Gravitational qubits

We report on the behaviour of two-level quantum systems, or qubits, in the background of rotating and non-rotating metrics and provide a method to derive the related spin currents and motions. The calculations are performed in the external field approximation.


INTRODUCTION
Spin effects in gravity straddle the boundary between quantum and classical physics. The difference between quantum and classical behaviour becomes particularly transparent with spin, which is relevant to low energy approaches to quantum gravity.
Quantum beats are an unequivocal indication that the system considered obeys the laws of quantum mechanics. Quantum systems with a two-dimensional Hilbert space are also called qubits. This definition is borrowed from quantum computing where two-level quantum systems play a predominant role. Qubits provide examples of systems that are genuinely quantum mechanical and, at the same time simple, can be studied within the confines of first quantization and are ideal in the study of relativistic gravity close to, or at the quantum level. There are gravitational qubits in the universe. Some of them are disussed below.
In what follows, use is made of the external field approximation [1,2] that treats gravity as a classical theory when it interacts with quantum particles. This approximation can be applied successfully to all those problems involving gravitational sources of weak to intermediate strength for which the full-fledged use of general relativity is not required [3,6,[8][9][10][11][12], it is encountered in the solution of relativistic wave equations and takes different forms according to the statistics obeyed by the particles [6,7,[13][14][15]. The approximation can also be applied to theories in which acceleration has an upper limit [16][17][18][19][20][21][22][23][24][25] and that allow for the resolution of astrophysical and cosmological singularities in quantum gravity [26,27]. It is of interest to those theories of asymptotically safe gravity that can be expressed as Einstein gravity coupled to a scalar field [28], and can produce results complementary to those of the method of space-time deformation [29].
By multiplying (3) on the left by (−iγ ν (x)∇ ν − m), we obtain the equation whose solutionΨ is exact to first order. The operatorΦ G (x) is defined aŝ and Ψ 0 (x) satisfies the usual free Dirac equation In (4) and (7), the path integrals are taken along the classical world line of the particle starting from an arbitrary reference point P . Only the path to O(γ µν ) needs to be known in the integrations indicated because (4) already is a first order solution. The positive energy solutions of (8) are given by where N = E+m 2E , u + u = 1,ū = u + γ 0 , u + 1 u 2 = u + 2 u 1 = 0 and σ = (σ 1 , σ 2 , σ 3 ) represents the Pauli matrices.. In addition φ can take the forms φ 1 and φ 2 where φ 1 = 1 0 , and φ 2 = 0 1 .
L αβ andk α are the angular and linear momentum operators of the particle. It follows from (6) and (4) that the solution of (1) can be written in the form [6] Ψ and also as where Φ T = Φ s + Φ G is of first order in γ αβ (x). The condition that both sides of the equation agree when the gravitational field vanishes accounts for the presence of the factor −1/2m on the r.h.s. of (11). On multiplying (1) on the left by (−iγ ν (x)D ν − m) and using the relations and we obtain the equation In (13) and (14), R αβλµ (z) = − 1 2 (γ αλ,βµ + γ βµ,αλ − γ αµ,βλ − γ βλ,αµ ) is the linearized Riemann tensor, R the corresponding Ricci scalar, and σ αβ ( By using Eq. (4), we also find Eq. (14) implies that the gyro-gravitational ratio of a massive Dirac particle is one, as found in [45][46][47]. The transformations of coordinates x µ → x µ +ξ µ , with ξ µ (x) small of first order, lead to the "gauge" transformations γ µν → γ µν − ξ µ,ν − ξ ν,µ . It is therefore necessary to show that Φ T in (11) is gauge invariant. In fact, on applying Stokes theorem to a closed spacetime path C and using (12), we find that Φ T changes by where Σ is a surface bound by C and J αβ is the total angular momentum of the particle. Eq. (16) shows that (10) and (11) are gauge invariant and confirms that, to first order in the gravitational field, the gyro-gravitational ratio of a Dirac particle is one. Use of (10), or (11) assures the correct treatment of both spin and angular momentum. The plan of this work is as follows. In Section 2 we discuss qubits represented by particles in accelerators. In Section 3 we derive the gravitational deflection of particles propagating in a gravitational background represented by the Lense-Thirring metric and obtain the contribution due to the rotation of the source. The neutrino helicity transitions are derived in Section 4 and some astrophysical consequences are discussed in Section 5. Spin currents and spin motion are presented in Section 6 and are followed by a summary.

