Difference between revisions of "Nonlinear Dynamics"
| Line 1: | Line 1: | ||
| − | === |
+ | === Physics of Complex Systems === |
| + | Rydberg molecules |
||
| − | Suppose we would be able to unleash the power of the quantum world in ways which would have been unthinkable only a few years ago. For instance, we could use quantum superposition, the possibility for a quantum bit to contain all conceivable and mutually excluding classical states in itself. Then, in a single computational step, we could realize the parallel processing of all these classical states, whose number grows exponentially with the number of classical bits involved, through the quantum state evolution of this single state. That is the vision of quantum parallelism, which is one of the driving forces of quantum computing, and at the same time one of the fastest growing areas of research in the last decade or so. These strategies have all been made possible with new techniques capable to produce, manipulate, and detect single quanta, such as photons, neutrons and electrons. |
||
| + | Rydberg atoms are true giants in the atomic world with sizes reaching 10 micrometer for n=300 and constitute mesoscopic entities. They allow to probe classical-quantum correspondence and the transition from the quantum to the classical world. They also provide a useful testing ground for concepts of non-linear dynamics since even moderate external (magnetic or electric) fields provide strong non-separable perturbations. Rydberg atoms have recently attracted attention also in the quest for building blocks of quantum information systems. Rydberg atoms have been proposed as a quantum phase register and as quantum gates employing the Rydberg dipole blockade. The latter makes use of the strong long-range dipole-dipole interactions which conditionally block excitations of atoms in the vicinity of a Rydberg atom already created. More generally, an ensemble of ultracold Rydberg atoms constitutes a strongly interacting many-body system far from the ground state. |
||
| − | Suppose someone claims that the chances of rain in Vienna and Budapest are 0.1 in each one of the cities alone, and the joint probability of rainfall in both cities is 0.99. Would such a proposition appear reasonable? Certainly not, for even intuitively it does not make much sense to claim that it rains almost never in one of the cities, yet almost always in both of them. The worrying question remains: which numbers could be considered reasonable and consistent? Surely, the joint probability should not exceed any single probability. This certainly appears to be a necessary condition, but is it a sufficient one? Boole, and much later Bell - already in the quantum mechanical context and with a specific class of experiment in mind - derived constraints on the classical probabilities from the formalization of such considerations. In a way, these bounds originate from the conception that all classical probability distributions are just convex sums of extreme ones, which can be characterized by two-valued measures interpretable as classical truth values. They form a convex polytope bounded by Boole-Bell-type inequalities. |
||
| + | Rydberg molecules are another interesting topic which can, for example, be used as a sensitive probe for the spatial distribution of ultracold gases. When a Rydberg atom is excited in a dense gas of atoms, one or more ground-state atom(s) can be found within the Rydberg electron orbit. For an attractive interaction between the quasi-free Rydberg electron and a ground-state atom an ultralong-range Rydberg molecule can be formed. |
||
| − | [[Image:Alice_eve_bob.jpg|Quantum cryptography Alice Eve Bob]] <br /> |
||
| + | Typically the interaction and the resulting binding energy are very small and a low kinetic energy of a ground state atom is required and the formation of Rydberg molecules is observed in an ultracold gas of atoms (of the order of 1 μK or less). Within the Born-Oppenheimer approximation, the molecular potential experienced by a ground state atom is approximately proportional to the squared wave function of Rydberg electron. This is because the interaction is larger at higher probability density of Rydberg electron (Fig.1). The vibrational levels are formed by trapping a ground state atom within these potential wells. |
||
| − | Fig.: Quantum cryptography using single-photon sources. (copyright) http://www.epfl.ch |
||
| − | Remarkable, quantum probability theory is entirely different from classical probability theory, as it allows a statistics of the joint occurrence of events which extends and violates Boole's and Bell's classical constraints. Alas, quantum mechanics does not violate the constraints maximally, quantum bounds fall just "in-between" the classical and maximal bounds. |
||
| + | Fig.1 : Molecular potential (black line) for a 5s38s 3S1 strontium atom together with the wavefunctions for the v = 0 (red solid line), v = 1 (green dashed line), and v = 2 (blue dot-dashed line) molecular vibrational states. The binding energy corresponding to each wave function is indicated by its axis. |
||
| + | For a Rydberg dimer the lowest energy vibrational state has a wavefunction well localized in space around Rn = 2 n2 (a.u.) (Fig.1). Thus, the probability of creating the dimer molecule will depend on the likelihood of initially finding a pair of ground-state atoms with the appropriate internuclear separation, Rn. By varying the principal quantum number of the Rydberg atom the wavefunction can be localized at different positions and a pair distribution at different distances can be probed. For example, the excitation of Rydberg dimers between n=30 and 50 leads to a pair correlation in the range of R = 90nm and 250nm. This technique has been applied to an ultracold gas of bosons as well as fermions and the exchange hole in the two-body correlation of a fermion gas has been observed. |
||
| − | === Steering Rydberg wave packets with ultrashort pulses === |
||
| + | The Rydberg molecule can also be extended to involve more than one ground-state atom. An excitation of Rydberg trimer in which two ground-state atoms are bound may open a possibility to probe a three-body correlation of ultracold gases. By increasing the density of a gas or the principal quantum number n, the number of ground state atoms within a Rydberg molecule can be increased. Currently, a Rydberg molecule involving up to about 10000 atoms has been created. |
