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Abstract
The study of transition probabilities in few-body systems and their long-range interactions using the Landau-Zener method could be helpful for solving a wide range of problems in fields of quantum simulations, Rydberg blockade, quantum gates, and dipole transition in Rydberg atoms. Here, the transition probabilities for a four-particle system in a square-shaped lattice are studied by solving the many-body Landau-Zener Hamiltonian in the next-nearest-neighbor approximation. It is observed that the complete transition occurs for the anti-ferromagnetic coupling while it is limited to a constant probability for the ferromagnetic coupling. These probabilities suppress when the magnetic field's energy rate is increased. We demonstrate that in the presence of the next-nearest-neighbor's interactions, the final probabilities’ behavior is different. Considering the next-nearest-neighbor's interactions, it is possible to achieve the complete transition in imperfect anti-ferromagnetic regions. At the same time, the complete transition does not occur in perfect anti-ferromagnetic regions for certain values of the sweeping rate. These results emphasize that it is necessary to consider the next-nearest-neighbors’ interactions in a many-body system. Furthermore, the implementation of this Hamiltonian in the dipole-dipole and van der Waals interaction reveals that a steeper decrease for dipole-dipole interaction.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A wide range of problems in quantum physics is originated from many-body systems and long-range interactions between their particles. The statistical and dynamical properties of these systems with long-range interactions have been widely investigated [1–3]. Meanwhile, the study of long-range and/or strong interactions in a spin system is significant due to their applications in quantum simulation, controllable quantum systems, and quantum gates [4–8]. In this regard, systems of neutral atoms and their interactions in the Rydberg blockade and anti-blockade regime could be useful to the implementation of quantum simulation [4,5]. In such systems, the ability to control long-range interactions of photons, dipole-dipole interactions, and the strong interactions between Rydberg atoms are essential where the distance between the atoms is an influential parameter [5,8–10].
Several methods have been introduced to study population transfer in a spin system [11,12]. In this area, the Landau-Zener (LZ) model has been extensively studied [13–17]. This model was first introduced by Landau and Zener [18,19]. Majorana also independently introduced a similar model [20]. LZ model is based on a time-dependent Hamiltonian, giving the transition probabilities of a two-level system by sweeping a magnetic field [21].
In recent years, the study of interactions in many-body systems by the Landau-Zener model has attracted the attention of researchers [22]. In Ref. [23], Rydberg atoms and dipole-dipole interactions are investigated by LZ transition. In Ref. [24], the probability densities are calculated for three and four-particle systems by LZ model. Classification of solvable Hamiltonians for N-states LZ systems is presented in Ref. [25]. LZ dynamics for a many-body system are experimentally investigated in coupled single-dimensional Bose liquids by Chen et al. [21]. I Ref. [26] the population dynamics and the ratio between the interchain and intrachain coupling strengths are studied in a many-body LZ system. The effects of ferromagnetic and anti-ferromagnetic couplings on population transfer by the standard LZ solution investigated in Refs. [27–30]. In the mentioned researches, the coupling strength values are generally considered to be equal for interactions between nearest and next-nearest-neighbors.
Here, the population transfer is studied by the LZ model for a four-particle coupled system in the shape of a square lattice. The effects of the next-nearest-neighbors’ interactions as a long-range interaction, especially in van der Waals and dipole-dipole interactions, are also investigated in this paper. It has been shown that in particular values of the sweeping rate of the magnetic field's energy, the next-nearest-neighbor disturbs the anti-/ferromagnetic behaviors. Calculating transition probability values in different ferromagnetic and anti-ferromagnetic regions lead to a better understanding of the effect of the next-nearest-neighbors’ interactions.
2. Theory
Figure 1 illustrates the idea of the interactions between nearest and next-nearest-neighbors in a four-particle system, where particle 1 is in the nearest neighborhood with particles 2 and 4, while particle 3 is in its next-nearest-neighborhood. The values and ratio of J1 and J2 are different for various materials, although the interactions between next-nearest-neighbors are usually smaller than the interactions between the first nearest neighbors.
Fig. 1. Periodic boundary condition in a four-particle system, particles 2 and 4 are closest neighbors to particle 1 (connected via blue line), while particle 3 (on the diameter) is in the next-nearest-neighborhood to particle 1 (connected via red line).
