We consider protocols for control of dipolar Bose-Einstein condensates where the critical role is played by the long-range anisotropic interatomic magnetic dipole-dipole interaction. The phase diagram of such a condensate has been explored theoretically and experimentally with certain values of the interatomic scattering length corresponding to superfluid and supersolid phases, where supersolidity appears as a modulation in the ground state density. Preparation of this modulated ground state is challenging, since excitations appear as a result of a finite-time evolution required to produce qualitative changes in the wavefunction density. To solve this problem we consider the time-dependent control of a dipolar Bose-Einstein condensate using shortcuts to adiabaticity techniques, concentrating on design of the time-dependent scattering length, a parameter of the system easily tunable by contemporary experiments. The first technique is the variational approach based on the Euler-Lagrange equations for a separable ansatz describing the evolution of the superfluid state. Secondly, we study the transition from superfluid to supersolid using a direct optimization protocol. We discuss the fidelity of the developed protocols in terms of the evolution time.
We consider protocols for control of dipolar Bose-Einstein condensates where the critical role is played by the long-range anisotropic interatomic magnetic dipole-dipole interaction. The phase diagram of such a condensate has been explored theoretically and experimentally with certain values of the interatomic scattering length corresponding to superfluid and supersolid phases, where supersolidity appears as a modulation in the ground state density. Preparation of this modulated ground state is challenging, since excitations appear as a result of a finite-time evolution required to produce qualitative changes in the wavefunction density. To solve this problem we consider the time-dependent control of a dipolar Bose-Einstein condensate using shortcuts to adiabaticity techniques, concentrating on design of the time-dependent scattering length, a parameter of the system easily tunable by contemporary experiments. The first technique is the variational approach based on the Euler-Lagrange equations for a separable ansatz describing the evolution of the superfluid state. Secondly, we study the transition from superfluid to supersolid using a direct optimization protocol. We discuss the fidelity of the developed protocols in terms of the evolution time.
We investigate the quench dynamics of quasi-one- and two-dimensional dipolar Bose-Einstein condensates of 164Dy atoms under the influence of a fast rotating magnetic field. The magnetic field thus controls both the magnitude and sign of the dipolar potential. We account for quantum fluctuations, critical to formation of exotic quantum droplet and supersolid phases in the extended Gross-Pitaevskii formalism, which includes the so-called Lee-Huang-Yang correction. An analytical variational ansatz allows us to obtain the phase diagrams of the superfluid and droplet phases. The crossover from the superfluid to the supersolid phase and to single and droplet arrays is probed with particle number and dipolar interaction. The dipolar strength is tuned by rotating the magnetic field with subsequent effects on phase boundaries. Following interaction quenches across the aforementioned phases, we monitor the dynamical formation of supersolid clusters or droplet lattices. We include losses due to three-body recombination over the crossover regime, where the three-body recombination rate coefficient scales with the fourth power of the scattering length (as) or the dipole length (add). For fixed values of the dimensionless parameter, ${\ensuremath{\epsilon}}_{dd}={a}_{dd}/{a}_{s}$, tuning the dipolar anisotropy leads to an enhancement of the droplet lifetimes.
In this work several aspects of dipolar Bose-Einstein condensates are investigated. These condensates have attracted much interest since their experimental realization and offer the chance to explore the novel, fascinating, and sometimes unexpected physics originating from the dipole-dipole interaction. For the theoretical description the time-dependent variational principle with an ansatz of coupled Gaussian wave packets is applied to the analysis of this form of matter. In order to describe physical systems, where the shape of the condensate wave function cannot be approximated adequately by origin-centered Gaussian functions, translational and rotational degrees of freedom of the particular wave packets are included. This provides not only a full-fledged alternative method to full-numerical grid calculations that are demanding in computation time, used for the determination of ground states and investigation of the real-time dynamics, but also allows for the calculation of excited states and the examination of stability properties and offers the possibility for a deeper physical understanding of dipolar condensates. The variational method is applied to three different physical systems. In the first one anisotropic quasi-two-dimensional solitons are investigated. Ground states and small excitations are presented and the full flexibility of the approach is utilized in the simulation of the collision of two solitons. Thereby simulations of collisions are performed, where initially the solitons are in the repelling side-by-side configuration and move towards each other with a specific momentum. It is shown that the results are in excellent agreement with full-numerical grid calculations. Collisions with various initial parameters such as the initial momentum and impact factor are performed, allowing to classify the qualitative interactions of two solitons and illuminating the effects of the solitons' two-dimensional nature on the collision process. In the second system dipolar Bose-Einstein condensates in a triple-well potential are investigated. This constitutes a well-suited model system for periodic optical potentials with important contributions of the non-local and anisotropic dipole-dipole interaction, which show a variety of effects such as self-organization and formation of patterns. Here, a macroscopic sample of dipolar bosons in the mean-field limit is addressed. The analysis goes beyond the calculation of ground states and clarifies the role of excited and metastable states in such systems. In particular, the formation of phases is found to originate from the interplay of several states with distinct stability properties. As some of the phases are formed by metastable states, special attention is paid to the characteristics of phase transitions in real time and the dynamical stabilization of the condensate. The dynamical simulations that support these results, constitute a way for an experimental observation of these findings. The third physical application is an open quantum system. Here, an external potential models gain and loss of particles and features parity with simultaneous time-reversal symmetry. For these dipolar condensates real eigenvalues of the nonlinear Hamiltonian are found, preserving the PT-symmetry of the external potential, as well as states that break this symmetry. A detailed stability analysis of these, in general complex states, is performed. The comparison with a similar system featuring a non-dipolar condensate demonstrates the role of the dipolar interaction for novel effects. Furthermore, the real-time dynamics is investigated, which reveals, among other effects, long-living oscillations of the two wells' population.