SPIN-ROTATION COUPLING IN ACCELERATORS
The spin-rotation effect described by Mashhoon is conceptually important, it extends our knowledge of rotational inertia to the quantum level and violates the principle of equivalence [30,32] that is well-tested at the classical level.
It has, of course been argued that the principle of equivalence does not hold true in the quantum world. This is the case for phase shifts in particle interferometers [44,48] and wave functions depend on the masses of the particles involved [49]. In addition, the equivalence principle does not apply in the context of the causal interpretation of quantum mechanics as shown by Holland [50]. Several models predicting quantum violations of the equivalence principle have also been discussed in the literature [51,52], also in connection with neutrino oscillations [53][54][55][56][57]. The Mashhoon term, in particular, yields different potentials for different particles and for different spin states and can not, therefore, be regarded as universal. It plays, nonetheless, an essential role in precise measurements of the g − 2 factor of the muon.
The experiment [58,59] involves muons in a storage ring. Muons on equilibrium orbits within a small fraction of the maximum momentum are almost completely polarized with spin vectors pointing in the direction of motion. As the muons decay, those electrons projected forward in the muon rest frame are detected around the ring. Their modulated angular distribution reflects the precession of the muon spin along the cyclotron orbits.
Our calculations use the covariant Dirac equation and are performed in the rotating frame of the muon and do not therefore require a relativistic treatment of inertial spin effects [63] . Then the vierbein formalism yields Γ i = 0 and where a i and ω i are the three-acceleration and three-rotation of the observer and in the chiral representation of the usual Dirac matrices. The second term in (17) represents the Mashhoon effect. The first term drops out. The remaining contributions to the Dirac Hamiltonian, to first order in a i and ω i , are [33,44] − ω · L + σ 2 .
All quantities in H are time-independent and are referred to a left-handed tern of axes rotating about the x 2 -axis in the clockwise direction of motion of the muons. The muon momentum is directed along the x 3 -axis which is tangent to the muon orbits. The magnetic field is B 2 = −B. Only the Mashhoon term then couples the helicity states of the muon. The remaining terms contribute to the overall energy E of the states, and we indicate by H 0 the corresponding part of the Hamiltonian.
Before decay the muon states can be represented as where |ψ + > and |ψ − > are the right and left helicity states of the Hamiltonian H 0 and satisfy the equation The total effective Hamiltonian is µ 0 represents the total magnetic moment of the muon and µ 0 is the Bohr magneton. We will neglect the presence of electric fields that also affect the muon spin. Their effects can be controlled in suitable ways [59].
The coefficients a(t) and b(t) in (19) evolve in time according to where M is the matrix and Γ represents the width of the muon and is not particularly relevant to what follows. Equations (21) and (22 describe a two-dimensional qubit. The non-diagonal form of M (when B = 0) implies that rotation does not couple universally to matter. M has eigenvalues and eigenstates The muon states that satisfy (21), and the condition |ψ(0) >= |ψ − > at t = 0, are The spin-flip probability is therefore The Γ-term in (24) accounts for the observed exponential decrease in electron counts due to the loss of muons by radioactive decay [59]. The term in square brackets represents the well known phenomenon of quantum beats which one should expect because muons are quantum systems. It also represents the characteristic behaviour of a two-dimensional qubit.
The spin-rotation contribution to P ψ−→ψ+ is represented by ω 2 which is the cyclotron angular velocity eB m [59]. The spin-flip angular frequency is then which is precisely the observed modulation frequency of the electron counts [64] (see also Fig. 19 of Ref. [59]) and yields the value ∆ of the energy level splitting. This result is independent of the value of the anomalous magnetic moment of the particle. It is therefore the spin-rotation coupling of that gives evidence to the g−2 term in Ω by exactly cancelling, in 2µB, the much larger contribution µ 0 that one would get if the fermion had no anomalous magnetic moment. The cancellation is made possible by the non-diagonal form of M and is therefore a direct consequence of the violation of the equivalence principle.
