Von Neumann spin measurements with Rashba fields

We show that dynamics in spin-orbit coupling field simulates the von Neumann measurement of a particle spin. We demonstrate how the measurement influences the spin and coordinate evolution of a particle by comparing two examples of such a procedure. First example is a simultaneous measurement of spin components, $\sigma _{x}$ and $\sigma _{y}$, corresponding to non-commuting operators, which cannot be accurately obtained together at a given time instant due to the Heisenberg uncertainty ratio. By mapping spin dynamics onto a spatial walk such a procedure determines measurement-time averages of $\sigma _{x}$ and $\sigma _{y}$, which already can be precisely evaluated in a single short-time measurement. The other, qualitatively different, example is the spin of a one-dimensional particle in a magnetic field. Here the outcome depends on the angle between the spin-orbit coupling and magnetic fields. These results can be applied to studies of spin-orbit coupled cold atoms and electrons in solids.


Introduction: The spin-orbit coupling Hamiltonian and von Neumann measurement.
Unusual properties [1,2,3,4,5], simplicity, and ability to manipulate the strength [6] put the spin-orbit coupling (SOC) in the focus of many research fields, where the mutual dependence of spin and coordinate motion is important [7]. For several decades, solids and solid-state structures were the systems to study the effects of spin-orbit coupling for electrons [8] and holes [9]. Recently, at least two completely new classes of systems with spin-orbit coupling were discovered and became new research fields. The first class is cold atoms in highly coherent laser fields [10,11,12,13,14,15,16]. For both types of particles, in addition to the SOC, an effective magnetic field can be produced optically. Studies of spin-orbit coupled cold atoms are related to the properties of new phases (see e.g. [17,18,19]) and macroscopic spin dynamics [20,21,22]. The other class is topological insulators [23], considered as promising elements for the spintronics applications.
The very general character of the spin-orbit coupling should have consequences for the fundamental quantum mechanics of single particles and their ensembles and stimulate the search for these consequences in experimentally realizable system. Motivated by the interest in the observability of the effects of spin-orbit coupling on the basic quantum mechanical level, here we show that and how it is directly related to the quantum measurement of spin 1/2 in terms of the procedure proposed by von Neumann. We begin with a general Hamiltonian (assume ≡ 1) of spin-orbit coupling linear in the two-dimensional particle momentum: wherek x = −i∂/∂x−A x andk y = −i∂/∂y−A y with corresponding components of gauge potential A x,y , M is the effective mass, and σ is the Pauli matrices pseudovector. Here α(t) is spin-orbit coupling parameter, in general time-dependent, ν is a two-dimensional vector of unit length, h x and h y are the unit vectors corresponding to the type of spinorbit coupling, ∆ is the Zeeman splitting, and b is the unit vector of the direction of magnetic field. The conventional Rashba coupling is given by ν = (1, 1) / √ 2, h x = y, h y = −x, while the Dresselhaus couping is given by ν = (1, 1) / √ 2, h x = x, and h y = −y (x and y are the corresponding coordinate system vectors). The value of α is in the range from 1-10 cm/s for cold atoms to 10 6 cm/s for electrons in semiconductor nanostructures and 10 7 -10 8 cm/s for and topological insulators, where the kinetic energy vanishes [23], and surfaces with extremely strong spin-orbit coupling [24,25,26,27].
