Magnetic monopoles in quantum adiabatic dynamics and the ... - CNRS

JOURNAL OF MATHEMATICAL PHYSICS 47, 092105 共2006兲

Magnetic monopoles in quantum adiabatic dynamics and the immersion property of the control manifold David Viennota兲 Observatoire de Besançon (CNRS UMR 6091), 41 bis Avenue de l’Observatoire, BP1615, 25010 Besançon Cedex, France 共Received 21 June 2006; accepted 29 July 2006; published online 20 September 2006兲

It is well known that the Berry phase of a cyclic adiabatic dynamical system appears formally as the flux of a magnetic field in the control parameter manifold. In this electromagnetic picture a level crossing appears as a Dirac magnetic monopole in this manifold. We make an extensive study of the magnetic monopole model of eigenvalue crossings. We show that the properties of the monopole magnetic field in the control manifold are determined by the immersion of the control manifold in a space given by the universal classifying theorem of fiber bundles. We give a detailed illustrative study of the simple but instructive case of a two level crossing of a system controlled by a two-dimensional manifold. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2345473兴

I. INTRODUCTION

In 1984, Berry Ref. 1 proved, in the context of the standard adiabatic approximation, that the wave function of a quantum dynamical system takes the form

冉冕

t

␺共t兲 = exp − iប−1

ជ 共t⬘兲兲dt⬘ − Ea共R

0

冕

t

0

冊

ជ 共t⬘兲兩⳵ 兩a,Rជ 共t⬘兲典dt⬘ 兩a,Rជ 共t兲典, 具a,R t⬘

共1兲

where Ea is a nondegenerate instantaneous eigenvalue which is isolated from the rest of the ជ 共t兲典. Rជ is a set of classical Hamiltonian spectrum and has the instantaneous eigenvector 兩a , R control parameters used to model the time-dependent environment of the system. The set of all configurations of Rជ is assumed to form a C⬁-manifold M. The important result is the presence of an extra phase term exp共−兰t0具a , Rជ 共t⬘兲兩⳵t⬘兩a , Rជ 共t⬘兲典dt⬘兲 called the Berry phase. Simon2 later found that the mathematical structure which models the Berry phase phenomenon is a principal bundle with base space M and with structure group U共1兲. If we eliminate the dynamical phase by a gauge transformation, which involves redefining the eigenvector at each time, then the expression 关Eq. ជ 共t兲 with gauge potential A 共1兲兲 is the horizontal lift of the curve C described by t 哫 R = 具a , Rជ 兩dM兩a , Rជ 典. If C is closed then the Berry phase exp共−养CA兲苸 U共1兲 is the holonomy of the horizontal lift. In 1984, Wilczek and Zee3 introduced the concept of a non-Abelian Berry phase in the context ជ 共t兲兲 be an M-fold degenerate instantaneous eigenvalue of the adiabatic approximation. Let Ea共R isolated from the rest of the spectrum and 兵兩a , i , Rជ 共t兲典其i=1,. . .,M be an orthonormal basis for the associated eigensubspace. If the initial state is ␺共0兲 = 兩a , i , Rជ 共0兲典, then the wave function is a兲

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David Viennot

冉冕

M

␺共t兲 = 兺 exp − iប−1 j=1

t

ជ 共t⬘兲兲dt⬘ Ea共R

0

冊关

兴

t ជ ជ 共t兲典, Te−兰0A共R共t⬘兲兲 ji兩a, j,R

共2兲

ជ 典, and T is the timewhere the matricial one-form A has the elements Aij = 具a , i , Rជ 兩dM兩a , j , R ordering operator. By elimination of the dynamical phase this expression becomes a horizontal lift of the curve C described by t 哫 Rជ 共t兲 into a principal bundle with base space M and structure group ជ 兲兲苸 U共M兲 is the holonomy of the horizontal lift, P being U共M兲. If C is closed then P exp共−养CA共R the path-ordering operator. ជ 共t兲兲其 More generally for a quantum dynamical system, let 兵Ea共R a苸I be a set of eigenvalues indexed by I, and isolated from the rest of the spectrum. Suppose that ␺共0兲 = 兩a , Rជ 共0兲典; the wave function is then