It is perhaps surprising that spin-rotation coupling as such has almost gone unnoticed for such a long time. It is however significant that its effect is observed in an experiment that has already provided crucial tests of quantum electrodynamics and a test of Einstein's time-dilation formula to better than a 0.1 percent accuracy.
Applications of these ideas to compound spin systems like heavy ions in accelerators can be found in [60][61][62].

GEOMETRICAL OPTICS OF SPIN-1/2 PARTICLES
In this Section we study the propagation of a spin-1/2 particle in the Lense-Thirring metric [65] represented, in its post-Newtonian form, by where and M, R, ω = (0, 0, ω) and J are mass, radius, angular velocity and angular momentum of the source. The vierbein field to O(γ µν ) is It can further isolate the gravitational contribution in (28) by writing e μ α ≃ δ μ α + h μ α . The components of the spin connection can be calculated using (2) and (28) and are and have the explicit form In what follows, use is made of the Dirac representation of the γμ, of the first derivative of Φ G with respect to x µ and of the second derivative where Γ α µν are the Christoffell symbols of the second type. For the Lense-Thirring metric and to O(γ µν ), these are In the geometrical optics approximation |∂ i γ µν | ≪ |kγ µν |, where k is the momentum of the particle, the geometrical phase Φ G is sufficient to reproduce the classical angle of deflection, as it should, but also some effects due to the angular velocity of rotation of the source.
The deflection angle δ is defined by where p ⊥ and p are the orthogonal and parallel components of the momentum with respect to the initial direction of propagation. In the weak field approximation tan δ ≃ δ and (34) reduces to where k = p is the unperturbed momentum and . It is clear from (7) and (10) that, once Ψ 0 (x) is chosen to be a plane wave solution of the flat spacetime Dirac equation, the geometrical phase of a particle of four-momentum k µ is given by The components of p ⊥ can be determined from the equation We consider the two cases of propagation along the z-axis, which is parallel to the angular momentum of the source, and along the x-axis, orthogonal to it. In both instances, the fermions are assumed to be ultrarelativistic, i.e.
We consider fermions starting from z = −∞ with impact parameter b ≥ R and propagating along x = b, y = 0. We find From (35), we finally obtain which is the deflection predicted by general relativity for photons, with corrections due to the fermion mass and to ω. In the limit z → ∞ Eq. (41) reduces to Propagation along x In this case φ 2 = 1 (i.e. φ 1,2 are eigenstates of σ 1 ) and the fermions start from x = −∞ with impact parameter b. For simplicity, we consider particles propagating in the equatorial plane z = 0, y = b. We find and It then follows that where the first term is the one predicted by general relativity. Contrary to the case of propagation along z, the contribution of ω does not vanish in the limit x → ∞. In fact, in this limit we get

NEUTRINO HELICITY TRANSITIONS
In what follows, it is convenient to write the left and right neutrino wave functions in the form where σ = (σ 1 , σ 2 , σ 3 ) represents the Pauli matrices. ν L,R are eigenvectors of σ · k corresponding to negative and positive helicity andν 0 L,R (k) ≡ ν † 0 L,R (k)γ0, ν † 0 L,R (k)ν 0 L,R (k) = 1. This notation already takes into account the fact that if ν ± are the helicity states, then we have φ 2 ≃ ν − , φ 1 ≃ ν + for relativistic neutrinos. The propagation is in vacuo.
In general, the spin precesses during the motion of the neutrino. This can be expected because of the presence of Φ s in Φ T .
