Many-Body Effects in a Composite Bosonic Josephson Junction

Abstract

In standard bosonic Josephson junctions (BJJs), particles tunnel between two single-well potentials linked by a finite barrier. The dynamics of standard BJJs have been extensively studied, both at the many-body and mean-field levels of theory. In the present work, we introduce the concept of a composite BJJ. In a composite BJJ, particles tunnel between two double-well potentials linked by a finite potential barrier between them. We focused on the many-body facets of quantum dynamics and investigate how the complex structure of the junction influences the tunneling. Employing the multiconfigurational time-dependent Hartree method for bosons, highly accurate many-boson wavefunctions were obtained, from which properties were computed. We analyzed the dynamics using the survival probability, the degree of fragmentation of the junction, and the fluctuations of the observables, and discuss how the many-boson tunneling behaved, and how it may be controlled, using the composite nature of the junction. A central result of this work relates to the degree of fragmentation of composite BJJs with different numbers of bosons. We provide strong evidence that a universal degree of fragmentation into multiple time-dependent modes takes place. Further applications are briefly discussed.

1. Introduction

Understanding the quantum tunneling of correlated systems is a fundamental problem of quantum mechanics since it is the main mechanism behind several phenomena observed in different disciplines of physics, ranging from condensed matter physics to quantum information and communication [1,2,3,4]. It is imperative to have a highly maneuverable system to explore the various features of many-body quantum tunneling and develop a deep understanding of the physics of the many-body quantum tunneling of correlated systems. In this context, it is to be noted that ultra-cold atomic systems and quantum gases have emerged as the preferred choices for the simulation of correlated quantum many-body systems.

In particular, trapped ultra-cold bosons in a symmetric double-well potential called the bosonic Josephson junction (BJJ) [5,6,7] provide a prototype for the Josephson effect originally predicted for the tunneling of a Cooper pair between two weakly linked superconductors [8]. Therefore, a BJJ provides deep insight into the tunneling dynamics of correlated quantum many-body systems. Naturally, it was the focus of a large amount of theoretical and experimental research [9,10,11,12,13,14,15]. The mean-field theory and the two-mode Bose–Hubbard (BH) model are commonly used for theoretically studying BJJ. While several important features of BJJ dynamics, such as the collapse and revival of the density oscillations [16] and the fragmentation [17,18,19], cannot be captured by the mean-field level of theory, the BH model can at best provide a qualitative description for such features and ultimately totally fails to capture the many-body features of the many-particle variances. This is due to not taking into account the role of the higher energy band. Therefore, to accurately describe the BJJ dynamics in particular and the dynamics of a correlated many-body system in general, at least when the higher energy bands participate, solving the many-body Schrödinger equation is necessary. In this context, our group developed a numerically exact many-body method called the multiconfigurational time-dependent Hartree method for bosons (MCTDHB), which takes into account all the participating energy bands.

In recent times, the MCTDHB has been thoroughly utilized in studying the BJJ dynamics in different situations, such as in 1D and 2D with repulsive and attractive interactions [20], with contact and finite-range interactions [6,21,22], and in the presence of an asymmetry in the trap [23,24]. Studying fragmentation and the uncertainty product of the many-particle position and momentum operators by the MCTDHB [25] has only reinforced the importance of using a full many-body model.

The many-particle uncertainty product of the BEC in a double well was shown to grow with time t as $t^2$ (up to the leading order in t). It was further shown that, contrary to the BH dimer, the full many-body dynamics of a BJJ for the repulsive and attractive interactions are not equivalent [20]. Also, the fragmentation of a BEC in both the symmetric and asymmetric double-well trap was shown to exhibit an indifference to the particle number N as long as the interaction parameter $\Lambda ={\lambda }_0(N-1)$ (with ${\lambda }_0$ being the strength of the interaction) remains unchanged. Moreover, resonantly enhanced tunneling, which was experimentally observed, was also demonstrated for an asymmetric BJJ with the MCTDHB [23]. On a different note, the Josephson effects were investigated in various complex systems, such as a two-component BEC [26], spinor condensates [27], polariton condensates [28], a fermionic superfluid [29], and a spin–orbit-coupled BEC [15].

