# Tunnelling of a composite particle in presence of a magnetic field

B Faulend, J Dragašević  Department of Physics, Faculty of Science, University
of Zagreb, Zagreb, Croatia<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a simple model of composite particle tunnelling through a
rectangular potential barrier in presence of magnetic field. The exact
numerical solution of the problem is provided and the applicability to real
physical situations is discussed. Some qualitative features of tunnelling with
no magnetic interaction are retained, but some new ones are also observed. The
resonance peaks in transmission spectrum generally do not reach $100\%$
transmission probability when the magnetic field is turned on. We observe
splitting and in some cases widening of transmission probability peaks. When
the width $b$ of area with magnetic field is large, we observe oscillations of
spin-flip probability with energy and $b$ which are caused by Larmor
precession of spin about the vector of magnetic field. For some values of
relevant parameters we also observe significant increase of tunnelling
probability for low energies in the single particle case.

##  1 Introduction

Quantum tunnelling of systems with internal structure is a process that occurs
frequently in many physical contexts. In case of particles without internal
structure, resonant tunnelling through double-humped potential barrier is a
well-known phenomenon with applications such as resonant tunnelling diodes.
Tunnelling of two particles in a bound state was first studied by Zakhariev
and Sokolov [1]. Saito and Kayanuma [2] studied tunnelling of a two particles,
with infinite square well binding potential, through a rectangular barrier,
numerically and showed that resonant tunnelling of composite particles can
occur in presence of a single potential barrier. Saito and Kayanuma also
studied the tunnelling of a Wannier exciton through a single barrier
heterostructure [3] and showed similar effects as in [2]. Their work has been
expanded by investigating more general binding potentials and barriers by
Pen’kov [4] [5], Flambaum and Zelevinsky [6], Goodvin and Shegelski [7] [8],
and Bacca and Feldmeier [9]. Bertulani et al. [10] showed that tunnelling
probability can be enhanced for systems with spin-like internal structure
which can transition between states of different energy.

Examples of composite particle tunnelling phenomena are tunnelling of
molecules [2] [7] [8] [11] [12] [13] [14] [15] [16] [17] [18], fusion of
loosely-bound nuclei [19] [10] [20] [21] [6] [9], tunnelling of excitons [3]
[22] [23] [24] and tunnelling of Cooper pairs [6]. There are experimental
studies of exciton tunnelling which show that excitons tunnel as a whole [25]
[26] and resonant tunnelling behaviour is also observed [26].

In this paper, we expand on previous works by considering tunnelling of
composite particle with two internal degrees of freedom: binding interaction
between the particles and spin. It can be expected that tunnelling of systems
with two or more particles with spin, in presence of a magnetic field will
occur in various situations, such as fusion of loosely-bound nuclei and
tunnelling of excitons. Experimental studies of exciton tunnelling in presence
of the magnetic field have already been conducted [27]. The objective of this
paper is to investigate this phenomenon using a simple model, and to expand
the existing results by examining the effects of an additional spin degree of
freedom on composite particle tunnelling. In our model, we consider a one-
dimensional two-particle system where one of the particles has spin $1/2$,
tunnelling through a rectangular potential barrier. The interaction between
particles is modelled as an infinite potential well, as in [2], while the
magnetic field is homogeneous. Although the results are obtained within this
simple model, we expect that their qualitative properties would be retained in
a more sophisticated three-dimensional model with realistic binding potential,
as in case of composite particle tunnelling without spin.

##  2 Description of the problem

As in Saito and Kayanuma [2] we consider a one dimensional two-particle system
tunnelling through a rectangular barrier of width $a$, as in Figure 1. For a
large enough barrier quantum tunnelling will occur with some probability which
we will calculate. Each of the particles has a mass $m$ and is point-like. One
of the particles has spin $1/2$. The potential of intrinsic interaction is
modelled as an infinite square potential well so the distance of the particles
is confined in range $[l-d/2,l+d/2]$ with an average distance of $l$. The
strength of binding can be controlled by the parameter $d$. The magnetic field
is confined to a region of width $b$ and has the same direction (the
$x$-direction) as the velocity of the particle.

Figure 1: Quantum tunnelling of a composite particle through a rectangular
potential barrier

The interaction between particles is given as:

$U(x)=\left\\{\begin{array}[]{ll}0,&l-d/2\leq x\leq l+d/2\\\
\infty,&\mbox{otherwise}\end{array}\right.$ (1)

The potential barrier is defined as:

$V(x)=\left\\{\begin{array}[]{ll}V_{0},&|x|\leq a/2\\\
0,&\mbox{otherwise}\end{array}\right.$ (2)

The magnetic interaction is defined as:

$f(x)=\left\\{\begin{array}[]{ll}u,&|x|\leq b\\\
0,&\mbox{otherwise}\end{array}\right.$ (3)

The positions of particles are $x_{1}$ and $x_{2}$. We define the centre of
mass coordinate as $x=(x_{1}+x_{2})/2$ and the relative coordinate
$y=x_{2}-x_{1}-l+d/2$. Now we can write the Hamiltonian of the system:

$H=-\frac{\hbar^{2}}{4m}\frac{\partial^{2}}{\partial
x^{2}}-\frac{\hbar^{2}}{m}\frac{\partial^{2}}{\partial
y^{2}}+U(y)+V(x-y/2)+V(x+y/2)-f(x-y/2)\sigma_{x}$ (4)

Tunnelling of a composite particle through a single square potential barrier
is equivalent to successive tunnelling through two potential barriers [2]. The
potential is equal to $V_{0}$ in the regions where $2x-a-l+d/2\leq y\leq
2x+a-l+d/2$ and $-2x-a-l+d/2\leq y\leq-2x+a-l+d/2$. Where these regions
coalesce (which happens if $l-d/2<a$) the potential is $2V_{0}$. We define the
potential $W(x,y)$ with these properties.

