Berry’s Phases for Arbitrary Spins Non-Linearly Coupled to External Fields. Application to the Entanglement of N>2 Non-Correlated One-Half Spins

Berry’s Phases for Arbitrary Spins Non-Linearly Coupled to External Fields. Application to the Entanglement of Non-Correlated One-Half Spins


We derive the general formula giving the Berry’s Phase for an arbitrary spin, having both magnetic-dipole and electric-quadrupole couplings with external time-dependent fields. We assume that the “effective” electric and magnetic fields remain orthogonal during the quantum cycles. This mild restriction has many advantages. It provides simple symmetries leading to useful selection rules. Furthermore, the Hamiltonian parameter space coincides with the density matrix space for a spin . This implies a mathematical identity between Berry’s phase and the Aharonov-Anandan phase, which is lost for . We have found, indeed, that new physical features of Berry’s phases emerge for integer spins . We provide explicit numerical results of Berry’s phases for . For any spin, one finds easily well-defined regions of the parameter space where instantaneous eigenstates remain widely separated. We have decided to stay within them and not do deal with non-Abelian Berry’s phases. We present a thorough and precise analysis of the non-adiabatic corrections with separate treatment for periodic and non-periodic Hamiltonian parameters. In both cases, the accuracy for satisfying the adiabatic approximation within a definite time interval is considerably improved if one chooses for the time derivatives of the parameters a time-dependence having a Blackman pulse shape. This has the effect of taming the non-adiabatic oscillation corrections which could be generated by a linear ramping. For realistic experimental conditions, the non-adibatic corrections can be kept below the 0.1 % level. For a class of quantum cycles, involving as sole periodic parameter the precession angle of the electric field around the magnetic field, the corrections odd upon the reversal of the associated rotation velocity can be cancelled exactly if the quadrupole to dipole coupling is chosen appropriately (“magic values”). The even ones are eliminated by taking the difference of the Berry Phases generated by two “mirror” cycles. We end by a possible application of the theoretical tools developed in the present paper. We propose a way to perform an holonomic entanglement of non-correlated one-half spins by performing adiabatic cycles governed by a Hamiltonian given as a non-linear function of the total spin operator , defined as the sum of the individual spin operators. The basic idea behind this proposal is the mathematical fact that any non-correlated states can be expanded into eigenstates of and .The same eigenvalues appear several times in the decomposition when but all these states differ by their symmetry properties under the -spin permutations. The case and is treated explicitly and a maximum entanglement is achieved.

03.65.Vf, 03.67.Bg, 03.75.Dg, 37.25.+k

I Introduction

About twenty five years ago, Geometry made a new entrance in Quantum Mechanics with the discovery of geometric phases [[1], [2], [3]]. These developments hinge upon the simple fact that all the physical information relative to an isolated system described by a pure quantum state is contained in the density matrix . This mathematical object has two important properties: i) it is invariant upon the abelian gauge transformation where is an arbitrary real number, ii) it satisfies the non-linear relation . The density matrix space has clearly a non-trivial topology. A geometric phase is acquired when the quantum system performs a time evolution along a closed circuit on , and satisfies at every instant the so-called “parallel transport” condition . The above terminology reflects the fact that the set of quantum vector states associated with a given density matrix can be viewed as the “fibre” of a linear “fibre bundle” constructed above the “base space” .

Our work deals mainly with adiabatic quantum cycles performed within a definite time interval . They are generated by an Hamiltonian depending on a set of parameters , assumed to be a system of coordinates for a differential manifold . In this way, an adiabatic quantum cycle generates a mapping of a closed circuit drawn upon the parameter space onto a closed loop upon the density matrix space . The Berry phases can then be viewed as the geometric phases associated with this particular class of quantum cycles.

At this point, a question arises naturally: how can one extract the geometric phase since is phase independent? The answer lies within the Superposition Principle of Quantum Mechanics: any linear combination of two quantum states and relative to a given quantum system, is an accessible state for the system. In the present context, such a construction will be achieved via the interaction of the system with specific classical radio-frequency fields, using the so-called Ramsey pulses [[4]]. The density matrix associated with is . The crossed contribution: contains all the information needed to obtain the difference of the geometric phases acquired by the states and during an adiabatic quantum cycle. In some experimental schemes considered in a forthcoming paper [[5]] the geometric phase acquired by will be a priori zero, and the measurement will yield directly the phase acquired by .

