# Magnetic field induced asymmetric splitting of the output signal

L. R. Rahul Biswas, Joydip Das and Bidhan Chandra Bag 111Author for
correspondence<EMAIL_ADDRESS>Department of
Chemistry, Visva-Bharati, Santiniketan 731 235, India

###### Abstract

In this paper we have investigated the dynamics of a damped harmonic
oscillator in the presence of an electromagnetic field. The transients for the
two dimensional harmonic oscillator imply about the modulation of the
frequency of the oscillator by the velocity dependent non conservative force
from an applied magnetic field. Except a special condition, the motion is in
general quasi periodic nature even in the absence of damping. Another
interesting finding is that the magnetic field may induce an asymmetric
splitting of the spectrum of the output signal with two peaks in the case of a
driven damped two dimensional harmonic oscillator. One more additional peak
may appear for the three dimensional case. In some cases the spectrum may have
similarity with the Normal Zeeman Effect. At the same time one may observe to
appear the anti resonance phenomenon even for the driven damped cyclotron
motion where the system with the purely non conservative force fields is
driven by an electric field. Finally, our calculation exhibits how the
magnetic field can modulate the phase difference (between input and output
signals) and the efficiency like quantity of the energy storing process. Thus
the present study might be applicable in the areas related to the refractive
index, the barrier crossing dynamics and autonomous stochastic resonance,
respectively.

Isotropic, anisotropic, charged particle, harmonic oscillator, magnetic field,
periodic force, electric field, resonance frequency

##  I Introduction

Although the study on the dynamics in the presence of an electromagnetic field
is an old issue but due to the following facts it may be still an important
issue in the recent technology. The investigation on the ion conducting
electrolytic materials is a key area in physics and chemistry scros ; gray ;
wright ; bruce ; armand ; angel ; lilley . The materials have potential
applications in a diverse range of all-solid-state devices, such as
rechargeable lithium batteries, flexible electrochromic displays and smart
windows scros . The properties of the electrolytes are tuned by varying
chemical composition to a large extent and hence are adapted to specific needs
angel ; lilley . High ionic conductivity is needed for optimizing the glassy
electrolytes in various applications. Then it would be very interesting if one
can tune the ionic conductivity according to specific need by a physical
method. In this context, very recent studies katsuki ; pere ; telang ; vdo ;
bag3 ; bag4 ; bag5 ; bag6 ; physa  show that the conductivity of an
electrolytic material can be tuned by an applied magnetic field. To tune the
conductivity of ions in the solid electrolytes the combination of both
magnetic field and time-dependent electric field may be an important choice.
Then one may be interested to know the basic dynamics as well as energetics of
an analytically solvable model like driven damped harmonic oscillator in the
presence of a magnetic field. The driven damped harmonic oscillator is a well
studied text book material feyn ; symon . At the same time, the dynamics of a
particle in the presence of both constant magnetic and electric fields is also
a well studied issue in the text book symon . This study has been extended in
different contextsextnmf ; extnmf1 . But to the best of our knowledge, the
study on the dynamics of periodically driven damped harmonic oscillator in the
presence of a magnetic field was not addressed. It does not mean that this
issue is not an important one. In other words, the model study may be a very
relevant one in the context of barrier crossing dynamics as mentioned above.
At the same time it may also be an important one to explain the refractive of
a dielectric material in the presence of a magnetic fieldrefi . Thus our
objective is to explore distinguishable feature (if any) of this dynamics
including the energetics and the resonance at the steady state. Then we start
with the transients of the two dimensional damped harmonic oscillator (having
same frequency along both $x$ and $y$-directions) in the presence of a
magnetic field. $x(t)$ and $y(t)$ are superposition of two periodic terms in
the absence of damping. Thus the motion may be quasi periodic in nature. Then
we determine the condition for simple periodic motion. At the same time we
determine how the frequencies of the periodic terms in the damped oscillation
may depend on the damping strength. These calculations corroborate to the
resonance conditions which are determined based on the steady state dynamics
for the driven system. Here we find that the magnetic field induces an
asymmetric splitting of the spectrum of the output signal with the finite
values of the amplitude at the resonance conditions. But the values of the
amplitudes at resonance condition become infinite in the absence of damping.
It is to be noted here that for this case, the phase shift between the input
and output signals at the resonance condition is similar as that of the driven
damped harmonic oscillator. It proofs indirectly that the finite value of the
amplitude at the resonance condition in the presence of damping is solely due
to the dissipation of energy. In other words, the phase shift has no
significant role in this context. Major points like these have been included
in the conclusion section.

Before leaving this section we would mention that in recent past the magnetic
field has been considered in different contexts such as barrier crossing
dynamics katsuki ; pere ; telang ; vdo ; bag3 ; bag4 ; bag5 ; bag6 ; physa ,
non Markovian dynamics of a Brwonian particle in the presence of a magnetic
field bag4 ; bag6 ; physa ; nmar ; nmar1 ; nmar4 ; jcp ; preha , stochastic
thermodynamics stoct , nonlinear dynamics nonlin  and others indmag ; fabio ;
jayn ; gelf . The present study may be relevant in some of these areas. In the
conclusion section we have mentioned about the possible applications of the
present investigation.

The outlay of the paper is as follows. In Sec.II, we have presented the
dynamics of a two dimensional harmonic oscillator in the presence of an
electromagnetic field. The steady state dynamics of a driven damped three
dimensional harmonic oscillator in the presence of a magnetic field has been
addressed in the next section. The paper is concluded in Sec.IV.

##  II Dynamics of a two dimensional harmonic oscillator in the presence of an
electromagnetic field

###  II.1 Transients: Implication of the resonance condition

The dynamics of a oscillating particle (with angular frequency $\omega$ and
mass $m$) in the presence of a magnetic field ${\bf B}=(0,0,B_{z})$ and the
frictional force can be described by

$m\dot{u}_{x}=-m\omega^{2}x+m\Omega u_{y}-\gamma u_{x}\;\;\;,$ (1)

and

$m\dot{u}_{y}=-m\omega^{2}y-m\Omega u_{x}-\gamma u_{y}\;\;\;.$ (2)

Here $u_{x}$ and $u_{y}$ correspond to the components of the velocity for the
motion in the $x$-$y$ plane. The magnetic force from the given field is
confined in this plane as implied in the above equations of motion. Here the
parameter, $\Omega=\frac{qB_{z}}{m}$ is the cyclotron frequency for the
rotational motion of the particle (with charge, $q$) which is driven by only
the magnetic force. The remaining parameter, $\gamma$ in Eqs. (1-2) measures
the damping strength. However, the above coupled equations of motion can be
solved using the transformation, $\xi=x+iy$ landau . Then we have

$\ddot{\xi}=-\omega^{2}\xi-\beta\dot{\xi}\;\;\;,$ (3)

where $\beta=\gamma+i\Omega$. This leads to have the solution of the above
equation as

$\xi(t)=\xi(0)e^{-\beta/2t}\cos\left(\sqrt{\omega^{2}-\beta^{2}/4}t\right)\;\;\;.$
(4)

Here $\xi(0)$ can be identified as, $x(0)+iy(0)$. However, the argument of
$\cos(\sqrt{\omega^{2}-\beta^{2}/4}t)$ (which may be represented by $\theta$)
restricts us to proceed further analytically. It can be expressed as

$\theta=\sqrt{A+iB}t\;\;\;.$ (5)

Here we have used $A=\omega^{2}-\gamma^{2}/4+\Omega^{2}/4$ and
$B=-\frac{\gamma\Omega}{2}$. If $\gamma$ as well as $B$ is zero then
decomposing the above solution one may read time dependence of $x(t)$ and
$y(t)$ as

$\displaystyle x(t)$ $\displaystyle=$
$\displaystyle\frac{x(0)}{2}\left[\cos\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}+\Omega/2\right)t\right\\}+\cos\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}-\Omega/2\right)t\right\\}\right]$
(6) $\displaystyle+$
$\displaystyle\frac{y(0)}{2}\left[\sin\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}+\Omega/2\right)t\right\\}-\sin\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}-\Omega/2\right)t\right\\}\right]\;\;\;,$

and

$\displaystyle y(t)$ $\displaystyle=$
$\displaystyle\frac{y(0)}{2}\left[\cos\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}+\Omega/2\right)t\right\\}+\cos\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}-\Omega/2\right)t\right\\}\right]$
(7) $\displaystyle-$
$\displaystyle\frac{x(0)}{2}\left[\sin\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}+\Omega/2\right)t\right\\}-\sin\left\\{\left(\sqrt{\omega^{2}+\Omega^{2}/4}-\Omega/2\right)t\right\\}\right]\;\;\;.$

The above relations (6-7) satisfy the initial condition. For further check,
one can show easily that Eq. (6-7) reduce to the expected results at the limit
$\Omega=0.0$. However, it is apparent in the above solutions that how the
coupling of the two dimensional motion through the cross effect of the
velocity dependent magnetic force modifies the respective amplitudes. Another
important point is to be noted here that $x(t)$ as well as $y(t)$ are
superposition of periodic terms with periods

$T_{1}=\frac{2\pi}{\sqrt{\omega^{2}+\Omega^{2}/4}+\Omega/2}\;\;\;,$ (8)

and

$T_{2}=\frac{2\pi}{\sqrt{\omega^{2}+\Omega^{2}/4}-\Omega/2}\;\;\;.$ (9)

Their ratio can be read as
$\frac{T_{2}}{T_{1}}=\frac{\omega^{2}+\Omega^{2}/2+\Omega\sqrt{\omega^{2}+\Omega^{2}/4}}{\omega^{2}}$.
If the ratio is an integer then motion would be a simple periodic one
otherwise the dynamics may be quasi periodic in nature. One may obtain the
condition for the integer ratio in the following way. Let $n\geq 1$ is a ratio
between the two periods. Then we have

$\Omega^{2}+2\Omega\sqrt{\omega^{2}+\frac{\Omega^{2}}{4}}-2(n-1)\omega^{2}=0\;\;\;.$
(10)

The solution of the above equation can be read as

$\Omega=\frac{(n-1)\omega}{\sqrt{n}}\;\;\;.$ (11)

The above relation suggests that for $n=1$, $\Omega=0$. It is a check of the
above calculation. For $\Omega=\omega$, $n$ is $\frac{3+\sqrt{5}}{2}\simeq
2.618033989....$. Thus the motion seems to be quasi periodic in nature for
$\Omega=\omega$. In Fig.1, we have demonstrated the feature of the dynamics
based on Eqs. (6-7) for different $\Omega$. It is fully consistent with Eq.
(11).

Figure 1: Plot of $u_{x}$ vs. $x$ for different values of n along with the
relevant parameter set, $\omega^{2}=2.0,x(0)=0.1,y(0)=0.0,u_{x}(0)=0.0$ and
$u_{y}(0)=0.0$ (Units are arbitrary)

We now consider another condition, $\gamma=0$ and $\omega=0$. Then from Eq.
(4) we have

$\displaystyle x(t)=\frac{x(0)}{2}\big{[}1+\cos(\Omega
t)\big{]}-\frac{y(0)}{2}\sin(\Omega t)\;\;\;,$ (12)

and

$\displaystyle y(t)=\frac{y(0)}{2}\big{[}1+\cos(\Omega
t)\big{]}-\frac{x(0)}{2}\sin(\Omega t)\;\;\;.$ (13)

If $y(0)=0$ then above equations (12-13) imply expected cyclotron motion with
radius $=x(0)/2$, frequency $=\Omega$ and speed
$=\\{x(0)\Omega\\}/2=radius\times cyclotronfrequency$. Similarly for $x(0)=0$
Eqs.(12-13) also imply expected cyclotron motion. These are the checks of the
present calculation. However, if both $x(0)$ and $y(0)$ are not zero then
center of the cycle will oscillate in time along a line in the $x-y$ plane.
Thus we may have a complicated cycloid type motion.