(a)

(b)

(c)

Figure 2: Diagrams of the potential-barrier structure are shown in (a) and
(b). For (a) $l-d/2>a$ and (b) $l-d/2<a$. In the dotted area in (a)
$W(x,y)=V_{0}$, shaded area $W(x,y)=2V_{0}$ and $W(x,y)=0$ otherwise.
$x_{L}=-(2a+2l+d)/4$, $x_{R}=(2a+2l+d)/4$. Diagram in (c) shows the structure
of the magnetic interaction potential. In the dotted area of (c)
$f(x-y/2)=\frac{2u}{d}$ and $f(x-y/2)=0$ elsewhere.

Hence, the Schrödinger equation is written as:

$\displaystyle\begin{split}-\frac{\hbar^{2}}{4m}\frac{\partial^{2}}{\partial
x^{2}}\Psi(x,y)-\frac{\hbar^{2}}{m}\frac{\partial^{2}}{\partial
y^{2}}\Psi(x,y)+U(y)\Psi(x,y)+\\\
+W(x,y)\Psi(x,y)-f(x-y/2)\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\Psi(x,y)=E\Psi(x,y)\end{split}$ (5)

We now expand the wave function $\Psi(x,y)$ in the basis of spin in the
$z$-direction and eigenfunctions of an infinite potential well:

$\Psi(x,y)=\begin{pmatrix}{\sum_{j=1}^{\infty}\psi_{j1}(x)\phi_{j1}(y)}\\\
{\sum_{j=1}^{\infty}\psi_{j2}(x)\phi_{j2}(y)}\end{pmatrix}=\sum_{j=1}^{\infty}\begin{pmatrix}{\psi_{j1}(x)}\\\
{\psi_{j2}(x)}\end{pmatrix}\phi_{j}(y)$ (6)

where

$\phi_{j}(y)=\sqrt{\frac{2}{d}}\sin\left(\frac{j\pi}{d}y\right)$ (7)

is the solution of the equation:

$-\frac{\hbar^{2}}{m}\frac{d^{2}}{dy^{2}}\phi_{j}(y)=\varepsilon_{j}\phi_{j}(y)$
(8)

with $\varepsilon_{j}=(\hbar/m)(j\pi/d)^{2}$ and the boundary condition
$\phi_{j}(0)=\phi_{j}(d)=0$.

By substituting the wave function in the Schrödinger equation by its expanded
form, and then multiplying by $\phi_{i}^{*}$, we obtain:

$\displaystyle\begin{split}\sum_{j=1}^{\infty}-\frac{\hbar^{2}}{4m}\phi_{i}^{*}\phi_{j}\frac{d^{2}}{dx^{2}}\begin{pmatrix}\psi_{j1}\\\
\psi_{j2}\end{pmatrix}-\frac{\hbar^{2}}{m}\begin{pmatrix}\psi_{j1}\\\
\psi_{j2}\end{pmatrix}\phi_{i}^{*}\frac{d^{2}}{dy^{2}}\phi_{j}+\begin{pmatrix}\psi_{j1}\\\
\psi_{j2}\end{pmatrix}\phi_{i}^{*}U(y)\phi_{j}+\\\
+\begin{pmatrix}\psi_{j1}\\\
\psi_{j2}\end{pmatrix}\phi_{i}^{*}W(x,y)\phi_{j}-\phi_{i}^{*}f(x-y/2)\begin{pmatrix}\psi_{j2}\\\
\psi_{j1}\end{pmatrix}\phi_{j}=\sum_{j=1}^{\infty}E\begin{pmatrix}\psi_{j1}\\\
\psi_{j2}\end{pmatrix}\phi_{i}^{*}\phi_{j}\end{split}$ (9)

We now integrate the whole equation from $-\infty$ to $\infty$ over the
variable $y$ and by using the orthonormality of eigenfunctions
$\int_{-\infty}^{\infty}\phi_{i}^{*}(y)\phi_{j}(y)dy=\delta_{ij}$, we obtain
the following equation by index:

