ARTICLES
PUBLISHED ONLINE: 22 MAY 2017 |
DOI: 10.1038/NPHYS4119
Measuring out-of-time-order correlations and
multiple quantum spectra in a trapped-ion
quantum magnet
Martin
Gärttner
1†
, Justin G. Bohnet
2†
, Arghavan Safavi-Naini
1
, Michael L. Wall
1
, John J. Bollinger
2
and Ana Maria
Rey
1
*
Controllable arrays of ions and ultracold atoms can simulate complex many-body phenomena and may provide insights into
unsolved problems in modern science. To this end, experimentally feasible protocols for quantifying the buildup of quantum
correlations and coherence are needed, as performing full state tomography does not scale favourably with the number of
particles. Here we develop and experimentally demonstrate such a protocol, which uses time reversal of the many-body
dynamics to measure out-of-time-order correlation functions (OTOCs) in a long-range Ising spin quantum simulator with
more than 100 ions in a Penning trap. By measuring a family of OTOCs as a function of a tunable parameter we obtain
fine-grained information about the state of the system encoded in the multiple quantum coherence spectrum, extract the
quantum state purity, and demonstrate the buildup of up to 8-body correlations. Future applications of this protocol could
enable studies of many-body localization, quantum phase transitions, and tests of the holographic duality between quantum
and gravitational systems.
T
ime reversal has fascinated and puzzled physicists for
centuries. In an iconic example, Josef Loschmidt argued that
the second law of thermodynamics would be violated by
time-reversing an entropy-increasing collision
1
. Ludwig Boltzmann
responded by formulating the probabilistic definition of entropy,
one of the cornerstones of statistical mechanics, and now a
fundamental concept in quantum information. Since the days of
Boltzmann and Loschmidt, the notion of time reversal has moved
from the arena of thought experiments into the laboratory, with
time reversal of non-interacting quantum systems in the form of
Hahn spin echoes
2
forming an essential part of nuclear magnetic
resonance (NMR)
3
and magnetic resonance imaging.
Recently, the experimental implementation of many-body time-
reversal protocols
4,5
in atomic quantum systems has attracted
attention
6–9
for their potential to quantify the flow of quantum
information in time and set bounds on thermalization times
10–13
,
which might also enable experimental tests of the holographic
duality between quantum and gravitational systems
6,14–17
. The
key quantities sought after are speci al types of out-of-time-order
correlation (OTOC) functions,
F(τ ) =h
ˆ
W
†
(τ )
ˆ
V
†
ˆ
W (τ )
ˆ
V i (1)
where
ˆ
W (τ ) = e
i
ˆ
Hτ
ˆ
W e
−i
ˆ
Hτ
, with
ˆ
H an interacting many-body
Hamiltonian and
ˆ
W and
ˆ
V two commuting unitary operators.
Physically, F(τ ) measures the ‘scrambling’ of quantum information
across the system’s many-body degrees of f reedom—for example,
how fast an initial local perturbation becomes inaccessible to
local probes
16
. Since Re[F(τ )] = 1 − h|[
ˆ
W (τ ),
ˆ
V ]|
2
i/2, F(τ )
encapsulates the degree by which the initially commuting operators
ˆ
W and
ˆ
V fail to commute at later times due to the interactions
generated by
ˆ
H, which we adopt as an operational definition
of scrambling.
Most theoretical studies of scrambling have focused on so-called
fast scramblers in ther mal states
10,11,16
, systems where the commuta-
tor grows exponentially at a rate exclusively determined by the tem-
perature. However, the scrambling behaviour of non-equilibrium
systems at zero temperature will depend on the microscopic param-
eters of the Hamiltonian. This largely unexplored topic can
provide valuable insights into the dynamics of interacting quantum
many-body systems.
