This thesis has two parts; the first part is a contribution to the research field of quantum measurement in quantum optics while the second part focuses on quantum thermodynamics for fermionic systems. \n \nThe aim of the research on quantum optics is to detect and subsequently characterize quantum states of light. Specifically, we focus on characterizing 1) entanglement between a two-level atom and superposition of coherent states (known as Bell cat state) 2) quantum superposition of coherent states (Schr\\"odinger cat states). The photon is the particle of light which carries quantum information; it is usually lost (destroyed) while being detected. Many physical implementations of quantum logic gate aim to encode quantum information processing into large registers of entangled qubits. However for these larger much distinguishable states, creating and preserving entanglement becomes difficult due to rapid onset of decoherence. Encoding quantum information on Schrodinger's cat states take advantage of a cavity resonators much larger Hilbert space, as compared with that of a two-level system. This architecture allows redundant qubit encodings that can simplify the operations needed to initialize, manipulate and measure the encoded information. For such a system to be viable as a quantum computing platform, efficient measurement of such encoded qubit observables must be possible. \n \nThe concept of quantum non demolition measurement was introduced to evade the problem of decoherence. Researchers now know through quantum theory that it is indeed possible to count photons in a given state of light without destroying them. This nondestructive measurement scheme is coined in the term ``quantum non-demolition measurement". We can extend the ideas of quantum nondemolition measurement scheme to detect a system made up of two or more quantum states (not necessarily states of light) that are combined based on the superposition principle. An example is the Schr\\"odinger's cat state which is a superposition of two coherent states of light of equal amplitudes but opposite phase. At this point, one is not only interested in counting photons, but in understanding the nature of the superposition, the possible problems and the different physical properties that follow. Ways to detect the Schr\\"odinger cat states and subsequently a Bell cat state (Schr\\"odinger cat entangled with a qubit) without significantly perturbing them are discussed. The method analyzed is the mode-invisibility measurement scheme earlier proposed to detect single Fock states and coherent states of light. The method gives a new insight to the known properties of Schr\\"odinger cat states and contributes to our understanding of the quantum-classical boundary problem. \n \nThe second part of the thesis falls in the research field of quantum thermodynamics and open quantum systems. Most problems in quantum thermodynamics have been explored in bosonic systems with little or less done in fermionic systems. Therefore the aim of this part of the thesis is to explore related quantum thermodynamical problems in fermionic systems. I begin by considering the problem of work extraction from noninteracting fermionic systems. For work to be extracted from the state of a quantum system, a unitary operation on the state must act to reduce the average energy of the system. Passive states are those states whose energy cannot be reduced through unitary transformation, that is work cannot be extracted via unitary transformations given only a single copy of the system. It follows that some passive states may have extractable work if several copies of the system is processed. Passive states for which no work can be extracted, no matter the number of available copies, are called completely passive states. An example is the thermal Gibbs state. Here, the limit for which multiple copies of passive states in fermionic systems can be activated for work extraction is studied. It was observed for n > 3 fermionic modes at the same frequency, the product state of n thermal states with different temperatures is not passive. This in principle implies that the construction of a heat engine in fermionic systems need access to three thermal baths at different temperature. This is unlike the bosonic system, where access to only two thermal baths are required. \n \nOn the other hand, while the product state of three thermal states of three fermionic modes at the same frequency but different temperatures is not passive, the unitary transformation required to extract work from the state is difficult to realize. A set of operations that are easier to realize are Gaussian unitaries which are generated by Hamiltonian that are at most quadratic in the system's operators. One may consider extracting work via the restricted class of Gaussian unitaries. Hence fermionic Gaussian passive states for which energy cannot be extracted using only Gaussian operations are characterized. \n \nThe last problem I investigate is that of understanding the dynamics of an open Markovian non-interacting fermionic system. I introduce a classification scheme for the generators of open fermionic Gaussian dynamics and simultaneously partition the dynamics along the following four lines: 1) unitary vs. non-unitary, 2) active vs. passive, 3) state-dependent vs. state-independent, and 4) single-mode vs. multi-mode. Unlike in the bosonic case where only eleven of these sixteen types of dynamics turn out to be possible, one observe only nine types of dynamics in the fermionic setting. Using this partition I discuss the consequences of imposing complete positivity on fermionic Gaussian dynamics. In particular, I show that completely positive dynamics must be either unitary (and so can be implemented without a quantized environment) or active (and so must involve particle exchange with an environment).