$\displaystyle\begin{split}-\frac{\hbar^{2}}{4m}\frac{d^{2}}{dx^{2}}\begin{pmatrix}\psi_{i1}\\\
\psi_{i2}\end{pmatrix}+(\varepsilon_{i}-E)\begin{pmatrix}\psi_{i1}\\\
\psi_{i2}\end{pmatrix}+\sum_{j=1}^{\infty}\begin{pmatrix}\psi_{j1}\\\
\psi_{j2}\end{pmatrix}\int_{-\infty}^{\infty}\phi_{i}^{*}W(x,y)\phi_{j}dy-\\\
-\sum_{j=1}^{\infty}\begin{pmatrix}\psi_{j2}\\\
\psi_{j1}\end{pmatrix}\int_{-\infty}^{\infty}\phi_{i}^{*}f(x-y/2)\phi_{j}dy=\begin{pmatrix}0\\\
0\end{pmatrix}\end{split}$ (10)

We now define the following functions:

$W_{ij}(x)=\int_{-\infty}^{\infty}\phi_{i}^{*}W(x,y)\phi_{j}dy$ (11)

$F_{ij}(x)=\int_{-\infty}^{\infty}\phi_{i}^{*}f(x-y/2)\phi_{j}dy$ (12)

Functions $W(x,y)$ and $f(x-y/2)$ are equal to zero outside of the regions
$[0,d]$ and $[2(x-b),2(x+b)]$ respectively. Thus, the he real limits of the
above integrals can be determined from Figures 2(a) and 2(b) for equation
(11), and from Figure 2(c) for equation (8). After substituting the identity
(7) into the equations, they can be solved analytically.

After some algebraic manipulation, and by defining the wave-number
$k_{j}=2\sqrt{m(E-\varepsilon_{j})/\hbar}$, the equations become:

$\frac{d^{2}}{dx^{2}}\begin{pmatrix}\psi_{i1}\\\
\psi_{i2}\end{pmatrix}+k_{j}^{2}\begin{pmatrix}\psi_{i1}\\\
\psi_{i2}\end{pmatrix}-\frac{4m}{\hbar^{2}}\sum_{j=1}^{\infty}\begin{pmatrix}\psi_{j1}W_{ij}(x)-\psi_{j2}F_{ij}(x)\\\
\psi_{j2}W_{ij}(x)-\psi_{j1}F_{ij}(x)\end{pmatrix}=\begin{pmatrix}0\\\
0\end{pmatrix}$ (13)

From now on we will call the $k$-th eigenstate of the internal mode channel
$k$. So, for an incident wave with energy $E$, where
$\varepsilon_{n}<E<\varepsilon_{n+1}$, propagating states in the region out of
the potential barrier can exist for channels up to $n$ (for both spin up and
spin down states). The reflection amplitude $R_{ln}$ and the transmission
amplitude $T_{ln}$ for the incident wave coming in channel $n$ and going out
in channel $m$ are calculated using the Method of Variable Reflection
Amplitude as shown in [28]. For an equation of the form:

$\frac{d^{2}}{dx^{2}}\psi_{n}(x)+k_{n}^{2}\psi_{n}(x)-\sum_{m=0}^{\infty}v_{nm}(x)\psi_{m}(x)=0$
(14)

which can be easily adjusted to solve our vector equation (13), by defining
the function $v_{nm}(x)$ accordingly and mapping the index $m$ over both our
pairs of indices $j1$ and $j2$, thus transforming our system of vector
equations (defined by index) with equation (13), into a system of scalar
equations and doubling their number.

If a plane wave form approaches the barrier from the left, in the $l$-th
channel, the formal solution of the equation (14) is:

$\psi_{ln}(x)=e^{ik_{n}x}\delta_{ln}+\sum_{m=0}^{\infty}\int_{-\infty}^{\infty}e^{ik_{n}|x-t|}v_{nm}(t)\psi_{lm}(t)dt$
(15)

From [28], the reflection an transmission amplitudes can be written in the
following form:

$R_{ln}=\frac{1}{2ik_{n}}\sum_{m=0}^{\infty}\int_{-\infty}^{\infty}e^{ik_{n}t}v_{nm}(t)\psi_{lm}(t)dt$
(16)

$T_{ln}=\delta_{ln}+\frac{1}{2ik_{n}}\sum_{m=0}^{\infty}\int_{-\infty}^{\infty}e^{-ik_{n}t}v_{nm}(t)\psi_{lm}(t)dt$
(17)

and further transformed into a system of differential equations:

$\frac{d}{dx}R_{ln}(x)=-\sum_{j=0}^{\infty}\frac{1}{2ik_{j}}\left(e^{ik_{j}x}\delta_{jn}+R_{jn}(x)e^{-ik_{j}x}\right)\sum_{m=0}^{\infty}v_{jm}(x)\left(e^{ik_{m}x}\delta_{lm}+R_{lm}(x)e^{-ik_{m}x}\right)$
(18)

$\frac{d}{dx}T_{ln}(x)=-\sum_{j=0}^{\infty}\frac{1}{2ik_{j}}T_{jn}(x)e^{ik_{j}x}\sum_{m=0}^{\infty}v_{jm}(x)\left(e^{ik_{m}x}\delta_{lm}+R_{lm}(x)e^{-ik_{m}x}\right)$
(19)

with the following boundary conditions:

$\displaystyle R_{ln}(x\rightarrow\infty)\rightarrow 0,\quad
R_{ln}(x\rightarrow-\infty)\rightarrow R_{ln}$ (20) $\displaystyle
T_{ln}(x\rightarrow\infty)\rightarrow\delta_{ln},\quad
T_{ln}(x\rightarrow-\infty)\rightarrow T_{ln}$ (21)