For further check, we consider the condition, $\Omega=0$. Then Eq. (4) reduces
to

$\displaystyle x(t)=x(0)e^{-(\gamma
t)/2}\cos\left(\sqrt{\omega^{2}-\gamma^{2}/4}t\right)\;\;\;,$ (14)

and

$\displaystyle y(t)=y(0)e^{-(\gamma
t)/2}\sin\left(\sqrt{\omega^{2}-\gamma^{2}/4}t\right)\;\;\;.$ (15)

These relations imply the expected damped oscillation at each direction. For
$\gamma=0$, the above relations reduce to the well known results. For direct
check, one may arrive to these results putting $\Omega=0$ and $\gamma=0$ in
Eq.(4).

We now consider the approximate solutions at the limit,
$\frac{B}{A}\rightarrow 0$, when both $\Omega$ and $\gamma$ may not be zero.
Then from Eq. (5) we have

$\theta\simeq\sqrt{A}\left(1+i\frac{B}{2A}\right)t\;\;\;.$ (16)

Using this relation in Eq.(4) one may have

$\displaystyle x(t)$ $\displaystyle\simeq$
$\displaystyle\frac{x_{0}}{4}p_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}+e^{-\frac{Bt}{2\sqrt{A}}}\right)-\frac{x_{0}}{4}q_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}-e^{-\frac{Bt}{2\sqrt{A}}}\right)$ (17)
$\displaystyle+$ $\displaystyle\frac{y_{0}}{4}s_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}-e^{-\frac{Bt}{2\sqrt{A}}}\right)-\frac{y_{0}}{4}r_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}-e^{-\frac{Bt}{2\sqrt{A}}}\right)\;\;\;$

and

$\displaystyle y(t)$ $\displaystyle\simeq$
$\displaystyle\frac{y_{0}}{2}p_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}+e^{-\frac{Bt}{2\sqrt{A}}}\right)-\frac{y_{0}}{4}q_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}-e^{-\frac{Bt}{2\sqrt{A}}}\right)$ (18)
$\displaystyle+$ $\displaystyle\frac{x_{0}}{4}s_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}-e^{-\frac{Bt}{2\sqrt{A}}}\right)-\frac{x_{0}}{4}r_{0}e^{-\frac{\gamma
t}{2}}\left(e^{\frac{Bt}{2\sqrt{A}}}-e^{-\frac{Bt}{2\sqrt{A}}}\right)\;\;\;$

where
$p_{0}=\cos\left\\{\left(\sqrt{A}+\frac{\Omega}{2}\right)t\right\\}+\cos\left\\{\left(\sqrt{A}-\frac{\Omega}{2}\right)t\right\\}$,
$q_{0}=\cos\left\\{\left(\sqrt{A}+\frac{\Omega}{2}\right)t\right\\}-\cos\left\\{\left(\sqrt{A}-\frac{\Omega}{2}\right)t\right\\}$,
$r_{0}=\sin\left\\{\left(\sqrt{A}+\frac{\Omega}{2}\right)t\right\\}+\sin\left\\{\left(\sqrt{A}-\frac{\Omega}{2}\right)t\right\\}$
and
$s_{0}=\sin\left\\{\left(\sqrt{A}+\frac{\Omega}{2}\right)t\right\\}-\sin\left\\{\left(\sqrt{A}-\frac{\Omega}{2}\right)t\right\\}$.
The above equations imply the damped oscillation which is composed of two
frequencies, $\sqrt{A}+\frac{\Omega}{2}$ and $\sqrt{A}-\frac{\Omega}{2}$,
respectively. Now one can check easily that they reduce to all the exact
results at the appropriate limits as mentioned above. Furthermore, from these
approximate solutions one may infer the damped cyclotron motion for
$A=\Omega^{2}/4-\gamma^{2}/4$.

Before leaving this part we would mention that the argument in the sinusoidal
function implies the condition at which the resonance may appear if the
dynamical system (1-2) is driven periodically. In the next subsection we will
check it based on the steady state dynamics.

###  II.2 Steady state dynamics: Resonance condition and energetics

We are now in a position to include an electric field (which is periodic in
time with the angular frequency, $\omega_{E}$) in Eqs. (1-2) to explore the
resonance condition and the related aspect. Let the electric field applied be
as follows

${\bf E}=\hat{i}E_{0x}\cos(\omega_{E}t)+\hat{j}E_{0y}\cos(\omega_{E}t)\;\;\;.$
(19)

Here $E_{x}(t)=E_{0x}\cos(\omega_{E}t)$ and $E_{y}(t)=E_{0y}\cos(\omega_{E}t)$
are the components of the applied electric field. Then the Eqs. (1-2) of
motion become

$m\ddot{x}=-m\omega^{2}x-m\gamma\dot{x}+m\Omega\dot{y}+qE_{0x}\cos(\omega_{E}t)\;\;\;,$
(20)

and

$m\ddot{y}=-m\omega^{2}x-m\gamma\dot{y}-m\Omega\dot{x}+qE_{0y}\cos(\omega_{E}t)\;\;\;.$
(21)

Thus the components of driving force are $F_{x}=qE_{0x}\cos(\omega_{E}t)$ and
$F_{y}=qE_{0y}\cos(\omega_{E}t)$, respectively. One may now choose the
following particular solutions for the above equations

$x(t)=a\cos(\omega_{E}t-\phi_{1})\;\;\;,$ (22)

and

$y(t)=b\cos(\omega_{E}t-\phi_{2})\;\;\;.$ (23)

Here $\phi_{1}$ and $\phi_{2}$ are two relevant phase constants. Using
relations (22-23) into Eqs. (20-21) we have

$\displaystyle\left\\{a\left(\omega^{2}-\omega_{E}^{2}\right)\cos\phi_{1}\right\\}\cos(\omega_{E}t)+\left\\{a\left(\omega^{2}-\omega_{E}^{2}\right)\sin\phi_{1}\right\\}\sin(\omega_{E}t)$
(24) $\displaystyle=$
$\displaystyle(a\gamma\omega_{E}\cos\phi_{1})\sin(\omega_{E}t)-(a\gamma\omega_{E}\sin\phi_{1})\cos(\omega_{E}t)-(b\Omega\omega_{E}\cos\phi_{2})\sin(\omega_{E}t)$
$\displaystyle+$
$\displaystyle(b\Omega\omega_{E}\sin\phi_{2})\cos(\omega_{E}t)+\frac{q}{m}E_{0x}\cos(\omega_{E}t)\;\;\;.$

and

$\displaystyle\left\\{b\left(\omega^{2}-\omega_{E}^{2}\right)\cos\phi_{2}\right\\}\cos(\omega_{E}t)+\left\\{b\left(\omega^{2}-\omega_{E}^{2}\right)\sin\phi_{2}\right\\}\sin(\omega_{E}t)$
(25) $\displaystyle=$
$\displaystyle(b\gamma\omega_{E}\cos\phi_{2})\sin(\omega_{E}t)-(b\gamma\omega_{E}\sin\phi_{2})\cos(\omega_{E}t)+(a\Omega\omega_{E}\cos\phi_{1})\sin(\omega_{E}t)$
$\displaystyle-$
$\displaystyle(a\Omega\omega_{E}\sin\phi_{1})\cos(\omega_{E}t)+\frac{q}{m}E_{0y}\cos(\omega_{E}t)\;\;\;.$

Comparing the coefficients of $\cos(\omega_{E}t)$ and $\sin(\omega_{E}t)$ in
the both sides of Eq. (24) we get the following relations

$a\left(\omega^{2}-\omega_{E}^{2}\right)\cos\phi_{1}=\frac{q}{m}E_{0x}-a\gamma\omega_{E}\sin\phi_{1}+b\Omega\omega_{E}\sin\phi_{2}\;\;\;,$
(26)

and

$a\left(\omega^{2}-\omega_{E}^{2}\right)\sin\phi_{1}=a\gamma\omega_{E}\cos\phi_{1}-b\Omega\omega_{E}\cos\phi_{2}\;\;\;.$
(27)

Similarly from Eq. (25) we get the following relations

$b\left(\omega^{2}-\omega_{E}^{2}\right)\cos\phi_{2}=\frac{q}{m}E_{0y}-b\gamma\omega_{E}\sin\phi_{2}-a\Omega\omega_{E}\sin\phi_{1}\;\;\;,$
(28)

and

$b\left(\omega^{2}-\omega_{E}^{2}\right)\sin\phi_{2}=b\gamma\omega_{E}\cos\phi_{2}+a\Omega\omega_{E}\cos\phi_{1}\;\;\;.$
(29)

Now from coupled Eqs. (26-29) we have

$a=\frac{\sqrt{H_{1}^{2}+H_{2}^{2}}}{H_{0}}\;\;\;,$ (30)

$b=\frac{\sqrt{H_{3}^{2}+H_{4}^{2}}}{H_{0}}\;\;\;,$ (31)

$\tan\phi_{1}=\frac{H_{2}}{H_{1}}\;\;\;.$ (32)

and

$\tan\phi_{2}=\frac{H_{4}}{H_{3}}\;\;\;.$ (33)

Here we have used

$H_{0}=\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}^{2}+4\gamma^{2}\Omega^{2}\omega_{E}^{4}\;\;\;,$
(34)

$H_{1}=\frac{q}{m}\left(\omega^{2}-\omega_{E}^{2}\right)\left[\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}E_{0x}+2\gamma\Omega\omega_{E}^{2}E_{0y}\right]\;\;\;,$
(35)

$H_{2}=\frac{q}{m}\omega_{E}\left[\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}\left(\gamma
E_{0x}-\Omega E_{0y}\right)+2\gamma\Omega\omega_{E}^{2}\left(\gamma
E_{0y}+\Omega E_{0x}\right)\right]\;\;\;,$ (36)

$H_{3}=\frac{q}{m}\left(\omega^{2}-\omega_{E}^{2}\right)\left[\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}E_{0y}-2\gamma\Omega\omega_{E}^{2}E_{0x}\right]\;\;\;,$
(37)

and

$H_{4}=\frac{q}{m}\omega_{E}\left[\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}\left(\gamma
E_{0y}+\Omega E_{0x}\right)-2\gamma\Omega\omega_{E}^{2}\left(\gamma
E_{0x}-\Omega E_{0y}\right)\right]\;\;\;.$ (38)

Thus the amplitudes of the particular solutions carry the interesting
signature of the magnetic field induced coupling of the two dimensional motion
through the similarity in their structures in terms of the relevant
parameters, frequencies of the harmonic oscillator and the driving field,
amplitudes of the driving fields and strength of the magnetic field,
respectively. $a$ becomes $b$ on replacement of $\Omega$ by $-\Omega$ and
interchange of position between $E_{0x}$ and $E_{0y}$. Thus even for
$E_{0x}=E_{0y}$, $a\neq b$. It is a signature of the magnetic force induced
breakdown of the equivalence between the motions along the two directions.
Eqs. (30-33) imply the role of another velocity dependent dissipative force in
this context. Similarly one may expect the signature of the breakdown of the
equivalence in terms of the phase shift between input and output signals.
Shortly we will discuss this in detail.

