to be measured by taking ∼N frequency spectra
and requires only global interactions and site-resolved measurements.
The spin-1/2 particles are represented by a
string of 171 Ybþ ions confined in a linear Paul trap.
The spin states j↓⟩z and j↑⟩z are encoded in the
magnetic-field-insensitive (mF ¼ 0) hyperfine states
of the ground electronic manifold (28). The spin-
spin couplings and effective magnetic fields derive
from lasers that globally illuminate the ion chain,
driving stimulated Raman transitions between
the spin states (14, 20). State initialization com-
prises optical pumping into the j↓↓↓⋯⟩z state
followed by a coherent rotation to polarize all
spins along the desired axis. After applying the
spin-spin couplings and the probe field(s) de-
scribed above, the individual spin states are read
out along any axis by performing a coherent ro-
tation from that axis to the measurement basis
j↓⟩z and j↑⟩z, then collecting state-dependent flu-
orescence onto a charge-coupled device imager
with site-resolving optics (20).
We measure the energy splittings in our spin
system using a weakly modulated transverse field
as a probe.
BðtÞ ¼ B0 þ Bpsinð2pnptÞ ð2Þ
When the probe frequency np is matched to the
energy difference jEa − Ebj between two eigen-
states ja⟩ and jb⟩, the field will drive transitions
between the two states if there is a nonzero
matrix element ⟨bjBðtÞ∑
isy i ja⟩ ≠ 0. For example,
in the weak-field regime BðtÞ << J0, the Hamil-
tonian eigenstates are symmetric combinations
of the sx eigenstates, and the matrix element
⟨bjBðtÞ∑
isy i ja⟩ is nonzero only when ja⟩ and jb⟩
differ by the orientation of exactly one spin.
In the weak-field regime, a transition at a sin-
gle frequency can easily be monitored, and its
stability can provide a good proxy for the entire
Hamiltonian. Each splitting depends on multi-
ple spin-spin couplings—for example, a transition
from j1111⋯⟩ to j0111⋯⟩, where j1⟩ (j0⟩) denotes
the sx eigenstate j↑⟩x (j↓⟩x), requires energy
DE ¼ 2ðJ1;2 þ J1;3 þ ⋯ þ J1;N Þ ð3Þ
These splittings are therefore sensitive to changes
in the motional mode structure or the laser inten-sities at each of the ions.
We demonstrate the mapping of individual ener-
gy splittings in the weak-field regime Bðt Þ=J0 << 1
in Fig. 1. The spins are prepared along the x
direction in j111⋯⟩, and a probe field correspond-
ing to BðtÞ ¼ ð100 Hz)sinð2pnptÞ is applied for
3 ms, which is sufficient to transfer more than
50% of the population between states, before mea-
suring along x. These parameters allow resolu-
tion of the energy differences in an eight-spin
system while still accommodating the few ms
decoherence time scale in our system (18).
Population transfer is clearly seen when np is
resonant with an energy splitting (e.g., Fig. 1, B
and C). We quantify the energy of a particular
state relative to the initial state by fitting Lorentz-
ians to the spectra (20). The spectral positions
are insensitive to state preparation and measure-
ment error, which affects only the contrast of these
resonances, as seen in Fig. 1C with N ¼ 18 spins.
A sequence of multiple probe frequencies (shown
in fig. S1) can be used to populate any desired
spin configuration with a global beam in no more
than ⌊N=2⌋ pulses. We can transfer population
into any of the 32 eigenstates of a five-spin system
by starting in either the j11111⟩ or j00000⟩ and
applying at most two pulses of the transverse
field. This system is small enough to also mea-
sure the entire relative energy spectrum, which
scales exponentially with system size. Starting
from the states j11111⟩, j00000⟩, j10101⟩, and j01010⟩
[the last two of which are prepared using an
adiabatic ramp of a transverse field (18)], we use
single and multiple frequency drives to measure
all possible energy splittings.
Figure 2 shows the measured spectrum of this
five-spin system, obtained by direct addition of
the measured energy splittings, compared to that
given by the interactions estimated from the same
data (as detailed below). An examination of the
full spectrum of a many-body quantum system is
generally difficult to achieve and shows the ver-
satility of this form of many-body spectroscopy.
We can also use modulated transverse fields to
prepare arbitrary coherent quantum states, which
can be used to probe many-body quantum dy-
namics (24). An example protocol for preparing
a specific spin configuration is shown in fig. S1.
Fig. 2. Reconstructed energy spectrum. In a system of five spins, the energy of each spin configuration
above the j10101⟩ ground state (colored points) is compared to the calculated energies (black lines).
Calculations are based on the spin-spin couplings estimated from the same energy measurements (inset).
Error bars include statistical errors and an estimate of systematic error due to experimental drifts.
Ji,j (Hz)
Ionindexi Ionindexj
641
479
376
79 266
222
174
546
374
359
225
232
215
482
417
326
245
217
504
414
325
268
599
458
326
500
408
646
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7
81
2
3
4
5
6
7
8
Ji,j (Hz)
Ionindexi Ionindexj
702
416
156 317
168
104
75
637
420
211
226
180
102
588
264
291
185 290
644
439
328
139
603
435
285
583
364
641
123456
7
81
2
3
4
5
6
7
8
1234567
0
200
400
600
800
r (distance between ions)
J
i
,
i
+r
(Hz
) α=0.54±0.06
α=0.85±0.07
α ≈ 0.54 α ≈ 0.85
AB C
Fig. 3. Experimentally determined spin-spin coupling profiles. The couplings were measured in a
system of eight spins for two sets of trap parameters, corresponding to a more long-range or more
short-range interaction profile. (A) and (B) depict the individual elements of the measured coupling
matrix. (C) plots measured average interactions against ion separation and shows fits to a power law
J0=ra. The error in a is an estimate of the standard error in the fit parameter; this takes into account the
errors in the Ji,j estimates based on fit error and statistical error in population measurements (20).
