Quantum Quasi-Zeno Dynamics: Transitions mediated by frequent projective measurements near the Zeno regime

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by  Thomas J. Elliott

Quantum Quasi-Zeno Dynamics: Transitions mediated by frequent projective
measurements near the Zeno regime
T. J. Elliott1,∗ and V. Vedral1,2,3,4
1Department of Physics, Clarendon Laboratory, University of Oxford,
Parks Road, Oxford OX1 3PU, United Kingdom
2Centre for Quantum Technologies, National University of Singapore, Singapore 117543
3Department of Physics, National University of Singapore, 2 Science Drive 3, 117551 Singapore
4Center for Quantum Information, Institute for Interdisciplinary
Information Sciences, Tsinghua University, 100084 Beijing, China
(Dated: July 28, 2016)
Frequent observation of a quantum system leads to quantum Zeno physics, where the system
evolution is constrained to states commensurate with the measurement outcome. We show that,
more generally, the system can evolve between such states through higher-order virtual processes
6
that pass through states outside the measurement subspace. We derive effective Hamiltonians to
1
0 describe this evolution, and the dependence on the time between measurements. We demonstrate
2 application of this phenomena to prototypical quantum many-body system examples, spin chains
and atoms in optical lattices, where it facilitates correlated dynamical effects.
l
u
J
I. INTRODUCTION In this article we demonstrate how these higher-order
7
processes, which we call quantum quasi-Zeno dynamics
2
(QqZD), arise from perturbative considerations of stan-
Reminiscent of the arrow paradox put forth by Zeno
] dardQZD.WefindeffectiveHamiltonianstodescribethe
h of Elea, concerning the apparent discrepancy in the mo-
evolution of the system, and suggest interpreting such
p tion of objects when they can at any and all instants
processes as virtual transitions, providing a simple il-
-
be observed to be stationary, the quantum Zeno effect
t lustrative example using a three-level system. We ex-
n (QZE) [1–3] argues that the act of continuously observ-
tendtheformalismtoencompasstime-dependentHamil-
a
ing a quantum state leads to a zero probability of evolv-
u tonians,non-equally-andstochastically-spacedmeasure-
ing away from the state, thus freezing the system evo-
q ments, and discuss how transitions to different measure-
[ lution. This effect has been extended to encompass the mentsubspacesmaybeincorporatedintothetreatment.
case of degenerate measurement subspaces, where multi- We then apply this formalism to exhibit how this phe-
2
plestatesofthesystempossessidenticaloutcomesofthe
v nomenon may manifest in two archetypal examples of
measuredobservable,suchthatevolutionwithinthissub-
4 many-body systems, spin chains and atoms in optical
2 space is unhindered by the measurement, a phenomenon lattices, where we show that the higher-order processes
6 called quantum Zeno dynamics (QZD) [4–6]. The QZE correspond to correlated dynamics in the system.
6 and QZD have been observed in a range of experimental
0 setups, including ions [7], photons [8], nuclear magnetic
.
1 resonance spins [9], atoms in microwave cavities [10], II. FUNDAMENTALS OF QUANTUM
0 Bose-Einstein condensates [11, 12], and Rydberg atoms QUASI-ZENO DYNAMICS
6 [13]. Therehasalsobeenmuchtheoreticalinterestinthe
1 field,particularlybecauseoftheopportunitiesofferedby
Consider a system evolving under Hamiltonian H.
:
v measurement-based control of a system [14–29]. Between measurements, the evolution of the quantum
i
X It has been shown that even when consecutive mea- state ρ after time t is described by the unitary opera-
surements are finitely spaced, the locking to a measure- tor U(t) = exp(−iHt), through ρ → UρU† [33] (we use
r
a mentsubspacecanstilloccur[30]. However,inthiscase, natural units (cid:126)=1). This system is subject to measure-
the description of the system evolution solely in terms ment from an external source, and we model the effect
(cid:80)
of this subspace is incomplete [31]; the finite time be- of a measurement of the observable A = jAjPj with
tweenmeasurementsallowshigher-orderprocessestooc- outcomeAk tomodifythestateaccordingtoρ→PkρPk,
cur, where the system first transitions away from the where Pk is the projector for the subspace containing all
measurement subspace, and then subsequently back in states with measurement eigenvalue Ak (see Appendix
to it before the next measurement, thus preserving the A).
value of the measured observable. Similar effects have In this formalism, we can describe the evolution of a
been predicted for continuous measurement in the quan- system subject to frequent measurement. For two con-
tum jump formalism [28, 32]. secutivemeasurementsatimeδtapart,astateρinitially
in eigenspace P of the measurement operator evolves
ρ → P(cid:48)U(δt)ρU†(δt)P(cid:48), where P(cid:48) is the subspace of
the measurement outcome. In the limit where Hδt is
∗Electronicaddress: [email protected] small,wecanexpandtheexponentialU(δt)≈1−iHδt−
2
H2δt2/2+O(δt3), to calculate the probability that the on timescales independent of δt. Because the QqZD cor-
measurementoutcomechanges. Theprobabilitythatthe rection is of a similar magnitude to the transition prob-
measurementresultsinavaluecorrespondingtosubspace ability, the quasi-Zeno deviation can become very non-
Q=(cid:54) P ishencegivenbyP(Q)=Tr(HρHQ)δt2+O(δt3). negligible especially when one considers long experimen-
Summing this over all measurement subspaces different tal runs, such as those where the measurement subspace
to P, we have the condition that for the probability of a changes during the trajectory (see Section IV).