By solving these equations for permitted channels (the sums will go from $0$
to $n$ instead of $\infty$) and in the specified range for the coordinate $x$
(the centre of mass coordinate in range $[-(2a+2l+d)/4,(2a+2l+d)/4]$), the
probability of reflection and transmission from channel $l$ to channel $n$ can
be calculated as follows:

$\displaystyle P_{r,l\rightarrow n}=\frac{k_{n}}{k_{l}}|R_{ln}(-\infty)|^{2}$
(22) $\displaystyle P_{t,l\rightarrow
n}=\frac{k_{n}}{k_{l}}|T_{ln}(-\infty)|^{2}$ (23)

These expressions can be generalised to take into account the different spin
states for each channel as explained before. Thus, there are $2n$ possible
incoming states and $2n$ possible outgoing states, and the probabilities for
every combination of those states can be calculated accordingly.

##  3 Results

(a)

(b)

(c)

(d)

Figure 3: The graphs show transmission probabilities from the first channel
for $a=1$, $b=1$, $d=5$, $l=5$ and (a) $u=0$, (b) $u=0.05$, (c) $u=0.15$, (d)
$u=0.05$. $P_{t,11}^{++}$ denotes a particle incoming in spin up state from
channel $1$ and outgoing in the same state, $P_{t,11}^{+-}$ denotes a particle
incoming in spin up state from channel $1$ and outgoing in spin down state.
The probabilities $P_{t,11}^{--}$ and $P_{t,11}^{-+}$ are the same as
$P_{t,11}^{++}$ and $P_{t,11}^{+-}$, respectively. The last graph shows the
lowest energy peak of graph (b) in greater detail.

(a)

(b)

Figure 4: The graphs show transmission probabilities around the lowest energy
peak for the same parameters as in Figure 3(a), except that for graph (a)
$u=0.001$ and for (b) $u=0.005$.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: The graphs show transmission probabilities from the first channel
for $a=1$, $d=5$, $l=3$, $u=0.05$ and (a) $b=1$, (b) $b=3.5$, (c) $b=8$, (d)
$b=15$, (e) $b=100$, (f) $b=200$.

(a)

(b)

(c)

(d)

Figure 6: The graphs show transmission probabilities for $a=1$, $d=7$, $l=5$,
$b=1$ and (a) $u=0.05$ from the first channel, (b) $u=0.15$ from the first
channel, (c) $u=0.05$ from the second channel, (d) $u=0.15$ from the second
channel. The second channel opens at the energy of around $0.6$ units for the
given parameters.

(a)

(b)

Figure 7: The graphs show transmission probabilities from the first channel
for $a=1$, $b=1$ (a) $d=0.5$, $l=0.5$, $u=0.05$, (b) $d=0.05$, $l=0.05$,
$u=0.05$.

(a)

(b)

Figure 8: The graphs show transmission probabilities from the first channel
for $a=1$, $u=0.05$, $d=0.05$, $l=0.05$ (a) $b=100$, (b) $b=200$.

(a)

(b)

Figure 9: The graphs show transmission probabilities from the first channel
for $a=0.3$, $u=0.05$, $l=0.05$, $d=0.05$, $b=100$ for different energy
ranges.

Equations (18) and (19) are solved numerically using the Runge-Kutta method of
order 8 for various values of parameters: width of the potential barrier $a$,
strength of the magnetic field $u$ and mean distance between the particles
$l$, width of the potential well $d$ and width of the area where the magnetic
field is present $b$. In all cases we used fixed values of parameters: $m=1$,
$\hbar=1$, $V_{0}=1$. Python code that calculates transmission and reflection
probabilities as functions of energy is provided in the appendix.

We checked the convergence of numerical results by reducing the maximum step
used in the Runge-Kutta algorithm by a factor of $5$. As expected, the results
for spin up and spin down are symmetric and the sum of all probabilities of
transmission and reflection for any given entry channel is $1$.

In Figure 3(a) we reproduce one of the results shown in [2] when the magnetic
field is absent, using the variable reflection amplitude method. In Figures
3(b) and 3(c) effects of the magnetic field are observed with other parameters
kept the same as in Figure 3(a). Resonances are wider and do not reach $100\%$
transmission probability. It is visible that peaks are divided due to
splitting of energy levels in presence of the magnetic field. This is more
pronounced for the lowest energy peak. However, the splitting of the second
peak is also clearly visible for stronger magnetic fields. As shown in Figure
3(d) spin transition is enhanced in the area of resonances.

In Figure 4 we examine the case of weak magnetic fields in the area of lowest
energy resonance, wtih $b=a$. Bertulani et al. conclude in [10] that an
extremely strong magnetic field is necessary to observe significant effects in
the single particle case. If we examine composite particle tunnelling such
effects are visible for much weaker fields in the area of resonances. In
Figure 4(a) the lowest energy resonance peak at $u=0.001$ already does not
reach $100\%$ transmission probability as it does when there is no magnetic
field present in Figure 3(a). Figure 4(b) shows that splitting of energy
levels happens for magnetic fields as low as $u=0.005$.

