Abstract
While batteries offer electronic source and sink for electronic devices, atomic analogues of source and sink and their theoretical descriptions have been a challenge in coldatom systems. Here we consider dynamically emerged local potentials as controllable source and sink for bosonic atoms. Although a sink potential can collect bosons in equilibrium and indicate its usefulness in the adiabatic limit, sudden switching of the potential exhibits low effectiveness in pushing bosons into it. This is due to conservation of energy and particle in isolated systems such as cold atoms. By varying the potential depth and interaction strength, the systems can further exhibit averse response, where a deeper emerged potential attracts less bosonic atoms into it. To explore possibilities for improving the effectiveness, we investigate what types of systemenvironment coupling can help bring bosons into a dynamically emerged sink, and a Lindblad operator corresponding to local cooling is found to serve the purpose.
Introduction
Recent advances in trapping and manipulating ultracold atoms in magnetic or optical potentials have brought new tools for studying nonequilibrium phenomena of manybody systems via quantum simulations^{1,2}. In contrast to conventional solid state materials, ultracold atoms provide more flexibility in their structures and are controllable over a broad range of parameters such as interactions and temperature^{3,4}. Those new techniques also provide opportunities for testing and verifying theories of transport properties in solid state devices and cold atom systems^{5,6,7,8,9,10,11,12}. Recently, the concept of atomtronics^{13,14,15,16} has drawn intense attention due to intriguing experimental and theoretical studies, including quantum point contact^{17,18}, atomic SQUID^{19,20,21,22,23}, transistor^{24}, capacitor^{25}, and open quantum systems^{26,27,28,29}. There is a bright future for atomtronics, and here we will address a challenging issue on driving atoms in atomtronic circuits via local manipulations.
While particle reservoirs like batteries play the role of source or sink in conventional electronic systems, atomic analogues of particle source or sink for atomtronics are highly desired. However, due to charge neutrality of atoms, one needs creative ways for supplying or removing atoms. It is possible to use atoms from a nearby trap as a source^{18,30} and remove atoms using photon or electron beams, which acts as a sink^{5,27,31}. Nevertheless, programmable atomtronic circuits may need dynamically generated sources or sinks. We will investigate whether local manipulations of the potential in a small region can act as a source or sink effectively in isolated systems modeling coldatoms. Recent experimental advance of creating an additional narrow dimple trap^{32,33,34} may provide realizations of the setups discussed here. Such a scheme is more suitable for bosons since Pauli exclusion principle may limit the amount of fermions allowed in a narrow region. For noninteracting and weakly interacting bosons, the ground state corresponds to a congregate of bosons in the deep potential. However, we will show that the quantum nature and energy conservation severely compromise the effectiveness of a dynamic sink from a sudden emergence of a deep potential. The BoseHubbard model (BHM) will be implemented and we alter the onsite energy of a selected single site to generate a sink. In equilibrium, a meanfield estimation of the maximal amount of bosons attracted into the sink qualitatively agrees with numerical simulations. The simulations show promising results in equilibrium with a large fraction of atoms in the sink. This ensures the effectiveness of local deep potentials as atomic sinks in the adiabatic limit where the potential changes slowly.
Atomtronic circuits, nevertheless, are expected to operate within finite durations. The finite hold time of atomic clouds also restricts the switching time of atomtronic devices. Therefore, we explore the opposite limit where a local potential is suddenly changed and analyze its effectiveness as a dynamical sink. As one will see shortly, a dynamically emerged source or sink acts poorly in providing or collecting quantum particles. The origin of the ineffectiveness comes from the wave nature of quantum particles and energy conservation. By ramping the sink potential to deeper depth, the number of bosons attracted into the sink may even decrease. The system exhibits averse response reminiscent of the negative differential conductivity in electronic and atomic systems^{5,35}, where a stronger driving field leads to a smaller current. The same conclusions are reached in a continuum model summarized in the Supplementary Information.