As a result, even at ∆ = 0, spin rotates around the axis which direction depends on k. The operator of particle's velocity depends on the orientation of the particle's spin, The velocity components do not commute with each other if the cross product of h x and h y is not zero. Even if it is zero, but h x , h y , and b are not collinear, the velocity components do not commute with the total Hamiltonian, complicating strongly the orbital dynamics. Now we can see a connection between Hamiltonian (1) and a quantum spin measurement. Consider a quantum system characterized by a multicomponent operator Ô x ,Ô y ,Ô z coupled to momentumP of another system, "pointer", as κ jiPjÔi , where κ ji is the corresponding coupling strength. This coupling is presented as (−i∂/∂X j )Ô i , whereX is the pointer position, and the Hamiltonian for the free pointer is H pnt (P,X). This coupling causes evolution of the quantities, characterized by operator O and, at the same time, makes the corresponding dynamics visible by mapping it on the pointer position in the coordinate space. Thus, by tracing the pointer position, one can expect tracing the motion of the operator components ofÔ i . In the case of spinorbit coupling,P is the particle momentum, andÔ represents the spin components, as can be seen from Eqs.(1)-(3). This simple observation, being the idea behind the von Neumann measurement, can have interesting consequences, including entanglement of the spin and coordinate degrees of freedom and, correspondingly, a spin dephasing in the measurement procedure, for experimentally realizable systems. We will describe these effects in this paper. Moreover, we will demonstrate, that the dynamics of the pointer due to the Hamiltonian H pnt (P,X), e.g., its kinetic energy, influences the measurement procedure and its accuracy. This approach corresponds to the measurements by solving dynamical models, with some of them recently reviewed in Ref. [28].
To perform a spin measurement, we choose the spin-orbit coupling α(t) switched on for a finite time interval making the Hamiltonian time-independent during the measurement. As a result, in the measurement procedure, evolution of the initial state Ψ(r|0), where r is the position, and Ψ(r|t) is the two-component spinor wavefunction is given by: For a translational invariant system of our interest, it is convenient to represent Eq. (6) in the form: Here D is the system dimensionality, G (r − r ′ |t) is the 2×2 Green function for the total Hamiltonian in Eq.(1) (see Ref. [29] for an example), and G (k|t) and Ψ(k|0) are the corresponding Fourier components. During the measurement time T , the von Neumann pointer, that is the coordinate of the particle, provides information about the motion of its spin components. Here the mapping onto the coordinate motion makes visible otherwise hidden spin dynamics over all possible Feynman paths in the spin subspace. The rest of the paper is organized as follows. In Section 2 this approach will be applied to a simultaneous measurement of noncommuting spin components for a two-dimensional particle with the Rashba spin-orbit coupling. We will show why the attempt of instantaneous measurement fails, and that the actual measured quantities are the averaged over the measurement time spin components. In Section 3 we study the von Neumann measurement and measurement-induced dephasing of a spin rotating in a magnetic field for a one-dimensional particle with non-commuting spin-orbit and Zeeman terms. In Section 4 we summarize the results and show possible extension and generalizations of the relation between spin-orbit coupling and spin measurement procedures.

Measurement, Feynman paths, and outcome
The Heisenberg uncertainty relation established the limit on precision of instantaneous measurement of observables corresponding to two non-commuting operators in terms of the expectation value of their commutator. Although this general statement is one of the basic properties of quantum motion, the measurement procedure itself is still an unresolved issue [30,31,32,33,34,35] even for momentum and coordinate observables. Arthurs and Kelly [30] considered two meters employed to measure jointly particle's position and momentum. The analogue of the Arthurs-Kelly experiment for non-commuting spin components was proposed in Ref. [32]. An implementation of such a measurement through coupling to radiation modes was considered in Ref. [34] and through coupling of quantum spin to classical Ising states, in Ref. [35]. However, the straightforward interpretation of proposed experiments is hardly possible. In this section we remind the reader the measurement scheme using spin-orbit coupling [36], where coordinate pointers are attached to non-commuting spin components. For this purpose we return to the Hamiltonian (1) and study spin dynamics of a single wavepacket in the absence of external magnetic field.
Without loss of generality, we choose the spin-orbit coupling in the conventional Rashba form. Neglecting the internal dynamics of the pointer (exact condition will be given further in the text) we obtain the Schrödinger equation with an initial condition: where Ψ(x, y|t) and ξ are two-component spinors (we will usually employ the representation of σ z − eigenstates), ψ 0 (x, y) is the initial wavepacket, and α ≡ α/ √ 2.