The four-particle system studied here has a sixteen–dimensional Hilbert space. Any of these identical two-state particles could be either in the ground state ($|\uparrow \rangle $) or the excited state ($|\downarrow \rangle $). All sixteen possible states could be described as below:
Here, the single-particle probability is defined by summation over probabilities of all states in which a specific particle is in the ground state ($|\uparrow \rangle $) or excited state ($|\downarrow \rangle $).
Transition probabilities for a spin system with four particles are calculated by solving Schrodinger's equation. The Hamiltonian is calculated by the summation of free particles Hamiltonian in a four-particle system and interaction Hamiltonians. The Hamiltonian of these free four-particles is inferred from Single-Particle LZ (SPLZ) Hamiltonian [13,31]:
In this paper, the interactions between the two neighbors are represented by two separate Hamiltonians to illustrate the interactions between nearest-neighborhoods and next-nearest- neighborhoods. Therefore, the total Hamiltonian is represented as:
HFPLZ is the Four-Particle LZ (FPLZ) Hamiltonian, while ${H_{SPLZ}}$ is a $2 \times 2$ matrix, HFPLZ is a $16 \times 16$ matrix. Hnn-int and Hnnn-int represent nearest-neighborhood and next-nearest-neighborhood interaction Hamiltonians, respectively.
In this work, the calculations are done using dimensionless parameters. The capped parameters are defined as dimensionless parameters, where $\hat{g} = 1,\,\hat{J} = {J / {g,\,}}\hat{\alpha } = {{\alpha {\hbar ^2}} / {{g^2}}},\,\widehat t = t/\tau$, and $\tau = \hbar /g$. It is necessary to mention that the system is assumed to be in its initial condition at t=-∞.
3. Results and discussion
Final transition probabilities of excited states (FTPE) are defined as follows for further investigations. This parameter is defined as the average value of 10% of the final values of transition probabilities for excited states with respect to time, which the mathematical approach is given in (Eq. (4)) [13,14,32]:
where E.S(t) represents the magnitude of probability for excited states (E.S) as a function of normalized time, and T is the total time. Equation (4) represents a simple mathematical method to calculate the average value of transition probabilities of excited states in the studied interval. Presented simulations in this section demonstrated that the transition probabilities are stable for final values of transition probabilities versus time. Final probabilities are stable in the range of about 5 to 15%. Therefore, 10% of the final values of transition probabilities are considered. The average value helps to have a reasonable estimation of the values of transition probabilities [13,14,32].3.1 Without the next-nearest neighbors’ interaction
Figure 2 shows the FTPE versus normalized coupling strength and normalized magnetic field's energy rate when the next-nearest-neighbors’ interactions are not included (${\hat{J}_2}$=0). Nearest neighbors’ normalized coupling strength values change between -2 and 2, where positive values of coupling strength demonstrate a ferromagnetic system, and negative values define an anti-ferromagnetic system. Values of $\hat{\alpha }$ changes between 0.5 and 1.5. This figure illustrates that for the negative values of coupling strengths, the transition is complete, but for the positive coupling strengths, the maximum of FTPE is about 0.6. This result is comparable to the Ref. [29], where it is analytically demonstrated that the transition probability tends to the constant value of 50% in a two-particle ferromagnetic system, while a complete transition is achievable in an anti-ferromagnetic system [29,30,33,34]. Furthermore, the magnetic field's energy rate does not change the FTPE effectively, and FTPE values are mainly affected by coupling strength values in the studied range of $\hat{\alpha }$. This is in agreement with Ref. [28]. In this reference, it is shown that the population transfer is almost independent of the magnetic field's energy rate at a slow-sweep regime.
Fig. 2. FTPE verses $\hat{J}$ and $\hat{\alpha }$ for a four-particle system while next-nearest-neighbor interactions are neglected.
It is necessary to mention that the results of a four-particle system when the next-nearest neighbors’ interactions are excluded (${\hat{J}_2}$=0) are similar to a three-particle system. In Ref. [35], the FTPE and dynamic diagrams for this condition were studied in detail.