change in the measurement value to occur to be negligi- In the standard QZD regime, the effective Hamilto-
ble, we require (cid:80) P(Q) (cid:28) 1, i.e. for any state ρ in P, nian is simply the Zeno Hamiltonian H(1), which, being
(cid:80) Tr(QHρH)δt2Q(cid:28) 1. This forms our ‘Zeno-locking’ Hermitian,leadstounitarydynamics. CZontrastingly,the
Q
requirement on timescales for the periods between mea- quasi-ZenoHamiltoniansarealternativelyHermitianand
surements. From here, we assume this condition is met. anti-Hermitian, and thus due to the second-order quasi-
After N (cid:29) 1 such measurements in a time τ = Nδt, ZenoHamiltonianH(2) beingnon-vanishingforanynon-
Z
each resulting in the same measurement value, with sub- trivial Hamiltonian and measurement operator when δt
space P, the system evolution can be approximated by is finite, in the quasi-Zeno regime the dynamics is non-
the effective evolution operator Ueff(τ) = exp(−iHeffτ) unitary. Instead, the dynamics of the system will tend
(see Appendix A for details) , with the corresponding towards the eigenstate(s) of H(2) with lowest eigenvalue
effective Hamiltonian Z
that can be accessed by the dynamics from the (quasi-
)Zeno Hamiltonians, and the decay in the norm of the
H =(cid:88)∞ (−iδt)k−1H(k), (1) state corresponds to the survival probability of remain-
eff k! Z ing in the measurement subspace.
k=1
Fromtheabove,wehavethattheprobabilityofatran-
where we define the quasi-Zeno Hamiltonians H(k) = sition out of the measurement subspace between times t
PH((I − P)H)k−1P. In the limit that δt →Z 0, and t+δt is given by
i.e. the standard QZD scenario, this evolution becomes P(P¯,t)=Tr(Hρ(t)H(I−P))δt2+O(δt3)
exp(−iH(1)τ),withH(1)beingthestandardZenoHamil-
Z Z ≈Tr(H(2)ρ(t))δt2.
tonian [6], recovering the QZD result. However, when δt Z
is small-but-finite, we have the more general QqZD sce-
Thus, the total survival probability of remaining in the
nario, where the quasi-Zeno Hamiltonians H(k) mediate initial measurement subspace after N measurements is
Z
kth-ordertransitionswheretheinitialandfinalstatesare givenby(cid:81)N (1−P(P¯,nδt). Forlongtimes,thesurvival
n=1
in the measurement subspace P, but intermediate states probabilitywilltendtozero,unlessthereisastatespace
are not. Because of the dependence of each quasi-Zeno which satisfies Tr(H(2)ρ) = 0. The second-order quasi-
Hamiltonian’s contribution to the effective Hamiltonian Z
Zeno evolution causes the system to tend towards this
on increasing powers of δt, each one is less significant
state, effecting a ‘natural selection’ of states, removing
than that of the previous order, and the intimate depen-
those for which the survival probability is lower, tending
dence of QqZD on the measurement timestep is evident.
towards a steady state space. More generally, the evo-
Indeed,astheprobabilityofameasurementoutcomebe-
lution tends towards effective steady states which min-
longing to a different subspace scales as δt2, in practice
imise the rate of higher-order processes the system un-
itislikelythatthesecond-orderquasi-ZenoHamiltonian
dergoes, and hence those with the largest survival prob-
HZ(2), also scaling as δt2 forms the only significant de- ability. Such effective steady states are fragile, as they
viation from standard QZD, with the higher-order H(k) have a non-zero transition probability, and so for longer
Z
forming corrections on top of this. times will eventually transition out of the measurement
Heuristically, wecanseethatsinceboththetransition subspace.
probabilityandthesecond-orderquasi-ZenoHamiltonian
have similar magnitude ∼ Tr(PHQHPρ)δt2, the QqZD
correction within a given subspace is of the same order III. INTERPRETATION AND EXAMPLE
astheprobabilitytotransitionoutofthesubspaceThus,
provided we are in the regime for which QqZD is valid Physically, the QqZD second-order terms involve a
(i.e. this quantity is much less than unity), the relative small-but-finite occupation of an intermediate state be-
magnitude of the correction to the state at the time at tween measurements, which is then removed by the pro-
whichatransitionultimatelyoccursisapproximatelyin- jection of the subsequent measurement, provided the
dependentofδt. Thesizeofδtisstillrelevanthowever,as locking of the measurement eigenvalue is maintained.
it governs the accuracy of the approximate effective evo- While this intermediate state is occupied, it can tran-
lutionoperator(moreaccurateforsmallerδt),thesizeof sition to other states as usual for the non-measurement
thecorrectionduetothehigher-orderquasi-ZenoHamil- case. These transitions can either be to states also out-
tonians(decreasingwithδt),andthetotaltimescalesover sideofthemeasurementsubspace(inwhichcaseoccupa-
whichtheQqZDevolutiontakesplace(longerforsmaller tion of these states is also removed by the next measure-
δt). Note that the standard Zeno dynamics takes place ment), or back in to the measurement subspace, but not
3
necessarily into the original state. Higher-order terms
involve transitions with additional intermediate states.