We are now in a position to dertermine the resonance conditions. It seems to
be difficult to find the resonance condition since the derivative of $a$ or
$b$ with respect to the driving frequency may correspond to an algebraic
equation which is not solvable analytically. Before going to predict the
approximate resonance condition, we show that Eqs. (30-33) reduce to the well
known results for $\Omega=0$. At this limit Eqs. (34-38) become

$H_{0}=\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right\\}^{2}\;\;\;,$
(39)

$H_{1}=\frac{q}{m}E_{0x}\left(\omega^{2}-\omega_{E}^{2}\right)\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right\\}\;\;\;,$
(40)

$H_{2}=\frac{q}{m}E_{0x}\gamma\omega_{E}\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right\\}\;\;\;,$
(41)

$H_{3}=\frac{q}{m}E_{0y}\left(\omega^{2}-\omega_{E}^{2}\right)\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right\\}\;\;\;,$
(42)

and

$H_{4}=\frac{q}{m}E_{0y}\gamma\omega_{E}\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right\\}\;\;\;.$
(43)

Then Eqs.(30-33) become

$a=\frac{qE_{0x}}{m\left[\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right]^{1/2}}\;\;\;,$
(44)

$b=\frac{qE_{0y}}{m\left[\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+\gamma^{2}\omega_{E}^{2}\right]^{1/2}}\;\;\;,$
(45)
$\tan\phi_{1}=\tan\phi_{2}=\frac{\gamma\omega_{E}}{\omega^{2}-\omega_{E}^{2}}\;\;\;.$
(46)

Eqs. (44-46) imply a very good check of the present calculation. For further
check, one may determine easily the following resonance condition from Eqs.
(44-45),

$\omega_{E}=\omega\left(1-\frac{\gamma^{2}}{2\omega^{2}}\right)^{1/2}\;\;\;.$
(47)

This condition is implied by the transient motion (14-15) in the presence of
the driving force. It reduces to another well known result,
$\omega_{E}=\omega$ for $\gamma=0$. Then
$a=\frac{qE_{0x}}{m\left(\omega^{2}-\omega_{E}^{2}\right)}$,
$b=\frac{qE_{0y}}{m\left(\omega^{2}-\omega_{E}^{2}\right)}$ and
$\phi_{1}=\phi_{2}=0$. For the direct check, one may arrive to these results
using $\gamma=\Omega=0$ in Eqs. (30-33). However, comparing Eqs. (45-46) we
find a contrast result in the presence of a magnetic field. The effect of
dissipative force may also appear in the numerator of the amplitude function
as a signature of the modulation of the frequency of the dynamics by the
field. In other words, the interference between the two driving components
through velocity dependent coupling results to appear $\gamma$, $\Omega$,
$E_{0x}$ and $E_{0y}$ in the numerator of the amplitude functions, $a$ and
$b$, respectively.

Now we have to determine the resonance condition in the presence of a magnetic
field. One may determine it applying a trick by inspection of both numerator
and denominator in Eqs. (30-31). At the weak damping limit when the resonance
phenomenon may appear then $H_{0}$ may be minimum around the following
condition,

$\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\Omega^{2}\omega_{E}^{2}=0\;\;\;,$
(48)

Thus the approximate resonance conditions may be read as

$\omega_{L}\simeq\sqrt{\omega^{2}+\frac{\Omega^{2}}{4}}-\frac{\Omega}{2}\;\;\;,$
(49)

and

$\omega_{R}\simeq\sqrt{\omega^{2}+\frac{\Omega^{2}}{4}}+\frac{\Omega}{2}\;\;\;.$
(50)

It is to be noted here that the above conditions are very closed to those
which are implied in Eqs. (17-18). To check the validity of our calculation we
have demonstrated the exact results (30-31) in Fig.2.

Figure 2: Plot of $a$ and $b$ vs. $\omega_{E}$ for different values of $\Omega$. (a) $\omega^{2}=2.0,\gamma=0.1$ and $E_{0x}=E_{0y}=0.25$ (b) $\omega^{2}=2.0,\gamma=0.1$ and $E_{0x}=E_{0y}=0.25$ (c) $\omega^{2}=2.0,\gamma=0.0$ and $E_{0x}=E_{0y}=0.25$ (d) $\omega^{2}=2.0,\gamma=0.0$ and $E_{0x}=E_{0y}=0.25$ (Units are arbitrary) Table 1: Comparison between theoretically calculated resonating frequencies and the exact result for the driven damped two dimensional harmonic oscillator Value of | Resonance at $\omega_{L}$ | Resonance at $\omega_{R}$
---|---|---
$\Omega$ | Theoretical | Exact | Theoretical | Exact
0.5 | 1.190 | 1.189 | 1.680 | 1.679
1.0 | 1.002 | 1.000 | 1.997 | 1.996
1.5 | 0.852 | 0.849 | 2.348 | 2.347
2.0 | 0.732 | 0.729 | 2.730 | 2.730

The resonance conditions according to this figure are compared with the
analytically calculated results in Table 1. It shows that there is a very good
agreement between the theoretical and the exact results. Another important
point is to be noted here that the Fig.2 exhibits an asymmetric splitting of
the spectrum with the output signal. The splitting is implied by the solutions
(17-18) which are composed of two frequencies. Following the above
approximation, the amplitudes at the resonance condition may be read as

$a\simeq\frac{q}{m}\frac{\sqrt{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}\left(\gamma
E_{0x}+2\Omega E_{0y}\right)^{2}+\omega_{E}^{2}\left[\gamma\left(\gamma
E_{0x}-\Omega E_{0y}\right)+2\Omega\left(\gamma E_{0y}+\Omega
E_{0x}\right)\right]^{2}}}{\gamma^{3}\omega_{E}^{2}+4\gamma\Omega^{2}\omega_{E}^{2}}\;\;\;,$
(51)

and

$b\simeq\frac{q}{m}\frac{\sqrt{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}\left(\gamma
E_{0y}-2\gamma\Omega E_{0x}\right)^{2}+\omega_{E}^{2}\left[\gamma\left(\gamma
E_{0y}+\Omega E_{0x}\right)-2\gamma\Omega\left(\gamma E_{0x}-\Omega
E_{0y}\right)\right]^{2}}}{\gamma^{3}\omega_{E}^{2}+4\gamma\Omega^{2}\omega_{E}^{2}}\;\;\;,$
(52)

The above equations imply that at the resonance condition, the amplitude may
decreases with increase in the resonating frequency. In panels (a) and (b) of
Fig.2 we have demonstrated nature of the asymmetric splitting for different
strength of the magnetic field.

We now check the fate of the asymmetric splitting in the absence of damping.
In the absence of dissipative force, Eqs. (30-33) become

$a=\frac{q\sqrt{E_{0x}^{2}\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+E_{0y}^{2}\Omega^{2}\omega_{E}^{2}}}{m\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\Omega^{2}\omega_{E}^{2}\right\\}}\;\;\;,$
(53)

$b=\frac{q\sqrt{E_{0y}^{2}\left(\omega^{2}-\omega_{E}^{2}\right)^{2}+E_{0x}^{2}\Omega^{2}\omega_{E}^{2}}}{m\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\Omega^{2}\omega_{E}^{2}\right\\}}\;\;\;,$
(54)

$\tan\phi_{1}=-\frac{E_{0y}\Omega\omega_{E}}{E_{0x}(\omega^{2}-\omega_{E}^{2})}$
(55)

and

$\tan\phi_{2}=\frac{E_{0x}\Omega\omega_{E}}{E_{0y}(\omega^{2}-\omega_{E}^{2})}\;\;\;.$
(56)

The denominator of the amplitude functions imply that the resonating
frequencies may be equal to those which are given by Eqs. (49-50). These are
consistent with Eqs. (6-7). Then amplitude becomes infinity as suggested by
Eqs. (51-52). To check this we have demonstrated Eqs. (30-31) in panels (c)
and (d) of Fig.2. It exactly corresponds to Eqs. (51-52). Thus in the absence
of damping, the asymmetric nature of the splitting is not clear as like as the
previous case.

Figure 3: Plot of $\phi_{1}$ and $\phi_{2}$ vs. $\omega_{E}$ for different
values of $\Omega$ along with the relevant parameter set, $\omega^{2}=2.0$ and
$E_{0x}=E_{0y}=0.25$. (a) $\gamma=0.0$ (b) $\gamma=0.1$ (Units are arbitrary).

We now consider the phase shift between the input and output signals. Using
Eqs. (32-33), the signature of the breakdown of the equivalence in terms of
the phase shift between the input and the output signal for the respective
directions has been demonstrated in Fig.3 both for dissipative and non-
dissipative systems. Panel (a) of this figure shows that the magnetic field
induces the phase shift even in the absence of damping. Here the magnetic
force takes the similar role as like as the dissipative one feyn ; symon .
This is explicit in Eqs.(55-56). Thus the phase shift at $\omega=\omega_{E}$,
may not depend on the strength of the applied magnetic field as a signature of
similarity between damping strength and MF. Now one may address the difference
between these as we expect from the equations of motion. The relevant points
are to be noted here. First, the phase shift may depend on the amplitude of
the driving force in the presence of the magnetic field. It is a sharp
contrast behavior compared to the case where the damping induces the phase
shift which is independent on the amplitude of the driving force (46). Eq.
(55) implies that the phase shift for the $x$-component motion is enhanced by
the motion of the other direction in the presence of opposition from the
driving force along the $x$-direction. Similarly, one may interpret Eq. (56).
Thus here the origin of the phase shift is the effect of the cross coupling
which may mimic the role of dissipative action in the same context. Then it is
expected that in the presence of damping force, the phase shift may depend on
the amplitude of the periodic electric field in a more complicated way as
implied in Eqs. (32-33) as well as panel (b) of Fig.3. One may notice here the
nature of change of the phase shits around the resonance conditions. It may be
useful to corroborate the appearance of the magnetic field induced anti
resonance phenomenon. Shortly we will consider this issue. Second, although
there is a phase difference (at the resonance condition in the absence of
damping force) between the input and outputs as like as the driven damped
harmonic oscillator but the amplitudes of the output signal is still infinite
as shown in panels (c) and (d) of Fig.2. The nature of divergence is quite
similar to the driven harmonic oscillator in the absence of damping when the
phase shift is zero. Thus the finite amplitude at the resonance condition in
the presence of damping for the driven harmonic oscillator may be due to the
dissipation of energy and here the phase shift may not have any role. It is an
indirect conclusion with the help of driven harmonic oscillator in the
presence of a magnetic field. Here the non zero phase shift gives the
indication that the resonating frequency may be different from the
periodically driven harmonic oscillator. However, as the amplitudes becomes
infinite at the resonance condition for this case, the asymmetric nature (as
expected from the numerator of the amplitude of the output signal) of the
splitting is not prominent in the panels (c) and (d) as like as the other
panels in the same figure. Thus the terms, $\gamma^{2}\omega_{E}^{2}$ and
$4\gamma^{2}\Omega^{2}\omega_{E}^{4}$ (which appears in the denominator of the
amplitude functions (30-31) due to dissipative action) are the leading
quantities to modulated the nature of the asymmetric splitting of the spectrum
of the output signal. In other words, one may observe a single peak instead of
two at relatively high strength of the applied magnetic field as implied in
panel (a) of Fig.2.

Figure 4: Plot of $a$ and $b$ vs. $\omega_{E}$ for different values of $\phi$
along with the parameter set, $\omega^{2}=2.0,\gamma=0.1$ and
$E_{0x}=E_{0y}=0.25$ (Units are arbitrary).