In Figure 5 we show the effect of varying width of the area where the magnetic
field is present. In the limit of small values of $b$ the results tend towards
the case when there is no magnetic field, i.e. the transmission probabilities
$P_{t,ij}^{+-}$ tend to zero, which is to be expected. However, as can be seen
from the progression of results in Figures 5(a)-5(f), as the range of the
field expands, $P_{t,ij}^{+-}$ and $P_{t,ij}^{++}$ exhibit oscillatory
behaviour that will later be examined more closely. The graph of total
transmission probability retains roughly the same shape. There are also
smaller oscillations on the curves visible in Figures 5(e) and 5(f).

In Figure 6 we can see a situation with multiple open channels. A
discontinuity is observed in $P_{t,11}^{++}$ when the second channel becomes
available. This is consistent with results from previous works [7]. We also
observe that the total probability of transmission is reduced when higher
channels open up, as in the case with no magnetic interaction [7].

In Figure 7 the case when the dimensions of the particle are smaller than
dimensions of the barrier is examined. When $l$ and $a$ are of the same order
of magnitude the effects of internal structure are clearly seen and in the
case when $l<<a$ the single particle limit is reproduced, as expected.

In Figure 8 we show the effects of widening of the magnetic field in the
single particle limit. There are two distinct oscillatory behaviours present
for transmission probabilities with and without spin-flip, with respect to
energy. The one with larger period doubles its frequency when the width of the
field is doubled and this behaviour continues for every value of $b$. This
effect can be explained by Larmor precession. Smaller oscillations also
shorten their period when $b$ increases. We can see that the overall
transmission probability $P_{t,11}=P_{t,11}^{++}+P_{t,11}^{+-}$ remains
roughly the same as in the single particle case (Figure 7(b)) invariant to
$b$, except for the small oscillations visible along the green line. There are
also beats present in Figure 8(b) along the graph of total transmission
probability which indicate a superposition of two oscillatory behaviours.

Finally, in Figure 9 we can see the remarkable properties of oscillatory
behaviours in the single particle case when the ratio $b/a$ is very large.
There is a peak in total transmission probability for very low energies and it
is not isolated, but accompanied by a large number of similar peaks, as shown
in a more detailed look of the low energy region in Figure 9(b). This
behaviour, however is highly dependent upon the ratio $b/a$ and is not always
enhanced by increasing it, however, for lower values it is always less
pronounced.

##  4 Discussion

The results show that the additional magnetic interaction has several
noticeable effects on tunnelling probability. When dimensions of the composite
particle are larger than dimensions of the barrier, the effects of magnetic
field and spin are most clearly observed around the transmission probability
resonances. In case with no magnetic field, the resonances reach $100\%$
transmission probability, which is no longer the case when the magnetic field
is turned on. Also, with the magnetic field present, the resonance peaks split
and each forms two peaks close to each other. The structure of these peaks is
generally nearly symmetric and depends on values of parameters $u$ and $b$, as
seen in Figures 3, 5 and 6(a). In realistic physical situations we can expect
that the interaction with magnetic field will either be present in the whole
space, i.e. an area much larger than the potential barrier, or localised in
the area of the potential barrier.

When we set the width of area with magnetic field $b=a\sim 1$, effects of
magnetic field are relatively small for field strengths $u<10^{-2}V_{0}$,
which can be expected in most situations. The splitting of transmission
probability peaks can be observed if $u\sim 10^{-2}V_{0}$ or larger, but even
with $u\sim 10^{-3}V_{0}$ the lowest energy transmission probability peak no
longer reaches $1$. Probability of transmission with spin-flip $P_{t}^{+-}$ is
much larger relatively to $P_{t}^{++}$ in the resonance area. With the opening
of higher channels, the structure of plots becomes more complicated, but no
qualitatively new effects are observed.

In cases with $l<a$, the magnetic field has small effects on total tunnelling
probability, except for very large values of $u\sim V_{0}$ which are not
realistic, or very large values of $b$. Generally, the effects of magnetic
field are more pronounced for smaller values of $a$ which is expected because
when the barrier is smaller, small splitting of energy levels becomes more
important. The effects of internal structure are observed when $l<a$ if $l$
and $a$ are of the same order of magnitude because the effective barrier as
seen by particle’s centre of mass is different than in the single particle
case: the barrier effectively becomes wider and lower.

If the magnetic field is present in a wide area ($b>>a$), several interesting
effects can be seen. The effects of magnetic interaction can be observed with
field strength which is an order of magnitude smaller than in case $b=a$. The
splitting of resonances becomes more pronounced, and the new peaks are farther
apart on the energy scale. Energy difference between the peaks is
approximately $2u$, which is equal to the energy difference between the
eigenstates of spin in the $x$-direction. Total transmission probability does
not change significantly compared to the case when $b=a$, but probability of
spin-flip does. When we calculate transmission and reflection probabilities
for large values of $b$, the obtained results do not converge to a constant
value for each probability with given parameters. Instead, the results for
$P_{t}^{++}$ and $P_{t}^{+-}$ exhibit a sort of oscillatory behaviour when we
examine the results for different values of parameter $b$, as seen in Figures
5, 8 and 9. This behaviour is observed even more clearly when we examine the
single particle limit, with $l<<a$. In that limit, the oscillations of spin-
flip probability with energy are also clearly seen. An additional effect
possible for some values of parameters in the single particle case is seen in
Figure 9 where we observe multiple transmission probability peaks in the low
energy range which can significantly increase the total tunnelling
probability.