␺共t兲 = 兺关Te−iប

ជ 共t 兲兲dt −兰t A共Rជ 共t 兲兲 −1兰t E共R ⬘ ⬘ 0 ⬘ 0

b苸I

兴ba兩b,Rជ 共t兲典,

共3兲

ជ 共t兲兲 = E 共Rជ 共t兲兲␦ and A共Rជ 兲 = 具a , Rជ 兩d 兩b , Rជ 典. When the matrices where we have the matrices E共R ab a ab ab M E and A do not commute then the mathematical structure which models the non-Abelian phase phenomenon is a principal composite bundle with base space M ⫻ R, where R models the time 共see Ref. 4兲. For each of the three preceding cases, an appropriate gauge potential A 苸 ⍀1共M , u共M兲兲 has been defined, to be associated with the principal bundle P with base space M and with structure group U共M兲共where M = 1 for the Berry’s orginal case; M is the degenerate order for Wilczek’s case, and M = cardI for the general case兲. In the one-dimensional case, the bundle curvature is defined by B = dMA and it satisfies the equation dMB = 0. It is easy to prove that this equation is the analog in M to half of the Maxwell equations. We can then identify B with a magnetic field existing in M and A with its magnetic potential. In this picture, for any closed curve C for which there exists a surface S 傺 M such that ⳵MS = C, the holonomy term can be written 共using the Stokes theorem兲 as the flux of the magnetic field B through S养CA = 兰SB. By using the expression for A, it is easy to show that

B = dMA = 2iI

⳵具a,Rជ 兩 ⳵兩a,Rជ 典 ␮ dR Ù dR␯ . ⳵R␮ ⳵R␯

共4兲

By using the closure relation one can write B = iI 兺

b⫽a

ជ 兩⳵ H共Rជ 兲兩b,Rជ 典具b,Rជ 兩⳵ H共Rជ 兲兩a,Rជ 典具a,R ␮ ␯ dR␮ Ù dR␯ , 2 ជ ជ 共E 共R兲 − E 共R兲兲 b

共5兲

a

where H共Rជ 兲 is the system Hamiltonian. We cannot strictly apply the one-dimensional adiabatic approximation if the eigenvalue Ea is not isolated from the rest of the spectrum. Nevertheless, the ជ 苸 M, formal use of this approximation in the case where two eigenvalues Ea and Ec cross at R ac produces the result B共Rជ ac兲 = + ⬁. This magnetic field divergence must be interpreted as a magnetic monopole at Rជ ac in M. In the M-dimensional case the field defined by the Cartan structure equation F = dMA + 21 关A , A兴 satisfies the Bianchi identity dMF + 关A , F兴 = 0. The Bianchi identity is analogous in M to half of the Yang-Mills equations. The eigenvalue crossings then appear as non-Abelian monopoles 共so-called colored monopoles in the case where the structure group is SU共3兲, see Refs. 5 and 6兲, as has already been pointed by Wilczek and Zee.3 This paper studies the properties of the adiabatic magnetic monopoles, with special attention to the monopole magnetic field distribution in M and on the apparent charge of the monopole. We show that these properties are determined by the immersion of M in a universal space associated with the topology of the adiabatic bundle.


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Adiabatic monopoles and the control manifold

The concept of the virtual magnetic monopole is today very important for nonrelativistic quantum physics. In 1986, Moody et al.7 showed the possibility of realizing a magnetic monopole via the precession of a diatomic molecule. Fang et al.8 showed that a trapped ⌳-type atom induced a magnetic monopole. Recently Zhong et al.9,10 observed experimentally a magnetic monopole in the crystal momentum space related to the anomalous Hall effect in the SrRuO3 crystal. These examples show the importance of understanding the properties of the adiabatic magnetic monopoles. Section II recalls the definition and the properties of the universal classifying space for the general case. Section III is devoted to the Dirac magnetic monopole theory, in particular its link with Berry’s original example of a geometric phase. A more complete discussion of the results reviewed in these sections can be found in Ref. 11. Originally M is a soft manifold 共i.e., one without a natural metric兲, because it is a set of classical control parameters of the environment which do not necessarily have the same physical nature. Section IV shows that M can be rigidified 共endowed with a metric兲 by an appropriate immersion. This rigidification is needed to describe the monopole magnetic field distribution in M. Section V is devoted to the analysis of the apparent monopole charge in M. Section VI generalizes the discussion to the case of the nonAbelian monopoles. In most of this paper we suppose that dim M = 2 and restrict our attention to the monopole 共level crossing兲 neighborhood. We discuss the effect of these assumptions in the last section. II. THE UNIVERSAL SPACE OF QUANTUM ADIABATIC DYNAMICS

