Engineering correlation and entanglement dynamics in spin systems

0.16 MB·English

by  T. S. Cubitt

Engineering correlation and entanglement dynamics in spin systems
T. S. Cubitt1,2 and J.I. Cirac1
1Max Planck Institut fu¨r Quantenoptik, Hans–Kopfermann Str. 1, D-85748 Garching, Germany
2Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
(Dated: January 9, 2007)
We show that the correlation and entanglement dynamics of spin systems can be understood in
terms of propagation of spin waves. This gives a simple, physical explanation of the behaviour
seeninanumberofrecentworks,inwhich alocalised, low-energy excitation iscreatedandallowed
to evolve. But it also extends to the scenario of translationally invariant systems in states far
from equilibrium, which require less local control to prepare. Spin-wave evolution is completely
determinedbythesystem’sdispersion relation, andthelattertypicallydependsonasmallnumber
ofexternal,physicalparameters. Therefore,thisnewinsightintocorrelationdynamicsopensupthe
7 possibilitynotonlyofpredictingbutalsoofcontrolling thepropagationvelocityanddispersionrate,
0
bymanipulating theseparameters. Wedemonstrate this analytically in a simple, example system.
0
2
PACSnumbers: 03.67.-a,03.67.Mn

n
a
J Correlations play a predominant role in the study of tionally invariant system, by tuning only simple, global,

0 spin systems. On the one hand, they characterize dif- physical parameters. This may be relvant for the op-
1 ferent phases of matter, and thus can help reveal the timal creation of entanglement in spin systems, as well

mechanisms underlying phase transitions. On the other, as contributing to a better understanding of how cor-
1
they are directly related to the entanglement between relations are created in dynamical processes, something
v
differentspins,whichcanbe exploitedbyapplicationsin whichcanbetestedexperimentallyinpresentsetups. We
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5 the field of quantum information processing. Whereas will show that, even with this severely limited control
0 so far, much of the work on correlations has focused on overthe system, the correlationspeed can be engineered
1 the static properties of equilibrium systems, an increas- whilst simultaneously keeping dispersion to a minimum,
0
ing interest in the corresponding dynamical properties so that correlations can be concentrated between partic-
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has developed over the last few years. The reasonis two ular spins. Indeed, by manipulating system parameters
0
/ fold. Firstly, new experimental setups, such as atoms in during the evolution, the speed can be adjusted at will,
h
optical lattices, have reached an unprecedented level of even to the extent of reducing it to zero, allowing corre-
p
control,allowingphysicalparametersto bechangeddur- lations to be frozen at a desired location.
-
t ingtheexperiments. Thustheoreticaldescriptionsofthe
n It is instructive to first consider the entanglementand
a time-dependent properties of such systems have become correlationpropagationdescribedin the referencesgiven
u important. Secondly, it has been recognized that the
above from a new perspective. In many of those works,
q
way entanglement is created and how it propagates are
: correlation propagation can be understood as follows.
v important fundamental questions in quantum informa-
The spin system is initially preparedin its groundstate.
i tiontheory. Inparticular,the answersmayinfluence the
X Alocalised,low-energyexcitationisthencreated(e.g.by
designofquantumrepeatersandnetworks,whosegoalis
r flipping one spin), and allowed to evolve. Since the low-
a to establish as much entanglement as possible between
energy excitations take the form of spin waves, the cor-
different nodes in the shortest possible time.
relation and entanglement dynamics can be understood
The time evolution of correlation functions in spin asnothing other thanpropagationof spinwaves. This is
systems has been studied recently in various scenarios, completely determined by the dispersion relation (given
mainly from a condensed matter physics perspective. In by the system’s spectrum). The form of the dispersion
Refs. [1, 2], two-point correlations were studied numer- relation will typically depend on external, physical pa-
ically, whereas in Ref. [3] their evolution in a critical rameters of the system (e.g. the strength of an external
modelwasstudiedanalyticallyusingconformalfieldthe- magnetic field). Thus already in these setups, we can
oretic methods. In all cases, correlations were seen to manipulate the external parameters to control the dis-
propagateatafinite speed. InRef.[4],aproofwasgiven persion relation, and hence control the propagation of
that correlations necessarily propagate at a finite speed. correlations. For example, changing the gradient of the
On the other hand, information and entanglement prop- dispersion relation will change the propagationspeed.
agation in spin systems has mostly been studied from a
However, the ground state will typically be highly-
quantum information perspective [5, 6, 7, 8].
