Sustained Quantum Coherence and Entanglement in the Avian Compass

2.6 MB·English

by  Erik Gauger

Quantum coherence and entanglement in the avian compass
Erik Gauger∗,1 Elisabeth Rieper∗,2,† John J. L. Morton,1,3 Simon C. Benjamin,2,1 and Vlatko Vedral2,3,4
1Department of Materials, University of Oxford, Parks Rd, Oxford OX1 3PH, UK
2Center for Quantum Technologies, National University of Singapore, Singapore
3Clarendon Laboratory, University of Oxford, Parks Rd, OX1 3PU, UK
4Department of Physics, National University of Singapore, Republic of Singapore
(Dated: September 30, 2009)
Tremendouseffortsareunderwaytobuildtechnologiesthatharnessthedeepquantumphenomena
of superposition and entanglement. These properties have proven fragile, often decaying rapidly
unless cryogenic temperatures are used. Could life have evolved to exploit such phenomena [1]?
Certain migratory birds have the ability to sense very subtle variations in the Earth’s magnetic
field[2]. Hereweuserecentexperimentalobservations[3]togetherwiththewelldeveloped‘radical
pair’ model of the avian compass [4], and employ a master equation with various decoherence
9
operators in order to examine the system’s vulnerability to environmental noise. Remarkably, the
0
roomtemperaturenoisetoleranceinthisnaturalsystemappearsgreaterthanthatofthebestman-
0
made molecular radical [5] or solid state singlet/triplet devices [6]. We find that entanglement,
2
thoughprobablynotanessentialfeatureofthisprocess,appearstopersisttotensofmicroseconds,
p or more.
e
S
Recentlyseveralauthorshaveraisedtheintriguingpos- All these experiments can be explained with the
0
3 sibility that living systems may use non-trivial quantum commonly-accepted Radical Pair (RP) model, see Fig.1.
effects to optimise some tasks. Studies range from the Here we employ the simplest RP model, considering the

] role of quantum physics in photosynthesis [7, 8, 9, 10] spins of two electrons [4, 24] and one nucleus of the
h
p and in natural selection itself [11], through to the obser- molecule. Absorption of a photon and subsequent trans-
- vation that ‘warm and wet’ living systems can embody fer of one electron to an acceptor part of the molecule,
t
n entanglement given a suitable cyclic driving [12]. In this gives rise to a radical pair initially in the singlet state.
a letter, we examine a particularly important form of nat- Due to the spatial separation it now becomes meaning-
u
ural information processing known as magnetoreception ful to talk about electron spin entanglement. Because of
q
– the ability to sense characteristics of the surrounding thenuclearinteractionwithoneoftheelectronspins,the
[
magneticfield. Thereareaseveralmechanismsbywhich singlet state is no longer an eigenstate of this Hamilto-

3
this sense may operate [2]. In certain species (including nian leading to an angle dependent singlet-triplet oscil-
v
somebirds[4],fruitflies[13,28]andevenplants[14]),the lation. Recombination occurs either from the singlet or
5
2 evidencesupportsaRadicalPair(RP)mechanism,which triplet state, leading to different chemical endproducts.
7 relies on the quantum evolution of a spatially-separated Theconcentrationofthoseproductsconstitutesamacro-
3 interacting pair of electron spins. This is supported by scopicchemicalsignal,correlatedtotheorientationofthe
.
6 resultsfromthefieldofspinchemistry[15,16,17,18,19]
0 and recent experiments which were able to demonstrate
9 a chemical compass [20].
0 B!.
: By manipulating a captive bird’s magnetic envi- Zeeman
v interaction
ronment and recording its response, one can make
i
X inferences about the mechanism of the magnetic sen-
r sor[3,21,22,23]. Specifically,EuropeanRobinsareonly
a sensitivetotheinclinationandnotthepolarizationofthe anisotropic
hyperfine
interaction
magnetic field [21], and this sensor is evidently activated
by photons entering the bird’s eye [22]. Importantly for
the present analysis, a very small oscillating magnetic
field can disrupt the bird’s ability to orientate [3, 23].
It is also significant that birds are able to ‘train’ to bird's eye retina
different field strengths, suggesting that the navigation
FIG.1: Schematicofthebird’seye. Thebackoftheeyecon-
sense is robust, and unlikely to depend on very special tainsnumerousmolecules[30],fixedwithspecificorientations
values for the parameter in the model [3]. but not necessarily ordered. In the simplest RP model, each
such molecule involves three crucial components (see inset):
there are two electrons, initially photo-excited to a singlet
state,andanuclearspinthatcouplestooneoftheelectrons.
Thiscouplingisanisotropic,sothatthemoleculehasadirec-
tionality to it.
∗Theseauthorshavecontributedequallytotheworkreportedhere.
2
molecule with the magnetic field.