||
| − | In recent years there has been increasing interest in the control and manipulation of atomic wave functions. The engineering of wave functions promises applications in many areas of physics, such as quantum computing, promotion of chemical reactions towards any preferable direction, or optimization of high harmonic generation. Theoretically any wave function can be formed as a coherent superposition of energy eigenstates. In practice, however, it is not an easy task to prepare a preselected target state experimentally. Thus there are increasing demands for establishing protocols to produce any preferable designer state starting from the states which are experimentally accessible. Recently a few protocols have been suggested to create and manipulate a wave packet. |
||
| + | For more information about this topic, please consult the following websites: |
||
| − | [[image:Kicked_atom.jpg|Poincaré surface for periodically kicked atom]] <br /> |
||
| + | http://dollywood.itp.tuwien.ac.at/~shuhei/ |
||
| − | Fig. 1: Poincaré surface of section for the periodically kicked atom by a train of kicks with <math>v = 1.095</math> and <math>\Delta p = -0.1</math>. A periodic orbit (blue dashed line) is located |
||
| + | https://tqd.itp.tuwien.ac.at/ |
||
| − | at the center (green cross) of main stable island (red) in the Poincaré surface. The |
||
| − | upper frame explains graphically how the periodic orbit can be stabilized. |
||
| + | Ultrashort pulses |
||
| + | Ultrashort pulses as first characterized at TU Wien are the ultimate tool to investigate the fundamentals of light-matter interactions and the meaning of time and time differences in quantum mechanics. |
||
| − | A Rydberg wave packet is a coherent superposition of highly excited atomic states, localized in phase space. Due to the relatively large time and spatial scale (<math>t</math> ~ <math>n^3</math> and <math>r</math> ~ <math>n^2</math>) of Rydberg atoms with quantum number n, Rydberg wave packets are known to be among the best explored quantum objects which approximately follow the dynamics of the corresponding classical particle and serve as benchmark for probing the crossover between classical and quantum dynamics. With recent advances in ultra-short pulse generation it has become possible to engineer wave packets using Rydberg atoms . Using such a Rydberg wave packet as the initial state, we have demonstrated a few protocols to steer such a Rydberg wave packet towards any preferable location in phase space or to manipulate the size of a wave packet using a train of short pulses, so-called half-cycle pulses (HCPs). |
||
| + | We have studied, e.g., the full two-electron dynamics of helium atoms irradiated by strong and short laser pulses to calculate the apparent time delay of photoemission from different initial states of the parent atom (together with MPQ Garching, TU Munich). To this end, we have solved the two-electron Schrödinger equation for helium and have successfully simulated so-called attosecond streaking and RABBITT experiments both of which measure the relative time delay of different electron groups reaching the detector. Our simulation results are used to determine time delays on an absolute scale and have allowed for a detailed look into photon-induced excitation processes in atoms. |
||
| + | Our simulation results were also used to work out the time evolution of the temporal build-up of interference structures in photoemission (Fano resonances, MPI Heidelberg). These interferences are caused by two alternative pathways to the same final state, e.g., direct emission vs. delayed emission via a metastable excited state. Pump-probe experiments with variable delay between the two pulses interrupt the build-up of the resonance and allow for taking snapshots of the process thereby showing the temporal evolution of the multipath interference. |
||
| − | The response of Rydberg atoms to a train of identical HCPs equispaced in time has been studied extensively revealing a wide variety of dynamical behaviors. Under the influence of a periodical train of kicks, the electron experiences a random sequence of energy transfers <math>\Delta E</math> leading to a random-walk behavior in energy space. On the other hand, by tuning the frequency of a train of kicks near the Kepler orbital frequency and setting the kick strength <math>\Delta p = - 2 p</math> to satisfy <math>\Delta E=0</math>, the motion of the electron can be synchronized with the periodic train and stabilized without any energy transfer. This motion is analogous to a tennis ball (electron) hitting a wall (nucleus as a scatterer). At each hit (kick) by a racket the tennis ball changes only its direction of motion <i>i.e.</i> <math> p_{after} = p_{before} + \Delta p = - p_{before}</math> when <math>\Delta p = - 2 p_{before}</math>. By hitting a ball with a proper frequency a periodic motion can be established. This idea of dynamical stabilization has been used to create a wave packet localized in phase space. The main stable island (red) seen in the Poincaré surface of section is a manifestation of a periodic motion and quasi-periodic trajectories surrounding it. Classical trajectories inside the island are kept trapped as long as a train of kicks is applied and the trajectories outside are spread out over whole phase space and eventually this unbounded motion (blue chaotic sea) leads to ionization. Another consequence of the island structure is that parts of the quantum wave function outside the islands get trimmed off by the periodic pulse and consequently the wave packet will be well localized inside the island. |
||
| + | Fig. 2: Temporal build-up of a two-path interference in the photoionization of helium. Results of our simulations are compared to experimental data. Asymptotically, the energy spectrum converges to the well-known Fano shape. |
||
| + | Another interesting application of attosecond science is the non-linear upconversion of photons of a strong incident laser pulse to high-order multiples of the original photon energy in a process called high-harmonics generation. Recently, upconversion factors of more than 1000 could be reached at TU Wien (Institute for Photonics) in this process. Our work helps to interpret the experimental results and to optimize this process aimed at generating XUV pulses of sub-as duration. |