###  II.3 Phase difference between the input signals: Anti resonance

In the earlier discussion we have noted that the interference between the two
driving components through the velocity dependent coupling may have important
consequences. Then we consider the phase difference between the two input
signals with the following driving electric field,

${\bf
E}=\hat{i}E_{0x}\cos(\omega_{E}t-\phi)+\hat{j}E_{0y}\cos(\omega_{E}t)\;\;\;.$
(57)

Then equations (20-21) of motion become

$m\ddot{x}=-m\omega^{2}x-m\gamma\dot{x}+m\Omega\dot{y}+qE_{0x}\cos(\omega_{E}t-\phi)\;\;\;,$
(58)

and

$m\ddot{y}=-m\omega^{2}x-m\gamma\dot{y}-m\Omega\dot{x}+qE_{0y}\cos(\omega_{E}t)\;\;\;.$
(59)

Choosing the particular solutions as like as given by Eqs. (22-23) and
following the above procedure we have

$a=\frac{\sqrt{H_{1}^{2}+H_{2}^{2}}}{H_{0}}\;\;\;,$ (60)

$b=\frac{\sqrt{H_{3}^{2}+H_{4}^{2}}}{H_{0}}\;\;\;,$ (61)

$\tan\phi_{1}=\frac{H_{2}}{H_{1}}$ (62)

and

$\tan\phi_{2}=\frac{H_{4}}{H_{3}}\;\;\;.$ (63)

It is to be noted here that $H_{0}$ remains same as given by Eq. (34) but Eqs.
(35-38) are modified as

$\displaystyle
H_{1}=\frac{q}{m}E_{0x}\Big{[}\left(\omega^{2}-\omega_{E}^{2}\right)^{2}$
$\displaystyle-$
$\displaystyle\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\Big{]}\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)\cos\phi-\gamma\omega_{E}\sin\phi\right\\}$
(64) $\displaystyle+$ $\displaystyle
2\frac{q}{m}\gamma\Omega\omega_{E}^{2}\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)E_{0y}-E_{0x}\Omega\omega_{E}\sin\phi\right\\}\;\;\;,$

$H_{2}=\frac{1}{\left(\omega^{2}-\omega_{E}^{2}\right)}\left(\frac{q}{m}E_{0x}H_{0}\sin\phi+\gamma\omega_{E}H_{1}-\Omega\omega_{E}H_{3}\right)\;\;\;,$
(65)

$\displaystyle
H_{3}=\frac{q}{m}\Big{[}\left(\omega^{2}-\omega_{E}^{2}\right)^{2}$
$\displaystyle-$
$\displaystyle\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\Big{]}\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)E_{0y}-E_{0x}\Omega\omega_{E}\sin\phi\right\\}$
(66) $\displaystyle-$ $\displaystyle
2\frac{q}{m}E_{0x}\gamma\Omega\omega_{E}^{2}\left\\{\left(\omega^{2}-\omega_{E}^{2}\right)\cos\phi-\gamma\omega_{E}\sin\phi\right\\}\;\;\;$

and

$H_{4}=\frac{\omega_{E}}{\left(\omega^{2}-\omega_{E}^{2}\right)}\left(\gamma
H_{3}+\Omega H_{1}\right)\;\;\;.$ (67)

Putting $\phi=0$ in Eqs. (64)-(67) one can show easily that Eqs. (60-63)
reduce to Eqs. (30-33). It constitutes an important check of the present
calculation.

Figure 5: Plot of $\phi_{1}$ and $\phi_{2}$ vs. $\omega_{E}$ for different
values of $\phi$ along with the relevant parameter set, $\omega^{2}=2.0$,
$\gamma=0.1$, $\Omega=1.0$ and $E_{0x}=E_{0y}=0.25$. (Units are arbitrary).

It is really difficult to find out the resonant condition from Eqs. (60-61).
But one may anticipate it by inspection of both numerator and denominator in
Eqs. (60-61) as before. The condition would be remain same as that of $\phi=0$
case since transients are same for both the situations. As a matter of fact
same denominator ($H_{0}$) appears in Eqs. (30-31) and Eqs. (60-61),
respectively. Thus the phase difference between the input signals may modulate
amplitude of the output signals (60-61) without affecting the resonance
condition as a signature of their interference through the velocity dependent
coupling. This has been demonstrated in panels (a) and (b) of Fig.4. Another
point is to be noted from this figure. Comparing it with Fig.2 we find that in
the asymmetric splitting process, the peak height at higher resonating
frequency may be greater than that of the other. An extreme case such as one
of the peaks may disappear as shown in panels (a) and (b). Finally, the
remaining panels (c) to (f) exhibits that the magnetic field may induce anti
resonance phenomenon. Thus for a certain phase difference between the input
signals the reduced amplitude at anti resonance can be regarded as due to
destructive interference or cancellation of forces acting on the oscillator.
In this context we have demonstrated variation of the relevant phase constants
$\phi_{1}$ and $\phi_{2}$ with the driving frequency in Fig.5. Comparing it
with Fig.3 we find that the nature of change of phase constants around the
resonating driving frequency corroborates the appearance of the anti
resonance. It is to be noted here that both the resonance and the anti
resonance phenomena may appear at the same driving frequency depending on the
phase difference between the input signals. This is consistent with the
transient motion as well as Fig.2. As the anti resonance phenomenon is driven
by the phase difference between the input signals then one may expect to
appear it at the relevant frequency which may correspond to a normal mode.
This is sharp contrast to the case of coupled oscillators having coupling
through the conservative force fieldantir . As a signature of this kind of
coupling the system may constitute one or more frequencies which are different
from the relevant normal mode(s) and anti resonance may appear at these
frequencies. For this case resonance and anti resonance may appear
alternatively in the output signal. Thus the magnetic field induced anti
resonance is a distinct one. It is to be noted here that there is a similarity
between the two case. In the case of velocity dependent coupling, if a
resonating frequency become anti resonating one then another frequency behaves
as a resonating one.

####  II.3.1 The energetics

Although the magnetic force does not work but it may modulate the dynamics of
a dynamical system through the modification of the characteristic frequency of
the system as implied in the transient dynamics. Then the work done on the
system by the driving force may depend on it. An indication regarding this
already we have shown through the investigation on amplitude of the
oscillation at the steady state. Thus we are in a position to calculate the
power ($P$) due to the work done by the driving force feyn ; symon . It is the
product between the force and the velocity. For the two dimensional motion the
power is given by

$P=F_{x}\frac{dx}{dt}+F_{y}\frac{dy}{dt}\;\;\;.$ (68)

It is an oscillating quantity. Then one may be interested to know the average
power, ($\langle P\rangle$). Using Eqs. (22-23) into the above equation we
obtain

$\langle
P\rangle=\frac{1}{2}m\gamma\omega_{E}^{2}\left(a^{2}+b^{2}\right)\;\;\;.$ (69)

Similarly one can determine the average energy as

$\langle
U\rangle=\frac{1}{2}m\omega_{E}^{2}\frac{1}{2}\left(a^{2}+b^{2}\right)+\frac{1}{2}m\omega^{2}\frac{1}{2}\left(a^{2}+b^{2}\right)\;\;\;.$
(70)

Eq. (22) implies that the first term corresponds to the stored mean potential
energy and the other one is due to the average kinetic energy.

Figure 6: Demonstration about average power, average stored energy and their
ratio. Common parameter set for panels (a), (b), (c) and (d) :
$\omega^{2}=2.0,\gamma=0.1$ and $E_{0x}=E_{0y}=0.25$. Common parameter set for
panels (e) and (f): $\omega^{2}=2.0$ and $\gamma=0.1$. (Units are arbitrary)

In Fig.6, we have shown that how the average power and the mean stored energy
depend on the driving frequency. Panel (a) shows that the asymmetric nature of
the splitting of the spectrum of the average power is not so prominent as for
the case of amplitude provided the phase difference between the two driving
component is zero. It is a consequence of the following fact. The amplitude at
the resonance condition decreases with increase in resonating frequency as the
frictional loss of energy becomes higher. The average power (69) depends on
both these quantities as implied by Eqs. (22-23, 68). Then the strongly
asymmetric splitting of the amplitudes results the work done per unit time at
lower resonating frequency with higher amplitude may be comparable to the same
at higher frequency having lower amplitude. But the panel (b) implies that the
spectrum of the stored mean energy may mimic the splitting pattern as that of
the amplitudes. It may happen because the stored mean potential energy is not
an explicit function of driving frequency (70) as like as the other part of
the average energy.

Now one may be interested to compare the stored energy ($\langle U\rangle$)
with the amount of work that is done in one cycle. Then the following relevant
quantity feyn  of interest is given by

$Q=2\pi\frac{\langle U\rangle}{\langle P\rangle
2\pi/\omega_{e}}=\frac{\omega^{2}+\omega_{E}^{2}}{2\gamma\omega_{E}}\;\;\;.$
(71)

Thus $Q$ is defined as $2\pi$ times the mean stored energy, divided by the
work done per cycle. It may be an very useful quantity at resonance condition.
In the presence of a magnetic field, for the resonance at lower frequency we
represent $Q$ by $Q_{L}$. Other one is denoted by $Q_{R}$. Similarly one may
represent $\langle P\rangle$ and $\langle U\rangle$ as $\langle P\rangle_{L}$
$\langle P\rangle_{R}$, $\langle U\rangle_{L}$ and $\langle U\rangle_{R}$,
respectively. Panels (c), (d), (e) and (f) exhibit how these quantities depend
on the different parameters. It is apparent here that at the resonance
condition, the process of storing of the energy may be more efficient as the
magnetic field becomes more stronger. It is also apparent here that although
the stored energy depends on the amplitudes of the driving components and the
phase difference between therm but the efficiency like quantity ($Q$) does not
depend on these parameters. But the efficiency depends on the parameters,
frequency of the oscillator and damping strength, respectively. This aspect
has been demonstrated in Fig.6.

####  II.3.2 Driven damped cyclotron motion

We now consider nature of the resonance phenomenon for the driven damped
cyclotron motion. For $\omega=0$, Eqs. (34)-(38) become

$H_{0}=\left\\{\omega_{E}^{4}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}^{2}+4\gamma^{2}\Omega^{2}\omega_{E}^{4}\;\;\;,$
(72)

$H_{1}=\frac{-q\omega_{E}^{2}}{m}\left[\left\\{\omega_{E}^{4}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}E_{0x}+2\gamma\Omega\omega_{E}^{2}E_{0y}\right]\;\;\;,$
(73)

$H_{2}=\frac{q}{m}\omega_{E}\left[\left\\{\omega_{E}^{4}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}\left(\gamma
E_{0x}-\Omega E_{0y}\right)+2\gamma\Omega\omega_{E}^{2}\left(\gamma
E_{0y}+\Omega E_{0x}\right)\right]\;\;\;,$ (74)

$H_{3}=\frac{-q\omega_{E}^{2}}{m}\left[\left\\{\omega_{E}^{4}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}E_{0y}-2\gamma\Omega\omega_{E}^{2}E_{0x}\right]\;\;\;,$
(75)

and

$H_{4}=\frac{q}{m}\omega_{E}\left[\left\\{\omega_{E}^{4}-\left(\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\right\\}\left(\gamma
E_{0y}+\Omega E_{0x}\right)-2\gamma\Omega\omega_{E}^{2}\left(\gamma
E_{0x}-\Omega E_{0y}\right)\right]\;\;\;.$ (76)

Making use of the above relations in Eqs. (30-31) we find again that it is
difficult to determine the resonance condition maximizing $a$ and $b$ with
respect to the driving frequency. Then one may adopt the previous trick. At
the weak damping limit when the resonance phenomenon may be appear then
$H_{0}$ in Eq. (72) may be minimum around the following condition,

$\omega_{E}^{4}-\Omega^{2}\omega_{E}^{2}=0\;\;\;,$ (77)

Thus the approximate resonance condition may be read as

$\omega_{E}\simeq\Omega\;\;\;.$ (78)

Another root as a solution of Eq. (78), $\omega_{E}=0$. It corresponds to the
diverging motion of the charged particle in the presence of a constant
electromagnetic field with ${\bf E}=\hat{i}E_{0x}+\hat{j}E_{0y}$. One may
verify it easily considering the relevant equations of motion. Thus we may
have two peaks at $\omega_{E}=0$ and $\omega_{E}=\Omega$, respectively. This
is implied by Eqs.(6-7). To check accuracy of the approximation we have
demonstrated the exact results in Fig.7. The resonance conditions according to
this figure compared with the analytically calculated results in Table 2. It
shows that there is a very good agreement between the approximate and the
exact results.