Bohm and Mostafazadeh showed in Refs. 12 and 13 that the Berry phase phenomena are related to the nonadiabatic geometric phases of cyclic dynamical quantum systems discovered by Aharonov and Anandan.14 This relationship is given by the universal classifying theorem of principal bundles 共see Refs. 15 and 16兲. Here we recall this theorem, and the associated induced geometric structure of quantum mechanics. A. The universal classifying theorem

Let B = 共B , X , G , ␲B兲 be a principal bundle with base space X, total space B, structure group G, and projection ␲B. We say that B is n-universal if for every cell n-complex K 共see Refs. 15 and 16兲, for every cell subcomplex L 傺 K, for every principal bundle B⬘ = 共B⬘ , K , G , ␲B⬘兲 and for every map h : 共␲B−1⬘共L兲 , L , G , ␲B⬘兲 → 共B , X , G , ␲B兲, there exists an extension of h to 共B⬘ , K , G , ␲B⬘兲 → 共B , X , G , ␲B兲. If B = 共B , X , G , ␲B兲 is 共n + 1兲-universal and if K is a cell n-complex then there exists a map f : K → X such that the following diagram commutes f *B

f*

←

␲ f *B ↓ K

B ↓ ␲B ,

f

→

X

where f *B is the bundle induced by f. X is called the universal manifold of K and f is called the universal map. In the case of the adiabatic bundle P, the situation is as follows. Suppose that the Hilbert space is n-dimensional. In the case of the one-dimensional adiabatic approximation, the universal manifold of P is the projective space CPn−1, and the universal map can be written: f:

M → CPn−1

ជ 哫兩a,Rជ 典具a,Rជ 兩 R

.

In the m-dimensional adiabatic approximation, the universal manifold is the Grassmanian manifold Gm共Cn兲 = U共n兲 / 共U共m兲 ⫻ U共m − n兲兲 and the universal map is


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David Viennot

M → Gm共Cn兲 f:R ជ哫

兺兩a,Rជ 典具a,Rជ 兩

a苸I

共see Ref. 13兲. In the following we will restrict our attention to CPn−1 and will analyze its structure.

B. The Kählerian structure of quantum dynamics

The physical reason which explains why CPn−1 is the universal space of quantum mechanics is the following. We know that the mathematical structure of quantum mechanics is a separable C-Hilbert space H which models the state space. The probabilistic interpretation of quatum mechanics states that a physical state ␺ is a probability amplitude, and so we must have 储␺储 = 1. In this way two proportional vectors represent the same physical state. Moreover, the phase of a vector does not give physical information; only the phase difference between two vectors has a physical meaning 共quantum interference theory兲. We then see that the Hilbert space H contains a lot of “nonphysical” information, so that it is not the most efficient structure to describe quantum mechanics. To define the “true” space of quantum mechanics we first consider H. If ␺ = ␣␾ with ␣ 苸 R+ and ␣ ⫽ 0 then the two vectors ␺ and ␾ define the same physical state 共they have only a different norm兲. This previous relation is an equivalence relation ⬃, which signifies that a physical state is an equivalence class. So the physical space is N = 共H \ 兵0其兲/ ⬃ ,

共6兲

which is called the space of normed rays. Suppose that the question of interference is ignored; then two vectors which are only different by a phase factor represent the same physical state. We define the group action of U共1兲 on N by "ei␸ 苸 U共1兲, " 关␺兴苸 N,

ei␸关␺兴 = 关ei␸␺兴.

共7兲

Since two vectors represent the same state if they belong to the same orbit, then the physical space is the homogeneous space of the orbits R = N/U共1兲,

共8兲

which is called the complex projective space. If H is finite dimensional, dim H = n, we have R = CPn−1 共it is a 共n − 1兲-dimensional C⬁-differential complex manifold兲. In the case n = 2 there exists an exceptional diffeomorphism CP1 ⯝ S3/S1 ⯝ S2 .