Although various intriguing many-body effects were already observed in one- dimensional ultra-cold bosonic ensembles, in all these studies, only the lowest band had a significant role in the investigated parameter regime. To explore the role of higher energy bands in the many-body dynamics in one spatial dimension, in this work, we investigated the many-body dynamics of an interacting bosonic system in a composite double well, which we call a composite bosonic Josephson junction (CBJJ). In a CBJJ, the system tunnels between two double-well potentials are connected via a barrier of finite height, which may be used to model Josephson heat oscillations [30]. The complex geometry of the system makes the many-body tunneling richer with the involvement of the higher energy bands. The tunneling dynamics of a BEC in multiwell potentials are also important due to its potential applications in atomtronics-enabled quantum technologies [31,32,33], as well as for providing a prototype for nonintegrable systems [34,35,36]. The dynamics of a BEC in a four-well [37,38,39] trap was already attempted theoretically. The trapping potential with double wells [11,40] and triple wells [41], as well as an optical lattice [42] (which may be conceived as a collection of shallow wells), are routinely realized in experiments. By the same token, it should be possible to experimentally realize composite double-well traps with the available technology. In particular, a quadrupole well system can be realized either by using four closely neighboring microtrap potentials or with a double-well trap in which two internal states are coupled by a Rabi laser [43,44].

To highlight the extent of the participation of the higher energy bands, we analyzed the variation in the survival probability, occupation numbers, and fluctuations as a function of time. Already for a BJJ, the survival probability is directly related to the experimentally observed population imbalance and also theoretically studied by a mean-field method. However, experimentally observed features of decay in the oscillations of population imbalance cannot be described by the mean-field method and requires a many-body theory, such as the MCTDHB [23]. It is obvious that fragmentation and depletion defined through the occupation numbers in higher orbitals can only be studied by a many-body method. Similarly, it is not possible even to capture the qualitative features of the many-particle position and momentum variances, which are measures of fluctuations in the system [25]. Therefore, all the quantities considered here have already shown distinct many-body features for a BJJ and are expected to show more prominent many-body signatures for a CBJJ, where the higher orbitals are expected to have a greater role due to the complexity of the geometry of the system.

The organization of this paper is as follows: Section 2 introduces the system studied here and gives an outline of the methodology used. We discuss our findings in Section 3. We summarize our main findings and draw our conclusions in Section 4. Further details of the methodology, as well as the numerical convergence of our results, are discussed in the Appendix A.

2. Formalism

To explore the role of the higher energy bands in the many-body dynamics of correlated systems, here we considered the post-quench many-body tunneling dynamics of a system of interacting ultra-cold bosons trapped in a composite double well (CBJJ) given by

$$V_{Trap}\left(x\right)=V_T\left(x\right)+V_{0}exp[-a{(x+2)}^2]+V_{0}exp[-a{(x-2)}^2]$$

(1)

where

$$V_T\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{(x+2)}^2,& x<-{\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{3}{2}}(1-x^2),& \left|x\right|\le {\displaystyle \frac{1}{2}}\\ {\displaystyle \frac{1}{2}}{(x-2)}^2,& x>{\displaystyle \frac{1}{2}}\end{array}\right..$$

(2)

Note that the trapping potential Equation (1) reduces to a symmetric double well (BJJ) for $V_0=0$ and gives localization in the four-well trap for a very large $V_0$. In this work, we varied $V_0$ from $V_0=0$ to a sufficiently large value $V_0=10$, which was enough to reveal the growth of many-body signatures in the dynamics. The typical shape of the composite double well is shown in Figure 1. Initially, the system was prepared in the ground state of

$$V_{Trap}\left(x\right)={\displaystyle \frac{1}{2}}{(x+2)}^2+V_{0}exp[-a{(x+2)}^2]$$

(3)

and at $t=0$, the system was quenched by suddenly changing the trap $V_{Trap}\left(x\right)$ from Equation (3) to Equation (1). We then simulated the post-quench out-of-equilibrium dynamics by solving the time-dependent many-body Schrödinger equation using the MCTDHB method [45,46]:

$$\begin{array}{c}\widehat{H}\Psi =i{\displaystyle \frac{\partial \Psi }{\partial t}},\end{array}$$

(4)

$$\begin{array}{c}\widehat{H}(x_1,x_2,\dots ,x_N)=\sum _{j=1}^N\widehat{h}\left(x_j\right)+\sum _{k>j=1}^N\widehat{W}(x_j-x_k).\end{array}$$

Here, $x_j$ is the coordinate of the j-th boson, $\widehat{h}\left(x\right)=\widehat{T}\left(x\right)+{\widehat{V}}_{Trap}\left(x\right)$ is the one-body Hamiltonian containing kinetic energy $T\left(x\right)$ and trapping potential $V_{Trap}\left(x\right)$ terms, and the pairwise interaction between the j-th and k-th bosons is given by $W(x_j-x_k)={\lambda }_0\delta (x_j-x_i)$, with ${\lambda }_0$ being the interaction strength. Dimensionless units are employed throughout this work by scaling the Hamiltonian by ${\displaystyle \frac{{\hslash }^2}{m{L}^2}}$, where L is the length of the CBJJ and m is the mass of the boson. We adopted natural units, where $\hslash =m=1$.

