With the ability to directly obtain the Wigner function and density matrix of photon states, quantum tomography (QT) has had a significant impact on quantum optics, quantum computing and quantum information. By an appropriate sequence of measurements on the evolution of each degreeof freedom (DOF), the full quantum state of the observed photonic system can be determined. The first proposal to extend the application of QT to reconstruction of complete quantum states of matter wavepackets had generated enormous interest in ultrafast diffraction imaging and pump-probe spectroscopy of molecules. This interest was elevated with the advent of ultrafast electron and X-ray diffraction techniques using electron accelerators and X-ray free electron lasers to add temporal resolution to the observed nuclear and electron distributions. However, the great interest in this area has been tempered by the illustration of an impossibility theorem, known as the dimension problem. Not being able to associate unitary evolution to every DOF of molecular motion, quantum tomography could not be used beyond 1D and categorically excludes most vibrational and all rotational motion of molecules. Here we present a theoretical advance to overcome the notorious dimension problem. Solving this challenging problem is important to push imaging molecular dynamics to the quantum limit. The new theory has solved this problem, which makes quantum tomography a truly useful methodology in ultrafast physics and enables the making of quantum version of a molecular movie. With the new theory, quantum tomography can be finally advanced to a sufficient level to become a general method for reconstructing quantum states of matter, without being limited in one dimension. Our new concept is demonstrated using a simulated dataset of ultrafast diffraction experiment of laser-aligned nitrogen molecules.
With the ability to directly obtain the Wigner function and density matrix of photon states, quantum tomography (QT) has had a significant impact on quantum optics, quantum computing and quantum information. By an appropriate sequence of measurements on the evolution of each degreeof freedom (DOF), the full quantum state of the observed photonic system can be determined. The first proposal to extend the application of QT to reconstruction of complete quantum states of matter wavepackets had generated enormous interest in ultrafast diffraction imaging and pump-probe spectroscopy of molecules. This interest was elevated with the advent of ultrafast electron and X-ray diffraction techniques using electron accelerators and X-ray free electron lasers to add temporal resolution to the observed nuclear and electron distributions. However, the great interest in this area has been tempered by the illustration of an impossibility theorem, known as the dimension problem. Not being able to associate unitary evolution to every DOF of molecular motion, quantum tomography could not be used beyond 1D and categorically excludes most vibrational and all rotational motion of molecules. Here we present a theoretical advance to overcome the notorious dimension problem. Solving this challenging problem is important to push imaging molecular dynamics to the quantum limit. The new theory has solved this problem, which makes quantum tomography a truly useful methodology in ultrafast physics and enables the making of quantum version of a molecular movie. With the new theory, quantum tomography can be finally advanced to a sufficient level to become a general method for reconstructing quantum states of matter, without being limited in one dimension. Our new concept is demonstrated using a simulated dataset of ultrafast diffraction experiment of laser-aligned nitrogen molecules.
When the dynamics of a quantum system of interest is known, an informationally complete set of observables is not needed for state reconstruction via tomographic techniques: letting the system evolve before performing the measurement allows one to effectively extend the available ways to probe the system. This idea leads to dynamical quantum tomography, whose feasibility we characterize for general quantum dynamics using Krylov-based methods. Specializing in Markovian ones, we also provide deterministic tests and randomized ones to effectively assess parametric dynamics. The limits of the methods are explored by comparing unitary and open dynamics when a single observable is available, and the set of observables whose expectation can be reconstructed from the available ones characterized. The framework is illustrated with applications to a spin chain (with or without dissipation) and an electron-nuclear system.
In the first part of the dissertation, we develop a theoretical approach to analyze nonlinear spectroscopy experiments based on the formalism of quantum state (QST) and process tomography (QPT). In it, a quantum system is regarded as a black box which can be systematically tested in its performance, very much like an electric circuit is tested by sending a series of inputs and measuring the corresponding outputs, but in the quantum sense. We show how to collect a series of pump-probe or photon-echo experiments, and by varying polarizations and frequency components of the perturbations, reconstruct the quantum state (density matrix) of the probed system for a set of different initial conditions, hence simultaneously achieving QST and QPT. Furthermore, we establish the conditions under which a set of two-dimensional optical spectra also yield the desired results. Simulations of noisy experiments with inhomogeneous broadening show the feasibility of the protocol. A spin-off of this work is our suggestion of a witness that distinguishes between spectroscopic time-oscillations corresponding to vibronic only coherences against their electronic counterparts. We conclude by noting that the QST/QPT approach to nonlinear spectroscopy sheds light on the amount of quantum information contained in the output of an experiment, and hence, is a convenient theoretical and experimental paradigm even when the goal is not to perform a full QPT. In the second part of the thesis, we discuss a methodology to study the electronic dynamics of complex molecular systems, such as photosynthetic units, in the framework of time-dependent density functional theory (TD-DFT). By treating the electronic degrees of freedom as the system and the nuclear ones as the bath, we develop an open quantum systems (OQS) approach to TD-DFT. We formally extend the theoretical backbone of TD-DFT to OQS, and suggest a Markovian bath functional which can be readily included in electronic structure codes.
Emission tomography provides a method for the characterization of quantum states of those degrees of freedom of material quantum systems that are coupled to electronic states of the system. In such systems, the time-dependent fluorescence spectrum contains complete information about the non-electronic degree of freedom. Phase-space or, in certain cases, density matrix representations of the state can be reconstructed from the spectrum using standard methods of computer-assisted tomography or linear algebra. The principles and implementation of this procedure are described in detail and applied to the specific example of the reconstruction of the quantum state of the vibrational mode of a diatomic molecule excited into a non-stationary wavepacket state.