Here we perform measurements of OTOCs with a quantum
simulator composed of more than 100 trapped ions
18
interacting
via all-to-all Ising interactions that can be reversed in time. This
Ising interaction allows us to study interesting entangled states
18–21
,
yet still operate in a regime where simulations on conventional
computers are feasible. Thus, our work is a first stepping stone
for exploring scrambling in initially pure quantum systems. Our
approach is modelled after the multiple quantum coherence (MQC)
protocol developed in the context of NMR
3,22,23
to quantify the
buildup of multi-particle coherences (off-diagonal elements of the
many-body density matrix). We show that this protocol, under
specific choices of the initial state (pure states in our experiment),
implements t he measurement of a family of OTOCs. C areful
comparison with theory allows us to us e the measurements as
a verific ation protocol to benchmark t he performance of the
quantum simulator, and to sensitively quantify different sources of
decoherence and imperfect control. In our experiment, which starts
with a pure product state, s crambling can be physically interpreted
as the process by which the information stored (or encoded) in
the initial state, through the interactions, is distributed over and
therefore stored in other many-body degrees of freedom of the
1
JILA, NIST and Depar tment of Physics, University of Colorado, 440 UCB, Boulder, Colorado 80309, USA.
2
National Institute of Standards and Technology
(NIST), Boulder, Colorado 80305, USA.
†
These authors contributed equally to this work.
*
e-mail:
arey@jila.colorado.edu
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ARTICLES
NATURE PHYSICS
DOI: 10.1038/NPHYS4119
system. Thus, it cannot be extracted by measurements of single-
particle observables. Instead it requires measurements of higher-
order correlations. The information is not lost, but requires reading
out the various degrees of freedom.
Future generalizations such as adding a spatially inhomogeneous
magnetic field or a periodic drive would allow one to experimentally
study scrambling behaviour in regimes intractable to theory,
to explore the possibility of fast scrambling in low-temperature
systems, and to investigate possible connections between chaos and
fast scrambling away from the semi-classical limit. The protocols
demonstrated here are widely applicable, and could be implemented
in a variety of other platforms w ith reversible dynamics, such
as linear ion chains
20,24
, ultracold atomic gases
4,5,25
, cold atoms
in optical cav ities
26–28
, Rydberg-dressed atoms
29
, superconducting
qubits
30
, and NMR systems
22
.
The general protocol is illustrated in Fig. 1a. For concreteness,
we consider the system of spin-1/2s, which we implement in our
trapped-ion experiment. The state of interest ˆρ(τ) is prepared by
evolving a fiducial state, ˆρ
0
, under an interacting Hamiltonian
ˆ
H for a time τ. In our experiment the initial density matrix is
ˆρ
0
=
|
+.. .+
ih
+.. .+
|
, where |+i=(
|
↑
i
+
|
↓
i
)/
√
2 and
ˆ
H is
a collective Ising model given by
ˆ
H
zz
=
J
N
X
i<j
ˆσ
z
i
ˆσ
z
j
(2)
where N is the number of spins and ˆσ
z
i
are Pauli spin operators.
Inverting the sign of
ˆ
H
zz
(by changing J to −J ) and evolving
again for time τ to the final state ˆρ
f
, implements the many-
body time reversal, which ideally takes the system back to the
initial state ˆρ
0
. If a state rotation
ˆ
R
x
(φ) =e
−i
ˆ
S
x
φ
, here ab out the x-
axis with
ˆ
S
x
=
1
2
P
i
ˆσ
x
i
, is inserted between the two halves of the
time evolution through a variable angle φ, the dependence of the
revival probability on this angle contains information about ˆρ(τ ).
In this work, we measure two different observables at the end of
the s equence, the collective magnetization along the x-direction,
h
ˆ
S
x
i=tr[
ˆ
S
x
ˆρ
f
], and the fidelity F
φ
(τ ) =tr[ˆρ
0
ˆρ
f
].
The magnetization provides a direct measurement of
2
N
h
ˆ
S
x
i=F
φ
(τ ) =h
ˆ
W
†
φ
(τ ) ˆσ
x
i
ˆ
W
φ
(τ ) ˆσ
x
i
i
0
(3)
for any i, with
ˆ
W
φ
(τ ) =e
i
ˆ
H
zz
τ
ˆ
R
x
(φ)e
−i
ˆ
H
zz
τ
. Here, h·i
0
denotes the
expectation value in state ˆρ
0
. The implementation is facilitated
by the fact that
ˆ
V
|
+
i
= ˆσ
x
i
|
+
i
=
|
+
i
. Moreover, single-spin
resolution is not necessary due to the permutation symmetry of our
system that directly maps ˆσ
x
i
to the global magnetization along x:
ˆσ
x
i
→(2/N)
ˆ
S
x
. In the absence of permutation symmetry, t he OTOC
measured by F
φ
(τ ) should be interpreted as the average over the
magnetization of each of the spins in the array.