The MCTDHB has already been extensively used in the literature [5,6,7,21,22,23,47,48,49,50,51,52,53,54,55,56,57,58]. A detailed discussion on the MCTDHB can be found in [46,57]. In this method, the ansatz is taken as the superposition of all possible $\left(\begin{array}{c}N+M-1\\ N\end{array}\right)$ configurations, obtained by distributing N bosons in M time-dependent single-particle orbitals ${\varphi }_k(x,t)$, i.e.,

$$\left|\Psi \left(t\right)\right.〉=\sum _{\overrightarrow{n}}{C}_{\overrightarrow{n}}\left(t\right)\left|\overrightarrow{n};t\right.〉,$$

(5)

where the occupations $\overrightarrow{n}=(n_1,n_2,\cdots ,n_M)$ preserve the total number of bosons N. Although M should be infinitely large for an exact calculation, one needs to truncate the series Equation (5) at a finite M in all practical numerical computations. In actual calculations, we keep on increasing M until we reach the convergence for M, thereby obtaining a numerically exact result. Here, we would like to point out that the flexibility of incorporating as high an M as required is an advantage of the MCTDHB over other popular many-body methods, such as the Bose–Hubbard model. Accordingly, we truncated Equation (5) at $M=4$ for our present work on the CBJJ [see Appendix A for the convergence with respect to M], whereas $M=2$ is enough for the BJJ [6,23]. This may give the impression that the number of orbitals M required for numerically exact results is equal to the number of wells present in the trapping potential. However, there is no such direct correlation between the two. The number of orbitals M required for achieving numerical convergence, and hence a numerically exact description of the system, depends on the degree of fragmentation of the condensate. In the quench dynamics, the system develops fragmentation with time depending on various factors, such as the nature of the interaction, its strength, and the trap geometry. However, for a weakly interacting system, it is routinely observed that the number of orbitals M required for convergence is equal to the number of wells in the trapping potential.

Without going into further details, here we just mention the working equations of the MCTDHB for determining the time-dependent coefficients $\left\{C_{\overrightarrow{n}}\left(t\right)\right\}$ and the time-dependent orbitals $\left\{{\varphi }_k(x,t)\right\}$:

$$\begin{array}{c}i\left|{\dot{\varphi }}_j\right.〉=\widehat{\mathbf{P}}\left[\widehat{h}\left|{\varphi }_j\right.〉+\sum _{k,s,q,l=1}^M{\left\{\rho \left(t\right)\right\}}_{jk}^{-1}{\rho }_{ksql}{\widehat{W}}_{sl}\left|{\varphi }_q\right.〉\right];\hfill \\ \widehat{\mathbf{P}}=1-\sum _{j^{\prime }=1}^M\left|{\varphi }_{j^{\prime }}\rangle \langle {\varphi }_{j^{\prime }}\right|\hfill \\ \mathbf{H}\left(t\right)C\left(t\right)=i{\displaystyle \frac{\partial C\left(t\right)}{\partial t}}.\hfill \end{array}$$

(6)

where $\rho \left(t\right)$ is the reduced one-body density matrix [see Equation (7) below], ${\rho }_{ksql}$ are the elements of the two-body reduced density matrix [see Equation (8)] [59,60], and $\mathbf{H}\left(t\right)$ is the Hamiltonian matrix $H_{\overrightarrow{n}{\overrightarrow{n}}^{\prime }}\left(t\right)=〈\overrightarrow{n};t\left|\widehat{H}\right|{\overrightarrow{n}}^{\prime };t〉$. Note that here, the working Equations (6) are expressed in terms of the reduced one-body and two-body reduced density matrices to produce a compact look. Given the normalized many-body wavefunction $\Psi \left(t\right)$, the reduced one-body density matrix is given as

$$\begin{array}{ccc}\hfill {\rho }^{\left(1\right)}\left(x_1\right|x_1^{\prime };t)& =& N\int d{x}_2\dots d{x}_N{\Psi }^{*}(x_1^{\prime },x_2,\dots ,x_N;t)\hfill \\ & & \times \Psi (x_1,x_2,\dots ,x_N;t)\hfill \\ & =& \sum _{j=1}^M{n}_j\left(t\right){\varphi }_j^{*NO}(x_1^{\prime },t){\varphi }_j^{NO}(x_1,t).\hfill \end{array}$$

(7)

