substantially overlapping and differently focused
views from multiple eyes, allowing the scallop to
improve visual acuity relative to the isolated
eye and potentially to determine the depth of
features in the environment. This would offset
the drawback of limited areas of well-focused vision in individual eyes.
The crystal morphology, multilayer structure,
and 3D shape of the scallop’s eye mirror are
finely controlled to produce functional images
on its two retinas. Understanding the strategies
that organisms use to control crystal morphology
and arrangement for complex optical functions
paves the way for the construction of novel bio-inspired optical devices (39, 40). In particular,
the resemblance of the scallop’s tiled, off-axis
mirror to the segmented mirrors of reflecting telescopes provides inspiration for the development
of compact, wide-field imaging devices derived
from this unusual form of biological optics.
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We thank C. Jones of Haven Diving Services for supplying the scallops
and for the photographs in Fig. 1, A and B, and A. Hirsch for her help
in the analysis of TEM images. We thank B. Geiger for the use of
DeltaVision fluorescence microscopy. Electron microscopy studies
were supported by the Irving and Cherna Moskowitz Center for Nano
and Bio-Nano Imaging at the Weizmann Institute of Science. This
work was supported by the Israel Science Foundation (grant
2012\224330*), the Crown Center of Photonics, and the I-CORE
(Israeli Center for Research Excellence) “Circle of Light.” B.A.P. is
the recipient of a Human Frontiers Cross-Disciplinary Postdoctoral
Fellowship. G.J. T. is supported by a stipend from Carl Tryggers
Stiftelse (CTS15:38). L.A. and S. W. are the incumbents of the
Dorothy and Patrick Gorman Professorial Chair of Biological
Ultrastructure and the Dr. Trude Burchardt Professorial Chair of
Structural Biology, respectively.
Materials and Methods
Figs. S1 to S11
9 February 2017; resubmitted 25 June 2017
Accepted 23 October 2017
Spectroscopic signatures of
localization with interacting photons
in superconducting qubits
P. Roushan,1*† C. Neill,2† J. Tangpanitanon,3† V. M. Bastidas,3† A. Megrant,1 R. Barends,1
Y. Chen,1 Z. Chen,2 B. Chiaro,2 A. Dunsworth,2 A. Fowler,1 B. Foxen,2 M. Giustina,1
E. Jeffrey,1 J. Kelly,1 E. Lucero,1 J. Mutus,1 M. Neeley,1 C. Quintana,2 D. Sank,1
A. Vainsencher,1 J. Wenner,2 T. White,1 H. Neven,1 D. G. Angelakis,3,4 J. Martinis1,2
Quantized eigenenergies and their associated wave functions provide extensive
information for predicting the physics of quantum many-body systems. Using a chain of
nine superconducting qubits, we implement a technique for resolving the energy levels of
interacting photons. We benchmark this method by capturing the main features of the
intricate energy spectrum predicted for two-dimensional electrons in a magnetic field—the
Hofstadter butterfly. We introduce disorder to study the statistics of the energy levels
of the system as it undergoes the transition from a thermalized to a localized phase. Our work
introduces a many-body spectroscopy technique to study quantum phases of matter.
Consider a system of interacting particles isolated from the environment initially pre- pared in a very low entropy state far from equilibrium. It is often observed that the system acts as its own thermal reservoir
and approaches the equilibrium state. In this ther-
mal phase, the system shows ergodic behavior,
wherein it uniformly explores all accessible states
over time. Recent works discuss the emergence
of another phase in certain parameter regimes
in which ergodicity breaks down and thermal
equilibrium becomes unattainable (1–8). This
phase is referred to as the many-body localized
(MBL) phase (9–16). The conventional quantum
phase transitions, e.g., from para- to ferromag-
netic, are characterized by changes in the ground
state of the system. However, the signature dif-
ferences between the thermalized and MBL phases
are in dynamical behaviors, indicating that the
transition involves change in the properties of
all many-body eigenstates of the system. Hence,
the physics goes beyond the ground state and
requires study of the entire energy spectrum, which
constitutes an experimental challenge.
In quantum physics, the quantized eigenenergies and their associated wave functions provide
extensive information for predicting the chemistry of molecules or physics of condensed-matter
systems. Creating local perturbations and recording
the subsequent response of the system as a function
of time can reveal the characteristic modes of
that system (17, 18). Our method for measuring the
energy spectrum of a Hamiltonian is based on this
1Google Inc., Santa Barbara, CA, USA. 2Department of Physics,
University of California, Santa Barbara, CA, USA. 3Centre for
Quantum Technologies (CQT), National University of Singapore,
Singapore. 4School of Electrical and Computer Engineering,
Technical University of Crete, Chania, Crete, Greece.
*Corresponding author. Email: firstname.lastname@example.org (P.R.);
†These authors contributed equally to this work.