The Effect of Fluctuations in the Observer’s Frame of Reference on Tunneling through a 1-Dimensional Double Square Potential Barrier

In this paper, particle tunneling through a 1-dimensional double square potential barrier is considered under fluctuations of an observer’s frame of reference (OFR). The concept of duration density—the proportion of the duration of a measurement that a particle has a particular energy—is adopted to formulate the average transmission probability of a particle for two types of fluctuations in the OFR. For single-period square-wave fluctuations, the average transmission probability is found to oscillate with a period of π or 2π as a function of the energy of the particle; the period of the oscillations rapidly switches from one value to another as a result of extremely small changes in the wavenumber configuration. For single-period sawtooth-wave fluctuations, the period of the oscillations in the transmission probability gradually increases from π to 2π as the wavenumber configuration changes. It is concluded, therefore that the average transmission probability has the potential to provide information about the nature of the fluctuations in an OFR.

Introduction

Tunneling is a microscopic phenomenon that results from the wave-particle nature of matter [1]–[3] by which a particle can pass through a potential barrier that would be impenetrable according to classical physics. Tunneling forms the basis of many concepts and phenomena in modern physics, including radioactive disintegration [4]–[6], electron tunneling [7], scanning tunneling microscopy [8], quantum dots [9], quantum computation [10], [11], and resonant tunneling through double barriers [12].

In classical mechanics, a particle that is incident on a potential barrier will be reflected if its energy is smaller than that of the barrier. In quantum mechanics, however, there is a small (often extremely small), but finite, probability that the particle can be found on the opposite side of the barrier due to its wave nature; as would be expected as a result of Heisenberg’s uncertainty principle [3]. Heisenberg’s uncertainty principle states that a measurement of the momentum of a particle can be performed to any degree of precision but that the size of the disturbance in the position of the particle will then be inversely proportional to the uncertainty in its momentum. When a particle is incident on a potential barrier, there will be uncertainty in its momentum throughout the period that measurements of the momentum of the particle are being made. The resulting disturbance to the particle’s position may then be greater than the width of the barrier. As a result, there is the possibility of finding the particle on the opposite side of the barrier—i.e., of the particle tunneling through the barrier. A one-dimensional (1-D) double square potential barrier (DSPB) is an idealized situation in which there is an exact solution for the probability of a particle being transmitted through two successive potential barriers. This solution can also provide an approximate solution for, and an intuitive understanding of, more realistic situations, such as the tunneling of an electron through quantum dots [9].

Previously, the tunneling of a particle through a DSPB has been quantitatively studied in terms of the transmission probability of the particle. In these studies, it was assumed that the observer’s frame of reference (OFR) remains constant—conventionally, the OFR was assumed to be stationary throughout the performance of measurements of the transmission probability. Thus, how much the transmission probability of a particle through a DSPB changes for a fluctuating OFR remains an open question.

Recently, it has been suggested that a novel observer effect resulting from a fluctuating OFR can occur for an Einstein solid [13]—a single-electron transistor [14]—tunneling through a 1-D single square potential barrier (SSPB) [15] and for blackbody radiation [16]. In the case of tunneling through a 1-D SSPB, details of the fluctuations in the OFR can potentially be derived from the transmission probability of a particle as this depends on the patterns of the fluctuations in the OFR. For a varying OFR, in the case of periodic square-wave fluctuations, the average transmission probability of a particle penetrating a 1-D SSPB rapidly increases just above the energy corresponding to the amplitude of the fluctuations in the OFR; in the case of periodic sawtooth-wave fluctuations, in contrast, it gradually increases above this energy. As for blackbody radiation, the specific intensity of cosmic microwave background radiation implies the existence of fluctuations in the OFR with an amplitude of the order of 1 $\mathrm{\mu }\text{eV}$.

