# Quantum Dynamics of the Avian Compass

###### Abstract

The ability of migratory birds to orient relative to the Earth’s magnetic field is believed to involve a coherent superposition of two spin states of a radical electron pair. However, the mechanism by which this coherence can be maintained in the face of strong interactions with the cellular environment has remained unclear. This Letter addresses the problem of decoherence between two electron spins due to hyperfine interaction with a bath of spin 1/2 nuclei. Dynamics of the radical pair density matrix are derived and shown to yield a simple mechanism for sensing magnetic field orientation. Rates of dephasing and decoherence are calculated ab initio and found to yield millisecond coherence times, consistent with behavioral experiments.

The ability of a migratory bird to orient itself relative to the Earth’s magnetic field is at once a familiar feature of everyday life and a puzzling problem of quantum mechanics. That birds have this ability is well established by a long series of behavioral experiments. However, the precise mechanism by which an organism may sense the orientation of the weak geomagnetic field remains unclear and theoretically problematic.

Although commonly referred to as the “avian compass,” an ability to
sense the local magnetic field orientation has been observed in every
major group of vertebrates, as well as crustaceans, insects, and a
species of mollusc Wiltschko and Wiltschko (2005); Muheim (2008). For
the majority of species, the primary compass mechanism appears to be
light-activated, with a few exceptions such as the sea turtle or the
subterranean mole ratMuheim (2008). In addition to a
light-activated compass located in the eye, migratory birds are
believed to possess a separate mechanism involving magnetite, with
possible receptors identified in the beakWiltschko *et al.* (2010), the
middle earWu and Dickman (2011) and the brain
stemWu and Dickman (2012), although the existence of a receptor in the
beak has been challenged in a recent studyTreiber *et al.* (2012).
This paper addresses the light-activated mechanism, which in addition
to being widespread is also well studied by a long series of
behavioral experiments, reviewed in
Muheim (2008); Wiltschko *et al.* (2011); Ritz (2011); Johnsen and Lohmann (2005); Able (1995).

The basic parameters of the compass mechanism may be probed by confining a
bird in a conical cage during its preferred migration period
Beck and Wiltschko (1983). The restless nocturnal hopping behavior, or
Zugunruhe, will tend to orient in the preferred migration direction, and the
effects of environmental parameters can be judged by whether they affect the
bird’s ability to orient. Such experiments have established that the compass
is light activated, with an abrupt cutoff between wavelengths 560.5 and 567.5
nm Muheim *et al.* (2002), and that birds are sensitive to the orientation
of magnetic field lines but not their polarity – they cannot distinguish
magnetic north from southWiltschko *et al.* (1972). Provocatively, a
recent experiment has found that an oscillatory magnetic field oriented
transverse to the static field can cause disorientation when it is narrowly
tuned to the Larmor frequency for an electron in the static field to flip its
spin. On resonance, an oscillatory field strength of 15 nT (Rabi frequency
Hz) is sufficient to cause
disorientationRitz *et al.* (2009, 2004).

Qualitatively, such experiments are well explained by a “radical
pair” model of the avian
compassSchulten *et al.* (1978); Ritz *et al.* (2000), in which the
magnetic field drives coherent oscillations between the
singlet state and the
triplet state of an electron
radical pair formed by absorption of a photon. If the singlet and
triplet states react to form distinguishable byproducts, or can be
otherwise distinguishedStoneham *et al.* (2012), monitoring the ratio of
the byproducts probes the time spent in each state, and thus the
oscillation frequency.

The radical pair model gives an excellent phenomenological description of the
avian compass, and predicts disorientation by an on-resonance oscillatory
field. However, it remains theoretically problematic, requiring that
coherence be maintained between different spin states for very long times
despite the presence of an environment which is very hostile to this. As
observed in Gauger *et al.* (2011), the slow spin flip time
(ms) implies that the process it disrupts must be
slower still. Gauger *et al.* (2011) and Bandyopadhyay *et al.* (2012)
use similar methods to infer coherence times of s. However,
the proteins and water molecules present in a cellular environment possess
large numbers of hydrogen nuclei, each of which interact with the radical pair
via the hyperfine interaction. Somehow, the necessary quantum information
must survive such interactions long enough to give a biologically useful
signal.

Previous work considering the radical pair compass in the presence of
decoherence includes Kominis *et al.* (2009); Jones and Hore (2010); Kominis (2011); Dellis and Kominis (2011), treating effects of rapid singlet
and triplet reaction rates on the evolution of the density matrix.
Decoherence due to hyperfine interactions has been treated in terms of an
effective magnetic field in Kavokin (2009), while
Cai *et al.* (2010); Tiersch and Briegel (2012); Cai *et al.* (2012a) consider a radical
pair interacting with a small number of nuclei. The related problem of
decoherence in a singlet/triplet quantum dot has been treated in
Johnson *et al.* (2005); Petta *et al.* (2005).

