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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Dynamical quantum correlations of Ising models on an arbitrary W911NF-12-1-0228
lattice and their resilience to decoherence
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Michael Foss-Feig,, Kaden R A Hazzard, , John J Bollinger, , Ana Maria
Rey, , Charles W Clark 5e. TASK NUMBER
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14. ABSTRACT
Ising models, and the physical systems described by them, play a
central role in generating entangled states for use in quantum metrology and
quantum information. In particular, ultracold atomic gases, trapped ion systems,
and Rydberg atoms realize long-ranged Ising models, which even in the absence
of a transverse eld can give rise to highly non-classical dynamics and long-range
15. SUBJECT TERMS
spin models decoherence Isig exact solution trapped ions squeezing
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a. REPORT b. ABSTRACT c. THIS PAGE ABSTRACT OF PAGES Ana Maria Rey
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Report Title
Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence
ABSTRACT
Ising models, and the physical systems described by them, play a
central role in generating entangled states for use in quantum metrology and
quantum information. In particular, ultracold atomic gases, trapped ion systems,
and Rydberg atoms realize long-ranged Ising models, which even in the absence
of a transverse eld can give rise to highly non-classical dynamics and long-range
quantum correlations. In the rst part of this paper, we present a detailed theoretical
framework for studying the dynamics of such systems driven (at time t = 0) into
arbitrary unentangled non-equilibrium states, thus greatly extending and unifying
the work of Ref. [1]. Speci cally, we derive exact expressions for closed-timepath
ordered correlation functions, and use these to study experimentally relevant
observables, e.g. Bloch vector and spin-squeezing dynamics. In the second part, these
correlation functions are then used to derive closed-form expressions for the dynamics
of arbitrary spin-spin correlation functions in the presence of both T1 (spontaneous
spin relaxation/excitation) and T2 (dephasing) type decoherence processes. Even
though the decoherence is local, our solution reveals that the competition between
Ising dynamics and T1 decoherence gives rise to an emergent non-local dephasing e ect,
thereby drastically amplifying the degradation of quantum correlations. In addition
to identifying the mechanism of this deleterious e ect, our solution points toward a
scheme to eliminate it via measurement-based coherent feedback.
REPORT DOCUMENTATION PAGE (SF298)
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Continuation for Block 13
ARO Report Number 61841.7-PH
Dynamical quantum correlations of Ising models ...
Block 13: Supplementary Note
© 2013 . Published in New Journal of Physics, Vol. Ed. 0 (2013), (Ed. ). DoD Components reserve a royalty-free, nonexclusive
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Approved for public release; distribution is unlimited.
Dynamical quantum correlations of Ising models on
an arbitrary lattice and their resilience to
3
1 decoherence
0
2
n M Foss-Feig1,2, K R A Hazzard3,4, J J Bollinger4, A M Rey3,4,
u
J and C W Clark1,2
2 1National Institute of Standards and Technology, Gaithersburg, MD, 20899, USA
2Joint Quantum Institute and the Department of Physics, University of Maryland,
]
s College Park, MD, 20742, USA
a
g 3JILA and the Department of Physics, University of Colorado, Boulder, CO
- 80309-0440, USA
t
n 4National Institute of Standards and Technology, Boulder, Colorado 80305, USA
a
u
q Abstract. Ising models, and the physical systems described by them, play a
.