correlated and difficult to prepare, and with the level
In contrast with previous work, we will consider to of local control required to create the local excitation
what extent it is possible to control the propagation and break the translational symmetry, more sophisti-
speed and dispersion of the correlations in a transla- cated quantum-repeater setups are possible. Also, it is
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notclearthatthe correlationswillremainlocalised;they spins down. The interactions are then switched on and,
are likely to disperse rapidly as they propagate. as this initial state is not an eigenstate of the Hamilto-
Therefore,wewillextendtheideatosystemsprepared nian (unless λ → ∞), the state evolves in time. The
in translationally invariant, easily created, uncorrelated initial state is the vacuum of the Majorana operators
initial states. For example, the fully polarisedstate with x (p ) = σzσx(y) obtained after applying just the
l l j j l
all spins aligned can be prepared by applying a large, Jordan-Wigner transformation, and is completely deter-
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external magnetic field. As the initial state will be far mined by its two-point correlation functions. In other
from the ground state, it will contain many excitations. words, the vacuum is a fermionic Gaussian state, and
The correlation dynamics is then the result of the prop- can be represented by its covariance matrix Γ =
m,n
agationand interference of a large number of spin waves 1h[r ,r ]i where r =x and r =p .
2 m n 2l−1 l 2l l
at many different frequencies. Nonetheless, we will show From the Heisenberg evolution equations, it is sim-
analytically that, at least for some simple models, the ple to show that any evolution governed by a quadratic
system can be engineered so that correlations propagate Hamiltonian corresponds to an orthogonal transforma-
in well-defined, localised wave packets, with little dis- tion of the covariance matrix. It is also clear that, as
persion. The external parameters can then be used to the Fourier and Bogoliubov transformations are canon-
control the propagation of these correlationpackets. ical (anti-commutation-relation-preserving) transforma-
In the following, we will consider a specific model tions of the Majorana operators, they similarly leave
which, despite its simplicity, is sufficiently rich to dis- GaussianstatesGaussian,andtheytoocanbeexpressed
play most of the features we are interested in. The as orthogonal transformations. Thus the time-evolved
model is simple enough to envisage implementing it ex- state of the system is given by a series of orthogonal
perimentally, for instance using atoms in optical lattices transformations of the fermionic vacuum:
or trapped ions. The XY–model for a chain of spin–
12 particles is described by the Hamiltonian HXY = Γ(t)=OΓvacOT, O=OFTTOBTogO(t). (1)
−1 ((1 + γ)σxσ + (1 − γ)σyσy + 2λσz), where
4 l l l+1 l l+1 l Thisisablock-Toeplitzmatrix,composedof2×2blocks
the σ’s arethe usualPaulioperatorsandthe sumis over
P G atdistancexfromthemaindiagonal. Inthethermo-
spin indices. The parameter λ can be interpreted as the x
dynamiclimitN →∞with 2πk →φandε →ε(φ)≡ε,
strength of a global, external magnetic field, whereas γ N k
controls the anisotropy of the interactions. π g g
This Hamiltonian can be brought into diagonal form Gx = dφ g 0 g1 , g0 =iSsin(φx)sin 2εt
by the well-known procedure [9] of applying Jordan- Z−π (cid:18) −1 0(cid:19)
(cid:0) (cid:1)
Wigner, Fourier and Bogoliubov transformations, giv- g =2CSsin(φx)sin2 εt ±cos(φx) C2+S2cos 2εt
±1
ing H = −i ε (γxγp − γpγx) with spectrum
XY 4 k k k k k k (cid:0) (cid:1) (cid:16) (cid:0) (cid:1)(cid:17)
ε = ((cos(2πk/N) − λ)2 + γ2sin2(2πk/N))1/2. The where C = (cos(φ) −λ)/ε(φ), S = γsin(φ)/ε(φ), and
k P
γx,p are Majorana operators, related to the more usual x = m−n. We can now calculate certain string corre-
k
Jordan-Wigner fermionic annihilation operator γ by lations, which are given directly by elements of the co-
k
γx = γ† +γ and γp = (γ† −γ )/i, and obey canoni- variance matrix. For example, σx( σz)σy =
k k k k k k n n<i<m i m
cal anti-commutation relations γa,γb =2δ δ . 1Γ = s π dφScos(φx+2sεt)/2.