In order to explore ways for improving the effectiveness of dynamic source and sink for cold atoms, we relax the isolatedsystem condition by considering environmental effects and extending the theoretical description to an opensystem approach. For atomtronic systems, external perturbations by light or atoms need to be introduced for significant environmental effects. For an open quantum system, one may describe the dynamics using the quantum master equation approach^{26}, which has been extremely successful in quantum optics^{36,37,38}. We followed this method and analyzed the effects of different Lindblad operators on bringing the particles into a dynamically generated sink. While a popular form of Lindblad operators in studying decoherence^{28,39,40,41} does not improve the amount of particles drawn into the sink, we explore a particular Lindblad operator inspired by a study of BoseEinstein condensate formation^{42,43,44} and find such an intervention draws substantially more particles into the sink. Implications of this particular Lindblad operator and possible experimental connections will be discussed. The quantum master equation approach complements the shortcuttoadiabaticity approach^{45}, where additional timedependent deformations of potentials bring the system to its adiabatic limit.
Results
Isolated quantum system
First, we study Bose gases in a one dimensional lattice potential with tunable onsite energy of selected sites. The system may be described by a singleband BHM model, whose Hamiltonian is given by
Here (b_{i}) is the boson creation (annihilation) operator on lattice site i, is the boson number operator on site i, and 〈i, j〉 represents nearest neighbors. We set ħ = 1 and the time unit is t_{0} = ħ/J. To simulate different setups with a sink, a source, or a combination of both, we consider different sequences of the timedependent local potential energy, V_{i}(t).
The ground state with or without a sink or a source can be found by the exact diagonalization (ED) method, and the dynamics can be monitored by using a similar technique. We simulate small systems up to L = 13 lattice sites and N = 11 bosons. For larger systems, the ED method is less practical and we rely on the density matrix renormalization group^{46,47,48} (DMRG) method.
Equilibrium ground state with a sink potential
We first investigate the ground state in the presence of a sink modeled by a deep potential on one site, as illustrated in Fig. 1a. For the noninteracting case with U = 0, it can be shown that a bound state exists when a single site potential is deeper than the half bandwidth 2J^{49}. For a system with L = 13 and N = 11, we use the ED to simulate the ground state wavefunction with a sink at site i (V_{i} = V) and V_{j≠i} = 0. In order to deal with finite size effects, we calculate the fidelity metric defined as
where the fidelity, , is the overlap between two normalized ground states obtained with a small change of the parameter Y. For finitesize systems, a decrease in the fidelity is a precursor to a crossover or quantum phase transition^{50,51}, and that corresponds to a peak in the fidelity metric. In our simulations we vary the potential depth when evaluating the fidelity. For a fixed potential depth V (or coupling constant U), we vary the interaction (or potential depth) and determine the critical point between the ground state with all particles in the sink and the ground state with one particle outside the sink. The fidelity metric shows a sharp peak when the configuration with all particles in the sink is no longer stable.
The critical interaction strength (or potential depth), when the configuration with all particles in the sink is no longer stable, is shown in Fig. 1b, which agrees well with a meanfield analysis shown in the Supplementary Information. A continuum model is also analyzed in the Supplementary Information and the results converge to the same conclusions. We also use the DMRG to study a larger system with L = 61 sites, N = 20 bosons, and the sink potential V = 38J. The fidelity metric shows a peak in Fig. 1c around U ≈ 2.1J which is close to the meanfield prediction of U_{c} = 2J when the ground state with all particles in the sink is no longer stable. In general, the number of particles in the sink potential decreases as the interaction becomes stronger. Moreover, the BHM may be driven across the Mott insulatorsuperfluid transition by manipulating the potential on one single site with suitable filling^{52}.
