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arXiv:1103.5585 · New Journal of Physics

The Fermi problem in discrete systems

Erez Zohar, Benni Reznik

Published Jul 29, 2011• 10 citations
arXivHTMLPDF

Abstract

The Fermi two-atom problem illustrates an apparent causality violation in quantum field theory that has to do with the nature of the built-in correlations in the vacuum. It has been a constant subject of theoretical debate and discussion in the last few decades. Nevertheless, although the issues at hand could in principle be tested experimentally, the smallness of such apparent violations of causality in quantum electrodynamics have prevented the observation of the predicted effect. In this paper, we show that the problem can be simulated within the framework of discrete systems that can be manifested, for instance, by trapped atoms in optical lattices or trapped ions. Unlike the original continuum case, the causal structure is no longer sharp. Nevertheless, as we show, it is possible to distinguish between ‘trivial’ effects due to ‘direct’ causality violations and the effects associated with Fermi's problem, even in such discrete settings. The ability to control externally the strength of the atom–field interactions enables us also to study both the original Fermi problem with ‘bare atoms’ and correction in the scenario that involves ‘dressed’ atoms. Finally, we show that, in principle, the Fermi effect can be detected using trapped ions.

DOI
10.1088/1367-2630/13/7/075016

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The Fermi two-atom problem illustrates an apparent causality violation in quantum field theory that has to do with the nature of the built-in correlations in the vacuum. It has been a constant subject of theoretical debate and discussion in the last few decades. Nevertheless, although the issues at hand could in principle be tested experimentally, the smallness of such apparent violations of causality in quantum electrodynamics have prevented the observation of the predicted effect. In this paper, we show that the problem can be simulated within the framework of discrete systems that can be manifested, for instance, by trapped atoms in optical lattices or trapped ions. Unlike the original continuum case, the causal structure is no longer sharp. Nevertheless, as we show, it is possible to distinguish between ‘trivial’ effects due to ‘direct’ causality violations and the effects associated with Fermi's problem, even in such discrete settings. The ability to control externally the strength of the atom–field interactions enables us also to study both the original Fermi problem with ‘bare atoms’ and correction in the scenario that involves ‘dressed’ atoms. Finally, we show that, in principle, the Fermi effect can be detected using trapped ions.
We use 40K atoms trapped in a cubic optical lattice to simulate the Fermi-Hubbard model. The work in this thesis focuses on investigating dynamics in the Fermi-Hubbard model and developing techniques for engineering Hamiltonians beyond the minimal Hubbard model. We discussed three experiments. In the first, we investigated the transport properties of a Fermi lattice gas by directly measuring the transport lifetime at various interaction strengths and temperatures. The resistivity is inferred from the measured transport lifetime. We observe anomalous transport behavior, which is analogous to bad-metal behavior in strongly correlated electronic materials. The second experiment presents the first realization of correlated, density-dependent tunneling in a Fermi-Hubbard optical lattice model by applied Raman laser fields. This correlated tunneling involves spin-flips and the generation of doublons, which have been observed experimentally. We also confirmed that the amplitude of correlated tunneling is suppressed when neighboring lattice sites are unoccupied. The last experiment explores the possibility to introduce long-range interactions for fermions trapped in optical lattices via Rydberg-dressed states. We developed a novel velocity-selective spectroscopy method to measure the transition between the 5P_1/2 and Rydberg states via electromagnetically induced transparency. This measurement is a first step toward inducing Rydberg-dressed interactions in optical lattices.
The distinction between vacuum-field and source-radiation effects in phenomena such as the Lamb shift, Casimir forces, and spontaneous emission relies on additional assumptions at the theoretical level, and an experimental approach was never considered feasible. Fermi’s two-atom problem, a Gedankenexperiment on how atoms interact with the electromagnetic field via vacuum and source radiation, provides key theoretical insight. Advances in ultrafast optics now enable experimental analogues using two laser pulses in a nonlinear crystal. Here we demonstrate the detection of vacuum- and source-radiation-induced correlations between two pulses, separated by their causal properties. Specifically, vacuum fluctuations and source radiation are shown to correlate distinct quadratures of near-infrared pulses, enabling individual probing via phase-sensitive detection. Our results experimentally verify the time-domain fluctuation-dissipation theorem at the quantum level and open avenues for studying quantum radiation effects in time-dependent media, including entanglement harvesting from the vacuum and quantum field detection in curved-space analogues. Two nearby atoms build up correlations due to two fundamentally different effects: individual interactions with the same vacuum fluctuations, and photon exchange. It is normally assumed that only the total effect is experimentally observable, and not the two individual contributions. Here, the authors challenge this in an experiment using an electro-optic analogue of Fermi’s two atoms setup.