||
| − | === Atoms in ultrashort laser fields === |
||
| + | Lately, attosecond experiments have also been performed on extended systems such as dielectrics, where the high target-atom density raises hopes to generate harmonics with larger intensity. Due to the inherent multi-particle nature of such systems, simplified models have been invoked that have given first qualitative insight in the light-driven electronic processes in dielectrics. However, we could show that such simplifications often fail to reproduce even the qualitative behavior of realistic systems let alone provide a quantitative prediction for any observable in experiment. We have taken the first steps to a multi-scale description of laser-solid interactions combining the microscopic electronic motion described by the Schödinger equation with the mesoscopic world of light propagation (Maxwell's equations). |
||
| − | Since the first working laser device was built by Maiman in 1960, the progress in laser technology has been tremendous. The intensity of the lasers has been increased by many orders of magnitude. Intensities reach presently well above 1020W/cm2, where plasma effects as well as relativistic effects are important. In the near future, laser intensities may even reach the critical field strength to directly produce positron-electron pairs. At the same time, the length of the shortest pulses has decreased by more than 10 orders of magnitude. While the first lasers had a pulse length of some 100ms, very short pulses can nowadays be produced through mode-locking. In 1990, Zewail et al. managed to generate pulses as short as several femtoseconds, which meant that snapshots of chemical reactions could be directly taken. This opened up the field of femto-chemistry. |
||
| + | Fig. 3: High harmonic spectrum induced in diamond by a linearly polarized few-cycle laser pulse as a function of the position along the propagation direction inside a 1 µm thick diamond crystal. |
||
| − | [[Image:Lenght_pulse.jpg|Pulse duration]] <br /> |
||
| − | Fig.: Decrease of pulse duration as a function of time. |
||
| + | Working together with experimental groups in Munich, Zurich, and Graz we have elucidated the interplay of different processes involved in light-solid and light-liquid interactions and have helped interpret fundamental experiments in the field possibly paving the way to light-driven electronics on the PHz scale. |
||
| − | The possibility of driving an atom by a femtosecond laser as well as the usage of high harmonics generation to producing the shortest pulses presently available challenges our current understanding of the processes taking place in the atom driven by the ultrashort electric field. Two different regimes can be distinguished: the multiphoton regime (high frequency and low intensity) and the tunneling regime (low frequency and high intensity). In the multiphoton regime many experimental (see for example [2]) and theoretical studies have been performed, which have led to a fairly complete understanding of the physical processes involved. In the tunneling regime, on the other hand, recent experiments with linearly polarized lasers have shown novel and previously unexplained structures in the momentum distribution of the photoionized electrons in rare gases. The so-called "double-hump" structure in the longitudinal momentum distribution has been identified as a rescattering process for double-ionization and as the interaction between the electron and the core for single ionization. |
||
| + | For more information about this topic, please consult the following websites: |
||
| − | We study the hydrogen atom driven by a linearly polarized laser field both classically and quantum mechanically. For the first approach we employ the classical trajectory Monte Carlo (CTMC) method including tunnel effects (CTMC-T). The electron is allowed to tunnel through the potential barrier whenever it reaches the outer turning point. Alternatively, the time-dependent Schrödinger equation is solved numerically by means of the generalized pseudo-spectral method. The process of detecting an electron of momentum [k\vec] can then be viewed as a projection of the wave function onto the Coulomb wave functions. |
||
| + | http://concord.itp.tuwien.ac.at/~lemell/ |
||
| + | https://tqd.itp.tuwien.ac.at/ |
||
| + | Dynamics of many-body systems |
||
| + | Quantum many-body systems are at the core of current interest both in experimental as well as in theoretical physics, since a better understanding of these systems bears a large potential for technology and applications. Table-top sources of coherent X-ray light from strongly driven gases of atoms might become available in near future. Highly accurate gravimeters are being constructed using the coherence of matter waves. Petahertz electronics is at reach based on the driving of currents in dielectrics by femtosecond laser pulses. All these applications share that the underlying effects originate from or exploit non-equilibrium quantum matter. Many novel experimental tools are being developed that allow for unprecedented driving for example by strong and ultrashort laser pulses, by bichromatic fields, or by structured light, and are being used to explore new effects and applications. |
||
| + | Fig.4: The two-particle reduced density matrix (left) and the cumulant measuring two-particle correlations (right) of the beryllium atom. |
||
| ⚫ | |||
