distribution of frag for the ergodic and many-body localized phases, respectively, is
PDGOEðrÞ ¼27 4 r þ r2 ð1 þ r þ r2Þ5=2 ;
PDPoissonðrÞ ¼ 2 ð1 þ rÞ2 ð4Þ
In Fig. 3C, we focus on D=J ¼ 1 and 5, showing the measured histograms with dots and the
numerical simulations with solid lines (21). The
dashed lines are plots of Eq. 4, providing the expected behavior in the thermodynamic limit
(number of sites Nq→∞), and for limiting values
of D=J. In contrast to these cases, the finite size of
our chain results in features that can be seen in
both data and simulation. When disorder is small,
the energy eigenstates are extended across the
chain (Fig. 4), and hence the energy levels repel
each other. Consequently, there are strong correlations between the levels, and an equidistant
distribution of levels would be favorable. When
D becomes larger, the eigenstates become localized in space and unaware of each other’s presence at a given energy, and level repulsion ceases.
Therefore, the levels independently distribute
themselves, showing a Poisson distribution in the
energy landscape. The exact realization of Poisson
distribution takes place only when J=D→0; in
our case, J=D ¼ 0:2, which is where the peak in
the histogram appears (ra = 0.2). Because the
Poisson distribution is a signature of independent
events, we conclude that the transition from ergodic
to localized phase is associated with vanishing
correlations in energy levels.
A key signature of the transition from ergodic
to MBL phase is the change in the localization
length of the system from being extended over
the entire system to localized over a few lattice
sites. This physics can be studied by measuring
the probability of each energy eigenstate being
present at each lattice site fPa;ng (21). In our
method, the frequencies of the FT signal give
the eigenenergies, and from the magnitude
of the FT terms, fPa;ng can be measured; for
instance, P9;6 is highlighted in Fig. 1C. In the
study of metal–insulator transitions (32, 33),
a common way to quantify the extension in
real space or energy landscape is via the second
moment of the probabilities, defined by partic-
ipation ratio (PR)
PRSpaceðaÞ ≡ 1=
P2 a;n; PREnergyðnÞ ≡ 1=
PRSpace indicates the number of sites over which
an energy eigenstate jfai has an appreciable magnitude. Similarly, PREnergy measures how many
energy eigenstates have a discernable presence on
lattice site n. Note that the first moments of the
probability distributions are normalization conditions
Pa;n ¼ 1 and
Pa;n ¼ 1.
Having demonstrated that we can fully resolve
the energy spectrum of the two-photon energy
manifold, we now extract fPa;ng. In Fig. 4A, we
plot PRSpace for various disorder strengths in
the order of increasing energy. In this energy
manifold, there are 36 single- ðe:g:;j001000100iÞ
and 9 double-occupancy states ðe:g:;j000020000i Þ,
¼ 45 energy levels. For low
disorder ðD=J < 1Þ, PRSpace is about 8, indicating
that almost all energy eigenstates are extended
over the entire chain of nine qubit lattice sites.
As the strength of disorder increases, the eigen-
states with their energies close to the edge of the
energy band start to shrink, whereas the eigen-
states with energies in the middle of the band
remain extended at larger disorders. This is sim-
ilar to the Anderson localization picture, in which
localization begins at the edges of the band, and
a mobility edge forms (the yellow hue) and ap-
proaches the center of the band as disorder be-
comes stronger (32). The existence of the mobility
edge in MBL has been theoretically questioned,
and proper investigation of it requires going to
larger systems and finite size scaling (34). Given
that numerical exact diagonalization is limited
to small systems, scaling up the experiment could
shed light on this matter and the general under-
standing of MBL (33, 35).
In Fig. 4B, we plot the PREnergy, which shows
that as the disorder becomes stronger, the number of eigenstates present at a given lattice site
is reduced, indicating that eigenstates are becoming localized on lattice sites. Furthermore,
with increasing disorder, the eigenstates are avoiding the edges of the chain, and more eigenstates
are present toward the center of the chain. The
changes in PRSpace and PREnergy are the fastest
near D=J ¼ 2, suggestive of a phase transition
that has been smeared out owing to finite-size
effects. Nevertheless, we emphasize that the
quantum phase transition to the MBL phase is
only defined in the thermodynamic limit (Nq→∞)
(15). Given the finite size of our system and the
presence of only two interacting particles, it is
interesting that we see finite-size precursors associated with the MBL phase transition.
It is worth noting that the signatures of localization presented in this work are common
between single-particle localization (Anderson)
and MBL. We refer to our observations as MBL
because of the existence of nonzero interaction U
whose value has been established through independent experiments. However, a decisive distinction could result from carefully engineered
pulse sequences, as suggested in (11).
Our work demonstrates that properties of various phases can be extracted by directly measuring
the energy levels of a system. It is interesting to
consider the application of this method to a device
with a few tens of qubits, where classical simulations will begin to fail. The technique presented here is scalable to more qubits but is
ultimately limited by the frequency broadening
that results from decoherence. For large systems,
the level spacing becomes exponentially denser
and the current approach needs to be revised;
promising methods are suggested in (35, 36).
Nevertheless, the valuable computational resource
that our platform offers resides in measuring the
dynamics of observables; those dynamics quickly
become intractable for classical computers as the
size of the system grows.
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1178 1 DECEMBER 2017 • VOL 358 ISSUE 6367 sciencemag.org SCIENCE
Fig. 4. Participation ratio and
mobility edges. In Eq. 2, we set
; 1Þ=2, J=2p ¼ 50 MHz,
which results in U=J ¼ 3:5. We
measure the evolution of
c2ðn; mÞ ¼ hsX nsX mi ; hs Y ns Y miþ
ihsX n sY mi þ ihs Y n sX mi for all pairs
of n; m∈f1; 2; :::; 9g as a function
of time for various strengths of
disorder D. From the magnitude of
the peaks seen in the FTof the data,
the probabilities relating the
positions of two-photon Fock states
to energy eigenstates fPa;ng are
extracted. See fig. S3 for details.
On the basis of those data, we
calculated (A) PRSpace and
(B) PREnergy using Eq. 5 and plotted
the results. The Emax ; Emin is the
width of the energy band at a given D.