Table 2: Comparison between theoretically calculated resonating frequency and the exact result for the driven damped cyclotron motion Value of $\Omega$ | Theoretical | Exact
---|---|---
0.5 | 0.50 | 0.48
1.0 | 1.00 | 0.99
1.5 | 1.50 | 1.49
2.0 | 2.00 | 1.99
Figure 7: Demonstration about amplitudes and anti resonance for the driven
damped oscillator. (a) and (b) Plot of $a$ and $b$ vs. $\omega_{E}$ for
different values of $\Omega$ and $\phi$ along with the paramete set:
$\omega^{2}=0.0,\gamma=0.1$ and $E_{0x}=E_{0y}=0.25$; (c) and (d) Plot of $a$
and $b$ vs. $\omega_{E}$ for different values of $\phi$ along with the
paramete set: $\omega^{2}=0.0,\gamma=0.1$ $\Omega=0.5$ and
$E_{0x}=E_{0y}=0.25$ (Units are arbitrary).

The remaining panels of Fig.7 imply that the anti resonance phenomenon may
appear even for a system with only non conservative force fields. For this
system both resonance and anti resonance phenomena may appear at the same
driving frequency depending upon the phase difference between the input
signals. Thus the underlying reason to appear the anti resonance for the
driven damped cyclotron motion may be the same as that of driven damped
harmonic oscillator in the presence of a magnetic field.

Before leaving this part we would mention here that making use of relations
(30-31) with Eqs. (72-76) into the previous subsection one may study the
energetics for the driven damped cyclotron motion.

##  III Steady state dynamics of a three dimensional harmonic oscillator in
the presence of an electromagnetic field: Similarity of the output spectrum to
the Normal Zeeman effect

We now consider the steady state dynamics for the general case, driven damped
three dimensional harmonic oscillator in the presence of a magnetic field
(${\bf B}=(B_{x},B_{y},B_{z})$) which may be at arbitrary direction. For this
case we choose a periodic electric field as

${\bf
E}=\hat{i}E_{0x}\cos(\omega_{E}t-\phi_{x})+\hat{j}E_{0y}\cos(\omega_{E}t-\phi_{y})+\hat{k}E_{0z}\cos(\omega_{E}t-\phi_{z})\;\;\;$
(79)

where $\phi_{x}$, $\phi_{y}$ and $\phi_{z}$ are the phase constants associated
with each component along respective directions. Then one may write the
equations of motion as

$m\ddot{x}=-m\omega_{x}^{2}x-m\gamma\dot{x}+m\Omega_{z}\dot{y}-m\Omega_{y}\dot{z}+qE_{0x}\cos(\omega_{E}t-\phi_{x})\;\;\;,$
(80)

$m\ddot{y}=-m\omega_{y}^{2}y-m\gamma\dot{y}+m\Omega_{x}\dot{z}-m\Omega_{z}\dot{x}+qE_{0y}\cos(\omega_{E}t-\phi_{y})\;\;\;,$
(81)

$m\ddot{z}=-m\omega_{z}^{2}z-m\gamma\dot{z}+m\Omega_{y}\dot{x}-m\Omega_{x}\dot{y}+qE_{0z}\cos(\omega_{E}t-\phi_{z})\;\;\;.$
(82)

Here we have used $\Omega_{x}=qB_{x}/m,\Omega_{y}=qB_{y}/m$ and
$\Omega_{x}=qB_{z}/m$. Increase in dimension of the system makes difficult to
study the dynamics at short time in the absence of electric field. However, to
characterize the steady state dynamics we choose the following particular
solutions to solve the above equations

$x(t)=a\cos(\omega_{E}t-\phi_{1})\;\;\;,$ (83)

$y(t)=b\cos(\omega_{E}t-\phi_{2})\;\;\;,$ (84)

and

$z(t)=c\cos(\omega_{E}t-\phi_{3})\;\;\;.$ (85)

Following the earlier case now one may determine the amplitudes and the phase
constants as

$a=\frac{\sqrt{A_{1}^{2}+A_{2}^{2}}}{D_{1}}\;\;\;,$ (86)

$b=\frac{\sqrt{B_{1}^{2}+B_{2}^{2}}}{D_{1}}\;\;\;,$ (87)

$c=\frac{\sqrt{C_{1}^{2}+C_{2}^{2}}}{D_{2}}\;\;\;.$ (88)

$\tan\phi_{1}=\frac{A_{2}}{A_{1}}\;\;\;,$ (89)

$\tan\phi_{2}=\frac{B_{2}}{B_{1}}\;\;\;,$ (90)

and

$\tan\phi_{3}=\frac{C_{2}}{C_{1}}\;\;\;.$ (91)

In these relations we have used

$A_{1}=\left(H_{c}H_{x}-H_{z}H_{2}\right)\left(H_{b}H_{c}+H_{3}H_{6}\right)+\left(H_{c}H_{y}+H_{z}H_{3}\right)\left(H_{c}H_{1}+H_{2}H_{6}\right)\;\;\;,$
(92)

$A_{2}=\left(H_{c}H_{x}^{\prime}-H_{z}^{\prime}H_{2}\right)\left(H_{b}H_{c}+H_{3}H_{6}\right)+\left(H_{c}H_{y}^{\prime}+H_{z}^{\prime}H_{3}\right)\left(H_{c}H_{1}+H_{2}H_{6}\right)\;\;\;,$
(93)

$B_{1}=\left(H_{c}H_{y}+H_{z}H_{3}\right)\left(H_{a}H_{c}+H_{2}H_{5}\right)-\left(H_{c}H_{x}-H_{z}H_{2}\right)\left(H_{c}H_{4}-H_{3}H_{5}\right)\;\;\;,$
(94)

$B_{2}=\left(H_{c}H_{y}^{\prime}+H_{z}^{\prime}H_{3}\right)\left(H_{a}H_{c}+H_{2}H_{5}\right)-\left(H_{c}H_{x}^{\prime}-H_{z}^{\prime}H_{2}\right)\left(H_{c}H_{4}-H_{3}H_{5}\right)\;\;\;,$
(95)

$C_{1}=\left(H_{b}H_{z}-H_{y}H_{6}\right)\left(H_{a}H_{b}+H_{1}H_{4}\right)+\left(H_{b}H_{x}+H_{y}H_{1}\right)\left(H_{b}H_{5}+H_{4}H_{6}\right)\;\;\;,$
(96)

$C_{2}=\left(H_{b}H_{z}^{\prime}-H_{y}^{\prime}H_{6}\right)\left(H_{a}H_{b}+H_{1}H_{4}\right)+\left(H_{b}H_{x}^{\prime}+H_{y}H_{1}\right)\left(H_{b}H_{5}+H_{4}H_{6}\right)\;\;\;,$
(97)

$D_{1}=\left(H_{a}H_{c}+H_{2}H_{5}\right)\left(H_{b}H_{c}+H_{3}H_{6}\right)+\left(H_{c}H_{1}+H_{2}H_{6}\right)\left(H_{c}H_{4}-H_{3}H_{5}\right)\;\;\;.$
(98)

$D_{2}=\left(H_{a}H_{b}+H_{1}H_{4}\right)\left(H_{b}H_{c}+H_{3}H_{6}\right)+\left(H_{b}H_{5}+H_{4}H_{6}\right)\left(H_{b}H_{2}-H_{1}H_{3}\right)\;\;\;,$
(99)

with

$\displaystyle H_{a}$ $\displaystyle=$
$\displaystyle\left(\omega_{x}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)+\Big{\\{}\gamma^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)$
(100) $\displaystyle-$
$\displaystyle\Omega_{y}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)-\Omega_{z}^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$

$\displaystyle H_{b}$ $\displaystyle=$
$\displaystyle\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)+\Big{\\{}\gamma^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)$
(101) $\displaystyle-$
$\displaystyle\Omega_{z}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)-\Omega_{x}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$

$\displaystyle H_{c}$ $\displaystyle=$
$\displaystyle\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}+\Big{\\{}\gamma^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)$
(102) $\displaystyle-$
$\displaystyle\Omega_{x}^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)-\Omega_{y}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$

$\displaystyle H_{x}$ $\displaystyle=$
$\displaystyle\frac{q}{m}\Big{\\{}E_{0x}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\cos\phi_{x}$
(103) $\displaystyle-$ $\displaystyle
E_{0x}\gamma\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\sin\phi_{x}$
$\displaystyle+$ $\displaystyle
E_{0y}\Omega_{z}\omega_{E}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\sin\phi_{y}$
$\displaystyle-$ $\displaystyle
E_{0z}\Omega_{y}\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\sin\phi_{z}\Big{\\}}\;\;\;,$

$\displaystyle H_{y}$ $\displaystyle=$
$\displaystyle\frac{q}{m}\Big{\\{}E_{0y}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\cos\phi_{y}$
(104) $\displaystyle-$ $\displaystyle
E_{0x}\Omega_{z}\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\sin\phi_{x}$
$\displaystyle-$ $\displaystyle
E_{0y}\gamma\omega_{E}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\sin\phi_{y}$
$\displaystyle+$ $\displaystyle
E_{0z}\Omega_{x}\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\sin\phi_{z}\Big{\\}}\;\;\;,$

$\displaystyle H_{z}$ $\displaystyle=$
$\displaystyle\frac{q}{m}\Big{\\{}E_{0z}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\cos\phi_{z}$
(105) $\displaystyle+$ $\displaystyle
E_{0x}\Omega_{y}\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\sin\phi_{x}$
$\displaystyle-$ $\displaystyle
E_{0y}\Omega_{x}\omega_{E}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\sin\phi_{y}$
$\displaystyle-$ $\displaystyle
E_{0z}\gamma\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\sin\phi_{z}\Big{\\}}\;\;\;,$

$\displaystyle H_{x}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{q}{m}\Big{\\{}E_{0x}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\sin\phi_{x}$
(106) $\displaystyle+$ $\displaystyle
E_{0x}\gamma\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\cos\phi_{x}$
$\displaystyle-$ $\displaystyle
E_{0y}\Omega_{z}\omega_{E}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\cos\phi_{y}$
$\displaystyle+$ $\displaystyle
E_{0z}\Omega_{y}\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\cos\phi_{z}\Big{\\}}\;\;\;,$

$\displaystyle H_{y}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{q}{m}\Big{\\{}E_{0y}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\sin\phi_{y}$
(107) $\displaystyle+$ $\displaystyle
E_{0x}\Omega_{z}\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\cos\phi_{x}$
$\displaystyle+$ $\displaystyle
E_{0y}\gamma\omega_{E}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\cos\phi_{y}$
$\displaystyle-$ $\displaystyle
E_{0z}\Omega_{x}\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\cos\phi_{z}\Big{\\}}\;\;\;,$