t central role in generating entangled states for use in quantum metrology and
a
m quantum information. In particular, ultracold atomic gases, trapped ion systems,
and Rydberg atoms realize long-ranged Ising models, which even in the absence
-
d of a transverse field can give rise to highly non-classical dynamics and long-range
n
quantum correlations. In the first part of this paper, we present a detailed theoretical
o
c framework for studying the dynamics of such systems driven (at time t = 0) into
[ arbitrary unentangled non-equilibrium states, thus greatly extending and unifying
1 the work of Ref. [1]. Specifically, we derive exact expressions for closed-time-
v path ordered correlation functions, and use these to study experimentally relevant
2
observables, e.g. Bloch vector and spin-squeezing dynamics. In the second part, these
7
1 correlation functions are then used to derive closed-form expressions for the dynamics
0 of arbitrary spin-spin correlation functions in the presence of both T (spontaneous
1
6. spin relaxation/excitation) and T2 (dephasing) type decoherence processes. Even
0 though the decoherence is local, our solution reveals that the competition between
3
IsingdynamicsandT decoherencegivesrisetoanemergentnon-local dephasingeffect,
1 1
thereby drastically amplifying the degradation of quantum correlations. In addition
:
v
to identifying the mechanism of this deleterious effect, our solution points toward a
i
X scheme to eliminate it via measurement-based coherent feedback.
r
a
CONTENTS 2
Contents
1 Introduction 2
2 Coherent dynamics in Ising models 4
2.1 Bloch vector dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Equal time correlation functions . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Unequal-time correlation functions . . . . . . . . . . . . . . . . . . . . . 10
3 Inclusion of dissipation 11
3.1 Description of the problem . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Dephasing (T decoherence) . . . . . . . . . . . . . . . . . . . . . . . . . 13
2
3.3 Spontaneous relaxation and excitation (T decoherence) . . . . . . . . . . 14
1
3.4 A simple application: under-damped to over-damped transitions . . . . . 17
3.5 Qualitative insights into decoherence in interacting many-body systems . 18
4 One-axis twisting in an open system 20
4.1 Spin Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2 Macroscopic superposition states . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Removing the effects of decoherence via measurement-base feedback . . . 24
5 Conclusions and future directions 26
AppendixA 27
AppendixB 30
1. Introduction
Interacting spin models provide a remarkably accurate description of a diverse set of
physicalsystems, rangingfromquantummagneticmaterials[2,3,4]toquantumdots[5],
nitrogenvacancycenters[6], superconductingqubitarrays[7], ultracoldatomicgases[8],
and trapped ions [9, 10]. Despite being relatively simple, and often admitting accurate
theoretical descriptions, they support a variety of complex equilibrium properties found
in real materials, e.g. emergent spatial ordering [2], quantum criticality [11], and
nontrivial topological phases [12, 13]. While the equilibrium physics of the simplest
quantumspinmodelsis, withmanynotableexceptions, fairlywellunderstood, thestudy
of driven, dissipative, and otherwise non-equilibrium behavior is comparatively full of
openquestions: Underwhatcircumstancescanequilibriumcorrelationssurvivecoupling
to a noisy environment [14, 15, 16]? To what extent do the concepts of criticality and
universality extend to dynamics and non-equilibrium steady-states [17, 18, 19, 20]?
Can interesting quantum-correlated states be stabilized by (rather than degraded by)
decoherence[21,22,23,24,25,26,27,28,29]? Inrecentyears,ithasbecomeincreasingly
apparent that non-equilibrium dynamics is ideally suited to investigation by quantum
CONTENTS 3
simulation[30], makingsuchquestionsespeciallytimelyandimportant. Moreover, there
are many examples where interesting non-equilibrium states of matter are more readily
achievable than low temperature equilibrium states in ultracold neutral gases [31], polar
molecules [32], and trapped ions [33].
With these motivations in mind, in this manuscript we develop a general formalism
for calculating unequal-time correlation functions of arbitrary-range Ising models driven
far out of equilibrium at time t = 0, thus establishing a comprehensive toolbox for the
description of non-equilibrium dynamics in a simple context. In addition to providing
a tractable example of quantum many-body spin dynamics, the Ising model is realized
to a good approximation in a variety of experimentally relevant systems. And, despite
its simplicity, Ising spin dynamics is known to be useful for the production of entangled
states with applications in quantum information and precision metrology [34]. Our
results constitute a unified approach to describing experiments aimed at producing
such states [35, 36, 37, 38, 39, 33], and facilitate a quantitative treatment of a variety
of unavoidable experimental complications, e.g. long-range (but not infinite-range)
interactions and initial-state imperfections.