| + | At the same time, theoretical tools to describe these systems are lagging behind those available for systems at rest. In our research, we address exactly this discrepancy. We use and develop novel theoretical approaches to describe quantum systems out of equilibrium. Starting point of our investigations into non-equilibrium quantum matter is the many-body time-dependent Schrödinger equation. The complexity of this equation increases exponentially with the number of particles in the system. It thus cannot be solved numerically exactly except for the simplest systems consisting of just a few particles. A simple estimate shows that storing the wavefunction of, e.g. , the lithium atom with reasonable resolution would exceed the storage capacity of current supercomputers not to mention performing calculations with it. One way around is to avoid the wave function and use a reduced object such as the particle density. The time-dependent density functional theory, e.g., follows these lines by using the particle density as the fundamental object. However, it suffers from unknown energy functionals because quantum correlations cannot be easily taken into account. We have developed a time-dependent quantum many-body approach that uses the two-particle reduced density matrix instead. In this way all two-particle correlations are incorporated. Since all fundamental interactions can be regarded as pairwise interactions, two-body correlations are the most important ones. With the new method, we were able to solve a wide variety of problems ranging from the multi-electron dynamics of atoms driven by strong and ultrashort pulses to quench dynamics of ultracold atoms in optical lattices. |
||
| + | |||
| + | For more information about this topic, please consult the following website: |
||
| + | https://tqd.itp.tuwien.ac.at/ |
||
| + | |||
| ⚫ | |||
| + | |||
| + | Novel nanomaterials and their heterostructures play a central role in current computational material science research. Recent experimental progress has seen an appearance of several two-dimensional solids with a wide range of electronic properties, ranging from the famous semi-metal graphene to strongly insulating hexagonal boron nitride. Modern synthesis techniques for the creation and manipulation of stable layers of two-dimensional crystals have now become well developed. Likewise, noble metal nanoclusters feature high catalytic reactivity and sharp plasmonic resonances that can be tailored to specific energies. By depositing and subsequent transfers, individual layers may be combined like a sandwich, stacking layers in a predefined sequence. The resulting van der Waals heterostructures exhibit interesting new effects that go beyond the physics accessible only by a single layer. Nanostructure devices composed of such components promise a wide range of applications, from highly efficient catalysts or solar cells to ultra-low-power nanoelectronics. |
||
| + | |||
| + | Our research focuses on simulating new nanostructure materials composing realistic nanodevices, including defects and impurities. The term nanodevice in this context refers to the typical device size, up to a few micrometers and containing millions of atoms, but still below mesoscopic dimensions. Consequently, quantum effects play an important role. The theoretical description of these systems thus poses a challenging multi-scale problem, requiring active method development. We simulate transport through nanodevices built by our experimental collaborators (for example at RWTH Aachen or ETH Zürich) as well as the electronic structure and properties of low-dimensional materials and their heterostructures. |
||
| + | Fig. 5: (left panel) Graphene nano-constriction sandwich device fabricated by the Stampfer group at RWTH Aachen, superposed with the image of a scattering state calculated using our tight-binding approach. Scale bar is 500 nm. (Middle panel) Conductance measurements for two different cool-downs (green and black) and our theory (blue). (Right panel) Local density of states in a graphene nano-constriction for energies close (upper panel) and far away (lower panel) from the Fermi energy. |
||
| + | |||
| + | |||
| + | Fig. 6: (a) Hexagonal graphene flake on single hBN layer (carbon: blue, nitrogen: green, boron: red) and substrate. The lattice mismatch generates a moiré pattern with unit length depending on the angle of rotation between the two layers (yellow diamond). (b) Graphene flake in an effective embedding potential which replaces hBN and the substrate - the system size is strongly reduced. (c,d) Density of states of graphene on hexagonal boron nitride, as a function of magnetic field for (c) experiment [Yu 2014] and (d) our theoretical simulations. Notice the linear structures (dashed lines) above and below the Dirac point that emerge due to superlattice effects. |
||
| + | |||
| + | A small lattice mismatch between adjacent layers, for example for graphene and hexagonal boron nitride, gives rise to regular, periodic moiré patterns [see Fig. 6(a)]. Even in structures composed of layers of the same material, twisting the layers with respect to each other will induce moiré potentials, whose periodicity sensitively depends on the twist angle. The resulting heterostructures feature altered electronic properties such as unconventional superconductivity or Mott-insulating phases. |
||
| + | From a broader scope, twist angles allow for modifying the band structure of the heterostructure in surprising ways, promising a pathway towards engineering of desired material properties. Unfortunately, theoretical treatment of the large unit cells of moiré patterns, including a substrate and adjacent functional layers makes a full ab-initio treatment challenging. In an ongoing collaboration with Allan McDonald (Houston, TX, USA), we are developing moiré potentials for twisted trilayer graphene as well as transition metal dichalcogenides. Using an effective moiré potential [see Fig. 6(b)], we can reproduce the observed fine structures resulting from the moiré, see Fig. 6(c,d). Using a quantum dot induced by an STM tip, our experimental collaborators at RWTH Aachen were also able to directly probe moiré potentials, which compared well to our simulations. |
||
| + | |||
| + | For more information about this topic, please consult the following websites: |
||
| + | http://dollywood.itp.tuwien.ac.at/~florian/ |
||
| + | https://tqd.itp.tuwien.ac.at/ |
||
| + | |||
| + | Non-Hermitian physics and complex scattering |
||
| + | |||