$\displaystyle H_{z}^{\prime}$ $\displaystyle=$
$\displaystyle\frac{q}{m}\Big{\\{}E_{0z}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\sin\phi_{z}$
(108) $\displaystyle-$ $\displaystyle
E_{0x}\Omega_{y}\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\cos\phi_{x}$
$\displaystyle+$ $\displaystyle
E_{0y}\Omega_{x}\omega_{E}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\cos\phi_{y}$
$\displaystyle+$ $\displaystyle
E_{0z}\gamma\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\cos\phi_{z}\Big{\\}}\;\;\;,$

$H_{1}=\Big{\\{}\gamma\Omega_{z}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)+\gamma\Omega_{z}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)-\Omega_{x}\Omega_{y}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$
(109)

$H_{2}=\Big{\\{}\gamma\Omega_{y}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)+\gamma\Omega_{y}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)+\Omega_{z}\Omega_{x}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$
(110)

$H_{3}=\Big{\\{}\gamma\Omega_{x}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)+\gamma\Omega_{x}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)-\Omega_{y}\Omega_{z}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$
(111)

$H_{4}=\Big{\\{}\gamma\Omega_{z}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)+\gamma\Omega_{z}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)+\Omega_{x}\Omega_{y}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$
(112)

$H_{5}=\Big{\\{}\gamma\Omega_{y}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)+\gamma\Omega_{y}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)-\Omega_{z}\Omega_{x}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;,$
(113)

and

$H_{6}=\Big{\\{}\gamma\Omega_{x}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)+\gamma\Omega_{x}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\left(\omega_{x}^{2}-\omega_{E}^{2}\right)+\Omega_{y}\Omega_{z}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\Big{\\}}\omega_{E}^{2}\;\;\;.$
(114)

One may now show easily that Eqs. (86)-(91) reduce to the results of the
previous section. It constitutes an important check of our calculation.

Figure 8: Plot of $a$, $b$ and $c$ vs. $\omega_{E}$ for different values of
$\Omega_{x}$, $\Omega_{y}$ and $\Omega_{z}$ alon with the parameter set:
$\omega_{x}^{2}=1.5,\omega_{y}^{2}=2.0,\omega_{z}^{2}=3.0,\gamma=0.1,E_{0x}=E_{0y}=E_{0z}=0.25$
and $\phi_{x}=\phi_{y}=\phi_{z}=0.0$ (Units are arbitrary).

We are now in a position to determine the resonance condition. The nature of
the amplitude functions implies that it is very difficult to have the
condition even applying the previous technique. According to the earlier
procedure, we have to solve the following equation,

$\displaystyle\left(\omega_{x}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}\bigg{[}\Big{\\{}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}-\Omega_{x}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}$
(115) $\displaystyle-$ $\displaystyle
2\Omega_{y}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}-\Omega_{z}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}\omega_{E}^{2}+\Omega_{x}^{2}\Omega_{y}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\omega_{E}^{4}$
$\displaystyle+$
$\displaystyle\Omega_{y}^{4}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\omega_{E}^{4}+\Omega_{y}^{2}\Omega_{z}^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{4}\Big{\\}}\times\Big{\\{}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}$
$\displaystyle-$ $\displaystyle
2\Omega_{x}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}-\Omega_{y}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}-\Omega_{z}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}\omega_{E}^{2}$
$\displaystyle+$
$\displaystyle\Omega_{x}^{4}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\omega_{E}^{4}+\Omega^{2}\Omega_{y}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\omega_{E}^{4}+\Omega_{z}^{2}\Omega_{x}^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{4}\Big{\\}}-\Omega_{x}^{2}\Omega_{y}^{2}\Big{\\{}\Omega_{x}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}$
$\displaystyle+$
$\displaystyle\Omega_{y}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}+\Omega_{z}^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)\omega_{E}^{2}-\left(\omega_{x}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{z}^{2}-\omega_{E}^{2}\right)^{2}\Big{\\}}^{2}\omega_{E}^{4}\bigg{]}=0\;\;\;,$

to find the values of $\omega_{E}$ for which $a$ and $b$ may be maximum.
Similarly, one may face difficulty to fine the conditions for which $c$ may be
maximum. Then we may be interested to find the resonance condition for
relatively simple case like $\omega_{x}=\omega_{y}=\omega_{z}=\omega$,
$\Omega_{x}\neq\Omega_{y}\neq\Omega_{z}\neq\Omega$. For this case to determine
the conditions for which $a$ and $b$ may be maximum we have to solve the
following equation,

$\displaystyle\omega_{E}^{16}-\left(a_{1}+a_{2}+8\omega^{2}\right)\omega_{E}^{14}+\left\\{28\omega^{4}+6\left(a_{1}+a_{2}\right)\omega^{2}+\left(a_{1}a_{2}-a_{2}a_{1}+b_{1}+b_{2}\right)\right\\}\omega_{E}^{12}$
(116) $\displaystyle-$
$\displaystyle\left\\{56\omega^{6}+15\left(a_{1}+a_{2}\right)\omega^{4}+4\left(a_{1}a_{2}-a_{2}a_{1}+b_{1}+b_{2}\right)\omega^{2}-\left(a_{1}b_{2}+a_{2}b_{1}+2a_{3}b_{3}\right)\right\\}\omega_{E}^{10}$
$\displaystyle+$
$\displaystyle\left\\{70\omega^{8}+20\left(a_{1}+a_{2}\right)\omega^{6}+6\left(a_{1}a_{2}-a_{2}a_{1}+b_{1}+b_{2}\right)\omega^{4}+2\left(a_{1}b_{2}+a_{2}b_{1}+2a_{3}b_{3}\right)\omega^{2}+\left(b_{1}b_{2}-b_{3}^{2}\right)\right\\}\omega_{E}^{8}$
$\displaystyle-$
$\displaystyle\left\\{56\omega^{6}+15\left(a_{1}+a_{2}\right)\omega^{4}+4\left(a_{1}a_{2}-a_{2}a_{1}+b_{1}+b_{2}\right)\omega^{2}-\left(a_{1}b_{2}+a_{2}b_{1}+2a_{3}b_{3}\right)\right\\}\omega^{4}\omega_{E}^{6}$
$\displaystyle-$
$\displaystyle\left\\{28\omega^{4}+6\left(a_{1}+a_{2}\right)\omega^{2}+\left(a_{1}a_{2}-a_{2}a_{1}+b_{1}+b_{2}\right)\right\\}\omega^{8}\omega_{E}^{4}-\left(a_{1}+a_{2}+8\omega^{2}\right)\omega^{12}\omega_{E}^{2}+\omega^{16}=0\;\;\;$

where $a_{1}=\Omega_{x}^{2}+2\Omega_{y}^{2}+\Omega_{z}^{2}$,
$a_{2}=2\Omega_{x}^{2}+\Omega_{y}^{2}+\Omega_{z}^{2}$,
$a_{3}=\Omega_{x}\Omega_{y}$,
$b_{1}=\Omega_{y}^{2}\left(\Omega_{x}^{2}+\Omega_{y}^{2}+\Omega_{z}^{2}\right)$,
$b_{2}=\Omega_{x}^{2}\left(\Omega_{x}^{2}+\Omega_{y}^{2}+\Omega_{z}^{2}\right)$
and
$b_{3}=\Omega_{x}\Omega_{y}\left(\Omega_{x}^{2}+\Omega_{y}^{2}+\Omega_{z}^{2}\right)$.
It seems to be difficult to solve the above equation. Similarly, one may face
difficulty to find the conditions for which $c$ may be maximum. Then we may
determine the resonance condition from the plot as shown in the panels (a),
(b) and (c) in Fig.8. It exhibits an interesting feature that in the process
of asymmetric splitting one more additional peak may appear around the
frequency of the relevant vibrational mode. To understand the appearance of
the three peaks we consider the simplest case,
$\omega_{x}=\omega_{y}=\omega_{z}=\omega$,
$\Omega_{x}=\Omega_{y}=\Omega_{z}=\Omega$. For this condition, Eqs. (92)-(99)
become

$A_{1}=\left(H_{x}H_{0}-H_{z}H_{2}\right)\left(H_{0}^{2}+H_{1}H_{2}\right)+\left(H_{y}H_{0}+H_{z}H_{1}\right)\left(H_{2}^{2}+H_{0}H_{1}\right)\;\;\;,$
(117)

$A_{2}=\left(H_{x}^{\prime}H_{0}-H_{z}^{\prime}H_{2}\right)\left(H_{0}^{2}+H_{1}H_{2}\right)+\left(H_{y}^{\prime}H_{0}+H_{z}^{\prime}H_{1}\right)\left(H_{2}^{2}+H_{0}H_{1}\right)\;\;\;,$
(118)

$B_{1}=\left(H_{y}H_{0}+H_{z}H_{1}\right)\left(H_{0}^{2}+H_{1}H_{2}\right)+\left(H_{x}H_{0}-H_{z}H_{2}\right)\left(H_{1}^{2}-H_{0}H_{2}\right)\;\;\;,$
(119)

$B_{2}=\left(H_{y}^{\prime}H_{0}+H_{z}^{\prime}H_{1}\right)\left(H_{0}^{2}+H_{1}H_{2}\right)+\left(H_{x}^{\prime}H_{0}-H_{z}^{\prime}H_{2}\right)\left(H_{1}^{2}-H_{0}H_{2}\right)\;\;\;,$
(120)

$C_{1}=\left(H_{z}H_{0}+H_{x}H_{1}\right)\left(H_{0}^{2}+H_{1}H_{2}\right)+\left(H_{y}H_{0}-H_{x}H_{2}\right)\left(H_{1}^{2}-H_{0}H_{2}\right)\;\;\;,$
(121)

$C_{2}=\left(H_{z}^{\prime}H_{0}+H_{x}^{\prime}H_{1}\right)\left(H_{0}^{2}+H_{1}H_{2}\right)+\left(H_{y}^{\prime}H_{0}-H_{x}^{\prime}H_{2}\right)\left(H_{1}^{2}-H_{0}H_{2}\right)\;\;\;,$
(122)

$D_{1}=D_{2}=D=\left(H_{0}^{2}+H_{1}H_{2}\right)^{2}-\left(H_{1}^{2}-H_{0}H_{2}\right)\left(H_{2}^{2}+H_{0}H_{1}\right)\;\;\;,$
(123)

with

$H_{0}=\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-\left(2\Omega^{2}-\gamma^{2}\right)\omega_{E}^{2}\;\;\;,$
(124)

$H_{1}=\Omega\omega_{E}^{2}\left(2\gamma-\Omega\right)\;\;\;,$ (125)

$H_{2}=\Omega\omega_{E}^{2}\left(2\gamma+\Omega\right)\;\;\;,$ (126)

$H_{x}=\frac{q}{m}E_{0x}\left(\omega^{2}-\omega_{E}^{2}\right)\;\;\;,$ (127)

$H_{y}=\frac{q}{m}E_{0y}\left(\omega^{2}-\omega_{E}^{2}\right)\;\;\;,$ (128)

$H_{z}=\frac{q}{m}E_{0z}\left(\omega^{2}-\omega_{E}^{2}\right)\;\;\;,$ (129)

$H_{x}^{\prime}=\frac{q}{m}\omega_{E}\left(\gamma E_{0x}-\Omega E_{0y}+\Omega
E_{0z}\right)\;\;\;,$ (130)