Perhaps the most interesting effect of magnetic interaction are these
oscillations of spin-flip probability. They are caused by Larmor precession of
spin. We expanded the composite particle spin in the $\sigma_{z}$ eigenbasis,
and the magnetic field is pointing in $x$-direction. Because of that choice,
we can set $\vec{A}=0$ along the path of composite particle which simplifies
the Schrödinger equation, and we also have transitions from spin-up to spin-
down state and vice versa. Spin-up and spin-down states in $z$-direction are
not the eigenstates of $\sigma_{x}$, so Larmor precession of spin happens when
the magnetic field is present. We solved the time-independent Schrödinger
equation, but we can always think of its solution as a limit of a time-
dependent wave packet initially travelling in $+x$-direction incident on a
potential barrier and area with magnetic field. It is clear that the total
angle of spin rotation is proportional to time spent in the area with magnetic
field, which means it is proportional to $b$ when $b>>a$, i.e. the only
interaction is magnetic and kinetic energy are constant for most of the time
spent travelling through the field. Also, for different energies, time of
travel through the area with field is different, so the total angle of
rotation is different. A nice confirmation of this is halving of the "period"
of oscillations in energy when $b$ is doubled in the single particle case, as
seen in Figure 8.

We give no direct explanation for smaller oscillations seen in Figures 5(e),
5(f), 8 and 9. They appear when the ratio of width of magnetic field and width
of potential barrier becomes large enough, roughly of the order of magnitude
$10^{2}$, and when other effects (i.e. resonant tunnelling) become less
noticeable. Hence, they are best visible in the single particle limit, but can
also appear when the compositeness of the particle is relevant (Figures 5(e)
and 5(f)). In some cases, those oscillations can significantly influence total
tunnelling probability as showed in Figure 9. This behaviour is reminiscent of
some sort of "nutation" superposed on Larmor precession, however, this effect
is yet to be studied more closely.

We also see that the effects of magnetic interaction are the greatest around
transmission resonances. In some cases, apart from splitting, widening of
resonances is also observed. The most likely explanation for this is that the
coupling between the magnetic interaction, the internal interaction of the
particle (infinite potential well in this case) and the interaction with
potential barrier has the strongest effects when the coupling between the
latter two interactions produces transmission resonances.

As noted in the introduction, tunnelling of composite particles occurs in
various physical contexts, and this simple model should be qualitatively
applicable to those situations where splitting of energy levels in magnetic
field is large enough. If we take a particle with magnetic moment $\mu_{B}$ as
an example, with $V_{0}$ and $a$ of order $10^{-1}$ eV and nm, respectively,
for magnetic field strength of $1$ T we get $u\sim 10^{-4}$ eV. This means
that the effects of magnetic field presence could be observable for realistic
field strengths and barriers for particles with large enough magnetic moment,
for example in experiments with exciton tunnelling and possibly in some
molecule tunnelling situations. The effects should be more apparent if
splitting of energies is very large, example of which is the giant Zeeman
effect.

##  5 Conclusion

We have studied tunnelling of a composite particle in presence of a magnetic
field. The exact numerical solution is provided within the context of a simple
model we used. Further work could include studying more realistic binding
potentials, three-dimensional version of the problem, and magnetic field
pointing in a general direction. These expansions would make the problem
significantly more complicated because orbital angular momentum and presence
of a magnetic vector potential would have to be taken into account. However,
we expect that qualitative features of our results would be retained as shown
in e.g. [8] for the case of composite particle tunnelling with no magnetic
field, but it is also possible that additional degrees of freedom would
significantly alter the results. Effects originating from Larmor precession
and splitting of energy levels in magnetic field would certainly be present in
more realistic models, but because of complexity of the results it is hard to
predict whether additional translational and rotational degrees of freedom
would result in some new effects which are not present in our one-dimensional
case. A topic of further research could also be the explanation of smaller
oscillations which are noticed when $b>>a$.

This work shows that, based on results obtained within a simple framework, the
main effects of additional magnetic interaction on composite particle
tunnelling are splitting and in some cases widening of resonance transmission
probability peaks which do not reach $100\%$ transmission probability when the
magnetic field is turned on, and oscillations of spin-flip probability with
energy and width of the magnetic field. It can also be expected that these
effects could be observed in realistic experiments with exciton tunnelling,
where composite particle tunnelling in presence of a magnetic field has
already been studied. When the splitting of energy levels caused by magnetic
field is more than 4 orders of magnitude smaller than barrier height, with
realistic barrier widths, the magnetic interaction can be neglected, which
would be the case in situations from nuclear physics.