| + | The propagation of waves is a topic spanning many spatial and temporal scales, from the fundamental quantum aspects of light-matter interaction to the scattering of radio waves in complex media like the earth’s atmosphere. On all these levels the input from theoretical physics is essential for understanding and controlling such wave phenomena. At the institute, considerable effort is being dedicated to two specific topics in the vast field of wave physics: (i) the non-Hermitian physics associated with waves in systems that are subject to both amplification (gain) and dissipation (loss) as well as (ii) the scattering of waves in disordered media. In both of these research areas, we collaborate strongly with various experimental teams; to come as closely as possible to the situation encountered in the laboratory, we employ numerical techniques and run simulations on the Vienna Scientific Cluster. |
||
| + | In the field of non-Hermitian physics (i), we are especially interested in controlling the scattering properties of systems by tailoring the spatial distribution of gain and loss in them. Using such an approach, we found, e.g., that a highly disordered system can be made completely transparent and even invisible by adding a tailored gain/loss distribution to it [see Fig. 7(a) for the case of a Gaussian laser beam]. Special attention also receives a particular non-Hermitian singularity, called an exceptional point, that leads to quite a number of fascinating phenomena that we could recently demonstrate in collaboration with different experimental groups in nano-photonics and in laser physics. |
||
| + | |||
| + | |||
| + | |||
| + | |||
| + | Fig. 7: (a) A Gaussian laser beam entering a disordered medium from the left gets scattered and builds up a highly complicated interference pattern (left panel). By adding a tailored distribution of gain and loss to this medium, the beam can propagate like in free space (right panel). (b) A specially designed laser beam that applies a well-defined torque onto the quadratic target in the middle, turning it in clockwise direction. |
||
| + | |||
| + | In the field of complex scattering (ii) we are pursuing the challenging goal of controlling how light propagates through disordered media. (Think here of the speckle patterns arising when directing a laser beam at a piece of paper.) How to deal with the complex interferences arising in this context is a challenging question arising in many fields of physics—from biomedical optics to observational astronomy. What comes to our advantage here is the fact that scattering is a deterministic process—at least for classical waves—such that the shape of an incident wave front determines how the wave will propagate through a medium. This insight forms the basis for a series of modern experiments that use spatial light modulators to characterize and to control light fields even in strongly disordered media. Our contributions to this newly emerging field of wave front shaping include, e.g., a concept to generate waves that follow a specific path across a disordered medium or that focus onto a designated point inside of it. Moreover, we also showed how to design waves in order to micro-manipulate a target embedded inside a disordered environment [see Fig. 7(b) for an example]. By tuning the incident wave front we could recently achieve the first realization of a random anti-laser, i.e. the time-reverse of a random laser in the sense that a random medium was shown to perfectly absorb a suitably engineered incoming wave front. |
||
| + | |||
| + | For more information about this topic, please consult the following website: |
||
| + | https://rottergroup.itp.tuwien.ac.at |
||
| − | A major aim in ballistic transport theory is to simulate and stimulate experiments in the field of phase-coherent electron conductance through nano-scaled semiconductor devices. However, even for two-dimensional quantum dots ("quantum billiards") the numerical solution of the Schrödinger equation has remained a computational challenge. This is partly due to the fact, that many of the most interesting phenomena occur in a parameter regime of either high magnetic field B or small de Broglie wavelength lD. |
||
| − | However interesting they may be, these parameter ranges are difficult to handle from a computational point of view. This is because in the ``semi-classical regime'' of small lD as well as in the ``quantum Hall regime'' of high magnetic fields B, the proper description of the transport process requires a large number of basis functions. As a result, the theoretical models which are presently being employed eventually become computationally unfeasible or numerically instable. |
||
| − | [[Image:MRGM_grid.jpg|Recursive Green's Function Method]] <br /> |
||
| − | Fig.: (a) Illustration of the conventional tight-binding discretization employed in the Recursive Green's Function Method for transport through a circular quantum dot with innite leads. Our modular approach as illustrated in (b) leads to increased efficiency in the numerical calculations. |
||
| − | At the Institute for Theoretical Physics an extension of the widely used Recursive Green's Function Method (RGM) was developed which can bypass several of the limitations of conventional techniques. Key ingredient of this approach is the decomposition of the scattering geometry into separable substructures ("modules") for which all the numerical procedures can be performed very effectively. All the modules are eventually connected with each other by means of matrix Dyson equations such that they span the entire scattering region. In this way we reach a high degree of computational efficiency. |
||
Revision as of 13:46, 11 May 2020
Physics of Complex Systems
Rydberg molecules
Rydberg atoms are true giants in the atomic world with sizes reaching 10 micrometer for n=300 and constitute mesoscopic entities. They allow to probe classical-quantum correspondence and the transition from the quantum to the classical world. They also provide a useful testing ground for concepts of non-linear dynamics since even moderate external (magnetic or electric) fields provide strong non-separable perturbations. Rydberg atoms have recently attracted attention also in the quest for building blocks of quantum information systems. Rydberg atoms have been proposed as a quantum phase register and as quantum gates employing the Rydberg dipole blockade. The latter makes use of the strong long-range dipole-dipole interactions which conditionally block excitations of atoms in the vicinity of a Rydberg atom already created. More generally, an ensemble of ultracold Rydberg atoms constitutes a strongly interacting many-body system far from the ground state.