System evolution with dynamically emerged sink
The equilibrium results suggest that a deep potential on one site may serve as a particle sink to collect bosons in the weakly interacting regime. As the sink potential can be tuned very slowly, it is expected that the system remains in the ground state after the sink potential emerges and all the particles stay in the sink if the ratio between the interaction and trap depth, U/V, falls below the critical value. However, the adiabatic limit would be less relevant for designing scalable atomtronic devices. The time scale of the emerged sink to satisfy the adiabatic limit in the noninteracting case can be shown to increase with the system size. Detailed discussions of the proper time scale and the results of different ramping times are provided in the Supplementary Information, where the results are shown to smoothly interpolate between an instantaneous quench and the adiabatic limit as the ramping time increases. We consider the following scenarios of instantaneous switching of the sink or source potentials: 1) a suddenly generated well potential at one edge or the center of the system as shown in Fig. 2a and 2e, 2) an initial well potential at one edge is suddenly lifted while a well potential appears at the other edge as a sourcesink combination discussed in the following section. The interactions are assumed to be uniform.
The dynamics after the sink or source potential is turned on may be interpreted as a response theory, where the driving field corresponds to the sink or source potential and the response may be the particle number difference in the sink site. Such a response theory is similar to the case where a magnetic field drives the magnetization or a mass current responds to a chemical potential difference. Although the mean field predictions work well in equilibrium, for dynamical systems the standard HartreeFock approximation ignores highorder correlations and becomes less reliable^{53}. Thus, numerical simulations which can monitor the dynamics more faithfully are implemented in this work. Moreover, the typical system size in ultracold experiments is usually of a few tens of lattice sites^{11}, so here we focus on finite systems of comparable size. Since it is easier to observe the dynamics when the motion is slow, we focus on lattice models because tunneling between adjacent sites slows down the dynamics^{11}. Nevertheless, the qualitative behavior is the same in lattice as well as continuum models as shown in the Supplementary Information. In this work, the system is prepared in the ground state of the initial Hamiltonian, then an instantaneous quench changes the local potential to induce dynamics. During the dynamics, the system is assumed not to exchange heat with the environment. Later on when we introduce the opensystem approach, the constraint will be relaxed.
We monitor the density distribution in real time with different values of the uniform coupling constant U in Eq. (1), and the results from the ED are shown in Fig. 2 for L = 13 and N = 11. The density profiles can be measured directly from absorption images, and the current can be inferred from the time derivative of the density^{11}. By quenching the sink potential to a constant value V = 3J, we observe different dynamics with different interacting strength, but in general the dynamically generated sink attracts much less particles when compared to its equilibrium counterpart. For example, in the case with a sink at the edge shown in Fig. 2a–d, less than 2% of the total particles flow into the suddenly emerged sink. The case with a sink at the center shown in Fig. 2e–h can attract more particles due to the initial inhomogeneous density distribution, but it is still far less than the equilibrium counterpart. The difference between the case with a sink at one edge and the case with one at the center is that stronger interaction strength pushes more particles towards the edge in the initial state. The results with different sink potential depth and various interaction strength are summarized in Fig. 2i and 2j. Here ΔN_{sink} is defined as the particle number difference in the sink between the initial value and the first peak in its evolution. (See Fig. 2b for example.) The continuum model analyzed in the Supplementary Information also exhibits similar ineffectiveness of a dynamically emerged sink and dependence of the sink location.
The reason of the low efficiency of the dynamically generated sink is mainly due to conservation of energy in isolated systems such as cold atoms. After the sink potential suddenly appears, the ground state of the initial uniform lattice becomes a relatively highenergy state of the new Hamiltonian with the sink potential. The lowenergy states in the presence of the sink should be those with particles localized inside the sink. When one particle hops into the sink potential, it will lower the energy by an amount of the order of V. Due to energy conservation, this energy loss has to be compensated by, for example, the kinetic or interaction energy. For noninteracting gases, the kinetic energy per particle is constrained by the bandwidth . Therefore, it is impossible for particles to accumulate in the sink when the loss of potential energy is much larger than the bandwidth representing the kinetic energy. Although similar analyses show that adding weakly repulsive interactions allows few more particles to flow into the sink potential, complexity arises in the strong interaction regime and will be discussed later.
