J. Phys.: Cond. Mat.: combining MCTDH-X with path-integral quantum Monte Carlo

A new paper that combines #MCTDHX and path-integral quantum Monte Carlo was published in Journal of Physics: Condensed Matter by A. Sakhel and collaborators from Jordan! Effect of zero-point motion on properties of quantum particles adsorbed on a substrate: https://iopscience.iop.org/article/10.1088/1361-648X/ad3095 .

PRL: Unidirectional atomic currents in a non-Hermitian momentum lattice

Dissipation is ubiquitous in quantum optical systems, and it can induce surprising consequences by bringing non-Hermiticity into the system. For example, even an extremely weak dissipation can stabilize high-energy steady states which are otherwise inaccessible in closed systems. This behavior has been predicted in different variations of Dicke-like models (see for instance this and this work). 

In this experimental work, we realize such a dissipative dynamics in a momentum lattice, where we observe a unidirectional atomic current tracing a pathway determined by the dissipation. During the experiments, we used MCTDH-X to guide, confirm, and interpret the experimental observations, more specifically by reproducing the dissipative dynamics in continuum. This is particularly useful in this specific experiment, because few mode expansions are inadequate due to the nature of the dynamics. A sequence of atomic configurations predicted by the Gross-Pitaevskii equation are shown in the figure, where a hopping in the momentum lattice can be clearly observed. Moreover, MCTDH-X can also capture the dephasing effects due to contact interaction between atoms.

Read more on the arXiv and PRL. The MCTDH-X simulations are discussed in details in the Supplemental Material.


The density distribution of the spinor atoms in real and momentum spaces in four different time points (second to fifth rows). The first and second columns are the real space densities of the spin-up and spin-down states, while the third and fourth columns are the momentum space densities. A outwards hopping is observed in the momentum space. 

SciPost Physics: Mott transition in a cavity-BEC system

In a cavity-BEC system, the light-matter interaction induces an interference field which makes the atoms self-organized into a checkerboard lattice. This phase transition is well-known to be mapped to the superradiant phase transition of a Dicke model. Upon self-organization, the atoms remain coherent and are said to be in a superfluid phase. However, as the self-organization becomes stronger and thus the atoms become more localized in each of the lattice sites, the correlations between atoms vanish, and the system enters a Mott insulator phase. From an effective Bose-Hubbard model point of view, this is because the tightening of the lattice sites not only reduces the tunneling strength, but also increases the on-site interaction.

As a continuation of our work from 2019, we implemented simulations in a two-dimensional system, which is experimentally much more relevant. We utilized a scheme to capture the essential Bose-Hubbard physics and bypass the computational difficulties by confining our atoms in an effective four-site model. This allowed us to correctly extract the momentum space distribution to identify the superfluid-Mott boundary. In collaboration with the experimental group of Hans Keßler and Andreas Hemmerich in Hamburg, we compared our MCTDH-X numerical results with their experimental data, and found quantitative agreement. This comparison is shown in the figure below.

In comparison to the conventional approach of a mapping to the Bose-Hubbard model, MCTDH-X has the advantage of a more adaptive wave function for capturing local densities at each lattice sites. With this work, we have thus demonstrated the capability of MCTDH-X as a reliable numerical tool for investigating Mott transitions.

Read more on SciPost.


PRA: accuracy of quantum simulators

Ultracold quantum simulators are used to recreate, i.e. simulate, the environment of a target quantum system with ultracold atoms. The atoms play the roles of the quantum particles we want to simulate. Optical laser traps can be used to spatially localize the atoms around discrete sites and introduce an effective lattice structure. Their interactions can be controlled, too, for instance with magnetic fields via Feshbach resonances. To obtain information about the system, we can shine a laser onto it and analyze the light that comes out. Whereas the original system is typically hard to realize directly in experiments or study with computational methods, ultracold quantum simulators consist of very stable and controllable components. This allows us for instance to realize lattices with no manufacturing defects, where the positions of each site and the interactions between the particles can be controlled with extreme precision, and where measurements are much easier to perform. Another advantage of quantum simulators is that they are, in principle, scalable to much larger systems that can be possibly simulated with classical computations. The fast development of quantum simulators also raises the fundamental issue of their accuracy and validity. If we want to make sure that they realize the target toy model and not a different one, we need to calibrate the parameters correctly. Furthermore, there might be regimes of the target quantum system that simply cannot be achieved with the quantum simulators because of technical limitations (not strong enough lasers, not dilute enough gases, or secondary processes between the actors that cannot be suppressed). For instance, it is known that the validity of quantum simulation can become reduced when the particles become strongly interacting, specially if the spatial range of such interactions is large.