0.40
We employ the Hamiltonian corresponding to the sys- k = 106 s-1
tem once the two electrons have become separated. The
anisotropic hyperfine tensor coupling the nucleus and
electron 1, is conveniently written in its diagonal basis eld 0.35 k = 105 s-1
A = diag(Ax,Ay,Az), and we assume an axially sym- yi
metric (or cigar-shaped) molecule with A = 10−5 meV et
z gl
and A = A = A /2. This is the simplest assumption n
thatcaxnprovyideuszwithdirectionality,andwehavecho- Si 0.30 k = 104 s-1
sen the general shape and magnitude of the tensor to be
consistent with [29]. The Hamiltonian is
reference, k = 106 s-1
H =Iˆ·A·Sˆ1+γB·(Sˆ1+Sˆ2), 0.25 oscillatory field on
0 π/8 π/4 3π/8 π/2
where Iˆ is the nuclear spin operator, Sˆ = (σ ,σ ,σ ) Angle θ
i x y z i
are the electron spin operators (i = 1,2), B is the mag-
FIG.2: Angular dependence of the singlet yield in the
netic field vector and γ = 21µ0g the gyromagnetic ratio presence of an oscillatory field. Thebluereferencecurve
with µ0 being Bohr’s magneton and g = 2 the g-factor. showsthesingletyieldobtainedintheEarth’smagneticfield
Thefactor1/2inthegyromagneticratioaccountsforthe with B = 47µT), which is independent of the decay rate k
0
fact that we have a spin one-half system, but we will use fork≤107 s−1. Forbettervisibility,thebluecurvehasbeen
Pauli matrices such as σ = diag{1,−1} etc. Here only shifted upwards by 0.001. The red curves show the singlet
z
fieldwhena150nTmagneticfieldoscillatingwithafrequency
one electron is coupled to one nucleus, whereas the re-
resonantwiththeZeemansplittingoftheuncoupledelectron
moteelectronissoweaklyinteractingthatwedescribeit
(1.316MHz)magneticfieldissuperimposedperpendicularto
asfree. Previousauthorshaveconsideredthecasewhere
the direction of the static field. We see that this only has
more than one nucleus couples to the system [3, 19, 31], an appreciable effect on the singlet yield once k is of order
IntheSupplementaryInformationweshowourbasiccon- 104 s−1. Inset: a European Robin ((cid:13)c David Jordan)
clusionsarenotaffectedbyvaryingthehyperfinetensor,
adding a second nuclear spin, or even by replacing the
term completely with a anisotropic electron g-factor.
Withtheusualdefinitonofsinglet|s(cid:105)andtripletstates
Generally, the magnetic field we employ is
|t (cid:105)intheelectronicsubspace,while|↑(cid:105)and|↓(cid:105)describ-
i
ingthestatesofthenuclearspin, wedefinethefollowing
B = B (cosϕsinϑ,sinϕsinϑ,cosϑ)
0 decay operators:
+ B cosωt(cosφsinθ,sinφsinθ,cosθ), (1)
rf
P =|S(cid:105)(cid:104)s,↑|
S,↑
whereB0 =47µTistheEarth’smagneticfieldinFrank- PT0,↑ =|T(cid:105)(cid:104)t0,↑|
furt [3], and the angles describe the orientation of mag-
P =|T(cid:105)(cid:104)t ,↑|
netic field to the basis of the HF tensor. B = 150 nT T+,↑ +
rf P =|T(cid:105)(cid:104)t ,↑|
is an additional oscillatory field only applied in our sim- T−,↑ −
ulations where explicitly mentioned. For resonant exci- andsimilarlyforthe‘down’nuclearstates. Thisgivesus
tation with the uncoupled electron spin, ¯hω = 2γB0, so a total of two singlet projectors and six triplet projec-
that ν =ω/(2π)=1.316MHz. tors. For simplicity and because this choice corresponds
BecauseoftheaxialsymmetryoftheHFtensorwecan exactly to the expression for singlet yield used in previ-
setϕ=0andfocusontheϑintherange[0,π/2]without ous literature, all eight projectors have the same decay
loss of generality. Furthermore, for the oscillatory field, rate Γ =k.
P
we set φ=0. For our model we start from a initial density matrix
Wemodelthedynamicsofthesystemwithaquantum ρ(0)correspondingtotheelectronsinapuresingletstate,
master equation (ME) approach. We employ operators and a completely mixed nuclear state, i.e.,
representing the relaxation processes; specifically, we in-
ρ(0)=id ⊗|s(cid:105)(cid:104)s|=|s,↓(cid:105)(cid:104)s,↓|+|s,↑(cid:105)(cid:104)s,↑|.
clude two ‘shelving states’ which represent the system n
having decayed either from an electron singlet state, or The decay to the two shelving levels is then described
from one of the triplet states. Ultimately one of these using a standard quantum ME with above decay opera-
two forms of relaxation will occur. The three spins span torswhicheffectivelydiscrimatesingletandtripletdecay
an8dimensionalHilbertspacetowhichwethereforeadd events
two further levels |S(cid:105) and |T(cid:105) for the singlet and triplet (cid:32) 8 (cid:33)
decay outcomes, respectively. The populations of these ρ˙ =−i[H,ρ] +k (cid:88)P ρP†− 1(cid:16)P†P ρ+ρP†P(cid:17) (2)
levels will then correspond the singlet and triplet yield. ¯h − i i 2 i i i
i=1
3
0.40 singletyield(i.e.theeventualpopulationofshelvingstate
|S(cid:105)) as a function of the angle between the Earth’s field
andthemolecularaxis. Consistentwiththeexperimental
work, we find that there is no effect at such weak fields
d0.35 when the oscillatory field is parallel to the Earth’s field.