Here, ${\varphi }_j^{NO}(x_1,t)$ are the time-dependent natural orbitals and $n_j\left(t\right)$ are the time-dependent natural occupation numbers. The natural occupations $n_j\left(t\right)$ are used to characterize the (time-varying) degree of the condensation in a system of interacting bosons [61] and satisfy ${\sum }_{j=1}^M{n}_j=N$. If only one macroscopic eigenvalue $n_1\left(t\right)\approx \mathcal{O}\left(N\right)$ exists, the system is condensed [61], whereas if there is more than one macroscopic eigenvalue, the BEC is said to be fragmented [62,63,64,65]. The diagonal of the ${\rho }^{\left(1\right)}\left(x_1\right|x_1^{\prime };t)$ gives the density of the system $\rho (x;t)\equiv {\rho }^{\left(1\right)}\left(x\right|x^{\prime }=x;t)$.

Similarly, the two-body density can be expressed as

$$\begin{array}{c}\hfill {\rho }^{\left(2\right)}(x_1,x_2|x_1^{\prime },x_2^{\prime };t)=\\ \hfill N(N-1)\int d{x}_3\dots d{x}_N{\Psi }^{*}(x_1^{\prime },x_2^{\prime },x_3,\dots ,x_N;t)\\ \hfill \times \Psi (x_1,x_2,x_3,\dots ,x_N;t).\end{array}$$

(8)

Therefore, the matrix elements of the two-body reduced density matrix are given by ${\rho }_{ksql}=〈\Psi \left|b_k^{†}{b}_s^{†}{b}_q{b}_l\right|\Psi 〉$, where $b_k$ and $b_k^{†}$ are the bosonic annihilation and creation operators, respectively.

3. Results and Discusson

In this section, we present our findings of this study on the many-body dynamics of an interacting bosonic gas in a composite double well, as defined in Equation (1). Our main purpose of this study was to explore the growing role of higher energy bands as the system’s complexity increased. Accordingly, as already mentioned above, we varied $V_0$ from $V_0=0$ (for which the system was two-fold fragmented for the $\Lambda =0.1$ considered here) to $V_0=10$, when the system became four-fold fragmented (see below), and highlighted the characteristics exclusively due to the participation of the higher energy bands. Additionally, we provide strong numerical evidence that the universality of the degrees of fragmentation, earlier reported for two-fold fragmentation [6,23], still took place for more than $M=2$ time-dependent modes.

3.1. Many-Body Dynamics

As mentioned earlier, initially we prepared the system in the ground state of the left well [Equation (3)] of the CBJJ. With increasing values of $V_0$, the ground states acquired increasingly complicated shapes, which made the dynamics more complex. To highlight this increasingly complex nature of the dynamics and the necessity of considering higher orbitals beyond $M=2$ to capture the same faithfully, here we present our findings of the temporal evolution of the survival probability, occupation numbers, and the many-particle position and momentum variances for various $V_0$ values while keeping $a=10$ fixed. Note that $V_0$ mainly controlled the height of the barrier in each well, whereas a primarily controlled the intra-well barrier width; see Figure 1. Therefore, the impact of $V_0$ on the fragmentation should be more prominent than a. Accordingly, we kept a fixed to some intermediate value $a=10$, where the intra-well barrier was neither very narrow nor very wide. The qualitative physics described in this work did not change with a.

Furthermore, it is desired to use the same reference time scale while comparing the many-body tunneling dynamics in the composite double well with various barrier heights $V_0$. While the Rabi period $t_{Rabi}$ provides the natural time scale of the system, it varies with the trap geometry. However, $t_{Rabi}$ still would be of the same order of magnitude for all cases. The Rabi period for the BJJ, $t_{Rabi}=132.498$ [6], suggests that the time period of the inter-band oscillations in the CBJJ should be of the order of a few 100. Accordingly, we scaled the time t by $t_0=100$ while comparing the dynamics for various trap geometry corresponding to different $V_0$. This time t in the dimensionless unit can be converted into a standard time unit (such as second) by multiplying t with the inverse of the scaling factor $m{L}^2/\hslash$ [see Section 2]. Considering $L=1$ μm and the mass of ⁸⁷Rb is $m=1.4431\times {10}^{-25}$ kg, this conversion factor turns out to be $m{L}^2/\hslash =1.37$ milliseconds.