We now study the helicity flip of one flavour neutrinos as they propagate in the gravitational field produced by a rotating mass. The neutrino state vector can be written as where |α| 2 + |β| 2 = 1 and λ is an affine parameter along the world-line. In order to determine α and β, we can write Eq. (10) as and |ψ 0 (λ) is a plane wave solution of (9). The latter can be written as Strictly speaking, |ψ(λ) should also be normalized. However, it is shown below that α(λ) is already of O(γ µν ), can only produce higher order terms and is therefore unnecessary in this calculation. From (49), (50) and (52) we obtain An equation for β can be derived in an entirely similar way. If we consider neutrinos which are created in the left-handed state, then |α(0)| 2 = 0, |β(0)| 2 = 1, and we obtain whereẋ µ = k µ /m. As remarked in [66],ẋ µ need not be a null vector if we assume that the neutrino moves along an "average" trajectory. We also find, to lowest order, where Γ β αµ are the usual Christoffel symbols, and In order to solve the evolution equations for α and β and complete the equations describing this two-dimensional qubit, one also needs the terms ν L |ẋ µ ∂ µT |ν L and ν R |ẋ µ ∂ µT |ν R of the usual qubit matrix M . In what follows, we compute the probability amplitude (56) for neutrinos propagating along the z and the x directions explicitly.

Propagation along z
For propagation along the z-axis, we have k 0 = E and k 3 ≡ k ≃ E(1 − m 2 /2E 2 ) and we choose y = 0, x = b. We find Summing up, and neglecting terms of O(m/E) 2 , Eq.(56) becomes The contributions to O((E/m) 2 ) vanish. As a consequence and the probability amplitude for the ν L → ν R transition is of O(m/E), as expected.
It also follows that In this qubit, the first term in (61) comes from the mass of the gravitational source. The second from the source's angular momentum and vanishes for r → ∞ because the contribution from −∞ to 0 exactly cancels that from 0 to +∞. In fact, if we consider neutrinos propagating from 0 to +∞, we obtain According to semiclassical spin precession equations [67], there should be no spin motion because spin and ω are parallel. The probabilities (61) and (62) mark therefore a departure from expected results. They yield however results that are small of second order. Both expressions vanish for m → 0, as it should because helicity is conserved [68]. It is interesting to observe that spin precession also occurs when ω vanishes [69,70]. In the case of (61) the mass contribution is larger when b < (r/R) 5r 2ω , which, close to the source, with b ∼ r ∼ R, becomes Rω < 5/2 and is always satisfied. In the case (62), the rotational contribution is larger if b/R < 2ωR/5 which restricts the region of dominance to a strip about the z-axis in the equatorial plane, if the source is compact and ω is relatively large.
In proximity of the source where the gravitational field is stronger and r ∼ b ∼ R the evolution equations for α and β are dα dλ ≈Dβ and dβ dλ ≈D * α,D is almost constant and α and β oscillate with frequency The contribution of the source rotation is therefore (ωb/c) 2 ∼ (v/c) 2 . The helicity oscillations discussed are, in principle, relevant in astrophysics because right-handed neutrinos are considered sterile. This point is discussed in the next section.
Propagation along x In this case, we put k 0 = E, k 1 ≡ k ≃ E(1 − m 2 /2E 2 ). The calculation can be simplified by assuming that the motion is in the equatorial plane with z = 0, y = b. We then have Summing up, and neglecting terms of O(m/E) 2 , Eq.(56) becomes The contributions to O((E/m) 2 ) again vanish and we get Integrating (66) from −∞ to x, we obtain and Obviously, the contribution of M is the same as for z-axis propagation. However, the two cases differ substantially in the behaviour of the term containing ω. In this case, in fact, the term does not vanish for r → ∞. If we consider neutrinos generated at x = 0 and propagating to x = +∞, we find The M term is larger when 2ωR 2 5b < 1. At the poles b ∼ R and the M term dominates because the condition ωR < 5/2 is always satisfied. The angular momentum contribution prevails in proximity of the equatorial plane. The transition probability vanishes at b = 2ωR 2 /5.
An altogether different type of qubit is represented by neutrino flavour oscillations. They have been discussed in the context of the Lense-Thirring metric in [6]. The qubit frequency is in this case proportional to ∆m 2 = m 2 2 − m 2 1 , where m 1 and m 2 are the masses of the neutrino mass eigenstates.

Neutrino conversion in supernovae
The results of the previous section may be applied to the propagation of a beam of neutrinos in vacuo. The presence of a medium is realized by means of a potential V . The neutrinos are massive and may therefore have a magnetic moment µ. In the presence of an external magnetic field B and of the Mashhoon term proportional to the angular velocity of the source Ω, the evolution equations become [35,71] where ν = ν R ν L and P ± = (1 ± σ 3 )/2 are the R and L projection operators. Eq.(70) leads to neutrino oscillations.