A huge amount of work has been triggered by the publication of M. Berry’s paper [[1]] on the phase acquired by a quantum state at the end of its adiabatic evolution along a closed cycle. Among the flurry of following papers, two selections have been presented in comprehensive reviews [[6], [7]] providing guides to the extensive literature on the subject. There also exist pedagogical presentations and a thorough review of manifestations giving many examples drawn from a great variety of physical systems [[8], [9], [10]]. More recently, Berry’s phase has become a topic of renewed interest, both theoretical and experimental, with regard to improved information processing and more specifically for its potential use in quantum computing [[11], [12]]. At the same time, its unwanted manifestations in fundamental precision measurements, have to be carefully investigated [[13], [14], [15]]. Conversely, in other settings, Berry’s phase might be just the right tool to detect still unobserved effects, such as parity violation in atomic hydrogen [[16], [17]].

This paper presents a detailed study of Berry’s phases resulting from adiabatic quantum cycles performed by an arbitrary spin, interacting non-linearly with external electromagnetic fields. We assume that the Hamiltonian, governing the time evolution, involves both linear and quadratic couplings of the spin operator . The addition of a quadratic spin coupling generates new features which, as we shall see, can show up for . The first purpose of this paper is to develop the formalism to calculate them, whatever the value of , integer or half-integer. In addition, we shall give explicit numerical results for several spin values. Throughout this paper, we shall deal with the following set of quadratic spin Hamiltonians:


We have found in the literature only a few papers dealing with the Berrry’s phase for a spin submitted to a time-varying quadrupole interaction [[18]],[[19]]. These deal with nuclear quadrupole resonance spectra in a magnetic resonance experiment involving sample rotation. However, the absence of any magnetic interaction is assumed, which leads to level-degeneracy. These papers generalize the Berry’s phase to the adiabatic transport of degenerate states. In this case, the geometric phase is replaced by a unitary matrix given by the Wilson loop integral of a non-abelian gauge potential where is the dimension of the eigenspace associated with a given degenerate quantum level [[20], [21], [22]]. In contrast, in our work we are interested in situations where the energy levels of the instantaneous Hamiltonian stay widely separated.

We impose on the model the extra constraint . As shown previously by one of us (C. B.) and G. Gibbons [[24], [25]], the parameter space of with is isomorphic to the space relative to spin-one states, namely,the complex projective plane . In [[25]], the Berry’s phase generated by the Hamiltonian (1) for , is found to be mathematically identical to the Aharonov-Anandan (A.A.) phase, if one uses an appropriate parametrization of . However, the physical contents are in general different, since, in contrast to the Berry’s phase, the A.A. phase is not restricted to adiabatic quantum cycles. Hereafter, we derive an expression for Berry’s phase relative to (1) for spins . In this case one loses the identity with the A.A. phase, which involves now closed circuits drawn upon the larger projective complex spaces .

It is convenient to introduce the rotation which brings the and axes along the and fields and the associated unitary transformation . This allows us to rewrite the spin Hamiltonian (1) as , with the dimensionless Hamiltonian given by . The parameter , combined with the Euler angles relative to leads to a convenient set of coordinates for . A quantum cycle within the time interval satisfies the boundary conditions , where and are arbitrary integers. The eigenstates of , are labeled with a magnetic number by requiring that the analytical continuation of towards coincides with the angular momentum eigenstates . Thanks to the constraint , has several discrete symmetries we shall use throughout this paper. The “” parity , associated with a -rotation around , is a good quantum number for . Therefore can mix states and if and only if is an even integer. This selection rule implies that can be written as a direct sum of even and odd blocks. This greatly simplifies the construction of the eigenstates of , and the Berry-phase determination.

