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Entangled states of dipolar bosons generated in a triple-well potential
by Arlei P. Tonel, Leandro H. Ymai, Karin Wittmann W., Angela Foerster, Jon Links
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We study the generation of entangled states using a device constructed from dipolar bosons confined to a triple-well potential. Dipolar bosons possess controllable, long-range interactions. This property permits specific choices to be made for the coupling parameters, such that the system is integrable. Integrability assists in the analysis of the system via an effective Hamiltonian constructed through a conserved operator. Through computations of fidelity we establish that this approach, to study the time-evolution of the entanglement for a class of non-entangled initial states, yields accurate approximations given by analytic formulae.
Published as SciPost Phys. Core 2, 003 (2020)
Author comments upon resubmission
List of changes
Response to Anonymous Report 1.
1. The referee is correct that the entanglement has been normalised. We have modified the figure labels and caption text to reflect this.
2. The following text has been included to answer the question raised by the referee.
From these results one can identify that panels (c) and (f) show the states with the most uniform probability distribution. This helps to understand why the variance, and the entanglement entropy, both increase with increasing number of particles $k$ in well 3 of the initial state. The decomposition of the state in terms of Fock states comprises an increasing number of components with increasing $k$. This correlates with the cases for $k=N/2$ ($N$ even), or $k=(N\pm 1)/2$ ($N$ odd) having the highest variance, as shown in Fig. 6, and the most entanglement, as shown in Fig. 8.
3. As mentioned in the original submission, the resonant tunneling regime is dependent not only on $U$, but also on $l$. To quote
When $UN/J$ is sufficiently large, but finite, these sets still provide accurate approximations for the bases. But as $l$ increases for $l\leq N/2$, or alternatively $l$ decreases for $l\geq N/2$, the threshold value of $UN/J$ which ensures well-separated bands increases.
To further clarify this point we have added the following text in relation to Fig. 9
Note that the right panels in Fig. 9, where $l=9$, are shown for a higher value of $U$ compared to $l=0$ shown in the left panels. As mentioned earlier, this higher value of $U$ is required in order to reach the resonant tunneling regime with well separated energy bands.
Response to Anonymous Report 2.
1. To clarify the issues raised by the referee, we have added the following text on page 2
That is, our aim in this work is to input non-entangled states and analyse the capacity of the device to produce entangled states of an ultracold quantum gas. Due to the integrability of the system, many results can be obtained through analytic formulae.
and on page 4
These Fock states are the most general non-entangled, number-conserving, pure states.
2. The atomtronic realisation of the model is now described in Appendix B. We agree with the referee that the issue of the robustness of the system with respect to deviations from the integrable point is a very important point to investigate. Moreover, as was discussed in  the breaking of integrability is advantageous in the control of the tunneling oscillations. This will also be the case for the generation of entanglement. The text on page 11 has been modified to read
One of these is to understand the mechanisms for controlling the entanglement generation through the breaking of integrability, specifically through the inclusion of external fields as in . Such an analysis of integrability breaking needs to encompass a study of the robustness of the system with respect to general deviations from integrable coupling. Some preliminary results are presented in , and this matter will be further investigated in a later, in depth, study.
3. The information requested by the referee has been included in Appendices A and B.
4. We do not consider the case of periodic boundary conditions for two reasons. The first is that periodic boundary conditions are not compatible with the experimental set up described in Appendix B. Secondly, the imposition of periodic boundary conditions breaks the integrability of the model. In particular, the operator $Q$ given by Eq. (2) does not commute with the Hamiltonian for periodic boundary conditions. The sentence
We remark that $Q$ does not commute with the Hamiltonian if periodic boundary conditions are imposed.
has been added on page 3.
Submission & Refereeing History
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