For fixed interaction strength and relatively weak sink depth, the maximal amount of particles drawn into the sink increases as the sink potential is quenched to larger values, which indicates an improvement of the effectiveness of a dynamical sink. However, the amount of particles in the sink decreases when the depth of the sink potential exceeds a critical value. This indicates that in the deep sink regime, the system exhibits averse response, where a deeper sink potential results in less particles in the sink. This averse response is similar to the negative differential conductivity (NDC)^{35}, where a stronger driving field leads to less current, and the NDC has been discussed and observed in coldatom experiments^{5,53}.
The issue on whether introducing interactions can improve the effectiveness of a dynamically emerged sink is complicated by several issues. For example, if the quenched sink potential depth is fixed and the interactions are set to different values, the dynamics depends on the sink location because the initial density profiles change with interactions. Moreover, increasing the interactions tend to reduce the maximal number of particles allowed in the sink due to the repulsion between particles. In general, if the sink is located at one edge, stronger interactions can lead to more particles in the sink. Figure 2i shows that there is an optimal potential depth for selected interaction strength. For the case with a sink quenched at the center, the interactions do not provide observable improvement as shown in Fig. 2j. The help of effectiveness from the interactions disappears as the interaction energy exceeds a critical value when a site with multiple interacting bosons leads to huge interaction energy, and the number difference in the sink, ΔN_{sink}, becomes insignificant.
To simulate larger systems, we use the tDMRG with L = 61 sites and N = 20 bosons and study the intermediatetime behavior before the matter wave due to the sudden appearance of the sink potential bounces back from the edges and exhibits finitesize effects. A lightcone structure can be observed in the time evolution of the density profile as shown in Fig. 3a. The effectiveness of the dynamically emerged sink, however, is not improved for larger systems due to conservation of energy. The number of particles in the sink is shown in Fig. 3b, which oscillates in time with a frequency depending on the quenched potential depth V. The longtime behavior of this quantity tends to approach a stationary value, so we take its longtime average and plot it in Fig. 3c. Clearly, averse response showing a decreased number of particles in the sink as the sink potential increases is observable as the potential depth exceeds a critical value depending on the interaction. Thus, the behavior of larger systems from the tDMRG qualitatively agrees with smaller systems calculated from the ED.
Transport in combined dynamic source and sink
Next, we consider a system with a sink potential initially at one end. Then, another sink potential appears on the opposite end and the initial potential vanishes simultaneously as illustrated in Fig. 4a. The initial potential well may be interpreted as a particle source because the ground state has an initial surplus of particles in the well which are pushed out and generating a mass current. This setup may be interpreted as a pair of dynamically generated source and sink, and the results are summarized in Fig. 4b–f. Initially, the well for reserving particles is located at the left end (the source site) with depth V, then the potential suddenly rises to zero while the sink potential appears on the right end (the sink site) with the same depth. We begin with the case where the interaction U is fixed and the depth of the source and sink potentials is varied to check if this dynamical process can induce a current through the system. This is indeed the case as one can see in Fig. 4b and 4c, where the initial surplus of particles on the left is transferred to the right at a later time. An interesting finding in the combined source and sink setup is illustrated in Fig. 4b. One can see that a few particles are transferred from the left source site to the right sink site. The number of particles transported is sensitive to the ratio between the sink potential depth and the interaction strength. Furthermore, the density evolution in the inset shows that the particles stay in the sink site after they arrive there. Thus, the combined dynamic source and sink shed light on controlling fewparticle transport across a quantum device with strongly interacting particles. This may be more difficult to demonstrate in conventional solid state systems^{54}.