Similarly, the fidelity, that is, many-body overlap with the initial
state, can be cast as an OTOC, where now
ˆ
V = ˆρ
0
is not unitary
but F
φ
(τ ) still measures t he failure of two operators to commute
following dynamical evolution (see Methods). Moreover, the fidelity
can be directly linked to t he s o-called multiple quantum intensities
I
m
(see Methods), which quantify the amplitudes of the off-diagonal
elements
3
, or coherences, of the density matr ix ˆρ(τ). The I
m
are
measured by the Fourier components of
F
φ
(τ ) =tr[ˆρ
f
ˆρ
0
]=tr[ˆρ(τ ) ˆρ
φ
(τ )]=
N
X
m=−N
I
m
(τ )e
−imφ
(4)
where ˆρ
φ
(τ ) =
ˆ
R
x
(φ) ˆρ(τ )
ˆ
R
†
x
(φ) (se e Methods). In contrast to
previous implementations in NMR spectroscopy, which typically
operate at effectively infinite temperature, here we consider a spin
system that is initially in a pure state at zero temperature.
Be yond the expected dec ay of the measured OTOCs for
increasing τ and fixed φ, studying the dependence of them on
the rotation angle φ thus reveals information about the buildup
of correlations and provides fine-grained information about the
many-body properties of the state ˆρ(τ). The value of the fidelity at
φ =0 mo d 2π als o provides a direct measurement of the purity of
the many-body spin state, F
0
(τ ) =tr[ˆρ(τ)
2
]. Note that the fidelity
measurement directly implements a many-body Loschmidt echo,
which is typically challenging to exp erimentally measure for systems
of more than ∼10 particles.
To clearly illustrate the dynamics of I
m
and their connection
to off-diagonal elements of the density matrix, we compute F
φ
(τ )
for a small system with N = 6 spins shown in Fig. 1b,c. At
τ =t
cat
=π
¯
hN /(4J ) a macroscopic super position (Schrödinger c at)
state along x is formed
31
, which is signalled in the MQC spectrum
by the cancellation of all I
m
except I
0
and I
±N
. Note that for this case
our scheme is equivalent to the interferometric cat-state verification
scheme realized with N ≤6 ions in Paul traps
19
.
Motivated by the MQC protocol we study the dynamics of the
Fourier amplitudes A
m
of the magnetizat ion
F
φ
(τ ) =
N
X
m=−N
A
m
(τ )e
−imφ
(5)
which probe the buildup of many-body correlations. One can show
that a non-zero A
m
(τ ) signals the buildup of at least m-body cor-
relations. In the case of the Ising model, where all terms in the
Hamiltonian commute with each other, A
m
(τ ) can only be non-
zero if the Hamiltonian directly couples a given spin to m −1 other
spins (see Methods and Supplementary Information). In Fig. 1d we
illustrate the I
m
and A
m
dynamics for N =48 in the absence of deco-
herence, showing the sequential buildup of higher-order coherences
and correlations. Even for the homogeneous Ising interaction, the
protocol reveals a rich structure in the many-body state, including
multiple revivals of coherences. The I
m
spread more rapidly than the
A
m
because the I
m
depends on the many-body overlapwith the initial
state, an N -body operator, which is more sensitive to the central
rotation than the mean spin, a single-body observable.
Our experimental demonstration uses t wo-dimensional (2D) ar-
rays of laser-cooled
9
Be
+
ions in a Penning trap, where the spins are
the valence electron spin states in t he B=4.46 T magnetic field
18,32,33
.
Arbitrary collec tive spin rotations are applied via microwave pulses
(see Fig. 2 and Supplementary Information). Long-range, tunable
spin interactions are engineered through a time-dependent optical
dipole force (ODF), characterized by a frequency µ
r
, that couples
the spins to the axial motional (phonon) modes of the ion crystal.