When the time between measurements decreases, the
maximumoccupationof the intermediatestateswill also
be decreased, and hence the rate of transitions out of
thesestateswillbelessened. Thisisreflectedintheform
of the effective Hamiltonians and their dependence on
δt. In the infinitely frequent measurement limit of QZD,
there is no occupation of the intermediate states, and
thus there are no transitions beyond first-order.
Because of the locking to the measurement subspace,
occupationoftheintermediatestatesisneverdirectlyob-
served. However, through the occurrence of transitions
that take place via these states, their temporary occu-
pation may be indirectly inferred. As a result of this,
and because the dynamics can be described by effective
Hamiltonians acting only on the measurement subspace,
allowing a description of the intermediate states to be
omitted, these states outside of the measurement sub-
space can be viewed as virtual states, and consequently,
FIG. 1: Transitions via virtual processes: The quasi-
that the transitions between states in the measurement
Zeno Hamiltonians give rise to transitions that occur via
subspace that pass through these virtual states can be
states outside the measurement subspace, which can be
seen as virtual processes. Similar transitions via such viewed as virtual processes. These can be compared concep-
virtual states are also present in the continuous, non- tually with Feynman diagrams, where instead of interactions
projective measurement case [28, 32], where they are occurring via virtual particles, we instead represent transi-
compared with Raman-like processes. tionsoccurringviaoccupationofvirtualstates. Singlevertex
We draw visual analogy with Feynman diagrams [34] processes(a)aremediatedbythestandardZenoHamiltonian
for these virtual processes (see Fig. 1). Feynman dia- H(1), while higher-order processes (b) with n vertices and
Z
grams, used as pictorial representations of interactions n−1 virtual states are mediated by the quasi-Zeno Hamil-
in high-energy physics, depict interactions as being me- tonian HZ(n). Transitions to a virtual state and back to the
diated by virtual particles. We can construct a similar initial state can be represented by loop diagrams (c), akin to
therepresentationofself-interactionwithFeynmandiagrams.
picture for the virtual processes of QqZD, where the in-
coming and outgoing lines are the initial and final states
inthemeasurementsubspace,theverticesarethetransi-
transverse field of strength λ, such that it has Hamilto-
tionsbetweenstates,andtheinternallinescorrespondto √
nian H = λSX = (λ/ 2)(|−1(cid:105)(cid:104)0|+|0(cid:105)(cid:104)1|+h.c.), and
occupation of the intermediate states. In this represen-
frequent measurement is made of the magnitude of its
tation,thenumberofverticescorrespondstothenumber
spin value (A=|SZ|). The corresponding projectors for
oftransitions,andhencetheorderofthequasi-Zenopro-
the measurement subspaces are P = |0(cid:105)(cid:104)0| for A = 0,
cess;transitionsdescribedbytheZenoHamiltonianHZ(1) and P1 =I−P0 =|−1(cid:105)(cid:104)−1|+|1(cid:105)(cid:104)01| for A=1.
have one vertex and no virtual states, as they do not re-
The Hamiltonian contains no direct transitions be-
quireoccupationoftheintermediatestates,whilsttransi-
tweenthe|−1(cid:105)and|1(cid:105)states, insteadrequiringthestate
tionsfromthesecond-orderquasi-ZenoHamiltonianHZ(2) to first go via the |0(cid:105) state. Thus, in such a setup in the
are represented by two vertices and one virtual state. standard QZD scenario, the Zeno Hamiltonian vanishes
Second-order processes that return back to the same ini- for both measurement subspaces; H(1) = 0. However,
tial state can be considered akin to self-interacting pro- Z
when the measurements are finitely frequently spaced,
cesses, giving rise to self-energy type contributions to
as in the QqZD regime presented here, the second-order
the Hamiltonian. We note however, that this analogy
quasi-Zeno Hamiltonian for the A = 1 subspace is non-
is intended as a graphical aid to interpret the transitions
zero; applying the appropriate projectors to obtain the
within the QqZD framework, and we are not proposing
second-order quasi-Zeno Hamiltonian, we find for the
thatthemathematicsoftheprocessesdescribedbyFeyn-
{|−1(cid:105),|1(cid:105)} subspace that it takes the form
man diagrams be directly mapped onto QqZD.