共9兲

Thus the space of a quantum two-level system 共a spin for example兲 is the sphere. Let ␺ 苸 H, with ␺ = 共␺0 , ␺1 , . . . , ␺n−1兲. We set wi = ␺i / ␺0 for all i = 1 , . . . , n − 1. The complex numbers 兵wi其i are called the homogeneous coordinates of ␺ in CPn−1. We can write the gauge potential by using these coordinates:

A=

=

␺ †d ␺ i ␺ †d ␺ − ␺ d ␺ † = ␺ †␺ 2 ␺ †␺

共10兲

i ¯␺␣d␺␣ − ␺␣d¯␺␣ 2 ¯␺ ␺␥

共11兲

␥


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J. Math. Phys. 47, 092105 共2006兲

¯ idwi − widw ¯i iw . j 2 1+w ¯ jw

=

共12兲

Einstein’s notation has been adopted for the sum, with the convention that the Greek indices take the values 0 , . . . , n − 1 and that the Latin indices take the values 1 , . . . , n − 1. The magnetic field 共the curvature of the universal bundle兲 can be written F = dA = i

¯ iw j − 共1 + w ¯ kw k兲 ␦ i j i w ¯ j. dw Ù dw ¯ lw l兲 2 共1 + w

共13兲

¯ kwk兲, K = 21 Ln共1 + w

共14兲

We introduce the function

where Ln is the multivalued complex logarithm. We also introduce the Dolbeault operators of the complex manifold ⳵ and ¯⳵ 共¯⳵ + ⳵ = d兲共a presentation of complex differential geometry can be found in Ref. 17兲. It is easy to see that ¯⳵⳵K =

¯ iw j − 共1 + w ¯ kw k兲 ␦ i j i 1w ¯j dw Ù dw 2 ¯ lw l兲 2 共1 + w

共15兲

and we then have F = 2i¯⳵⳵K. We recognize here the structure of a Kählerian geometry 共see Ref. 17兲; K is the Kähler potential and F is the Kähler form of CPn−1. We know that a Kählerian manifold is endowed with a natural metric, which in this case is the Fubini-Study metric

␩ = dl2 =

¯ kw k兲 ␦ i j − w ¯ iw j i 共1 + w ¯ j. dw dw l 2 ¯ lw 兲共1 + w

共16兲

This Kählerian structure of quantum mechanics was indicated by Anandan and Aharonov in Ref. 18. III. MAGNETIC MONOPOLE

We have seen that the divergence of the magnetic field of M relates a level crossing to the presence of a magnetic monopole in the electromagnetic picture. We recall here the theory of the Dirac magnetic monopole19 and also the direct relationship between this theory and the Berry phase phenomenon. A. The Dirac magnetic monopole theory

We consider a magnetic monopole with magnetic charge g, at the position 共0, 0, 0兲 in a three-dimensional space. The first aspect of the Dirac model is that we do not consider R3 as being the fundamental manifold, but as a foliation 共S2 , R+兲, where S2 is a sphere centered on the monopole and R+ is the foliation parameter space which describes the radius of the sphere. We consider the U共1兲-principal bundle with base space S2. We know that an atlas of S2 must have at least two local charts. We choose the following charts: UN = 兵共␪, ␸兲, ␪ 苸关0, ␲/2 + ⑀兴, ␸ 苸关0,2␲关其, US = 兵共␪, ␸兲, ␪ 苸关␲/2 − ⑀, ␲兴, ␸ 苸关0,2␲关其. In these charts of the north and south hemispheres 共the choice of the equator S1 being arbitrary兲 ⑀ is a small parameter which is used to ensure the nonvanishing intersection of the charts. On S2 we introduce the local potential AN = ig共1 − cos ␪兲d␸ ,

共17兲


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AS = − ig共1 + cos ␪兲d␸ .

共18兲

By using the vector formalism of R3 we have

ជ N = g共1 − cos ␪兲 eជ , A ␸ r sin ␪

共19兲

ជ S = − g共1 + cos ␪兲 eជ . A ␸ r sin ␪

共20兲

共We should note that AN is singular if we extend it at ␪ = ␲ and AS is singular if we extend it at ␪ = 0.兲 The magnetic field 共the curvature兲 is then F = dA. We compute the magnetic flux through S2; by using Stokes theorem we find

冕

F

= lim

冉冕

dAN +

冉冕

AN +

⌽=

⑀→0

= lim

⑀→0

=

共21兲

S2

UN

dAS

共22兲

冕冊

共23兲

US

⳵UN

AS

⳵US

冕

AN − AS

共24兲

冕

2gd␸

共25兲

S1

=

冕冊

S1

=4␲g.