The driven spin-dependent motion, combined with the Coulomb
force, mediates the spin–spin interaction. Laser cooling and optical
pumping al low us to initialize the spins in a pure, coherent collective
spin state with fidelity >99.9% (ref. 34), and initialize the motional
modes with an average thermal occupation of six quanta, set by the
Doppler cooling limit.
To implement the reversible Ising dynamics, we operate in a
regime where the spins couple to a single phonon mode, the
collective centre-of-mass (COM) mode at frequency ω
z
. Although
there are N axial phonon modes in the cr yst al, the COM mode
is well-resolved for the ODF detuning from the COM mode
δ =µ
r
−ω
z
used here
33
, justify ing the single-mode approximation.
Then the spin–phonon dynamics are given by
31,35
ˆ
H
I
=−
Ω
0
2
√
N
N
X
j=1
ˆa
0
e
iδτ
+ˆa
†
0
e
−iδτ
ˆσ
z
j
(6)
where Ω
0
is proportional to the ODF and ˆa
0
(ˆa
†
0
) is the
annihilation(creation) operator for the COM mode phonons.
In general, the spins will be coupled to the phonon mode, except at
782
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NATURE PHYSICS DOI: 10.1038/NPHYS4119
ARTICLES
Time
Prepare
−π/2
Detect
ττ
|
↓
...
↓
〉
a
bc
x
y
z
φ
0 2/5 1/2 2/3 4/5 1
Fourier amplitude |A
m
| of F
φ
(τ)
0 0.1 1
d
ρ
0
^ ρ
f
^ρ(τ)^ ρ
φ
(τ)^
R
y
^
π/2
R
y
^
R
x
(φ)
^
H
zz
^
−H
zz
^
0.0
0.2
0.4
0.6
0.8
1.0
Fidelity
0
π/2 π 3π/2 2π
Rotation angle φ
0.0
0.5
1.0
0.0
0.2
0.4
Coherence I
m
0.0
0.2
0.4
−6 −4 −2 0 2 4 6
m
τ = 0.1t
cat
τ/t
cat
τ = t
cat
τ = 0.3t
cat
−40
−20
0
20
40
m
−40
−20
0
20
40
m
Fourier amplitude I
m
of
φ
(τ)
Figure 1 | Illustration of the many-body echo scheme. a, Experimental sequence. The global −π/2 rotation
ˆ
R
y
about the y-axis prepares an initial state with
all spins pointing along the x-axis, and enables a measurement in this same basis. The generalized Bloch spheres illustrate the evolution of the state
(Husimi distribution). In the case of φ =0 (blue) the spins return to the initial state, while for φ =π/2 (green) the overlap of the final state ˆρ
f
with the
initial state is small. b, Fidelity signal for an idealized case with N =6 spins and different evolution times τ given in c. c, The Fourier transforms of the fidelity
signals of b. The Fourier amplitudes are identical to the MQCs I
m
, which quantify the coherence of the state ˆρ(τ ). The small squares on the right show the
absolute values of the density matrix elements of ˆρ(τ ) in the basis of symmetric Dicke states. Thus, I
m
is the sum of the squares of all matrix elements at a
distance m from the diagonal. The times are given in units of the time to reach the Schrödinger cat state t
cat
=π¯hN/(4J). d, Simulated dynamics of the
Fourier amplitudes of fidelity, I
m
, and magnetization, A
m
, for purely coherent evolution of 48 ions, illustrating complementary probes of the flow of quantum
information. The vanishing odd Fourier components are not shown.
partic ular decoupling times τ
n
=2πn/δ for an integer n (Fig. 2c and
Supplementary Information). Here we always choose |δ|=2πn/τ ,
ensuring spins and phonons decouple. This guarantees that the
dynamics matches that of the Ising Hamiltonian in equation (2)
with uniform couplings J (δ)/
¯
h =Ω
2
0
/(2δ), and leads to different
values of the coupling constant J at different interaction times
τ . The detuning-dependent coupling enables us to implement a
many-body echo of the spin dynamics by inverting the sign of δ.