To further illustrate these virtual processes, we use λ2
a very basic toy model consisting of the simplest sys- H(2) = (|−1(cid:105)(cid:104)−1|+|1(cid:105)(cid:104)1|+|1(cid:105)(cid:104)−1|+|−1(cid:105)(cid:104)1|), (2)
Z 2
temthatcanexhibitnon-trivialQqZD:athree-statesys-
tem where two states possess a degenerate measurement where the first two terms are ‘self-energy’ type contribu-
eigenvalue. Consider such a system, say a spin-1 parti- tions from the system transitioning to |0(cid:105) and back in to
cle, with states {|−1(cid:105),|0(cid:105),|1(cid:105)}, where the label signifies theinitialstate,whilethelattertermsgiverisetotransi-
the SZ value of the state. The particle is subject to a tions between the |SZ| = 1 states, again by sequentially
4
undergoing two transitions to and from state |0(cid:105). This evolution,andimposingappropriatetime-ordering. This
intermediate state |0(cid:105) is never observed to be occupied, generalisationalsoallowsforsystemswherethemeasure-
and thus the transitions appear as a virtual processes. ment timestep depends on a stochastic process (for ex-
Thissimpleexamplecanbestraightforwardlysolvedto ample, the decay of a particle) to be described. If the
find the steady state to which QqZD drives the system. varianceinthetimestepsforsuchaprocessissufficiently
The eigenvalues of Eq. (2) are 0 and λ2, with associated narrow, an ‘average’ trajectory could be considered by
√
eigenstates |−(cid:105) = (|−1(cid:105) − |1(cid:105))/ 2 and |+(cid:105) = (|−1(cid:105) + calculating the moments (cid:104)δtn(cid:105), with an average number
√
|1(cid:105))/ 2 respectively. Thus, according to standard QZD, of measurements (cid:104)N(cid:105)=τ/(cid:104)δt(cid:105).
√
a system initialised in the state |−1(cid:105) = (|+(cid:105)+|−(cid:105))/ 2 Thus far, we have taken the measurements to occur
subject to such a measurement will remain in this state, sufficientlyclosetogetherthatthemeasurementoutcome
whileincontrast,QqZDpredictsthesystemevolutionto can be assumed to be constant. However, it is possi-
√
be (exp(−λ2tδt/2)|+(cid:105)+|−(cid:105))/ 2, the decay in the norm bletorelaxthiscondition,stillwithmeasurementoccur-
representing the probability to remain in the measure- ringmuchmorefrequentlythanchangestothemeasured
mentsubspace. Thus,accordingtoQqZD,thelong-term value,anddescribethesystemevolutionbyastraightfor-
evolution of the system is towards the state |−(cid:105), pro- wardextensiontotheQqZDformalism. Betweenchanges
vided the system remains in the same measurement sub- inthemeasurementvalue,thesystemisdescribedbythe
space, which occurs with probability 1/2. Interestingly, appropriate QqZD effective evolution operator for this
thisfinalstateisadarkstateoftheoriginalHamiltonian subspace. When the measurement eigenvalue changes,
(H|−(cid:105)=0), and hence once this steady state is reached, fromavaluecorrespondingtosubspacewithprojectorP
the system will remain in it even if the measurement is to that of subspace with projector P(cid:48), the change in the
no longer performed. stateisρ→P(cid:48)HPρPHP(cid:48),fromtheleadingtermO(δt2)
As three-level systems are routinely realised in a vari- allowing transitions out of the measurement subspace.
etyofexperimentalsetups,thisexamplemayalsoprovide After the measurement, the system is again described
ausefulschematicforaninitialexperimentaldemonstra- by a QqZD effective evolution operator, but now that
tion of QqZD. corresponding to the new subspace. In an experimental
run, one can simply determine when this change in sub-
space occurs by observing when the measurement value
IV. FURTHER GENERALISATIONS changes.
In theabove, wetook thetime between measurements
V. APPLICATION TO MANY-BODY SYSTEMS
to be equal for simplicity. Generalisation to non-equal
timestepsbetweenmeasurementsisstraightforward. The
formofthequasi-ZenoHamiltoniansareunchanged, but Many-body systems often possess interesting proper-
the dependence on δt now leads to differing strengths of ties that are described by observables dependent on the
theH betweeneachmeasurementintheeffectiveHamil- collective state of multiple particles. Different configura-
Z
tonian. Wecanmodifytheeffectiveevolutiontoaccount tions of particles can still result in the same system-wide
for this by including a product over effective evolutions measurement value for this observable; such configura-
for all the different measurement timesteps. Taking δt tions hence correspond to the same measurement sub-
j
as the time between measurements j − 1 and j, with space. This makes many-body systems a suitable arena
(cid:80)N δt =τ, this can be written for QqZD, and we shall here provide examples of many-
j=1 j
body systems, along with associated observables formed
N of linear functions of the occupation numbers of the sys-
Ueff(τ)= (cid:89)e(cid:80)∞k=1 (−iδkt!j)kHZ(k). tem modes, showing how this can lead to correlated dy-
namics.