After some manipulations involving Group Theory techniques, Berry’s phase is written as a loop integral in the parameter space :


where is the average value of along the field direction. This quantity is obtained by taking the gradient of the eigenenergies of with respect to . We have performed an explicit calculation of the eigenvalues of for the spins and 4 for values of running from 0 to 2, 1.4 and 1.2 respectively (Fig.1). A superficial look at the above expression of may give the impression that our result is, after all, not so different from the case of a pure dipole coupling. However, there are specific features associated with the quadrupole coupling which appear more easily for special cycles where and are the only time varying parameters. The particular case , is especially instructive to this respect. We recall that Berry’s phase associated with a pure dipole coupling is vanishing for an arbitrary cycle with state. Performing the above simple cycle with the state and assuming a constant quadrupole coupling, , one obtains from a look at curves of Fig. 1 the quite remarkable result

Our motivation for this paper is to make contact with experiment, having in mind the spectacular progress of atomic interferometry. One has then to face the problem of the practical realization of a quadrupole coupling, having a magnitude comparable to the magnetic dipole one. For alkali atoms this is unrealistic if the Stark shift arises from a static E-field. The ac Stark shift [[23]] induced by a light beam is much more flexible. However, in most cases, reaching values of close to 1 requires the tuning of the beam frequency to be so close to an atomic line that the ac Stark effect induces an instability of the “dressed” atomic ground state. In typical cases this implies stringent constraints upon the duration of the Berry’s cycle. As a result, the question of the validity of the adiabatic approximation becomes crucial. This has led us to devote a full section of this paper to the precise evaluation of the non-adiabatic corrections. Our theoretical analysis is illustrated by numerical results for a few relevant cases. It provides guide lines for our forthcoming paper, devoted to experimental proposals.

Our analysis of the non-adiabatic corrections proceeds in two steps. We first deal with the corrections associated with the time derivatives of the Euler angles . A convenient approach is the study of the Berry’s cycle in the rotating frame attached to the , and fields, by performing upon the laboratory quantum state the unitary transformation . The corresponding Hamiltonian is obtained by adding to the extra term , where is the magnetic field generated by the Coriolis effect. The longitudinal component , gives rise to a pure dynamical phase shift at the end of the Berry’s cycle. As expected it incorporates at its lowest order contribution with respect to . When is the sole varying Euler parameter, the higher order terms in gives the complete set of non-adiabatic corrections. In addition, we show that the odd-order ones vanish exactly if one chooses for the “magic” value . On the other hand the transverse component , proportional to , presents risks: it involves a linear combination of the spin operators and and induces a mixing with opposite -parity states ( ), possibly nearly degenerate unless stringent constraints are imposed upon . As regards the non-adiabatic corrections associated with , we concentrate our attention to the ramping process of from to , the Euler angles keeping fixed values. Our method can be viewed as an extension of the rotating frame approach. We introduce the unitary transformation which makes diagonal within the basis. The time evolution equation acquires an extra non-diagonal matrix which has the same effect as an rf pulse with sharp edges if increases linearly with . This leads to rather large oscillating non-adiabatic corrections exhibited in our work (see Fig.5). The standard procedure to tame them out is to give to a Blackman pulse shape [[27]].

In the last section, we propose an “holonomic” procedure for the entanglement of non-correlated 1/2-spins (or N Qbits.) The basic tools are Berry’s cycles generated by a Hamiltonian, formally identical to , except that, now, is meant to be the total spin operator of the spins, The method is based on a known mathematical property: any non-correlated 1/2-spin state can be expanded into a sum of and eigenstates. A given eigenvalue of will appear several times if , but all the angular momentum states have different symmetry properties under permutations of the spins, which leave invariant by construction. Thus the states , organized into an orthogonal basis, behave as if they were associated with isolated spins. An initial non-correlated state, written as the sum , is transformed at the end of Berry’s cycle into . With an appropriate choice of the cycle, we have been able to achieve maximum entanglement for .

Ii The instantaneous eigenfunctions of for a given adiabatic cycle

In this section we construct the instantaneous eigenfunctions of for an arbitrary adiabatic cycle. The result is put under a form well adapted to the calculation of Berry’s phase by group theoretical methods. Our method applies to both integer and half-integer spins.

ii.1 Instantaneous spin Hamitonian. Symmetry properties

As a preliminary step, it is convenient to study the particular field configuration where and are along the - and -axes respectively, , and write:


where the term , that plays no role in the calculation of Berry’s phase has been omitted. In other words, is the spin Hamiltonian in the frame attached to the fields and , ignoring their time-dependence. For the explicit calculations to be performed in the cases , it is convenient to introduce the dimensionless Hamiltonian :


It is important to note that is invariant under three transformations. The first, , corresponds to the reflexion with respect to the plane, which changes into and , but has no effect on


The second transformation, , is the product of the reflexion with respect to the plane by the time reversal operation. The transformation of the spin operator under obeys the relations:


The third transformation is a rotation of around the axis, which changes and into and respectively, while leaving unaltered.


