Measurement Protocol for the Entanglement Spectrum of Cold Atoms

Abstract

Entanglement, and, in particular, the entanglement spectrum, plays a major role in characterizing many-body quantum systems. While there has been a surge of theoretical works on the subject, no experimental measurement has been performed to date because of the lack of an implementable measurement scheme. Here, we propose a measurement protocol to access the entanglement spectrum of many-body states in experiments with cold atoms in optical lattices. Our scheme effectively performs a Ramsey spectroscopy of the entanglement Hamiltonian and is based on the ability to produce several copies of the state under investigation, together with the possibility to perform a global swap gate between two copies conditioned on the state of an auxiliary qubit. We show how the required conditional swap gate can be implemented with cold atoms, either by using Rydberg interactions or coupling the atoms to a cavity mode. We illustrate these ideas on a simple (extended) Bose-Hubbard model where such a measurement protocol reveals topological features of the Haldane phase.

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Plain Language Summary

Entanglement is fundamental to our understanding of many-body quantum systems, and it is a key concept underlying a plethora of phenomena such as topological properties. Entanglement is also responsible for the complexity of simulating quantum many-body physics on a classical computer. One of the most powerful theoretical tools to characterize entanglement in a quantum many-body system is the so-called entanglement spectrum, which characterizes the statistical properties of a (reduced) quantum state. Thus far, however, because of the lack of an experimentally applicable measurement scheme, it has been used only as a theoretical concept. Here, we build on versatile tools to control and manipulate cold atoms available in current experiments, and we develop a protocol to measure the entanglement spectrum.

We propose to use Ramsey-type spectroscopy to access the spectrum of a many-body quantum state. Our protocol is based on the ability to produce several copies of the state being investigated and to couple these copies between one other and with an auxiliary atom. We show how these required ingredients can be implemented with cold atoms held in an optical lattice using Rydberg interactions and tunnel coupling. In particular, we make use of recently developed techniques to address atoms in optical lattices in a site-resolved way. We show how these tools can be combined to access the largest eigenvalues of the quantum state, and we find that the spectral resolution of our measurement scheme is determined by the total number of copies used.

We expect that our findings will extend the tools available in quantum gas experiments and elevate the entanglement spectrum from a fundamental but purely theoretical concept to a quantity that is measurable in the laboratory.