmas subject to external electromagnetic fields are studied in this paper within the theoretical framework of [1]. The latter work focusses on the global residual symmetry of electrodynamics, the displacement symmetry, which has been shown to be realized in the Wigner-Weyl mode in the electron plasma. The so-called zero-modes of the gauge field, that is, zero-momentum “photons”, play a crucial role. These dynamical quantum mechanical variables describe spatially constant, time-dependent electric fields with three polarization states. The concomitant symmetry is spontaneously broken for free or weakly interacting electromagnetic fields; this explains the vanishing mass of photons, in the sense that they can be interpreted as Goldstone bosons.In a plasma, however, the interactions between center of mass and zero-modes causes the symmetry to be realized in the Wigner-Weyl mode. This is explicitly demonstrated for a plasma of nonrelativistic electrons using periodic boundary conditions and assuming some uniformly distributed background charge. Its response to an external homogeneous electric field is also addressed in [1].Here we will apply this formalism to an ion plasma and extend it to the two-species plasma. Such a plasma serves as a model for the “free” ions, for example, sodium, pottassium, calcium, and chloride, present in the intra- and extracellular fluids of the human body. In particular, we want to investigate the plasma response to external electromagnetic fields with various waveforms. If such a response is nontrivial, it may indicate a mechanism for a possible biological effect of radio frequency fields used in, for example, communication. Such mechanisms are hardly known up to now. Since the plasma frequency is crucial for the response, we estimate these frequencies for the ions in the intra- and extracellular fluids.Further theoretical developments concern additional interactions. To this end, we use the Heisenberg picture of quantum mechanics. In realistic systems, damping and fluctuations are of course expected. In order to include such effects, we extend the plasma with a surrounding medium—thebath. It is shown that temperature does not affect the induced electric field in this extended model whereas it changes the energy.The possibility of effects of electromagnetic fields in biological systems has been addressed from a theoretical physics perspective in [2]. The following remarks elucidate the connection between this seminal study and our present work. We have found a long-range effect, in the sense that zero-mode oscillations are constant in space, and a collective effect, in the sense that the CM motion of the ions plays a crucial role. The screening of charges by ions is already mentioned in [2]. It is also noted that plasma modes of unattached electrons can be a source of electric vibrations. The paper [2] discusses the interaction of polarization waves with unattached ions as well. There is, of course, a vast amount of literature on biological effects of electromagnetic radiation. Some more recent examples are [3–7]. It is not the purpose of this study to discuss these papers; a recent review, including many references, is given in [8].The outline of this paper is as follows. First we introduce the theoretical description of the periodic ion plasma and the coupling of an external electromagnetic field, that is, essentially the formalism of [1]. Secondly, the plasma response is calculated for linearly polarized electromagnetic fields with a number of relevant waveforms. In Section 4, the results are extended to general bandpass signals. Estimates for realistic ion plasma frequencies within intra- and extracellular fluids are given next. Section 6 deals with the two-species plasma. Additional interactions are introduced in Section 7. The time-dependent energy of the plasma coupled to an external field is calculated as well. Then a surrounding medium, the bath, is included in the plasma model; Section 8 also considers temperature effects. Finally, we present some conclusions, further discussion, and an outlook.


## 2. Ion Plasma

For the complete description of the periodic plasma we refer to [1]. Note that the original derivation deals with electrons, implying that we need to change the relevant physical parameters like mass, charge and density. To neutralize the charge of the ions, a uniformly distributed background charge is assumed—in our application corresponding to other ions.


### 2.1. Formalism and Plasma Oscillations

In the following we restrict ourselves to the essential part of the dynamics needed to demonstrate the typical effects. The relevant Hamiltonian is given by(1)H0=12Nm(P⃗-NeA⃗0)2+12V(Π⃗0)2.
It describes the center of mass X⃗ of N ions, conjugate momentum P⃗, in interaction with the zero-mode of the gauge field A⃗0. The conjugate momentum of this zero-mode Π⃗0 is proportional to a spatially constant electric field E⃗0=-Π⃗0/V. The ion mass is denoted by m, its charge by e and V=L3 is the quantization volume; recall that periodic boundary conditions are imposed. Relative motion, the complete radiation field and their couplings are described by their respective Hamiltonians.The displacement symmetry, a relic of the original gauge symmetry, is characterized by the operators(2)D⃗0=NeX⃗-Π⃗0,Ω(n⃗)=exp(-i2πeLD⃗0·n⃗),
where n⃗ is integer, thereby respecting the boundary conditions. The transformations Ω shift the zero-mode and accordingly the center of mass momentum. They leave the Hamiltonians invariant, in particular,(3)Ω(n⃗)H0Ω†(n⃗)=H0.
This residual symmetry is extensively discussed in [1].Here we proceed to the known eigenfunctions of (1):(4)H0Ψn⃗,m⃗(X⃗,A⃗0)=ℰm⃗Ψn⃗,m⃗(X⃗,A⃗0)
which are given by(5)Ψn⃗,m⃗(X⃗,A⃗0)=1Vei(2π/L)n⃗·X⃗ψm⃗(A⃗0-2πNeLn⃗).
The nonstandard notation for the energy ℰ is chosen in order to avoid confusion with electric fields. The functions ψm⃗ are harmonic oscillator wave functions; the ground state reads(6)ψ0⃗(a⃗)=(Vωpπ)3/4exp(-12Vωpa⃗2),
with plasma frequency(7)ωp2=Ne2Vm.
The energy eigenvalues depend on the principal quantum number mp but are independent of center of mass motion, that is, infinitely degenerated:(8)ℰm⃗=ωp(32+mp);
we have taken ℏ=1. The equilibrium position of the harmonic oscillator, however, is determined by the center of mass momentum. The eigenfunctions Ψn⃗,m⃗ are center of mass momentum eigenstates as well. They are no eigenfunctions of the displacement operator D⃗0. These are obtained by the linear combinations
(9)ΨX⃗0,n⃗,m⃗(X⃗,A⃗0)=∑k⃗1Vei(2π/L)(n⃗+Nk⃗)·(X⃗-X⃗0)×ψm⃗(A⃗0-2πNeL(n⃗+Nk⃗)),
which are also eigenfunctions of the Hamiltonian, degenerated with (5). Once more, we refer to [1] for further discussion. It is only noted that the expectation value of the current operator J⃗=(1/2Nm){P⃗-NeA⃗0,δ(x⃗-X⃗)} vanishes for center of momentum eigenstates as well as for the gauge invariant states:(10)J⃗n⃗,m⃗(x⃗)=〈Ψn⃗,m⃗|J⃗|Ψn⃗,m⃗〉=0,J⃗X⃗0′,X⃗0,n⃗,m⃗(x⃗)=〈ΨX⃗0′,n⃗,m⃗|J⃗|ΨX⃗0,n⃗,m⃗〉=0.


### 2.2. Plasma in Homogeneous Electric Field

In order to assess the effects of pulsed electromagnetic fields in biological systems, we extend the work of [1] on a plasma in an external field. It is assumed that the electric field is spatially constant—this appears to be a good approximation as long as the involved wavelengths are not too small. An arbitrary time dependence E⃗(t) is allowed. Such a field couples to the center of mass of ions and the Hamiltonian becomes time dependent:(11)H0(t)=12Nm(P⃗-Ne(A⃗0+A⃗(t)))2+12V(Π⃗0)2,
where the electric field follows from the vector potential:(12)E⃗(t)=-ddtA⃗(t).
The time-dependent Schrödinger equation is supplemented with the initial condition:(13)Ψn⃗,m⃗(X⃗,A⃗0,t=0)=Ψn⃗,m⃗(X⃗,A⃗0);
compare (5). Therefore it is solved with the ansatz(14)Ψn⃗,m⃗(X⃗,A⃗0,t)=1Ve-iφ(t)eik⃗n⃗·X⃗eiA⃗0·d⃗(t)×ψm⃗(A⃗0-1Nek⃗n⃗-a⃗(t))
with k⃗n⃗=(2π/L)n⃗. Matching the various A⃗0 dependences yields the explicit solution for a⃗ [1]. This is sufficient to calculate the expectation values of the total electric field in the plasma and the current density. Modifying the notation of [1], we explicitly obtain(15)E⃗tot(t)=E⃗(t)+E⃗0=E⃗(t)+ωpImZ⃗[E⃗],(16)J⃗(t)=eVmReZ⃗[E⃗],
where
(17)Z⃗[E⃗]=∫0tdτeiωp(τ-t)E⃗(τ).
The external field thus induces a current in the plasma; note that the electromagnetic current density follows as J⃗em=eNJ⃗. Concomitantly, the external electric field gets modified. In Section 3, this will be studied in more detail for various wave forms. The initial condition may be altered to the gauge invariant wave function (6). It can be checked that the obtained expectation values do not change.The obtained expectation values satisfyJ⃗em=-dE⃗0/dt, consistent with the remaining nontrivial Maxwell equation in the zero-mode sector. Finally, we mention a simple classical model for electron plasma oscillations [9]. If an external electric field is coupled to the electron gas, the model yields similar results for the classical field and current.


## 2.1. Formalism and Plasma Oscillations

In the following we restrict ourselves to the essential part of the dynamics needed to demonstrate the typical effects. The relevant Hamiltonian is given by(1)H0=12Nm(P⃗-NeA⃗0)2+12V(Π⃗0)2.
It describes the center of mass X⃗ of N ions, conjugate momentum P⃗, in interaction with the zero-mode of the gauge field A⃗0. The conjugate momentum of this zero-mode Π⃗0 is proportional to a spatially constant electric field E⃗0=-Π⃗0/V. The ion mass is denoted by m, its charge by e and V=L3 is the quantization volume; recall that periodic boundary conditions are imposed. Relative motion, the complete radiation field and their couplings are described by their respective Hamiltonians.The displacement symmetry, a relic of the original gauge symmetry, is characterized by the operators(2)D⃗0=NeX⃗-Π⃗0,Ω(n⃗)=exp(-i2πeLD⃗0·n⃗),
where n⃗ is integer, thereby respecting the boundary conditions. The transformations Ω shift the zero-mode and accordingly the center of mass momentum. They leave the Hamiltonians invariant, in particular,(3)Ω(n⃗)H0Ω†(n⃗)=H0.
This residual symmetry is extensively discussed in [1].Here we proceed to the known eigenfunctions of (1):(4)H0Ψn⃗,m⃗(X⃗,A⃗0)=ℰm⃗Ψn⃗,m⃗(X⃗,A⃗0)
which are given by(5)Ψn⃗,m⃗(X⃗,A⃗0)=1Vei(2π/L)n⃗·X⃗ψm⃗(A⃗0-2πNeLn⃗).
The nonstandard notation for the energy ℰ is chosen in order to avoid confusion with electric fields. The functions ψm⃗ are harmonic oscillator wave functions; the ground state reads(6)ψ0⃗(a⃗)=(Vωpπ)3/4exp(-12Vωpa⃗2),
with plasma frequency(7)ωp2=Ne2Vm.
The energy eigenvalues depend on the principal quantum number mp but are independent of center of mass motion, that is, infinitely degenerated:(8)ℰm⃗=ωp(32+mp);
we have taken ℏ=1. The equilibrium position of the harmonic oscillator, however, is determined by the center of mass momentum. The eigenfunctions Ψn⃗,m⃗ are center of mass momentum eigenstates as well. They are no eigenfunctions of the displacement operator D⃗0. These are obtained by the linear combinations
(9)ΨX⃗0,n⃗,m⃗(X⃗,A⃗0)=∑k⃗1Vei(2π/L)(n⃗+Nk⃗)·(X⃗-X⃗0)×ψm⃗(A⃗0-2πNeL(n⃗+Nk⃗)),
which are also eigenfunctions of the Hamiltonian, degenerated with (5). Once more, we refer to [1] for further discussion. It is only noted that the expectation value of the current operator J⃗=(1/2Nm){P⃗-NeA⃗0,δ(x⃗-X⃗)} vanishes for center of momentum eigenstates as well as for the gauge invariant states:(10)J⃗n⃗,m⃗(x⃗)=〈Ψn⃗,m⃗|J⃗|Ψn⃗,m⃗〉=0,J⃗X⃗0′,X⃗0,n⃗,m⃗(x⃗)=〈ΨX⃗0′,n⃗,m⃗|J⃗|ΨX⃗0,n⃗,m⃗〉=0.