Ultracold atomic systems are also well suited to the controlled inclusion of
dissipation, prompting a number of theoretical proposals to exploit dissipation for the
creation of interesting quantum states [40, 19, 21, 22]—remarkably, such ideas are
already coming to experimental fruition [24, 41]. Verifying that these experimental
systems behave in the expected manner in the presence of dissipation, however, is
extremely challenging, in large part due to the numerical complexity of simulating
dynamics in open quantum systems and the scarcity of exact solutions. The Ising
model, especially as implemented in trapped ion experiments [39, 42, 33], poses a unique
opportunity to study the effects of dissipation in a controlled and, as we will show,
theoretically tractable setting. In the absence of dissipation, an important issue in the
Isingmodeliswhethergroundstatecorrelationssurvivetheapplicationofanequilibrium
coherent drive that does not commute with the interactions—i.e. a transverse field. In
the dissipative Ising model an analogous question can be posed: How does the system
respond to being driven incoherently by processes that do not commute with the Ising
interaction? Our formalism for the calculation of unequal-time correlation functions
allows us to definitively answer this question.
Quite surprisingly, non-equilibrium dynamics in the Ising model remains solvable
in the presence of non-commuting dissipation [1], even for completely arbitrary spatial
dependence of the Ising couplings (and therefore in any dimension). This manuscript
substantially extends the groundwork laid in Ref. [1], where the quantum trajectories
technique was used to obtain a closed-form solution of non-equilibrium dynamics for a
special class of initial states. The present work not only provides a more direct, unified,
and comprehensive exposition of the relevant theory, but also generalizes those results
to include a broader class of initial conditions and observables, and applies the solutions
to a number of experimentally relevant problems (most of which had previously been
explored only by numerical or approximate techniques, if at all). We also develop a
CONTENTS 4
clear physical picture of the interplay between coherent interactions and spontaneous
spin flips, which reveals that T decoherence is much more detrimental to entanglement
1
generation than might be naively expected. However, our solution also points toward a
measurement-based feedback scheme that can mitigate its detrimental effects.
The organization of the manuscript is as follows. In Sec. 2 we consider the coherent
(Hamiltonian) far-from-equilibrium dynamics of an Ising model with arbitrary spin-
spin couplings. Our results comprise a unified framework for calculating unequal-time
correlation functions starting from arbitrary unentangled pure states. As special cases,
these results will be applied to calculating Bloch-vector dynamics, arbitrary equal time
spin-spin correlation functions, and two-time dynamical correlation functions. These
results are substantially more general than any already available in the literature
[43, 44, 34, 45, 1], and will help quantify the quantum-enhanced precision in metrology
experiments using trapped ions and ultracold neutral atomic gases. In Sec. 3
we consider the effect of Markovian decoherence on this dynamics, incorporating
dephasing, spontaneous excitation, and spontaneous relaxation. Because the excitation
and relaxation processes do not commute with the Ising dynamics, including them is
especially nontrivial: we work in the interaction picture of the Ising Hamiltonian and
incorporatethemastime-dependentperturbations. Termsintheperturbativeexpansion
are evaluated using the tools laid out in Sec. 2, and by summing the perturbation theory
to all orders we obtain an exact (closed-form) description of the dissipative dynamics
of arbitrary two-point correlation functions. This is in stark contrast to the behavior
of a coherently driven Ising model, where such a perturbative expansion cannot in
general be resummed. An interesting feature revealed by our exact solution is that the
spin dynamics undergoes an oscillatory-to-damped transition at a critical dissipation
strength, which—in the absence of a coherent drive—cannot occur at the single-particle
or mean-field level. This feature, along with the more general structure of our solution,
is demonstrated by solving for spin dynamics in a nearest-neighbor Ising model. We
conclude this section by casting our solution in terms of a clear physical picture, in
which T decoherence (spontaneous excitation/relaxation), through its interplay with
1
the Ising dynamics, gives rise to an emergent non-local dephasing process. Section 4
applies the solution to calculating experimentally relevant observables in a dissipative
version of the one-axis twisting model. We show that the emergent dephasing discussed
in Sec. 3 severely diminishes the precision enhancement achievable compared to that
obtained in the absence of decoherence. However, we also show that, in special cases,
thisdegradationcanbepreventedbyameasurement-basedfeedbackmechanism. InSec.