Rydberg molecules are another interesting topic which can, for example, be used as a sensitive probe for the spatial distribution of ultracold gases. When a Rydberg atom is excited in a dense gas of atoms, one or more ground-state atom(s) can be found within the Rydberg electron orbit. For an attractive interaction between the quasi-free Rydberg electron and a ground-state atom an ultralong-range Rydberg molecule can be formed. Typically the interaction and the resulting binding energy are very small and a low kinetic energy of a ground state atom is required and the formation of Rydberg molecules is observed in an ultracold gas of atoms (of the order of 1 μK or less). Within the Born-Oppenheimer approximation, the molecular potential experienced by a ground state atom is approximately proportional to the squared wave function of Rydberg electron. This is because the interaction is larger at higher probability density of Rydberg electron (Fig.1). The vibrational levels are formed by trapping a ground state atom within these potential wells.
Fig.1 : Molecular potential (black line) for a 5s38s 3S1 strontium atom together with the wavefunctions for the v = 0 (red solid line), v = 1 (green dashed line), and v = 2 (blue dot-dashed line) molecular vibrational states. The binding energy corresponding to each wave function is indicated by its axis.
For a Rydberg dimer the lowest energy vibrational state has a wavefunction well localized in space around Rn = 2 n2 (a.u.) (Fig.1). Thus, the probability of creating the dimer molecule will depend on the likelihood of initially finding a pair of ground-state atoms with the appropriate internuclear separation, Rn. By varying the principal quantum number of the Rydberg atom the wavefunction can be localized at different positions and a pair distribution at different distances can be probed. For example, the excitation of Rydberg dimers between n=30 and 50 leads to a pair correlation in the range of R = 90nm and 250nm. This technique has been applied to an ultracold gas of bosons as well as fermions and the exchange hole in the two-body correlation of a fermion gas has been observed.
The Rydberg molecule can also be extended to involve more than one ground-state atom. An excitation of Rydberg trimer in which two ground-state atoms are bound may open a possibility to probe a three-body correlation of ultracold gases. By increasing the density of a gas or the principal quantum number n, the number of ground state atoms within a Rydberg molecule can be increased. Currently, a Rydberg molecule involving up to about 10000 atoms has been created.
For more information about this topic, please consult the following websites: http://dollywood.itp.tuwien.ac.at/~shuhei/ https://tqd.itp.tuwien.ac.at/
Ultrashort pulses
Ultrashort pulses as first characterized at TU Wien are the ultimate tool to investigate the fundamentals of light-matter interactions and the meaning of time and time differences in quantum mechanics. We have studied, e.g., the full two-electron dynamics of helium atoms irradiated by strong and short laser pulses to calculate the apparent time delay of photoemission from different initial states of the parent atom (together with MPQ Garching, TU Munich). To this end, we have solved the two-electron Schrödinger equation for helium and have successfully simulated so-called attosecond streaking and RABBITT experiments both of which measure the relative time delay of different electron groups reaching the detector. Our simulation results are used to determine time delays on an absolute scale and have allowed for a detailed look into photon-induced excitation processes in atoms. Our simulation results were also used to work out the time evolution of the temporal build-up of interference structures in photoemission (Fano resonances, MPI Heidelberg). These interferences are caused by two alternative pathways to the same final state, e.g., direct emission vs. delayed emission via a metastable excited state. Pump-probe experiments with variable delay between the two pulses interrupt the build-up of the resonance and allow for taking snapshots of the process thereby showing the temporal evolution of the multipath interference.
Fig. 2: Temporal build-up of a two-path interference in the photoionization of helium. Results of our simulations are compared to experimental data. Asymptotically, the energy spectrum converges to the well-known Fano shape.