This paper gives an analytic treatment of the preservation and decay of coherence for a radical pair interacting with a bath of spin 1/2 nuclei. A long lived component of the quantum information is identified, and shown to yield a simple and robust compass mechanism. Design considerations for an efficient compass are identified, and the coherence lifetime is shown to be consistent with lifetimes inferred from behavioral experiments. Atomic units are used throughout.

The evolution of the reduced density matrix for a radical pair interacting with a Markovian bath is given by the Lindblad master equation

(1) |

As a nonzero field drives states , with and , with out of degeneracy with the states, they can be omitted from this equation, although they are necessary to derive the dephasing rates . is the Lindblad superoperator corresponding to projection operator , and

(2) |

is the Zeeman Hamiltonian for the radical pair, where results
from interactions between the electrons and their immediate surroundings,
is the projection of the field onto some axis preferred
by this interaction, and . Both the receptor involved in the avian compass
Liedvogel and Mouritsen (2010); Hogben *et al.* (2009); Nießner *et al.* (2011) and
the origin of
Cai (2011); Cai *et al.* (2012a, b) are currently
unidentified; in the absence of specific knowledge, this paper simply assumes
is order 1.

As derived in the Supplementary Material, the dephasing induced by the hyperfine bath is given by two parameters, which can be found analytically. In Eq. 1, and , where is large for moderate field strengths and is zero for some orbital symmetries. Mapping the density matrix to a Bloch sphere according to , , and , where are Pauli matrices, the component of the Bloch vector will decay rapidly when is large, while the x and y components will behave as damped harmonic oscillators,

(3) |

where

(4) |

When , the system is overdamped, , and and decay more slowly as the dephasing term grows. These dynamics are illustrated in Figure 1.

Although arising from a different source, these dynamics are similar to the
quantum Zeno regime treated in Kominis *et al.* (2009); Dellis and Kominis (2011),
where fast singlet or triplet reaction rates take the place of rapid
dephasing, and to Walters (2012), where the long lived coherences
occur in photosynthetic molecules. Because the component of the Bloch
vector decays rapidly, the symmetry group of the long lived information is
rather than .

The value of
derived in the Supplementary Materials can be found for a cellular
environment by assuming a density of hydrogen nuclei equal to that of
liquid water. For B=50 T, is the number
of nuclei within radius bohr, at which the hyperfine
interaction equals the Zeeman interaction in magnitude. The dynamics
are thus strongly overdamped, with a dephasing lifetime
ps and a coherence lifetime
ms, somewhat longer than the
s-s inferred in
Gauger *et al.* (2011); Bandyopadhyay *et al.* (2012). It is this slow
loss of coherence which allows for a biologically useful signal.

As the dynamics of the radical pair are strongly overdamped, the Bloch vector will not precess about the z axis as in the original radical pair model. Rather, a Bloch vector in the equatorial plane will be frozen in place and evolve only due to decoherence. For an initially pure singlet state, , basis, so that , basis. Loss of coherence thus manifests itself as a transfer of population from singlet to triplet at a rate which varies as . Identical logic applies if the initial state is a triplet. A compass, then, requires only that the triplet reaction rate be sufficiently large to prevent backwards population transfer. Assuming such a rate, the ratio of triplet to singlet byproducts is in the , or in the

(5) |

where is the singlet reaction rate. Consistent with behavioral experiments, the compass signal is sensitive to the alignment of magnetic field lines but not their polarity.

While the identity of the avian compass receptor remains unknown, a number of design considerations may be inferred from Eq. 5 and from the dephasing dynamics derived in the Supplementary Material .

One such consideration relates to the mechanism of detecting
the formation of triplet states. While the original radical pair
model proposed a spin sensitive chemical reaction, this is not an
essential feature of the model, and more recent papers
Stoneham *et al.* (2012) have proposed that physical detection of the
triplet states may be advantageous. A possible mechanism for such
detection can be seen in Table 3 in the
Supplementary Material, which
shows that dephasing due to nuclei distant from the radical pair will
result in population transfer from state to states
, with lifetime 33 ps. As the states have
nonzero magnetic moments, they are easily distinguishable from the
states by physical means. Because equilibration between the
populations of states , and is
rapid, detection of any triplet state will suffice for the purposes of
the compass mechanism.