This thesis has two parts; the first part is a contribution to the research field of quantum measurement in quantum optics while the second part focuses on quantum thermodynamics for fermionic systems. \n \nThe aim of the research on quantum optics is to detect and subsequently characterize quantum states of light. Specifically, we focus on characterizing 1) entanglement between a two-level atom and superposition of coherent states (known as Bell cat state) 2) quantum superposition of coherent states (Schr\"odinger cat states). The photon is the particle of light which carries quantum information; it is usually lost (destroyed) while being detected. Many physical implementations of quantum logic gate aim to encode quantum information processing into large registers of entangled qubits. However for these larger much distinguishable states, creating and preserving entanglement becomes difficult due to rapid onset of decoherence. Encoding quantum information on Schrodinger's cat states take advantage of a cavity resonators much larger Hilbert space, as compared with that of a two-level system. This architecture allows redundant qubit encodings that can simplify the operations needed to initialize, manipulate and measure the encoded information. For such a system to be viable as a quantum computing platform, efficient measurement of such encoded qubit observables must be possible. \n \nThe concept of quantum non demolition measurement was introduced to evade the problem of decoherence. Researchers now know through quantum theory that it is indeed possible to count photons in a given state of light without destroying them. This nondestructive measurement scheme is coined in the term ``quantum non-demolition measurement". We can extend the ideas of quantum nondemolition measurement scheme to detect a system made up of two or more quantum states (not necessarily states of light) that are combined based on the superposition principle. An example is the Schr\"odinger's cat state which is a superposition of two coherent states of light of equal amplitudes but opposite phase. At this point, one is not only interested in counting photons, but in understanding the nature of the superposition, the possible problems and the different physical properties that follow. Ways to detect the Schr\"odinger cat states and subsequently a Bell cat state (Schr\"odinger cat entangled with a qubit) without significantly perturbing them are discussed. The method analyzed is the mode-invisibility measurement scheme earlier proposed to detect single Fock states and coherent states of light. The method gives a new insight to the known properties of Schr\"odinger cat states and contributes to our understanding of the quantum-classical boundary problem. \n \nThe second part of the thesis falls in the research field of quantum thermodynamics and open quantum systems. Most problems in quantum thermodynamics have been explored in bosonic systems with little or less done in fermionic systems. Therefore the aim of this part of the thesis is to explore related quantum thermodynamical problems in fermionic systems. I begin by considering the problem of work extraction from noninteracting fermionic systems. For work to be extracted from the state of a quantum system, a unitary operation on the state must act to reduce the average energy of the system. Passive states are those states whose energy cannot be reduced through unitary transformation, that is work cannot be extracted via unitary transformations given only a single copy of the system. It follows that some passive states may have extractable work if several copies of the system is processed. Passive states for which no work can be extracted, no matter the number of available copies, are called completely passive states. An example is the thermal Gibbs state. Here, the limit for which multiple copies of passive states in fermionic systems can be activated for work extraction is studied. It was observed for n > 3 fermionic modes at the same frequency, the product state of n thermal states with different temperatures is not passive. This in principle implies that the construction of a heat engine in fermionic systems need access to three thermal baths at different temperature. This is unlike the bosonic system, where access to only two thermal baths are required. \n \nOn the other hand, while the product state of three thermal states of three fermionic modes at the same frequency but different temperatures is not passive, the unitary transformation required to extract work from the state is difficult to realize. A set of operations that are easier to realize are Gaussian unitaries which are generated by Hamiltonian that are at most quadratic in the system's operators. One may consider extracting work via the restricted class of Gaussian unitaries. Hence fermionic Gaussian passive states for which energy cannot be extracted using only Gaussian operations are characterized. \n \nThe last problem I investigate is that of understanding the dynamics of an open Markovian non-interacting fermionic system. I introduce a classification scheme for the generators of open fermionic Gaussian dynamics and simultaneously partition the dynamics along the following four lines: 1) unitary vs. non-unitary, 2) active vs. passive, 3) state-dependent vs. state-independent, and 4) single-mode vs. multi-mode. Unlike in the bosonic case where only eleven of these sixteen types of dynamics turn out to be possible, one observe only nine types of dynamics in the fermionic setting. Using this partition I discuss the consequences of imposing complete positivity on fermionic Gaussian dynamics. In particular, I show that completely positive dynamics must be either unitary (and so can be implemented without a quantized environment) or active (and so must involve particle exchange with an environment).