Another interesting application of attosecond science is the non-linear upconversion of photons of a strong incident laser pulse to high-order multiples of the original photon energy in a process called high-harmonics generation. Recently, upconversion factors of more than 1000 could be reached at TU Wien (Institute for Photonics) in this process. Our work helps to interpret the experimental results and to optimize this process aimed at generating XUV pulses of sub-as duration. Lately, attosecond experiments have also been performed on extended systems such as dielectrics, where the high target-atom density raises hopes to generate harmonics with larger intensity. Due to the inherent multi-particle nature of such systems, simplified models have been invoked that have given first qualitative insight in the light-driven electronic processes in dielectrics. However, we could show that such simplifications often fail to reproduce even the qualitative behavior of realistic systems let alone provide a quantitative prediction for any observable in experiment. We have taken the first steps to a multi-scale description of laser-solid interactions combining the microscopic electronic motion described by the Schödinger equation with the mesoscopic world of light propagation (Maxwell's equations).
Fig. 3: High harmonic spectrum induced in diamond by a linearly polarized few-cycle laser pulse as a function of the position along the propagation direction inside a 1 µm thick diamond crystal.
Working together with experimental groups in Munich, Zurich, and Graz we have elucidated the interplay of different processes involved in light-solid and light-liquid interactions and have helped interpret fundamental experiments in the field possibly paving the way to light-driven electronics on the PHz scale.
For more information about this topic, please consult the following websites: http://concord.itp.tuwien.ac.at/~lemell/ https://tqd.itp.tuwien.ac.at/
Dynamics of many-body systems Quantum many-body systems are at the core of current interest both in experimental as well as in theoretical physics, since a better understanding of these systems bears a large potential for technology and applications. Table-top sources of coherent X-ray light from strongly driven gases of atoms might become available in near future. Highly accurate gravimeters are being constructed using the coherence of matter waves. Petahertz electronics is at reach based on the driving of currents in dielectrics by femtosecond laser pulses. All these applications share that the underlying effects originate from or exploit non-equilibrium quantum matter. Many novel experimental tools are being developed that allow for unprecedented driving for example by strong and ultrashort laser pulses, by bichromatic fields, or by structured light, and are being used to explore new effects and applications.
Fig.4: The two-particle reduced density matrix (left) and the cumulant measuring two-particle correlations (right) of the beryllium atom. At the same time, theoretical tools to describe these systems are lagging behind those available for systems at rest. In our research, we address exactly this discrepancy. We use and develop novel theoretical approaches to describe quantum systems out of equilibrium. Starting point of our investigations into non-equilibrium quantum matter is the many-body time-dependent Schrödinger equation. The complexity of this equation increases exponentially with the number of particles in the system. It thus cannot be solved numerically exactly except for the simplest systems consisting of just a few particles. A simple estimate shows that storing the wavefunction of, e.g. , the lithium atom with reasonable resolution would exceed the storage capacity of current supercomputers not to mention performing calculations with it. One way around is to avoid the wave function and use a reduced object such as the particle density. The time-dependent density functional theory, e.g., follows these lines by using the particle density as the fundamental object. However, it suffers from unknown energy functionals because quantum correlations cannot be easily taken into account. We have developed a time-dependent quantum many-body approach that uses the two-particle reduced density matrix instead. In this way all two-particle correlations are incorporated. Since all fundamental interactions can be regarded as pairwise interactions, two-body correlations are the most important ones. With the new method, we were able to solve a wide variety of problems ranging from the multi-electron dynamics of atoms driven by strong and ultrashort pulses to quench dynamics of ultracold atoms in optical lattices.
For more information about this topic, please consult the following website: https://tqd.itp.tuwien.ac.at/
Quantum transport through nanostructures
Novel nanomaterials and their heterostructures play a central role in current computational material science research. Recent experimental progress has seen an appearance of several two-dimensional solids with a wide range of electronic properties, ranging from the famous semi-metal graphene to strongly insulating hexagonal boron nitride. Modern synthesis techniques for the creation and manipulation of stable layers of two-dimensional crystals have now become well developed. Likewise, noble metal nanoclusters feature high catalytic reactivity and sharp plasmonic resonances that can be tailored to specific energies. By depositing and subsequent transfers, individual layers may be combined like a sandwich, stacking layers in a predefined sequence. The resulting van der Waals heterostructures exhibit interesting new effects that go beyond the physics accessible only by a single layer. Nanostructure devices composed of such components promise a wide range of applications, from highly efficient catalysts or solar cells to ultra-low-power nanoelectronics.
Our research focuses on simulating new nanostructure materials composing realistic nanodevices, including defects and impurities. The term nanodevice in this context refers to the typical device size, up to a few micrometers and containing millions of atoms, but still below mesoscopic dimensions. Consequently, quantum effects play an important role. The theoretical description of these systems thus poses a challenging multi-scale problem, requiring active method development. We simulate transport through nanodevices built by our experimental collaborators (for example at RWTH Aachen or ETH Zürich) as well as the electronic structure and properties of low-dimensional materials and their heterostructures. Fig. 5: (left panel) Graphene nano-constriction sandwich device fabricated by the Stampfer group at RWTH Aachen, superposed with the image of a scattering state calculated using our tight-binding approach. Scale bar is 500 nm. (Middle panel) Conductance measurements for two different cool-downs (green and black) and our theory (blue). (Right panel) Local density of states in a graphene nano-constriction for energies close (upper panel) and far away (lower panel) from the Fermi energy.