If the reference of energy of a particle is equal to an OFR, then it is possible to investigate an observer effect that is induced by fluctuations in the OFR. In this study, the average transmission probability of a particle tunneling through a 1-D DSPB [17] was investigated for a fluctuating OFR. For pedagogical purposes, the average transmission probability is formulated for two types of fluctuations by adopting the concept of duration density—what proportion of the duration of the measurements the particle has a particular energy. The two types of fluctuations considered are periodic square-wave fluctuations and periodic sawtooth-wave fluctuations. Under single-period square-wave fluctuation, the average transmission probability is found to oscillate with a period of $\mathrm{\pi }$ or $2\mathrm{\pi }$ as a function of the energy of the particle, depending on the wavenumber configuration; this period rapidly switches from $\mathrm{\pi }$ to $2\mathrm{\pi }$ as a result of extremely small changes in the wavenumber configuration. Under single-period sawtooth-wave fluctuation, the average transmission probability again oscillates with a period of $\mathrm{\pi }$ or $2\mathrm{\pi }$ but gradually switches from $\mathrm{\pi }$ to $2\mathrm{\pi }$ as the wavenumber configuration changes. Therefore, it is concluded that the average transmission probability for a particle of given energy has the potential to provide information about the pattern of fluctuations of an OFR.

Average Transmission Probability through a 1-D DSPB

Fig. 1a represents a 1-D DSPB, where $V_1$ and $2d_1$ are the potential energy and width of the barrier 1, respectively; $V_{\text{w}}$ and $b$ are the potential energy and width of the well, respectively; $E$ is the energy of a particle; $E_{\text{ref}}(t)$ is the energy of an OFR at time $t$. ($V_2$ and $2d_2$ are similarly defined for barrier 2). In this study, $E_{\text{ref}}(t)$ is assumed to be constant during the time interval between $t$ and $t+dt$, where d$t\ll \mathrm{\Delta }\text{t}(=t_f-t_i)$, with $t_i$ and $t_f$ being the initial and final times, respectively. A particle approaches from left and is incident on barrier 1 at $\text{x}\ll -d_1$; it then tunnels through the barrier and is detected by an observer located far from the 1-D DSPB at $\text{x}\gg d_1+b+2d_2$.

Fig. 1. (a) Diagram of a 1-D DSPB and (b) The probability of transmission of a particle, $1-|R_0{|}^2$, through the 1-D DSPB in the case of a stationary OFR, plotted as a function of $E$. See the details in text.

For a particle with energy $E$, where $E_{\text{ref}}(t)\le E<V_{1(2)}$, the wave function of the particle within the 1-D DSPB at time $t$ is $\mathrm{\Psi }(x,t)=\phi (x)e^{-iEt/\hslash }$, where $\phi (x)$ is expressed as

(1) $$\phi (x)=\{\begin{array}{l}\begin{array}{c}{e}^{ikx}+R{e}^{-ikx}\mathrm{for}-\infty <x<-d_1\\ A_1{e}^{-{\beta }_{1}x}+B_1{e}^{{\beta }_{1}x}\mathrm{for}-d_1\le x<d_1\end{array}\hfill \\ C_{\text{w}}{e}^{-i{k}_{\text{w}}x}+D_{\text{w}}{e}^{i{k}_{\text{w}}x}\mathrm{for}{d}_1\le x<d_1+b\hfill \\ A_2{e}^{-{\beta }_{2}x}+B_2{e}^{{\beta }_{2}x}\mathrm{for}{d}_1+b\le x<d_1+b+2d_2\hfill \\ T{e}^{iqx}\mathrm{for}{d}_1+b+2d_2\le x<\infty \hfill \end{array}$$

Here, $k=\sqrt{2mE}/\hslash$, $k_{\text{w}}=\sqrt{2m(E-V_w)}/\hslash$, $q=\sqrt{2mE(t)}/\hslash$. ${\beta }_{1(2)}=\sqrt{2m(V_{1(2)}-E)}/\hslash$ is the damping rate for a particle that passes through potential barrier 1(2). Here, $m$ is the mass of the particle; $\hslash$ the Planck’s constant divided by $2\mathrm{\pi }$; $E\left(t\right)=E-E_{\text{ref}}\left(t\right)$ is the measured energy for $E$ at time $t$, and $R$ and $T$ are the particle’s reflection and transmission coefficients, respectively. In matrix form, $R$ and $T$ can be expressed as follows: where $M_j=m_j{e}^{i{\theta }_j}$. Here, $j=\text{L}11$, $\text{L}12$, $\text{L}21$, $\text{L}22$, $\text{R}11$, $\text{R}12$, $\text{R}21$, and $\text{R}22$.