Recently, experimental and theoretical studies have been actively conducted on the supersolid state of a Bose-Einstein condensate (BEC) with magnetic dipole-dipole interaction (DDI).However, most of these studies have employed a single-component BEC, whose stabilization relies on the beyondmean-field effect with high density, and thus its lifetime is restricted.In this work, we propose a supersolid state of a two-component dipolar BEC, in which a weaker DDI can form the supersolid structure, stabilizing the system without the beyond-mean-field effect.As a result, such a supersolid has a longer lifetime, which enables the observation of long-time dynamics.Here we numerically demonstrate the dynamics of periodic droplet-structure formation triggered by sudden or gradual change of a parameter.We find that the generated droplets have the phase coherence, confirming dynamical formation of supersolid.
The formation of quantized vortices above a certain critical trap rotation frequency serves as a hallmark signature of superfluidity. Based on the beyond mean-field framework, crucial for the formation of exotic supersolid and droplet states, we investigate dynamical protocols for vortex nucleation in the superfluid and supersolid states of a dipolar Bose-Einstein condensate (BEC) at a trap rotation frequency lower than the conventional critical rotation frequency. We find that the critical rotation frequency depends not only on the dipole-dipole interaction strength, but also on the polarization direction of the external magnetic field. Interestingly, in the superfluid state, the critical rotation frequency increases with the tilt angle of the magnetic field, whereas in the supersolid state, increasing the tilt angle lowers the critical threshold. Leveraging these characteristics of dipolar BECs, along with the robustness of vortices with respect to changes in the polarization direction and s-wave scattering length, we demonstrate three dynamical protocols for vortex nucleation, even when rotating below the critical rotation frequency, viz., (i) varying the s-wave scattering length, (ii) changing the polarizing angle, and (iii) successive modulation of both the scattering length and polarizing angle. These dynamical vortex seeding protocols could serve as important benchmarks for future experimental studies.
To engineer the wavefunction of a Bose–Einstein condensate is to exert control over both the density and phase of the Bose–Einstein condensate order parameter. Having the ability to engineer the condensate order parameter down to the smallest length scale relevant to condensate dynamics—the healing length scale—would enable the study of new combinations of topological defects and may pave the way to using Bose–Einstein condensates as versatile, precise quantum simulators. In this thesis, we present a new wavefunction engineering technique which reaches the sub-optical healing length scale. Influenced by magnetic resonance imaging, we name this technique magnetic resonance control. This technique uses time-varying coupling between internal spin states of a spinor Bose–Einstein condensate within a magnetic field gradient to address spatial regions of the condensate, enabling control over both the phase and the density of the condensate order parameter down to the healing length scale. Techniques already exist to engineer condensate wavefunctions, but not with such a fine degree of control. These techniques primarily rely on either the intensity variation of a laser beam, which limits the resolution to the diffraction limit (larger than the typically sub-optical healing length scale), or the adiabatic inversion of a magnetic trapping potential, which can not be easily changed to produce a variety of structures in the condensate wavefunction. To develop our magnetic resonance control technique, we simulate a spinor condensate in one dimension with time-dependent coupling between spin states and time-dependent external magnetic field gradients using the Gross–Pitaevskii equation. We show that magnetic resonance control can engineer a single black soliton using experimentally feasible parameters. A black soliton is an ideal target state to select for this demonstration because engineering such a state requires control over both the phase and density of the condensate with healing-length resolution. We demonstrate that magnetic resonance control can be extended to engineer more complicated target states by simulating the creation of multiple solitons in a condensate, with control over the initial positions and trajectories of the solitons. When magnetic resonance control is applied to Bose–Einstein condensates in the laboratory, it will be necessary to have an imaging system capable of resolving the fine structures created. As an alternative to high-cost, custom-manufactured lenses, and in-vacuum optical systems, I have designed and bench-tested an objective lens with a high numerical aperture (0.36) and a long working distance (35 mm) consisting of standard catalogue lenses. Using 780 nm light, suitable for imaging rubidium condensates, this objective can achieve a resolution of 1.3 μm across a diffraction-limited field of view of 360 μm through a 5 mm thick glass window of a science cell. By changing the spacing between the lens elements, this objective lens can compensate for the aberrations produced by a glass window up to 15 mm thick, and by changing the aperture size the objective becomes suitable for diffraction-limited monochromatic imaging on the D2 line of all the alkalis. Before performing proof-of-principle magnetic resonance control experiments on real Bose–Einstein condensates, we needed to construct an experimental apparatus capable of producing spinor Bose–Einstein condensates. In this thesis I summarise my main contributions to this group endeavour, including: constructing the ultra-high vacuum system; supervising the bakeout of our vacuum system; designing and aligning the optical systems to produce laser beams of different, tunable frequencies to trap, cool, and image rubidium gas; trapping a cloud of rubidium atoms in a magneto-optical trap; constructing laser beam shutters, photodetectors, and photodetector signal filters; and designing and constructing our side imaging and top imaging systems. Using our spinor Bose–Einstein condensate apparatus, I performed the first proof-of-principle magnetic resonance control experiments. With a pair of condensates side-by-side, separated by 200 μm, we can use magnetic resonance control to invert the spin of one condensate only, while leaving the other condensate unaffected. We anticipate magnetic resonance control being used in the laboratory to engineer the first black soliton in a Bose–Einstein condensate. Looking beyond solitons, magnetic resonance control could have applications to the field of magnon spintronics, and extending the technique to higher dimensions could enable the study of exotic topological defects such as spin knots in a quantum fluid.