3.2 With the next-nearest-neighbor's interaction
The magnetic behavior of the system is mainly defined based on the sign of the first neighbors’ coupling strength values. Here, the sign of the next-nearest-neighbors could indicate if the system is either perfect anti-/ferromagnetic or not. Positive signs of coupling strength in neighborhoods (${\hat{J}_1} > 0$ and ) demonstrate perfect ferromagnetic behavior of the system while ${\hat{J}_1} < 0$ and ${\hat{J}_2} > 0$ identify the perfect anti-ferromagnetic system. In this work, imperfect ferromagnetic and imperfe ${\hat{J}_2} > 0$ct ${\hat{J}_2} < 0$ anti-ferromagnetic systems are respectively considered for coupling strengths ${\hat{J}_1} > 0,\,{\hat{J}_2} < 0$ and ${\hat{J}_1} < 0$,. It could be inferred that the magnetic order of a system (for both ferromagnetic and anti-ferromagnetic) is strengthened when the next-nearest-neighbor coupling strength is positive (${\hat{J}_2} > 0$), and it is weakened when the next-nearest-neighbor coupling strength is negative (${\hat{J}_2} < 0$).
Figure 3 shows the system’s transition probabilities versus time. In this figure, the normalized magnetic field's energy rate is assumed to be a constant value ($\hat{\alpha } = 1$), where ${\hat{J}_1}$ demonstrates normalized coupling strength for the nearest neighborhoods, and ${\hat{J}_2}$ represents the normalized coupling strength of the next-nearest-neighborhoods. The normalized time range is chosen between -40 and 40 to guarantee a stable transition. As mentioned before, the system is in its ground states at $\hat{t} = $-40, becoming stable while transferred to the excited state at $\hat{t} = $40. When the coupling strength values in the nearest neighborhood are negative (${\hat{J}_1} < 0$), a complete transition is observed [Figs. 3(a) and 3(b)]. As could be seen in Fig. 3(c), the transition probability is about 90%. In this figure, the coupling strength of the nearest neighborhood is positive (${\hat{J}_1} > 0$), and the value of coupling strength in the next-nearest-neighborhood is negative (${\hat{J}_2} < 0$). However, when both coupling strength values are positive, the transition probability is reduced to 0.5. It could be realized that the transition probabilities are dependent on the sign of coupling strength values in the small values of the magnetic field (slow-sweep regime). As a practical instance, the process of suppression in a single-molecule magnet by ferromagnetic interactions is studied in Ref. [33], which is extremely dependent on the sign of coupling strength. Figure 3 indicates that the next-nearest-neighbor's interactions can affect transition probabilities in the slow-sweep regime.
Fig. 3. Transition probabilities versus normalized time for $\hat{\alpha } = 1$ and (a)${\hat{J}_1} ={-} 1,{\hat{J}_2} ={-} 1$, (b) ${\hat{J}_1} ={-} 1,{\hat{J}_2} = 1$, (c) ${\hat{J}_1} = 1,{\hat{J}_2} ={-} 1$, (d) ${\hat{J}_1} = 1,{\hat{J}_2} = 1$.
Figure 4 shows the transition probabilities for $\hat{\alpha } = 7$ observing the effects of increasing the magnetic energy sweeping rate. The coupling strength values are considered for various conditions of the anti-ferromagnetic and ferromagnetic. Contrary to the slow-sweep range, the complete transition only occurs in the case of the imperfect anti-ferromagnetic. When coupling strength values in both neighborhoods have positive signs (perfect ferromagnetic), the transition probabilities are almost similar to the slow-sweep range (Fig. 3), while the transition probabilities are suppressed to almost 50% when the coupling strength values in one of the neighborhoods are negative [28]. It is also evident that the transition probability reaches a stable state faster for the fast-magnetic energy sweeping.
Fig. 4. Transition probabilities versus normalized time for $\hat{J}\, \cdot \hat{\alpha } = 7$ and (a) ${\hat{J}_1} ={-} 1,{\hat{J}_2} ={-} 1$, (b) ${\hat{J}_1} ={-} 1,{\hat{J}_2} ={-} 1$, (c) ${\hat{J}_1} = 1,{\hat{J}_2} ={-} 1$, (d) ${\hat{J}_1} = 1,{\hat{J}_2} = 1$.