This thesis fathoms the possibilities and limitations of measurements in quantum optics. After a short introduction to the general aspects of measurements in quantum mechanics, it presents three examples of a measurement process in concrete applications to quantum optics. For one, while operators describing quantities of interest might have a simple theoretical form, it is often hard to actually implement a measurement of their expectation values experimentally. This situation is encountered in the first part which is devoted to the measurement of quantum statistical properties of a light beam. In contrast to classical optics, these statistics are not fully determined by first- and second-order correlations. Instead, higher-order correlations, usually described by non-Hermitian operators, are necessary to characterize the quantum state completely. A particular example is the N-photon Fock state, for which only a measurement of the Nth-order correlations allows for a full characterization. Here, we present a new method how to measure an important class of Nth-order correlations for arbitrary N in a single spatial mode with two polarizations with limited experimental resources. The second part is again centered around the realization of specific measurement operators for photon correlation measurements. However, in contrast to the previous chapter, these measurements are not an end in their own, but are designed to create specific quantum states as a consequence of the measurement process: A suitable measurement of photons emitted by a number of atoms allows to transfer the atoms into long-lived entangled states. We find that with this method it is possible to entangle two remote atomic qubits to an arbitrary and well-defined degree. Furthermore, we show how to generate two families of quantum states in an arbitrary number of remote atomic qubits: one family consists of all -- symmetric and non-symmetric -- total angular momentum eigenstates in N remote qubits, the other is a family of cluster states. It is also systematically investigated whether the technique of quantum state engineering by projective measurements allows for the creation of any arbitrary state. We find explicit solutions for the case of two and three qubits, while it is shown that no solution exist for N > 6. We conclude the section by proving that with some changes in the setup, this measurement-based quantum state engineering technique allows to create the same states among the scattered photons themselves rather than among the emitting atoms. The last part is devoted to the amount of information one is able to gain about a given system when performing a measurement. The corresponding limits of information are investigated at the example of the duality that arises in a two-way interferometer between the visibility of the interference pattern and the knowledge about the path of the interfering object. We show that there exist correlations between the phase difference the interfering object acquires on its way through the interferometer and the amount of which-way information retrievable from certain observables of the which-way detector. In this way, the which-way information becomes a phase-dependent quantity. In particular, we find that for certain values of the phase shift, the amount of extractable which-way information can be larger than allowed by phase-independent measurement. This property is put to use to find the optimal observable for the which-way detector readout in dependence on the phase of the interfering object. Following this strategy, we are able to gain more which-way information than previously thought possible.
After they have interacted, quantum particles generally behave as a single nonseparable entangled system. The concept of entanglement plays an essential role in quantum physics. We have performed entanglement experiments with Rydberg atoms and microwave photons in a cavity and tested quantum mechanics in situations of increasing complexity. Entanglement resulted either from a resonant exchange of energy between atoms and the cavity field or from dispersive energy shifts affecting atoms and photons when they were not resonant. With two entangled particles (two atoms or one atom and a photon), we have realized new versions of the Einstein-Podolsky-Rosen situation. The detection of one particle projected the other, at a distance, in a correlated state. This process could be viewed as an elementary measurement, one particle being a ‘‘meter’’ measuring the other. We have performed a ‘‘quantum nondemolition’’ measurement of a single photon, which we detected repeatedly without destroying it. Entanglement is also essential to understand decoherence, the process accounting for the classical appearance of the macroscopic world. A mesoscopic superposition of states (‘‘Schrodinger cat’’) gets rapidly entangled with its environment, losing its quantum coherence. We have prepared a Schrodinger cat made of a few photons and studied the dynamics of its decoherence, in an experiment which constitutes a glimpse at the quantum/classical boundary. We have also investigated entanglement as a resource for the processing of quantum information. By using quantum two-state systems (qubits) instead of classical bits of information, one can perform logical operations exploiting quantum interferences and taking advantage of the properties of entanglement. Manipulating as qubits atoms and photons in a cavity, we have operated a quantum gate and applied it to the generation of a complex three-particle entangled state. We finally discuss the perspectives opened by these experiments for further fundamental studies.