## 2.2. Plasma in Homogeneous Electric Field

In order to assess the effects of pulsed electromagnetic fields in biological systems, we extend the work of [1] on a plasma in an external field. It is assumed that the electric field is spatially constant—this appears to be a good approximation as long as the involved wavelengths are not too small. An arbitrary time dependence E⃗(t) is allowed. Such a field couples to the center of mass of ions and the Hamiltonian becomes time dependent:(11)H0(t)=12Nm(P⃗-Ne(A⃗0+A⃗(t)))2+12V(Π⃗0)2,
where the electric field follows from the vector potential:(12)E⃗(t)=-ddtA⃗(t).
The time-dependent Schrödinger equation is supplemented with the initial condition:(13)Ψn⃗,m⃗(X⃗,A⃗0,t=0)=Ψn⃗,m⃗(X⃗,A⃗0);
compare (5). Therefore it is solved with the ansatz(14)Ψn⃗,m⃗(X⃗,A⃗0,t)=1Ve-iφ(t)eik⃗n⃗·X⃗eiA⃗0·d⃗(t)×ψm⃗(A⃗0-1Nek⃗n⃗-a⃗(t))
with k⃗n⃗=(2π/L)n⃗. Matching the various A⃗0 dependences yields the explicit solution for a⃗ [1]. This is sufficient to calculate the expectation values of the total electric field in the plasma and the current density. Modifying the notation of [1], we explicitly obtain(15)E⃗tot(t)=E⃗(t)+E⃗0=E⃗(t)+ωpImZ⃗[E⃗],(16)J⃗(t)=eVmReZ⃗[E⃗],
where
(17)Z⃗[E⃗]=∫0tdτeiωp(τ-t)E⃗(τ).
The external field thus induces a current in the plasma; note that the electromagnetic current density follows as J⃗em=eNJ⃗. Concomitantly, the external electric field gets modified. In Section 3, this will be studied in more detail for various wave forms. The initial condition may be altered to the gauge invariant wave function (6). It can be checked that the obtained expectation values do not change.The obtained expectation values satisfyJ⃗em=-dE⃗0/dt, consistent with the remaining nontrivial Maxwell equation in the zero-mode sector. Finally, we mention a simple classical model for electron plasma oscillations [9]. If an external electric field is coupled to the electron gas, the model yields similar results for the classical field and current.


## 3. Fields and Current in Ion Plasma

In this section, we take a linearly polarized external electric fieldE⃗=E(t)e⃗. The results are then obviously proportional to the constant unit vector e⃗,E⃗tot=Etot(t)e⃗,J⃗=J(t)e⃗, and will therefore be expressed in the amplitudes Etot(t) and J(t).


### 3.1. Constant and Harmonic Fields

As in [1] we first consider two simple cases, which enable an analytical calculation of electric field and current in the ion plasma. A time-independent electric field E(t)=E0 yields a harmonic total field:(18)Etot(t)=E0cosωpt.
The expectation value of current density is given by
(19)J(t)=eE0Vmsinωptωp.
Pure harmonic time dependencies with plasma frequency are obtained. On the average, the external field is shielded.For a periodic external fieldE(t)=E0cosωt one obtains the total field:(20)Etot(t)=E0(ωp2ωp2-ω2cosωpt-ω2ωp2-ω2cosωt),
and the current(21)J(t)=E0eVm(ωpωp2-ω2sinωpt-ωωp2-ω2sinωt).
For low frequencies, ω≪ωp, the external field is shielded. The total field is harmonic with frequency ωp. This is not the case for high frequencies, ω≫ωp: the external field is not modified and the induced current is very small. An interesting resonance mechanism appears in the region ω≃ωp. This is shown in Figure 1, where external and total field as well as the current are depicted. Note that the total field appears to be amplitude modulated. For small frequency difference δω=|ωp-ω| this can indeed be analytically verified. The modulation frequency is given by (1/2)δω and an amplitude amplification factor (ωp+ω)/2δω is obtained.(a) External and total fields. (b) Current density,f=0.95 MHz, fp=1.0 MHz.
(a)(b)


### 3.2. Pulsed Fields

For the study of possible bioelectromagnetic effects, a number of specific waveforms are selected. Pulsed signals are of particular interest; typical values for the parameters are used. We present results for total field and current density in the plasma for the various pulsed signals. The integrations cf. (17) have been computed numerically. Figure 2 shows the response to the Gaussian pulse:(a) External and total fields,fp=3 GHz. (b) Current density.
(a)(b)(22)E(t)=E0exp(-π(tT)2)
with T= 166 ps, for a plasma frequency fp=3GHz. The result is equivalent to exciting a harmonic oscillator—in our case the zero-mode gauge field. Next, in Figure 3(a), the effects of pulse repetition are shown. The results for the modulated Gaussian pulseExternal and total fields: (a) Gaussian pulse, repetition;fp=3 GHz; (b) on/off switching, fp=1.0 MHz.
(a)(b)(23)E(t)=E0sinωtexp(-π(tT)2),
with T= 332 ps, ω=4π GHz, are not essentially different (cf. Figure 4). On the other hand, if the plasma frequency is not in the frequency band of such a pulse one obtains no effects for small ωp and shielding for large ωp; see Figure 5. For completeness, we also include amplitude modulation with modulation frequency 0.1 MHz. Shielding and the absence of effects appear in the respective frequency regions. Here we depict the resonance effects for field and current in Figure 6. Figure 7 shows a resonant plasma response for ultra wideband (wideband high-power microwaves) signals:(a) External and total fields,fp=3 GHz. (b) Current density.
(a)(b)(a) No effects,fp≪3 GHz. (b) Shielding fp≫3 GHz.
(a)(b)(a) External and total fields,fp≈fc=1 MHz. (b) current.
(a)(b)(a) External and total fields,fp=3 GHz. (b) Current density.
(a)(b)(24)E(t)=E0texp(-π(tT)2)
with T= 0.2 ns. Another example of a broadband signal is the nuclear electromagnetic pulse (NEMP)
(25)E(t)=E0(e-αt-e-βt):
with α=4.0·106s-1 and β=4.76·108s-1. The resonant plasma response is shown in Figure 8. Note the relatively low plasma frequency used in this NEMP calculation. The results for narrow band high-power microwaves, GSM and chirp signals resemble those of a continuous harmonic wave which is switched on and off. The concomitant plasma response, in particular the electric field, is shown in Figure 3(b).(a) External and total fields,fp=20 MHz. (b) Current density
(a)(b)


## 3.1. Constant and Harmonic Fields

As in [1] we first consider two simple cases, which enable an analytical calculation of electric field and current in the ion plasma. A time-independent electric field E(t)=E0 yields a harmonic total field:(18)Etot(t)=E0cosωpt.
The expectation value of current density is given by
(19)J(t)=eE0Vmsinωptωp.
Pure harmonic time dependencies with plasma frequency are obtained. On the average, the external field is shielded.For a periodic external fieldE(t)=E0cosωt one obtains the total field:(20)Etot(t)=E0(ωp2ωp2-ω2cosωpt-ω2ωp2-ω2cosωt),
and the current(21)J(t)=E0eVm(ωpωp2-ω2sinωpt-ωωp2-ω2sinωt).
For low frequencies, ω≪ωp, the external field is shielded. The total field is harmonic with frequency ωp. This is not the case for high frequencies, ω≫ωp: the external field is not modified and the induced current is very small. An interesting resonance mechanism appears in the region ω≃ωp. This is shown in Figure 1, where external and total field as well as the current are depicted. Note that the total field appears to be amplitude modulated. For small frequency difference δω=|ωp-ω| this can indeed be analytically verified. The modulation frequency is given by (1/2)δω and an amplitude amplification factor (ωp+ω)/2δω is obtained.(a) External and total fields. (b) Current density,f=0.95 MHz, fp=1.0 MHz.
(a)(b)


## 3.2. Pulsed Fields

For the study of possible bioelectromagnetic effects, a number of specific waveforms are selected. Pulsed signals are of particular interest; typical values for the parameters are used. We present results for total field and current density in the plasma for the various pulsed signals. The integrations cf. (17) have been computed numerically. Figure 2 shows the response to the Gaussian pulse:(a) External and total fields,fp=3 GHz. (b) Current density.
(a)(b)(22)E(t)=E0exp(-π(tT)2)
with T= 166 ps, for a plasma frequency fp=3GHz. The result is equivalent to exciting a harmonic oscillator—in our case the zero-mode gauge field. Next, in Figure 3(a), the effects of pulse repetition are shown. The results for the modulated Gaussian pulseExternal and total fields: (a) Gaussian pulse, repetition;fp=3 GHz; (b) on/off switching, fp=1.0 MHz.
(a)(b)(23)E(t)=E0sinωtexp(-π(tT)2),
with T= 332 ps, ω=4π GHz, are not essentially different (cf. Figure 4). On the other hand, if the plasma frequency is not in the frequency band of such a pulse one obtains no effects for small ωp and shielding for large ωp; see Figure 5. For completeness, we also include amplitude modulation with modulation frequency 0.1 MHz. Shielding and the absence of effects appear in the respective frequency regions. Here we depict the resonance effects for field and current in Figure 6. Figure 7 shows a resonant plasma response for ultra wideband (wideband high-power microwaves) signals:(a) External and total fields,fp=3 GHz. (b) Current density.
(a)(b)(a) No effects,fp≪3 GHz. (b) Shielding fp≫3 GHz.
(a)(b)(a) External and total fields,fp≈fc=1 MHz. (b) current.
(a)(b)(a) External and total fields,fp=3 GHz. (b) Current density.
(a)(b)(24)E(t)=E0texp(-π(tT)2)
with T= 0.2 ns. Another example of a broadband signal is the nuclear electromagnetic pulse (NEMP)
(25)E(t)=E0(e-αt-e-βt):
with α=4.0·106s-1 and β=4.76·108s-1. The resonant plasma response is shown in Figure 8. Note the relatively low plasma frequency used in this NEMP calculation. The results for narrow band high-power microwaves, GSM and chirp signals resemble those of a continuous harmonic wave which is switched on and off. The concomitant plasma response, in particular the electric field, is shown in Figure 3(b).(a) External and total fields,fp=20 MHz. (b) Current density
(a)(b)


## 4. General Bandpass Signals

The results obtained so far suggest that there are three regions in the frequency domain, governing the response to general bandpass signals. In order to confirm this, the cosine transform of a real electric field is used [10]:(26)Ec(ω)=2π∫0∞dtE(t)cosωt.
The inverse transformation is given by(27)E(t)=2π∫0∞dωEc(ω)cosωt.
Straightforward integration leads to the following expression for Z:(28)Z(t)=-12i2π∫0∞dωEc(ω)×{1ωp+ω(eiωt-e-iωpt)+1ωp-ω(e-iωt-e-iωpt)}.
For the total field we obtain(29)Etot(t)=E(t)-2π∫0∞dωEc(ω)ωp2ωp2-ω2×{cosωt-cosωpt}.This result is generally valid. Let us now consider bandpass signals. This means that the nonvanishing frequency components of the electric field are prominently centered around a carrier frequencyωc with bandwidth B. It implies that we can approximate the frequency integrals:(30)∫0∞dωEc(ω){⋯}≈∫ωc-B/2ωc+B/2dωEc(ω){⋯}.
Now the division into three frequency regions can be made. For ωp≫ωc it follows from (29) that the external field is shielded. On the other hand, we see that for ωc≫ωp the electric field is almost not influenced by the plasma Etot(t)≈E(t). Most interesting is the region ωp≈ωc, where resonance effects appear(31)Etot(t)≈E(t)-2π∫ωc-B/2ωc+B/2dωEc(ω)ωp2ωp2-ω2×{cosωt-cosωpt}.
Equivalent results can be derived by employing the sine-transform which is also defined in [10].


## 5. Estimates of Plasma Frequencies

As mentioned in the introduction, a possible application of this framework is the effect of electromagnetic fields on biological systems. It is known that intracellular as well as extracellular fluid contains ions, in particular sodium, potassium, calcium, and cloride. Here we estimate the concomitant plasma frequencies for realistic concentrations. The latter is usually given as molar concentration or molarity denoting the number of moles per liter; the symbol for mole/liter is M. In Table1 typical concentrations are listed. The  corresponding plasma frequencies can be computed with (7); a convenient approximate expression is ωp2≈1.0·1027×q2(MC/A), with (atomic) mass number A, molar concentration MC, and the integer charge q in terms of the elementary charge. Thus q=2 for calcium; for the other ions q=1. This leads to the estimated plasma frequencies fp=ωp/2π shown in Table 2.Table 1
Ion densities.
IonSymbolAtomic and mass numberIntracellularExtracellularSodiumNa+Z=11,A=2310 mM140 mMPotassiumK+Z=19,A=39160 mM4 mMCalciumCa2+Z=20,A=400.1μM2 mMChlorideCl-Z=17,A=355 mM110 mMTable 2
Plasma frequencies.
IonSymbolAtomic and mass numberIntracellularExtracellularSodiumNa+Z=11,A=23100 GHz370 GHzPotassiumK+Z=19,A=39320 GHz50 GHzCalciumCa2+Z=20,A=400.5 GHz70 GHzChlorideCl-Z=17,A=3560 GHz280 GHz


## 6. Two-Species Plasma

In this section we extend the theory to a plasma with two species of ions. Their charges areeq1 and -eq2, where e now denotes the elementary charge and q1,q2 are positive integers. The respective numbers of particles are denoted by N1,N2; we furthermore introduce the charges Q1=eN1q1 and Q2=eN2q2. If Q1=Q2, no additional background charge needs to be assumed. Otherwise we still have to neutralize the net charge by a uniform background distribution [1].