The frequency of oscillation is then Ω ⊥ /2 where Ω ⊥ is the component of the angular velocity normal to the neutrino trajectory. In particular, if a beam of neutrinos consists of N L (0) particles at z = 0, the relative numbers of ν L and ν R at z will be These oscillations are interesting because the ν R 's, if they exist, do not interact. They would therefore provide an energy dissipation mechanism with possible astrophysical implications. The conversion rate is not large for galaxies and white dwarfs. In fact one can obtain from (71) N R ∼ 10 −6 N L (0) for galaxies of size L for which Ω ⊥ L ∼ 200km/s. Similarly, for white dwarfs for which Ω ⊥ ∼ 1.0s −1 , one finds N R ∼ 10 −4 N L (0). On the other hand, the ν L 's diffuse out of a canonical neutron star in a time 1 to 10s, during which they travel a maximum distance 3 × 10 9 cm between collisions. This and the fact that for a millisecond pulsar the conversion rate ν L → ν R is ∼ 0.5 at distances L ∼ 5 × 10 6 cm suggest that the dynamics of the star could be affected by such a cooling mechanism. Indeed the star may even cool too rapidly at higher rotational speeds for a pulsar to form. The magnetic moment of the neutrino does not appear in the calculations because magnetic spin-flip rates of magnitude comparable to (71) would require magnetic moments in excess of the value µ ∼ 10 −19 µ B mν 1eV predicted by the standard model.
The behaviour of neutrinos in a medium is modified by a potential V that vanishes for ν R 's. In the core of a supernova V can be written as [72] V (ν e ) = 14eV ρ ρ c y( r, t), where y( r, t) ≡ 3Y e ( r, t) + 4Y νe ( r, t) − 1, the Y 's represent the lepton fractions present and ρ c = 4 × 10 14 g/cm 3 . For supernovae V can be large, of the order of several electron volts, and rotation may be neglected. Only the acceleration term in (71) need be considered and the effective Hamiltonian then has the form [71] where and a ⊥ is the component of the acceleration transverse to the neutrino trajectory. If the initial state is pure ν L and the number of particles in this state is N 0 at z = 0, then the corresponding numbers of ν L and ν R at z are N L = N 0 cos 2 (Ωz) + cos 2 (2θ a ) sin 2 (Ωz) , where >h a ⊥ 2c , spin precession is strongly suppressed and the flux of particles at z consists mainly of ν L 's. The conversion takes place at resonance if V = c 4 δm 2 2E . Summarizing, the components of acceleration transverse to the particle path couple to its spin. This and the Mashhoon term applied to massive neutrinos, produce ν L ↔ ν R oscillations which may have macroscopic effects if the ν R 's are sterile, as frequently assumed. In fact ν L → ν R conversion by rotation-spin coupling may help to explain why pulsars of period shorter than a millisecond are relatively rare.

SPIN CURRENTS
The realization that the flow of spin angular momentum can be separated from that of charge has recently stimulated intense interest in fundamental spin physics [73], particularly in view of its applications [74,75].
In this section we study the generation and control of spin currents by rotation and acceleration [3]. In this context the fundamental tool still is the covariant Dirac equation.
We use the first order solutions of (1) that have the form where Ψ 0 (x) is a solution of (8) and the operatorT is given by (10), or (11). When acceleration and rotation are present, γ µν is given by [33,44] where a and Ω represent acceleration and rotation respectively. To first order the tetrad is given by from which the spinorial connection can be calculated in the usual way. The result is Γ i = 0 and Γ 0 = − 1 2 a i σ0î− 1 2 Ω·σI. For electrons, u 1 corresponds to the choice φ = φ 1 and u 2 to φ = φ 2 . Substituting in (9), one finds the spinors u 1 and u 2 . These are not eigenspinors of the matrix Σ 3 = σ 3 I and do not, therefore, represent the spin components in the z-direction. They become however eigenspinors of Σ 3 when k 1 = k 2 = 0, or when k = 0 (electron rest frame).