In order to comprehensively investigate the effects of the next-nearest-neighbors on the system population transfer, the FTPE versus coupling strength for the nearest and next-nearest-neighbors and different rates of the magnetic field energies ($\hat{\alpha }$) are illustrated in Fig. 5. This figure demonstrates that by increasing the magnetic field energy rate, the FTPE values suppresses. This is in agreement with Ref. [28]. This reference expresses that the full population transfer is achievable by slowing down the sweep rate of the magnetic field's energy.
Fig. 5. FTPE versus nearest and next-nearest- neighbor interactions and (a) $\hat{\alpha } = 1$, (b) $\hat{\alpha } = 7$, (c) $\hat{\alpha } = 13.8$. Solid and dashed lines are for dipole-dipole and van der Waals interactions, respectively.
Figure 5 shows that the FTPE values decrease from 0.6 at $\hat{\alpha }$=1 to 0.4 at $\hat{\alpha }$=7 and 0.2 at $\hat{\alpha }$=13.8 for ${\hat{J}_1} > 0$ and ${\hat{J}_2} > 0$, that means FTPE values are greatly affected by $\hat{\alpha }$ values in a perfect ferromagnetic system. Indeed, the extended areas of the complete transition shrink as the magnitude of $\hat{\alpha }$ increases and the population transfer is limited to the imperfect anti-ferromagnetic areas. FTPE values collapse as $\hat{\alpha }$ values increase, and no transitions are observed in the perfect ferromagnetic regions. In other words, as the sweeping rate increases, the transition probability in various regions is reduced and limited.
As could be seen in Fig. 5, in the areas of the perfect ferromagnetic, the FTPE is restricted to the constant value of the transition probability by considering a specific value of the magnetic energy sweep rate. This behavior can be compared to a two-particle system in which the transition probability of a ferromagnetic system tends to a constant value [28,29].
Figure 5 shows that the complete and robust population transfer occurs in anti-ferromagnetic systems for a slow sweep rate while the complete transition does not arise in a perfect ferromagnetic system. This behavior could be ascribed to the total field jump in a ferromagnetic and anti-ferromagnetic system. This jump is negative for an anti-ferromagnetic coupling, and the system could tunnel through the resonance region. Therefore, the transition occurs, while the total field jump is positive for a ferromagnetic coupling. In this condition, the particles passed through the resonance region, and the possibility of increasing the system energy is very low; the system loses its chance to flip. Accordingly, the complete transition does not happen in such systems [28].
Comparing Figs. 2 and 5(a), it is clear that considering the next-nearest-neighbor reveals the significant areas (${\hat{J}_1} < 0,\,{\hat{J}_2} < 0$ and ${\hat{J}_1} > 0,\,{\hat{J}_2} < 0$), and it is momentous because of extensive areas of complete transition in the imperfect anti-ferromagnetic coupling.
In imperfect ferromagnetic areas ( and ${\hat{J}_2} < 0$), an ${\hat{J}_1} > 0$ oscillating behavior is shown in Fig. 5(a). This behavior is smoothened by increasing the value of $\hat{\alpha }$, and it vanishes when $\hat{\alpha }$=13.8. In Fig. 5(b), a periodic behavior is also observed for the imperfect anti-ferromagnetic region. A smooth periodic behavior starts to appear at $\hat{\alpha }$=7, which becomes noticeable at $\hat{\alpha }$=13.8 [Fig. 5(c)]. These oscillating behaviors could be attributed to the weakening of the system magnetic order when ${\hat{J}_2} < 0$. These oscillations are similar to Ref. [29]. This reference shows that the transition probability slightly oscillates in an anti-ferromagnetic coupling at some values $\hat{\alpha }$.
As could be seen in Fig. 5, in most imperfect areas, the complete transition occurs for the slow-sweep rate [Fig. 5(a)]. The weaker magnetic order in imperfect areas may have an essential role in increasing the transition probabilities. Increasing the magnetic field energy rate, the full transition is limited to the imperfect anti-ferromagnetic areas [Fig. 5(b)]. In other words, the complete transition is no longer observed in the perfect antiferromagnetic regions, while in the same conditions, we can see a complete transition in the imperfect anti-ferromagnetic regions. This behavior affirms that the investigation of the next-nearest- neighbor coupling is necessary for many-body systems.