$H_{y}^{\prime}=\frac{q}{m}\omega_{E}\left(\gamma E_{0y}-\Omega E_{0z}+\Omega
E_{0x}\right)\;\;\;,$ (131)

and

$H_{z}^{\prime}=\frac{q}{m}\omega_{E}\left(\gamma E_{0z}-\Omega E_{0x}+\Omega
E_{0y}\right)\;\;\;.$ (132)

To determine the resonance condition we now follow the previous section. At
the weak damping limit when the resonance phenomenon may appear then $D$ may
be minimum around the following condition,

$\left(H_{0}^{2}+H_{1}H_{2}\right)^{2}-\left(H_{1}^{2}-H_{0}H_{2}\right)\left(H_{2}^{2}+H_{0}H_{1}\right)=0\;\;\;.$
(133)

where

$H_{0}=\left(\omega^{2}-\omega_{E}^{2}\right)^{2}-2\Omega^{2}\omega_{E}^{2}\;\;\;,$
(134)

$H_{1}=-\Omega^{2}\omega_{E}^{2}\;\;\;,$ (135)

and

$H_{2}=\Omega^{2}\omega_{E}^{2}\;\;\;,$ (136)

Rearranging Eq. (133) we have

$H_{0}\left(H_{0}-H_{1}+H_{2}\right)\left\\{H_{0}\left(H_{0}-H_{1}+H_{2}\right)+3\Omega^{4}\omega_{E}^{4}\right\\}=0\;\;\;.$

It implies the following resonance conditions

$\omega_{m}=\omega\;\;\;,$ (137)

$\omega_{L}=\sqrt{\omega^{2}+\frac{\Omega^{2}}{2}}-\sqrt{\frac{\Omega^{2}}{2}}\;\;\;$
(138)

and

$\omega_{R}=\sqrt{\omega^{2}+\frac{\Omega^{2}}{2}}+\sqrt{\frac{\Omega^{2}}{2}}\;\;\;$
(139)

These conditions are consistent with panels (a), (b) and (c) in Fig.9. We
compare the above conditions with the exact results in Table 3. It shows a
fair agreement between them.

Figure 9: Plot of $a$, $b$ and $c$ vs. $\omega_{E}$ for the parameter set: $\omega^{2}=2.0,\gamma=0.1$, $E_{0x}=E_{0y}=E_{0z}=0.25$ and $\phi_{x}=\phi_{y}=\phi_{z}=0.0$ (Units are arbitrary). Table 3: Comparison between theoretically calculated resonating frequency and the exact result for the driven damped isotropic harmonic oscillator Value of | Resonance at $\omega_{L}$ | Resonance at $\omega_{m}$ | Resonance at $\omega_{R}$
---|---|---|---
$\Omega$ | Theoretical | Exact | Theoretical | Exact | Theoretical | Exact
0.5 | 1.104 | 1.039 | 1.412 | 1.409 | 1.811 | 1.919
1.0 | 0.874 | 0.789 | 1.412 | 1.409 | 2.288 | 2.529
1.5 | 0.707 | 0.620 | 1.412 | 1.409 | 2.828 | 3.230
2.0 | 0.586 | 0.500 | 1.412 | 1.409 | 3.414 | 3.970

Another point is to be noted here. It is apparent in the resonance conditions
that in the presence of a magnetic field, a vibrational motion may be composed
of three frequencies and one ($\omega_{m}$) of them may not depend on the
applied field as a signature of the velocity dependent coupling in the
equation of motion. This is also true even for the case,
$\omega_{x}=\omega_{y}=\omega_{z}=\omega$,
$\Omega_{x}\neq\Omega_{y}\neq\Omega_{z}\neq\Omega$ as shown in panels (d), (e)
and (f) of Fig.9. Thus the appearance of three peaks for the isotropic
harmonic oscillator has a similarity with the Normal Zeeman Effect in the
sense that the middle peak does not depend on the applied magnetic field. One
may find all the peaks making use of oscillating electric field which may be
unpolarized or polarized at any of the three directions. However, for the
driven damped isotropic harmonic oscillator in the presence of a magnetic
field along the $z$-direction, Eqs. (138-139) reduce to Eqs. (49-50) and one
may find three peaks at the frequencies given by Eqs. (137, (49-50). This is
also similar to the Normal Zeeman Effect. To avoid any confusion we would
mention here that if the driven damped three dimensional anisotropic
oscillator experiences a magnetic field along the $z$-direction then the
characteristics of the spectrum also may mimic the Normal Zeeman Effect.
Shortly we will show that one may determine approximately the field dependent
resonating frequencies. For these cases the motion along the $z$-direction is
decoupled from the motion in $x$-$y$ plane. Then three peaks may appear for
the polarized electric field which may lying either in $x$-$z$ or $y$-$z$
plane. Of course one may find all the three peaks making use of unpolarized
electric field. Finally, it is to be noted here that the position of the three
peaks may depend on the direction of the applied field (if it is not along the
$z$-direction) for the three dimensional anisotropic oscillator as implied in
Fig.8.

We are now in a position to demonstrate the effect of phase difference among
the components of the input signal on the output signal. Fig.10 has been
included in this context. Again this figure implies that the phase difference
between the input signals may modulate amplitude of the output signals as a
signature of the interference between the input signals through the velocity
dependent coupling. As a result of that the special features for the two
dimensional case is also continued for three dimensional motion. One more
additional feature, anti resonance may appear at nearby the resonating
frequency for the latter one.

Figure 10: Plot of $a$, $b$ and $c$ vs. $\omega_{E}$ for different values of
$\phi_{x},\phi_{y}$ and $\phi_{z}$ along with the parameter set:
$\omega_{x}^{2}=1.5,\omega_{y}^{2}=2.0,\omega_{z}^{2}=3.0,\gamma=0.1$ and
$E_{0x}=E_{0y}=E_{0z}=0.25$ (Units are arbitrary).

Before leaving this section we show that one may determine the resonance
condition for the driven damped anisotropic oscillator at relatively simple
situation such as $\omega_{z}=0$, $\Omega_{x}=\Omega_{y}=0$ and
$\Omega_{z}=\Omega$. Then the relevant amplitudes and phases of the output
signal can be read from Eqs. (86, 87, 89, 90) as

$a=\frac{\sqrt{H_{1}^{2}+H_{2}^{2}}}{H_{0}}\;\;\;,$ (140)

$b=\frac{\sqrt{H_{3}^{2}+H_{4}^{2}}}{H_{0}}\;\;\;,$ (141)

$\tan\phi_{1}=\frac{H_{2}}{H_{1}}\;\;\;,$ (142)

and

$\tan\phi_{2}=\frac{H_{4}}{H_{3}}\;\;\;.$ (143)

Here we have used

$H_{0}=H_{0x}H_{0y}+\gamma^{2}\Omega^{2}\omega_{E}^{4}\left(\omega_{x}^{2}+\omega_{y}^{2}-2\omega_{E}^{2}\right)^{2}\;\;\;,$
(144)

$H_{1}=\frac{q}{m}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left\\{E_{0x}H_{0y}+E_{0y}\gamma\Omega\omega_{E}^{2}\left(\omega_{x}^{2}+\omega_{y}^{2}-2\omega_{E}^{2}\right)\right\\}\;\;\;,$
(145)

$H_{2}=\frac{q}{m}\omega_{E}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left\\{E_{0x}H_{0y}\gamma-
E_{0y}H_{0x}\Omega+\gamma\Omega\omega_{E}^{2}\left(\omega_{x}^{2}+\omega_{y}^{2}-2\omega_{E}^{2}\right)\left(E_{0y}\gamma+E_{0x}\Omega\right)\right\\}\;\;\;,$
(146)

$H_{3}=\frac{q}{m}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\left\\{E_{0y}H_{0x}-E_{0x}\gamma\Omega\omega_{E}^{2}\left(\omega_{x}^{2}+\omega_{y}^{2}-2\omega_{E}^{2}\right)\right\\}\;\;\;,$
(147)

$H_{4}=\frac{q}{m}\omega_{E}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left\\{E_{0y}H_{0x}\gamma+E_{0x}H_{0y}\Omega+\gamma\Omega\omega_{E}^{2}\left(\omega_{x}^{2}+\omega_{y}^{2}-2\omega_{E}^{2}\right)\left(E_{0y}\Omega-
E_{0x}\gamma\right)\right\\}\;\;\;.$ (148)

with

$H_{0x}=\left(\omega_{x}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)+\gamma^{2}\omega_{E}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)-\Omega^{2}\omega_{E}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\;\;\;,$
(149)

and

$H_{0y}=\left(\omega_{y}^{2}-\omega_{E}^{2}\right)^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)+\gamma^{2}\omega_{E}^{2}\left(\omega_{x}^{2}-\omega_{E}^{2}\right)-\Omega^{2}\omega_{E}^{2}\left(\omega_{y}^{2}-\omega_{E}^{2}\right)\;\;\;,$
(150)

It is to be noted here that for the purpose of the determination of the
resonance condition we have chosen no phase difference between the driving
components. However, one can show that for $\omega_{x}=\omega_{y}$, Eqs.
(140-143) reduce to Eqs. (30-33). It is an important check for this section.

Figure 11: Plot of $a$, $b$ and $c$ vs. $\omega_{E}$ for the parameter set:
$\gamma=0.1$, $E_{0x}=E_{0y}=E_{0z}=0.25$ and $\phi_{x}=\phi_{y}=\phi_{z}=0.0$
(Units are arbitrary).

Following the earlier calculation we now find the resonance condition. Since
the resonance phenomenon may appear at $\gamma\rightarrow 0$ then the
amplitude of the output signal may be maximum around the following condition

$\left(\omega_{x}^{2}-\omega_{E}^{2}\right)\left(\omega_{y}^{2}-\omega_{E}^{2}\right)-\Omega^{2}\omega_{E}^{2}=0\;\;\;.$
(151)

If $\omega_{y}=0$ then the solution of the above can be read as

$\omega_{E}=\sqrt{\omega_{x}^{2}+\Omega^{2}}\;\;\;.$ (152)

and

$\omega_{E}=0\;\;\;.$ (153)

Similarly for $\omega_{x}=0$ we have

$\omega_{E}=\sqrt{\omega_{y}^{2}+\Omega^{2}}\;\;\;.$ (154)

and

$\omega_{E}=0\;\;\;.$ (155)

The above Eqs. (152-155) imply that for the one dimensional harmonic
oscillator in the presence of a magnetic field (which is perpendicular to the
direction of the oscillator) only one resonance peak may appear as shown red
curve in panel (a) of Fig.10. For the given parameter set for this curve, the
location of the peak at $\omega_{E}=2.345$. This is very much consistent with
the theoretical value. To avoid any confusion we would mention here that the
peak at $\omega_{E}=0$ as implied by Eq.(153) is corresponding to the
diverging motion in the presence of the constant electric field. One may
verify it easily considering the relevant equations of motion. However, Fig.10
is a typical demonstration of steady state dynamics of driven damped
anisotropic oscillators. It shows that if the field is tilted from the
perpendicular direction then one more additional peak may appear. Panels (b)
and (c) imply that the additional peak appears as a signature of a complex
motion whose projection in $y-z$ plane is like a damped driven cyclotron
motion. Then continuation of the discussion for a higher dimensional
oscillator is straight forward.