Equation (8) describes spin 1/2 coupled to two von Neumann pointers [37] with positions x and y, respectively, in attempt to measure spin components σ y and σ x simultaneously. Considering first measurement of a single spin component, σ y , by choosing for a moment H so = −i α∂ x σ y , offers a useful insight. First, we present initial state as Ψ(x, y|0) = ψ 0 (x, y) (ζ 1 |1 y + ζ 2 | − 1 y ) , where |1 y and | − 1 y are the eigenstates of σ y with the corresponding eigenvalues. The operator exp(− αt∂ x σ y ) splits this initial state into two components traveling along the x-axis with opposite speeds α as ζ 1 ψ 0 (x − αt, y)|1 y + ζ 2 ψ 0 (x + αt, y)| − 1 y . If the wave packets are well separated, and the particle is found at a location x, x/ αt approximates the value of σ y = ±1. The accuracy of the approximation depends on the width of the ψ 0 (x, y) -for a very narrow initial distribution we would only have x ≈ αt or x ≈ − αt, realizing a conventional von Neumann measurement of a single spin component.
However, our case is more complicated. Since σ x and σ y , and, thus, v y and v x , do not commute, the pointer in Eq.(8) does not have a well defined two-component velocity.
To study its motion, we slice the time interval [0, T ] into L subintervals ǫ = T /L, and take the limit by the Lie-Trotter formula [38] where with σ i |m i = m|m i and i = x, y, corresponding to all possible, hidden in the absence of spin-orbit coupling, virtual Feynman walks on an infinite lattice x(j x ) = j x αǫ, y(j y ) = j y αǫ, j x , j y = . . . − 1, 0, 1, . . . reminiscent of the Feynman's checkerboard for a Dirac electron [39]. In every time step the particle moves forwards or backwards along the x-and y-axis, and its position at the end of the measurement, t = T , is determined by the differences, ∆n x and ∆n y , between the numbers of forward and backward steps taken in each direction or, more precisely, by the interference between all paths sharing the same ∆n x , and ∆n y (see Fig.1(a)). Next we assign values m i (l) = ±1, l = 1, . . . , L to σ i in each step and define: As a result, finding the pointer at a location (x, y) determines time averages of the spin components, σ x T and σ y T , defined for the spin-space Feynman paths ( Fig.1(b)) as: to an accuracy determined by the position spread of the initial ψ 0 (x, y). In this Figure, we characterize both spins by a single function The inverse is given by: m x (l) = signM(l), and m y (l) = (−1) M(l) signM(l).

Transition amplitudes and expectation values of observables
To study the details of the measurement procedure, we begin with introducing the radius R so ≡ αT and note that a particle initially localized at the origin would never leave the 'allowed' circle r ≡ (x 2 + y 2 ) 1/2 ≤ R so . To study the coupled evolution, we define the initial state as the spatial Gaussian, corresponding to the particle release from the ground state of a harmonic potential: and the matrix U(x, y|T ; ψ 0 ), such that at the end of the measurement (in the following we omit the explicit dependence on the initial state ψ 0 (x, y)) Using Eqs. (6) and (7) for the Hamiltonian corresponding to Eq.(8), we find that in the cylindrical coordinates U(r, ϕ|T ) is a Hermitian matrix with where J n (z) is the Bessel function of the first kind of order n and ϕ is the angle between r and the x-axis [40]. As a results, the initial state shows a two-dimensional spread and after the measurement produces the density Ψ † (r, ϕ|T )Ψ(r, ϕ|T ) = |Ψ 2 1 (r, ϕ|T )| + |Ψ 2 2 (r, ϕ|T )|, concentrated in a ring of a radius R so , width ≈ w and dependent on the angle ϕ. The large argument asymptotes of the Bessel functions [41] J n (kr) ∼ 2 πkr show that in the limit w → 0 the integrals become singular as r approaches R so , and the measurement is possible only with a finite accuracy at nonzero width. It is worth mentioning that an additional obstacle to a highly accurate measurement appears due to a finite mass of the particle, where packet spreads with the characteristic speed of the order of v sp = 1/wM. Thus, the conditions of the precise measurement can be formulated as: (i) α ≫ v sp , to assure that the wavepacket dynamics is due to the spin-orbit coupling, not due to the broadening resulting from the finite M, and (ii) αT ≫ w to assure that the split of the wavepacket at time T is sufficient to perform the measurement.
Three implementations of this procedure can be considered. 1) For electron in semiconductor structures with typical α ∼ 10 6 cm/s, that is, in conventional units, α ∼ 10 meVnm, and M ∼ 5 × 10 −29 g, condition α ≫ v sp can be achieved for w > 100 nm, and corresponding T > 10 −11 s.