(Note that and have the same effect on pseudo-vectors but opposite effects on vectors.)

Let us now discuss some consequences of these invariance properties. To this end, let us introduce the eigenvectors of the Hamiltonian , together with their eigenenergies :


All along this work the eigenenergies are supposed non-degenerate. In the limit , the states have to coincide with the angular momentum states : and . The relations (5) imply that the quantum average of relative to , i.e. the polarization of the quantum state , lies along the z-axis:


The invariance under and requires that the off-diagonal elements of the alignement tensor vanish:

A third important consequence of the above invariance properties is that the eigenvectors may be taken as real. This can be verified by noting that the matrix associated with in the angular momentum basis , using the standard phase convention, is real and symmetric, so that its eigenvectors are real.

Equation (7) implies that the operator associated with the rotation of around , , commutes with the Hamiltonian . For convenience, let us introduce the “m-Parity ”, . The m-Parity of the angular momentum eigenstate state is . Within the angular momentum basis, the Hamiltonian can then be written as the direct sum of two matrices acting respectively on the states even and odd with respect to the operator :


As a conclusion, we would like to stress that the field orthogonality condition plays an essential role in making the mathematical problem tractable for spins . Otherwise, the problem would become rapidly complicated and any insight into Berry’s phase physics gets blurred by the algebra.

ii.2 The instantaneous eigenfunctions of

Since we are going to use group theory arguments, it is appropriate to recall some basic facts about the rotation group in Quantum Mechanics. One introduces the unitary operator associated with the rotation acting on the spin state vectors:


where stands for the rotation around the unit vector by an angle . This operator provides a unitary representation of the rotation group in the sense that it obeys the multiplication rule: , together with the unitarity relation . Applying the above rule to the case where is an infinitesimal rotation, one derives the important relation:


which expresses the fact that the unitary transformation rotates the spin observables.

Let us associate with the orthogonal vectors and the trihedron which can be constructed by applying the rotation to the fixed coordinate trihedron , with denoting the usual Euler angles:


To ensure the validity of the adiabatic approximation for the quantum cycles generated by , we shall assume that the Euler angles are slowly varying functions of time. More precisely, we shall require that their time derivatives , - together with the time derivative of the Stark-Zeeman coupling ratio - are much smaller than the rate , where stands for the minimum distance between the energy levels of . The adiabatic cycle within the time interval is specified by the boundary conditions involving the two finite integers and :


Note that in contrast to the periodic variables and , and recover their initial values at the end of the cycle.

To proceed it is convenient to introduce the unitary operator associated with the rotation , given by:


Since belongs to a unitary representation of the rotation group, it can be written, using equation (15), as the following operator product:


Using the relation (12) with and the identity , it is straightforward to derive the important relation:


As a consequence, the wave functions defined as


are instantaneous eigenfunctions of with eigenvalues :


We would like to stress that the instantaneous wave functions have, by construction, a well defined phase since, as shown previously, the state vectors can all be taken as real.

Iii Explicit calculation of the eigenfunction parameters for Berry’s phase evaluation

Figure 1: Reduced energies and polarizations versus for and 4. Since , intersection of the curves with the vertical axis indicate their labeling. The curves exibit the symmetry relations and , derived in the text. For there is a clear evidence for the degeneracies expected when .

We present for and 4, explicit calcutations of the eigenenergies and the polarization of the eigenstates, which, for symmetry reason, is along the direction . (In this section, for the sake of simplicity we use a unit system where ). We construct the matrix associated with within the angular momentum basis:. It is convenient to rewrite the Hamiltonian in terms of the operators and

Using textbook formulae, for the matrix elements of in the basis it is easy to write in matrix form. Using the invariance under the symmetry introduced in subsection II.A, can be written as a direct sum of two matrices acting respectively upon the states with even and odd values of , one of order and the other of order depending on the parity of S. For and 2 one finds easily:


The Hamiltonians and as well as and are given in Appendix B.