### 6.1. Hamiltonian and Eigenfunctions

The zero-mode Hamiltonian (cf. (1)) is now given by(32)H0=12V(Π⃗0)2+12N1m1(P⃗1-Q1A⃗0)2+12N2m2(P⃗2+Q2A⃗0)2
with particle CM coordinates X⃗1,X⃗2, conjugate momenta P⃗1,P⃗2, and masses m1 and m2. The displacement operator is modified as(33)D⃗0=Q1X⃗1-Q2X⃗2-Π⃗0.
It generates the symmetry transformation(34)Ω(n⃗)=exp(-i2πeLD⃗0·n⃗),
which leaves the Hamiltonian invariant. The time-independent Schrödinger equation is solved with the translationally invariant ansatz:(35)Ψ(X⃗1,X⃗2,A⃗0)=1Veip⃗1·X⃗11Veip⃗2·X⃗2ψ(A⃗0)
with p⃗1=(2π/L)n⃗1 and p⃗2=(2π/L)n⃗2. Again we obtain harmonic oscillator wave functions ψl⃗:(36)ψ(A⃗0)=ψl⃗(A⃗0-Q1M2p⃗1-Q2M1p⃗2M2Q12+M1Q22),
and plasma frequency(37)ωp2=M2Q12+M1Q22M1M2V.
We have introduced the masses M1=N1m1 and M2=N2m2. The energy eigenvalues follow as(38)ℰl⃗,n⃗1,n⃗2=ωp(32+lp)+(Q2p⃗1+Q1p⃗2)22(M2Q12+M1Q22).
In contrast to the one-ion plasma, there are nonvanishing contributions of the center of mass momenta. Concomitantly, the eigenstates support nonvanishing expectation values of the current density J⃗n⃗1,n⃗2,l⃗(x⃗)(39)〈Ψn⃗1,n⃗2,l⃗|12N1m1{P⃗1-Q1A⃗0,δ(x⃗-X⃗1)}+12N2m2{P⃗2+Q2A⃗0,δ(x⃗-X⃗2)}|Ψn⃗1,n⃗2,l⃗〉=1VQ1+Q2M2Q12+M1Q22(Q2p⃗1+Q1p⃗2).
The constructed eigenfunctions transform under displacements as(40)Ω(k⃗)Ψn⃗1,n⃗2,l⃗=Ψn⃗1-Q1k⃗,n⃗2+Q2k⃗,l⃗.
Just like in the one-component plasma there is an infinite degeneracy:(41)H0Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0)=ℰl⃗,n⃗1+Q1k⃗,n⃗2-Q2k⃗Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0)=ℰl⃗,n⃗1,n⃗2Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0).
This enables the construction of gauge invariant eigenstates:(42)ΨX⃗01,X⃗02,n⃗1,n⃗2,l⃗G=∑k⃗Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0).
Indeed one can verify that(43)Ω(j⃗)ΨX⃗01,X⃗02,n⃗1,n⃗2,l⃗G=e-i(2π/L)j⃗·(Q1X⃗01-Q2X⃗02)×ΨX⃗01,X⃗02,n⃗1,n⃗2,l⃗G.


### 6.2. External Homogeneous Field

Again we study the coupling of a spatially constant electric field with arbitrary time dependence (12). The time-dependent Hamiltonian follows as(44)H0(t)=12V(Π⃗0)2+12M1(P⃗1-Q1(A⃗0+A⃗(t)))2+12M2(P⃗2+Q2(A⃗0+A⃗(t)))2.
The initial condition is taken as
(45)Ψn⃗1,n⃗2,l⃗(X⃗1,X⃗2,A⃗0,t=0)=Ψn⃗1,n⃗2,l⃗(X⃗1,X⃗2,A⃗0).
The time-dependent Schrödinger equation can be solved with an ansatz analogous to (14):(46)Ψn⃗1,n⃗1,l⃗(X⃗1,X⃗2,A⃗0,t)=1Ve-iφ(t)eip⃗1(n⃗1)·X⃗1×eip⃗2(n⃗2)·X⃗2eiA⃗0·d⃗(t)ψl⃗(A⃗0-q⃗n⃗1,n⃗2-a⃗(t)),
where q⃗n⃗1,n⃗2=(M2Q12+M1Q22)-1(Q1M2p⃗1-Q2M1p⃗2). After some algebra, one verifies that the previous differential equation for a⃗(t) is not modified in the two-species case. The differential equation for the phase φ(t), however, is changed. For the evaluation of the expectation values of electric field and current one does need the explicit solution of the phase. The resulting current can be written as(47)J⃗(t)=1VQ1+Q2M2Q12+M1Q22(Q2p⃗1+Q1p⃗2)+M1Q2-M2Q1VM1M2(a⃗(t)+A⃗(t)),
and total field is given by(48)E⃗tot(t)=E⃗(t)-da⃗(t)dt,
with (cf. (17) and [1])
(49)a⃗(t)=ωp∫0tdτA⃗(τ)sinωp(τ-t).
Thus we obtain that the response of the two-species plasma is not essentially different from the one-ion case. The plasma frequency is changed according to (37) and the induced additional current has a different prefactor. It is finally mentioned that one may also impose the initial condition analogous to (45) in terms of gauge invariant states (42) without changing these results.


## 6.1. Hamiltonian and Eigenfunctions

The zero-mode Hamiltonian (cf. (1)) is now given by(32)H0=12V(Π⃗0)2+12N1m1(P⃗1-Q1A⃗0)2+12N2m2(P⃗2+Q2A⃗0)2
with particle CM coordinates X⃗1,X⃗2, conjugate momenta P⃗1,P⃗2, and masses m1 and m2. The displacement operator is modified as(33)D⃗0=Q1X⃗1-Q2X⃗2-Π⃗0.
It generates the symmetry transformation(34)Ω(n⃗)=exp(-i2πeLD⃗0·n⃗),
which leaves the Hamiltonian invariant. The time-independent Schrödinger equation is solved with the translationally invariant ansatz:(35)Ψ(X⃗1,X⃗2,A⃗0)=1Veip⃗1·X⃗11Veip⃗2·X⃗2ψ(A⃗0)
with p⃗1=(2π/L)n⃗1 and p⃗2=(2π/L)n⃗2. Again we obtain harmonic oscillator wave functions ψl⃗:(36)ψ(A⃗0)=ψl⃗(A⃗0-Q1M2p⃗1-Q2M1p⃗2M2Q12+M1Q22),
and plasma frequency(37)ωp2=M2Q12+M1Q22M1M2V.
We have introduced the masses M1=N1m1 and M2=N2m2. The energy eigenvalues follow as(38)ℰl⃗,n⃗1,n⃗2=ωp(32+lp)+(Q2p⃗1+Q1p⃗2)22(M2Q12+M1Q22).
In contrast to the one-ion plasma, there are nonvanishing contributions of the center of mass momenta. Concomitantly, the eigenstates support nonvanishing expectation values of the current density J⃗n⃗1,n⃗2,l⃗(x⃗)(39)〈Ψn⃗1,n⃗2,l⃗|12N1m1{P⃗1-Q1A⃗0,δ(x⃗-X⃗1)}+12N2m2{P⃗2+Q2A⃗0,δ(x⃗-X⃗2)}|Ψn⃗1,n⃗2,l⃗〉=1VQ1+Q2M2Q12+M1Q22(Q2p⃗1+Q1p⃗2).
The constructed eigenfunctions transform under displacements as(40)Ω(k⃗)Ψn⃗1,n⃗2,l⃗=Ψn⃗1-Q1k⃗,n⃗2+Q2k⃗,l⃗.
Just like in the one-component plasma there is an infinite degeneracy:(41)H0Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0)=ℰl⃗,n⃗1+Q1k⃗,n⃗2-Q2k⃗Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0)=ℰl⃗,n⃗1,n⃗2Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0).
This enables the construction of gauge invariant eigenstates:(42)ΨX⃗01,X⃗02,n⃗1,n⃗2,l⃗G=∑k⃗Ψn⃗1+Q1k⃗,n⃗2-Q2k⃗,l⃗(X⃗1-X⃗01,X⃗2-X⃗02,A⃗0).
Indeed one can verify that(43)Ω(j⃗)ΨX⃗01,X⃗02,n⃗1,n⃗2,l⃗G=e-i(2π/L)j⃗·(Q1X⃗01-Q2X⃗02)×ΨX⃗01,X⃗02,n⃗1,n⃗2,l⃗G.


## 6.2. External Homogeneous Field

Again we study the coupling of a spatially constant electric field with arbitrary time dependence (12). The time-dependent Hamiltonian follows as(44)H0(t)=12V(Π⃗0)2+12M1(P⃗1-Q1(A⃗0+A⃗(t)))2+12M2(P⃗2+Q2(A⃗0+A⃗(t)))2.
The initial condition is taken as
(45)Ψn⃗1,n⃗2,l⃗(X⃗1,X⃗2,A⃗0,t=0)=Ψn⃗1,n⃗2,l⃗(X⃗1,X⃗2,A⃗0).
The time-dependent Schrödinger equation can be solved with an ansatz analogous to (14):(46)Ψn⃗1,n⃗1,l⃗(X⃗1,X⃗2,A⃗0,t)=1Ve-iφ(t)eip⃗1(n⃗1)·X⃗1×eip⃗2(n⃗2)·X⃗2eiA⃗0·d⃗(t)ψl⃗(A⃗0-q⃗n⃗1,n⃗2-a⃗(t)),
where q⃗n⃗1,n⃗2=(M2Q12+M1Q22)-1(Q1M2p⃗1-Q2M1p⃗2). After some algebra, one verifies that the previous differential equation for a⃗(t) is not modified in the two-species case. The differential equation for the phase φ(t), however, is changed. For the evaluation of the expectation values of electric field and current one does need the explicit solution of the phase. The resulting current can be written as(47)J⃗(t)=1VQ1+Q2M2Q12+M1Q22(Q2p⃗1+Q1p⃗2)+M1Q2-M2Q1VM1M2(a⃗(t)+A⃗(t)),
and total field is given by(48)E⃗tot(t)=E⃗(t)-da⃗(t)dt,
with (cf. (17) and [1])
(49)a⃗(t)=ωp∫0tdτA⃗(τ)sinωp(τ-t).
Thus we obtain that the response of the two-species plasma is not essentially different from the one-ion case. The plasma frequency is changed according to (37) and the induced additional current has a different prefactor. It is finally mentioned that one may also impose the initial condition analogous to (45) in terms of gauge invariant states (42) without changing these results.


## 7. Plasma: Additional Interactions

### 7.1. Heisenberg Picture

Thus far we have studied the ion plasma using the Schrödinger picture of quantum mechanics. From now on, we exploit the Heisenberg picture since it has turned to be more convenient for our calculations. Earlier results for the plasma are readily reproduced and extended by a calculation of the energy. Then we reinvestigate the two-species plasma within this framework. The plasma model extended with an additional interacting degree of freedom is studied as well; it can be seen as the prelude to include a bath [11]—as will be done in Section 8.In the Heisenberg picture states in the Hilbert space are time-independent whereas operatorsO(t) all depend on time. Their equation of motion is governed by the Hamiltonian:(50)dO(t)dt=i[H,O(t)]+∂O(t)∂t.
Expectation values follow as(51)〈O(t)〉=〈Ψ0∣O(t)∣Ψ0〉,
where Ψ0 denotes the time-independent state. For the Hamiltonian one of course obtains(52)dHdt=∂H∂t.
Below we focus on solving the equation of motion for the electric field operator.


### 7.2. Plasma

#### 7.2.1. Formalism

We start by analyzing the one-component plasma in the Heisenberg picture. The Hamiltonian (cf. (1)) can be rewritten as
(53)H0=12Vωp2(A⃗0-1NeP⃗)2+12V(Π⃗0)2.
Next we define the creation and annihilation operators for k=1,2,3(x,y,z) as(54)ak†=12Vωp(Πk0+iVωpAk0),ak=12Vωp(Πk0-iVωpAk0).
By means of [Πk0,Al0]=-iδkl one readily verifies the basic commutation rule [ak,al†]=δkl. Expressing the zero-mode vector potential and electric field in the a-operators(55)Πk0=12Vωp(ak+ak†),Ak0=i2Vωp(ak-ak†)
yields for the Hamiltonian(56)H0=ωp(ak†ak+32)-iωpNe12Vωp(ak-ak†)Pk+12NmP⃗2.
Note that we use the summation convention. For the CM momentum one gets(57)dP⃗dt=i[H0,P⃗]=0.
Consequently, the momentum is conserved and we can replace the operator by its eigenvalue P⃗=k⃗n⃗. We again impose periodic boundary conditions. At this point one can derive the equations of motion for the creation and annihilation operators and solve them. We proceed, however, by defining the “shifted” operators:(58)bk†=ak†-iαk,bk=ak+iαk
with α⃗=(1/Ne)(Vωp/2)k⃗n⃗. Note that we omit the n-dependence of α⃗ in our notation. The new operators obviously fulfill [bk,bl†]=δkl. The electric field is given by(59)Πk0=12Vωp(bk+bk†).
In terms of the shifted operators the Hamiltonian reads(60)H0=ωp(bk†bk+32).
One recognizes the Hamiltonian of a three-dimensional harmonic oscillator. The equation of motion for bk follows as(61)dbkdt=i[H0,bk]=-iωpbk
with solution(62)bk(t)=bk(0)e-iωpt.
For the adjoint operator one obtains(63)bk†(t)=bk†(0)eiωpt.
If we choose as state Ψ0 one of the eigenstates of H0, of course with the above specified CM momentum k⃗n⃗, then the expectation value of the electric field vanishes.