j=1
Considering only the terms up to O(δt) in the to-
tal evolution, we can neglect those arising from the A. Spin Chains
non-commutativity of effective Hamiltonians for differ-
ing timesteps, as they occur at higher-order, suppressed
For the first example, we consider an array of spins
by a factor δt −δt , and hence approximate U (τ) =
j j(cid:48) eff in a chain [35], where each pair of neighbouring spins is
exp(−iH(1)τ −(cid:80)N H(2)δt2/2). coupled by an exchange interaction S+S− , such that
Z j=1 Z j i i+1
With the evolution written explicitly in terms of each the full system Hamiltonian is
measurementtimestep,wecanalsoclearlyseehowtime-
dependent Hamiltonians may be incorporated into the H =−J(cid:88)S+S−
i j
formalism, at least for cases where they can be treated
(cid:104)ij(cid:105)
asbeingapproximatelypiecewiseconstantbetweenmea-
surements, by generalising the quasi-Zeno Hamiltonians where J is the coupling strength of the interaction and
H(k)(t)=PH(t)((I−PH(t))k−1P in the above effective (cid:104)ij(cid:105) indicates that i and j are spins on neighbouring
Z
5
a
i a
ii
(cid:43)
ii(cid:76)i
(cid:76)
b (cid:76)
i
b
(cid:76)
(cid:76)
ii
ii(cid:76)i (cid:43) (cid:45)
(cid:76)
(cid:76)
FIG. 2: Correlated processes in spin chains: Multiple
config(cid:76)urations of spins (a) belong to the same measurement FIG(cid:76). 3: Correlated atomic processes: (a) Frequent, con-
subspace. Processes that ultimately preserve the measure- sistent measurement of the occupation of a set of sites facil-
ment can take place, at (bi) first- or (bii,iii) higher-order. itates correlated tunnelling of atoms over the boundaries of
Here,themeasurementissignifiedbythetotalmagnetisation the measured region, while (b) measuring occupation num-
of the green regions. berdifferencesgivesrisetopairprocess-likeeffects. Coloured
boxesindicatethemeasuredregions(greenpositive,blueneg-
ative), and arrows the same colour indicate correlated tun-
sites. Further terms that can be added to this Hamil- nelling events.
tonian which we do not consider here in our examples
arebiasesduetoexternalfields/anisotropiesλSX,Y,Z and
spin-spin interactions along the Z axis J˜SZSZ. We take
i j separation of the two pairs, and hence these processes
the total magnetisation (the sum of SZ values) of a set
can be correlated over long distances; this then resem-
of sites as our measurement, such that the subspaces are
bles a superexchange interaction [36, 37], but with the
definedbystateswiththesamemagnetisationinthisre-
potential for longer separation between the pairs.
gion [Fig. 2(a)]. Dynamics changing this magnetisation
areforbiddenbytheZeno-locking,andthusthestandard
Zeno Hamiltonian contains only spin-exchange between
neighbouring spins with either both, or neither, in the
measured regions. That is, we can write the standard
B. Atoms in Optical Lattices
Zeno Hamiltonian as
H(1) =−J( (cid:88) S+S−+ (cid:88) S+S−), (3)
Z i j i j Analogousprocessescanbeconsideredforatomsinan
(cid:104)i∈A,j∈A(cid:105) (cid:104)i∈B,j∈B(cid:105) optical lattice. In state-of-the-art setups, lattices con-
tainingbosonicatomshavebeengeneratedinsideoptical
whereAisthesetofsitesinthemeasuredregion(s),and
cavities [38, 39], and these setups allow for linear func-
B the unmeasured sites.
tionsoftheatomicoccupationofeachsitetobemeasured
However, the higher-order quasi-Zeno Hamiltonians
through the leakage of light from the cavity after having
mediate correlated spin-exchange events, where multiple
been scattered by the atoms [27, 40]. In the absence of
pairs of spins flip approximately simultaneously between
measurement, and with negligible cavity backaction, the
measurements, conserving the total magnetisation mea-
atoms behave according to the Bose-Hubbard Hamilto-
sured. The second-order quasi-Zeno Hamiltonian can be
nian [41] H = −J(cid:80) b†b +U(cid:80) b†b†b b , where b is
written (cid:104)ij(cid:105) i j i i i i i i
the bosonic annihilation operator for an atom localised
H(2) =J2 (cid:88) (S+S−S−S++S−S+S+S−). (4) at site i, J parameterises the rate of atomic hopping be-
Z i j k l i j k l
tweenneighbouringsites,andU isthestrengthofon-site
(cid:104)i∈A,j∈B(cid:105)
(cid:104)k∈A,l∈B(cid:105) interactions between atoms.
These processes mediate two such correlated exchanges, Measurement of functions of atomic occupation num-
involving only pairs that straddle the measurement re- bers of lattice sites controls the allowed tunnelling pro-
gion boundaries, and can be of two forms: in the first, cesses b†b , forbidding those that change the measure-
i j
bothexchangeshappenbetweenthesamepair(i.e.i=k mentvalue,andcorrelatingsetsoftunnellingeventsthat
and j = l), but in opposite directions, thus leaving the together preserve it (analogous effects occur for continu-
individual spins unchanged; in the second, the two pairs ousmeasurementwithquantumjumps[28]). InFig.3(a)
aredistinct, withoneexchangeincreasingthetotalmag- we illustrate how measuring the total occupation of a
netisation of the measurement region, while the other central region mediates long-range correlated tunnelling
decreases it. These processes are illustrated in Fig. 2(b). events across the boundaries of the region. The Zeno
In the latter case, there is no restriction on the spatial Hamiltonian is given by (again with A indicating the
6
measured sites, and B the unmeasured) to an effective chemical potential and nearest-neighbour
density-density interaction for these boundary sites.