共26兲

This is, effectively, the flux of a central field with charge g. We introduce the transition function gNS = e2ig␸, and we then have AN = AS + 共gNS兲−1dgNS .

共27兲

To have a single transition one should require that gNS共␸ = 0兲 = gNS共␸ = ␲兲 Û 2g 苸 Z. This is the Dirac quantization condition. By the foliation we extend this result to all of R3 and we have AN = − ig

AS = ig

ydx − xdy , r共r + z兲

ydx − xdy , r共r − z兲

共28兲

共29兲


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F = ig

xdy Ù dz + ydz Ù dx + zdx Ù dy r3

共30兲

with r = 冑x2 + y 2 + z2. By using the Hodge duality we find the correct expression in the vector ជ = grជ / r3. formalism for the magnetic field: F N On the z axis A is singular at z 艋 0 and AS is singular at z 艌 0. We call these two semilines the Dirac strings. We note that by a gauge transformation 共with another choice of equator兲 the Dirac strings rotate in the space.

B. The magnetic monopole of a simple two-level system 1 2

We consider the system originally considered by Berry a spin magnetic field, described by the Hamiltonian

H共xជ 兲 = xi␴i =

冉

x3

x1 − ix2

x1 + ix2

− x3

冊

particle interacting with a

共31兲

.

The eigenvalues of H共xជ 兲 are E±共xជ 兲 = ± r = ± 冑共x1兲2 + 共x2兲2 + 共x3兲2. We thus have a level crossing at xជ = 0ជ . The eigenvector associated with E+ is

兩 + ,xជ 典N =

1

冑2r共r + x 兲 3

冉

r+x

3

x1 + ix2

冊

冢冣 cos

=

␪ 2

␪ e sin 2

,

共32兲

i␸

where ␪ = arctan 冑共x1兲2 + 共x2兲2 / x3 and ␸ = arctan x2 / x1 are, together with r, the spherical coordinates of R3. With another convention for the matrix representation, the eigenstate is

兩 + ,xជ 典S =

1

冑2r共r − x3兲

冉

1

x − ix r − x3

2

冊

冢冣 e−i␸ cos

=

sin

␪ 2

␪ 2

.

共33兲

We compute the adiabatic gauge potential i x2dx1 − x1dx2 i = 共1 − cos ␪兲d␸ , 2 r共r + x3兲 2

共34兲

i x2dx1 − x1dx2 i = − 共1 + cos ␪兲d␸ . 3 2 r共r − x 兲 2

共35兲

AN = N具+ ,xជ 兩d兩 + ,xជ 典N = −

AS = S具+ ,xជ 兩d兩 + ,xជ 典S =

We recognize here the magnetic potential associated with a magnetic monopole of charge 21 at xជ = 0ជ 苸 R3. We see that a level crossing leads to the equations appropriate to a Dirac magnetic monopole. Let us consider that r is constant; the manifold describing the system is the sphere S2 = CP1 with coordinate system 共␪ , ␸兲. We see that the present representation is in fact the universal model of quantum mechanics. We designate S2 by the term “universal manifold,” and R3 foliated by S2 by the term “generalized universal space.”


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FIG. 1. 共Color online兲 Representations of two levels with respect to the control manifold M, the immersions being defined ជ 兲 = R1, f 2共Rជ 兲 = R2, f 3共Rជ 兲 = 0, right: f 1共Rជ 兲 = R1, f 2共Rជ 兲 = R2, f 3共Rជ 兲 = 1. For the left panel, the monopole is in f共M兲, by, left: f 1共R ជ 兲 = 0. In the right panel, the monopole is not in f共M兲, but f共M兲 passes by the neighborhood and we have a crossing for f共R ជ of the monopole for R = 0; at this position we see an avoided crossing.

IV. THE ADIABATIC MAGNETIC MONOPOLE FIELD ASSOCIATED WITH A TWO LEVEL CROSSING A. Immersion of the control manifold, level crossings, and avoided level crossings

ជ 兲, with We consider an adiabatic quantum dynamical system described by the Hamiltonian H共R Rជ 苸 M. The control manifold M is supposed to be two dimensional. We consider a crossing between two nondegenerate eigenvalues E1 and E2 of H共Rជ 兲. In the neighborhood U of this crossing we consider the effective Hamiltonian associated with this crossing: ជ 苸 U, "R

ជ兲 = Heff共R

冉

ជ兲 f 3共R

ជ 兲 − if 2共Rជ 兲 f 1共R

ជ 兲 + if 2共Rជ 兲 f 1共R

ជ兲 − f 3共R

冊

.