For me asur ing magnetization and fidelity, we collect the global
ion fluorescence scattered from the Doppler cooling laser on the
cycling transition for ions in
|
↑
i
, after applying a π/2 rotation
of the spins. We count the total number of photons collected
on a photomultiplier tube (PMT) in a detection period, typically
t
c
=5 ms. From the independently calibrated photons collected per
ion, we can infer the state populations, N
↑
and N
↓
. After averaging
over many experimental trials, between 500 and 800, we calculate
the expectation values h
ˆ
S
z
i=h
ˆ
N
↑
i−N/2. To measure the f idelity,
we distinguish the single state with all ions in
|
↓
i
, which does
not scatter from the cooling las er, from all other states. Any ion
fluorescence indicates the system is no longer in the initial state. The
fidelity is the fraction of experimental trials that result in measuring
the state
|
↓...↓
i
(see Supplementary Information).
Figure 3 shows the measured fidelity F as a function of the
angle φ for different evolution times τ in an ar ray of 48 ions.
The measurements at φ = 0 and 2π give the state purity, while
the π-periodic oscillations encode information on the buildup of
MQCs. The pulse sequence in Fig. 3a follows Fig. 1, whereas in
Fig. 3b, an additional π-rotation has been inserted in the middle
of each e volution period τ to suppress some forms of decoherence.
We ext ract the coherences I
m
, shown in Fig. 3c, as the Fourier
components of the fidelity in Fig. 3b. We see a clear buildup of the
two-body (I
2
), and then four-body (I
4
) coherences with increasing
interaction time. Odd components are zero within statistical error,
consistent with the fact that the coherences are generated by the
Ising interaction, which can be viewed as only flipping pairs of spins.
All the measurements are in go od agreement with theory
calculations (solid lines) that account for independently calibrated
sources of decoherence. Off-resonant light scattering is the
dominant decoherence mechanism in the system. Because the
fidelity measures a projection onto a single many-body state, it
decays with a rate approximately NΓ , where Γ is the single-particle
decoherence rate. This causes a fast decay of I
0
as exp(−N Γ τ ).
However, Fig. 3c shows that I
0
decays as exp(−N Γ τ )I
(pure)
0
, where
the algebraic dec ay I
(pure)
0
≈ 1/(1 + J
2
τ
2
) (see Supplementary
Section 3) signals the buildup of higher-order coherences seen also
in the fully coherent case. Other sources of decoherence includeslow
drifts in the magnetic field
36
and COM mode frequency fluctuations,
which the MQC can distinguish. Figure 3a reveals the degree to
which the COM axial mode phonons cannot be decoupled from
the spins due to uncertainty in the COM mode frequenc y ω
z
. The
impact of residual spin–phonon coupling, arising f rom fluctuations
in ω
z
, is more pronounced at φ = π than φ = 0. In contrast,
slow magnetic field noise causes a reduction of the fidelity around
φ =0(2π), but has no effect at φ =π, allowing us to benchmark
these two imperfections independently. For the data in Fig. 3b,
where the sequence includes an additional π rotation to suppress
errors from slow drifts in the magnetic field and COM mode
frequency, the full t he ory collapses to a solution that includes only
off-resonant light scattering as the sole decoherence mechanism
(dashed line).
Single-body observables, like the collective magnetization, are
much less sensitive to decoherence, and provide an alternative way
to experimentally measure the sequential buildup of higher-order
correlations induced by spin–spin interactions. In Fig. 4, we show
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ARTICLES
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DOI: 10.1038/NPHYS4119
Axial phonon mode
a
f
odf
f
odf
+ µ
r
c
0.0
−6 −4
−δ/2π δ/2π
−2 0
Drive detuning (kHz)
Magnetization S
x
/(N/2)
246
0.2
0.4
0.6
0.8
1.0
b
50 μm
|
↑
〉
|
↓
〉
Side view
B
B
Bottom view
x
z
V
z
z
y
z
y
x
Figure 2 | Phonon-mediated, reversible spin–spin coupling in a
Penning trap. a, (Left) Illustration of Penning trap cross-section. Ions (blue
circles) are confined axially to a single 2D plane (shown in b) with static
electric fields from potentials on the electrodes (gold). Rotation of the ions
in the axial magnetic field B produces radial confinement from the Lorentz
force. A pair of detuned ODF beams (green) interfere and form a travelling
wave optical lattice, producing spin-dependent COM mode excitations that
couple the spins to the axial phonon mode. Shown here are two of (2N +1)
excitations: all ions in
|
↑
i
(purple) and all in
|
↓
i
(orange). (Right) The
phonon wave packets experience equal and opposite displacement in the
axial potential V
z
. Spin-dependent motion, along with the Coulomb
interaction, generates the spin–spin coupling. b, Rotating frame image of 2D
array of
9
Be
+
ions, integration time 2.1 s. c, Residual spin–phonon coupling
for drive frequencies away from the decoupling points ±δ appears as a
decrease in the magnetization measured after the experimental sequence
from Fig. 1, with φ =π, and without inverting
ˆ
H
zz
. Here τ =200 µs. Note
that decoupling points appear at ±δ with +δ giving an anti-ferromagnetic
interaction, and −δ giving a ferromagnetic interaction used for the time
reversal of the
ˆ
H
zz
dynamics.