5 we summarize our results and pose a number of unanswered questions that would be
interesting to address in future work.
2. Coherent dynamics in Ising models
Our goal is to develop a unified strategy for describing the dynamics of a collection of
spin-1/2 particles interacting via Ising couplings and initially (at time t = 0) driven far
CONTENTS 5
out of equilibrium. In the absence of a magnetic field, the most general form for an
Ising model is
‡
= J σˆzσˆz, (1)
H jk j k
j<k
(cid:88)
where σˆz are z Pauli matrices and the indices j,k label lattice sites located at spatial
j
positions r . The coupling constants J are left completely arbitrary, and hence there
j jk
is not necessarily any notion of dimensionality. In many physical realizations of this
Hamiltonian, such as trapped ions, neutral atoms, or Rydberg atoms, the couplings
exhibit a roughly power-law spatial dependence, J = J r r −ζ.
jk j k
| − |
Because there is no transverse field (a term h σˆx), the eigenstates of can
∝ j j H
alwaysbechosentobesimultaneouseigenstatesofalltheσˆz (witheigenvaluesσz = 1).
(cid:80) j j ±
As a result, the partition function Z(β) = tr e−βH (with β the inverse temperature)
is identical to that of a classical Ising model, and the equivalence of all equilibrium
(cid:2) (cid:3)
properties follows. This is the sense in which the Ising model without a transverse
field is often said to be “classical” (even though it is a quantum Hamiltonian acting
on vectors in a Hilbert space). In passing we note that classical Ising models can, of
course, be highly nontrivial: for example, disordered or frustrated couplings give rise to
classical glassiness [46, 47, 48]. Out of equilibrium, however, this notion of classicality
is inapplicable. While a thermal density matrix ρ(β) = Z−1e−βH commutes with ,
H
the density matrix describing some non-equilibrium initial conditions will not in general
commute with the Hamiltonian, and nontrivial dynamics will ensue. This dynamics—
whichhasnodirectanalogueintheclassical Isingmodel—isgenericallycharacterizedby
thegrowthofentanglement,leadinginsomespecialcasestospin-stateswithapplications
in quantum information and precision metrology[34].
Everywhere in this manuscript, we assume the system starts in a pure state that
is a direct product between the various spins [Fig. 1(a)]. The most general such state
§
can be specified by choosing spherical angles θ and φ describing the orientation of the
j j
spin at each site j [Fig. 1(b)]. Defining
θ θ
f (1) = e−iφj/2cos j, f ( 1) = eiφj/2sin j, (2)
j j
2 − 2
and states σ that are eigenstates of σˆz with eigenvalues σ = 1, such a state can be
| j(cid:105) j j ±
written
Ψ(t = 0) = ψ (3)
j
| (cid:105) | (cid:105)
j
(cid:79)
= f (σ ) σ . (4)
j j j
| (cid:105)
(cid:79)j (cid:88)σj
Foruniformθ = θ andφ = φ,thestate Ψ(0) isfrequentlyencounteredinexperiments
j j
| (cid:105)
Note that many references studying long-ranged Ising models—i.e. those for which J is an
‡ j ij
extensive quantity—often normalize the interaction by dividing by the number of spins . We drop
N(cid:80)
this constant here to avoid cluttering the notation.
Unentangled but mixed initial density matrices can be easily accounted for by suitable averaging of
§
the expressions given over the initial ensemble.