2) For topological insulators, α ∼ 10 8 cm/s ( α ∼ 1 eVnm,) with infinite M, and, therefore, the first condition α ≫ v sp is always satisfied. However, at the moment, it is difficult to prepare and control individual electron states is these systems.
3) For cold bosonic and fermionic atoms, taking α ∼ 10 cm/s ( α ∼ 10 −6 Kµm, with the relevant energy scale measured in Kelvin and the distance in micron) and fermion 40 K as an example, we obtain w > 10 −4 cm and T > 10 −5 s. We mention here that coherent many-particle spin-orbit coupled Bose-Einstein condensates with the pseudospin 1/2 such as 87 Rb can be an interesting system for the proposed measurement. Condensates with a weak effective interatomic interaction are preferable for this purpose since with the increase in this interaction, the wave packet spread may be controlled by the interatomic repulsion or attraction rather than by the initial width. As a result, the corresponding spreading rate should be taken in the above criteria of accurate measurement.
For a highly accurate measurement with w ≪ R so main contributions to the integrals in (18), (19) come from the domain where 1/R so ≪ k ∼ 1/w. Replacing the Bessel functions by their asymptotes (20), and neglecting contribution of rapidly oscillating (∼ cos(R so k)) terms in the integrand yields demonstrating that spin-coordinate entanglement is, in fact, spin-angle entanglement. For a small w, the radial function F (r|T ), obtained in Ref. [36], has maximum and a minimum close to r = R so , rapidly decreases for r > R so , and exhibits a somewhat slower decay for r < R so , as shown in Fig.2. Figure 1 corresponds to a less accurate measurement, where R so and w are of the same order of magnitude.
Although the probability density is concentrated in a narrow vicinity of R so , the broad angular distribution in Eq.(21) appeared as a result of the measurement, has a relatively broad distribution of moments: x 2 (T ) = R 2 so /2, where σ y (0) and σ x (0) are expectation values of corresponding spin components at t = 0. The ϕ−dependent spin orientations at t = T were presented in Ref. [36].
To conclude this section, switching a Rashba coupling over a short time T simulates a von Neumann measurement of two non-commuting spin components. Here, the particle itself plays the role of a pointer which correlates its position, (x, y), with the time average of the corresponding spin components, σ x T and σ y T evaluated along Feynman paths defined for the two spin variables (Fig.1). There are infinitely many trajectories which share the same σ x T and σ y T and lead to the same pointer position. We stress here that since these time averages are not instantaneous values of the spin components, they can be measured simultaneously to any desired accuracy. Indeed, the Feynman paths shown in Fig.1 have no intrinsic time scale and can be cut into arbitrarily small ǫ/T → 0 pieces. Thus, even in the pulse limit T → 0, under condition αT = const one does not attain unique instantaneous values of the two spin components: no matter how short is T , all paths shown in Fig.1(a) contribute to the transition amplitude (17), densely filling the area with r ≤ R so , and no measurement can catch σ x and σ y simultaneously. Thus, the fact that the particle can choose an infinite set of completely different Feynman paths at any short measurement time forbids us to define sharp instantaneous values for the non-commuting spin components.

Measured quantity, Feynman paths, and Green functions
Next we consider measurement of a spin of a particle in a one-dimensional system, where the velocity operator and the Zeeman term do not commute. Such a measurement can be realized in semiconductor quantum wires [42] and narrow waveguides for the Bose-Einstein condensates [43,44] with implemented spin-orbit coupling. We take the one-dimensional version of Hamiltonian (1) as: and assume that at t = 0, Ψ(x|t) = ψ 0 (x)ξ, and b = (sin θ, 0, cos θ). We neglect the diamagnetic effects of vector potential A x,y in a one-dimensional system. In addition, we mention that for cold atoms [10,11,12,13,14,15,16] a synthetic Zeeman-like coupling can be realized without diamagnetic terms. Here, the spin-orbit coupling term αkσ z satisfies the condition of the von Neumann measurement and the resulting dynamics simulates the quantum measurement [45] of a single spin component, σ z in this case. Despite apparent simplicity, this system demonstrates rather nontrivial behavior, presented below. Although, unlike in the previous Section, only one component, is measured, the spin is not static, but undergoes a precession due to the Zeeman term in (25). With the SOC turned on, each component of the particle's momentum adds an additional field along the z-axis. As a result, the spin moves in a magnetic field, whose direction and magnitude are fundamentally uncertain. The velocity term ασ z given by the commutator i[Ĥ 1D , x], does not commute with the Hamiltonian, and wavepacket nonuniformly spreads even if the particle's mass is infinite. We begin with the general properties of this dynamics and then discuss expectation values of observables in relation to the von Neumann measurement procedure.