el
yi 1/Γ = 1 ms Therefore for our analysis we set the oscillatory field to
et be perpendicular. The results are shown in Figure 2.
gl
n We conclude that if the oscillating field is to disorient
Si0.30 the bird, as experiments showed, then the decay rate k
reference should be approximately 104 s−1 or less. For higher val-
1/Γ = 100 μs ues of k (shorter timescales for the overall process) there
1/Γ = 10 μs with noise is no time for the weak oscillatory field to significantly
0.25 perturb the system; it relaxes before it has suffered any
0 π/8 π/4 3π/8 π/2
effect. Such a value for the decay rate is consistent with
Angle θ
the long RP lifetimes in certain candidate cryptochrome
molecules found in migratory birds [25].
FIG. 3: Angular dependence of the singlet yield in
the presence of noise. A) All curves were obtained using Taking the value k = 104 s−1, we are able to move to
k = 104. The blue curve provides a reference in the absence the primary question of interest: how robust this mech-
ofnoiseandtheredcurvesshowthesingletyieldfordifferent anism is against environmental noise. There are several
noise rates given. It is apparent from the plot that a noise reasons for dehorence. For example dipole interaction,
rate Γ > 0.1k has a dramatic effect on the magnitude and
electron-electron distance fluctuations and other parti-
contrast of the singlet yield.
cles’ spin interaction with the electrons will cause deco-
herence. We describe generic environmental noise with
a standard Lindblad ME technique [26], where Eqn. 2
NotethatEqn.(2)doesnotcontainanyenvironmental above is extended with a dissipator as follows:
noise, though this does not alter the estimates of the RHS of (cid:88) (cid:18) 1(cid:16) (cid:17)(cid:19)
decay rate k (see Supp. Information). In the previ- ρ˙ = + Γ L ρL†− L†L ρ+ρL†L (3)
Eqn.2 i i i 2 i i i i
ous literature on the RP model of the avian compass,
i
it has been common to employ a Liouville equation to
Noise operators L are σ , σ , σ for each electron
model the dynamics. In fact, a term-by-term compar- i x y z
spin individually (i.e. tensored with identity matrices
ison of the evolution of the terms of the density ma-
for the nuclear spin and the other electron spin). This
trix readily confirms that this former approach and our
gives a total of six different noise operators L and we
MEareexactlyequivalentintheabsenceofenvironmen- i
use the same decoherence rate Γ for all of them. We are
tal noise. Both agree with the singlet yield integral,
Φ = (cid:82)∞(cid:104)ψ−|Tr (ρ(t))|ψ−(cid:105)ke−ktdt, another commonly now in a position to determine the approximate level of
0 n noise which the compass may suffer, by finding the mag-
usedquantityfromthepriorliterature, whensingletand
nitude of Γ for which the angular sensitivity fails. This
tripletreactionratesareequal. Specifically,theultimate
is shown in Fig. 3. Conservatively, we can say that when
population of our singlet ‘shelf’ |S(cid:105) then corresponds to
Γ ≥ k, the angular sensitivity is highly degraded. This
Φ. However, when we presently wish to introduce vari-
is remarkable, since it implies the decoherence time of
ous kinds of noise operator, in particular the pure phase
the two-electron compass system is of order 100 µs or
variant, then the ME approach provides a more intuitive
more [32]! To provide context for this number, we note
framework.
that the best laboratory experiment involving preserva-
We now wish to determine an appropriate choice for
tion of a molecular electron spin state has accomplished
our parameter k in Eqn. 2. In Ref. 3, the authors re-
a decoherence time of 80µs [5].
port that a perturbing magnetic field of frequency of
Itisinterestingtoask,whatisthesignificanceofentan-
1.316 MHz (i.e. the resonance frequency of the ‘remote’
glementbetweenthespinsintheaviancompass? Having
electron) can disrupt the avian compass. They note that
inferred approximate values for the key parameters, we
thisimmediatelyimpliesaboundonthedecayrate(since
can plot an appropriate entanglement measure over the
thefieldwouldappearstaticforsufficientlyrapiddecay).
course of the process, from the initial singlet generation
Hereweaimtorefinethisboundonk byconsideringthe
to the eventual decay. The metric we use is negativity:
oscillating magnetic field strength which suffices to com-
pletelydisorientthebird’scompass,i.e. 150nT. (Indeed, ||ρTA||
even a 15 nT field was reported as being disruptive, but N(ρ)=
2
to be conservative in our conclusions we take the larger
value here.) To model this effect, we activate the oscilla- where ||ρTA|| is the trace norm of the partial transpose
tory field component defined in Eqn. 1 and examine the of the system’s density matrix. The transpose is applied
4
1.0 1.0 QIPIRC (No. GR/S82176/01) for support. JJLM and
SCB thank the Royal Society for support. JJLM thanks
Entanglement (negativity)
Excited state population St. John’s College, Oxford. VV acknowledges financial
0.8 0.8
support from the Engineering and Physical Sciences Re-
n search Council, the Royal Society and the Wolfson Trust
y o
vit 0.6 0.6 ati in UK.
gati pul We note that in the final stages of preparation of this
Ne 0.4 0.4 Po manuscript, a related work has appeared which consid-
× ers a chemical magnetometer in the context of quantum
2 control [27].