(2) $$\begin{array}{r}\left[\begin{array}{c}1\\ R\end{array}\right]=\left[\begin{array}{cc}{M}_{\text{T}11}& M_{\text{T}12}\\ M_{\text{T}21}& M_{\text{T}22}\end{array}\right]\left[\begin{array}{c}T\\ 0\end{array}\right]=\left[\begin{array}{cc}{M}_{\text{L}11}& M_{\text{L}12}\\ M_{\text{L}21}& M_{\text{L}22}\end{array}\right]\left[\begin{array}{cc}{e}^{-i{\theta }_{\text{w}}b}& 0\\ 0& e^{i{\theta }_{\text{w}}b}\end{array}\right]\left[\begin{array}{cc}{M}_{\text{R}11}& M_{\text{R}12}\\ M_{\text{R}21}& M_{\text{R}22}\end{array}\right]\left[\begin{array}{c}T\\ 0\end{array}\right],\end{array}$$

Because of the need for continuity at the boundaries $\text{x}=-d_1$, $\text{x}=d_1$, $\text{x}=d_1+b$, and $\text{x}=d_1+b+2d_2$, the probability of the particle being transmitted through the 1-D DSPB, which is $1-|R{|}^2$, can be expressed as where

(3) $$\begin{array}{r}1-{\left|R\right|}^2=\frac{q}{k{\left|M_{\text{T}11}\right|}^2},\end{array}$$

(4) $$\begin{array}{r}{\left|M_{\text{T}11}\right|}^2={(m_{\text{L}11}{m}_{\text{R}11}-m_{\text{L}12}{m}_{\text{R}21})}^2+2m_{\text{L}11}{m}_{\text{R}11}{m}_{\text{L}12}{m}_{\text{R}21}[1+\cos (2k_{\text{w}}b+{\theta }_{\text{L}12}+{\theta }_{\text{R}21}-{\theta }_{\text{L}11}-{\theta }_{\text{R}11})].\end{array}$$

Here, and

(5) $$\begin{array}{r}{m}_{\text{L}11}=\sqrt{\frac{1}{4}{(1+\frac{k_{\text{w}}}{k})}^2{\cosh }^2(2{\beta }_1{d}_1)+\frac{1}{4}{\left(\frac{k{k}_{\text{w}}-{{\beta }_1}^2}{k{\beta }_1}\right)}^2{\sinh }^2(2{\beta }_1{d}_1)},\end{array}$$

(6) $$\begin{array}{r}{m}_{\text{R}11}=\sqrt{\frac{1}{4}{(1+\frac{\text{q}}{k_{\text{w}}})}^2{\cosh }^2(2{\beta }_2{d}_2)+\frac{1}{4}{\left(\frac{k_{\text{w}}\text{q}-{{\beta }_2}^2}{k_{\text{w}}{\beta }_2}\right)}^2{\sinh }^2(2{\beta }_2{d}_2)},\end{array}$$

(7) $$\begin{array}{r}{m}_{\text{L}12}=\sqrt{\frac{1}{4}{(\frac{k_{\text{w}}}{k}-1)}^2{\cosh }^2(2{\beta }_1{d}_1)+\frac{1}{4}{\left(\frac{k{k}_{\text{w}}+{{\beta }_1}^2}{k{\beta }_1}\right)}^2{\sinh }^2(2{\beta }_1{d}_1)},\end{array}$$

(8) $$\begin{array}{r}{m}_{\text{R}21}=\sqrt{\frac{1}{4}{(\frac{\text{q}}{k_{\text{w}}}-1)}^2{\cosh }^2(2{\beta }_2{d}_2)+\frac{1}{4}{\left(\frac{k_{\text{w}}\text{q}+{{\beta }_2}^2}{k_{\text{w}}{\beta }_2}\right)}^2{\sinh }^2(2{\beta }_2{d}_2)},\end{array}$$

(9) $$\begin{array}{r}{\theta }_{\text{L}11}=-{\tan }^{-1}[\frac{k{k}_{\text{w}}-{{\beta }_1}^2}{{\beta }_1(k+k_{\text{w}})}\tanh (2{\beta }_1{d}_1)]+(k+k_{\text{w}})d_1,\end{array}$$