Following the approach from the previous Section, and neglecting for a moment the kinetic energy, we slice the interval [0, T ] into L sub-intervals of a length ǫ, apply the Lie-Trotter formula now to exp{−iǫ[αkσ z + ∆ (b · σ) /2]}, and introduce virtual Feynman paths where σ z takes a value s j = ±1 at each ǫ-interval, j = 1, 2, . . . L. The exact propagator can be expressed as a path sum. Taking L → ∞ we obtain where the operator translates the initial state by a distance α σ z T T . If ψ 0 (x) is completely localized, |ψ 2 0 (x)| = δ(x), from the final position of the pointer x one can accurately deduce the value of σ z T = x/αT . A spread in the initial positions leads to a finite error in the determination of σ z T . Finally, including the pointer kinetic energy leads to spreading of the initial wave packet ψ 0 (x) in the measurement time. The corresponding Feynman checkerboard in the (t, x) space mapping spin motion on spatial dynamics is shown in Fig.3(a).
We begin with two simple limiting cases. Case 1. The spin-orbit coupling and Zeeman terms commute, with θ = 0 as an example. The Green function is diagonal in the spin space. Here g 0 (x|t) = (M/2πit) 1/2 exp(iMx 2 /2t) is the free particle propagator [39,46], which in the limit of infinite M tends to the Dirac δ(x)−function. For a nonzero sin θ, Green function (31) is valid for short time t ≪ 1/ ∆, where the effect of magnetic field on the spin precession is still weak. Case 2. Without spin-orbit coupling, α = 0, the Green function factorizes into a free propagator in the coordinate space, and a spin part describing Larmor precession: No spin measurement can be done here: although all virtual Feynman paths in Fig.3(b) are possible and interfere, they cannot be mapped on the x−coordinate motion.
In the general case of non-commuting spin-orbit and Zeeman terms, the particle can choose all the paths shown in Fig. 3(a), where x(T ) − x(0) = α σ z T , same as discussed for the two-dimensional motion. This dynamics is complicated due to spin precession in magnetic field, leading to nonzero sum of contributions from different paths. The precession stops once a steady spin state with σ y (∞) = 0 is reached, although the wavepacket continues to move and to spread.

Finite-time measurement and coupled spin-coordinate evolution
To focus on the effect of spin-orbit coupling, we consider (if not explicitely stated otherwise) the particle of infinite mass, avoiding the packet broadening due its initial kinetic energy. The infinite mass condition is essentially the first requirement of a precise measurement, α ≫ v sp , formulated in Section 2. To illustrate the origin of the complicated character of the spin dynamics even in the infinite mass limit, we present the continuity equation: where ρ(x|t) = Ψ † (x|t)Ψ(x|t) is the probability density, and j(x|t) = Ψ † (x|t)σ z Ψ(x|t) is the spin-determined flux. The resulting local velocity strongly varies with time and coordinate leading, it turn, to the nontrivial dependence of probability and current densities. We begin with the snapshots of the probability and spin densities showing the role of the width of the packet as the measurement tool. Although our approach is general, in calculations we use (as in Section 2) the Gaussian initial wave function in the coordinate and momentum spaces: and the initial spin state: We use Eq.(7) with Hamiltonian (25) and corresponding Green functions (28)- (29) to obtain time and coordinate-dependent wave functions. Figure 4 shows the distribution of density and velocity for dimensionless time t = 2π and different packet width. Figure 5 shows the effect of packet spread, that is internal evolution of the pointer due to the kinetic energy in the Hamiltonian. The spread of the pointer state decreases the accuracy of the measurement since it considerable decreases the available range of momenta. If the mass of the particle is infinite, the range of momenta is of the order of 1/w. If the spread velocity v sp = 1/wM is nonzero, at large t the momentum spread decreases as 1/ √ wv sp t, and, therefore, fine details of the Green function become gradually smeared, and the spin measurement accuracy decreases with time since the displacement of the particle is determined not only by the spin dynamics, but also by the wave packet's spreading. This statement can be understood with the following optical analogy. If v sp = 0, the Green function is seen through a magnifying glass with a given resolution, smearing its finer details. For nonzero v sp , the Green function is seen through a diffraction grating with a relatively large period, which increases with time, smearing the details to even greater extent.