0.2 0.2
0.0 0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30
† Electronic address: [email protected]
Time (ms)
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[3] T. Ritz et al.: Magnetic Compass of Birds Is Based on a
the Earth’s field and the molecular axis in π/4, although the
MoleculewithOptimalDirectionalSensitivity,Biophysical
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BasedMagnetoreceptioninBirds,BiophysicalJournal78,
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in CS , J. Chem. Phys. 124, 014508, (2006).
2
partitioning between the electron, on one side, and the
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Clearly, the initial singlet state is maximally entangled. [7] G. S. Engel et al.: Evidence for wavelike energy trans-
Under noise, entanglement falls off at a faster rate than fer through quantum coherence in photosynthetic systems,
Nature 446, 782-786 (2007).
the decay of population from the excited state.
[8] M. Mohseni, P. Rebentrost, S. Lloyd, A. Aspuru-Guzik:
The noise model above is of the most general kind,
Environment-assistedquantumwalksinphotosyntheticen-
representing a unrestricted degradation of the system’s
ergy transfer, J. Chem. Phys. 129, 174106, (2008).
stateduetoenvironmentalinteractions. Wefurtherhave [9] M.B. Plenio, S.F. Huelga: Dephasing assisted transport:
found that the compass mechanism is almost immune Quantum networks and biomolecules, New. J. Phys. 10,
to pure phase noise (typically the most aggressive noise 113019 (2008).
in artificial systems). Even starting from a fully de- [10] M.Sarovar,A.Ishizaki,G.R.Fleming,K.BirgittaWha-
ley, Quantum entanglement in photosynthetic light har-
phased state (|s(cid:105)(cid:104)s|+|t (cid:105)(cid:104)t |)/2, the compass operates
0 0 vesting complexes, arXiv: 0905.3787, (2009).
well. However, in the Supplementary Information we
[11] S.Lloyd: Aquantumofnaturalselection,NaturePhysics
show that if such noise were naturally present at a high
5, 164, (2009).
level in the compass (exceeding the generic noise level [12] J.-M.Cai,S.Popescu,H.J.Briegel:Dynamicalentangle-
Γ by more than an order of magnitude) then it would ment in oscillating molecules, arXiv:0809.4906, (2008).
renderthebirdimmunetotheweakoscillatorymagnetic [13] R. J. Gegear, A. Casselman, S. Waddell, S. M. Reppert:
fields of Ref. [3]. Thus the sensitivity to oscillatory fields Cryptochrome mediates light-dependent magnetosensitiv-
ity in Drosophila, Nature 454, 1014 (2008).
implies that both amplitude and phase, and thus entan-
[14] M. P. Galland, T. Ritz, R. Wilschko, and W.
glement, are indeed protected within the avian compass
Wilschko:Magnetic intensity affects cryptochrome-
on timescales exceeding tens of microseconds. It is not
dependent response in Arabidopsis thaliana, Planta. 225,
clear why such remarkable protection occurs, but given 615 (2006).
the widely-accepted RP model together with the recent [15] C.R.Timmel,K.B.Henbest: A study of spin chemistry
experimental data [3], this conclusion follows. in weak magnetic fields, Phil Trans Roy Soc London A
We thank Earl Campbell, Chris Rodgers, Peter Hore 362, 2573-2589, (2004).
[16] Y. Liu et al.:Magnetic field effect on singlet oxygen pro-
and Kiminori Maeda for stimulating discussions. We
ductioninabiochemicalsystem,Chem.Commun.,174-176,
thank the National Research Foundation and Ministry
(2005).
of Education of Singapore for support. EMG acknowl-
[17] T. Miura, K. Maeda, T. Arai: The Spin Mixing Process
edges support from the Marie Curie Early Stage Train- ofaRadicalPairinLowMagneticFieldObservedbyTran-
ing network QIPEST (MEST-CT-2005-020505) and the sient Absorption Detected Nanosecond Pulsed Magnetic
5
Field Effect, J. Phys. Chem. A 110, 4151-4156, (2006). Supporting Material
[18] C. T. Rodgers: Magnetic Field Effects in Chemical Sys-
tems, Pure Appl. Chem., 81, No.1, 87-111, (2009).
[19] C.T.Rodgers,P.J.Hore: Chemical magnetoreception in
birds: Theradicalpairmechanism,PNAS,1062,353-360,
THE TERMINOLOGY AND TECHNIQUES USED
(2009).
IN THE MANUSCRIPT
[20] K. Maeda et al.:Chemical compass model of avian mag-
netoreception, Nature, 453, 387, (2008).