3.1.1. Survival Probability

The survival probability $p_L\left(t\right)$, which is a measure of the density oscillations between the composite double wells, can be defined as

$$p_L\left(t\right)={\int }_{-\infty }^0\mathrm{d}x{\displaystyle \frac{\rho (x;t)}{N}},$$

(9)

where $\rho (x;t)$ is the density in the left composite well. It is closely connected to the population imbalance, which is routinely studied in experiments. In Figure 2, we plot the survival probability $p_L\left(t\right)$ in the left well for different values of $V_0$. For $V_0=0$, we obtained the standard BJJ that comprised of two symmetrical wells connected through a barrier, and accordingly, $p_L\left(t\right)$ also exhibited the usual collapse of density oscillations with time. As $V_0$ was increased from zero, a hump appeared in either well of the BJJ and it turned into the composite BJJ. We observed that an irregularity appeared in the oscillations of the $p_L\left(t\right)$ for $V_0\ne 0$. Furthermore, the irregularities were enhanced with the increase in $V_0$. With a further increase in $V_0$, we observed two distinct oscillations for $V_0=10$. The high-frequency oscillations indicate the tunneling between the two adjacent shallow wells. In contrast, the low-frequency oscillations, which appeared as the envelope to the high-frequency oscillations, could be attributed to the tunneling of the whole system between the two composite double wells. The emergence of two distinct frequencies in the tunneling dynamics implies atomic transitions between two distinct sets of energy levels, which means more than one energy band was involved. While the fast oscillations with a short time period corresponded to inter-band atomic transitions, the slower oscillations with a longer time period were due to the intra-band atomic transitions. To corroborate this, ideally one should calculate the many-body energy levels of the CBJJ. However, calculating even the low-lying many-body excited states of such systems is extremely challenging. While efforts to develop a linear response theory to the MCTDHB, called LR-MCTDHB [66], are being made to calculate the low-lying excitations, such computations are quite time-consuming, and achieving numerical convergence can be too challenging for complex systems, such as the CBJJ. However, at least for weak interactions, the appearance of two distinct oscillations can be understood from the non-interacting picture. At the lower part of the spectrum of a composite double well, the closely lying energy levels form a band, while in the upper part of the spectrum, the effect of the barriers is less pronounced and the spectrum is similar to that of a wide harmonic well. Therefore, there will be two different types of oscillations arising from atomic transitions between the energy levels of an energy band and between energy levels belonging to two different energy bands. In particular, for $V_0=10$, the intra-band time period for transitions between the energy levels of the lowest energy band was about $T_{intra-band}\sim 500$, while the intra-band transitions had a time period $T_{inter-band}$ of the order 10 for the transitions between the lowest band and the first higher band. This sharp difference in the time scale of the two oscillations is clearly visible in Figure 2. As $V_0$ was reduced, the gaps between the energy bands reduced, which decreased the difference in the time scale between $T_{inter-band}$ and $T_{intra-band}$. Therefore, the two oscillations in Figure 2 were not distinctly noticeable for smaller barrier heights $V_0$. The decay in the density oscillations is a purely many-body phenomenon and cannot be captured by mean-field theory [23]. The different decay rates of the density oscillations depending on the values of $V_0$ come from the initial shape of the ground orbital.

3.1.2. Occupation Numbers

Fragmentation of the BEC provides information about the relative macroscopic populations in higher orbitals. The connection between the damping of the density oscillations and the fragmentation for a BJJ is already well demonstrated in Ref. [5]. Naturally, our findings for the $p_L\left(t\right)$ mentioned above demand a study of the fragmentation as a function of time for various heights of the hump $V_0$. This would help us to correlate the changes in the density oscillations with the participation of higher energy bands.

The development of fragmentation in the system is characterized by the time evolution of the natural occupations per particle ${\displaystyle \frac{n_i}{N}}$. The system is said to be condensed if ${\displaystyle \frac{n_1}{N}}\sim \mathcal{O}\left(1\right)$. On the other hand, the system is fragmented if ${\displaystyle \frac{n_i}{N}}\sim \mathcal{O}\left(1\right)$ for more than one orbital.