To make a comparison with the analysis of the previous Section, we calculate the expectation values of coordinate-and spin-related observables to see the relationship between them. We begin with tracing the following quantities: for the expectation value of coordinate x(t) and width of the packet w(t), respectively. Figure 6(a) presents expectation value x(t) for different initial width of the packet and the time-dependent packet width in the inset. For any width at short times we obtain x(t) = αt, when all possible paths are in the vicinity of the straight line in Fig.3(a). After some time, dependent on w, the spin explores other Feynman walks, and the dependence becomes asymptotically linear with the main term x(t) = α σ z (∞) t. The inset shows the increase in the packet width with time, where different displacements x − x ′ = αt σ z t ≤ αt are possible due to the spin precession in magnetic field. The x(t)-dependences here split and the packets start to broaden at dimensionless t close to π, where the spin makes a half-turn. We emphasize that the packet broadening and nonlinear x(t) −dependence are solely due to the noncommutativity of the Zeeman and the Rashba terms.

Spin decoherence and long measurement
To better understand the coupled coordinate-and spin dynamics we consider the evolution of spin components by tracing the quantities calculated with the spin density matrixρ(t): With the increase in t, the different evolution of spinor components Ψ 1 (x|t) and Ψ 2 (x|t) produces a spin-coordinate entanglement and makes the initially pure spin state with trρ 2 (0) = 1, a mixed one with trρ 2 (t) < 1. As a result, the spin subsystem experiences a decoherence in the measurement process, and spin moves from the initial position at the Bloch sphere i σ i (0) 2 = 2trρ 2 (0) − 1, to the inner part of the corresponding Bloch ball, where this sum is less than one. The spin-dependent velocity in a system with Hamiltonian (25) determines the general relation between expectation values of spin and coordinate as v(t) ≡ d x(t) /dt = α σ z (t) . We consider three time-dependent observables: σ ≡ σ z cos θ + σ x sin θ ≡ ( σ · b) , σ y , and σ ⊥ ≡ σ z sin θ − σ x cos θ ≡ ( σ × b) y , where the explicit tdependence is omitted for brevity. Figure 6(b) shows the spiral behavior of σ y and σ ⊥ , strongly dependent on the packet width. The decreasing with time radius of the spiral here corresponds to the decoherence in the spin subspace. The final stationary values depend on the width of the packet, that is on the accuracy of the measurement. For a narrow packet, the displacement of the stationary point from the initial one is relatively small and increases, same as the maximum radius of the spiral, with the initial width w. Now we consider asymptotic values of spin components to see the steady states, t → ∞, produced by a long measurement. At long times, C and S defined above Eq.(28) as C ≡ cos t q 2 α 2 + ∆ 2 /2 and S ≡ sin t q 2 α 2 + ∆ 2 /2 , become rapidly oscillating, on the scale of the order of 1/tα, functions of momentum q. As a result, in calculations of integrals containing bilinear forms of these functions one can use semiclassical integration rule by substituting in the integrands C 2 and S 2 with 1/2 and CS with zero [47]. Then, by using Eqs. (6) and (7), with the Green function (28), (30), (29), initial state (35), and definition of observables (39), we obtain σ (∞) in the form: cos β [q 2 cos θ + q (cos 2 θ + 1) + cos θ] + sin β sin θ cos φ(q cos θ + 1) (q + cos θ) 2 + sin 2 θ .