共36兲

Heff is obtained by a partitioning technique,20 arising from a quantum KAM method,21 an adiabatic elemination method,21 or a Bloch wave operator method.20,22 Comparing the Hamiltonian 关Eq. 共36兲兴 with the Hamiltonian Eq. 共31兲 of the universal model, we see that we have a map ជ兲 f from the control manifold M to the generalized universal space R3, with f共R 1 ជ 2 ជ 3 ជ 3 = 共f 共R兲 , f 共R兲 , f 共R兲兲苸 R . In order to respect the geometric framework we suppose that f is a C⬁-map. Let f * : TRជ M → T f共Rជ 兲R3 be the push-forward map 共which is a linear map between two ជ 兲. We suppose that ker f = 兵0其共f is an injective map兲, and f is then an vector spaces for a fixed R *

*

immersion of M in R3 共we will discuss this assumption later兲. There is no reason to suppose that f is an embedding; several points of M can be associated with the same point in the universal space and we will use this possibility in the later discussion. ជ 苸 M if and only if f共Rជ 兲 = 0. Although only one magnetic We have a level crossing at R 0 0 monopole is present in the generalized universal space R3, it is possible that several monopoles ជ其 ជ ជ ជ 共crossings between E1 and E2兲 exist at several points 兵R i i=1,. . .,n if f共R1兲 = f共R2兲 = ¯ = f共Rn兲 = 0. An avoided crossing manifests itself in this geometric analysis by virtue of the fact that the immersed manifold f共M兲 does not include 0 but passes through a neighborhood of 0, i.e., "Rជ ជ 兲 ⫽ 0, but $Rជ ជ 兲储苸 U, f共R and $⑀ a small positive constant such that 储f共R 0 0 ជ 兲兲2 + 共f 2共Rជ 兲兲2 + 共f 3共Rជ 兲兲2 = ⑀ and "Rជ 苸 U, Rជ ⫽ Rជ , 储f共Rជ 兲储 ⬎ ⑀ 共Rជ is a local minimum of = 共f 1共R

冑

0

0

0

0

0

0

the vector norm兲. Note that 2⑀ is the energy gap of the avoided crossing 共Fig. 1兲. B. The Riemannian structure of the control manifold

The control manifold M is not endowed with a natural metric. The Euclidian metric defined ជ ·Q ជ = R␮Q␯␦ does not have a physical meaning, because the control by the scalar product R ␮␯ parameters can have different physical natures. However, we know that the universal manifold is endowed with a metric which is natural for the quantum mechanics, the Fubini-Study metric. We will use this metric and the immersion f to rigidify the manifold M.


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The universal manifold is a sphere S2 ⯝ CP1 considered as a sheaf of the foliation of R3. Let ␺ = 共␺0 , ␺1兲 = ␺0共1 , w1兲 be a vector of the space spanned by the eigenvectors of the Hamiltonian 关Eq. 共36兲兴. w1 苸 C is the coordinate on CP1. The Fubini-Study metric of CP 共to be closer to the usual electromagnetic formalism, we have removed the factor i in order to have real fields in place of purely imaginary fields兲 is

␩=

¯ 1w 1兲 − w ¯ 1w 1 ¯ 1dw1 共1 + w dw ¯ 1dw1 = dw . 1 2 ¯ 1w 兲 ¯ 1w 1兲 2 共1 + w 共1 + w

共37兲

This expression is precisely the conformal representation of the Riemannian metric of the sphere 共see Ref. 23兲. The Kählerian structure of CP1 is equivalent to the Riemannian structure of S2. We can then write, with the standard coordinate system of S2,

␩ = r2 sin2 ␸d␪2 + r2d␸2 .

