Shannon information entropies for rectangular multiple quantum well systems with constant total lengths
1. IntroductionShannon in his classical paper[1] reported the fundamental limits to signal processing operations such as reliable storing and communicating data and limits to compressing data. Thereafter, this theory was found to have other applications such as cryptography, quantum computing, thermal physics, atomic and molecular structure, etc. This is the so-called Shannon entropy, which is a measure of a random variable uncertainty and can also become the expected value of the information existing in a message.
During the last few decades, the Shannon entropies have been analytically calculated for low-lying states of some quantum systems with potentials such as the Morse potential,[2,3] harmonic oscillator,[4,5] the Pöschl–Teller potential,[2,6] modified Yukawa potential,[7] and the Rosen–Morse potential.[8] On the other hand, the Shannon entropies for other complicated quantum systems such as the position-dependent mass Schrödinger equation with null potential,[9] quantum dots,[10] double-well potential,[11] finite well,[12] and infinite circular well[13] have also been investigated. Relevant studies about this topic were also conducted in Refs. [14–17].
In addition to their intrinsic interest, the entropic uncertainty relations[4] have been widely used to reproduce the charge and momentum densities of the atomic and molecular systems[18] and also to investigate the squeezing localization.[19] The electronic localization properties of the system can be used for designing low-dimensional electron nano-devices and also electromagnetic waveguides with predetermined transport properties. The most widely studied potential models are one-dimensional (1D) quasi-periodic systems, such as the slowly varying potential,[20] Harper model,[21] and the complex quasi-periodic potential ones.[22]
Periodic structures of layers of two or more materials can construct a superlattice. The thickness of each layer is usually several nanometers. Different types of building blocks such as Fibonacci[23] and Pascal-type[24] possess some physical properties that are more appropriate for developing quantum devices. Our approach to the building of a superlattice or multiple quantum well system is based on retaining the total length of the structure constant and simultaneously changing the number of quantum wells. In this way, the smaller nano-device sizes in addition to better physical properties are accessible. So far, we have studied the optical properties of constant total effective length multiple quantum well systems,[25] the physical properties of GaN/AIN constant total effective radius multi-well quantum rings under well number variation effects,[26] and the optical properties of two-electron[27] and impurity including[28] GaN/AlN constant total effective radius multi-shell quantum rings and dots. And we also have studied the miniband formation scenario in GaN/AlN constant total effective length multiple quantum dots.[29]
It should be recognized that the above-mentioned studies of the Shannon entropy focus on the soluble quantum systems,[2–13] which means that the solutions of those quantum systems are analytical and consequently the wave function in momentum space can be calculated by the Fourier transform. In the present work, we study the quantum information entropies for rectangular multiple quantum well systems with a constant total length, in which the solution cannot be obtained analytically. We will obtain the position and momentum information entropy densities and the Shannon entropies for some constant total effective length multiple quantum wells through a numerical approach. We discuss the effects of the numbers of wells (1, 2, 3, 4, 15, and 22) and confined potential depth on the entropies and entropy densities for the ground state and the 1st, 2nd, 6th, and 50th states.
2. FormalismWe start with a 1D time-independent Schrödinger equation
where the geometrical confining potential
V(
r) is defined as,
[21–25]Here,
Vconf is a constant relative conduction band offset and the notation
i represents the
i-th well or barrier in the studied system. In this work, we assume that the well width and the barrier width are equal. The energies
En and the normalized wave functions
can be calculated through a numerical discretization technique.
[30]The position and momentum space information entropies Sx and Sp are related to the Shannon information entropy densities
and
defined as[8,9,11]
The position and momentum space information Shannon entropies Sx and Sp are written respectively as
where
ψ(
x) is the normalized wave function in position space and can be obtained through the numerical solution to Eq. (
1). The
φ(
p) is the normalized wave function in momentum space and can be calculated by the Fourier transform as
In this work, however, we calculate it by using the fast Fourier transform (FFT) method. Note that because of the logarithmic factors in the integrals, Shannon entropies (
4) become rather difficult to solve analytically even for quantum systems. As a result, we have to perform the integrals numerically for the present case. The entropic uncertainty relation obtained by Beckner, BBM, is given by
where
D denotes the spatial dimension and here we take
D = 1.
3. Results and discussionWe investigate the constant total effective length multiple quantum well systems with equally spaced quantum wells and barriers. If the total system length in our system is 60 nm, then the width of each well and barrier in a double quantum well system with constant 60 nm in total length is given by 60/5=12nm, in which there are two wells and three barriers, which are equally spaced between each other.