The eigenvalues are obtained in a standard way by solving the two polynomial equations:


The eigenvalue equations for and can be found in the appendix. Mathematica codes yield explicit expressions for the eigenenergies . Although the formulae are rather complex, they lead to very accurate numerical results for all the values of of interest. The accuracy is better than . This result has been checked using numerical Mathematica codes, obtained directly from the matrix expression. To calculate the polarization we have used the fact that it is given by the derivative of the eigenenergy with respect to (Hellmann-Feynman theorem). Writing the eigenvalue of as: , with , one gets immediately:


The results are given in Fig.1 for both the energies and the polarizations. The eigenenergies can be labelled unambiguously since for small values of the eigenvalues are equal to , up to corrections of the order of . In Fig.1 there is evidence for near-degeneracies of opposite-parity level pairs: is approaching when for , where is an even integer such that . This was expected since in this limit is dominated by the term . The convergence is much slower for since the positive term term has then to fight against a negative Zeeman effect. There are degenerate doublets with , as expected in the limit . In addition, by looking at the even-odd (or odd-even) pairs, one sees clearly that the pair converges, without crossing, to the degenerate doublet The next lower pair ends as the degenerate doublet having the energy and so on until one reaches the isolated level , which has no possibility other than converging towards the non-degenerate level with energy . While this behaviour is clearly exhibited in the simple case , only its two first steps are clearly apparent for and 4 , but we have verified that the above picture is valid for all values of by computing the ratios when .

A striking feature of the case in Fig.1 is the symmetry of the plotted curves under the transformation , involving both the eigenenergies and the polarizations: , To prove these symmetry properties in the general case, it is convenient to perform upon the spin system a rotation of angle around the axis, . By introducing the spin unitary operator associated with the rotation , one can write the transformation law for the Hamiltonian :

Applying to both sides of the eigenvalue equation , one can write:


The state is an eigenstate of with the eigenenergy . It coincides, up to a phase factor, with the eigenstate , since the effect of the rotation is to flip the spin component along the axis. This completes the proof that:

with the similar relation for as a consequence of Eq. (23).

In addition, the quantum-averaged polarizations satisfy the sum rule


valid for any value of S and . It is readily derived using the fact that can be evaluated as the partial derivative of the energy with respect to (Eq. (23)) and noting the general structure of the polynomial equation whose solutions provide the eigenenergies.

To end this section we would like to say a few words concerning the state which is of particular interest for integer spins . The fact that a second-order Stark effect can induce a polarization in an initially unpolarized state was somewhat unexpected. To gain greater insight we have performed a perturbation computation in powers of . Although one has to go to third order to find a non-zero effect the final result is given by the rather elegant formula:


It gives a rather accurate result for if but the domain of validity gets smaller for S=3: . Finally we would like to point out the smooth behaviour of ) for in the vicinity of and the remarkably simple exact values taken by the reduced energy and the polarization at , namely and . At the time of writing, it is unclear whether this is a mere numerical accident or an indication of something more profound.

Iv Berry’s phase for adiabatic cycles generated by acting on arbitrary spins

We start this section by a mini-review introducing the basic physical and mathematical features of Berry’s phase concept. In particular we derive the general formula giving the Berry’ phase in terms of the instantaneous eigenfunctions of the time-dependent Hamiltonian generating the adiabatic quantum cycles. Introducing in this formula, the results of the previous section and relying upon Group Theory arguments, we perform an explicit construction of Berry’s phase relative to , valid for arbitrary spins. The final result is expressed as a loop integral in the space, using as coordinates the Euler angles and the parameter . We show that for the case a circular loop drawn upon a spherical subspace of lead to a loop integral very different from that of the case which involves a magnetic monopole Bohm-Aharonov phase.

iv.1 The Berry’s phase as a physical observable and topological concept: a mini-review