In our recent work, we addressed precisely these questions with the concrete example of simulating a one-dimensional dipolar Bose-Hubbard model, in which bosons interact on a lattice via longer-ranged dipole-dipole interactions. These are the interactions that emerge between electric dipoles, i.e. particles with spatially separate positive and negative electric charges. Since electrons in solids interact via Coulomb interactions, which are also long-ranged, dipolar Hubbard and Bose-Hubbard models have the potential of providing an accurate description of the physics underpinning real high-temperature superconductors. One way to quantum simulate the dipolar Bose-Hubbard model is by employing ultracold polar molecules instead of ultracold atoms. NaCs (“Sodium-Caesium”) mixtures provide an excellent platform for this. NaCs are bialkali molecules formed of magnetoassociated sodium and caesium atoms and have one of the largest dipole moments currently achievable in experiments. In our paper, we studied the regime of validity of an ultracold quantum simulator composed of a one-dimensional gas of dipolar bosons like NaCs molecules in an optical lattice. We did so by running numerical simulations of both the quantum simulator itself, described as a continuum system with the software MCTDH-X, and the lattice toy model obtained by employing the QuSpin python library. We performed a full quantitative comparison between the continuum and the lattice descriptions by not only comparing energies and density distributions, but also by calculating direct overlaps between the continuum and lattice many-body wave functions. This allowed us to directly compare very different descriptions of the same phenomenon.

Our results show that in regimes of strong dipole-dipole interactions and high densities the continuum system indeed fails to recreate the desired single-band Bose-Hubbard lattice model. The discrepancy is not just quantitative, but also qualitative. For instance, whereas the single-band Bose-Hubbard model should have particles in the center of the lattice (see figure), the quantum simulator exhibits empty sites. Two-band Bose-Hubbard models become necessary to reduce the discrepancy, but quantitative deviations in the density profile still remain. This indicates that the quantum simulator is actually simulating a Bose-Hubbard model with many more bands than what we expected. Our study highlights the role of strong dipole-dipole interactions in generating physics beyond lowest-band descriptions, and demonstrates that such regimes require careful considerations when benchmarking the validity of quantum simulators. Furthermore, it offers a blueprint for a full-scale, quantitative analysis between continuum and lattice methods, since our analysis can be applied to more general ultracold quantum systems. In the future, we plan to extend our analysis to more diverse kinds of quantum simulators, for instance in two or three dimensions or with attractive interactions instead of repulsive ones.

The density profile of 6 ultracold bosons strongly interacting through dipole-dipole repulsions in two one-dimensional optical lattices of different potential depths (top panel: 5 times the recoil energy, bottom panel: 10 times). The dotted orange lines delineates the optical lattice. The solid blue line is the density obtained with continuum calculations to model the exact behavior of the ultracold quantum simulator. The dashed red line is the result from the single-band Bose-Hubbard model that the continuum system is supposed to simulate. As we can see, there are evident discrepancies, specially for the shallower optical lattice. The discrepancies are not just quantitative but also qualitative (e.g. the single-band lattice model gives a small population in the central site that is not present in the continuum). To obtain better agreement with the continuum calculations, we need to consider an extension of the Bose-Hubbard model with two-bands. The corresponding density is shown by the dash-dotted green curve. With this additional band, the lattice results behave qualitatively similarly to the continuum ones, although quantitative discrepancies can still be seen.
Read more on Physical Review A or on the arxiv.

PRA: crystallisation via cavity mediated interactions

In quantum mechanics, particles can be divided into bosons and fermions depending on their exchange statistics. Bosons can be swapped arbitrarily, while swapping two fermions produces a phase of minus one. This gives rise to very different density distributions. Bosons can occupy the same quantum mechanical state as exemplified in the famous Bose-Einstein condensation, where all the particles coalesce in the same quantum state and acquire macroscopic characteristics. Fermions instead obey the Pauli exclusion principle, whereby they can never occupy the same state and can therefore never condense (they instead tend to form lattice structures). 

When we confine quantum gases in lower-dimensional setups, however, things become more fuzzy. A gas of bosons with strong contact repulsions confined in one dimension (a so-called Tonks-Girardeau gas) can “fermionize” and mimic the Pauli principle: the bosonic real-space density distribution becomes indistinguishable from that of free fermions. To achieve the strong repulsions necessary to trigger fermionization, some sort of confinement is typically used, such as with optical lattices obtained by using standing waves of lasers. Things become even more interesting when the repulsions are long ranged, as in the case of dipolar particles. In a gas of dipolar bosons, for instance, the short-range part of the interactions pushes the particles first into a Tonks-Girardeau state. When the interaction strength is increased further, the long-range part impacts also the momentum-space density distribution, giving it fermionic character.