Second, it can be seen that the sensitivity of the compass mechanism depends greatly upon the form taken by the dephasing superoperators. An upper limit for the sensitivity of the compass mechanism may be found by considering the contrast between North/South and East/West alignment

(6) |

Here the contrast is independent of and , depending only upon the ratio of and . Figure 1 illustrates the decay of the Bloch vector for both an efficient (high contrast) and an inefficient (low contrast) compass. As is large relative to , it follows that an efficient compass receptor must have small or zero.

From table 5 in the Supplementary Material, it can be seen that will be small only in the case that it is zero by symmetry. Here the rate of dephasing due to for a nucleus far from the radical pair is inversely proportional to , the rate of decay for correlations in the environment. Thus, it is likely that an efficient compass will employ an excited state with cylindrical symmetry, thereby eliminating this term.

Similar logic can be used to compare the loss of contrast resulting from an oscillatory field tuned to the Larmor frequency with that seen in behavioral experiments. Here the oscillatory field may flip the spin of one electron in the radical pair, thereby populating states with . As the populations of , and equilibrate rapidly, the final triplet populations will be indistinguishable from those produced by the compass mechanism. As derived in the Supplementary Materials, the rate of such spin flips is . Adding this rate to and setting yields a new equation for the contrast

(7) |

which is plotted as a function of in Figure
2. Consistent with Ritz *et al.* (2004), Figure
2 shows a rapid loss of contrast as
grows from 1 to 10 nT – precisely the range in which
experiment shows a crossover from oriented to disoriented behavior.
Some inconsistency with experiment can be seen if the static field
strength is doubled – while experiment shows disoriented behavior for
=(100 T, 15 nT) and oriented behavior for (50
T, 5 nT), Figure 2 shows higher contrast
for the first case than for the second.

When the static field is doubled in the absence of an oscillatory
field, behavioral experiments Wiltschko *et al.* (2006) show temporary
disorientation lasting less than an hour, indicating that the
biological signal is affected by the field strength, but the
ability to orient is not. Here the contrast in Eq. 6 is
unaffected by the change in field strength, while the visibility
depends on the ratio of to . For
a migratory bird, which is exposed to a range of field strengths, it
may thus be advantageous to have some means of controlling , so
that the same receptor could give usable visibility at a variety of
field strengths.

The avian compass described in this paper represents a unique example of a quantum mechanical process which not only survives but is actually sustained by interaction with a surrounding bath. Although the precise identity of the receptor or receptors involved in the avian compass remains unknown, simple geometrical assumptions allow information sufficient for numerical comparison with experiment to be derived from first principles. The proposed mechanism requires neither unique properties nor elaborate manipulation of the radical pair state, and the biologically observable signal is distinctive and easy to interpret. The avian compass thus represents a simple model system for the emerging and still largely unexplored role of quantum mechanics in biological processes.

## Appendix A Supplementary Material: Dephasing Rates

The decoherence of a spin system due to interactions with a surrounding spin bath is one of the central theoretical problems associated with the avian compass. It is also a longstanding open problem in its own right Schlosshauer (2007). This Supplementary Materials section considers the decay of density matrix components arising due to hyperfine interactions between two spin 1/2 electrons and a surrounding bath of spin 1/2 nuclei. Dephasing due to the bath is treated within the Born-Markov approximation – the spin state of each nucleus is assumed to be in thermal equilibrium with the rest of the bath, to bear no memory of the previous states of the system or the bath, and to cause decoherence in the central spin system independently of the other nuclei in the bath. Having found rates of decay due to individual nuclei, rates due to the bath as a whole are found by performing a volume integral over all space assuming a constant density of nuclei per unit volume.

The hyperfine interaction between a single nucleus and a radical electron pair is given by

(8) |

where the dependence of the hyperfine interaction means that distant electrons interact with effectively distinct reservoirs, while proximate electrons interact with the same nuclei with comparable strength. As selection rules will be important in this derivation, note that the nuclear spin has different angular character than an magnetic vector field, so that the spin states coupled in this treatment may differ from effective field approaches.

Rather than treating the interactions between the nucleus and each electron separately, it is convenient to reexpress Eq. 8 in terms of the sum and the difference of the two spins. If the distance between the radical pair and a particular nucleus is large relative to the spatial extent of the radical pair and the distance between the two electrons, the hyperfine interaction with that nucleus can be broken up into two components having different angular character. Writing the spatial coordinates of the electrons as and and assuming that and , the hyperfine interaction with each nucleus can be written as the sum of a symmetric term and an antisymmetric term

(9) |

where to leading order in the small parameters and

(10) |

and

(11) |

Note that and are eigenstates of , with eigenvalues and , while states and are eigenstates of with eigenvalues .

