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Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)

Ultrashort pulses characterization by quantum state tomography

Stefan Lerch, André Stefanov

Published Jun 1, 2017• 0 citations
Publisher

Abstract

Summary form only given. The goal of any pulse characterization protocol is to describe how the spectral components of the electric field are changing in time. Apart from linear methods such as streak cameras, characterization techniques based on nonlinear processes have been developed. Those processes require broad spectral acceptance bandwidths, i.e. to be capable of resolving ultrashort features of the pulse. For many of those techniques, the pulses are reconstructed by iterative or deconvolution algorithms where the electric field is assumed to be fully coherent and do not account for the possibility of reduced coherence from pulse to pulse, or within a single pulse. In general, the chronocyclic Wigner function should be reconstructed in order to provide full information about an ensemble of partially coherent pulses. To make a parallel with quantum mechanics, most quantum systems cannot be considered as closed and interact with the environment, and subsequently, undergo decoherence. Therefore, a noisy quantum state cannot be described by a single pure state only but as a classical mixture of pure states.

DOI
10.1109/cleoe-eqec.2017.8086690

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Summary form only given. The goal of any pulse characterization protocol is to describe how the spectral components of the electric field are changing in time. Apart from linear methods such as streak cameras, characterization techniques based on nonlinear processes have been developed. Those processes require broad spectral acceptance bandwidths, i.e. to be capable of resolving ultrashort features of the pulse. For many of those techniques, the pulses are reconstructed by iterative or deconvolution algorithms where the electric field is assumed to be fully coherent and do not account for the possibility of reduced coherence from pulse to pulse, or within a single pulse. In general, the chronocyclic Wigner function should be reconstructed in order to provide full information about an ensemble of partially coherent pulses. To make a parallel with quantum mechanics, most quantum systems cannot be considered as closed and interact with the environment, and subsequently, undergo decoherence. Therefore, a noisy quantum state cannot be described by a single pure state only but as a classical mixture of pure states.
The fascinating nature of light has driven many discoveries in physics, from the theory of special relativity to quantum mechanics. As a carrier of information, light is at the core of high-speed communication over the internet and facilitates the study of numerous physical properties of materials by the interaction of light and matter. Advances in generating and detecting ultrashort light pulses enabled the spectroscopic investigation of matter with unprecedented detail. Meanwhile, the development of more precise and sensitive measurements, leveraging quantum properties of light, has expanded the experimental frontiers of the physical world. Naturally, the question arises: Could the combination of these two breakthroughs push the frontiers even further? A promising candidate to achieve this combination is electro-optic sampling, an electromagnetic-field measurement comprising an up-conversion and a phase-space measurement based on homodyne detection. 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In this thesis, we establish the theoretical foundations for a comprehensive tomographic reconstruction of pulsed quantum states of light, encompassing both the statistics and dynamics of the state. Additionally, we introduce tools to simplify the description of the nonlinear interaction underlying electro-optic sampling and in turn the tomography setups presented here. As a first step, we investigate the application of electro-optic sampling to the time-local reconstruction of various pulsed quantum states of light on ultrafast timescales. We identify two types of noise, which impact the quality of the reconstruction: shot noise, arising from the nonlinear interaction in electro-optic sampling, and thermal noise, due to the localization of the (correlated) quantum state within a small space-time volume during the measurement. To improve the reconstruction, we propose to use a strong probe pulse, suppressing the shot noise, and to select an appropriate detection frequency band, reducing thermal noise. Thus, the proposed tomography enables the characterization of quantum states in terahertz frequency range on ultrafast timescales. However, the time-local nature of the measurement imposes a trade-off between the reconstruction of the quantum state in phase space and the time domain. In a second step, we propose a reconstruction method based on two-time field-correlation measurement, which we call correlation tomography, able to overcome the restriction to time-local states. While the tomography of quantum states in an optimal pulse basis has been demonstrated, it requires prior knowledge about the temporal structure of the quantum state. We introduce an algorithm for reconstructing