As shown in Fig. 4d with large potential depth, there is very little flow of particles from the source site (on the left edge) after the potential energy is lifted. Therefore, observable fewparticle transport only occurs when the initial potential depth is not too deep and there are only slightly more particles in the source site than in other sites. When the depth exceeds a critical value depending on the interaction strength, fewer particles can flow even when more particles are initially in the source site, and this demonstrates another example of averse response. Figure 4e shows the particle number at the source site as a function of time with different potential depth, and one can see the averse response when a deeper initial potential pushes less particles out of it after the potential is lifted. This averse response is general for various interaction strengths. One can observe it in the difference of the initial and latertime particle numbers on the source site in Fig. 4f.
By a similar analysis using energy conservation, the initial interaction energy of the particles on the source site can be much larger than the kinetic energy limited by the bandwidth . Thus, the system cannot compensate for the loss of interaction energy when particles leave the source site. Recent experiments^{55,56} showing slowing down of particle transport in interacting bosonic and fermionic systems may be partly related to this energy conservation constraint. We remark that here we consider a singleband BHM, but in a more general model there may be more than one energy level inside a deep potential. Nevertheless, one may confine the analysis of a dynamic sink by focusing on the highest energy state in the sink potential (whose energy is still lower than the states outside the sink). The lowerenergy states in the sink are less relevant for satisfying energy conservation due to their larger energy differences with the initially uniform state. Therefore, multistate effects effectively reduce to the singlestate potential considered here, and the results should be qualitatively the same.
Open Quantum System Approach
So far the results show that it is very challenging to induce significant transport in an isolated system by only dynamically manipulating the potential in a small region. It is possible, however, to introduce environmental effects via external lightatom or atomatom interactions. Here we will relax the isolation condition and investigate whether transport can be enhanced by external effects. A system under external influence may be modeled by opensystem approaches, which have been studied extensively and discussed in many areas of physics, especially quantum optics^{37,38,57} and spin systems^{26,27,28,58}. Here we implement a commonly used approach in open quantum systems and summarize the key approximations which simplify the description of dynamics in the Supplementary Information.
After considering the BornMarkov approximation, we arrive at the master equation, which takes the Lindblad form^{36,37,38}
Here L_{j} is a Lindblad operator, and γ is a parameter characterizing the coupling between the system and environment. In general, the inclusion of Lindblad operator leads to energy exchange between the system and environment^{28}. We have tested some Lindblad operators discussed in the literature. For example, the local density operators have been implemented with the set of the Lindblad operators {L_{j}} set to {n_{l}, l ∈ [1, L]}. This is one type of commonly used operators for introducing decoherence, for example, in dephasing of hardcore lattice bosons^{5,28}. This type of Lindblad operators is Hermitian and it leads to localization of atoms on each single site, which corresponds to spreading of the spatially localized particles over all possible states in quasimomentum space. Hence, the kinetic energy decays to zero during the process^{28,59}. According to a recent study^{60}, using local density operators as the Lindblad operators can overcome the NDC of interacting bosons in a threesite potential. In that study, the initial state corresponds to an inhomogeneous density distribution with a single empty site and two adjacent sites with finite density. Since the local density operators favor a uniformly distributed density in the Lindblad equation, this process can dynamically fill the empty site as the system experiences decoherence.
The local density operators, however, do not attract particles into a dynamically emerged sink because they favor uniform density distributions. To verify whether a selected type of Lindblad operators can enhance the effectiveness of a dynamically emerged sink, we run simulations using the ED method for small systems. To connect to the isolatedsystem results, we consider weak coupling between the system and environment in the sense that γ is smaller than the rate V/ħ determined by the quenched sink potential. The results according to the Lindblad master equation are summarized in Fig. 5.