[21] W.Wiltschko,R.Wiltschko: MagneticcompassofEuro- The terminology used in the main paper arises from
pean robins Science, 176, 6264, (1972). the foundations of quantum mechanics and has more re-
[22] W. Wiltschko, R. Wiltschko: Magnetic compass ori- cently become important in the field of quantum infor-
entation in birds and its physiological basis Naturwis-
mation (QI). Thus we describe the avian compass and
senschaften, 89:445-452, (2002).
its dynamics in terms of quantum coherence, environ-
[23] T. Ritz, P. Thalau, J.B. Philips, R. Wiltschko, W.
mental decoherence, and entanglement; moreover we use
Wiltschko:Resonance effects indicate a radical-pair mech-
anism for avian magnetic compass, Nature, 429, 177, thequantummasterequationtechniquetomodelthesys-
(2004). tem’s dynamics. These terms and techniques are rather
[24] K. Schulten, C. E. Swenberg, A. Weller,A biomagnetic differenttothoseoftheestablishedradicalpair(RP)and
sensory mechanism based on magnetic field modulated co- aviancompasscommunities,thereforeitisappropriateto
herent electron spin motion, Z. Phys. Chem. NF111, 1,
justify their use and relate them to the established ter-
(1978).
minology.
[25] Liedvogel et al.:Chemical Magnetoreception: Bird Cryp-
tochrome 1a Is Excited by Blue Light and Forms Long- The primary conclusion of the paper concerns the ex-
Lived Radical-Pairs, PLoS ONE, 10 e1106, (2007). tenttowhichquantumcoherenceisprotectedinthecom-
[26] M. A. Nielsen and I. C. Chuang Quantum Computation pass system. By this term, we refer to the purity of the
and Quantum Information, Cambridge Cambridge Uni- system, which is degraded by interactions with the en-
versity Press (2000).
vironment because these will generally act as a kind of
[27] J.-M. Cai, G. G. Guerreschi, H. J. Briegel: Quan-
measurementofthestate(forexample,bytakingaquan-
tum control and entanglement in a chemical compass ,
tumofenergyfromthestateandthusleavingitinaspe-
arXiv:0906.2383, (2009).
[28] T. Yoshii, M. Ahmad, C. Helfrich-Forster: Cryp- cific low-energy state). Since the ‘outcome’ of this mea-
tochromeMediatesLight-DependentMagnetosensitivityof surement is lost to the environment, the system’s state
Drosophila’s Circadian Clock,Plos One (2009). becomesamixtureofthemultiplepossiblestates–gener-
[29] O. Efimova, P. J. Hore:Evaluation of nuclear quadrupole ically this loss of purity is called decoherence.
interactionsasasourceofmagneticanisotropyintherad-
In the applied QI field, much effort is dedicated to the
ical pair model of the avian magnetic compass, Molecular
preservation of quantum coherence. In order for quan-
Physics, 107, 665-671 (2009).
tumcoherencetobemaintained,itisessentialtoprevent
[30] I. A. Solov’yov, K.Schulten: Magnetoreception through
Cryptochrome May Involve Superoxide, Biophysical Jour- all kinds of interaction, including those that would flip
nal, 96, 4804 - 4813 (2009). spins, allowing them to relax and lose energy, and also
[31] C.T. Rodgers:Magnetic Field Effects in Chemical Sys- thoseinteractionswhichwouldmerelyalterthephasere-
tems, PhD Thesis, Oxford (2007). lationshipsinthestate(dephasingnoise). Thuswhilethe
[32] One could assume the bird to be more easily perturbed
conceptofspinrelaxationisatypicalsignatureofgeneral
by the oscillatory field (Fig. 2), and obtain a larger k.
coupling to the environment, we must also examine the
However,thatsameassumptionofhighsensitivityshould
effect of pure phase noise; the latter alone can suffice to
then be applied to the noise analysis (Fig. 3) and in fact
the two assumptions would cancel to give the same ba- completely degrade a pure state such as the singlet |s(cid:105)
sic estimate for the decoherence rate. This cancellation is into a completely incoherent mixture of |s(cid:105) and |t (cid:105) and
0
robust, being valid over an order of magnitude in k. would therefore undermine our conclusion. In order to
examine this point, and establish whether full coherence
isindeedmaintained, itisnecessarytoconstructaequa-
tion for the dynamics of the system where pure phase
noise alone occurs. This is most naturally done with
the quantum master equation (QME) formalism and the
corresponding Lindblad noise operators – however, it is
importanttonote(asmentionedinthemainpaper)that
the QME is actually identical to the more conventional
Liouvilleequationinthecaseofzeroenvironmentalnoise.
WeselecttheQMEasthemostsuitabletoolforexamin-
ing specific decoherence models, including both general
spin relaxation and those that specifically cause only de-
phasing.
6
Finallywenotethat,becausetheRPinvolvestwoelec-
0.40
trons, and these are spatially separated, it is meaningful reference
1/Γ = 1 ms
to use the quantity ‘entanglement’ as one specific char- 1/Γz = 100 µs
acterization of their mutual state. The established avian 1/Γzz = 10 µs
compassliteraturewouldtendtorefertotherelatedcon- 0.35
d
cept of spin correlation, however, entanglement is subtly el
yi
andimportantlydifferentinthatitcapturespreciselythe et
gl
non-classicalcorrelationsbetweenspins–itisimpossible, n
Si
for example, to create entanglement between two sites 0.30
simplybyexchangingclassicalinformationastohowthe
spins should be prepared. We chose to plot the entan-
glement in the avian compass because it is a key quan-
tity (arguably, the essential quantity) in the quantum 0.25
0 π/8 π/4 3π/8 π/2
mechanics, and therefore a reader from that community Angle θ
will naturally wonder about this aspect. However, we
find that while entanglement does persist for a remark- FIG. 5: Angular dependence of the singlet yield at k =
ably long time, this is likely to be merely an implication 104 s−1 in the presence of the oscillatory field for different
of the long coherence time rather than a key property of pure dephasing rates Γz. This is to be compared with the
k=104 s−1 line in Fig. 2 of the main paper. See text for an
the compass mechanism.
explanation.