#### 7.2.2. Plasma in External Electric Field

Once again we couple an external homogeneous electric field (12). The Hamilton operator is modified as(64)H0(t)=12Vωp2(A⃗0+A⃗(t)-1NeP⃗)2+12V(Π⃗0)2.
The CM momentum remains a conserved quantity; thus we insert P⃗=k⃗n⃗. Next we introduce the creation and annihilation operators ak†,ak and, subsequently, the shifted operators bk† and bk. Then we arrive at(65)H0(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗2(t).
Thus we get for the time development:(66)dbkdt=i[H0,bk]=-iωpbk-ωp12VωpAk(t),
which can be solved using the method of “variation of constants.” In the solution of the homogeneous equation bk(t)=bk(0)exp(-iωpt), we replace bk(0) by a function of time βk(t). This yields(67)dβkdt=-ωp12VωpeiωptAk(t),
which can be readily integrated. In this way we obtain(68)bk(t)=e-iωpt(bk(0)-ωp12Vωp∫0tdτeiωpτAk(τ)),bk†(t)=eiωpt(bk†(0)-ωp12Vωp∫0tdτe-iωpτAk(τ)),
which immediately gives(69)Πk0=12Vωp(bk(0)e-iωpt+bk†(0)eiωpt)-Vωp2∫0tdτcosωp(t-τ)Ak(τ).
Let us choose the state Ψ0 as eigenstate of H0. It implies for the expectation value of the total electric field:(70)E⃗tot(t)=E⃗(t)-1V〈Π⃗0〉=E⃗(t)+ωp2∫0tdτcosωp(t-τ)A⃗(τ).
This result is equivalent to (15) and (17) as can be verified by means of integration by parts.


#### 7.2.3. Energy Considerations

The expectation value of the time-dependent Hamiltonian (65) can be conveniently calculated in the Heisenberg picture. The initial state Ψ0 is taken as above with harmonic oscillator quantum numbers m1,m2,m3. Omitting this calculation redundant CM-label n⃗, we thus compute the energy ℰ(t), as(71)ℰ(t)=〈H0(t)〉=〈Ψ0∣H0(t)∣Ψ0〉=〈ψm⃗∣H0(t)∣ψm⃗〉.Using the time-developments (68), we obtain(72)ℰ(t)=ωp(mp+32)+12Vωp2×(ωp2Z⃗*[A⃗]·Z⃗[A⃗]+2ωpA⃗(t)·ImZ⃗[A⃗]+A⃗(t)·A⃗(t))
(cf. (17)). Integration by parts yields(73)ωpZ⃗[A⃗]=-i(A⃗(t)+Z⃗[E⃗]).
With this relation we can write the energy in terms of the external field as(74)ℰ(t)=ωp(mp+32)+12Vωp2Z⃗*[E⃗]·Z⃗[E⃗].
This result can also be expressed in the expectation values of electric field (15) and (electromagnetic) current (16) as(75)ℰ(t)=ωp(mp+32)+12V2NmJ⃗2(t)+12V(E⃗tot(t)-E⃗(t))2=ωp(mp+32)+V2ωp2J⃗em2(t)+12VE⃗0(t)2.
The time-dependent terms can be interpreted as the classical energy, expressed in classical current and field, induced by the external field. Note that this additional energy, just as current and field, does not depend on the quantum numbers of the initial state.


### 7.3. Two-Species Plasma

Using the Heisenberg picture, the two-species plasma is reanalyzed. The Hamiltonian (32) is rewritten as(76)H0=12V(Π⃗0)2+12ωp2(A⃗0)2-Q1M1P⃗1·A⃗0+Q2M2P⃗2·A⃗0+12M1P⃗12+12M2P⃗22,
where the plasma frequency ωp is given by (37). In terms of the creation and annihilation operators defined in (54), we get(77)H0=ωp(ak†ak+32)+12M1P⃗12+12M2P⃗22-i2Vωp(Q1M1(P⃗1)k-Q2M2(P⃗2)k)(ak-ak†).
Both CM operators commute with the Hamiltonian. Therefore, we can replace the operators by their respective eigenvalues, that is, p⃗1 and p⃗2:(78)H0=ωp(ak†ak+32)-iβk(ak-ak†)+12M1p⃗12+12M2p⃗22
with β⃗=(1/2Vωp)((Q1/M1)p⃗1-(Q2/M2)p⃗2). Introducing the shifted operators(79)bk=ak+iγk,bk†=ak†-iγk,
where γ⃗=β⃗/ωp=(1/2)Vωp((Q1M2p⃗1-Q2M1p⃗2)/(M2Q12+M1Q22)), diagonalizes the Hamiltonian
(80)H0=ωp(bk†bk+32)+(Q2p⃗1+Q1p⃗2)22(M2Q12+M1Q22).
Its energy eigenvalues obviously agree with those calculated in the Schrödinger picture (38).The inclusion of a homogeneous external electric field yields additional terms in the Hamiltonian:(81)H0(t)=H0+12Vωp2A⃗(t)2+Vωp2A⃗(t)·A⃗0-Q1M1P⃗1·A⃗(t)+Q2M2P⃗2·A⃗(t).
The CM momenta also commute with the modified Hamiltonian and therefore we replace them by their eigenvalues p⃗1 and p⃗2. Next we introduce the creation and annihilation operators ak,ak† and, eventually, bk and bk†. Straightforward algebra then leads to(82)H0(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗2(t),
which is formally identical to (65)—except for the fact that the plasma frequency is altered. Note that the plasma frequency appears also in the definition of the creation and annihilation operators. Herewith our previous results for the two-species plasma are confirmed.


### 7.4. Plasma: Additional Interaction and External Field

In this section, we include an additional degree of freedom which interacts with the zero-mode in the one-component plasma model. The interaction is in principle taken from [11] where a bath is coupled to a system, as will be studied below as well. Since we want to respect the displacement symmetry, however, a modification is necessary. We also couple the external homogeneous electric field from the onset. The complete Hamiltonian is thus taken as(83)H(t)=H0(t)+Hb+HI(t),
where H0(t) is given by (64) and Hb represents a simple harmonic oscillator:(84)Hb=12p⃗2+12ω12q⃗2.
The original interaction term [11] reads(85)HI0=gωpω1V(p⃗·Π⃗0+Vωpω1q⃗·A⃗0)
with coupling constant g. It has been actually been introduced as a simpler expression in terms of creation and annihilation operators. This term, however, explicitly breaks the displacement symmetry. The corresponding symmetric (gauge invariant) interaction is given by(86)HI=gωpω1V(p⃗·Π⃗0+Vωpω1q⃗·(A⃗0-1NeP⃗)).
Thus the additional degree of freedom couples  not only to the zero-mode but also to the CM of the ions. Coupling the external electric fields finally yields(87)HI(t)=gωpω1V×(p⃗·Π⃗0+Vωpω1q⃗·(A⃗0+A⃗(t)-1NeP⃗)).
We proceed as above, that is we introduce creation and annihilation operators for the zero-mode, substitute for the CM momentum its eigenvalue k⃗n⃗ and replace the ak,ak† by their shifted operators bk and bk†. Moreover, for the additional degree of freedom we define(88)ck=12ω1(pk-iω1qk),ck†=12ω1(pk+iω1qk),
which obey [ck,cl†]=δkl. After some computations one obtains for the Hamiltonian(89)H(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗(t)2+ω1(ck†ck+32)+g(bkck†+bk†ck)+ig12Vωp(ck-ck†)Ak(t),
where the original interaction term [11] is recognized.The Heisenberg equations of motion follow as(90)dbk(t)dt=i[H,bk]=-iωpbk-igck-ωp12VωpAk(t),dck(t)dt=i[H,ck]=-iω1ck-igbk-g12VωpAk(t).
Note that different Cartesian components do not couple. In matrix form we have(91)ddt(bkck)=-i(ωpggω1)(bkck)-Ak(t)12Vωp(ωpg).
The evolution operator exp𝒰t, where 𝒰 denotes the appearing 2×2 matrix, which does not depend on the Cartesian index k, of the homogeneous differential equations can be explicitly calculated. First, one obtains for the frequency eigenvalues(92)ω±=12(ωp+ω1)±12(ωp-ω1)2+4g2.
The corresponding eigenvectors read(93)1N(gω+-ωp),1N(ω--ω1g),
with normalization N=(ω+-ωp)2+g2=(ω--ω1)2+g2; note that one has ωp-ω+=ω--ω1. Herewith the evolution operator is constructed as(94)exp𝒰t=1N2(g2e-iω+t+ω̃2e-iω-tgω̂e-iω+t+gω̃e-iω-tgω̂e-iω+t+gω̃e-iω-tω̂2e-iω+t+g2e-iω-t)
where ω̃ denotes ω--ω1 and ω̂  denotes ω+-ωp. The solutions of the inhomogeneous  differential equations are given by(95)(bk(t)ck(t))=exp𝒰t(bk(0)ck(0))-12Vωp∫0tdt′exp[𝒰(t-t′)]Ak(t′)(ωpg).
The expression for Π⃗0(t) trivially follows.Finally, we specify the time-independent state as a product of eigenstates of the uncoupled Hamiltonians of the zero-modes and the additional degree of freedom respectively. Calculating the corresponding expectation value of the electric field operator yields after integrating by parts(96)E⃗tot(t)=E⃗(t)+ωpN2(g2I⃗(ω+)+(ω--ω1)2I⃗(ω-)),
where we have defined(97)I⃗(ξ)=∫0tdt′sinξ(t′-t)E⃗(t′).
In comparison with the plasma response without additional interaction, there appear two relevant frequencies which are determined by the original plasma frequency, the oscillator frequency and the coupling constant. The response at such a frequency is similar to the original one. Thus two resonances appear. This is explicitly seen by considering a harmonic external field E⃗(t)=E⃗0cosωt. Performing the integrals analytically gives for this case(98)E⃗tot(t)=E⃗0[g2N2ω+ωpω+2-ω2cosω+t+(ω--ω1)2N2ωpω-ω-2-ω2cosω-t+(1-g2N2ω+ωpω+2-ω2-(ω--ω1)2N2ω-ωpω-2-ω2)×cosωtdd].


## 7.1. Heisenberg Picture

Thus far we have studied the ion plasma using the Schrödinger picture of quantum mechanics. From now on, we exploit the Heisenberg picture since it has turned to be more convenient for our calculations. Earlier results for the plasma are readily reproduced and extended by a calculation of the energy. Then we reinvestigate the two-species plasma within this framework. The plasma model extended with an additional interacting degree of freedom is studied as well; it can be seen as the prelude to include a bath [11]—as will be done in Section 8.In the Heisenberg picture states in the Hilbert space are time-independent whereas operatorsO(t) all depend on time. Their equation of motion is governed by the Hamiltonian:(50)dO(t)dt=i[H,O(t)]+∂O(t)∂t.
Expectation values follow as(51)〈O(t)〉=〈Ψ0∣O(t)∣Ψ0〉,
where Ψ0 denotes the time-independent state. For the Hamiltonian one of course obtains(52)dHdt=∂H∂t.
Below we focus on solving the equation of motion for the electric field operator.


## 7.2. Plasma

### 7.2.1. Formalism

We start by analyzing the one-component plasma in the Heisenberg picture. The Hamiltonian (cf. (1)) can be rewritten as
(53)H0=12Vωp2(A⃗0-1NeP⃗)2+12V(Π⃗0)2.
Next we define the creation and annihilation operators for k=1,2,3(x,y,z) as(54)ak†=12Vωp(Πk0+iVωpAk0),ak=12Vωp(Πk0-iVωpAk0).
By means of [Πk0,Al0]=-iδkl one readily verifies the basic commutation rule [ak,al†]=δkl. Expressing the zero-mode vector potential and electric field in the a-operators(55)Πk0=12Vωp(ak+ak†),Ak0=i2Vωp(ak-ak†)
yields for the Hamiltonian(56)H0=ωp(ak†ak+32)-iωpNe12Vωp(ak-ak†)Pk+12NmP⃗2.
Note that we use the summation convention. For the CM momentum one gets(57)dP⃗dt=i[H0,P⃗]=0.
Consequently, the momentum is conserved and we can replace the operator by its eigenvalue P⃗=k⃗n⃗. We again impose periodic boundary conditions. At this point one can derive the equations of motion for the creation and annihilation operators and solve them. We proceed, however, by defining the “shifted” operators:(58)bk†=ak†-iαk,bk=ak+iαk
with α⃗=(1/Ne)(Vωp/2)k⃗n⃗. Note that we omit the n-dependence of α⃗ in our notation. The new operators obviously fulfill [bk,bl†]=δkl. The electric field is given by(59)Πk0=12Vωp(bk+bk†).
In terms of the shifted operators the Hamiltonian reads(60)H0=ωp(bk†bk+32).
One recognizes the Hamiltonian of a three-dimensional harmonic oscillator. The equation of motion for bk follows as(61)dbkdt=i[H0,bk]=-iωpbk
with solution(62)bk(t)=bk(0)e-iωpt.
For the adjoint operator one obtains(63)bk†(t)=bk†(0)eiωpt.
If we choose as state Ψ0 one of the eigenstates of H0, of course with the above specified CM momentum k⃗n⃗, then the expectation value of the electric field vanishes.


