In general, subjecting a left-right symmetric state
to a global resonant probe field prepares symmetric superpositions of states that exhibit a degree of
entanglement, although this is difficult to detect
without individual rotations for readout.
We generate entanglement by subjecting an
initial state j111⋯11⟩ to multiple frequencies simultaneously, such that all of the possible transitions
are driven equally. After an appropriate time, the
system will ideally be driven into a W-type state of
the form
jYW⟩¼ 1ffiffiffiffi N p ;eif0j011⋯11⟩þeif1j101⋯11⟩þ
⋯ þ eif1 j111⋯01⟩ þ eif0 j111⋯10⟩
;
ð4Þ
where the phases fi depend on the relative phase
of the applied modulation frequencies. Entanglement can then be detected using global measurements of the magnetization along various
directions of the Bloch sphere (29). In particular,
we use a witness operator
Wss ¼ ðN − 1Þð< J2 x > − < Jx >2Þ þ N2 −
<J2 y >−<J2 z > ð5Þ
where Jg ≡ 1 2
N
∑
i¼1
sg i (with appropriate phases) and
angle brackets denote ensemble averages. This
spin-squeezing observable will always be positive
for separable states, so measurement of a neg-
ative value certifies that at least two particles are
entangled. We prepare an entangled state of four
spins by applying two simultaneous frequencies
of the modulated transverse field to the state
j1111⟩ with an appropriate relative phase for
1.8 ms and measure the resulting state along
the Bloch sphere directions x, y, and z to obtain
the witness shown above. This certifies that the
full state is entangled. Moreover, individual spin-state imaging allows us to trace over any given
spin or pair of spins and apply the witness to this
reduced density matrix. Table S1 displays these
data, which is consistent with entanglement in
every possible reduced state.
Many-body spectroscopy using a transverse
probe field further enables determination of
each individual spin-spin coupling Ji;j. Using
only N þ 1 scans of the probe frequency, we can
measure
;
N
2
;
¼ NðN − 1Þ=2 energy splittings
and thus determine the entire interaction matrix
of
;
N
2
;
couplings (e.g., Eq. 3). For example, one
scan probes the state j1111⋯⟩ and yields the N
energy splittings to the single-defect states. Then,
as in fig. S1, N additional scans starting from each
single-defect state determine N−1 further energy
splittings. In total, these N þ1 scans yield N2
measurements [N from the first probe scan and
NðN − 1Þ from the rest]. Because of the parallel
processing enabled by imaging individual spin
states, the number of measurements to evaluate
all spin-spin couplings scales only linearly with
the system size (20). We perform this verification
protocol on a system of eight spins with two dif-
ferent interaction ranges. We can measure the
full interaction matrix with five frequency scans;
because of the left-right symmetry, single-defect
states are populated in pairs and only four scans
are necessary to probe all eight of the defect states.
In contrast, mapping all 2N energies (as done
above with five spins) would already require ~100
scans for an eight-spin system. The obtained matrix
agrees well with theory; roughly 70% of measured
interactions match the prediction within 1s stan-
dard error. We observe a distinction in the cou-
pling matrices for differently chosen ranges of
spin-spin interactions (Fig. 3).
Finally, we probe energy levels at nonzero
transverse field B0, including near the critical
region B0 ≈ < J >. Determining the critical energy gap D, at which the energy difference
between the ground and lowest coupled excited
states is minimized, is useful because this parameter determines the ability to perform an
adiabatic sweep of the transverse field (19, 30).
However, measuring the critical gap is difficult
in general because of the inability to measure or
even know the instantaneous eigenbasis.
The protocol described in Fig. 1 is effective
even when there is a small dc field B0 (Fig. 4, A
and B) but breaks down near the critical region.
However, for a finite-size ferromagnetic system,
measurements along a different axis of the Bloch
sphere (here, ˇ x þ ˇ y) allow us to still observe transitions from the ground to the first coupled excited state near the critical gap (Fig. 4, C and D).
As shown in Fig. 4E, these experiments allow us
to map the lowest coupled excited state from
B0 ¼ 0 beyond the critical energy gap D. The
downward drift in energies near B0 ¼ 0 can be
attributed to drifts in laser and trap parameters
as the experiments progressed from higher to
lower fields. An alternative protocol, which follows
the time evolution after a quench, has recently
been proposed for measuring the critical gap
and may scale better for larger systems (31).
Our technique will no doubt benefit from
further refinements borrowing from the extensive literature of spectroscopic methods developed in other fields, such as nuclear magnetic
0.0 0.5 1.0 1.5 2.0
1
2
3
4
5
6
7
8
0.0
0.2
0.4
0.6
0.8
1.0
Rescaled population in |11111111 x or |11111111 Φ
Population in |11111111 Φ after driving from
adiabatically ramped ground state
Population in |11111111 x after driving from |11111111 x
B0 = 0.4 kHz B0 = 0.8 kHz
B0 = 1.4 kHz B0 = 1.8 kHz
Probe frequency (kHz) Probe frequency (kHz)
Probe frequency (kHz) Probe frequency (kHz)
AB
CD
E
AB CD
∆
Transverse field offset, B0 (kHz)
Pro
be
fr
equ
en
cy
(
k
Hz
)
12345678
0.3
0.4
0.5
0.6
0.7
0.8
12345678
0.1
0.2
0.3
0.4
0.5
12345678
0.2
0.24
0.28
0.32
12345678
0.1
0.2
0.3
0.4
Fig. 4. Critical regime. (A to D) Populations in a polarized state versus
modulation frequency of the transverse field at four different values of the
offset field B0. Coloring is according to the rescaling scheme used in (E). In
(A) and (B), we subject the state j11111111⟩ to the modulated field, then
measure its population. In (C) and (D), we prepare the ground state via an
adiabatic ramp, subject it to the modulated field, and then measure the
population in j↑↑↑↑↑↑↑↑⟩f. (E) Rescaled populations in j11111111⟩ (left of the dashed line) or j↑↑↑↑↑↑↑↑⟩f (right of the dashed line) versus static field
offset B0 and modulation frequency. Calculated energy levels, based on measurements of trap and laser parameters, are overlaid as thin white lines, and
the lowest coupled excited state as a thick red line, showing the critical gap D at position C. The energy of the ground state is always taken to be zero.