(10) $$\begin{array}{r}{\theta }_{\text{R}11}=-{\tan }^{-1}[\frac{k_{\text{w}}q-{{\beta }_2}^2}{{\beta }_2(k_{\text{w}}+q)}\tanh \left(2{\beta }_2{d}_2\right)]+(k_{\text{w}}+q)d_2,\end{array}$$

(11) $$\begin{array}{r}{\theta }_{\text{L}12}=-{\tan }^{-1}[\frac{k{k}_{\text{w}}+{{\beta }_1}^2}{{\beta }_1(k_{\text{w}}-k)}\tanh \left(2{\beta }_1{d}_1\right)]+\pi +(k-k_{\text{w}})d_1,\end{array}$$

(12) $$\begin{array}{r}{\theta }_{\text{R}21}={\tan }^{-1}[\frac{k_{\text{w}}q+{{\beta }_2}^2}{{\beta }_2(q-k_{\text{w}})}\tanh \left(2{\beta }_2{d}_2\right)]+\pi -(k_{\text{w}}-q)d_2.\end{array}$$

For an OFR with energy $E_{\text{ref}}\left(t\right)$, where $E_{\text{ref}}\left(t\right)\le E<V_{1(2)}$, the average transmission probability through the 1-D DSPB during the time interval $\mathrm{\Delta }\text{t}$, $1-|R_{\text{avg}}{|}^2$, can be expressed as

(13) $$\begin{array}{r}1-|R_{\text{avg}}{|}^2=\frac{1}{\mathrm{\Delta }t}{\int }_{t_i}^{t_f}\frac{q}{k{\left|M_{\text{T}11}\right|}^2}dt.\end{array}$$

Stationary OFR

For an observer within a stationary frame of reference ($E_{\text{ref}}(t)=0$), the transmission probability $1-|R_0{|}^2$ can be expressed as

(14) $$\begin{array}{r}1-{\left|R_0\right|}^2=\frac{1}{{\left|M_{\text{T}11}\right|}^2},\end{array}$$

Fig. 1b shows the probability of transmission through a 1-D DSPB for a particle with a mass equal to that of an electron in the case of a stationary OFR; the probability is shown as a function of $E$. For $V_1=V_2=$ 3 $\text{ eV}$, $d_1=d_2=0.1$ $\text{ nm}$, and $b=2$ $\text{ nm}$, the value of $1-|R_0{|}^2$ oscillates with a period of $2\mathrm{\pi }$. Here, black and gray lines denote $1-|R_0{|}^2$ for $V_{\text{w}}=1.999$ $\mathrm{\mu }\text{eV}$ and −1.999 $\mathrm{\mu }\text{eV}$, respectively. The gray line is shifted by 0.05 for clality. The size of the envelope of these oscillations gradually increases with $E$, reaching a maximum value of 1, i.e., resonant tunneling through the 1-D DSPB then takes place. Note that because of the ${\theta }_{\text{L}12}+{\theta }_{\text{R}21}$ in the cosine term of the (4) this term must have a phase of 0 for the wavenumber configurations $k_{\text{w}}<k$ and $k_{\text{w}}>k$.

Fluctuating OFR

Fig. 2a shows $E\left(t\right)$ as a function of $t$ for a discretely fluctuating $E_{\text{ref}}\left(t\right)$; here $t_i$ and $t_f$ correspond to $t_0$ and $t_{\text{n}}$, which are the start and end points of the sequence $t_0$, $t_1$, ···, $t_{\text{n}-1}$, $t_{\text{n}}$, respectively. The time interval between $t_0$ and $t_{\text{n}}$ is $\mathrm{\Delta }t$. The energy of the particle within the time interval $\mathrm{\Delta }{t}_{\text{s}}=t_{\text{s}}-t_{\text{s}-1}$, $s=1,2,\cdot \cdot \cdot ,\text{n}-1,\text{n}$, is constant and given by $E_j$, where $j=-2,-1,0,1$. The energy of a stationary OFR is defined as $E_0$. Fig. 2b shows a different representation of $E\left(t\right)$, plotted in terms of increasing energy with the time sequence reordered as $t_0$^*, $t_1$^*, ···, $t_{\text{n}-1}$^*, $t_{\text{n}}$^*. In this part of the figure, $g_{\text{j}}$ shows the portion of measurement time corresponding to each energy $E_j$; i.e., the length of time for which the particle has energy $E_j$, divided by $\mathrm{\Delta }\text{t}$. The profile of $g_j$ is displayed as a function of $E\left(t\right)$ in Fig. 2c. In terms of $g_j$, $1-|R_{\text{avg}}{|}^2$ can be expressed as