### 7.2.2. Plasma in External Electric Field

Once again we couple an external homogeneous electric field (12). The Hamilton operator is modified as(64)H0(t)=12Vωp2(A⃗0+A⃗(t)-1NeP⃗)2+12V(Π⃗0)2.
The CM momentum remains a conserved quantity; thus we insert P⃗=k⃗n⃗. Next we introduce the creation and annihilation operators ak†,ak and, subsequently, the shifted operators bk† and bk. Then we arrive at(65)H0(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗2(t).
Thus we get for the time development:(66)dbkdt=i[H0,bk]=-iωpbk-ωp12VωpAk(t),
which can be solved using the method of “variation of constants.” In the solution of the homogeneous equation bk(t)=bk(0)exp(-iωpt), we replace bk(0) by a function of time βk(t). This yields(67)dβkdt=-ωp12VωpeiωptAk(t),
which can be readily integrated. In this way we obtain(68)bk(t)=e-iωpt(bk(0)-ωp12Vωp∫0tdτeiωpτAk(τ)),bk†(t)=eiωpt(bk†(0)-ωp12Vωp∫0tdτe-iωpτAk(τ)),
which immediately gives(69)Πk0=12Vωp(bk(0)e-iωpt+bk†(0)eiωpt)-Vωp2∫0tdτcosωp(t-τ)Ak(τ).
Let us choose the state Ψ0 as eigenstate of H0. It implies for the expectation value of the total electric field:(70)E⃗tot(t)=E⃗(t)-1V〈Π⃗0〉=E⃗(t)+ωp2∫0tdτcosωp(t-τ)A⃗(τ).
This result is equivalent to (15) and (17) as can be verified by means of integration by parts.


### 7.2.3. Energy Considerations

The expectation value of the time-dependent Hamiltonian (65) can be conveniently calculated in the Heisenberg picture. The initial state Ψ0 is taken as above with harmonic oscillator quantum numbers m1,m2,m3. Omitting this calculation redundant CM-label n⃗, we thus compute the energy ℰ(t), as(71)ℰ(t)=〈H0(t)〉=〈Ψ0∣H0(t)∣Ψ0〉=〈ψm⃗∣H0(t)∣ψm⃗〉.Using the time-developments (68), we obtain(72)ℰ(t)=ωp(mp+32)+12Vωp2×(ωp2Z⃗*[A⃗]·Z⃗[A⃗]+2ωpA⃗(t)·ImZ⃗[A⃗]+A⃗(t)·A⃗(t))
(cf. (17)). Integration by parts yields(73)ωpZ⃗[A⃗]=-i(A⃗(t)+Z⃗[E⃗]).
With this relation we can write the energy in terms of the external field as(74)ℰ(t)=ωp(mp+32)+12Vωp2Z⃗*[E⃗]·Z⃗[E⃗].
This result can also be expressed in the expectation values of electric field (15) and (electromagnetic) current (16) as(75)ℰ(t)=ωp(mp+32)+12V2NmJ⃗2(t)+12V(E⃗tot(t)-E⃗(t))2=ωp(mp+32)+V2ωp2J⃗em2(t)+12VE⃗0(t)2.
The time-dependent terms can be interpreted as the classical energy, expressed in classical current and field, induced by the external field. Note that this additional energy, just as current and field, does not depend on the quantum numbers of the initial state.


## 7.2.1. Formalism

We start by analyzing the one-component plasma in the Heisenberg picture. The Hamiltonian (cf. (1)) can be rewritten as
(53)H0=12Vωp2(A⃗0-1NeP⃗)2+12V(Π⃗0)2.
Next we define the creation and annihilation operators for k=1,2,3(x,y,z) as(54)ak†=12Vωp(Πk0+iVωpAk0),ak=12Vωp(Πk0-iVωpAk0).
By means of [Πk0,Al0]=-iδkl one readily verifies the basic commutation rule [ak,al†]=δkl. Expressing the zero-mode vector potential and electric field in the a-operators(55)Πk0=12Vωp(ak+ak†),Ak0=i2Vωp(ak-ak†)
yields for the Hamiltonian(56)H0=ωp(ak†ak+32)-iωpNe12Vωp(ak-ak†)Pk+12NmP⃗2.
Note that we use the summation convention. For the CM momentum one gets(57)dP⃗dt=i[H0,P⃗]=0.
Consequently, the momentum is conserved and we can replace the operator by its eigenvalue P⃗=k⃗n⃗. We again impose periodic boundary conditions. At this point one can derive the equations of motion for the creation and annihilation operators and solve them. We proceed, however, by defining the “shifted” operators:(58)bk†=ak†-iαk,bk=ak+iαk
with α⃗=(1/Ne)(Vωp/2)k⃗n⃗. Note that we omit the n-dependence of α⃗ in our notation. The new operators obviously fulfill [bk,bl†]=δkl. The electric field is given by(59)Πk0=12Vωp(bk+bk†).
In terms of the shifted operators the Hamiltonian reads(60)H0=ωp(bk†bk+32).
One recognizes the Hamiltonian of a three-dimensional harmonic oscillator. The equation of motion for bk follows as(61)dbkdt=i[H0,bk]=-iωpbk
with solution(62)bk(t)=bk(0)e-iωpt.
For the adjoint operator one obtains(63)bk†(t)=bk†(0)eiωpt.
If we choose as state Ψ0 one of the eigenstates of H0, of course with the above specified CM momentum k⃗n⃗, then the expectation value of the electric field vanishes.


## 7.2.2. Plasma in External Electric Field

Once again we couple an external homogeneous electric field (12). The Hamilton operator is modified as(64)H0(t)=12Vωp2(A⃗0+A⃗(t)-1NeP⃗)2+12V(Π⃗0)2.
The CM momentum remains a conserved quantity; thus we insert P⃗=k⃗n⃗. Next we introduce the creation and annihilation operators ak†,ak and, subsequently, the shifted operators bk† and bk. Then we arrive at(65)H0(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗2(t).
Thus we get for the time development:(66)dbkdt=i[H0,bk]=-iωpbk-ωp12VωpAk(t),
which can be solved using the method of “variation of constants.” In the solution of the homogeneous equation bk(t)=bk(0)exp(-iωpt), we replace bk(0) by a function of time βk(t). This yields(67)dβkdt=-ωp12VωpeiωptAk(t),
which can be readily integrated. In this way we obtain(68)bk(t)=e-iωpt(bk(0)-ωp12Vωp∫0tdτeiωpτAk(τ)),bk†(t)=eiωpt(bk†(0)-ωp12Vωp∫0tdτe-iωpτAk(τ)),
which immediately gives(69)Πk0=12Vωp(bk(0)e-iωpt+bk†(0)eiωpt)-Vωp2∫0tdτcosωp(t-τ)Ak(τ).
Let us choose the state Ψ0 as eigenstate of H0. It implies for the expectation value of the total electric field:(70)E⃗tot(t)=E⃗(t)-1V〈Π⃗0〉=E⃗(t)+ωp2∫0tdτcosωp(t-τ)A⃗(τ).
This result is equivalent to (15) and (17) as can be verified by means of integration by parts.


## 7.2.3. Energy Considerations

The expectation value of the time-dependent Hamiltonian (65) can be conveniently calculated in the Heisenberg picture. The initial state Ψ0 is taken as above with harmonic oscillator quantum numbers m1,m2,m3. Omitting this calculation redundant CM-label n⃗, we thus compute the energy ℰ(t), as(71)ℰ(t)=〈H0(t)〉=〈Ψ0∣H0(t)∣Ψ0〉=〈ψm⃗∣H0(t)∣ψm⃗〉.Using the time-developments (68), we obtain(72)ℰ(t)=ωp(mp+32)+12Vωp2×(ωp2Z⃗*[A⃗]·Z⃗[A⃗]+2ωpA⃗(t)·ImZ⃗[A⃗]+A⃗(t)·A⃗(t))
(cf. (17)). Integration by parts yields(73)ωpZ⃗[A⃗]=-i(A⃗(t)+Z⃗[E⃗]).
With this relation we can write the energy in terms of the external field as(74)ℰ(t)=ωp(mp+32)+12Vωp2Z⃗*[E⃗]·Z⃗[E⃗].
This result can also be expressed in the expectation values of electric field (15) and (electromagnetic) current (16) as(75)ℰ(t)=ωp(mp+32)+12V2NmJ⃗2(t)+12V(E⃗tot(t)-E⃗(t))2=ωp(mp+32)+V2ωp2J⃗em2(t)+12VE⃗0(t)2.
The time-dependent terms can be interpreted as the classical energy, expressed in classical current and field, induced by the external field. Note that this additional energy, just as current and field, does not depend on the quantum numbers of the initial state.


## 7.3. Two-Species Plasma

Using the Heisenberg picture, the two-species plasma is reanalyzed. The Hamiltonian (32) is rewritten as(76)H0=12V(Π⃗0)2+12ωp2(A⃗0)2-Q1M1P⃗1·A⃗0+Q2M2P⃗2·A⃗0+12M1P⃗12+12M2P⃗22,
where the plasma frequency ωp is given by (37). In terms of the creation and annihilation operators defined in (54), we get(77)H0=ωp(ak†ak+32)+12M1P⃗12+12M2P⃗22-i2Vωp(Q1M1(P⃗1)k-Q2M2(P⃗2)k)(ak-ak†).
Both CM operators commute with the Hamiltonian. Therefore, we can replace the operators by their respective eigenvalues, that is, p⃗1 and p⃗2:(78)H0=ωp(ak†ak+32)-iβk(ak-ak†)+12M1p⃗12+12M2p⃗22
with β⃗=(1/2Vωp)((Q1/M1)p⃗1-(Q2/M2)p⃗2). Introducing the shifted operators(79)bk=ak+iγk,bk†=ak†-iγk,
where γ⃗=β⃗/ωp=(1/2)Vωp((Q1M2p⃗1-Q2M1p⃗2)/(M2Q12+M1Q22)), diagonalizes the Hamiltonian
(80)H0=ωp(bk†bk+32)+(Q2p⃗1+Q1p⃗2)22(M2Q12+M1Q22).
Its energy eigenvalues obviously agree with those calculated in the Schrödinger picture (38).The inclusion of a homogeneous external electric field yields additional terms in the Hamiltonian:(81)H0(t)=H0+12Vωp2A⃗(t)2+Vωp2A⃗(t)·A⃗0-Q1M1P⃗1·A⃗(t)+Q2M2P⃗2·A⃗(t).
The CM momenta also commute with the modified Hamiltonian and therefore we replace them by their eigenvalues p⃗1 and p⃗2. Next we introduce the creation and annihilation operators ak,ak† and, eventually, bk and bk†. Straightforward algebra then leads to(82)H0(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗2(t),
which is formally identical to (65)—except for the fact that the plasma frequency is altered. Note that the plasma frequency appears also in the definition of the creation and annihilation operators. Herewith our previous results for the two-species plasma are confirmed.


## 7.4. Plasma: Additional Interaction and External Field

In this section, we include an additional degree of freedom which interacts with the zero-mode in the one-component plasma model. The interaction is in principle taken from [11] where a bath is coupled to a system, as will be studied below as well. Since we want to respect the displacement symmetry, however, a modification is necessary. We also couple the external homogeneous electric field from the onset. The complete Hamiltonian is thus taken as(83)H(t)=H0(t)+Hb+HI(t),
where H0(t) is given by (64) and Hb represents a simple harmonic oscillator:(84)Hb=12p⃗2+12ω12q⃗2.
The original interaction term [11] reads(85)HI0=gωpω1V(p⃗·Π⃗0+Vωpω1q⃗·A⃗0)
with coupling constant g. It has been actually been introduced as a simpler expression in terms of creation and annihilation operators. This term, however, explicitly breaks the displacement symmetry. The corresponding symmetric (gauge invariant) interaction is given by(86)HI=gωpω1V(p⃗·Π⃗0+Vωpω1q⃗·(A⃗0-1NeP⃗)).
Thus the additional degree of freedom couples  not only to the zero-mode but also to the CM of the ions. Coupling the external electric fields finally yields(87)HI(t)=gωpω1V×(p⃗·Π⃗0+Vωpω1q⃗·(A⃗0+A⃗(t)-1NeP⃗)).
We proceed as above, that is we introduce creation and annihilation operators for the zero-mode, substitute for the CM momentum its eigenvalue k⃗n⃗ and replace the ak,ak† by their shifted operators bk and bk†. Moreover, for the additional degree of freedom we define(88)ck=12ω1(pk-iω1qk),ck†=12ω1(pk+iω1qk),
which obey [ck,cl†]=δkl. After some computations one obtains for the Hamiltonian(89)H(t)=ωp(bk†bk+32)+iωp12Vωp(bk-bk†)Ak(t)+12Vωp2A⃗(t)2+ω1(ck†ck+32)+g(bkck†+bk†ck)+ig12Vωp(ck-ck†)Ak(t),
where the original interaction term [11] is recognized.The Heisenberg equations of motion follow as(90)dbk(t)dt=i[H,bk]=-iωpbk-igck-ωp12VωpAk(t),dck(t)dt=i[H,ck]=-iω1ck-igbk-g12VωpAk(t).
Note that different Cartesian components do not couple. In matrix form we have(91)ddt(bkck)=-i(ωpggω1)(bkck)-Ak(t)12Vωp(ωpg).
The evolution operator exp𝒰t, where 𝒰 denotes the appearing 2×2 matrix, which does not depend on the Cartesian index k, of the homogeneous differential equations can be explicitly calculated. First, one obtains for the frequency eigenvalues(92)ω±=12(ωp+ω1)±12(ωp-ω1)2+4g2.
The corresponding eigenvectors read(93)1N(gω+-ωp),1N(ω--ω1g),
with normalization N=(ω+-ωp)2+g2=(ω--ω1)2+g2; note that one has ωp-ω+=ω--ω1. Herewith the evolution operator is constructed as(94)exp𝒰t=1N2(g2e-iω+t+ω̃2e-iω-tgω̂e-iω+t+gω̃e-iω-tgω̂e-iω+t+gω̃e-iω-tω̂2e-iω+t+g2e-iω-t)
where ω̃ denotes ω--ω1 and ω̂  denotes ω+-ωp. The solutions of the inhomogeneous  differential equations are given by(95)(bk(t)ck(t))=exp𝒰t(bk(0)ck(0))-12Vωp∫0tdt′exp[𝒰(t-t′)]Ak(t′)(ωpg).
The expression for Π⃗0(t) trivially follows.Finally, we specify the time-independent state as a product of eigenstates of the uncoupled Hamiltonians of the zero-modes and the additional degree of freedom respectively. Calculating the corresponding expectation value of the electric field operator yields after integrating by parts(96)E⃗tot(t)=E⃗(t)+ωpN2(g2I⃗(ω+)+(ω--ω1)2I⃗(ω-)),
where we have defined(97)I⃗(ξ)=∫0tdt′sinξ(t′-t)E⃗(t′).
In comparison with the plasma response without additional interaction, there appear two relevant frequencies which are determined by the original plasma frequency, the oscillator frequency and the coupling constant. The response at such a frequency is similar to the original one. Thus two resonances appear. This is explicitly seen by considering a harmonic external field E⃗(t)=E⃗0cosωt. Performing the integrals analytically gives for this case(98)E⃗tot(t)=E⃗0[g2N2ω+ωpω+2-ω2cosω+t+(ω--ω1)2N2ωpω-ω-2-ω2cosω-t+(1-g2N2ω+ωpω+2-ω2-(ω--ω1)2N2ω-ωpω-2-ω2)×cosωtdd].