SENSITIVITY TO OSCILLATORY FIELD IN THE
PRESENCE OF PURE DEPHASING wereplacethepreviouslydefinednoiseoperatorsofL of
i
Eq. (3) by appropriate dephasing operators as follows:
Interestingly, if we begin the simulation with a com- wetreattheremoteelectronandtheeletronnuclearspin
pletely dephased state: (|s(cid:105)(cid:104)s|+|t (cid:105)(cid:104)t |)/2, the classical subsystem separately. Within both subsystems, we de-
0 0
correlations are still sufficient for achieving adequate an- fine dephasing operators
gularvisibilityandneitherquantumphasecoherencenor  
entanglementseemstobeaprerequisitefortheefficiency 1 (cid:88)
Zi = √  |λj(cid:105)(cid:104)λj|−|λi(cid:105)(cid:104)λi|
of the avian compass. 2
j(cid:54)=i
To explore this idea further, we would like to study
1
‘pure dephasing’ occurring during the singlet-triplet in- = √ (I −2|λ (cid:105)(cid:104)λ |), (4)
4 i i
2
terconversion. In essence we use energy conserving noise
operators, Eqn. (4), which are known to be the domi- where{|λ (cid:105)}arethesetofnormalisedeigenvectorsofthis
i
nant source of decoherence in so many other artificially subsystem. This results in two dephasing operators for
made quantum systems. By applying this specific noise, the remote electron (these can be combined to a single
we confirm that the compass mechanism’s performance σ operator rotated with the field) and four operators
z
is essentially immune, while of course the coherence of for the eletron nuclear spin subsystem. Each of these
the quantum state of the electrons would be degraded. dephasing operators corresponds to fluctuations of one
One might be inclined to conclude that, if pure de- of the (subsystem’s) energy levels.
phasing noise is indeed dominant, then the avian com- Strikingly, the singlet yield is entirely unaffected by
pass need not protect quantum coherence for the long thisparticularkindofnoise,i.e. itisentirelyindependent
time scales suggested in the main paper. But crucially, of the dephasing rate Γ . Thus, a curve obtained with
z
we also show that if such noise were naturally present at this model coincides perfectly with the reference curve
a high level in the compass (exceeding the generic noise of Fig. 3 of the main paper. However, we show in the
levelΓbymorethananorderofmagnitude)thenitwould following that the dephasing rate of this model can be
renderthebirdimmunetotheweakoscillatorymagnetic at most ten times faster than the generic noise rate to
fields studied by Ritz et al. [1]. Thus the sensitivity to retain sensitivity to the oscillatory field.
oscillatory fields implies that both amplitude and phase, Fig. 5showsthesingletyieldasafunctionofθ fordif-
and thus entanglement, are indeed protected within the ferent pure dephasing rates Γ . Pure phase noise would
z
avian compass on timescales exceeding tens of microsec- actuallyprotectthecompassfromtheharmfuleffectofan
onds. applied oscillatory field (by suppressing the Rabi oscilla-
Since the electron spin singlet state is not an eigen- tions caused by such a field). We see that an aggressive
state of the Hamiltonian, the dephasing operators will pure dephasing rate of 1/Γ = 10 µs almost completely
z
bedifferentfromtheonesmixingthephaseofthesinglet recovers the reference curve (corresponding to a noise-
andtripletstatewithintheelectronicsubspace. Instead, free system without oscillatory field).
7
GENERALITY OF THE MODEL parameters in Fig. 7, confirming that the qualitative
behaviour is still the same under these assumptions.
Europeanrobinscanadjusttodifferentstrengthsofthe
magneticfield. Thisimplies,thattheirchemicalcompass
cannot depend on the absolute value of the the electron
spins hyperfine (HF) coupling. We here argue that the
Markovian noise model
conclusionpresentedinthemainpaperisnotonlyrobust
against variations in the field strength, but also does not
depend on the specifics of the anisotropic interaction be- The noise model we employ is Markovian, implying
tween electron 1 and its local environment. (Of the two that the information lost to the environment will not re-
electrons constituing the RP, only one of them can be turn to the system (as it would, wholly or partially, if
significantly coupled to nuclear spin(s) because the res- the environment was entirely constituted by other iso-
onance effect [1] occurs at exactly the frequency of the latedqubitsthatcouldeffectivelyactasamemory). This
Zeeman splitting of an unperturbed electron.) Markovianassumptionisthenaturalchoicetomakefora
room temperature system whose detailed structure and
environmental interactions are unknown. One can im-
mediately think of ‘likely suspects’ for decoherence in
‘Disc-shaped hyperfine tensor’
the present system that would be Markovian: for exam-
ple, modulation in the electron-electron separation due
Wehaveexplicitlycheckedthatourconclusionsdonot
tovibrationsinthesupportingmatrix. Butmoreimpor-
rely on the specifics of the HF tensor given in the main
tantly,whileonecanreasonablyspeculatethattheremay
text. Specifically neither a weaker or stronger coupling,
also be non-Markovian processes occuring, between the
nor a different HF tensor symmetry changes our obser-
core system and other degrees of freedom that are well
vations about the shortest time scale required for the
isolatedfromthetrue‘bath’degreesoffreedom,thisdoes
processtobesensitivetotheoscillatoryfield. Neitherdo
not undermine our result. In essence, one would ‘draw a
these factors change the maximal tolerable environmen-
dotted line’ around the larger system, comprised of the
tal noise rates to maintain a pronounced signal contrast.