In our work, we achieved a completely new route to fermionization by using infinite-range interactions that are mediated by photons in an optical cavity. The basic mechanism of operation is rather straightforward: if we place a Bose-Einstein condensate in a cavity (egbetween two highly reflective mirrors) and shine a laser onto it, the photons will scatter on the atoms. The geometry of the setup can be chosen such that most of the light scatters within the cavity. Once the light gets reflected at the boundaries of the cavity, the scattering process can repeat itself. The photons can therefore couple with many different atoms many times. The effect is that the atoms will interact with each other through the mediation of the photons, and because of the many roundtrips of the light within the cavity, this interaction can be modelled as being infinite-ranged. The strength of the external laser then controls the intensity of the interactions.

In our work, we used MCTDH-X to simulate the cavity-atom system and solve the many-body Schrödinger equation to obtain its ground state properties. We found that, as the intensity of the external laser (and correspondingly the cavity-mediated interaction strength) is progressively increased, the bosons do indeed fermionize. In contrast to dipolar gases, however, in this case the fermionization of both real and momentum space observables happens simultaneously. Our approach also shows a very exciting interplay between contact and infinite-range interactions. It is possible to reduce the strength of the contact interactions needed to trigger fermionization by one order of magnitude if the cavity-mediated interactions are increased accordingly!

We expect the implementation of fermionization with cavities to have many other practical advantages. The cavity setup can be experimentally more flexible. It is for instance amenable to extensions to a multimode configuration. A multimode cavity set-up would shift the interaction range from infinite towards long-range, and would allow to investigate the effect of tuning the interaction range. As mentioned above, typical fermionization experiments use an optical lattice to artificially increase the interactions between the bosons. In our setup, as the interactions are instead mediated by the cavity, the optical lattice becomes superfluous. This means that it could instead be employed to investigate other phenomena with greater flexibility. For instance, by using a lattice incommensurate with the cavity wavelength, one could probe the interplay between fermionization and quasiperiodic structures. In a system where the pump wavelength is incommensurate with an additional optical lattice along the cavity direction it is namely possible to realize a Bose glass phase, an insulating state that contains rare “puddles” of superfluidity.

Read more on Physical Review A or on the arxiv.


(a) The cavity setup of our simulations: a Bose-Einstein condensate (blue) is placed into a high-finesse optical cavity (grey) and irradiated with an external laser (thick red arrow). Some of the photons scatter within the cavity (red vertical lines) and create effective infinite-range interactions between the atoms. (b) Sketch of the phase diagram of the system as a function of laser intensity η and cavity detuning Δc (the fermionization regime appears at large values of η). (c) Onset of fermionization (black region) as a function of laser intensity η and particle contact interaction strength g.


SciPost Physics: Pauli crystal melting

Pauli crystals are ordered geometric structures that appear when noninteracting fermionic gases – such as 8Li – are cooled to extremely low temperatures and confined by optical traps. Under these conditions, the quantum nature of the atoms in the gas dominate, and the crystal structure emerges because each atom in the gas repels its neighbors due to the Pauli exclusion principle. Pauli crystals are notoriously difficult to realize because of the very high degree of control needed in the experiments, and were definitively observed for the first time only in 2020 (see the original Physical Review Letter article or this Physics outreach article). When the conditions are not perfect, the Pauli crystal gets deformed – a phenomenon called melting. Pauli crystal melting has been observed in experiments, too, but the mechanism that leads to it remains unclear.

We have addressed this question by studying the melting dynamics of a few-particle fermionic system as a function of periodic driving and experimental imperfections in the optical trap (anisotropy and anharmonicity). To do so, we employed a combination of numerical simulations with MCTDH-X and Floquet theory. Surprisingly, we revealed that the melting of Pauli crystals is not simply a direct consequence of heating up the system, but is instead more related to the trap geometry, and to the population and behaviour of the time-periodic quantum states (Floquet states). We showed that the melting is absent in traps without imperfections and triggered only by a sufficiently large shaking amplitude in traps with imperfections.

Our study sheds light into the geometric and dynamical mechanisms that lead to melting and should help devise experimental protocols that prolong the lifetime of Pauli crystals. Reducing geometric distortions of the trap will substantially increase the stability and duration of the Pauli crystal phase. Our results also highlight that many-body correlations offer a very rich phenomenology to explore and that dynamical excitations play a crucial role for the stability of crystalline phases of matter of quantum gases.


The evolution of a Pauli crystal (star-like structure in panels a) as a function of time in a shaken optical trap. In a perfect harmonic trap, shown in the top panels, the Pauli crystal remains stable for a long time. In a trap with imperfections (such as anharmonicities), shown in the bottom panels, the crystals becomes progressively deformed and eventually loses its geometric structure – a phenomenon called melting.