## 8. Plasma, Bath and External Field

Finally, we extend our plasma model in order to describe the expected damping and fluctuations in realistic systems. To this end, we explicitly include the surrounding medium, orbath, as part of the system. In other words, additional interactions are taken into account; in principle we follow the approach of van Kampen [11]. In order not to overload the notation we consider only one Cartesian component of the complete model. It is assumed that the bath degrees of freedom as well as the external electric field couple to this component. The two other components then trivially decouple. Therefore the Cartesian index is omitted from now on. The essential physics evidently is captured in this way. Alternatively, one may consider the total Hamiltonian of [1] completely describing the plasma in interaction with the dynamical electromagnetic fields. This is, however, beyond the scope of our study.


### 8.1. Plasma and Bath

The Hamilton operator consists of three parts:(99)H=12Vω02(A0-1NeP)2+12V(Π0)2+HB+HI,
where the first terms describe again the zero-modes of the internal electromagnetic field coupled to the CM momentum of the ions. Here we denote the plasma frequency by ω0. The free bath Hamiltonian is taken as the sum of a number of harmonic oscillators with frequencies ωn, that is,(100)HB=12∑n(pn2+ωnqn2)=∑nωn(an†an+12)
with coordinates qn and momenta pn,[pn,qm]=-iδnm. The annihilation and creation operators are defined as(101)an=12ωn(pn-iωnqn),an†=12ωn(pn+iωnqn).
It follows that [an,am†]=δnm. We also define such operators for the zero-mode:(102)a0=12Vω0(Π0-iVω0A0),a0†=12Vω0(Π0+iVω0A0).
Van Kampen defines the interaction as [11](103)HI0=∑ngn(ana0†+an†a0)=1Vω0∑ngnωn(pnΠ0+Vω0ωnqnA0)
with coupling strengths gn. This interaction Hamiltonian is not invariant under displacements. Just as in the previous section, we therefore insert the symmetric “minimal substitution” term and use the interaction(104)HI=1Vω0∑ngnωn(pnΠ0+Vω0ωnqn(A0-1NeP))=∑ngn(ana0†+an†a0)-iNe12Vω0∑ngn(an-an†)P.
Thus the bath interacts with the CM degree of freedom as well. The Hamiltonian can now be now written as(105)H=ω0(a0†a0+12)-iω0Ne12Vω0(a0-a0†)P+∑nωn(an†an+12)+∑ngn(ana0†+an†a0)-iNe12Vω0∑ngn(an-an†)P+12NmP2
and can be analyzed analogous to the procedure of [11].


### 8.2. Inclusion of an External Electric Field

We proceed by including a homogeneous external field (12); the Hamiltonian becomes explicitly time-dependent:
(106)H(t)=ω0(a0†a0+12)-iω0Ne12Vω0(a0-a0†)P+iω012Vω0(a0-a0†)A(t)-Vω02NePA(t)+12Vω02A2(t)+∑nωn(an†an+12)+∑ngn(ana0†+an†a0)-iNe12Vω0∑ngn(an-an†)P+12NmP2+i12Vω0∑ngn(an-an†)A(t).
Since the CM coordinate is cyclic, [H,P]=0, the conjugate momentum P is a constant of motion P→kj=2nj/L. After inserting this in the Hamiltonian, we define the “shifted” operators b0 and b0†:(107)a0=b0-iα,a0†=b0†+iα,
with α=α(j)=(kj/Ne)(1/2)Vω0. Some algebra leads to(108)H(t)=ω0(b0†b0+12)+iω012Vω0(b0-b0†)A(t)+12Vω02A2(t)+∑nωn(an†an+12)+∑ngn(anb0†+an†b0)+i12Vω0∑ngn(an-an†)A(t).
The equations of motion for the Heisenberg operators follow as(109)db0(t)dt=i[H,b0]=-iω0b0-i∑mgmam-ω012Vω0A(t),dan(t)dt=i[H,an]=-iωnan-ignb0-gn12Vω0A(t).
The homogeneous differential equations are implicitly solved in [11]:(110)b0(t)=U(t)b0(0)+∑mVm(t)am(0),an(t)=Wn(t)b0(0)+∑mSnm(t)am(0).
The corresponding evolution operator is given by(111)expℳt=(U(t)V⃗tr(t)W⃗(t)S(t)).
The matrix ℳ is implicitly defined in (109). For b0(t) we consequently get as solution of the inhomogeneous differential equation:(112)b0(t)=U(t)b0(0)+∑mVm(t)am(0)-12Vω0η(t),
where we have defined(113)η(t)=ω0∫0tU(t-t′)A(t′)dt′+∑mgm∫0tVm(t-t′)A(t′)dt′.
The zero-mode electric field operator follows as(114)Π0=12Vω0(b0+b0†).
We take as time-independent state Ψ=Ψ0·ψB, with Ψ0 being an eigenstate of the plasma Hamiltonian and ψB being  an eigenstate of the bath Hamiltonian, that is, the product of the shifted zero-mode oscillator, the plane wave and bath harmonic oscillator wave functions. Then we get for the expectation value of the total electric field:(115)Etot=-1V〈Π0〉+E(t)=12ω0(η(t)+η*(t))+E(t).


### 8.3. Evolution Operator

Let us address the solution of the equations of motion (cf. (109)) in some more detail and explicitly construct the evolution operator. In principle we follow the approach of van Kampen [11].


#### 8.3.1. Exact Developments

The set of (109) is a solved with the ansatze-iλt for the time dependence of all modes. This leads to the eigenvalue equation:(116)λ-ω0=∑ngn2λ-ωn
with a sequence of solutions λν. Given the number of bath oscillators, their frequencies, and the coupling constants, they can be determined numerically. The corresponding eigenvectors are denoted by X⃗ν=(X0ν,Xnν) and fulfill(117)Xnν=gnλν-ωnX0ν.
The first component X0ν follows from the imposed normalization condition; it explicitly reads(118)1=X0ν2+∑nXnν2=X0ν2{1+∑ngn2(λν-ωn)2}.
The eigenvectors become orthonormal:(119)X⃗ν·X⃗μ=X0νX0μ+∑nXnνXnμ=δμν,
and the closure relations(120)∑νX0ν2=1,∑νXnνXmν=δmn,∑νX0νXnν=0
can be shown to hold true.The operatorsb0 and an are linear combinations of the normal modes:(121)b0(t)=∑νcνX0νe-iλνt,an(t)=∑νcνXnνe-iλνt
with superposition constants cν. Using orthonormality (119), they follow from the initial values as(122)cν=X0νb0(0)+∑nXnνan(0).
This indeed yields the solution (110) with the explicit expressions:(123)U(t)=∑νX0ν2e-iλνt,Vn(t)=∑νX0νXnνe-iλνt,Wm(t)=Vm(t),Smn(t)=∑νXnνXmνe-iλνt.
Thus, we have constructed the evolution operator (111). Note that closure (120) guarantuees the necessary property exp[ℳ(t=0)]=ℐ. These exact results can be used in order to calculate the expectation value of the electric field as outlined above. Note that the bath degrees of freedom are eventually eliminated by our choice of the state. Since a realistic number of bath oscillators is huge, the numerical computations are expected to be time/memory consuming and therefore challenging. We have not yet attempted to perform this task.


#### 8.3.2. Approximate Evolution Operator

In [11] further developments on the operator level preclude the elimination of the bath degrees of freedom in the way we have done. In order to proceed, an approximate, simpler form of the evolution operator is derived. The appearing summation over many bath modes is effectively done. This approximation may be also be interesting for our purposes because the numerical work mentioned above is considerably reduced. We therefore present a slightly modified derivation of the results in [11]; here we do not explicitly need the concept of a “strength function.” It starts by defining the function(124)G(z)=z-ω0-∑ngn2z-ωn.
Its zeros are the eigenfrequencies λν of the total system (cf. (116)). At such a zero one gets for the derivative(125)G'(λν)=1+∑ngn2(λν-ωn)2=1X0ν2,
where the normalization condition (118) has been used. Using complex function theory yields for t>0(126)U(t)=-12πi∫-∞∞e-ixt+ϵtG(x+iϵ)dx,
where, as usual, ϵ is small and positive. This is still exact. At this point additional assumptions are made. First, ϵ is small compared to the scale over which the coupling constants vary. This can be formulated in terms of the abovementioned strength function [11]. Secondly, ϵ is large compared to the distance between the bath frequencies ωn. Finally, and “more seriously,” it is assumed that the interaction is weak, meaning small couplings gn (and, concomitantly, small strength function). This implies that G(x+iϵ) can be approximated by replacing x by ω0 in the summation term of (124). The result can be written as(127)G(x+iϵ)≈x-ω0′+iΓ
with shifted frequency ω0′(128)ω0′=ω0+∑ngn2(ω0-ωn)(ω0-ωn)2+ϵ2,
and damping Γ(129)Γ=ϵ(1+∑ngn2(ω0-ωn)2+ϵ2).
Hence one obtains(130)U(t)≈e-iω0′t-Γt,
and the same approximations yield(131)Vn(t)=Wn(t)≈gnω0′-ωn-iΓ{e-iω0′t-Γt-e-iωnt}.


### 8.4. Finite Temperature Considerations

#### 8.4.1. Initial State—Occupation Number

Instead of selecting the initial state as product of the respective eigenstates of the plasma and bath Hamiltonians, one may assume the bath att=0 to be in thermal equilibrium with the temperature T [11]. The plasma initial state is supposed to be unaltered. The corresponding expectation values of an operator O are generally found as(132)〈O〉=Tr(Oe-βH)Tr(e-βH)
with inverse temperature β-1=kT. Recall that the trace of an operator can be computed using any complete orthonormal set of states |α〉:(133)Tr(O)=∑α〈α|O|α〉.
The denominator of (132) is the well-known partition function of the canonical ensemble. For the chosen bath, a collection of harmonic oscillators, it can readily be calculated using the harmonic oscillators states:(134)Tr(e-βHB)=∏ne-(1/2)βωn1-e-βωn.
The average occupation number of the bath degrees of freedom follows as(135)〈an†(0)am(0)〉=δmn1eβωn-1.
With this result and (110) van Kampen computes the expectation value of a0†(t)a0(t) (corresponding to the degree of freedom in the system, in our case the zero-mode with operators b0†(t),b0(t)) and obtains a nontrivial temperature dependence [11].


#### 8.4.2. Electric Field

We are concerned with the expectation value of the electric field operator, which is proportional toa0(t)+a0†(t). In principle, the field may get temperature dependent contributions due to the bath—as in the previous example. These possibly contributing terms, however, are linear in the operators an†(0) and an(0) (cf. (107) and (110)). The thermal equilibrium expectation values of the bath annihilation operators vanish:(136)〈an〉=Tr(ane-βHB)Tr(e-βHB)=0.
This can be understood by once more inserting the eigenstates of the bath Hamiltonian HB and formally expanding it. Each individual term in the trace contains an odd number of annihilation operators because of the additional operator an. Consequently, each expectation value in all eigenstates is zero and this yields a vanishing trace. Analogously, we find that 〈an†〉=0. Therefore, no temperature dependence in the expectation value of the electric field is present in this model. The results are equal to those obtained with an eigenstate of the bath Hamiltonian as initial state. This result is almost trivial for zero external electric field. It is also valid, however, for nonzero, spatially constant, time-dependent external electric fields. In particular, the induced electric field—stemming from the inhomogeneous terms in the equations of motion and in principle determined by (113)—is not modified.