共38兲

We know that the Riemannian metric of S2 centered on 0 in R3 is obtained as being the metric induced by the Euclidian metric of R3. We write ␦ for this metric 共␦ij = 0 if i ⫽ j or =1 if i = j兲. We see that the rigidification of M is obtained by the metric induced by the immersion of M in R3 endowed with ␦. The natural metric of M is then g = g␮␯dR␮dR␯ = ␦ij

⳵fi ⳵f j ␮ ␯ dR dR . ⳵R␮ ⳵R␯

共39兲

ជ 兲 = f i共Rជ 兲␴ where 兵␴ , ␴ , ␴ 其 are the Pauli matrices. Since Moreover, we can write Heff共R i 1 2 2 we obtain

1 2 tr共␴i␴ j兲 = ␦ij

冉

冊

1 ⳵Heff ⳵Heff . g␮␯ = tr 2 ⳵R␮ ⳵R␯

共40兲

We have supposed that f is an immersion, i.e., that f * is injective. If this is not the case, then the bilinear form g defined by Eq. 共39兲 is not a metric in a rigorous mathematical sense. g is then not positively defined 共if f * is not injective, then there exists an isotropic tangent vector of M, i.e., $X␮⳵␮ 苸 TM such that g␮␯X␮X␯ = 0兲. In this case, by an abuse of language, we will continue to call g a metric, and we will continue to consider that

冕冑 C

g ␮␯

d␩␮ d␩␯ ds ds ds

is the length of C parametrized by s 哫 ␩共s兲 in M. This abuse of language is standard in physics; for example, in the context of special relativity we call a metric the bilinear form defined by ␩␮␯ = 0 if ␮ ⫽ ␯, ␩ii = 1 if i = 1 , 2 , 3 and ␩00 = −1. The Minkwoski metric ␩ is also not positively defined and is not a metric in the mathematical sense. C. Magnetic field of the control manifold

In the universal space R3 we have the gauge potential 共we choose one of the two conventions兲共to be closer to the usual electromagnetic formalism, we have removed the factor i in order to have real fields in place of purely imaginary fields兲 A=

1 x2dx1 − x1dx2 苸 ⍀ 1R 3 2 r共r − x3兲

共41兲

and the magnetic monopole field is B = dA =

1 x1dx2 Ù dx3 + x2dx3 Ù dx1 + x3dx1 Ù dx2 苸 ⍀ 2R 3 2 r3

共42兲


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David Viennot

By using the identity

冢冣 ␦1i

dxi ↔ ␦2i ,

␦3i

we can write A 苸 ⍀1R3 as being a vector field of R3 : Aជ . We recall the definition of the Hodge star operator associated with a metric q in a n-dimensional manifold X: *q:⍀rX → ⍀n−rX *q共dxi1 Ù ¯ Ù dxir兲 =

冑兩det q兩共n − r兲!

qi1 j1 ¯ qir jr⑀ j1,. . .,jndx jr+1 Ù ¯ Ù dx jn ,

where ⑀ is the Levi-Civita symbol, qijq jk = ␦ik and det q = q1i1q2i2 . . . qnin⑀i1i2. . .in. In R3 we have simply *␦ : ⍀2R3 → ⍀1R3, with *␦共dx1 Ù dx2兲 = dx3. Then *␦B 苸 ⍀1R3 and we ជ = curl ជ Aជ . can consider it as being a magnetic vector field Bជ . The relation B = dA is then B Consider the pullback f * : ⍀*R3 → ⍀*M. The gauge potential in M is f *A = A i

⳵fi ␮ dR . ⳵R␮

共43兲

Note that we can compute f *A directly with the eigenvector of 关Eq. 共36兲兴:

ជ 兩d 兩1,Rជ 典 = f *共具+ ,xជ 兩d兩 + xជ 典兲 ជ ជ . f *A = 具1,R M x=f共R兲

共44兲

As f * is a chain map for the exterior differential, then the magnetic field is f *B = f *dA = dM f *A = Bij

⳵fi ⳵f j 1 ⳵fi ⳵f j ␮ ␯ dR Ù dR = 2B dR Ù dR2 . ij ⳵R␮ ⳵R␯ ⳵R1 ⳵R2

共45兲

In the same way that in R3 the field associated with the usual electromagnetic formalism is not f B 苸 ⍀2M but *g f *B 苸 ⍀0M, we see that *

*g共dR1 Ù dR2兲 = 冑兩det g兩g1␮g2␯⑀␮␯

共46兲

= 冑兩det g兩共g11g22 − g12g21兲

共47兲

= 冑兩det g兩det g−1

共48兲

=

=

冑兩det g兩

共49兲

det g sgn共det g兲

冑兩det g兩

.