Integrating over the spatial component of the wavefunction now leaves the hyperfine interaction in the form of a spin operator, and the coefficients of the dot products in Eqs. 10 and 11 as functions of the nuclear coordinates alone. Writing and , where is the angle between and ,

(12) |

If is separable, with well defined and ,

(13) |

where and . Note that the integral introduces a selection rule. Recalling that has angular character , with dependent upon the orientation, the Wigner-Eckart theorem gives

(14) |

where is a reduced matrix element and if and , but otherwise. Setting eliminates this term by symmetry.

Having performed these integrals, matrix elements for both components of the hyperfine interaction have the form , where for the symmetric component and for the antisymmetric component. Matrix elements of the dot product can be evaluated using a Clebsch-Gordan expansion (Woodgate (1983) Eq. 9.33)

(15) |

so that

(16) |

where the bras and kets represent eigenstates with quantum numbers . Eigenkets and corresponding indices for the basis are given in Table 1, and for in Table 2. Here the states, although losing degeneracy with the subspace in a nonzero magnetic field, must be included for the sake of second order terms in Eq. 19.

In addition to the hyperfine interaction, the system will evolve due to the influence of the Zeeman Hamiltonian , given by

(17) |

in the basis diagonalizing , and

(18) |

in the basis diagonalizing . Here the Zeeman terms involving the nuclear magneton, smaller than the Bohr magneton by a factor of , have been omitted.

Rates of dephasing due to the combined effects of the Zeeman Hamiltonian and the hyperfine interaction may now be found by calculating the evolution of the density matrix in the interaction picture

(19) |

where over short times

(20) |

Here indexes electronic states, nuclear states, and the states of the nucleus’s local environment, while ignores and includes the perturbation. For nuclei close to the radical pair, the hyperfine interaction will dominate the Zeeman interaction, so that and . For distant nuclei, the Zeeman term dominates, so that and . Rather than calculate directly, which would require diagonalizing anew for every value of , the Hamiltonian exponential is approximated by the split operator method Bandrauk and Shen (1991)

(21) |

so that

(22) |

The integrand of Eq. 19 is now given by the product of a large number of matrix exponentials multiplying the density matrix, so that each element of is given by a semi-infinite integral time integral of a large number of Fourier components. These integrals can be evaluated by imposing the Born and Markov approximations, so that

(23) |

and if , so that both and are diagonal with respect to the nuclear spin state. Here the Markov approximation is imposed by multiplying the integrand by a delta function inside the time integral, rather than simply replacing with as in Blum (2012).

The time integrals over the various Fourier components can now be evaluated using a dimensionless integral. Setting , where and are dimensionless,

(24) |

In the limit that oscillates slowly relative to the timescale on which the bath becomes Markovian, the above integral becomes

(25) |

where . Here, Eq. 24 is used for integrals over oscillating Fourier terms in Eq. 19, while Eq. 25 is used for integrals over constant terms.

Having found in terms of , the decaying components of the density matrix and their associated decay rates may be found by solving an eigenvalue equation. Tables 3, 4, 5, and 6 give these rates to second order in and first order in for both the symmetric and the antisymmetric hyperfine components, in the limits that and .

For the present purpose of calculating dynamics of the avian compass, two such rates in particular must be found. These correspond to the integrals over all space of , the rate of decay for coherence terms , , and , and , the rate of decay for population imbalances and in Table 3. As these rates apply when , the volume integral will be performed over all space outside a sphere of radius , where is the radius at which . Substituting , , , and , where is the density of nuclei per unit volume, yields integrated rates of decay for elements , , and and for elements and . and can be found by recalling that and , yielding bohr for a magnetic field of 50 T. Assuming a density of protons equal to that of liquid water yields . Thus, it is apparent that and is given by ps, so that any population transferred to the state will quickly equilibrate with the populations of states . , so that the system is strongly overdamped. The timescale for decay of population imbalances

### a.1 Rabi oscillation in the limit of strong dephasing

In Ritz *et al.* (2004), an oscillatory magnetic field tuned to the Larmor
frequency for an electron in the static geomagnetic field was found to
cause disorientation in European robins. In the body of the paper,
this was attributed to electrons flipping their spin due to the
oscillatory field, creating an alternate pathway for the formation of
triplet state population from an initial singlet state which does not
depend on the orientation of the compass molecule. Because the
populations of triplet states , and
equilibrate very rapidly, the population of triplet
states created in this way will be indistinguishable from those
created by the compass mechanism.

In the absence of dephasing, the rate of spin flips due to Hamiltonian

(26) |

Writing