the results of the magnetization OTOC measurement sequence,
which shows a buildup of Fourier amplitudes, A
m
, up to m =8,
observable even for N =111. These measurements also allow us
to benchmark the quality of our quantum simulator by comparing
to theor y predictions with no adjusted parameters. Here, the
dashed lines are obtained by solving t he pure spin model including
only spontaneous emission decoherence (see Supplementary
Information), showing agreement in both the φ-dependent signal
(Fig. 4a) and its Fourier transform (Fig. 4b). Accounting for static
magnetic field noise largely explains the remaining discrepancy
at small angles (solid lines in Fig. 4a). Comparison of the data to
theory predictions with no decoherence (Fig. 4c) confirms that the
decay of the Fourier amplitudes at long times is not a decoherence
effect but a consequence of many-body interactions which induce
a decreas e of low-m components with a corresponding buildup of
high-m components. Since the observed dynamics is dominated
by the coherent evolution under the Ising interaction, these results
suggest that the observed features can be explained only by the
formation of quantum correlations.
In summary, we have shown that many-body Loschmidt echo
sequences are powerful tools to measure OTOCs and quantify
the degree of coherence in quantum simulators, with an explicit
demonstration for ions in a Penning trap. In particular, we
studied OTOCs involving variable angle spin rotations. The Fourier
components with respect to the rotation angle (I
m
and A
m
) show
a buildup of many-bo dy coherence and correlations, indicating
I
0
e
−NΓτ
a
b
c
0.3 ms
0.5 ms
0.7 ms
0.0
0.1
0.2
0.3
0.4
0.52 ms
0.72 ms
1 ms
0 π/2 π 3π/2 2π
0.0
0.1
0.2
Rotation angle φ
0 π/2 π 3π/2 2π
Rotation angle φ
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.0 0.5 1.0 1.5 2.0 2.5
τ (ms) τ (ms)
Coherence
J/ħ (ms
−1
) J/ħ (ms
−1
)
I
2
I
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00
0.01
0.02
0.0 0.5 1.0 1.5 2.0 2.5
Fidelity Fidelity
Figure 3 | Measured fidelity and coherence spectrum of N = 48 ions.
a,b, Dependence of the fidelity F
φ
(τ ) on the rotation angle φ. The
experimental sequence in b includes an additional π pulse in the middle of
each evolution period τ . The dashed lines are simulations including
off-resonant light scattering as the only source of decoherence, with
Γ =62 s
−1
. The solid lines include effects of COM mode and magnetic field
fluctuations, with COM mode frequency fluctuations 1
COM
/ω
z
=
8.0×10
−5
r.m.s., and magnetic field noise 1
B
/B =0.32 ×10
−9
r.m.s.