core electron pair together with the non-Markovian envi-
Herewedescribeoneofthevariantswehavestudied: a
ronment,callingthattheaviancompass(althoughnoting
‘disc-shaped’HF-couplingtensorwithcouplingstrengths
that only the core part plays an active role). One would
that are different from the ones presented in the main
conclude that this larger system is, again, remarkably
paper. The largest coupling here is half as strong as the
well insulated from the environment.
one presented in the text: A = 0.5×10−5 meV, A =
x y
A /6,A =A .
x z x
Fig. 6 shows that we obtain exactly the same quali-
tative behaviour as for the parameters used in the main
text, the only difference being in the different shape of RP PAIR MODEL WITH 2 NUCLEAR SPINS
the singlet yield curve (as expected for a different geom-
etry of HF tensor). The oscillatory field does not have
In the model described in the main paper, there is a
an influence on the singlet yield for k >104 s−1, while a
single nuclear spin coupled to one of the electrons. How-
noise rate of Γ > 0.1k leads to a dramatic reduction of
ever, previous publications have studied the case where
contrast.
more than one nuclear spin is present [1, 3, 4]. There-
fore, it is interesting to check whether the addition of a
nuclear spin will alter our conclusions. The Hamiltonian
Anisotropic g-factor now gains an additional coupling term,
Insteadofcouplingtonuclearspin(s),ananisotropicg-
H =Iˆ ·A ·Sˆ +Iˆ ·A ·Sˆ +γB·(Sˆ +Sˆ ),
factor for one of the electrons could also lead to a singlet 1 1 1 2 2 1 1 2
yieldwithanangulardependence[2]. Wehereshowthat
our main conclusion remains valid under this different where A is the HF tensor of the main text and A =
1 2
Hamiltonian of the RP. 2/3A . Here we choose a second rugby shaped HF ten-
1
For illustration purposes, we consider a prounounced sor oriented parallel to the first. Note that we have also
anisotropy: theg-factorinz-directionis0.8×2,i.e. 80% considered different relative coupling strengths and ge-
of the free electronic g-factor of g = 2, while in x and ometries (such as a pancake shaped and rugby shaped
y-direction we have 0.3×2. A smaller anisotropy works tensor),butagain,thesechoicesdonotinfluenceourcore
as well, but gives less overall signal contrast (even in conclusions. Results for the particular choice of parame-
the unperturbed) scenario. We present results for these ters described above are shown in Fig. 8.
8
0.40 0.40
reference reference
k = 105 s-1 1/Γ = 1 ms
k = 104 s-1 1/Γ = 100 µs
0.35 0.35
d d
el el
yi yi
et et
gl gl
n n
Si Si
0.30 0.30
0.25 0.25
0 π/8 π/4 3π/8 π/2 0 π/8 π/4 3π/8 π/2
Angle θ Angle θ
FIG. 6: Results for a disc-shaped HF tensor. The graph on the left is precisely analogous to Fig. 2 of the main text, showing
angular dependence of the singlet yield in the presence of the oscillatory field for different decay rates k. The graph on the
rightcorrespondstoFig. 3ofthemainpaper,showingangulardependenceofthesingletyieldinthepresenceofenvironmental
noise[obtainedwithEq. (5)]fordifferentnoiseratesΓ. Althoughthecurveshapesdiffer,thelevelsofcontrastarethealmost
the same as the ‘cigar’ shaped HF model in the main paper.
0.50 0.50
0.45
0.45
reference
1/Γ = 1 ms
et yield 0.40 kkre ==fe 11re00n54c sse--11 et yield 0.40 1/Γ = 100 µs
gl gl
Sin Sin 0.35
0.35
0.30
0.30 0.25
0 π/8 π/4 3π/8 π/2 0 π/8 π/4 3π/8 π/2
Angle θ Angle θ
FIG. 7: Results for a model in which an anisotropic g-factor replaces the role of the nucleus to break the RP symmetry. The
graph on the left is precisely analogous to Fig. 2 of the main text, showing angular dependence of the singlet yield in the
presence of the oscillatory field for different decay rates k. The graph on the right corresponds to Fig. 3 of the main paper,
showing angular dependence of the singlet yield in the presence of environmental noise for different noise rates Γ. Although
thecurveshapesdiffer, remarkablythelevelsofcontrastremainsimilartothoseoftheconventionalmodelinthemainpaper.