Read more on SciPost or on arxiv.


PRL: Dipolar Bosonic Crystal Orders via Full Distribution Functions


Quantum Simulation of Crystal Formation with Ultracol Dipolar Bosons

The quantum properties underlying crystal formation can be replicated and investigated with the help of ultracold atoms. Our recent paper in Physical Review Letters shows how the use of dipolar atoms enables even the realization and precise measurement of structures that have not yet been observed in any material.

Crystals are ubiquitous in nature. They are formed by many different materials – from mineral salts to heavy metals like bismuth. Their structures emerge because a particular regular ordering of atoms or molecules is favorable, because it requires the smallest amount of energy. A cube with one constituent on each of its eight corners, for instance, is a crystal structure that is very common in nature. A crystal’s structure determines many of its physical properties, such as how well it conducts a current or heat or how it cracks and behaves when it is illuminated by light. But what determines these crystal structures? They emerge as a consequence of the quantum properties of and the interactions between their constituents, which, however, are often scientifically hard to understand and also hard measure.

To nevertheless get to the bottom of the quantum properties of the formation of crystal structures, scientists can simulate the process using Bose-Einstein condensates – trapped ultracold atoms cooled down to temperatures close to absolute zero or minus 273.15 degrees Celsius. The atoms in these highly artificial and highly fragile systems are extremely well under control. With careful tuning, the ultracold atoms behave exactly as if they were the constituents forming a crystal. Although building and running such a quantum simulator is a more demanding task than just growing a crystal from a certain material, the method offers two main advantages: First, scientists can tune the properties for the quantum simulator almost at will, which is not possible for conventional crystals. Second, the standard readout of cold-atom quantum simulators are images containing information about all crystal particles. For a conventional crystal, by contrast, only the exterior is visible, while the interior – and in particular its quantum properties – is difficult to observe.

Our paper demonstrates that a quantum simulator for crystal formation is much more flexible when it is built using ultracold dipolar quantum particles. Dipolar quantum particles make it possible to realize and investigate not just conventional crystal structures, but also arrangements that were hitherto not seen for any material. The study explains how these crystal orders emerge from an intriguing competition between kinetic, potential, and interaction energy and how the structures and properties of the resulting crystals can be gauged in unprecedented detail.

Read more at PRL and at arXiv.org .

See this page for the press release and this video with an interview about the paper by its lead author Budhaditya Chatterjee.


Quantum Science and Technology: MCTDH-X software tutorial paper published


We introduce and describe the multiconfigurational time-depenent Hartree for indistinguishable particles (MCTDH-X) software, which is hosted, documented, and distributed at http://ultracold.org.

We give an introduction to the MCTDH-X software via an easy-to-follow tutorial with a focus on accessibility. The illustrated exemplary
problems are hosted at http://ultracold.org/tutorial and consider the physics of a few interacting bosons or
fermions in a double-well potential.

A complete set of input files and scripts to redo all computations in this paper is provided at http://ultracold.org/data/tutorial_input_files.zip, accompanied by tutorial videos at https://tinyurl.com/tjx35sq

Read more at Quantum Science and Technology and arXiv.

RMP Colloquium on MCTDH-X et al.


In this Colloquium, the wavefunction-based Multiconfigurational Time-Dependent Hartree approaches to the dynamics of indistinguishable particles (MCTDH-F for Fermions and MCTDH-B for Bosons) are reviewed. MCTDH-B and MCTDH-F or, together, MCTDH-X are methods for describing correlated quantum systems of identical particles by solving the time-dependent Schrödinger equation from first principles. 

We highlight some applications to instructive and experimentally-realized quantum many-body systems: the dynamics of atoms in Bose-Einstein condensates in magneto-optical and optical traps and of electrons in atoms and molecules.

We discuss the current development and frontiers in the field of MCTDH-X: theories and numerical methods for indistinguishable particles, for mixtures of multiple species of indistinguishable particles, the inclusion of nuclear motion for the nonadiabatic dynamics of atomic and molecular systems, as well as the multilayer and second-quantized-representation approaches, and the orbital-adaptive time-dependent coupled-cluster theory are discussed.

Read more at Review of Modern Physics and arXiv:1908.03578 [cond-mat.quant-gas]

PRA: Superfluid -- Mott insulator transition of ultracold superradiant bosons in a cavity

Superfluid--Mott insulator transition of ultracold superradiant bosons in a cavity

We analyze the rich physics of the superfluid-to-Mott-insulator transition of ultracold interacting bosons in an optical cavity.

We explore the whole phase diagram of the system and underpin our MCTDH-X results with analytical considerations.

Read more at arxiv and Phys. Rev. A. .