Fig. 6: (a) Hexagonal graphene flake on single hBN layer (carbon: blue, nitrogen: green, boron: red) and substrate. The lattice mismatch generates a moiré pattern with unit length depending on the angle of rotation between the two layers (yellow diamond). (b) Graphene flake in an effective embedding potential which replaces hBN and the substrate - the system size is strongly reduced. (c,d) Density of states of graphene on hexagonal boron nitride, as a function of magnetic field for (c) experiment [Yu 2014] and (d) our theoretical simulations. Notice the linear structures (dashed lines) above and below the Dirac point that emerge due to superlattice effects.
A small lattice mismatch between adjacent layers, for example for graphene and hexagonal boron nitride, gives rise to regular, periodic moiré patterns [see Fig. 6(a)]. Even in structures composed of layers of the same material, twisting the layers with respect to each other will induce moiré potentials, whose periodicity sensitively depends on the twist angle. The resulting heterostructures feature altered electronic properties such as unconventional superconductivity or Mott-insulating phases. From a broader scope, twist angles allow for modifying the band structure of the heterostructure in surprising ways, promising a pathway towards engineering of desired material properties. Unfortunately, theoretical treatment of the large unit cells of moiré patterns, including a substrate and adjacent functional layers makes a full ab-initio treatment challenging. In an ongoing collaboration with Allan McDonald (Houston, TX, USA), we are developing moiré potentials for twisted trilayer graphene as well as transition metal dichalcogenides. Using an effective moiré potential [see Fig. 6(b)], we can reproduce the observed fine structures resulting from the moiré, see Fig. 6(c,d). Using a quantum dot induced by an STM tip, our experimental collaborators at RWTH Aachen were also able to directly probe moiré potentials, which compared well to our simulations.
For more information about this topic, please consult the following websites: http://dollywood.itp.tuwien.ac.at/~florian/ https://tqd.itp.tuwien.ac.at/
Non-Hermitian physics and complex scattering
The propagation of waves is a topic spanning many spatial and temporal scales, from the fundamental quantum aspects of light-matter interaction to the scattering of radio waves in complex media like the earth’s atmosphere. On all these levels the input from theoretical physics is essential for understanding and controlling such wave phenomena. At the institute, considerable effort is being dedicated to two specific topics in the vast field of wave physics: (i) the non-Hermitian physics associated with waves in systems that are subject to both amplification (gain) and dissipation (loss) as well as (ii) the scattering of waves in disordered media. In both of these research areas, we collaborate strongly with various experimental teams; to come as closely as possible to the situation encountered in the laboratory, we employ numerical techniques and run simulations on the Vienna Scientific Cluster. In the field of non-Hermitian physics (i), we are especially interested in controlling the scattering properties of systems by tailoring the spatial distribution of gain and loss in them. Using such an approach, we found, e.g., that a highly disordered system can be made completely transparent and even invisible by adding a tailored gain/loss distribution to it [see Fig. 7(a) for the case of a Gaussian laser beam]. Special attention also receives a particular non-Hermitian singularity, called an exceptional point, that leads to quite a number of fascinating phenomena that we could recently demonstrate in collaboration with different experimental groups in nano-photonics and in laser physics.
Fig. 7: (a) A Gaussian laser beam entering a disordered medium from the left gets scattered and builds up a highly complicated interference pattern (left panel). By adding a tailored distribution of gain and loss to this medium, the beam can propagate like in free space (right panel). (b) A specially designed laser beam that applies a well-defined torque onto the quadratic target in the middle, turning it in clockwise direction.
In the field of complex scattering (ii) we are pursuing the challenging goal of controlling how light propagates through disordered media. (Think here of the speckle patterns arising when directing a laser beam at a piece of paper.) How to deal with the complex interferences arising in this context is a challenging question arising in many fields of physics—from biomedical optics to observational astronomy. What comes to our advantage here is the fact that scattering is a deterministic process—at least for classical waves—such that the shape of an incident wave front determines how the wave will propagate through a medium. This insight forms the basis for a series of modern experiments that use spatial light modulators to characterize and to control light fields even in strongly disordered media. Our contributions to this newly emerging field of wave front shaping include, e.g., a concept to generate waves that follow a specific path across a disordered medium or that focus onto a designated point inside of it. Moreover, we also showed how to design waves in order to micro-manipulate a target embedded inside a disordered environment [see Fig. 7(b) for an example]. By tuning the incident wave front we could recently achieve the first realization of a random anti-laser, i.e. the time-reverse of a random laser in the sense that a random medium was shown to perfectly absorb a suitably engineered incoming wave front.
For more information about this topic, please consult the following website: https://rottergroup.itp.tuwien.ac.at