k l k,l a,b i 2n−1,2m−1 s=±1 −π (cid:10) Q (cid:11)
Ultimately, we are interested in “connected” spin– Althoughtheevolutionofthestringcorrelationsispro-
(cid:8) (cid:9) P R
spincorrelationfunctions,forexampletheZZcorrelation duced by the collective dynamics of a large number of
function C (n,m)=hσzσz i−hσzihσz i, in which the excitations, this expression has a simple, physical inter-
ZZ n m n m
“classical” part of the correlation is subtracted. These pretation: it is the equation for two wave packets with
are related to the localisable entanglement L(n,m) (the envelope S/2 propagating in opposite directions along
maximum average entanglement between two spins n thechain,accordingtoadispersionrelationgivenbythe
and m that can be extracted by local measurements system’sspectrumε(φ). Thiswave-packetinterpretation
on all the others [10]): the natural figure of merit for allows us to make quantitative predictions as to how the
quantum repeaters. In particular, for spin–1 systems, dynamicswillbeaffectedifthesystemparametersγ and
2
L(n,m) ≥ C(n,m) for any connected spin–spin correla- λ are modified. Specifically, modifying the parameters
tion function C [11]. However,we will start by consider- will change the dispersion relation, changing the group
ing the simpler, albeit less well-motivated, string corre- velocity of the correlation packets, as well as the rate at
lation functions such as S (i,j) = hσx( σz)σxi whichtheydisperse. (Thewave-packetenvelopesalsode-
XX i i<k<j k j
(important for revealing “hidden order” in certain mod- pendonthe systemparameters,sothe relevantregionof
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els [12]). Their behaviour will give insight into the more the dispersion relation may also change.) Thus by vary-
importantspin–spincorrelations,andwe willuse similar ing only global physical parameters, we can control the
techniques to calculate both. speed at which correlations propagate.
Assume the spin chain is initially in some completely Does this hold true for the more interesting spin–spin
separable, uncorrelated state, such as the state with all correlations? We will show analytically that they have
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a similar wave-packet description, although in terms of
multiple packets propagating simultaneously. This will
allow us to predict the behaviour of the spin–spin corre-
lationdynamicsfordifferentvaluesofthesystemparam-
eters. In particular, we will show that the correlations
can be made to propagate in well-defined packets whose
speed can be engineered by tuning the system parame-
ters. Moreover,the propagationspeed can be controlled
as the system is evolving, so we can speed up or slow
FIG. 1: For γ = 1.1, λ = 2, all the wave-packet envelopes
down the packets, even to the extent of reducing the
from Eq. (2) (non-red curves, inset) are similar in form, cen-
speed to zero. We confirm our predictions by numeri-
tred around a nearly linear region of the dispersion relation
cally evaluating the analytic expressions. with gradient ≈ 2 (red curve, inset). Thus the correlations
Let us now calculate the spin–spin connected correla- Czz(x,t) (indicated by the shading, main plot) propagate in
tion function using the covariance matrix derived above. well-defined packetsat a speed given bythegradient.
We have σz = x p , so the ZZ connected correla-
n n n
tion function is given by C (x) = hx p ihp x i −
ZZ n m n m
hx x ihp p i, where we have used Wick’s theorem to though at the expense of increased dispersion. The nu-
n m n m
expand the expectation value of the product of four Ma- merical results of Fig. 2 show precisely this behaviour.
joranaoperatorsintoasumofexpectationvaluesofpairs
[13, 14]. The latter are given by covariance matrix ele-
ments, resulting in the following analytic expression for
the correlation function:
C (x,t)2 = (2)
zz
π S 2
dφ scos(φx−2sεt)
2
(cid:18)Z−π s=±1 (cid:19)
X(cid:0) (cid:1)
π 1 2
+ dφCS sin(φx)− sin(φx+2sεt) FIG. 2: For γ = 10, λ = 0.9, the wave-packet envelopes
2
(cid:18)Z−π s=±1 (cid:19) are spread over the entire frequency range (non-red curves,
(cid:0) X (cid:1) inset). However, the dispersion relation (red curve, inset) is
π S2 2
− dφ C2cos(φx)+ cos(φx+2sεt) . almostlinearforwavenumbersnottoonearπ,withgradients
2 ±18. As the envelopes are symmetric about π, most of the
(cid:18)Z−π s=±1 (cid:19)
(cid:0) X (cid:1) correlations Czz(x,t) (shading, main plot) still propagate at
a well-defined speed ≈ 18, faster than in Fig. 1 (note scale),
Although more complicated than the string correla-
though as expected they also show more dispersion.
tions, this expression also describes wave packets evolv-
ing according to the same dispersion relation ε(φ), al-
beit multiple packets with different envelopes propagat- An even more interesting possibility is controlling
ing and interfering simultaneously (three in each direc- the correlation packets as they propagate. If the sys-
tion). In many parameter regimes, broad (in frequency- tem parameters are changed continuously in time, the
space) wave packets and a highly non-linear dispersion XY-Hamiltonian becomes time-dependent, and the or-
relation will cause the correlations to rapidly disperse thogonal evolution operator O(t) in Eq. (1) is given
and disappear. However, we can find regimes in which by a time-ordered exponential O(t) = T[eR0tdt′A(t′)] ≡
the wave packets are located in nearly linear regions of lim ⌊t/h⌋eA(nh). (A is a time-dependent, anti-
h→0 n=1
the dispersionrelation,andmaintaintheir shapeasthey symmetric matrix determined by the Hamiltonian.) In
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propagate. For example, at γ =1.1 and λ= 2, all three general,the time-ordering is essential. But if the system
wave packets of Eq. (2) are nearly identical, and reside parameters change slowly in time, dropping it will give
in an almost-linearregionof the dispersion relationwith a good approximation to the evolution operator. The
gradient roughly equal to 2, as shown in Fig. 1 (inset). state at time t is then just given by evolution under the
Thespin–spincorrelationdynamicswillthereforeinvolve time-average (up to t) of the Hamiltonian. If we remain
well-defined correlation packets propagating at a speed in a parameter regime for which the relevant region of
dx/dt = 2, dispersing only slowly as they propagate. the dispersion relation is nearly linear, adjusting the pa-
Fig.1showsthe resultofnumericallyevaluatingEq.(2), rameters changes the gradient without significantly af-
which clearly confirms the predictions. fecting its curvature or the form of the wave packets.