#### 8.4.3. Energy

Finally, we derive the expression for the expectation value of the Hamilton operator (109), interpretable as the time-dependent energy ℰ(t). The initial state is chosen as above, meaning that the bath is in thermal equilibrium at t=0. The harmonic oscillator state is labelled here with quantum number n0. In addition to the time-development of b0 given by (112) and (113), we also need(137)an(t)=Wn(t)b0(0)+∑mSnm(t)am(0)-12Vω0ζn(t)
with the definition(138)ζn(t)=ω0∫0tWn(t-t′)A(t′)dt′+∑mgm∫0tSnm(t-t′)A(t′)dt′.
For the occupation number one then gets(139)〈b0†(t)b0(t)〉=n0U*(t)U(t)+∑mVm*(t)Vm(t)(eβωm-1)-1+12Vω0η*(t)η(t).
If we omit the contribution due to the external field, that is taking η=0, we indeed reproduce the abovementioned result of [11]. The expectation value of the complete Hamiltonian 〈H(t)〉 follows as(140)ℰ(t)=ω0{n0U*(t)U(t)+∑mVm*(t)Vm(t)(eβωm-1)-1+12Vω0η*(t)η(t)+12∑m}+12Vω02{{12}i(η*(t)-η(t))A(t)+A2(t)}+12Vω0A(t)∑ngni(ζn*(t)-ζn(t))+∑nωn{n0Wn*(t)Wn(t)+12Vω0ζn*(t)ζn(t)+12}+∑ngn{12n0(Wn*(t)U(t)+Wn(t)U*(t))+12Vω0(ζn*(t)η(t)+ζn(t)η*(t))}+∑nm{{12}gn(Snm*(t)Vm(t)+Snm(t)Vm*(t))+ωnSnm*(t)Snm(t)(Snm*(t)Vm(t)+Snm(t)Vm*(t)){12}}(eβωm-1)-1.
This result is exact and can be implemented in an algorithm. Alternatively, one may use the approximations given above (cf. (130) and  (131)) supplemented with the analogous expression for S. The corresponding approximation for the occupation number is explicitly given in [11].


## 8.1. Plasma and Bath

The Hamilton operator consists of three parts:(99)H=12Vω02(A0-1NeP)2+12V(Π0)2+HB+HI,
where the first terms describe again the zero-modes of the internal electromagnetic field coupled to the CM momentum of the ions. Here we denote the plasma frequency by ω0. The free bath Hamiltonian is taken as the sum of a number of harmonic oscillators with frequencies ωn, that is,(100)HB=12∑n(pn2+ωnqn2)=∑nωn(an†an+12)
with coordinates qn and momenta pn,[pn,qm]=-iδnm. The annihilation and creation operators are defined as(101)an=12ωn(pn-iωnqn),an†=12ωn(pn+iωnqn).
It follows that [an,am†]=δnm. We also define such operators for the zero-mode:(102)a0=12Vω0(Π0-iVω0A0),a0†=12Vω0(Π0+iVω0A0).
Van Kampen defines the interaction as [11](103)HI0=∑ngn(ana0†+an†a0)=1Vω0∑ngnωn(pnΠ0+Vω0ωnqnA0)
with coupling strengths gn. This interaction Hamiltonian is not invariant under displacements. Just as in the previous section, we therefore insert the symmetric “minimal substitution” term and use the interaction(104)HI=1Vω0∑ngnωn(pnΠ0+Vω0ωnqn(A0-1NeP))=∑ngn(ana0†+an†a0)-iNe12Vω0∑ngn(an-an†)P.
Thus the bath interacts with the CM degree of freedom as well. The Hamiltonian can now be now written as(105)H=ω0(a0†a0+12)-iω0Ne12Vω0(a0-a0†)P+∑nωn(an†an+12)+∑ngn(ana0†+an†a0)-iNe12Vω0∑ngn(an-an†)P+12NmP2
and can be analyzed analogous to the procedure of [11].


## 8.2. Inclusion of an External Electric Field

We proceed by including a homogeneous external field (12); the Hamiltonian becomes explicitly time-dependent:
(106)H(t)=ω0(a0†a0+12)-iω0Ne12Vω0(a0-a0†)P+iω012Vω0(a0-a0†)A(t)-Vω02NePA(t)+12Vω02A2(t)+∑nωn(an†an+12)+∑ngn(ana0†+an†a0)-iNe12Vω0∑ngn(an-an†)P+12NmP2+i12Vω0∑ngn(an-an†)A(t).
Since the CM coordinate is cyclic, [H,P]=0, the conjugate momentum P is a constant of motion P→kj=2nj/L. After inserting this in the Hamiltonian, we define the “shifted” operators b0 and b0†:(107)a0=b0-iα,a0†=b0†+iα,
with α=α(j)=(kj/Ne)(1/2)Vω0. Some algebra leads to(108)H(t)=ω0(b0†b0+12)+iω012Vω0(b0-b0†)A(t)+12Vω02A2(t)+∑nωn(an†an+12)+∑ngn(anb0†+an†b0)+i12Vω0∑ngn(an-an†)A(t).
The equations of motion for the Heisenberg operators follow as(109)db0(t)dt=i[H,b0]=-iω0b0-i∑mgmam-ω012Vω0A(t),dan(t)dt=i[H,an]=-iωnan-ignb0-gn12Vω0A(t).
The homogeneous differential equations are implicitly solved in [11]:(110)b0(t)=U(t)b0(0)+∑mVm(t)am(0),an(t)=Wn(t)b0(0)+∑mSnm(t)am(0).
The corresponding evolution operator is given by(111)expℳt=(U(t)V⃗tr(t)W⃗(t)S(t)).
The matrix ℳ is implicitly defined in (109). For b0(t) we consequently get as solution of the inhomogeneous differential equation:(112)b0(t)=U(t)b0(0)+∑mVm(t)am(0)-12Vω0η(t),
where we have defined(113)η(t)=ω0∫0tU(t-t′)A(t′)dt′+∑mgm∫0tVm(t-t′)A(t′)dt′.
The zero-mode electric field operator follows as(114)Π0=12Vω0(b0+b0†).
We take as time-independent state Ψ=Ψ0·ψB, with Ψ0 being an eigenstate of the plasma Hamiltonian and ψB being  an eigenstate of the bath Hamiltonian, that is, the product of the shifted zero-mode oscillator, the plane wave and bath harmonic oscillator wave functions. Then we get for the expectation value of the total electric field:(115)Etot=-1V〈Π0〉+E(t)=12ω0(η(t)+η*(t))+E(t).


## 8.3. Evolution Operator

Let us address the solution of the equations of motion (cf. (109)) in some more detail and explicitly construct the evolution operator. In principle we follow the approach of van Kampen [11].


### 8.3.1. Exact Developments

The set of (109) is a solved with the ansatze-iλt for the time dependence of all modes. This leads to the eigenvalue equation:(116)λ-ω0=∑ngn2λ-ωn
with a sequence of solutions λν. Given the number of bath oscillators, their frequencies, and the coupling constants, they can be determined numerically. The corresponding eigenvectors are denoted by X⃗ν=(X0ν,Xnν) and fulfill(117)Xnν=gnλν-ωnX0ν.
The first component X0ν follows from the imposed normalization condition; it explicitly reads(118)1=X0ν2+∑nXnν2=X0ν2{1+∑ngn2(λν-ωn)2}.
The eigenvectors become orthonormal:(119)X⃗ν·X⃗μ=X0νX0μ+∑nXnνXnμ=δμν,
and the closure relations(120)∑νX0ν2=1,∑νXnνXmν=δmn,∑νX0νXnν=0
can be shown to hold true.The operatorsb0 and an are linear combinations of the normal modes:(121)b0(t)=∑νcνX0νe-iλνt,an(t)=∑νcνXnνe-iλνt
with superposition constants cν. Using orthonormality (119), they follow from the initial values as(122)cν=X0νb0(0)+∑nXnνan(0).
This indeed yields the solution (110) with the explicit expressions:(123)U(t)=∑νX0ν2e-iλνt,Vn(t)=∑νX0νXnνe-iλνt,Wm(t)=Vm(t),Smn(t)=∑νXnνXmνe-iλνt.
Thus, we have constructed the evolution operator (111). Note that closure (120) guarantuees the necessary property exp[ℳ(t=0)]=ℐ. These exact results can be used in order to calculate the expectation value of the electric field as outlined above. Note that the bath degrees of freedom are eventually eliminated by our choice of the state. Since a realistic number of bath oscillators is huge, the numerical computations are expected to be time/memory consuming and therefore challenging. We have not yet attempted to perform this task.


### 8.3.2. Approximate Evolution Operator

In [11] further developments on the operator level preclude the elimination of the bath degrees of freedom in the way we have done. In order to proceed, an approximate, simpler form of the evolution operator is derived. The appearing summation over many bath modes is effectively done. This approximation may be also be interesting for our purposes because the numerical work mentioned above is considerably reduced. We therefore present a slightly modified derivation of the results in [11]; here we do not explicitly need the concept of a “strength function.” It starts by defining the function(124)G(z)=z-ω0-∑ngn2z-ωn.
Its zeros are the eigenfrequencies λν of the total system (cf. (116)). At such a zero one gets for the derivative(125)G'(λν)=1+∑ngn2(λν-ωn)2=1X0ν2,
where the normalization condition (118) has been used. Using complex function theory yields for t>0(126)U(t)=-12πi∫-∞∞e-ixt+ϵtG(x+iϵ)dx,
where, as usual, ϵ is small and positive. This is still exact. At this point additional assumptions are made. First, ϵ is small compared to the scale over which the coupling constants vary. This can be formulated in terms of the abovementioned strength function [11]. Secondly, ϵ is large compared to the distance between the bath frequencies ωn. Finally, and “more seriously,” it is assumed that the interaction is weak, meaning small couplings gn (and, concomitantly, small strength function). This implies that G(x+iϵ) can be approximated by replacing x by ω0 in the summation term of (124). The result can be written as(127)G(x+iϵ)≈x-ω0′+iΓ
with shifted frequency ω0′(128)ω0′=ω0+∑ngn2(ω0-ωn)(ω0-ωn)2+ϵ2,
and damping Γ(129)Γ=ϵ(1+∑ngn2(ω0-ωn)2+ϵ2).
Hence one obtains(130)U(t)≈e-iω0′t-Γt,
and the same approximations yield(131)Vn(t)=Wn(t)≈gnω0′-ωn-iΓ{e-iω0′t-Γt-e-iωnt}.


## 8.3.1. Exact Developments

The set of (109) is a solved with the ansatze-iλt for the time dependence of all modes. This leads to the eigenvalue equation:(116)λ-ω0=∑ngn2λ-ωn
with a sequence of solutions λν. Given the number of bath oscillators, their frequencies, and the coupling constants, they can be determined numerically. The corresponding eigenvectors are denoted by X⃗ν=(X0ν,Xnν) and fulfill(117)Xnν=gnλν-ωnX0ν.
The first component X0ν follows from the imposed normalization condition; it explicitly reads(118)1=X0ν2+∑nXnν2=X0ν2{1+∑ngn2(λν-ωn)2}.
The eigenvectors become orthonormal:(119)X⃗ν·X⃗μ=X0νX0μ+∑nXnνXnμ=δμν,
and the closure relations(120)∑νX0ν2=1,∑νXnνXmν=δmn,∑νX0νXnν=0
can be shown to hold true.The operatorsb0 and an are linear combinations of the normal modes:(121)b0(t)=∑νcνX0νe-iλνt,an(t)=∑νcνXnνe-iλνt
with superposition constants cν. Using orthonormality (119), they follow from the initial values as(122)cν=X0νb0(0)+∑nXnνan(0).
This indeed yields the solution (110) with the explicit expressions:(123)U(t)=∑νX0ν2e-iλνt,Vn(t)=∑νX0νXnνe-iλνt,Wm(t)=Vm(t),Smn(t)=∑νXnνXmνe-iλνt.
Thus, we have constructed the evolution operator (111). Note that closure (120) guarantuees the necessary property exp[ℳ(t=0)]=ℐ. These exact results can be used in order to calculate the expectation value of the electric field as outlined above. Note that the bath degrees of freedom are eventually eliminated by our choice of the state. Since a realistic number of bath oscillators is huge, the numerical computations are expected to be time/memory consuming and therefore challenging. We have not yet attempted to perform this task.


## 8.3.2. Approximate Evolution Operator

In [11] further developments on the operator level preclude the elimination of the bath degrees of freedom in the way we have done. In order to proceed, an approximate, simpler form of the evolution operator is derived. The appearing summation over many bath modes is effectively done. This approximation may be also be interesting for our purposes because the numerical work mentioned above is considerably reduced. We therefore present a slightly modified derivation of the results in [11]; here we do not explicitly need the concept of a “strength function.” It starts by defining the function(124)G(z)=z-ω0-∑ngn2z-ωn.
Its zeros are the eigenfrequencies λν of the total system (cf. (116)). At such a zero one gets for the derivative(125)G'(λν)=1+∑ngn2(λν-ωn)2=1X0ν2,
where the normalization condition (118) has been used. Using complex function theory yields for t>0(126)U(t)=-12πi∫-∞∞e-ixt+ϵtG(x+iϵ)dx,
where, as usual, ϵ is small and positive. This is still exact. At this point additional assumptions are made. First, ϵ is small compared to the scale over which the coupling constants vary. This can be formulated in terms of the abovementioned strength function [11]. Secondly, ϵ is large compared to the distance between the bath frequencies ωn. Finally, and “more seriously,” it is assumed that the interaction is weak, meaning small couplings gn (and, concomitantly, small strength function). This implies that G(x+iϵ) can be approximated by replacing x by ω0 in the summation term of (124). The result can be written as(127)G(x+iϵ)≈x-ω0′+iΓ
with shifted frequency ω0′(128)ω0′=ω0+∑ngn2(ω0-ωn)(ω0-ωn)2+ϵ2,
and damping Γ(129)Γ=ϵ(1+∑ngn2(ω0-ωn)2+ϵ2).
Hence one obtains(130)U(t)≈e-iω0′t-Γt,
and the same approximations yield(131)Vn(t)=Wn(t)≈gnω0′-ωn-iΓ{e-iω0′t-Γt-e-iωnt}.