In the present PhD work, we focus on a specific class of quantum states of light: the non-Gaussian states. These states have the particularity of exhibiting Wigner functions with some negative values. This quantum feature is a necessary condition to perform some quantum computation task; furthermore it is also useful for various other applications, including quantum communication and metrology. Different strategies can be used to generate these states. Here, we start from Gaussian states produced by optical parametric oscillators in the continuous wave regime, (i.e. single-mode and two-mode squeezed vacuum states). The non-Gaussian feature can only be obtained by non-linear phenomena (over-quadratic Hamiltonian). In our case, the non-linearity is induced by photon-counting-based measurements (also called non-Gaussian measurements). This study is mainly divided into two parts. First, the generation of non-classical states associated with two types of qubit encoding: the single-photon state, used for quantum information with discrete variables, and the coherent state superposition (the so-called optical Schrodinger cat state), used for quantum information with continuous variables. These two states have then been used to perform some quantum information protocols. The first one addresses the problem of single-photon entanglement witness, and the other the generation of entanglement between the two encodings (also called hybrid entanglement).
This thesis is devoted to quantum information processing using strongly correlated many-body systems, and we have focused our interest on systems realized with ultracold atoms in optical lattices. We have proposed different protocols based either on the free dynamics followed by a particular initial state -out of the ground-state manifold-, or on the non-trivial quantum dynamics generated by the optimal modulation of optical lattices. Our protocols demonstrate the potential of strongly correlated systems to perform short-distance quantum communication. Protocols based on dynamical manipulations of the optical lattices have also been extended to two-dimensional lattices to prepare decoherence-free cluster states. We have also proposed a novel scheme combining free dynamics and continuous measurements, which lead to the Quantum Zeno effect, for quantum information processing in spin chains. Despite of the recent progress in using atoms trapped in optical lattices to prepare and manipulate strongly correlated systems, the precise detection and experimental characterization of such states is a major challenge in modem physics. We have exploited some of the proposed protocols to analyze strongly correlated systems from the quality of their performance when implementing a quantum information task. Our approach provides a new insight into the study of these complex and fascinating systems. Regarding the detection, we have also addressed the fundamental problem of inferring the state of a given quantum system. First, we have studied the optimal state estimation of the most fundamental quantum system, a two-level system or qubit. Second, we have also considered and proposed new methods to tackle the experimental challenge of detecting the many-body systems prepared with ultracold atoms in optical lattices. We have extended the applicability of the noise-correlation method to detect a mesoscopic Schrodinger cat many-body state. Despite the success of this method, it has the drawback that it demolishes the physical system. To overcome such limitation, we have proposed an . extension of the well-known Faraday effect at the quantum level, which yields a quantum non-demolition method to detect strongly correlated systems.