Then for $\omega_{x}=\omega_{y}=0$ one may show easily from Eq. (151) that

$\omega_{E}=\Omega^{2}\;\;\;.$ (156)

Thus it constitutes an important check for the present approximation
calculation. We now consider the condition when both $\omega_{x}$ and
$\omega_{y}$ are not zero. If $\omega_{y}\ll\omega_{E}$ then one may show
easily from Eq. (151) that

$\omega_{R}\simeq\sqrt{\omega_{x}^{2}+\Omega^{2}}\;\;\;.$ (157)

Similarly for $\omega_{x}\ll\omega_{E}$ we get

$\omega_{R}\simeq\sqrt{\omega_{y}^{2}+\Omega^{2}}\;\;\;.$ (158)

Table 4: Comparison between theoretically calculated resonating frequency and the exact result for the driven damped an-isotropic harmonic oscillator. For the relation, $\omega_{R}\simeq\sqrt{\omega_{x}^{2}+\Omega^{2}}$, $\omega_{x}^{2}=2.0$ and $\omega_{y}^{2}=0.1$. Similarly for $\omega_{R}\simeq\sqrt{\omega_{y}^{2}+\Omega^{2}}$, $\omega_{x}^{2}=0.1$ and $\omega_{y}^{2}=3.0$. Value | Resonance at $\omega_{R}\simeq\sqrt{\omega_{x}^{2}+\Omega^{2}}$ | Resonance at $\omega_{R}\simeq\sqrt{\omega_{y}^{2}+\Omega^{2}}$
---|---|---
of $\Omega$ | Theoretical | Exact | Theoretical | Exact
0.5 | 1.500 | 1.500 | 1.803 | 1.799
1.0 | 1.732 | 1.740 | 2.000 | 2.000
1.5 | 2.061 | 2.069 | 2.291 | 2.299
2.0 | 2.449 | 2.460 | 2.646 | 2.649
Figure 12: Plot of $a$ and $b$ vs. $\omega_{E}$ for different values of $\Omega$ at various limits of $\omega_{x}^{2}$ and $\omega_{y}^{2}$: (a) $\omega_{x}^{2}=2.0,\omega_{y}^{2}=0.1$, (b) $\omega_{x}^{2}=0.1,\omega_{y}^{2}=3.0$, (c) $\omega_{x}^{2}=50.0,\omega_{y}^{2}=3.0$, and (d) $\omega_{x}^{2}=2.0,\omega_{y}^{2}=50.0$. Other common parameters are: $E_{0x}=E_{0y}=0.25$ (Units are arbitrary). Table 5: Comparison between theoretically calculated resonating frequency and the exact result for the driven damped an-isotropic harmonic oscillator. For the relation, $\omega_{L}\simeq\frac{\omega_{x}\omega_{y}}{\sqrt{\omega_{x}^{2}+\Omega^{2}}}$, $\omega_{x}^{2}=50.0$ and $\omega_{y}^{2}=3.0$. Similarly for $\omega_{L}\simeq\frac{\omega_{x}\omega_{y}}{\sqrt{\omega_{y}^{2}+\Omega^{2}}}$, $\omega_{x}^{2}=2.0$ and $\omega_{y}^{2}=50.0$. Value | Corresponding to the left peak
---|---
of | Resonance at $\omega_{L}\simeq\frac{\omega_{x}\omega_{y}}{\sqrt{\omega_{x}^{2}+\Omega^{2}}}$ | Resonance at $\omega_{L}\simeq\frac{\omega_{x}\omega_{y}}{\sqrt{\omega_{y}^{2}+\Omega^{2}}}$
$\Omega$ | Theoretical | Exact | Theoretical | Exact
0.5 | 1.723 | 1.730 | 1.407 | 1.409
1.0 | 1.715 | 1.709 | 1.400 | 1.399
1.5 | 1.707 | 1.689 | 1.393 | 1.379
2.0 | 1.698 | 1.659 | 1.387 | 1.360

Again for $\omega_{x}\gg\omega_{E}$, we have from Eq. (151)

$\omega_{L}\simeq\frac{\omega_{x}\omega_{y}}{\sqrt{\omega_{x}^{2}+\Omega^{2}}}\;\;\;.$
(159)

Similarly for $\omega_{y}\gg\omega_{E}$, we get

$\omega_{L}\simeq\frac{\omega_{x}\omega_{y}}{\sqrt{\omega_{y}^{2}+\Omega^{2}}}\;\;\;.$
(160)

To check the accuracy of our calculation we have demonstrated the exact
results (140-143) in Fig.12. The resonance conditions according to this figure
are compared with the analytical results in Table 4 and 5. These show that
there is a very good agreement between the approximate and the exact results.
Further more, one may explain any distinguishable feature such as the peak
height of asymmetric spectrum of amplitudes considering the interference
between the input signal.

###  III.1 The energetics:

Extension of Eqs. (69-71) for three dimensional harmonic oscillator can be
read as

$\langle
P\rangle=\frac{1}{2}m\gamma\omega_{E}^{2}\left(a^{2}+b^{2}+c^{2}\right)\;\;\;,$

$\langle
E\rangle=\frac{1}{2}m\left(\omega_{x}^{2}+\omega_{E}^{2}\right)\frac{1}{2}a^{2}+\frac{1}{2}m\left(\omega_{y}^{2}+\omega_{E}^{2}\right)\frac{1}{2}b^{2}+\frac{1}{2}m\left(\omega_{z}^{2}+\omega_{E}^{2}\right)\frac{1}{2}c^{2}\;\;\;$

and

$Q=\frac{\left(\omega_{x}^{2}+\omega_{E}^{2}\right)a^{2}+\left(\omega_{y}^{2}+\omega_{E}^{2}\right)b^{2}+\left(\omega_{z}^{2}+\omega_{E}^{2}\right)c^{2}}{2\gamma\omega_{E}\left(a^{2}+b^{2}+c^{2}\right)}\;\;\;$

Making use of Eqs. (86-88) into the above relations one may find the role of
magnetic field in the process of energy storing.

##  IV Conclusion

In the present study we have considered various aspects of driven damped
harmonic oscillator in the presence of a magnetic field. Our investigation
includes the following points.

(A) Dynamics of a two dimensional harmonic oscillator (having same frequency
along both the directions) in the presence of magnetic and dissipative forces:

(i) The exact solution of the equations of motion in the absence of damping
exhibits a time dependent position which is composed of two frequencies as a
signature of the cross effect of the non conservative magnetic force. Then we
have determined the condition for a simple periodic motion.

(ii) We have determined the solutions of equations of motion at weak damping
limit which exhibits that the damped oscillation is a composed of two
frequencies. These solutions reduce to all the standard results for the cases
like damped harmonics oscillator and cyclotron motion, respectively.

Thus the above cases may imply the resonance conditions for the periodically
driven damped harmonic oscillator in the presence of a magnetic field.
Regarding this the following major points are given below.

(B) Dynamics at steady state for the periodically driven damped two
dimensional harmonic oscillator in the presence of a magnetic field:

(i) We have determined the relevant amplitudes and phase constants for the
oscillation at steady state. The values of the amplitudes at resonance
condition become infinite in the absence of damping. It is to be noted here
that for this case, the phase shift between the input and output signals at
the resonance condition is similar as that of the driven damped harmonic
oscillator. It proofs indirectly that the finite value of the amplitude at the
resonance condition in the presence of damping is solely due to the
dissipation of energy. In other words, the phase shift has no significant role
in this context.

(ii) Magnetic field induces an asymmetric splitting of the spectrum of output
signal with two peaks.

(iii) The relevant resonance conditions has been determined with a very good
approximation which describes how the resonating frequency depends on the
magnetic field and the force constant of the harmonic oscillator. This leads
to have the resonance condition for the driven damped cyclotron motion.

(iv) There is a magnetic field induced phase shift between the input and
output signals bears a clear signature of the magnetic field induced breakdown
of the equivalence of the two dimensional motion.

(v) Using the phase difference between the components of the driving field one
may modulate the amplitude of the oscillation at the steady state. As a
consequence of the interference between the two driving components through the
velocity dependent coupling the following special features may appear. (a) The
peak height at higher frequency may be higher compared to the other (b) One of
the two pics may disappear for a certain phase difference. (c) The anti
resonance phenomenon may occur. Even it may appear for a driven damped
cyclotron motion where the system with purely non conservative force fields is
driven by an electric field. It is to be noted here that for these cases both
resonance and anti resonance phenomena may appear at the same driving
frequency depending upon the phase difference between the input signals. But
for a three dimensional anisotropic oscillator the anti resonance may appear
at near by the resonating frequency.

(vi) Our calculation shows that the stored average energy closely mimics the
variation of amplitude as a function of driving frequency but the average
power does not follow this pattern. At the resonance condition, the latter
decreases monotonically with increase in the strength of the applied field but
the former pass through a minimum. For a given damped harmonic oscillator the
efficiency like quantity in the energy storing process does not depend on the
parameters related to driving electric field. But it depends on the damping
strength and the frequency of the oscillator, respectively.

(C) Dynamics at steady state for the periodically driven damped three
dimensional harmonic oscillator in the presence of a magnetic field at
arbitrary direction:

(i) For this very general case, we have calculated the relevant amplitudes and
the phase constants for the steady state dynamics. Here we find that magnetic
field may induce one more additional peak compared to the previous case. It is
interesting to be noted here that the position of the additional peak for the
isotropic harmonic oscillator does not depend on the strength and the
direction of the applied magnetic field . Thus this observation has a
similarity with the Normal Zeeman Effect. Again it is to be noted here that if
a driven damped isotropic harmonic oscillator experiences a magnetic field
along $z$-direction then one may find three peaks and the phenomenon is also
similar to the Normal Zeeman Effect. To avoid any confusion we would mention
here that if the driven damped three dimensional an-isotropic oscillator
experiences a magnetic field along the $z$-direction then the characteristics
of the spectrum also may mimic the Normal Zeman Effect. Finally, it is to be
noted here that the position of the three peaks may depend on the direction of
the applied field (if it is not along the $z$-direction) for the three
dimensional an-isotropic oscillator

(ii) The generalization of the problem restricts us to determine the relevant
resonance conditions only for the special cases like (1) the periodically
driven damped isotropic harmonic oscillator in the presence of a magnetic
field whose all the components are same, (2) the periodically driven damped
two dimensional harmonic oscillator (having different force constant along
each direction) whose motion is confined in $x-y$ plane in the presence of a
magnetic field along $z$-direction and (3) the periodically driven damped one
dimensional harmonic oscillator in the presence of a magnetic field which is
perpendicular to the direction of the harmonic motion.

(iii) Finally, we have determine how the quantities like the phase shift, the
average power and the average stored energy depend on the direction of the
magnetic field, the damping strength and the frequencies of the three
dimensional oscillator.

Before leaving this section we would mention the possible applications of the
present study. In this context one may address the following issues. First,
effect of magnetic field on the vibrational resonance. Second, the magnetic
field may induce a phase shift between the input and output signals even in
the absence of damping. Thus investigation on the modulation of the refractive
index of a material by virtue of the Lorentz force may be an worthy issue.
Third, modulation of the frequency as well as energy of a harmonic oscillator
by an applied magnetic field clearly requires a detail study on the thermally
activated barrier crossing dynamics in the presence of an electromagnetic
field. One may be interested to check whether the asymmetric splitting of the
spectrum of the barrier crossing rate constant occurs or not in the presence
of the field. Finally, in a recent study shmpre  we have shown that a colored
noise can recognize the character of a dynamical system in terms of autonomous
stochastic resonance which caries the sense of the dynamical resonance. Thus
investigation on the effect of magnetic field on the autonomous stochastic
resonance may be an worthy issue. These issues are in progress and shortly may
appear elsewhere.

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