The appropriate way to determine whether there is transfer of angular momentum between the external non-inertial field and the electron spin is to use the third rank spin current tensor [76] that in Minkowski space satisfies the conservation law S ρµν , ρ = 0 when all γ αβ (x) vanish and yields in addition the expected result S ρµν = 1 2ū 0 σμνu 0 in the rest frame of the particle. Writing σ µν (x) ≈ σμν + h μ τ στν + h ν τ σμτ , using the relation Φ G,µν = k α Γ α µν and substituting (76) and (11) in (79) one obtains, to O(γ αβ ), It is therefore possible to separate S ρµν in inertial and non-inertial parts. The first term on the r.h.s. of (80) gives the result expected when k = 0 and the external field vanishes. From (80) one finds where terms containing Γ 0,0 = 0 and ∂ α ∂ β h μ ν = 0 have been eliminated. It therefore follows that the external field invalidates the simple conservation law ∂ ρ S ρµν = 0 and that there is in this case continual interchange between spin and orbital angular momentum. The result is entirely similar to that found for external electromagnetic fields [76]. Thus, in principle, one can use non-inertial fields to generate spin currents. In the rest frame of the particle and when Ω = (0, 0, Ω), one finds and ∂ i S i13 = ∂ i S i23 = 0. In (82) u 0 corresponds to u 1 . The direct coupling of the non-inertial field to the particle's spin current violates the law ∂ ρ S ρµν = 0. Conservation is restored if the parameters Ω, or a 2 , or both vanish.

Spin motion
Qubits appear in the actual spin motion. In general, transfer of angular momentum between external and noninertial fields occurs when the operatorT has some non-diagonal matrix elements. If in fact at time t = 0 a beam of electrons is entirely of the u 2 variety, at time t the fraction of u 1 is | u 1 |T |u 2 | 2 . The last expression becomes, along the electron world line, where, as usual,ẋ µ = k µ /m and λ. From [6] one can see that A useful way to visualise the spin motion of under the action of rotation and acceleration follows from the Mashhoon term H M = −Ω · s, where s = σ 2 , and from 1 2 a i σ0î = i 2 a i αî. The two interaction terms lead to the first order equation of motion [77] Note that A µ , introduced by writing Φ T = Φ S + Φ G + Φ EM , where Φ EM = e x P dz λ A λ (z), does not contribute to (85) because u 1 |u 2 = 0. Note also and that the terms ie(h μ α γαk µ + γμΦ G,µ )A ν k ν m drop out, in the particle rest frame, on account of u 1 |γμ|u 2 = 0. No mixed effects of first order in rotation or acceleration and first order in the electromagnetic field are therefore present in this calculation. This applies to all terms containing the magnetic field B, like the Zeeman term, and electric fields, like the spin-orbit interaction, that are present in the lowest order Dirac Hamiltonian that can be derived from (1) [44]. To O(γ µν ), contributions to (85) from the electromagnetic field are present in the actual determination of the electron's path, as stated above.
From (85) one obtains where k 0 ≡ E. The parameter k µ corresponds to the electron four-momentum when Ω = 0 and a = 0. Some general conclusions can be drawn from (87). i) If k 3 = 0, the particles move in the (x, y)-plane. If, in addition, Ω 1 = Ω 2 = 0, then A 12 = 0 only if a 3 = 0. ii) If, however, a is also due to rotation, the conditions Ω 1 = Ω 2 = 0 imply a 3 = 0 and therefore A 12 = 0. This is the relevant case of motion in the (x, y)-plane with rotation along an axis perpendicular to it. One cannot therefore have a rotation induced spin current in this instance.
iii) Even for k = 0 one can have A 12 = 0 if one of Ω 1 and Ω 2 does not vanish. This is a direct consequence of the spin rotation interaction or Mashhoon term contained in (1) [20,21,44].
A few examples are discussed below.