H(1) =−J( (cid:88) b†b + (cid:88) b†b )+U(cid:88)b†b†b b ,
Z i j i j i i i i
(cid:104)i∈A,j∈A(cid:105) (cid:104)i∈B,j∈B(cid:105) i
(5) We use a small-scale simulation to demonstrate these
thus allowing tunnelling between pairs of sites where ei- effects,usingtheschemeofEqs.(5)and(6)andFig.3(a).
therbothorneitherareinthemeasuredregion,andleav- We simulate this setup, with 2 atoms distributed across
ingtheon-siteinteractionsunaffected. Thesecond-order 4lattice sites, withno interparticleinteractions (U =0),
quasi-Zeno Hamiltonian is given by andthetotaloccupationofthecentraltwositesmeasured
at timesteps Jδt = 10−2 (well within the Zeno-locking
HZ(2) =J2 (cid:88) (b†ibjb†kbl+b†jbib†lbk). (6) regime),fixingtheiroccupationat1atoms: N2+N3 =1
(cid:104)i∈A,j∈B(cid:105) (see Appendix B for further details). We calculate the
(cid:104)k∈B,l∈A(cid:105) evolution of the system for the QqZD effective Hamilto-
nian, the exact evolution, and the standard QZD evolu-
This mediates correlated pairs of tunnelling events be-
tion on a trajectory where the measurement outcome is
tween site pairs that straddle the boundaries of the
unchanging [Fig. 4]. We see that there is a very close
two regions A and B, preserving the total occupation
agreement between the effective (a) and exact evolu-
of sites within region A. When the two site pairs are
tion (c), as shown by their difference (d). They exhibit
identical (that is, i = l and j = k), the associated
the tunnelling between the central sites (as per standard
terms can be re-expressed as effective chemical potential
QZD), as well as the transfer of atoms mediated by the
and nearest-neighbour density-density interaction terms
quasi-Zeno dynamics between the two outer sites due to
J2(2n n +n +n ). Inthenon-interactinglimit(U =0),
i j i j correlated tunnelling, and the convergence to a (set of)
this can be mapped onto to the spin chain scenario
steady state(s). In contrast, the standard QZD evolu-
Eqs. (3) and (4) considered above, through a Holstein-
tion (e) completely fails to capture the correlated tun-
Primakoff transformation [35].
nelling and convergence to a steady state, showing only
We can also consider a scenario where the measure-
the tunnelling between the central sites. We also show
mentisofthedifferenceofoccupationnumbersatdiffer-
(b) the survival probability for the system to remain in
entsites. AsillustratedinFig.3(b),suchameasurement
theZenosubspace;weseethatwhilethefullconvergence
scheme can give rise to correlated events resembling pair
tothesteadystatetakesalongtime(withsuchtrajecto-
processes, where tunnelling events in to or out of a par-
riesoccurringwithlowprobability),theadditionalquasi-
ticular site/region can only occur in pairs, in order to
Zenodynamicscanstilltakeplaceontimescalesforwhich
preserve the measurement value. As before, the Zeno
Zeno-locking is maintained with a high probability. The
Hamiltonian contains the processes where tunnelling oc-
lower probability to reach the steady state (in compar-
curs between sites that are both in the same region, as
ison to the three-state example given in Section III) is
well as the on-site interactions:
inpartbecauseofthecompetitionbetweenthefirst-and
H(1) =−J( (cid:88) b†b + (cid:88) b†b + (cid:88) b†b ) second-orderdynamics,asthesystemcanbeinstatesfor
Z i j i j i j which (second-order) quasi-Zeno dynamics do not take
(cid:104)i∈A,j∈A(cid:105) (cid:104)i∈B,j∈B(cid:105) (cid:104)i∈C,j∈C(cid:105) place(e.g.|1,1,0,0(cid:105)),butleakagefromthemeasurement
(cid:88)
+U b†b†b b , subspace still can; the standard Zeno dynamics causes
i i i i
transitions between such states and the states for which
i
the quasi-Zeno dynamics do take place (e.g. |1,0,1,0(cid:105)).
where we now label the three regions as: A measured
(positive contribution); B unmeasured; and C measured
(negativecontribution). Thecorrespondingsecond-order
The primary quantitative disagreement between the
quasi-Zeno Hamiltonian is
exactandapproximateeffectiveevolutionisincapturing
H(2) =J2( (cid:88) b†b b†b + (cid:88) b†b b†b the first-order processes. This is because of the discrep-
Z i j k l i j k l ancybetweentheexactbinomialseries,andtheapproxi-
(cid:104)i∈A,j∈B(cid:105) (cid:104)i∈A,j∈B(cid:105)
mateeffectiveexponentialpowerseries. Naively,onecan
(cid:104)k∈C,l∈B(cid:105) (cid:104)k∈B,l∈A(cid:105)
+ (cid:88) b†ibjb†kbl+ (cid:88) b†ibjb†kbl+h.c.). aorfgtuheetlhairsgeesrtroprotsosibblee oOc(cHupZ(1a)t2itoδnt)of(utphetositae)m, abxeicmauusme
(cid:104)i∈B,j∈C(cid:105) (cid:104)i∈A,j∈C(cid:105)
for each quasi-Zeno Hamiltonian the discrepancy is in
(cid:104)k∈C,l∈B(cid:105) (cid:104)k∈C,l∈A(cid:105)
its associated second-order term in the evolution, thus
The first term in H(2) mediates the pair process-like ef- making the Zeno Hamiltonian H(1) responsible for the
Z Z
fects, while the other terms correspond to the simulta- primary level of error. However, the convergence to a
neous crossing in opposite directions across the bound- steady-state supresses the dynamics, and so curtails this
aries of each pairing of regions. As before, when these error to some maximum value due to the damping of the
boundarypairsareatthesamelocation,thisisequivalent accessible state space in time by H(2).