I
I
I
interactions
time 6
~.....---+------+-~•
0 t
(h\
z
Figure 1. The only restriction on the initial state is that it be unentangled (i.e.
a product state over the various sites in the spin model). For example, one can
imagine (a) an initial state with some slow spatial variations in the spin angles due
to inhomogeneities in the pulse strength of a Ramsey-type e..xperiment. The notation
used to characterize the state of any one spin is shown on the Bloch sphere in (b).
implementing Ramsey spectroscopy [49, 33, 32], and spatially varying angles could be
used, for example, to describe the effects of defects or excitation inhomogeneities in such
e>..'Periments [50, 51, 52].
Essentially all properties of the non-equilibrium dynamics are contained in unequal
time correlation functions of the spin operators of and Q-j (these subsume, of course,
the time evolution of all equal-time correlation functions). We focus first on the case
o-f
where only operators occur
·a-;:
g = (Tc (Q -j;; (t~) ·. (ti)Q-t (tm) · · · cl~~ (tl)) J · (5)
Here a, b = ±, and the time dependence of the operators is given in the Heisenberg
pictme of 1l
(6)
The time-ordering operator Tc orders aU operators along a closed-time-path C shown
in Fig. 2, with times t occurring on the forward path and times t* occurring along the
backwards path. This closed-time path ordering occurs naturally, for instance, in any
perturbative treatment of additional non-commuting terms in the Hamiltonian. In Sec.
3 we encounter this situation when treating a dissipative coupling to an environment,
but the same. structure occurs for coherent couplings, e.g. a transverse field h l:i aj.
Our goal in what follows is to obtain analytic expressions for such correlation functions
in full generality, and then apply them to calculating a variety of experimentally relevant
quantities. In order to describe g concisely, it is useful to define a variable ai on each
site such that CXj = 1 if there. are no occurrences of the operators aj in g, and CXj = 0
a-;=±
otherwise. Now we recognize that if an operator occurs in g one or more times,
the operator Q-j (appearing in the time evolution operator) is forced to take on a well
defined value aj(t) at all points in time (see Fig. 2). As a result, we can rewrite the
correlation function g as
l
JC(t))) IT
g = \ exp ( - i dt [pi + a.iJ, (7)
J
CONTENTS 7
8-j(tl) 8-J; (t2)
:.:
) • t
8-j (t;)' I
'UJ(t)
ak(t)
It; It,
0 10* 0 10*
tj
tl t2 tl
- 1 - 1
Figure 2. Graphical representation of a sample correlation function g =
(Tc (aj(t3)&;(t2)af(ti))). Here aj = ak = 0, al"#j,k = 1, and the time-dependent
functions oJ(t) and ak(t) are shown in the bottom two panels.
where
Pi = (1 - ai)h (O"j(O)] fJ (O"j(O*)] , (8)
J
(t = 0* marking the end of the backwards trajectory, Fig. 2), is the complex conjugate
off, fc is a time integral that runs along the closed-time path, and
L
K(t) = ~ Jik [(1 - ai)(1 - a~c)O"j(t)O"~(t) + 2(1 - ai)a~cO"j(t)&~ + aja~c&j&~] . (9)
j,k
The first thing to notice is that, since fc dt = 0, the final time-independent term in K
vanishes. Since this is the only non separable term inK, the remaining time evolution is
straightforward to compute. It is helpful to define the following parameters that depend
on the functions O"j (t )
1
L
<p~c - lj~c(1 - ai)a~c dt O"j(t) (10)
j,k c
1
L
fJ - ~ Jj~c(1 - ai)(1 - a~c) dt o-j(t)O"~(t), (11)
j,k c
in terms of which
1 L
dt K(t) = fJ + <pk&~. (12)
c
k
The time-ordered correlation functions of Eq. (7) can now be compactly written
II II
g = e-it'J (1hl e-i'PiOJ l'!f;i) (Pi+ ai) (13)
j j
II
= e-it'J [Pj + gj(rpi)], (14)
j
with
(15)
(gj will be useful momentarily). The equivalence between Eqs. (13) and (14) can
be understood by explicitly comparing the expressions inside the product for the two