Consider an electron wave packet moving along the x-axis of a frame rotating about the same axis. Then a = 0 and the remaining parameters are k = (k, 0, 0) and Ω = (Ω, 0, 0). While the electron propagates along x, u 1 and u 2 propagate in opposite directions along x because of (85) and (86). For a beam the spin current generated by rotation is therefore I s = I ↓ −I ↑ , with obvious meaning of the symbols. One finds P 2→1 = | iΩ(E+m) 4m t 0 dt 1 − k 2 (E+m) 2 | 2 = ( Ωt 2 ) 2 which holds for t ≤ 2/Ω on account of the requirement P 2→1 ≤ 1.
where R is the radius of the circle described by the wave packet in the plane z = 0. The motion of the center of mass of the wave packet is helical and so is the motion of the spin components which propagate, however, in opposite directions giving rise to a spin current.
In the case of motion occurring in the plane z = 0 (k 3 = 0) and Ω = (Ω, 0, 0), one also gets a = (−Ω 2 x, 0, 0) and where k 1 = k cos ωt, k 2 = k sin ωt and ω = eB mγ is the cyclotron frequency of the electrons along the circular path determined by the constant magnetic field B. Substituting (88) in (83) one finds which holds for all t for which P 2→1 ≤ 1. While both spin up and down electrons move on a circle of radius R about the z-axis, they propagate in opposite directions because of the spin-rotation coupling, thus generating a spin current. Summarizing, rotation and acceleration can be used to generate and control spin currents. This follows from the covariant Dirac equation and its exact solutions to O(γ µν ). To this order, external electromagnetic fields can be taken into account through the particle motion. The transition amplitude for the conversion spin-up to spin-down is proportional to A 12 and is expressed as a function of Ω, a and of the electron four-momentum k µ before the onset of rotation and acceleration. The same expression suggests criteria for the generation of spin currents and the transfer of momentum and angular momentum to them. No energy can, of course, be transferred from the non-inertial fields to the spin currents as long as γ µν remains stationary.
The particular forms of (87) discussed above provide additional examples of gravitational qubits.

SUMMARY
Gravitational qubits are the simplest quantum systems that can be employed to study gravity. They exist in the laboratory and astrophysical conditions. Attention has been focussed on spin-1/2 fermions because of their simple eigenstate structure. Spin-flip transitions occur in nature frequently. We have considered here some of those that are characterized by sustained oscillations.
To the laboratory belong particles rotating in accelerators. Their quantum beats refer to spin oscillations mediated by the Mashhoon spin-rotation interaction. They have been observed and play an important role in important measurement of the anomalous magnetic moment of the muon. Section 3 is entirely devoted to the calculation of the deflection of fermions in a gravitational background described by the Lense-Thirring metric. The interest is limited here the action of rotation on the particle spin. The procedure can be also applied to bosons. Basically, the part of the deflection that does not depend on rotation is that predicted by general relativity. Rotation of the source yields in general additional corrections which are due to the particle spin and are therefore quantum mechanical. These, for instance, are present in the deflection equations (41) and (46), but with a noticeable difference: in the case of (46) the contribution of the source angular momentum does not vanish in the limit x → ∞.
The neutrino helicity oscillations in vacuo described in Section 4 are small, but intriguing because ν L evolve into ν R which are sterile. This energy dissipation mode could be relevant to compact astrophysical objects. The introduction in Section 5 of a medium in which neutrinos propagate increases the ν L → ν R transition probability if the medium potential is attuned to the difference of the mass squared of the neutrino mass eigenstates. The fact that pulsars of period shorter than a millisecond are rare, lends support to the dissipation mechanism implied by the ν L → ν R conversion by rotation-spin coupling.
Spin currents and spin motion are introduced in Section 6. It is shown that gravitational fields violate the usual formulation of the spin conservation law. Some particular qubits are discussed together with the associated spin currents. We also consider the case of fermions accelerated to the maximal acceleration limits contemplated by Caianiello [16][17][18]. The calculations [78] confirm that continual interchange between spin and angular momentum can occur in this instance, but only if the acceleration is time-dependent. This requires a transfer of energy from a very compact star, or a black hole and the particle. Even in the case, uniform acceleration produces no observable effects on the particle spin, in agreement with [79].
Near-neighbour momenta oscillations of a gas of particles in a gravitational field are discussed in [80].