(Methods). Note that for each interaction time τ the detuning is chosen
so that δ =2π/τ (a) or δ =4π/τ (b). In each case, the spin–spin coupling
also varies as J/¯h =Ω
2
0
/(2δ) where Ω
0
=7,850 s
−1
. c, Fourier amplitudes
of fidelity (b) as a function of time. Solid lines are simulations including all
known decoherence processes. I
2
and I
4
clearly show the buildup of
higher-order MQCs. Odd coherences and coherences m ≥6 are zero
within the statistical error. For I
0
, decoherence induced decay (dashed)
and approximate analytic curve (dotted, see text) are shown. The data
points at τ =0.3 and 0.9 (not shown in b) have been added. The longest
measured evolution time of τ =1 ms corresponds to 6.5% of t
cat
(see Fig. 1d). All error bars denote the statistical error of 1 standard
deviation (s.d.) of the mean.
scrambling of quantum information. O ur experimental results are
described well by a theory model which accounts for all known
sources of decoherence (photon scattering, magnetic field noise, and
spin–phonon coupling), allowing us to benchmark the performance
of our trapped-ion quantum simulator.
784
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NATURE PHYSICS DOI: 10.1038/NPHYS4119
ARTICLES
a
b
c
0.3 ms
0.5 ms
0.7 ms
0.9 ms
0
π/2 π 3π/2 2π
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rotation angle φ
Magnetization OTOC F
φ
(τ)
Theory
0.0 0.2 0.4 0.6 0.8 1.0 1.2
012345
Experiment
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0
2
4
6
8
10
012345
τ (ms) τ (ms)
0.0 0.2 0.4 0.6 0.8 1.0 1.2
τ (ms)τ (ms)
Fourier components m
0.0
0.2
0.4
0.6
0.8
1.0
Ideal
Magnified
0
2
4
6
8
10
Ideal
0.0 0.5 1.0 1.5 2.0
0
10
20
30
40
Fourier components m
J/ħ (ms
−1
) J/ħ (ms
−1
)
Figure 4 | Probing scrambling through magnetization dynamics.
a, Dependence of the normalized component F
φ
(τ ) =(2/N)h
ˆ
S
x
iof the total
spin on the rotation angle φ, measured in an array of N =111(2) ions. Lines
are the solutions of the full master equation with (solid) and without
(dashed) magnetic field noise, where 1
B
/B =0.32 ×10
−9
r.m.s. The effect
of COM mode fluctuations is negligible here. Error bars denote the
statistical error of 1 s.d. of the mean. b, Fourier amplitudes A
m
as a function
of time. In the theory plot, the case without magnetic field noise (dashed
lines in a) was used. The interaction parameter varies as J/¯h=Ω
2
0
/(2δ)
where Ω
0
=7,450 s
−1
and Γ =91 s
−1
. The longest measured evolution
time of τ =1.2ms corresponds to 7.3% of t
cat
. c, Ideal case for N =111,
neglecting all decoherence effects. This corresponds to the lower panel of
Fig. 1d. The box in the left panel shows the experimentally accessed region
which is magnified in the right panel.
The characteristic features of A
m
s reported in this work demon-
strate a high level of control over the coherent many-body dy-
namics achieved by our trapped-ion quantum simulator and are
fully consistent with the buildup of quantum correlations. Al-
though currently the latter can be only indirectly inferred from the
measurements, it is supported by previous benchmarking of the
system using standard entanglement witnesses such as spin squeez-
ing
18
. We expect future work to derive formal connections be-
tween entanglement and scrambling, and to construct strict bounds
that witness entanglement directly from I
m
and A
m
measurements.
Although the current experimental system realizes a model
amenable to classical simulations, we envisage experiments going
beyond this limit—for example, by adding a spatially inhomoge-
neous magnetic field or preparing the system in non-symmetric
or impure initial states, such as thermal states. These general-
izations will allow us to explore the dynamics of OTOCs and
characterize scrambling in unexplored regimes and under condi-
tions where fast scrambling can occur. Furthermore, the ability to
time-reverse the dynamics will allow enhanced phase estimation
without single-particle detection resolution
5,29,37
, investigations of
quantum phase transitions
38
, criticality
39
, thermalization in nearly
closed quantum systems
13,40
and the exploration of the quantum-
classical boundary
41
—for example, observation of the violation of
Leggett–Garg inequalities
42
.
After the completion of this work, we became aware of measure-
ments of OTOCs using four spins in an NMR system
43
.
Methods
Methods, including statements of data availability and any
associated accession codes and references, are avai lable in the
online version of this paper.
Received 6 December 2016; accepted 30 March 2017;
published online 22 May 2017
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