We have also given a physical explanation for the fact that probabilities of
transmission $P_{t}^{++}$ and $P_{t}^{+-}$ do not converge to a single value
in the limit $b>>a$. It is interesting that spin-flip probability depends
strongly on width of an area with magnetic field presence, independently of
ratio $b/a$. This is a clear manifestation of Larmor precession in a time-
independent model. The splitting of transmission probability peak is expected
as a consequence of splitting of energy levels in a magnetic field. The fact
that new peaks do not reach $100\%$ transmission probability can be explained
as a consequence of the specific nature of coupling between magnetic
interaction with other interactions, so the conditions for $100\%$
transmission probability are no longer satisfied when it is present.

## Acknowledgements

We would like to thank M. Milin, also M. T. Cvitaš from the University of
Zagreb and K. Pisk from Ruđer Bošković Institute for useful discussions and
comments.

##  6 Appendix

###  6.1 Python code for generating graphs seen in the figures

⬇

from scipy.integrate import solve_ivp

import math

import cmath

import matplotlib.pyplot as plt

import numpy as np

a = 1\.

b = 1\.

d = 5\.

l = 5\.

u = 0.05

dot = 800

ran = 1

h = 1\.

m = 1\.

V0 = 1\.

konst = (4 * m / (h * h))

def W_funk(W0, g1, g2, n1, n2):

if n1 == n2:

return W0 * ((g2 / d) - (1 / (n1 * math.pi)) * (

math.sin(n1 * math.pi * g2 / d) *

math.cos(n1 * math.pi * g2 / d))) - W0 * (

(g1 / d) - (1 / (n1 * math.pi)) * (

math.sin(n1 * math.pi * g1 / d) *

math.cos(n1 * math.pi * g1 / d)))

else:

return (W0 / (math.pi * (n1 - n2))) * (

math.sin((math.pi * (n1 - n2) * g2) / d) \-

math.sin((math.pi * (n1 - n2) * g1) / d)) - (

W0 / (math.pi * (n1 + n2))) * (

math.sin((math.pi * (n1 + n2) * g2) / d) \-

math.sin((math.pi * (n1 + n2) * g1) / d))

def W(cnt1, cnt2, y):

y1 = 2 * y - a - l + d / 2

y2 = 2 * y + a - l + d / 2

y3 = -2 * y - a - l + d / 2

y4 = -2 * y + a - l + d / 2

Y = [y1, y2, y3, y4]

for pom in range(4):

if Y[pom] < 0: Y[pom] = 0

if Y[pom] > d: Y[pom] = d

W = 0

if Y[0] == Y[1] and Y[2] == Y[3]:

W = 0

elif Y[0] == Y[1]:

W = W_funk(V0, Y[2], Y[3], cnt1 + 1, cnt2 + 1)

elif Y[2] == Y[3]:

W = W_funk(V0, Y[0], Y[1], cnt1 + 1, cnt2 + 1)

elif Y[0] == Y[2] and Y[1] == Y[3]:

W = W_funk(2 * V0, Y[0], Y[1], cnt1 + 1, cnt2 + 1)

elif (Y[1] > Y[0] >= Y[3] > Y[2]) or (Y[3] > Y[2] >= Y[1] > Y[0]):

W = W_funk(V0, Y[0], Y[1], cnt1 + 1, cnt2 + 1) \+

W_funk(V0, Y[2], Y[3], cnt1 + 1, cnt2 + 1)

elif Y[1] >= Y[3] > Y[0] >= Y[2]:

W = W_funk(2 * V0, Y[0], Y[3], cnt1 + 1, cnt2 + 1) \+

W_funk(V0, Y[3], Y[1], cnt1 + 1, cnt2 + 1) \+

W_funk(V0, Y[2], Y[0], cnt1 + 1, cnt2 + 1)

elif Y[3] >= Y[1] > Y[2] >= Y[0]:

W = W_funk(2 * V0, Y[2], Y[1], cnt1 + 1, cnt2 + 1) \+

W_funk(V0, Y[1], Y[3], cnt1 + 1, cnt2 + 1) \+

W_funk(V0, Y[0], Y[2], cnt1 + 1, cnt2 + 1)

return W

def F_funk(cnt1, cnt2, y):

f = 0

if (l - d / 2) < 2 * (y - b) < 2 * (y + b) < (l + d / 2):

f += W_funk(u, 2 * (y - b), 2 * (y + b), cnt1 + 1, cnt2 + 1)

elif (l - d / 2) < 2 * (y - b) < (l + d / 2) < 2 * (y + b):

f += W_funk(u, 2 * (y - b), (l + d / 2), cnt1 + 1, cnt2 + 1)

elif 2 * (y - b) < (l - d / 2) < 2 * (y + b) < (l + d / 2):

f += W_funk(u, (l - d / 2), 2 * (y + b), cnt1 + 1, cnt2 + 1)

elif 2 * (y - b) < (l - d / 2) < (l + d / 2) < 2 * (y + b):

f += W_funk(u, (l - d / 2), (l + d / 2), cnt1 + 1, cnt2 + 1)

return f

def G(cnt1, cnt2, y):

if cnt1 < c and cnt2 < c:

return konst * W(cnt1, cnt2, y)

if cnt1 < c <= cnt2:

return -konst * F_funk(cnt1, cnt2 - c, y)

if cnt1 >= c > cnt2:

return -konst * F_funk(cnt1 - c, cnt2, y)

if cnt1 >= c and cnt2 >= c:

return konst * W(cnt1 - c, cnt2 - c, y)

def sys(y, R):