If the sink is quenched at one edge, the local density operators push some particles into the sink when they make the density uniform as shown in Fig. 5a and b, but this type of Lindblad operators cannot drive the majority of bosons into the sink. For the same reason, this process even reduces the particles in a dynamical sink quenched at the center of system because the initial ground state has higher density at the center and the particles spread out to reach a uniform density distribution. Thus, the local density operators are useful for bringing an initially state with an inhomogeneous density distribution in real space into a final state with a uniform density distribution. For a dynamically emerged sink, we are searching for Lindblad operators that work the other way around.
Another scheme^{42,43} of Lindblad operators leads to a coherent driving of bosons into a BoseEinstein condensate (BEC) by implementing the Lindblad operators {L_{j}} with { (b_{p} − b_{q}), where p, q are neighboring sites}. This coherent driving is designed to produce a steady state^{59} of BEC, which is the only dark state of the proposed Lindblad operators. In other studies, this type of Lindblad operators is further extended to certain manybody systems to create pairing states^{28,61}. Inspired by the latter scheme, we consider a Lindblad operator acting only on the sink site and its neighbor, so {L_{j}} is set to {, k = sink site}, which is nonHermitian and introduces dissipation. Figure 5c clearly shows that this type of Lindblad operators not only helps the system draw more particles into the sink potential dynamically, but it also minimizes the back scattering due to the quenched potential. As the potential depth increases, the sink accommodates more particles and the number of particles in the sink can be as large as 80% of the total particles as shown in Fig. 5d.
Therefore, the local Lindblad operator is efficient in bringing the system into the vicinity of its equilibrium configuration of the new Hamiltonian with an emerged sink. Interestingly, the dependence of the amount of particles attracted into the sink in the longtime limit on the systemenvironment coupling is nonmonotonic, as shown in Fig. 5e. There is a maximal value of γ where particles can be efficiently drawn into the sink. When the coupling increases further, the amount of particles attracted into the sink decreases. Nonmonotonic dependence of particle transport on systemenvironment coupling has also been discussed in fermionic systems allowing exchange of particles between the system and environment^{62}.
When more weaklyinteracting particles are brought into a deep sink, the overall energy of the system should decrease due to the sink potential. Indeed, as shown in Fig. 5f the energy of the system decreases during the dynamics governed by the Lindblad equation with the local operator, which indicates a net flow of energy out of the system. Thus, the local Lindblad operator may be considered as modeling a local cooling process, which may be of experimental interest as singlesite cooling techniques have been developed recently^{63,64}.
Yet another route for improving the effectiveness of the dynamically generated sink is, at least theoretically, to approximate the systemenvironment interaction as a relaxation process in the Liouvillevon Neumann equation^{37}. The equation of motion of the system is modeled as
with ρ_{eq} determined from the ground state of the final Hamiltonian with a dynamically emerged sink. The relaxation time τ_{s} is usually treated as a phenomenological parameter. The essence of relaxation approximation is to guide the system in the longtime limit toward a desired target state, which in this study is the ground state of the final Hamiltonian. Whether the target state is the genuine longtime limit of the full dynamics is beyond the relaxation approximation. Moreover, the relaxation approximation suffers some other drawbacks. For instance, the total particle number may not be strictly conserved during the dynamics. In contrast, the two aforementioned Lindblad operators and their equations respect particle conservation during the evolution as shown in the Supplementary Information. Secondly, the relaxation approximation may not guarantee the semipositivity of the density matrix during the dynamics^{37,38}. Last but not least, this method requires a prior knowledge of a final target state rather than a set of local Lindblad operators. When considering atomtronic devices as assemblies of various local elements, the Lindblad master equation approach would be more versatile.