Z
7
FIG. 4: Simulation of correlated many-body dynamics: (a) Simulation of an effective Hamiltonian for atoms in an
optical lattice, for the scheme of Eqs. (5) and (6) and Fig. 3(a), showing first- and second-order processes. Colour map shows
averagesiteoccupation. (b)SurvivalprobabilityforthesystemtoremaininthesameZenosubspace. (c)Exactevolutionofthe
same system. (d) Difference between results for exact and effective evolution; the primary error is in capturing the first-order
dynamics, and scales linearly with δt (see main text). (e) Evolution under the Zeno Hamiltonian of QZD for the same system
failstocapturethecorrelatedprocesses. Simulationsuse4sites,2atoms,δtJ =10−2,U =0,withmeasurementimposingthe
constraint N +N =1. Initial state |1,1,0,0(cid:105).
2 3
8
VI. DISCUSSION Appendix A: Derivation of Quantum quasi-Zeno
Dynamics
We have shown how, beyond the freezing of the ob-
Here we provide details on the derivation of QqZD,
served value of a frequently repeated measurement of a
and clarify assumptions made about the system evolu-
system manifest by QZE/QZD, it is possible for dynam-
tion. Firstly, we justify modeling the system as evolving
ics to still take place across different states in the mea-
under unitary evolution between measurements. This is
surement subspace, even in the absence of direct tran-
true in general for an isolated, closed quantum system.
sitions between them, via higher-order virtual processes
Appealingtotheso-calledchurchofhigherHilbertspace
that arise through transitions that take the system tem-
(CHHS) [48], an ancilla can be appended to the system
porarily out of the measurement subspace, without al-
toaccountfortheeffectofanenvironment,suchthatthe
tering the consistent outcome of the measurement value.
total system-ancilla evolution is unitary, even if the sys-
We developed this QqZD formalism, and derived effec-
tem dynamics alone is not. If the measurement outcome
tive Hamiltonians to describe the system evolution. We
depends only on the system state, and is independent of
generalised to incorporate measurements with non-equal
the ancilla state, then the inclusion of the ancilla does
timesteps and time-dependent Hamiltonians, and how
not affect the QqZD result - one has simply to trace out
the state and evolution of the system change when the
the ancilla from the QqZD evolution in the same man-
measurementvaluechanges. Weshowedthatthisregime
ner as usual for recovering the system dynamics from a
generates correlated dynamics in many-body systems.
CHHS treatment. A second simplification made in our
treatment is that we treat the measurement as von Neu-
Whilst being relatively simple both mathematically
mann projections. This is in keeping with the simple
andconceptually,thisregimehaspreviouslybeenlargely
derivations of QZE and QZD [1, 6], which have subse-
unexplored, despite the abundance of possibilities for
quentlybeenextendedtomoregeneralsettings,including
which it lays the foundations. The field of dissipative
coupling to external ‘measurement devices’ [49]. These
dynamics, where the interactions between a system and
treatments that incorporate the measurement device re-
its environment can be exploited to manipulate the dy-
coverQZEwhenthetimetakenfortheexternaldeviceto
namics of a system, and to prepare particular states of
measure the system state is much shorter than the sys-
the system, has seen a lot of interest [25, 26, 42–44], as
tem dynamics. It has also been shown that even when
has the very related field of using designed measurement
the measurements are not perfectly projective, QZE can
as the source of dissipation for quantum system engi-
still persist [30]. Thus, we expect when these realistic
neering [27, 29, 45–47]. This work extends these ideas,
concerns are incorporated into our simplified picture of
as by eliminating particular processes at first-order only,
measurement, the results should be preserved.
whilst preserving them at second-order (or higher), can
In deriving the effective evolution for QqZD, we make
leadtotheemergenceofcorrelateddynamics, asdemon-
use of some important properties of projectors; they are
strated here. An experimental realisation of this regime
idempotent and mutually orthogonal (P P = P δ ),
wouldpotentiallybelesstaxingthansimilarexperiments j k j jk
and together span the entire Hilbert space ((cid:80) P = I)
of standard QZE and QZD, as the requirement on the j j
time between measurements is less stringent. The pos- [50].