We present our results in Figure 3. It was found that at $t=0$, for all the values of $V_0$, only the ground orbital was occupied. This signified that although the shape of the ground state was modified due to different values of $V_0$, the coherency of it was unchanged. At $t>0$, we found that the system gradually became fragmented for all $V_0$. Meanwhile, for the BJJ ($V_0=0$), the system became two-fold fragmented, and the degree of fragmentation increased for $V_0\ne 0$. For the BJJ ($V_0=0$), after a sufficiently long time, the system became two-fold fragmented with ${\displaystyle \frac{n_1}{N}}\approx 60\%$ and ${\displaystyle \frac{n_2}{N}}\approx 40\%$, while the occupations in all higher orbitals were negligibly small. Therefore, the BJJ, with the interaction strength considered, could be accurately described with only $M=2$ orbitals. However, the occupations in the third and fourth orbitals started increasing with an increase in $V_0$ at the cost of occupations in the first and second orbitals. Meanwhile, for $V_0=2.5$, we saw ${\displaystyle \frac{n_3}{N}}\approx 5\%$, and it grew to over ${\displaystyle \frac{n_3}{N}}\approx 20\%$ for $V_0=10$. Similarly, ${\displaystyle \frac{n_4}{N}}$, which was very small for $V_0=2.5$, grew to ${\displaystyle \frac{n_4}{N}}\approx 5\%$ for $V_0=10$. Accordingly, the occupations in the first orbital reduced from ${\displaystyle \frac{n_1}{N}}\sim 60\%$ for $V_0=0$ to ${\displaystyle \frac{n_1}{N}}\sim 50\%$ for $V_0=2.5$, ${\displaystyle \frac{n_1}{N}}\sim 45\%$ for $V_0=5$, and ${\displaystyle \frac{n_1}{N}}\sim 37\%$ for $V_0=7.5$. With a further increase in $V_0$ to $V_0=10$, there was no significant variation in ${\displaystyle \frac{n_1}{N}}$. Similarly, ${\displaystyle \frac{n_2}{N}}$ first increased from ${\displaystyle \frac{n_2}{N}}\sim 40\%$ for $V_0=0$ to ${\displaystyle \frac{n_2}{N}}\sim 50\%$ for $V_0=2.5$, and then gradually decreased to ${\displaystyle \frac{n_2}{N}}\sim 45\%$ for $V_0=5$, ${\displaystyle \frac{n_2}{N}}\sim 35\%$ for $V_0=7.5$, and finally to ${\displaystyle \frac{n_2}{N}}\sim 32\%$ for $V_0=10$. So, initially, the second orbital primarily gained population at the cost of the first orbital, and both orbitals achieved similar occupations for $V_0=2.5$. With a further increase in $V_0$, particles were transferred to the third and fourth orbitals from the first two orbitals. So, for $V_0=5$, the occupations in the first and second orbitals became closer and, at the same time, occupations in the third and fourth orbitals also increased. While there was a further reduction in the occupations in both of the first two orbitals, more particles transferred from the second orbital, which resulted in reduced occupations in the second orbital. Finally, as $V_0$ increased to $V_0=10$ from $V_0=7.5$, there was a further redistribution of particles between the second, third, and fourth orbitals so that ${\displaystyle \frac{n_1}{N}}$ remained practically the same. Therefore, it is evident that there was a redistribution of the particles between the orbitals, and finally, for significantly large $V_0$, there was a more equitable distribution of the population. This indicates the greater role played by the higher orbitals in the tunneling dynamics of the CBJJ ($V_0\ne 0$).

Regarding this connection, it is instructive to note that the density oscillations died out in the BJJ as the system became correlated. However, the damping slowed down for the CBJJ, even though the CBJJ was more fragmented than the BJJ. This may be attributed to the delay in the growth of fragmentation in the CBJJ for smaller values of $V_0$. But for $V_0\ge 7.5$, the damping was slower, even though the correlation in the system grew faster than the BJJ. This shows the complicated relationship between the damping of density oscillations and the growth of the correlation in the system in the case of the CBJJ.

3.1.3. Many-Particle Variances

Next, we considered the many-particle position and momentum variances of the system, which is a measure of the quantum resolution of measurement of any observable. Although these cannot be measured easily, these are fundamental quantities due to their connection with the uncertainty principle. Contrary to fragmentation, which reflects only the relative occupations, many-body variances are dependent on the actual number of fragmented atoms. Therefore, these are expected to bear more prominent signatures of the actual occupations in higher orbitals, and thereby, may throw more light on the role of higher orbitals in the dynamics.

One can calculate the variance per particle ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{A}}^2\left(t\right)$ for any many-body operator $\widehat{A}={\sum }_{j=1}^N\widehat{a}\left(x_j\right)$, which is obtained from the single-particle Hermitian operator $\widehat{a}\left(x_j\right)$ by

$$\begin{array}{c}{\displaystyle \frac{1}{N}}{\Delta }_{\widehat{A}}^2\left(t\right)={\displaystyle \frac{1}{N}}\left[\langle \Psi \left(t\right)|{\widehat{A}}^2|\Psi \left(t\right)\rangle -\langle \Psi \left(t\right)|\widehat{A}{|\Psi \left(t\right)\rangle }^2\right]\end{array}$$