Discussion
Energy and particle conservation in isolated systems like cold atoms imposes constraints on their transport phenomena and leads to challenges on how to dynamically store or transfer particles. Although we present results from onedimensional systems, the mechanisms behind those phenomena are the quantum nature and energy/particle conservation and they should be general in higher dimensional systems with similar setups. In equilibrium, the systems discussed here show promising capability of accommodating particles in a sink potential. The equilibrium results guarantee the functionality of a dynamically generated sink in the adiabatic limit, but it may not be particularly useful in atomtronic circuitry requiring short switching times. For a sudden switch of the sink or source potential, our simulations show a lack of effectiveness to drive the particles into the equilibrium distribution, and this demonstrates another stark contrast between atomtronic and electronic systems^{11}. A dynamically generated sink can even lead to averse response where an increase of the potential depth attracts less particles into it. Nevertheless, fewparticle transport could still be observable in interacting systems with combined dynamical source and sink. The results of closed systems are analyzed with zerotemperature initial states, and the interaction can be tuned via external magnetic fields^{3}. The depth of sink or source potentials can be controlled in experiments^{34}, and the averse response presented here should be observable in the density profile.
To explore how external effects can help improve the effectiveness of a dynamically emerged source or sink, we test the masterequation approach with various kinds of Lindblad operators. While the commonlyused local density operators favor a uniform density distribution after time evolution, it is not helpful in the design of dynamical source and sink potentials. Instead, a local Lindblad operator showing local cooling behavior is found to significantly improve the effectiveness of a dynamical sink. This observation suggests that a combination of local cooling/heating and sitewise manipulations will have a bright future in making dynamical or programmable sinks or sources in atomtronic devices.
Method
Exact diagonalization
The Láczos procedure^{65,66} can calculate a few targeted states of a Hamiltonian which is sufficiently sparse, and a similar technique can be used to calculate real time dynamics. The method uses the Krylovspace approach^{67,68} to approximate the timeevolution operator , which evolves the wavefunction from time t to time t + dt according to
More specifically, the Krylov subspace spanned by the vectors
are orthogonalized with respect to the previous two vectors in the set, which leads to the Lánczos vectors
with the coefficients and .
For a given time t, we use as the first basis in the Krylov subspace. The matrix is composed of the Lánczos vectors in the form Thus, the Hamitonian can be expressed by the tridiagonal matrix
This procedure is exact if the number of the Láczos vector used, n, is equal to the dimension of the total Hilbert space of the Hamiltonian . However, it is possible to obtain results with high accuracy by taking just a few Láczos vectors and a small dt, and the error of the Euclidean norm of the wavefunction is controllable^{67,69}. Here we use 20 Lanczos vectors and a time step dt = 0.01t_{0}, and the estimated error is around 10^{−10}.
Density matrix renormalization group
The tDMRG simulations have been applied to larger systems in and out of equilibrium. For the simulations of equilibrium systems, we keep up to 150 states (bond dimension) and maintain the truncation error^{70} below 10^{−11}. For outofequilibrium dynamics, we decompose the evolution operator using the secondorder SuzukiTrotter formula^{71,72} and evolve the ground state obtained from the static DMRG algorithm by the timedependent DMRG^{48,73,74} (tDMRG). During the simulations of timedependent systems, the entanglement entropy increases drastically^{48}. We manage to keep the truncation error below 10^{−8} but do not keep more than 1000 states. The simulation used Extreme Science and Engineering Discovery Environment^{75}.
Additional Information
How to cite this article: Lai, C.Y. and Chien, C.C. Challenges and constraints of dynamically emerged source and sink in atomtronic circuits: From closedsystem to opensystem approaches. Sci. Rep. 6, 37256; doi: 10.1038/srep37256 (2016).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
We are grateful to Malcolm Boshier, Roland Winston, Kevin Mitchell, ChenLung Hung, and David Weld for many useful discussions. This work used the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI1053575.
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University of California Merced, School of Natural Sciences, Merced, 95343, USA
 ChenYen Lai
 & ChihChun Chien
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C.C.C. conceived the idea, and C.Y.L. performed the numerical calculation. C.C.C. and C.Y.L. wrote the manuscript and contributed equally to this work.
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The authors declare no competing financial interests.
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Correspondence to ChenYen Lai.
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