high-dimensional multimode Gaussian quantum states, which are completely characterized by the covariance matrix of the quadratures of fully distinguishable pulses. The reconstruction is achieved by orthogonalizing the overlapping time windows of the correlation measurement in post-processing. We compare a homodyne-based implementation to electro-optic sampling and develop an optimization scheme for the latter, to mitigate the shot noise in the non-perturbative regime of strong nonlinear interactions. Furthermore, we derive the full-counting statistics of the field-correlation measurement and theoretically demonstrate the reconstruction of pulsed Fock states. This opens an avenue to the full tomography of highly multimode non-Gaussian states. In the third and last step, we derive a formalism to simplify the description of the up-conversion required in electro-optic sampling. If a pulsed quantum states is up-converted and localized in a small space-time region, the transformation of the localized modes during the nonlinear interaction corresponds to an open quantum system, rendering the standard tools of quantum optics inapplicable. By introducing the generalized Bloch-Messiah decomposition, we simplify the input-output relation of the nonlinear interaction to a set of effective modes relevant for subsequent processing, such as detection. Using this framework, we derive an analytic phase-space input-output relation for the nonlinear interaction of pulsed quantum states, applicable to open quantum systems. We study the resulting dynamics using two example input states: a Fock state and a two-mode squeezed state. Based on the von Neumann entropy of the output state, we develop an intuitive understanding of the generalized Bloch-Messiah decomposition. The rich structure of pulsed quantum states of light can be a powerful resource, but also challenging to operate with, in theory as well as in experiment. The characterization methods and theoretical tools developed in this thesis, tailored to pulsed quantum light and the nonlinear interaction thereof, can pave the way to the application of this resource to fundamental research as well as quantum technologies.
The characterization of the complex spectral amplitude, that is, the spectrum and spectral phase, of single-photon-level light fields is a crucial capability for modern photonic quantum technologies. Since established pulse characterization techniques are not applicable at low intensities, alternative approaches are required. Here, we demonstrate the retrieval of the complex spectral amplitude of single-photon-level light pulses through measuring their chronocyclic Q −function. Our approach draws inspiration from quantum state tomography by exploiting the analogy between quadrature phase space and time-frequency phase space. In the experiment, we perform time-frequency projections with a quantum pulse gate (QPG), which directly yield the chronocyclic Q −function. We evaluate the complex spectral amplitude from the measured chronocyclic Q −function data with maximum likelihood estimation (MLE), which is the established technique for quantum state tomography. The MLE yields not only an unambigious estimate of the complex spectral amplitude of the state under test that does not require any a priori information, but also allows for, in principle, estimating the spectral-temporal coherence properties of the state. Our method accurately recovers features such as jumps in the spectral phase and is resistant against regions with zero spectral intensity, which makes it immediately beneficial for classical pulse characterization problems.
The two-time nonstationary correlation function is defined and applied to the problem of characterizing pulses whose amplitude and phase vary randomly from pulse to pulse. Using a linear-systems approach developed in earlier work, four simple interferometric techniques for measuring this quantity are described and analyzed. We show that these four methods offer simple and direct algorithms for reconstructing the correlation function and, in the case of a single pulse or a coherent ensemble of pulses, return the electric field amplitude and phase.
The development of key devices and systems in quantum information technology, such as entangled particle sources, quantum gates and quantum cryptographic systems, requires a reliable and well-established method for characterizing how well the devices or systems work. We report our recent work on experimental characterization of pulsed entangled photonic states and photonic quantum channels, using the methods of state and process tomography. By using state tomography, we could reliably evaluate the states generated from a two-photon source under development and develop a highly entangled pulsed photon source. We are also devoted to characterization of single-qubit and two-qubit photonic quantum channels. Characterization of typical single-qubit decoherence channels has been demonstrated using process tomography. Characterization of two-qubit channels, such as classically correlated channels and quantum mechanically correlated channels is under investigation. These characterization techniques for quantum states and quantum processes will be useful for developing photonic quantum devices and for improving their performances.