Fig. 2. (a) Measured values of the particle energy $E$ at time $t$, $E\left(t\right)(=E-E_{\text{ref}}(t))$ in the case of discrete fluctuations in the OFR, (b) $E\left(t\right)$ as a function of the rearranged time set of $t_0$^*, $t_1$^*,···, $t_{\text{n}-1}$^*, $t_{\text{n}}$^*, and (c) Profile of $g_j$ shown in (b), plotted as a function of $E\left(t\right)$. See the details in text.

where $\sum _j{g}_{\text{j}}=1$.

(15) $$\begin{array}{r}1-|R_{\text{avg}}{|}^2={\sum }_j\frac{g_{\text{j}}q(E_{\text{j}})}{k(E_{\text{j}}){|M_{\text{T}11}(E_{\text{j}})|}^2},\end{array}$$

For a continuously fluctuating $E_{\text{ref}}\left(t\right)$, in the limit of $\mathrm{\Delta }{E}_{\text{j}}(=E_{\text{j}+1}-E_{\text{j}})\to 0$, $g_j$ can be considered equal to the duration density, $g$, which is defined as the measurement time for $E\left(t\right)$ within $dE$ divided by $\mathrm{\Delta }\text{t}$. Thus, $1-|R_{\text{avg}}{|}^2$ can be expressed as

where $\int gdE=1$.

(16) $$\begin{array}{r}1-|R_{\text{avg}}{|}^2=\int \frac{gq}{k{\left|M_{\text{T}11}\right|}^2}dE,\end{array}$$

Average Transmission Probability through a 1-D DSPB for Periodic Square-Wave Fluctuations in the OFR

Given a periodic $E_{\text{ref}}(t)=\{\begin{array}{ll}-{\epsilon }_1& \text{for}0\le t<{\tau }_1/2\\ {\epsilon }_1& \text{for}{\tau }_1/2\le t<{\tau }_1\end{array}$ [shown as the left-hand part of Fig. 3a], the measured energy of the particle at time t, $E(t)$, can be expressed as

Fig. 3. (a) Representation of a fluctuating frame of reference at time $t$, $E_{\text{ref}}(t)$, for the case of periodic square-wave fluctuations with an amplitude and period are ${\mathrm{\epsilon }}_1$ and ${\mathrm{\tau }}_1$, respectively, (b) Average transmission probability for a particle, $1-|R_1{|}^2$, through the 1-D DSPB in the case of half-period square-wave fluctuations in the OFR, and (c) Average transmission probability for a particle, $1-|R_1{|}^2$, through the 1-D DSPB in the case of single-period square-wave fluctuations in the OFR. See the details in text.

where ${\mathrm{\epsilon }}_1$ and ${\tau }_1$ are the amplitude and period of the periodic square-wave fluctuations, respectively. The energy of stationary OFR, $E_{\text{ref}}$, is taken to be equal to 0.

(17) $$\begin{array}{r}E(t)=\{\begin{array}{ll}E+{\epsilon }_1& \text{for}0\le t<{\tau }_1/2\\ E-{\epsilon }_1& \text{for}{\tau }_1/2\le t<{\tau }_1\end{array},\end{array}$$

