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arXiv:1504.04194 · New Journal of Physics

Continuous matrix product state tomography of quantum transport experiments

G Haack, A Steffens, J Eisert, R Hübener

Published Nov 6, 2015• 12 citations
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Abstract

In recent years, a close connection between the description of open quantum systems, the input–output formalism of quantum optics, and continuous matrix product states (cMPS) in quantum field theory has been established. The latter constitute a variational class of one-dimensional quantum field states and have been shown to provide an efficient ansatz for performing tomography of open quantum systems. So far, however, the connection between cMPS and open quantum systems has not yet been developed for quantum transport experiments in the condensed-matter context. In this work, we first present an extension of the tomographic possibilities of cMPS by demonstrating the validity of reconstruction schemes based on low-order counting probabilities compared to previous schemes based on low-order correlation functions. We then show how fermionic quantum transport settings can be formulated within the cMPS framework. Our procedure, via the measurements of low-order correlation functions only, allows us to gain access to quantities that are not directly measurable with present technology. Emblematic examples are high-order correlations functions and waiting time distributions (WTD). The latter are of particular interest since they offer insights into short-time scale physics. We demonstrate the functioning of the method with actual data, opening up the way to accessing WTD within the quantum regime.

DOI
10.1088/1367-2630/17/11/113024

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In recent years, a close connection between the description of open quantum systems, the input–output formalism of quantum optics, and continuous matrix product states (cMPS) in quantum field theory has been established. The latter constitute a variational class of one-dimensional quantum field states and have been shown to provide an efficient ansatz for performing tomography of open quantum systems. So far, however, the connection between cMPS and open quantum systems has not yet been developed for quantum transport experiments in the condensed-matter context. In this work, we first present an extension of the tomographic possibilities of cMPS by demonstrating the validity of reconstruction schemes based on low-order counting probabilities compared to previous schemes based on low-order correlation functions. We then show how fermionic quantum transport settings can be formulated within the cMPS framework. Our procedure, via the measurements of low-order correlation functions only, allows us to gain access to quantities that are not directly measurable with present technology. Emblematic examples are high-order correlations functions and waiting time distributions (WTD). The latter are of particular interest since they offer insights into short-time scale physics. We demonstrate the functioning of the method with actual data, opening up the way to accessing WTD within the quantum regime.
In recent years, a close connection between the description of open quantum systems, the input–output formalism of quantum optics, and continuous matrix product states (cMPS) in quantum field theory has been established. The latter constitute a variational class of one-dimensional quantum field states and have been shown to provide an efficient ansatz for performing tomography of open quantum systems. So far, however, the connection between cMPS and open quantum systems has not yet been developed for quantum transport experiments in the condensed-matter context. In this work, we first present an extension of the tomographic possibilities of cMPS by demonstrating the validity of reconstruction schemes based on low-order counting probabilities compared to previous schemes based on low-order correlation functions. We then show how fermionic quantum transport settings can be formulated within the cMPS framework. Our procedure, via the measurements of low-order correlation functions only, allows us to gain access to quantities that are not directly measurable with present technology. Emblematic examples are high-order correlations functions and waiting time distributions (WTD). The latter are of particular interest since they offer insights into short-time scale physics. We demonstrate the functioning of the method with actual data, opening up the way to accessing WTD within the quantum regime.
We introduce the concept of quantum field tomography, the efficient and reliable reconstruction of unknown quantum fields based on data of correlation functions. At the basis of the analysis is the concept of continuous matrix product states (cMPS), a complete set of variational states grasping states in one-dimensional quantum field theory. We innovate a practical method, making use of and developing tools in estimation theory used in the context of compressed sensing such as Prony methods and matrix pencils, allowing us to faithfully reconstruct quantum field states based on low-order correlation functions. In the absence of a phase reference, we highlight how specific higher order correlation functions can still be predicted. We exemplify the functioning of the approach by reconstructing randomized cMPS from their correlation data and study the robustness of the reconstruction for different noise models. Furthermore, we apply the method to data generated by simulations based on cMPS and using the time-dependent variational principle. The presented approach is expected to open up a new window into experimentally studying continuous quantum systems, such as those encountered in experiments with ultra-cold atoms on top of atom chips. By virtue of the analogy with the input–output formalism in quantum optics, it also allows for studying open quantum systems.
The fertile new field of quantum information theory is inspiring new ways to study correlated quantum systems by providing fresh insights into the structure of their Hilbert spaces. One of the latest developments in this direction was the extension of the ubiquitous matrix-product-state constructions, epitomized by the density-matrix renormalization-group algorithm, to continuous space-time; so as to be able to describe low-dimensional field theories within a variational approach. Following the earlier success achieved for bosonic theories, we present the first implementation of a continuous matrix product state (cMPS) for spinfull non-relativistic fermions in 1D. We propose a construction of variational matrices with an efficient parametrization that respects the translational symmetry of the problem (without being overly constraining) and readily meets the regularity conditions that arise from removing the ultraviolet divergences in the kinetic energy. We tested the validity of our approach on an interacting spin-1/2 system with spin imbalance. We observe that the ansatz correctly predicts the ground-state magnetic properties for the attractive spin-1/2 Fermi gas, including a phase-oscillating pair correlation function in the partially polarized regime (the 1D correlate of the FFLO state). We shall also discuss how to generalize the cMPS ansatz to other situations.
We present a way of constructing continuous matrix product states (cMPS) for coupled fields. The cMPS is a variational ansatz for the ground state of quantum field theories in one dimension. Our proposed scheme is based on a physical interpretation in which the cMPS class can be produced by means of a dissipative dynamic of a system interacting with a bath. We study the case of coupled bosonic fields. We test the method with previous DMRG results in coupled Lieb-Liniger models. Besides, we discuss a novel application for characterizing the Luttinger liquid theory emerging in the low-energy regime of these theories. Finally, we propose a circuit QED architecture as a quantum simulator for coupled fields.
The experimental realization of large-scale many-body systems in atomic-optical architectures has seen immense progress in recent years, rendering full tomography tools for state identification inefficient, especially for continuous systems. To work with these emerging physical platforms, new technologies for state identification are required. Here we present first steps towards efficient experimental quantum-field tomography. Our procedure is based on the continuous analogues of matrix-product states, ubiquitous in condensed-matter theory. These states naturally incorporate the locality present in realistic physical settings and are thus prime candidates for describing the physics of locally interacting quantum fields. To experimentally demonstrate the power of our procedure, we quench a one-dimensional Bose gas by a transversal split and use our method for a partial quantum-field reconstruction of the far-from-equilibrium states of this system. We expect our technique to play an important role in future studies of continuous quantum many-body systems.
The central topic of this thesis is on the fermionic continuous matrix product states (cMPS) variational ansatz for many-body quantum systems in a one-dimensional (1D) continuum.The matrix product states (MPS) is a variational ansatz which has been tremendously successful for nearly two decades in efficiently approximating groundstate properties of low-dimensional quantum lattice systems.The MPS is intrinsically designed for discrete lattice systems, however, and the cMPS has been introduced in 2010 as a continuum analogue of the MPS, enabling the possibility to directly tackle systems of quantum gases and liquids without resorting to discretization of space.