## 8.4. Finite Temperature Considerations

### 8.4.1. Initial State—Occupation Number

Instead of selecting the initial state as product of the respective eigenstates of the plasma and bath Hamiltonians, one may assume the bath att=0 to be in thermal equilibrium with the temperature T [11]. The plasma initial state is supposed to be unaltered. The corresponding expectation values of an operator O are generally found as(132)〈O〉=Tr(Oe-βH)Tr(e-βH)
with inverse temperature β-1=kT. Recall that the trace of an operator can be computed using any complete orthonormal set of states |α〉:(133)Tr(O)=∑α〈α|O|α〉.
The denominator of (132) is the well-known partition function of the canonical ensemble. For the chosen bath, a collection of harmonic oscillators, it can readily be calculated using the harmonic oscillators states:(134)Tr(e-βHB)=∏ne-(1/2)βωn1-e-βωn.
The average occupation number of the bath degrees of freedom follows as(135)〈an†(0)am(0)〉=δmn1eβωn-1.
With this result and (110) van Kampen computes the expectation value of a0†(t)a0(t) (corresponding to the degree of freedom in the system, in our case the zero-mode with operators b0†(t),b0(t)) and obtains a nontrivial temperature dependence [11].


### 8.4.2. Electric Field

We are concerned with the expectation value of the electric field operator, which is proportional toa0(t)+a0†(t). In principle, the field may get temperature dependent contributions due to the bath—as in the previous example. These possibly contributing terms, however, are linear in the operators an†(0) and an(0) (cf. (107) and (110)). The thermal equilibrium expectation values of the bath annihilation operators vanish:(136)〈an〉=Tr(ane-βHB)Tr(e-βHB)=0.
This can be understood by once more inserting the eigenstates of the bath Hamiltonian HB and formally expanding it. Each individual term in the trace contains an odd number of annihilation operators because of the additional operator an. Consequently, each expectation value in all eigenstates is zero and this yields a vanishing trace. Analogously, we find that 〈an†〉=0. Therefore, no temperature dependence in the expectation value of the electric field is present in this model. The results are equal to those obtained with an eigenstate of the bath Hamiltonian as initial state. This result is almost trivial for zero external electric field. It is also valid, however, for nonzero, spatially constant, time-dependent external electric fields. In particular, the induced electric field—stemming from the inhomogeneous terms in the equations of motion and in principle determined by (113)—is not modified.


### 8.4.3. Energy

Finally, we derive the expression for the expectation value of the Hamilton operator (109), interpretable as the time-dependent energy ℰ(t). The initial state is chosen as above, meaning that the bath is in thermal equilibrium at t=0. The harmonic oscillator state is labelled here with quantum number n0. In addition to the time-development of b0 given by (112) and (113), we also need(137)an(t)=Wn(t)b0(0)+∑mSnm(t)am(0)-12Vω0ζn(t)
with the definition(138)ζn(t)=ω0∫0tWn(t-t′)A(t′)dt′+∑mgm∫0tSnm(t-t′)A(t′)dt′.
For the occupation number one then gets(139)〈b0†(t)b0(t)〉=n0U*(t)U(t)+∑mVm*(t)Vm(t)(eβωm-1)-1+12Vω0η*(t)η(t).
If we omit the contribution due to the external field, that is taking η=0, we indeed reproduce the abovementioned result of [11]. The expectation value of the complete Hamiltonian 〈H(t)〉 follows as(140)ℰ(t)=ω0{n0U*(t)U(t)+∑mVm*(t)Vm(t)(eβωm-1)-1+12Vω0η*(t)η(t)+12∑m}+12Vω02{{12}i(η*(t)-η(t))A(t)+A2(t)}+12Vω0A(t)∑ngni(ζn*(t)-ζn(t))+∑nωn{n0Wn*(t)Wn(t)+12Vω0ζn*(t)ζn(t)+12}+∑ngn{12n0(Wn*(t)U(t)+Wn(t)U*(t))+12Vω0(ζn*(t)η(t)+ζn(t)η*(t))}+∑nm{{12}gn(Snm*(t)Vm(t)+Snm(t)Vm*(t))+ωnSnm*(t)Snm(t)(Snm*(t)Vm(t)+Snm(t)Vm*(t)){12}}(eβωm-1)-1.
This result is exact and can be implemented in an algorithm. Alternatively, one may use the approximations given above (cf. (130) and  (131)) supplemented with the analogous expression for S. The corresponding approximation for the occupation number is explicitly given in [11].


## 8.4.1. Initial State—Occupation Number

Instead of selecting the initial state as product of the respective eigenstates of the plasma and bath Hamiltonians, one may assume the bath att=0 to be in thermal equilibrium with the temperature T [11]. The plasma initial state is supposed to be unaltered. The corresponding expectation values of an operator O are generally found as(132)〈O〉=Tr(Oe-βH)Tr(e-βH)
with inverse temperature β-1=kT. Recall that the trace of an operator can be computed using any complete orthonormal set of states |α〉:(133)Tr(O)=∑α〈α|O|α〉.
The denominator of (132) is the well-known partition function of the canonical ensemble. For the chosen bath, a collection of harmonic oscillators, it can readily be calculated using the harmonic oscillators states:(134)Tr(e-βHB)=∏ne-(1/2)βωn1-e-βωn.
The average occupation number of the bath degrees of freedom follows as(135)〈an†(0)am(0)〉=δmn1eβωn-1.
With this result and (110) van Kampen computes the expectation value of a0†(t)a0(t) (corresponding to the degree of freedom in the system, in our case the zero-mode with operators b0†(t),b0(t)) and obtains a nontrivial temperature dependence [11].


## 8.4.2. Electric Field

We are concerned with the expectation value of the electric field operator, which is proportional toa0(t)+a0†(t). In principle, the field may get temperature dependent contributions due to the bath—as in the previous example. These possibly contributing terms, however, are linear in the operators an†(0) and an(0) (cf. (107) and (110)). The thermal equilibrium expectation values of the bath annihilation operators vanish:(136)〈an〉=Tr(ane-βHB)Tr(e-βHB)=0.
This can be understood by once more inserting the eigenstates of the bath Hamiltonian HB and formally expanding it. Each individual term in the trace contains an odd number of annihilation operators because of the additional operator an. Consequently, each expectation value in all eigenstates is zero and this yields a vanishing trace. Analogously, we find that 〈an†〉=0. Therefore, no temperature dependence in the expectation value of the electric field is present in this model. The results are equal to those obtained with an eigenstate of the bath Hamiltonian as initial state. This result is almost trivial for zero external electric field. It is also valid, however, for nonzero, spatially constant, time-dependent external electric fields. In particular, the induced electric field—stemming from the inhomogeneous terms in the equations of motion and in principle determined by (113)—is not modified.


## 8.4.3. Energy

Finally, we derive the expression for the expectation value of the Hamilton operator (109), interpretable as the time-dependent energy ℰ(t). The initial state is chosen as above, meaning that the bath is in thermal equilibrium at t=0. The harmonic oscillator state is labelled here with quantum number n0. In addition to the time-development of b0 given by (112) and (113), we also need(137)an(t)=Wn(t)b0(0)+∑mSnm(t)am(0)-12Vω0ζn(t)
with the definition(138)ζn(t)=ω0∫0tWn(t-t′)A(t′)dt′+∑mgm∫0tSnm(t-t′)A(t′)dt′.
For the occupation number one then gets(139)〈b0†(t)b0(t)〉=n0U*(t)U(t)+∑mVm*(t)Vm(t)(eβωm-1)-1+12Vω0η*(t)η(t).
If we omit the contribution due to the external field, that is taking η=0, we indeed reproduce the abovementioned result of [11]. The expectation value of the complete Hamiltonian 〈H(t)〉 follows as(140)ℰ(t)=ω0{n0U*(t)U(t)+∑mVm*(t)Vm(t)(eβωm-1)-1+12Vω0η*(t)η(t)+12∑m}+12Vω02{{12}i(η*(t)-η(t))A(t)+A2(t)}+12Vω0A(t)∑ngni(ζn*(t)-ζn(t))+∑nωn{n0Wn*(t)Wn(t)+12Vω0ζn*(t)ζn(t)+12}+∑ngn{12n0(Wn*(t)U(t)+Wn(t)U*(t))+12Vω0(ζn*(t)η(t)+ζn(t)η*(t))}+∑nm{{12}gn(Snm*(t)Vm(t)+Snm(t)Vm*(t))+ωnSnm*(t)Snm(t)(Snm*(t)Vm(t)+Snm(t)Vm*(t)){12}}(eβωm-1)-1.
This result is exact and can be implemented in an algorithm. Alternatively, one may use the approximations given above (cf. (130) and  (131)) supplemented with the analogous expression for S. The corresponding approximation for the occupation number is explicitly given in [11].


## 9. Epilogue

The possible effects of electromagnetic fields on biological systems, for example, human beings, are a topical subject. Although possibly relevant physics has already been addressed some time ago [2], there are hardly biophysical mechanisms known and accepted to date. This paper may describe a coupling phenomenon with consequences, in the sense that the internal electric field can be much larger than the external one. It is based on the fact that there are “free” ions in intracellular and extracellular fluid. Consequently, one may describe these ions as a low-density plasma.The response of an ion plasma to external electromagnetic fields has therefore been investigated in this study. To this end, the theory developed in [1] for electron plasmas has been applied. It focusses on the zero-momentum photons, center of mass motion and spatially constant electric fields in periodic structures. Typical plasma oscillations show up. Since the length scale of our problem is much smaller than the wavelengths of typical external fields, this approach is appropriate. The basic quantum mechanical resonance mechanism of [1], demonstrated for harmonic fields, remains essentially the same. We have shown that resonances appear for several selected waveforms, as well as for general bandpass signals. The appearing resonance and plasma frequencies, however, are quite different because the pertinent physical parameters, density and mass, of the ions differ from those of electrons. We have explicitly estimated these frequencies for the relevant ions and found them to be in the region 0.5 GHz–400 GHz.The chosen formulation of the plasma is translationally invariant [1] and, concomitantly, a homogeneously distributed background charge is assumed. Oppositely charged ions in the intra- and extracelluar fluids are supposed to be related to this background. In our study, we have extended this framework to a two-species plasma because it appears to be more appropriate for our biophysical application. Such a plasma actually may or may not be electrically neutral; in the first case a background charge distribution is absent. The effects of coupling an external electric field to the two-species plasma are found to be essentially the same as in the simple plasma.Further investigations of the effects of electromagnetic fields are done by switching to the Heisenberg picture of quantum mechanics, which has turned out to be more convenient for our purposes. Earlier results are readily reproduced. The time-dependent energy of the plasma is also calculated and expressed in induced field and current. The plasma is extended with an additional degree of freedom, interacting with the zero-mode photon. The relic of the original gauge symmetry, that is the displacement symmetry [1], dictates an additional interaction involving the CM motion of the ions. The calculations yield two resonance frequencies instead of one.The chosen plasma model is certainly not completely realistic in the sense that damping and fluctuations, expected in real systems, are not present. This is also evident from the underlying complete theory [1]: we have restricted ourselves to the zero-mode Hamiltonian and ignored the other Hamiltonians describing relative motion, finite momentum photons and additional couplings. It is beyond the purpose of this research to study the effects of these extra degrees of freedoms; such investigations are actually initiated in [1]. Instead we choose the approach of [11] to include a surrounding medium or bath. An exact expression for the internal electric field is derived in terms of an evolution operator. In order to generate numerical results, however, further algorithmic implementation is necessary. This is planned for the nearby future. In the present study, an approximate weak coupling form [11] of the response is constructed. The resonance frequency is shifted and damping indeed appears. Finally, it is shown that assuming thermal equilibrium, the expectation value of the electric field is not changed. The energy of the complete system is found to depend on the temperature.The conclusion of this work is that in the chosen plasma model and its extensions resonance effects appear, which induce currents and generate large internal electric fields. The energy of the plasma is increased as well. Quantum mechanical plasma oscillations are responsible for this physical amplifying mechanism. Note that the response is linear and no demodulation of the external field takes place. On the other hand, a resonant harmonic field becomes amplitude modulated within the plasma. Whether such a large electric field has consequences in biological systems is, of course, an open question. It is furthermore not clear that the plasma model indeed grasps the essential physics of ions in intra- and extracellular fluids responding to external electromagnetic fields. Nevertheless we believe that we have found an interesting possibility of unexpected electromagnetic effects in biophysics.



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*Source: 592051-2010-01-13.xml*