For investigation of the van der Waals and dipole-dipole interactions, their trajectories are presented by the solid and dashed lines, respectively, in Fig. 5. The ratios of separation for next nearest neighbor to nearest neighbor for our system is 21/2; therefore, the ratios of ${{{{\hat{J}}_1}} / {{{\hat{J}}_2}}}$ are 8 for van der Waals interaction (due to 1/R6 dependent potential) and 23/2 for dipole-dipole interaction (due to 1/R3 dependent potential) [4]. These lines indicate that as the value of $\hat{\alpha }$ increases, the FTPE decreases for both interactions. Furthermore, it could predict that for a mixed system includes both van der Waals and dipole-dipole interactions, the FTPE lies inside the region enclosed by the mentioned two lines.
In order to investigate the variation of van der Waals and dipole-dipole interactions in more detail, the FTPE versus nearest-neighbor interactions for three values of the sweeping rate of the magnetic field’s energy is plotted in Fig. 6.
Fig. 6. FTPE versus nearest-neighbor interactions for (a) $\hat{\alpha } = 1$, (b) $\hat{\alpha } = 7$, (c) $\hat{\alpha } = 13.8$, and both van der Waals and dipole-dipole interactions.
As could be seen in all parts of Fig. 6, the transition probabilities are significantly distinct for the positive and negative values of the ${\hat{J}_1}$. For the slow sweep rate [Fig. 6(a)] and negative nearest neighbor interaction (and therefore the next nearest interaction) for both dipole-dipole and van der Waals cases, the near complete transition occurs. By increasing the interaction energy (for all parts o${\hat{J}_2}$f Fig. 6), the FTPE decreases and decreasing is faster for dipole-dipole interaction rather than van der Waals interaction. The larger the decreasing rate for dipole-dipole interaction rather than van der Waals interaction could be understood by comparing their second nearest-neighbor interactions. Because the dipole-dipole interaction is longer range interaction rather than van der Waals so its ${\hat{J}_2}$ is greater than for van der Waals in same ${\hat{J}_1}$, therefore increasing the ${\hat{J}_1}$ causes more increasing in ${\hat{J}_2}$ for dipole-dipole interaction and hence the more decreasing of the FTPE. Furthermore, this figure indicates that as the sweeping rate of the magnetic field’s energy increases, the FTPE reduces and moves away from the complete transition, and for van der Waals interaction, the reduction is more significant.
4. Conclusion
In this paper, coupling strength for nearest and next-nearest neighborhoods in a square-shaped four-particle system was studied. The transition probabilities of the excited states (FTPE) were calculated using Landau-Zener method.
Initially, next-nearest-neighbors’ interactions were ignored. In this system, the FTPE was maximum for the anti-ferromagnetic coupling while it was about 60% for the ferromagnetic coupling. This behavior could be attributed to the total field jump and the chance of tunneling for any coupling. Moreover, it was shown that this system was nearly independent of the sweep rates of the magnetic field's energy.
Finally, FTPE was studied for a four-particle system with the interaction of next-nearest-neighbors, and variation of FTPE for dipole-dipole and van der Waals interactions were specifically discussed. In this case, the FTPE was different for the various magnetic field's energy rates; somehow, the transition probability increased by slowing down the sweeping rate. In the presented results, the coupling strength values for both neighborhoods changed, and four regions were observed in FTPE graphs. It was shown when the next-nearest-neighbor's interaction is considered in a square-shaped lattice, the complete transition occurs in the imperfect anti-ferromagnetic areas, while it does not happen in the perfect anti-ferromagnetic areas for the specific values of the sweeping rate. In other words, the long-range and next-nearest-neighbors’ interactions and their transition probabilities play an essential role in quantum simulations, and it is necessary for the investigation of next-nearest-neighbors’ interaction.
The study of transition probabilities by FTPE for van der Waals and dipole-dipole interactions revealed that the transition probabilities for dipole-dipole interaction case are decreasing faster than van der Waals interactions by increasing the interaction strength.
Disclosures
The authors declare no conflicts of interest.
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