We propose a feasible experimental test of a 1D version of the Fermi problem using superconducting qubits. We give an explicit nonperturbative proof of strict causality in this model, showing that the probability of excitation of a two-level artificial atom with a dipolar coupling to a quantum field is completely independent of the other qubit until signals from it may arrive. We explain why this is in perfect agreement with the existence of nonlocal correlations and previous results which were used to claim apparent causality problems for Fermi's two-atom system.
Non-Markovian time evolution of quantum systems is a challenging problem, often mitigated by employing numerical methods or making simplifying assumptions. In this work, we address this problem in waveguide quantum electrodynamics (QED) by developing a diagrammatic approach, which performs fully analytical non-Markovian time evolution of single-photon states. By revisiting Fermi's two-atom problem, we tackle the impeding question of whether the rotating-wave approximation violates causality in single-photon waveguide QED. Afterward, we introduce and prove the no upper half-plane poles (no-UHP) theorem, which connects the poles of scattering parameters to the causality principle. Finally, we visualize the time-delayed coherent quantum feedback mediated by the field and discuss the Markovian limit for microscopically separated qubits where short-distance causality violations occur and the emergence of collective decay rates in this limit. Our diagrammatic approach is a method to perform exact and analytical non-Markovian time evolution of multiemitter systems in waveguide QED.
Direct experimental access to some of the most intriguing quantum phenomena is not granted due to the lack of precise control of the relevant parameters in their naturally intricate environment. Their simulation on conventional computers is impossible, since quantum behaviour arising with superposition states or entanglement is not efficiently translatable into the classical language. However, one could gain deeper insight into complex quantum dynamics by experimentally simulating the quantum behaviour of interest in another quantum system, where the relevant parameters and interactions can be controlled and robust effects detected sufficiently well. Systems of trapped ions provide unique control of both the internal (electronic) and external (motional) degrees of freedom. The mutual Coulomb interaction between the ions allows for large interaction strengths at comparatively large mutual ion distances enabling individual control and readout. Systems of trapped ions therefore exhibit a prominent system in several physical disciplines, for example, quantum information processing or metrology. Here, we will give an overview of different trapping techniques of ions as well as implementations for coherent manipulation of their quantum states and discuss the related theoretical basics. We then report on the experimental and theoretical progress in simulating quantum many-body physics with trapped ions and present current approaches for scaling up to more ions and more-dimensional systems.
Over the past three decades or so, experiments of the type called Gedanken have become real. Recall the Schrödinger quote from Chapter 8: “… we never experiment with just one electron or atom or (small) molecule.” This is no longer true. We can do experiments involving single atoms or molecules and even on single photons, and thus it becomes possible to demonstrate that the “ridiculous consequences” alluded to by Schrödinger are, in fact, quite real. We have already discussed some examples of single-photon experiments in Chapter 6, and in Chapter 10 we shall discuss experiments performed with single atoms and single trapped ions. In the present chapter, we shall elaborate further on experimental tests of fundamentals of quantum mechanics involving a small number of photons. By fundamental tests we mean tests of quantum mechanics against the predictions of local realistic theories (i.e. hidden variable theories). Specifically, we discuss optical experiments demonstrating violations of Bell's inequalities, violations originally discussed by Bell in the context of two spin-one-half particles. Such violations, if observed experimentally, falsify local realistic hidden-variable theories. Locality refers to the notion, familiar in classical physics, that there cannot be a causal relationship between events with space-like separations. That is, the events cannot be connected by any signal moving at, or less than, the speed of light; i.e. the events are outside the light-cone. But in quantum mechanics, it appears that nonlocal effects, effects seemly violating the classical notion of locality in a certain restricted sense, are possible.