(10)

$$\begin{array}{c}\equiv {\Delta }_{\widehat{a},density}^2\left(t\right)+{\Delta }_{\widehat{a},MB}^2\left(t\right),\\ {\Delta }_{\widehat{a},density}^2\left(t\right)=\int dx{\displaystyle \frac{\rho (x;t)}{N}}{a}^2\left(x\right)-{\left[\int dx{\displaystyle \frac{\rho (x;t)}{N}}a\left(x\right)\right]}^2,\\ {\Delta }_{\widehat{a},MB}^2\left(t\right)={\displaystyle \frac{{\rho }_{1111}\left(t\right)}{N}}{\left[\int dx|{\varphi }_1^{NO}{(x;t)|}^{2}a\left(x\right)\right]}^2\\ -(N-1){\left[\int dx{\displaystyle \frac{\rho (x;t)}{N}}a\left(x\right)\right]}^2\\ +\sum _{jpkq\ne 1111}{\displaystyle \frac{{\rho }_{jpkq}\left(t\right)}{N}}\left[\int dx{\varphi }_j^{*NO}(x;t){\varphi }_k^{NO}(x;t)a\left(x\right)\right]\\ \times \left[\int dx{\varphi }_p^{*NO}(x;t){\varphi }_q^{NO}(x;t)a\left(x\right)\right].\end{array}$$

Accordingly, in Figure 4a, we present our results for the many-particle position variance, while Figure 4b depicts the results for the many-particle momentum variance. For $V_0=0$, we again obtained the oscillatory growth of ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$ with time followed by the saturation as the density oscillation collapsed. For $V_0\ne 0$, we obtained overall behavior that was quite similar to the BJJ, but the main difference lay in the details. For a small value of $V_0=2.5$, the growth rate of ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$ became slower, which was consistent with the fragmentation dynamics. However, the fluctuations in the saturation value were very high for such a small value of $V_0$. This was expected because all three barriers were of similar heights [see Figure 1], and therefore, the system was delocalized over all four wells. The fluctuations were expected to be at a maximum when all three barriers were of the same height, which should occur for $V_0=1.5$. With a further increase in $V_0$, the growth rate of ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$ increased but still remained slower than the BJJ. Also, the fluctuations reduced drastically for higher values of $V_0$.

The fluctuations in ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$ about the saturation value were the manifestation of the breathing oscillations within the two wells of each of the compound wells. For a small intra-well barrier height $V_0$, large breathing oscillations between the two wells of each compound well led to large fluctuations in ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$. With increasing $V_0$, the breathing oscillations became damped, which led to smaller fluctuations in ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$ for higher values of $V_0$.

The dynamics of atoms will have a more prominent impact on the many-particle momentum variance. Naturally, we again observed fluctuations in ${\displaystyle \frac{1}{N}}{\Delta }_{{\widehat{P}}_X}^2$ around a mean value for the BJJ ($V_0=0$). For a small $V_0$, the barrier heights of both the inter-well and intra-well barriers were again of similar heights, which resulted in large intra-well breathing oscillations, in addition to inter-well tunneling oscillations. Accordingly, we observed very large fluctuations in ${\displaystyle \frac{1}{N}}{\Delta }_{{\widehat{P}}_X}^2$. With a further increase in $V_0$, the damping in the intra-well breathing oscillations resulted in the gradual reduction in the fluctuations of ${\displaystyle \frac{1}{N}}{\Delta }_{{\widehat{P}}_X}^2$. For such values of $V_0$, ${\displaystyle \frac{1}{N}}{\Delta }_{{\widehat{P}}_X}^2$ fluctuated around a mean value, which initially increased slightly with time before it stabilized to a saturation value. This saturation value also depended on $V_0$ and first increased before it gradually decreased with increasing $V_0$.

3.2. Universality of Fragmentation

Since the participation of the higher energy bands significantly affected the growth of the fragmentation in the system, it is natural to ask about its impact on the universality of the degree of fragmentation with respect to N. It is a novel phenomenon predicted for the BJJ dynamics when only a single band plays the dominant role in the dynamics. Therefore, next, we examined the degree of fragmentation of the CBJJ when the second band (in addition to the lowest band) also played a significant role in the dynamics. We present our results for different $V_0$ values in Figure 5. The panels in the left column show ${\displaystyle \frac{n_i}{N}}$ for the first two orbitals, while those on the right column exhibit the same for the third and fourth orbitals.

For the symmetric double well ($V_0=0$), we obtained the usual two-fold fragmentation with the ${\displaystyle \frac{n_1}{N}}\sim 60\%$ and ${\displaystyle \frac{n_2}{N}}\sim 40\%$ for all N, keeping $\Lambda$ fixed at $0.1$. Although there were no macroscopic occupations in the higher orbitals for the BJJ, we still observed the depletion of the same order in the third (${\displaystyle \frac{n_3}{N}}\sim {10}^{-3}$) and fourth (${\displaystyle \frac{n_4}{N}}\sim {10}^{-4}$) orbitals for all N.

