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Quantum state tomography of 1D resonance fluorescence

Ingrid Strandberg

Published Jan 1, 2017• 1 citation

Abstract

Tomography is the name under which all state reconstruction techniques are denoted,\none of the most recognized examples being medical tomography. Quantum\nstate tomography is a procedure to determine the quantum state of a physical system.\nBy performing homodyne measurements on resonance fluorescence from an\nartificial atom coupled to a one-dimensional transmission line, its quantum state\ncan be reconstructed. Resonance fluorescence is one of the simplest setups that\nresults in non-classical states of light. If these states are non-classical in the sense\nthat they have a negative Wigner function, they can be used as a computational\nresource for quantum computing.\nThere are many different approaches to quantum computing. Some, like gate\nbased quantum computing using discrete variables like qubits, have been extensively\nresearched, both theoretically and experimentally. There exists and alternative\napproach: continuous variable quantum computing. The continuous variables\nwe will be concerned with are the components of the electromagnetic field that\nconstitute the resonance fluorescence.\nThere are different parameters that affect the nature of the resonance fluorescence,\nfor example, the number of transmission lines the atom is coupled to, or\nthe strength of the driving field. In this work, we develop the tools necessary to\nnumerically simulate homodyne detection of resonance fluorescence for different\nsets of parameters, and reconstruct the quantum state as well as calculating the\nWigner negativity.

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Tomography is the name under which all state reconstruction techniques are denoted,\none of the most recognized examples being medical tomography. Quantum\nstate tomography is a procedure to determine the quantum state of a physical system.\nBy performing homodyne measurements on resonance fluorescence from an\nartificial atom coupled to a one-dimensional transmission line, its quantum state\ncan be reconstructed. Resonance fluorescence is one of the simplest setups that\nresults in non-classical states of light. If these states are non-classical in the sense\nthat they have a negative Wigner function, they can be used as a computational\nresource for quantum computing.\nThere are many different approaches to quantum computing. Some, like gate\nbased quantum computing using discrete variables like qubits, have been extensively\nresearched, both theoretically and experimentally. There exists and alternative\napproach: continuous variable quantum computing. The continuous variables\nwe will be concerned with are the components of the electromagnetic field that\nconstitute the resonance fluorescence.\nThere are different parameters that affect the nature of the resonance fluorescence,\nfor example, the number of transmission lines the atom is coupled to, or\nthe strength of the driving field. In this work, we develop the tools necessary to\nnumerically simulate homodyne detection of resonance fluorescence for different\nsets of parameters, and reconstruct the quantum state as well as calculating the\nWigner negativity.

Quantum tomography is a procedure to determine the quantum state of a physical system, or equivalently, to estimate the expectation value of any operator. It consists in appropriately averaging the outcomes of the measurement results of different observables, obtained on identical copies of the same system. Alternatively, it consists in maximizing an appropriate likelihood function defined on the same data. The procedure can be also used to completely characterize an unknown apparatus. Here we focus on the electromagnetic field, where the tomographic observables are obtained from homodyne detection.

This review covers the latest developments in continuous-variable quantum-state tomography of optical fields and photons, placing a special emphasis on its practical aspects and applications in quantum-information technology. Optical homodyne tomography is reviewed as a method of reconstructing the state of light in a given optical mode. A range of relevant practical topics is discussed, such as state-reconstruction algorithms (with emphasis on the maximum-likelihood technique), the technology of time-domain homodyne detection, mode-matching issues, and engineering of complex quantum states of light. The paper also surveys quantum-state tomography for the transverse spatial state (spatial mode) of the field in the special case of fields containing precisely one photon.

Introduction Quantum optics provides a complete description of the resonant interaction of light and a material system. The light scattered as a result of this interaction – the resonance fluorescence – provides a means to monitor and affect the material system's dynamics and also allows for the generation of nonclassical states of the electromagnetic field. Initial theoretical studies characterizing resonance fluorescence addressed fundamental questions regarding the necessity of quantizing the electromagnetic field. This is a particularly illustrative period in optical physics where multiple theories existed that might describe resonance fluorescence – semi-classical optical Bloch equations [1, 2] or quantum electrodynamics [4, 3] – and experimental evidence was necessary to identify the appropriate theoretical framework. Indeed, experiments in 1974 and 1977 investigating the resonance fluorescence lineshape [5, 6] and the photon statistics of the scattered light [7] vindicated the full quantum electrodynamical description of light-matter interaction over other models. In the subsequent 30 years, resonance fluorescence has been used as a high-resolution spectroscopic probe of a variety of physical systems, as a means to investigate quantum coherence in atomic gases and, with the current interest in quantum information science (QIS), as a tool for quantum-state preparation, measurement and control in trapped ions [8]. During this same time period, advances in material science have ushered in an era where it is possible to grow and fabricate solid-state devices at length scales commensurate with the quantum confinement of electrons and holes. It is only recently that resonance fluorescence has been observed from the prototypical artificial nanostructure exhibiting three-dimensional quantum confinement – a single semiconductor quantum dot (QD) [9].

In this work, a continuous variable quantum tomography is presented. This is done using two approaches, one using simulated data and another one using data from measurements performed in the laboratory. The Radon inverse transform, under the approximation of filtered back projection, alongside the details on the implementa- tion of balanced homodyne detection needed to perform the tomography are clearly explained. Density matrices in the quadrature and number representation are also obtained from the Wigner function. Additionally, complete documentation of the computational implementation and a user-friendly interface for the tomography is reported

Quantum computation depends on quantum entanglement, a correlation between subsystems that cannot occur classically. A variety of theoretical measures exist for quantifying the degree entanglement in such schemes, all of which are functions of the system density matrix. How can the entanglement be measured experimentally? Using quantum tomography techniques developed for two photon entangled states, the density matrix can be reconstructed from the appropriate experimental data. In this case the state tomography gives the complete characterization of the physical system (for the relevant degree of freedom, such as spin or polarization). It gives information on both the degree of nonclassical correlation, that is entanglement, as well as the amount of decoherence in the system. In this proceedings we discuss the general state tomography procedure required to characterize a few qubit quantum computer, for any architecture.