Single-Period Square-Wave Fluctuations

For an observer in a fluctuating frame of reference that fluctuates according to a half-period square-wave pattern, Fig. 3b displays the transmission probability for a particle with a mass equal to that of an electron, $1-|R_1{|}^2$, as a function of $E\ge {\epsilon }_1$ for $V_1=V_2=3$ $\text{ eV}$, $d_1=d_2=0.1$ $\text{ nm}$, $b=2$ $\text{ nm}$, $V_{\text{w}}=1.999$ $\mathrm{\mu }\text{eV}$, and ${\mathrm{\epsilon }}_1=2$ $\mathrm{\mu }\text{eV}$. Black and gray lines represent $1-|R_1{|}^2$ for $g=\mathrm{\delta }(E(t)-E-{\mathrm{\epsilon }}_1)$ and $g=\mathrm{\delta }(E(t)-E+{\mathrm{\epsilon }}_1)$, respectively, where $\mathrm{\delta }$ is the Dirac delta function. Similar to the results for $1-|R_0{|}^2$ described above, the value of $1-{\left|R_1\right|}^2$ also oscillates with a period of $2\mathrm{\pi }$, and the envelope of this value gradually increases with $E$, saturating at a value of 1. Notably, the gray line leads the black line by a phase difference of $\mathrm{\pi }$ because, in this case, the ${\theta }_{\text{L}12}+{\theta }_{\text{R}21}$ in the cosine term of the (4) provides an additional phase of $\mathrm{\pi }$ for the wavenumber configuration $k_{\text{w}}<k$ and $k_{\text{w}}>q$ relative to the configuration $k_{\text{w}}<k$ and $k_{\text{w}}<q$.

Single-Period Square-Wave Fluctuations

For an observer in a fluctuating frame of reference that fluctuates according to a single-period square-wave pattern, $g=0.5\mathrm{\delta }(E(t)-E+{\mathrm{\epsilon }}_1)+0.5\mathrm{\delta }(E(t)-E-{\mathrm{\epsilon }}_1)$ (as shown in the right-hand part of Fig. 3a). In Fig. 3c, the transmission probability of a particle with a mass equal to that of an electron, $1-|R_1{|}^2$, is displayed as a function of $E\ge {\epsilon }_1$ for $V_1$ = $V_2$ = 3 $\text{ eV}$, $d_1$ = $d_2$ = 0.1 nm, $b$ = 2 nm, and ${\mathrm{\epsilon }}_1$ = 2 $\mathrm{\mu }\text{eV}$. It can be seen that if $V_{\text{w}}=1.999$ $\mathrm{\mu }\text{eV}$, $1-|R_1{|}^2$ oscillates with a period of $\mathrm{\pi }$ (shown as the black line); the size of the corresponding envelope gradually increases with $E$, saturating at a value of 0.5 because, in this case, the wavenumber configurations $k_{\text{w}}<k$, $k_{\text{w}}>q$ and $k_{\text{w}}<k$, $k_{\text{w}}<q$ exist simultaneously. The former configuration provides the additional phase of $\mathrm{\pi }$ and the latter the phase of $0$, which, together, give oscillations in $1-|R_1{|}^2$ that have a period of $\mathrm{\pi }$. However,at $V_{\text{w}}=2.001\mathrm{\mu }\text{eV}$ (represented by the gray line), the oscillations have a period of $2\mathrm{\pi }$ because then the only possible wavenumber configuration is $k_{\text{w}}<k$ and $k_{\text{w}}<q$. These oscillations are, thus, clearly distinguishable from the oscillations in $1-|R_0{|}^2$.

Long-Period Square-Wave Fluctuations

For an observer in a frame of reference that fluctuates according to a long-period square-wave fluctuation, the transmission probability, $1-|{R_1}^{{ }^{\prime }}{|}^2$, can be expressed as

where $t_i=0$ and $t_f\gg {\tau }_1$.

(18) $$\begin{array}{r}1-|{R_1}^{{ }^{\prime }}{|}^2\approx 1-|R_1{|}^2,\end{array}$$

Average Transmission Probability through a 1-D DSPB for Periodic Sawtooth-Wave Fluctuations in the OFR

Given a periodic $E_{\text{ref}}\left(t\right)={\epsilon }_2-{\displaystyle \frac{2{\epsilon }_2}{{\tau }_2}}t$ for $0\le t<{\tau }_2$ [as shown in the left-hand part of Fig. 4a], the measured energy of the particle at time t, $E(t)$, can be expressed as

Fig. 4. (a) Representation of a fluctuating frame of reference, $E_{\text{ref}}(t)$, for the case of periodic sawtooth-wave fluctuations with an amplitude and period of ${\mathrm{\epsilon }}_2$ and ${\mathrm{\tau }}_2$, respectively, (b) Average transmission probability of a particle, $1-|R_2{|}^2$, through the 1-D DSPB in the case of single-period sawtooth-wave fluctuations in the OFR. See the details in text.

where ${\mathrm{\epsilon }}_2$ and ${\tau }_2$ are the amplitude and period of the periodic sawtooth-wave fluctuations, respectively.