We can study limiting cases of Eq.(41). First limit is the weak spin-orbit coupling, that is a broad initial packet, w ≫ α/ ∆. Here the range of possible momenta goes to zero, one can substitute q = 0 in the fraction in the integral to obtain σ (∞) = cos θ cos β +sin θ cos φ sin β. For a strong spin-orbit coupling, that is for a narrow packet, one can neglect lower than q 2 powers of q and obtain σ ∞) = cos θ cos β.
It is instructive to compare the obtained asymptotic behavior with that intuitively expected in a simple case of commuting Zeeman and spin-orbit couplings, that is θ = 0. Here the overlap of the functions Ψ 1 (x|t) = ψ 0 (x−αt) and Ψ 2 (x|t) = ψ 0 (x+αt) vanishes on the time scale of the order of w/α, and the spin state becomes mixed as can be seen in Eq. (40). The expectation values of spin components σ z = cos β, and σ y = 0 are time-independent. The x−component changes from σ x (0) = sin β to σ x (∞) = 0, in agreement with (41) and (42). The spirals in Fig.6 are transformed into projection of a log connecting points (sin β, 0, cos β) and (0, 0, cos β) on the Bloch sphere and inside the Bloch ball, correspondingly.

Conclusions and possible extensions
In summary, we have shown that motion in the spin-orbit coupling fields simulates a von Neumann measurement of spin-1/2 system. The spin-orbit dynamics maps spin motion onto a spin-dependent coordinate walk, thus making distinguishable otherwise hidden spins virtual histories. The accuracy of the measurement depends on the resources available in the momentum space, that is on the spatial width of the initial state. We considered two examples of such a procedure: simultaneous measurement of two noncommuting spin components and measurement of a spin rotating in external magnetic field. In the first case, since the virtual Feynman paths can be divided into infinitely small pieces in the time domain, the produced angular density distribution is timeindependent at any measurement duration. As a result, any attempt of instantaneous measurement of non-commuting spin components fails, and the averages of the spin components over the measurement time are observed with the accuracy determined by the width of the initial wavepacket. In the second case the average of a single spin component corresponding to spin-orbit coupling axis for a particle moving in one dimension is measured. We showed that even if the mass of the particle is infinite, the initial packet broadens due to spin-orbit coupling, the motion in the coordinated space is complicated, spin state becomes mixed rather than pure, and decoherence in the spin subspace occurs.
To conclude, we mention several extensions and generalizations of the von Neumann measurement procedure seen in the physics of spin-orbit coupling.
First extension can be related to the manifestation of the Zeno effect [45], that is slowing down the dynamics of a constantly measured system. Recent direct calculations indeed demonstrated that in the presence of strong driving electric fields [48], or strong spin-orbit coupling [49] the dynamics of the system becomes much slower than expected from the linear response approach. It would be of interest to see the relation between these results and the Zeno effect.
Second extension can be related to generalizations for other systems and Hamiltonians. Recent development in producing synthetic spin-orbit coupling allows one to realize the three-dimensional spin-orbit coupling [50] in the (k · σ) form. Therefore, an attempt of von Neumann measurement of all three spin components in a single experiment providing another realization of full qubit monitoring [51] can be done. Similar problem is spin-orbit coupling in a system with SU(3) symmetry as can be realized in cold atomic gases [52]. Here the linear in momentum Hamiltonian is expressed is terms of eight generators of the SU(3) group rather than simply in terms of spin 1 axis projections. As a result, a problem of spin component measurements becomes more complicated than in the case of spin 1/2, where the number of the SU(2) group generators is the same as the number of the coordinate axes.
Another example is provided by holes in two-dimensional semiconductor structures, presenting the realization [1] of the k 3 rather than linear in k Rashba model. Although the von Neumann measurement assumes the linear coupling, a similar qualitative analysis can be done here. Taking into account that the spin-orbit splitting here is proportional to γk 3 , where γ is the coupling constant [9], the conditions of accurate measurement can be reformulated as γM > w and γT > w 3 , and, therefore, a narrow packet (small w) is needed for this purpose. The realization of this measurement requires a separate discussion.