For a finite $V_0$, the system was four-fold fragmented. However, we still found that over time, ${\displaystyle \frac{n_i}{N}}$ for all four orbitals reached a fixed value for various N corresponding to the same $\Lambda$ for all $V_0$ studied here. However, as discussed above, the final values of ${\displaystyle \frac{n_i}{N}}$ changed with an increase in $V_0$, and the particles of the system became more evenly distributed between the four orbitals with reduced occupations in the lower orbitals and increased occupations in the higher orbitals. Therefore, even though the details of the fragmentation varied for various trap geometries, the systems exhibited the universality of fragmentation for all four orbitals with respect to N corresponding to a fixed $\Lambda =0.1$, irrespective of the value of $V_0$. However, we noticed that the pace of the appearance of the universality of fragmentation depended on $V_0$. As we switched on the internal barrier $V_0$, for a small value of $V_0$, it took longer to observe the universality of fragmentation than the symmetric double well $(V_0=0)$. Then, as $V_0$ was increased, the universality appeared faster. Furthermore, for a fixed value of $V_0$, the system fragmented later for larger N. This was expected, as the many-body effects were reduced with increased N while keeping $\Lambda$ fixed.

In our earlier study [23], we demonstrated that the universality of fragmentation is preserved in an asymmetric BJJ. So, our present study, in combination with the earlier findings, indicates that the universality of fragmentation with respect to N is neither limited to systems where only two orbitals play a dominant role nor is susceptible to the geometry of the double-well potential. Thus, it is indeed a robust many-body effect.

4. Conclusions

In this work, we studied the many-body tunneling dynamics of an interacting Bose gas in a composite double well. The composite double well was formed by merging two deformed harmonic wells that had a hump at their center. The complex geometry of the system manifested in the rich many-body tunneling dynamics that involved higher energy bands. We examined the complexity of the dynamics by analyzing the time evolution of survival probability, occupation numbers, and many-particle variances. All these quantities were previously used to analyze the many-body features of BJJ dynamics.

We found an irregularity in the density oscillations for a very small hump and then two distinct oscillations for a bigger hump. While the fast high-frequency oscillation was due to the tunneling through the hump within the same well, the low-frequency tunneling was due to the tunneling between the two wells. The emergence of two frequencies in the tunneling dynamics indicates the transitions between two different sets of energy levels, and thereby, the involvement of more than one energy band.

Indeed, we observed (see Appendix A) that at least four orbitals were required for the accurate description of the many-body dynamics of the system, contrary to the standard BJJ dynamics, where two orbitals are sufficient.

The correlation between the damping of the density oscillation and the growth of the fragmentation is already well established. We observed that the same correlations still existed, even for a composite BJJ. However, the degree of fragmentation, as well as the growth of the fragmentation, significantly differed in the two cases. While the BJJ became two-fold fragmented, the CBJJ was four-fold fragmented. On the other hand, the growth of fragmentation became slower in the CBJJ, except for a very high hump $V_0$. Accordingly, the damping of the density oscillations became slower in the CBJJ. Furthermore, the occupations of different orbitals ${\displaystyle \frac{n_i}{N}}$ clearly showed the more prominent participation of the third and fourth orbitals in the case of the CBJJ. All these observations were further strengthened by this study of ${\displaystyle \frac{1}{N}}{\Delta }_{\widehat{X}}^2$ and ${\displaystyle \frac{1}{N}}{\Delta }_{{\widehat{P}}_X}^2$, which exhibited higher fluctuations but slower growth.

Another interesting finding of this study was that the universality of the fragmentation with respect to the occupation in an orbital corresponding to a fixed interaction strength $\Lambda$ was not limited to systems where only two orbitals played dominant roles. Furthermore, not only universality was observed for orbitals with macroscopic occupations but even orbitals with microscopic occupations also had the same order of occupations with respect to different N values but with a fixed $\Lambda$. These features were observed for all $V_0$, which reaffirmed that the universality of fragmentation is a global many-body phenomenon.

Thus, we saw that depending on the system, the role of a different number of orbitals in the dynamics became dominant. Therefore, it is necessary to have a many-body method capable of taking into account all relevant orbitals for an accurate description of the complex many-body dynamics. The MCTDHB method is an important and promising many-body technique in this direction. The present work can inspire us to investigate the tunneling dynamics of even more intriguing ground states of bosonic and fermionic ensembles.