To understand in detail the relation between unitary quantum theory that describes our world at the microscopic scale and thermodynamics, which was long believed only to apply to macroscopic objects, is one of the most interesting and long-standing problems in physics. Recently, this problem has received renewed attention, in particular from the community of quantum information theory, but also from the field of statistical mechanics, inspired from stochastic thermodynamics. These results suggest that thermodynamics is also relevant for individual quantum systems, provided that they can be brought into contact with thermal baths. In this thesis, I use recently developed tools to provide new results both on fundamental questions, but also on questions which are of practical relevance for potential miniaturized thermal machines. In terms of fundamental questions, the results in this thesis contribute to understanding how the basic laws of thermodynamics and statistical mechanics can be understood directly from unitary quantum mechanics. In particular, I discuss and answer the following questions: i) How can we quantify the third law of thermodynamics using information theoretic methods? ii) How can we quantify the "thermodynamic value" of a state-transition in quantum systems? iii) How can we axiomatically characterize the non-equilibrium free energy and relative entropy? iv) How can we justify statistical ensembles from an operational perspective, without having to introduce either some probability measures or an information theoretic entropy measure? v) How can we understand the equilibration of closed quantum systems, how long does it take and how difficult is it to avoid? In the second part of the thesis I discuss in detail how experimental restrictions, which become important at the quantum scale, influence the ultimate thermodynamic bounds for thermal machines. In particular, the results in this thesis provide thermodynamic bounds on work-extraction and efficiencies of thermal machines in situations where 1.) an experimenter only has limited field strengths available, 2.) an experimenter cannot control the interactions between particles, but external fields arbitrarily well, and 3.) situations in which a small quantum system can only be strongly coupled to heat baths. These bounds are tight and I provide explicit examples illustrating the different behaviours. Finally, I come back to a classic problem in statistical physics: The emergence of spontaneous symmetry breaking. Here, I provide general and rigorous new results that show how symmetry-breaking stationary states emerge from fluctuations in order parameters in dissipative lattice models.
In this thesis, experimentally relevant aspects and open questions of quantum information processing and measurement in circuit quantum electrodynamics have been investigated theoretically. Circuit quantum electrodynamics is a relatively young field combining superconducting transmission line resonators on-chip with superconducting quantum bits which serve as artificial atoms. Remarkable experiments have underlined the prospects of circuit QED as a possible architecture for quantum information processing as well as a framework within which quantum optics experiments can be performed. In contrast to their optical counterparts, these experiments reach the ultra-strong coupling limit and allow e.g. the generation of Fock states, the study of decoherence and the observation of quantum jumps. We present the physics of superconducting circuits based on the Josephson junction as a non-linear circuit element. We discuss the various types of quantum bits (qubits) before turning to an introduction on cavity QED and conclude the introduction with a brief presentation of the most relevant experiments and a short introduction to the principles and prospects of quantum information processing. Building on the pioneering theoretical and experimental work in which up to three qubits have been integrated on a chip and successfully coupled via a cavity mode, we propose a scheme to overcome the natural limits concerning scalability in these systems. Usually, scalability is severely limited by the resonance width of the qubit transitions and the attainable frequency range for both qubit transition and cavity resonance frequencies. This limits the present setups to at most ten qubits. As a possible solution, we propose and investigate a cross-bar grid layout of cavities with qubits at the intersections. It turns out that this setup, allowing for up to about 1000 qubits, offers significant advantages over e.g. nearest-neighbor coupling schemes. In addition, it can serve as a building block for a truly scalable scheme for fault-tolerant quantum information processing with superconducting circuits by combining 7x7 arrays in a staircase manner. Furthermore, we discuss and simulate different possibilities to improve upon the suggested scheme by e.g. using resonant gates and optimal control theory. Another focus of the present thesis is measurement physics. In order to investigate e.g. the possibilities of using a circuit QED system to dispersively detect single microwave photons or to generate multi-qubit entangled states, we introduce in a pedagogical manner the basics of stochastic master equations which generate quantum trajectories before turning to concrete applications. Quantum trajectories have proven to be a state-of-the-art tool to analyze measurement situations in a very realistic manner by giving access to both the measurement record as well as the internal quantum dynamics conditioned on this record. Using the method of quantum trajectories as generated by stochastic master equations, we discuss a scheme to efficiently generate multi-qubit entangled states in a very flexible way. The method is based on a collective, dispersive qubit readout. By this measurement, a suitable product state of N qubits can be reliably converted into e.g. an N-qubit Greenberger-Horne-Zeilinger or W state. We propose and discuss the scheme and investigate the effects of decoherence and parameter spread as they would be encountered in experiment. Last but not least, we propose and fully analyze a scheme to dispersively detect single itinerant microwave photons. While experiments to generate single microwave photons have already been successfully performed, their single-shot quantum non-demolition detection has not been possible so far. The scheme presented in this thesis closes this gap in the circuit QED toolbox and reaches detection efficiencies of 30% only limited by fundamental quantum mechanics in this case the Quantum Zeno effect. This detection scheme could be used in quantum communication or quantum cryptography applications.