OSCILLATORY FIELD SENSITIVITY IN THE tion of finite noise leads to a suppression of the harmful
PRESENCE OF NOISE effect of the oscillatory field, and thus would lead one to
inferastillmoredramaticcoherencetime. Therefore,by
setting the noise to zero we make the more conservative
In the first stage of the argument in the main paper,
assumption.
we show how the bird’s sensitivity to field angle is de-
graded by an applied oscillatory field (Fig. 2, main pa-
per). There we set the environmental noise operators to
zero – this is of course unrealistic, and one might worry
that finite noise could ultimately lead to less dramatic
conclusions regarding protection of quantum coherence. We confirm this by employing the following master
However,hereweshowthatthereverseistrue: theaddi- equation (which is exactly Eq. (3) of the main paper),
9
0.40 reference 0.40 reference
kk == 110054 ss--11 11//ΓΓ == 11 0m0s µs
d d 0.35
el el
yi yi
et 0.35 et
gl gl
n n
Si Si
0.30
0.30
0.25
0 π/8 π/4 3π/8 π/2 0 π/8 π/4 3π/8 π/2
Angle θ Angle θ
FIG. 8: Results for two nuclear spins coupled to electron 1. The graph on the left is precisely analogous to Fig. 2 of the main
text, showing angular dependence of the singlet yield in the presence of the oscillatory field for different decay rates k. The
graph on the right corresponds to Fig. 3 of the main paper, showing angular dependence of the singlet yield in the presence
of environmental noise for different noise rates Γ. Although the curve shapes differ somewhat, nevertheless the parameters
corresponding to serve degradation remain the same as those in the main paper (k=104 s−1, Γ=100 µs).
while at the same time applying the oscillatory field: tumcoherence. Therefore,anynoiseactingontheremote
(cid:32) 8 (cid:33) electron will decohere the spin and thus suppress rather
ρ˙ = −i[H,ρ]+k (cid:88)P ρP†− 1(cid:16)P†P ρ+ρP†P (cid:17) than enhance the effect of the oscillatory field. This can
¯h i i 2 i i i i also be seen in the right panel of Fig. 9: the red curves
i=1
are very similar but not quite identical, the agreement
+ (cid:88)6 Γ (cid:18)L ρL†− 1(cid:16)L†L ρ+ρL†L (cid:17)(cid:19). (5) is better for the green curves and almost perfect for the
i i i 2 i i i i blue one.
i=1
The left panel of Fig. 9 corresponds to Fig. 2 of the
mainpaperwithsuchadditionally appliednoiseatarate
of Γ = 0.1k. Note that the noise rate is adjusted with
k, as we wish to concentrate on the combined effect of † Electronic address: [email protected]
the oscillatory field and ‘mild noise’ over the timescale [1] T. Ritz et al.: Magnetic Compass of Birds Is Based on a
of the process. We infer that the singlet yield contrast MoleculewithOptimalDirectionalSensitivity,Biophysical
is only significantly reduced when k <105 s−1, meaning Journal, 96 3451-3457, (2009).
[2] I.A. Solov’yov, D. E. Chandler, K. Schulten:Magnetic
the addition of noise to Fig. 2 in the main paper does
FieldEffectsinArabidopsisthalianaCryptochrome-1,Bio-
not influence our conclusion.
physical Journal, 92, 2711-2726 (2007).
Toestablishthatthisconclusionisrobustwithrespect
[3] C. T. Rodgers, P.J. Hore: Chemical magnetoreception in
to the particular choice of Γ = 0.1k, we now investigate birds: Theradicalpairmechanism,PNAS,1062,353-360,
different noise rates for a fixed value of k = 105 s−1. (2009).
In the right panel of Fig. 9, we show a pair of curves [4] C.T.Rodgers:MagneticFieldEffectsinChemicalSystems,
for each noise rate: one obtained with and one without PhD Thesis, Oxford (2007).
the oscillatory field. We make two observations: firstly,
the two curves of each pair nearly coincide, indicating
that there is only a small sensitivity to the oscillatory
field indepedent of the noise rate. Secondly, the sensitiv-
ity decreases even more when the system is subjected to
more aggressive noise.
In fact, this second observation is not surprising: we
would not expect the additional noise terms to lead to
a greater sensitivity to the oscillatory field. The singlet
yield signal is corrupted by spin flips of the remote elec-
tron in the presence of the oscillatory field, which essen-
tially causes the spin to Rabi flop, thus requiring quan-
10
0.35 0.35
k = 106 s-1 Γ = 0.05 k
k = 105 s-1 Γ = 0.1 k
k = 104 s-1 Γ = 0.5 k
d d
el el
yi yi
et 0.30 et 0.30
gl gl
n n
Si Si
0.25 0.25
0 !/8 !/4 3!/8 !/2 0 π/8 π/4 3π/8 π/2
Angle " Angle θ
FIG.9: Left: singletyieldasafunctionofθ inthepresenceoftheresonantoscillatoryfieldwithadditionallyappliednoiseat
a rate Γ=0.1k (see text). Right: The singlet yield as a function of θ, this time with k fixed at k=105 s−1 and for different
noise rates. For each noise rate, there are two curves, one with and one without the oscillatory field. However, as explained in
the text, these can only be properly distinguished for the case of weak noise (red curves).