In this introductory subsection, we are going to follow, in several places, a presentation due to the late Leonard Schiff, in tribute of its memory. He gave in few pages of its venerable textbook [[26]] a correct and precise treatment of the adiabatic approximation, involving a non-integrable phase. To study the adiabatic quantum cycles generated by the Schrödinger equation

it is convenient to expand the solution in terms of the eigenstates of :


The first term in (28) is a phase that vanishes for , but is otherwise arbitrary. Its value will be determined by the reasoning leading to the adiabatic approximation. The second term,


known as the “dynamical phase”, produces a contribution which cancels in the wave equation. We shall take as the initial condition : or, in other words, . In this case we can replace the exact Schrödinger equation by the system of differential equations involving the expansion coefficients :


Since the adiabatic condition requires that whatever , a necessary condition to ensure its validity is to cancel the coefficient of in the r.h.s of equation (30). This is achieved if we make the following choice for the “gauge ” :


Later on, we shall see that this non integrable gauge is a basic ingredient in the mathematical expression of Berry’s phase in terms of the instantaneous wave functions . The next step to validate the adiabatic approximation is to find appropriate conditions allowing the sum to remain below a predefined level for . This task will be performed in details in Sec. V but for the moment, let us assume that it is achieved. Within the adiabatic approximation, the solution of the Schrödinger equation , with the initial condition , is then given by:


We can now calculate the phase shift of the wave function at the end of the adiabatic cycle:


where we have made Berry’s phase stand out on the r.h.s of the above equation. Using the equation (31), one gets immediately the basic formula giving in terms of the instantaneous wave functions :


It is crucial to note that the “dynamical phase ” and obey different scaling laws under the transformation, , involving an arbitrary real parameter , while keeping invariant the Euler angles and the dimensionless parameter . If one remembers that , one sees immediately that is multiplied by , while , being geometric, is unchanged. In principle, this scaling difference could be used to separate the two phases. However, a more practical way to isolate Berry’s phase consists in measuring the phase for a given adiabatic cyclic evolution and that associated with the “image” circuit obtained by performing on the Hamiltonian parameters the transformations: , while keeping the other two unchanged. The two competing phases are transformed as , so that the dynamical phase can be eliminated by subtraction.

To end this mini-review, we would like to give, within the present context, a simplified description of the topological interpretation of the Berry’s Phase, due to Simon [[2]]. We have just shown that is a physical state obeying the Schrödinger equation within the adiabatic approximation. For our purpose, it is convenient to introduce the vector state and to calculate the differential form taken along the adiabatic loop:

The evolution of the state vector along the closed loop is then said to satisfy the “parallel transport” condition . If the state is injected into the general formula for the Berry’s phase, one immediately finds that


in the case of parallel transport.

Let us now give a rather elementary introduction to the mathematical concepts behind the above notion of “parallel transport” applied to the evolution of quantum states. This arises rather naturally from a linear fiber bundle interpretation of Quantum Mechanics. The linear fiber bundle associated with the quantum state space is constructed from the “base space” of the “pure ” state density matrix . Assuming for simplicity that the states have unit norms, one finds that satisfies the simple nonlinear relation . This implies clearly that has a non-trivial topology. The “fiber” is the one-dimensional space associated with a definite . The vector states of the fiber are given by , where is an arbitrary phase and a representative state of the fiber. The infinitesimal variation during the quantum cycle is said to be “vertical” if it takes place along the fiber: , where is an infinitesimal -number. Conversely, it is called “horizontal” or “parallel” if . The fact that the Berry’s phase can be viewed as a displacement respective to the fiber, associated with - in our case a phase shift resulting from a parallel transport along a closed loop drawn upon the base space - emphasizes its topological character.

iv.2 The Berry’s phase as a loop integral in the Hamiltonian parameter space

Our starting point is the formula (34) giving Berry’s phase for quantum adiabatic cycles associated with the instaneous wave function .

The fundamental property of is its invariance under the gauge transformation , where is an arbitrary real function. The density matrix of an isolated quantum system has clearly the same gauge invariance property. The adiabatic approximation allows us to make a mapping of the Hamitonian parameter space onto the density matrix space. As a consequence, could also be viewed as a line integral along a closed path drawn in the density matrix space.

A group theoretical derivation To evaluate the expression (34), it is convenient to write in terms of :