F = [0j] * 8 * c * c

Fa = np.array(F)

Ra = np.array(R)

Rm = Ra.reshape(4 * c, 2 * c)

Fm = Fa.reshape(4 * c, 2 * c)

y1 = 2 * y - a - l + d / 2

y2 = 2 * y + a - l + d / 2

y3 = -2 * y - a - l + d / 2

y4 = -2 * y + a - l + d / 2

Y = [y1, y2, y3, y4]

for pom in range(4):

if Y[pom] < 0: Y[pom] = 0

if Y[pom] > d: Y[pom] = d

for i in range(2 * c):

for j in range(2 * c):

for cnt1 in range(2 * c):

konst1 = (-1 / (2j * k[cnt1 % c]))

konst2 = cmath.exp(-1j * k[cnt1 % c] * y)

konst3 = cmath.exp(1j * k[cnt1 % c] * y)

for cnt2 in range(2 * c):

konst4 = cmath.exp(-1j * k[cnt2 % c] * y)

konst5 = cmath.exp(1j * k[cnt2 % c] * y)

Fm[i][j] += konst1 * (konst3 * (i == cnt1) + konst2 *

Rm[i][cnt1]) * (G(cnt1, cnt2, y) * (

konst5 * (j == cnt2) + konst4 * Rm[cnt2][j]))

Fm[2 * c + i][j] += konst1*(konst2 * Rm[2 * c + i][cnt1]) *

(G(cnt1, cnt2, y) * (

konst5 * (j == cnt2) + konst4 * Rm[cnt2][j]))

return Fm.flatten()

Tgg11 = [0] * dot

Tgg12 = [0] * dot

Tgd11 = [0] * dot

Tgd12 = [0] * dot

T11 = [0] * dot

en = [0] * dot

uk = [0] * dot

for cnt in range(dot):

n = 7

c = 0

E = (h * h * math.pi * math.pi / (m * d * d)) + (cnt + 1) * ran * V0 / dot

xl = -1 * (2 * max(b,a) + 2 * l + d) / 4

xr = (2 * max(b,a) + 2 * l + d) / 4

ep = [0] * n

k = [0] * n

for i in range(1, n + 1):

ep[i - 1] = h * h * i * i * math.pi * math.pi / (m * d * d)

if ep[i - 1] < E:

c = i

k[i - 1] = (2 / h) * math.sqrt(m * (E - ep[i - 1]))

en[cnt] = (E - ep[0]) / V0

init_ar = np.zeros(8 * c * c) + 0j

init_m = init_ar.reshape(4 * c, 2 * c)

for red in range(2 * c):

init_m[2 * c + red][red] += 1

init = init_m.flatten()

res = solve_ivp(sys, (xr, xl), init, method=’DOP853’, max_step=0.3)

f_moment = len(res.t) - 1

res_f = [0j] * len(res.y)

for pom in range(len(res.y)):

res_f[pom] += res.y[pom][f_moment]

res_a = np.array(res_f)

res_m = res_a.reshape(4 * c, 2 * c)

Tgg11[cnt] += abs(res_m[2 * c][0]) * abs(res_m[2 * c][0])

if c > 1:

Tgd11[cnt] += abs(res_m[2 * c + 2][0]) *

abs(res_m[2 * c + 2][0])

Tgg12[cnt] += abs(res_m[2 * c + 1][0]) *

abs(res_m[2 * c + 1][0]) * k[1] / k[0]

Tgd12[cnt] += abs(res_m[2 * c + 3][0]) *

abs(res_m[2 * c + 3][0]) * k[1] / k[0]

else:

Tgd11[cnt] += abs(res_m[2 * c + 1][0]) *

abs(res_m[2 * c + 1][0])

T11[cnt] = Tgd11[cnt] + Tgg11[cnt]

for pom in range(2 * c):

uk[cnt] += k[pom % c] / k[0] * (abs(res_m[pom][0]) *

abs(res_m[pom][0]) + abs(

res_m[2 * c + pom][0]) * abs(res_m[2 * c + pom][0]))

print(cnt)

fig, ax = plt.subplots(figsize=[7, 7])

plt.plot(en, Tgg11, label=r’$P_{t,␣11}^{++}$’)

plt.plot(en, Tgg12, label=r’$P_{t,␣12}^{++}$’)

plt.plot(en, Tgd11, label=r’$P_{t,␣11}^{+-}$’)

plt.plot(en, Tgd12, label=r’$P_{t,␣12}^{+-}$’)

# plt.plot(en, T11, label=r’$P_{t, 11}$’)

# plt.plot(en, uk, label="uk")

plt.xlabel(r’$(E-\epsilon_1)/V_0$’, fontsize=18)

plt.ylabel(r’$P_t$’, fontsize=18)

plt.xticks(fontsize=15)

plt.yticks(fontsize=15)

plt.legend(fontsize=18)

plt.show()

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