Quantum state tomography provides a powerful and versatile framework for characterizing quantum states across diverse quantum systems. However, tomography is still largely restricted to systems with a few degrees of freedom, while for continuous systems, it is typically tailored to quantum-optical platforms. For other many-body systems described by continuous quantum fields such as ultracold atomic gases, tomographic technique is not as well-established. The present thesis addresses this gap by providing new approaches to measurement and tomography of ultracold Bose gases trapped in one (1D) and two dimensions (2D). These systems serve as analog quantum field theory (QFT) simulators and exhibit novel physics absent in higher dimensions. We begin by systematically assessing state-of-the-art method for extracting the relative phase field in parallel 1D Bose gases with matter-wave interference. In particular, we derive a formula for the systematic phase shift due to longitudinal dynamics, a factor that is often ignored in previous modeling of the measurement. We further examine how the single-shot readout error propagates to correlation and distribution functions of the relative phase. Then, we broaden the scope of experimentally accessible quantities by developing a new methodology to extract the common phase field---the sum of the phase fields of the two parallel gases---from the already available density ripple data. We rigorously benchmark this new readout with numerical simulations and experimental data in 1D systems and further generalize it to 2D systems. By combining the relative and common phases information, we improve Gaussian state tomography in parallel 1D Bose gases and apply it to probe new physical phenomena in quantum field simulators. Specifically, we focus on two applications: (i) field-space entanglement detection at finite temperatures and out of equilibrium, and (ii) probing non-linear relaxation in near-integrable quantum systems. This thesis bridges theory and experiment in the simulation of QFTs and reveals new opportunities for exploring many-body quantum effects in ultracold bosonic gases.
Quantum state tomography (QST) refers to any method that allows one to reconstruct the accurate representation of a quantum system based on data obtainable from an experiment. In this paper, we concentrate on theoretical methods of quantum tomography, but some significant experimental results are also presented. Due to a considerable body of literature and intensive ongoing research activity in the field of QST, this overview is restricted to presenting selected ideas, methods, and results. First, we discuss tomography of pure states by distinguishing two aspects—complex vector reconstruction and wavefunction measurement. Then, we move on to the Wigner function reconstruction. Finally, the core section of the article is devoted to the stroboscopic tomography, which provides the optimal criteria for state recovery by including the dynamics in the scheme. Throughout the paper, we pay particular attention to photonic tomography, since multiple protocols in quantum optics require well-defined states of light.
Ultrafast pulses allow observation of molecular dynamics with femtosecond time resolution through pump probe experiments. However, averaging over an ensemble of molecules tends to wash out phase sensitive information, necessary to probe quantum effects, due to the intrinsic inhomogeneity in molecular conformations, orientations and interactions that lead to unique potential energy landscapes for each molecule. It is therefore important to go beyond the ensemble average when looking at quantum dynamics of organic systems at room temperature, and resolve the behaviour of specific molecules on an individual basis. In this thesis, we show the creation, manipulation and observation of ultrafast coherent effects in single molecules at room temperature, and resolve a certain measure of environmental influence on the specific dynamics of each molecule. Moreover, we apply this insight to investigate a functional light harvesting biosystem, and lay the basis for a technique that has the time and space resolution required to observe these systems in vivo. In chapter 1, we introduce the concepts and techniques the research in this thesis is built on. In chapter 2, we treat the possibility of controlling ultrafast pulses at the high-NA diffraction limit, and come to conclusions about the procedure to follow there that hold for all pulse-shaping experiments. We show in proof of principle experiments that we can control the ultrafast characteristics of optical pulses in nanometric excitation volumes. In chapters 3 and 4 we report the creation, detection and control of ultrafast quantum dynamics in single organic molecules at room temperature. We show that manipulation of superposition states is possible in these systems within a coherence dephasing time of ~50 fs. This leads to the first observation of rabi-oscillations in room temperature single molecules, to ultrafast operation of an organic qubit, and to the creation of non-stationary superposition states (vibrational wavepackets). We probe the influence of the local environment on the composition and dynamics of these wavepackets and show we can optimize the state preparation protocol for each individual molecule in its own nanoenvironment, leading to high fidelity coherent control. In these chapters we lay out the proof of principle work of detecting the quantumdynamics of a complex system in interaction with its environment at room temperature. In chapter 5 we discuss application of these techniques to the investigation of long lived coherence in photosynthetic systems. We show that electronic coherence between different rings of the LH2 system persists to time scales of 100s of femtoseconds at room temperature. Moreover we show that the energy transfer pathways in LH2 adapt to environmentally induced changes in the molecule and that the nature of the transfer remains coherent for each pathway, providing strong evidence that coherent energy transfer is the optimum process for energy transfer in photosynthesis. Finally, in chapter 6 we take the technical development one step further and report on the creation of a framework based on plasmonic antennas that allows for control of the amplitude-phase characteristics in nanometric sized hotspot fields. We show for the first time that the ultrafast characteristics of plasmonic hotspots can directly be engineered through design of the plasmonic system and experimentally demonstrate two much-anticipated examples: a sub-diffraction resolution phase shaper and an ultrafast plasmonic switch for pump probe experiments. The results presented in this thesis form the first creation and observation of ultrafast coherent dynamics in individual molecular systems at room temperature. This is a necessary step to be able to do true quantum tomography in complex systems, resolve the influence of the environment on molecular dynamics, and investigate the physics that determines evolutionary optimization and functionality in biomolecules.