At the first step, in Fig. 1 we plot some normalized wave functions as a function of position x for a four-well system. This figure clearly shows that the curves are consistent with the node theorem of wave functions, i.e., the 3rd excited state has three nodes. Then, in order to study the entropy distributions for the quantum wells and barriers, we plot the position and momentum space entropy densities for different circumstances. In Fig. 2 we present the variations of the position and momentum entropy densities with position x for a single quantum well. The total system length is assumed to be 60 nm. The blue solid line, green dashed line, black dotted line, and purple dash–dotted line represent the position and momentum entropy densities for the normalized ground state, 1st, 2nd, and 6th excited states, respectively. Panels (a)–(f) illustrate different confined potential depths. Note that in all systems the position entropy densities for the ground state, 1st, and 2nd excited states are sensible only inside the quantum well. However, for the systems each with a small depth of the quantum well, e.g. panel (a), the maximum position entropy densities occur inside the barriers (see the purple dash-dotted line in panel (a)). Also, notice that by increasing the well depth (see panels (c) and (e)), the position entropy density of the 6th excited state in the barrier decreases and its value in the quantum well increases. Comparing the position entropy densities with the corresponding momentum entropy densities, it is found that for the smaller FWHM of the position entropy density, the FWHM of the momentum entropy density becomes larger and vice versa. To see this clearly, for the position entropy density of the ground state, we compare the solid blue lines in panels (a) and (b) and those in panels (e) and (f). This fact is also true for the position and momentum entropy densities of other excited states. In the meantime, by increasing the potential depth the FWHM of the position entropy density decreases, but the FWHM of the momentum entropy density increases.
On the other hand, to observe the effects of the confined potential on the position and momentum entropy density of a high excited state for a single quantum well, we plot the variations of the position and momentum entropy densities with position x for a single quantum well in Fig. 3. The total system length is again assumed to be 60 nm. The black solid line shows the position and momentum entropy densities of the 50th excited state. figure 3(a)–3(f) show the cases of different confined potential depths. It is interesting to find that in Fig. 3(a), where the confined potential depth is small (20 meV), the position entropy density of the 50th state is distributed uniformly inside the total length of the system. However, by increasing the potential depth (see Figs. 3(c) and 3(e)), this uniform distribution shall be broken and gradually the position entropy density inside the well decreases. A comparison between the position and corresponding momentum entropy densities shows that if the position entropy density is distributed inside a wide position space interval, the corresponding momentum entropy density will be localized in a narrow interval of the momentum space. However, for an intermediate confined potential depth (see Figs. 3(c) and 3(d)), the momentum space entropy density has the widest interval of distribution. Finally, we see that by increasing the confined potential depth, the frequency of the position entropy density oscillation inside the quantum barriers decreases, while the frequency of the position entropy density oscillation inside the quantum wells increases.
Now, we increase the number of wells to 2 and obtain the results as shown in Figs. 4 and 5. In our double quantum well system we keep the 60 nm total length constant (60 nm). Unlike the scenario in Fig. 2, in this system the position entropy densities for all the ground state, 1st, 2nd, and other excited states are localized inside the quantum wells. For double well systems with small confined potential depth panel (Fig. 5(a)), the distributions of the position entropy densities inside the quantum wells are identical. Notice the different lines inside the two wells. However, by increasing the confined potential depth to 200 meV (Fig. 5(c)), these identical distributions between quantum wells are destroyed. In Fig. 5(c), the maximum position entropy density of the 1st excited state occurs in the left quantum well, maximum position entropy density of the 2nd excited state occurs in the right quantum well and maximum position entropy density of the 6th excited state occurs in the left well. This scenario is also true for Fig. 5(e) with larger confined potential depth (1000 meV). In the meantime, for the corresponding momentum information entropies we find that although a position entropy density may be asymmetric with respect to the origin (one quantum well may have larger position entropy density, e.g., the one represented with the black dotted line in the panel (c)), the corresponding momentum information entropy is symmetric with respect to the origin.
We now study the effects of the confined potential on the position and momentum entropy densities of the 50th excited state for the double quantum well system. Unlike the scenario in Fig. 3(a), the distributions of the position entropy of small confined potential depth (20 meV) of the 50th excited state for a double quantum well system in Fig. 5(a) are not completely uniform. However, the value of the position information entropy inside the quantum wells is smaller than inside the quantum barriers. But by increasing the confined potential depth up to an intermediate value (200 meV, Fig. 5(c)), we find that the value of the position entropy density is uniform inside the whole system except inside the central quantum barrier in which its value is larger. Our previously mentioned issue about the frequency of position entropy density oscillation for a single quantum well system is still true for the double quantum well system. Here, however, there is another fact. By increasing the number of wells (comparing Fig. 3(e) with Fig. 5(e)), the frequency of position entropy density oscillation decreases inside the barriers and increases inside the quantum wells. Finally, we note that in Figs. 5(a), 5(c), and 5(e) the amplitude of the position entropy density inside the central quantum barrier increases with confined potential depth increasing.