(19) $$\begin{array}{r}E\left(t\right)=E-{\epsilon }_2+\frac{2{\epsilon }_2}{{\tau }_2}t,\text{for}0\le t<{\tau }_2,\end{array}$$

Single-Period Sawtooth-Wave Fluctuations

For an observer in a fluctuating frame of reference whose fluctuations take the form of single-period sawtooth waves, $g=1/2{\mathrm{\epsilon }}_2$ between $E-{\mathrm{\epsilon }}_2$ and $E+{\mathrm{\epsilon }}_2$; otherwise it is 0 [as shown in the right-hand part of Fig. 4a]. In Fig. 4b, the transmission probability of a particle with a mass equal to that an electron, $1-|R_2{|}^2$, is displayed as a function of $E\ge {\epsilon }_2$ for $V_1=V_2=3$ $\text{ eV}$, $d_1=d_2=0.1$ $\text{ nm}$, $b=2$ $\text{ nm}$, and ${\mathrm{\epsilon }}_2=4$ $\mathrm{\mu }\text{eV}$, where the gray, black and light gray lines denote $1-{\left|R_2\right|}^2$ for $V_{\text{w}}$ = $1.001\mathrm{\mu }\text{eV}$, $0.901\mathrm{\mu }\text{eV}$, $0.001\mathrm{\mu }\text{eV}$, respectively. It can be seen that at $V_{\text{w}}=0.001$ $\mathrm{\mu }\text{eV}$, the oscillations in $1-|R_2{|}^2$ have a period of $\mathrm{\pi }$ because its situation corresponds to the wavenumber configuration $k_{\text{w}}<k$ and $k_{\text{w}}>q$; the amplitude of these oscillations is 0.5, as is the case for single-period square-wave fluctuations. As $V_{\text{w}}$ increases, the peak values of the oscillations in $1-{\left|R_2\right|}^2$ that have a phase of $\mathrm{\pi }$ gradually decrease from 0.5 to saturate at a value to 0; however, the peak values of the oscillations whose peak have a phase of $2\mathrm{\pi }$ gradually increase from 0.5 and saturate at a value of 1. Thus, these oscillations are also clearly distinguishable from those in both $1-|R_0{|}^2$ and $1-|R_1{|}^2$.

Long-Period Sawtooth-Wave Fluctuations

For an observer in a fluctuating frame of reference where the oscillations can be described by long-period sawtooth-wave fluctuations, the transmission probability, $1-|{R_2}^{{ }^{\prime }}{|}^2$, can be expressed as

where $t_i=0$ and $t_f\gg {\tau }_2$.

(20) $$\begin{array}{r}1-|{R_2}^{{ }^{\prime }}{|}^2\approx 1-|R_2{|}^2,\end{array}$$

Conclusion

In this paper, the tunneling of a particle through a 1-D DSPB in cases where the OFR exhibited periodic fluctuations was described, and the average transmission probability for a particle was determined. Based on the assumption that the reference of energy of the particle was equal to the OFR and using the concept of duration density, the average transmission probability was formulated for two types of fluctuations: periodic square-wave fluctuations and periodic sawtooth-wave fluctuations. For single-period square-wave fluctuations or in the long-term limit for this type of fluctuation, when plotted against the energy of the particle, the average transmission probability was found to oscillate with a period of $\mathrm{\pi }$ or $2\mathrm{\pi }$, depending on the wavenumber configuration; the period rapidly switches from $\mathrm{\pi }$ to $2\mathrm{\pi }$ as a result of small changes in the wavenumber configuration. For single-period sawtooth-wave fluctuations or in the long-term limit for this type of fluctuation, the average transmission probability gradually changed from $\mathrm{\pi }$ to $2\mathrm{\pi }$ as a result of a change in the wavenumber configuration. It can be concluded, that a plot of the average transmission probability through a 1-D DSPB against the energy of an incident particle provides clear basis for characterizing the pattern of fluctuations in an OFR.