Wecanengineeradifferentcorrelationspeedbychang- Thus, to good approximation, slowly adjusting the pa-
ing the parameters. For instance,forγ =10andλ=0.9 rametersshouldcontrolthe speedofthe wavepacketsas
we predict a higher propagation speed dx/dt ≈ 18, al- they propagate, allowing us to speed them up and slow
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themdown. Numericallyevaluatingthetime-orderedex- systembyabruptlychangingtheparameters,freezingthe
ponential shows this is indeed possible (Fig. 3). correlations at that location, as shown in Fig. 4.
FIG. 3: Starting from γ = 1.1, λ = 2 as in Fig. 1, γ and FIG.4: Thesystemisinitiallyallowedtoevolvewithγ0 =0.9
λ are smoothly changed to move from the red to the green λ0 = 0.5, then “quenched” at time t1 = 20 to γ1 = 0.1,
dispersion relation (inset), increasing the correlation speed. λ1 =10. Someofthecorrelationsarefrozenattheseparation
(x ≈ 20) they reached at t1. Others propagate according to
thenew dispersion relation, or are “reflected”.
Clearlyitwouldbeusefultobeabletostopthecorrela-
tions once they reacha desiredlocation. One way would
We have shown that entanglement and correlation
be to simply switch off the interactions. But strictly
propagation in many spin-model setups can be under-
speakingthiswouldrequiremorecontrolthanisprovided
stood in terms of propagation of spin waves, and have
bythe twoparametersdefinedinthe Hamiltonian(there
introducedthe idea of controllingthe dynamics via their
is no value of γ for which all interaction terms vanish),
dispersionrelation,bymanipulatingexternalparameters
and may be difficult in physical implementations. If the
of the system. Although this is in principle possible
spin model were realised in a solid-state system, for ex-
for almostany spinsystem, preparinga single-excitation
ample, switchingoff the interactionswouldlikely involve
initial state would require control over individual spins.
fabricating an entirely new system. In any case, we will
Therefore, we have analysed in detail the more complex
show that switching off the interactions is not necessary
case of systems preparedin uncorrelated,translationally
in order to freeze correlations at a specific location.
invariantinitial states, which typically contain many ex-
Instead of changing the parameters continuously, we
citations. We haveshownfor anexample modelthatthe
now consider changing them abruptly. The time-evolved
dynamics can be described by a small number of corre-
covariance matrix in this scenario can be calculated an-
lation wave packets, and that the control afforded by a
alytically by the same methods as used above. Suppose
few external, physical parameters is sufficient to allow
the initial system parameters γ and λ are suddenly
0 0
detailed control over the propagation of correlations.
changed to γ and λ at time t . The spin–spin correla-
1 1 1
tions will initially evolve according to Eq. (2), as before.
After time t , the evolution becomes more complicated.
1
The analogue of Eq. (2) separates into a sum of wave
packets evolving in four different ways: those that ini- [1] F.Igl´oiandH.Rieger,Phys.Rev.Lett.85,3233(2000).
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1
according to −ε , those that only start evolving at t , (2006).
1 1
and those that that undergo no further evolution after [4] S. Bravyi, M. B. Hastings, and F. Verstraete, quant-
ph/0603121 (2006).
t (Fig. 4). For t > t , the terms whose evolution is
1 1 [5] T. J. Osborne and N. Linden, Phys.Rev. A. 69, 052315
“frozen” at time t are given by
1 (2004).
[6] S. Bose, Phys. Rev.Lett. 91, 207901 (2003).
Ct1(x,t)2 =
z [7] M. Christandl, N. Datta, A. Ekert, and A. J. Landahl,
1 π 2 Phys. Rev.Lett. 92, 187902 (2004).
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0 1 0 1 1 0 1 0
2Z−π s=±1 ! A. 71, 052308 (2005).
X [9] S. Sachdev,Quantum Phase Transitions (CUP,2001).
2
1 π [10] F.Verstraete,M.Popp,andJ.I.Cirac,Phys.Rev.Lett.
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X
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