For more illustrations, we increase the number of wells up to 3 and 4. Then, we repeat the Figs. 2 and 3. The results are displayed in Figs. 6, 7, 8, and 9. Unlike the scenarios in Figs. 3 and 5, in Fig. 7, the maximum amplitude of the position entropy density is not related to the central quantum barrier. Here, there are two maxima due to the quantum barrier, which are localized at the two ends of the system. The situation for a 4-well system (see Fig. 9) is again different and two other quantum barriers have the maximum amplitudes. Roughly speaking, there is not a general statement about the barrier which has a maximum position entropy density. Each desired quantum system has to be analyzed individually. Figure 8(e) is also interesting. In this figure, the position entropy densities of different four states (ground state, the 1st, 2nd, and 6th excited states) each are localized inside a distinct quantum well.
In addition, we explore two extra systems with larger numbers of wells i.e., 15 and 22. The results are shown in Figs. 10, 11, 12, and 13, respectively. At this stage, the number of wells is so large that the over system exhibits a behavior like a single quantum well with a length of 60 nm. But, there are additional oscillations in the position entropy densities as indicated with the blue solid line in Figs. 10 and 12. This behavior is very similar to that of the ground state position entropy density (blue solid line) in Fig. 2(a), in which the position entropy density is in the Gaussian form inside the quantum well. To see this similarity more clearly, one may compare the position entropy densities associated with other excited states as shown with dashed, dotted, and dash-dotted lines in Figs. 2(a), 10(a), and 12(a). In the meantime, it is clear from Figs. 10(a), 10(c), 10(e), 12(a), 12(c), and 12(e) that by increasing the confined potential depth the value of the position entropy density inside the barriers vanishes. This is because the probability of the particle staying in the barriers with infinite height is infinitesimal. Finally, we note that because the position entropy density is widely distributed inside the total position effective length, the momentum entropy density is localized in a narrow interval of the momentum in the momentum space.
Finally, in order to study the behaviors of the position and momentum entropies and also to explore whether the BBM inequality is satisfied, we plot the variations of the position entropies with confined potential depth ‘Vconf’ as is shown in Fig. 14. Figure 14(b) is the same as Fig. 14(a) except the plot of momentum entropy. figure 14(c) also shows the sum of the momentum and position entropies as a function of the confined potential depth “Vconf”. As can be seen clearly from Figs. 14(a) and 14(b), as Vconf increases the position entropies decrease and at the same time the momentum entropies increase. However, the single well system has much smaller position and momentum entropies. To check the BBM inequality, we examine Fig. 14(c). From Figs. 14(a) and 14(b) we see that when one entropy increases the other entropy decreases. However, their sum stays above a lower bound with a value (1 + ln(π)) = 2.1447. Thus, the BBM inequality is shown to be satisfied.
4. ConclusionsIn this work we investigate the position and momentum information entropy densities in addition to position and momentum information entropies of constant total effective length multiple quantum well systems. We find that in single quantum wells, for all well depths the ground state, 1st and 2nd excited state, the position entropy densities are sensible only within the well, while for the 6th excited state, by increasing the well depth, the position entropy density in the barriers decreases and its value in the quantum well increases. For the small FWHM of the position entropy density, we have increased the FWHM of the momentum entropy density and vice versa. For the single quantum well, by increasing the potential depth, the 50th state position entropy density within the well gradually decreases. Here, by increasing the potential depth, the FWHM of the position entropy density decreases and the FWHM of the momentum entropy density increases. For an intermediate confined potential depth of 200 meV, the momentum space entropy density has the widest interval of distribution. By increasing the potential depth, the frequency of the position entropy density oscillation inside the quantum barriers decreases and the frequency of the position entropy density oscillation inside the quantum wells increases. By increasing the number of wells, the frequency of position entropy density oscillation decreases within the barriers and it increases inside the quantum wells. In a four-well quantum system we are able to localize the ground state, 1st, 2nd, and 6th excited state position entropy densities inside a distinct quantum well. Finally, we find that the BBM inequality is satisfied. Thus, by modifying the number of wells and the potential depth we can describe the localization positions and values of the position and momentum information entropies.