sible obstacles we foresee are the need to maintain the As noted in the main text, the effect on state ρ of uni-
coherenceofthesystemforsufficientlylongtimestowit- tary evolution followed by a projective measurement is
ness the higher-order effects, and that for verification of described by ρ → PU(δt)ρU†(δt)P. Defining U1(δt) =
these effects, a second observable must be measured at PU(δt), this can be written ρ→U (δt)ρU†(δt). Follow-
1 1
the start and end of the protocol that can distinguish ing N such sets of evolution and measurement in a total
states in the measurement subspace. time τ = Nδt, with each measurement outcome in the
same subspace P, we can describe the resulting system
evolution by
ρ→U (δt)ρU†(δt),
N N
where U (δt) = U (δt)N = (PU(δt))N. Expanding
N 1
Acknowledgements U(δt) as a power series in terms of the Hamiltonian H,
we hence have
We thank Wojciech Kozlowski and Igor Mekhov for (cid:18) (cid:18) δt2 (cid:19)(cid:19)N
U (δt)= P 1−iHδt−H2 +O(δt3) . (A1)
discussions. TJE is funded by an EPSRC DTA. VV ac- N 2
knowledges funding from the National Research Foun-
dation (Singapore), the Ministry of Education (Singa- In the full QZD limit, where δt→0 and N →∞, this
pore), EPSRC, the Templeton Foundation, the Lever- binomial expansion is exactly equal to the exponential
hulme Trust, the Oxford Martin School, and Wolfson of the Zeno Hamiltonian U (τ) → exp(−iH(1)τ), giv-
N Z
College, University of Oxford. ing the standard QZD result. Close to, but outside of
9
this limit, we can still approximately describe this evo- Appendix B: Simulation Details
lution by an exponential, but with the inclusion of the
higher-order terms added perturbatively. These correc-
Asstatedinthemaintext,wesimulatethescenarioof
tions can be found by examining the difference between
Eqs. (5) and (6) and Fig. 3(a) in Fig. 4, for 2 atoms dis-
the exact evolution, and the evolution under the Zeno
tributedacross4latticesites. Thechosenparametersare
Hamiltonian HZ(1). Specifically, focussing on the O(δt2) Jδt=10−2 and U =0, with measurement of the central
term in the power series expansion, we see that the ex-
two sites fixing N +N =1, and initial state |1,1,0,0(cid:105).
act evolution contains the term −PH2Pδt2/2, whereas 2 3
We take the outcome of each measurement to be consis-
an expansion of the exponential of the Zeno Hamilto-
tentwiththisvalue;thatis,wepost-selectthetrajectory
nian yields −H(1)2δt2/2. We use the difference between in which there are no jumps to other subspaces.
Z
these two to motivate our definition of the second-order Applying this specific case to Hamiltonians Eqs. (5)
quasi-Zeno Hamiltonian: HZ(2) = PH(I − P)HP, and and (6), the Zeno Hamiltonian contains only the tun-
the requisite correction is given by −H(2)δt2/2. More- nelling terms between sites 2 and 3:
Z
over, we then define the general quasi-Zeno Hamiltonian
as HZ(k) = PH((I−P)H)k−1P to account for the cor- H(1) =−J(b†b +b†b ). (B1)
rections at higher orders of δt. These corrections then Z 2 3 3 2
lead to the definition of the effective Hamiltonian, which
contains the QqZD corrections to the Zeno Hamiltonian: Thesecond-orderquasi-ZenoHamiltoniancontainsterms
where atoms tunnel 1 → 2 and 3 → 4 in a correlated
Heff =(cid:88)∞ (−iδkt!)k−1HZ(k), mtuannnneellrin(gasnadcrtohsestrheveesrasme eprboacrersies)r,inasopwpeollsiatesdciorrercetliaotnesd:
k=1
as previously stated in Eq. (1). Taking this corrected H(2) =J2(2(b†b b†b +b†b b†b )
Z 1 2 3 4 2 1 4 3
effective Hamiltonian, we proceed as with the standard
+b†b b†b +b†b b†b
QZDcaseandexponentiateittogivetheeffectiveevolu- 1 2 2 1 2 1 1 2
tionoperatorU (τ)=exp(−iH τ),andtheassociated +b†b b†b +b†b b†b ).
eff eff 3 4 4 3 4 3 3 4
systemevolutionattimeτ =NδtafterN measurements
is hence
This can be rearranged and rewritten in terms of
ρ→U (τ)ρU† (τ). the effective chemical potentials and nearest-neighbour
eff eff
density-density interactions:
We note that this replacement of the binomial series by
an exponential is approximate, and becomes exact only H(2) =J2(2(b†b b†b +b†b b†b )
in the limit δt → 0 (for fixed τ). However, as we are Z 1 2 3 4 2 1 4 3
+2(n n +n n )+n +n +n +n ). (B2)
in the limit |Hδt| (cid:28) 1, this approximation is still very 1 2 3 4 1 2 3 4
faithfultotheexactevolution,ascanbewitnessedinour
simulation.
Theconstraintimposedbyhavingthecentraltwosites’
The kth-order quasi-Zeno Hamiltonian mediates kth-
occupation fixed at 1 atom reduces the size of the acces-
order transitions, where the initial and final states are
sible state space. There are 2 possible states accessible
in the measurement subspace P, while all the interme-
to these two sites (|1,0(cid:105) and |0,1(cid:105)). Similarly, given that
diate states are not. The absence of processes in these
the total number of atoms is also fixed at 2, the outer
Hamiltonianswhichhaveintermediatereturntothemea-
two sites 1 and 4 also have their occupation fixed at 1
surement subspace is due to such terms already arising
atom, and thus may be any of the same 2 states, giv-
from products of lower-order quasi-Zeno Hamiltonians;
ing a total state space of dimension 4 for the whole sys-
asthehigher-orderquasi-ZenoHamiltoniansareintended
tem. This is then very amenable to exact calculations.
as corrections to the evolution described by the lower-
We calculate the effective and exact evolution by using
orderquasi-ZenoHamiltoniansthisisnotsurprising. For
Eqs. (B1) and (B2) with Eqs. (1) and (A1) respectively,
example,thetermPHPHP canbeobtainedfromH(1)2, and the standard Zeno evolution by inputting Eq. (B1)
Z
while PH(I−P)HPHP =H(2)H(1). into U (τ)=exp(−iH(1)τ).
Z Z QZD Z
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