共50兲

As g is a Riemannian metric 共because it is induced by the Euclidian metric兲 det g ⬎ 0, and then *g f *B = 2Bij

⳵fi ⳵f j 1 . ⳵R1 ⳵R2 冑det g

共51兲

Consider the two-coform


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N=

⳵fi ⳵f j ⳵ ⳵ . 1 2 i Ù ⳵R ⳵R ⳵x ⳵x j

共52兲

ជ = * N is a vector normal to the immersed manifold f共M兲. We compute the norm It is clear that N ␦ ជ: of N ជ 储2 = ␦ 共* N兲k共* N兲l 储N kl ␦ ␦ = ␦kl

共53兲

⳵fi ⳵f j k ⳵fn ⳵fm l ⑀ ⑀ ⳵R1 ⳵R2 ij ⳵R1 ⳵R2 nm

=共␦in␦ jm − ␦im␦ jn兲

共54兲

⳵fi ⳵f j ⳵fn ⳵fm , ⳵R1 ⳵R2 ⳵R1 ⳵R2

共55兲

where we have used the following property of the Levi-Civita symbol: ⑀ijk⑀nml␦kl = ␦in␦ jm − ␦im␦ jn 共see Ref. 24兲. We also have the result: det g = g1␮g2␯⑀␮␯ = ␦in

共56兲

⳵fi ⳵fn ⳵ f j ⳵ f m ␮␯ ⑀ 1 ␮ ␦ jm ⳵R ⳵R ⳵R2 ⳵R␯

= ␦in␦ jm

冉

⳵fi ⳵f j ⳵fn ⳵fm ⳵fm ⳵fn − ⳵R1 ⳵R2 ⳵R1 ⳵R2 ⳵R1 ⳵R2

=共␦in␦ jm − ␦im␦ jn兲

⳵fi ⳵f j ⳵fn ⳵fm . ⳵R1 ⳵R2 ⳵R1 ⳵R2

共57兲

冊

共58兲

共59兲

ជ 储 = 冑det g. Let uជ = 共1 / 冑det g兲N ជ be the unit normal vector to f共M兲. The density We see that 储N N ជ through f共M兲 is then of the flux of B ជ · uជ = B N

1

冑det g

具B,N典 = 2Bij

⳵fi ⳵f j 1 . ⳵R1 ⳵R2 冑det g

共60兲

Here 具.,.典 is the duality product between tangent and cotangent spaces 共i.e. 具dxi , ⳵ / ⳵x j典 = ␦ij兲. To summarize, we have the results

ជ · uជ . *g f *dA = *g f *B = *gdM f *A = B N

共61兲

The magnetic field in M共*g f *B兲 is equal to the density of the flux of the magnetic field of R3 through the immersed manifold f共M兲. D. Examples

The charts of the magnetic field dM f *A can be used to analyze the adiabatic properties of a quantum dynamical system, since this field makes monopoles appear at the eigenvalue crossings and since it is proportional to the nonadiabatic transitions. It is sometimes difficult to interpret these charts; the immersion property of the control manifold can help us with that interpretation. In order to illustrate the effects of the control manifold immersion we consider three simple examples of three level systems. To simplify the analysis our chosen method of producing an effective Hamiltonian is the partioning method by adiabatic elimination.21 Let H be the total Hamiltonian matrix with the form


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David Viennot

冉

H=

冊

H00 H10 , H01 H11

共62兲

where Hij are matrix blocs, H00 being the bloc corresponding to the considered crossing. The effective Hamiltonian after the adiabatic elimination process is 共63兲

−1 Heff = H00 − H01H11 H10 .

Moreover we can redefine Heff → Heff − 共trHeff / trI兲I 共I is the identity matrix兲 in order to have a traceless effective Hamiltonian. The first example we study is the Hamiltonian

H1共u, ␪兲 =

冢

0

0 uei␪

0

1

1

1

1

ue

−i␪

冣

共64兲

.

This Hamiltonian corresponds to the RWA description of a three level atom interacting with two lasers such that the first laser is quasiresonant with the transition 兩1典 to 兩3典 and the second laser is quasiresonant with the transition 兩2典 to 兩3典共兩i典 is the bare atom basis, the basis in which the matrix 共Eq. 共64兲 is written兲. u and ␾ are the amplitude and the phase of the first laser and constitute the control parameters; its polarization is constant. The second laser is constant with an amplitude equal to 1 in reduced units. For u = 0,

冢冣 0 0 0

H1共0, ␪兲 = 0 1 1 0 1 1

has the eigenvalues 兵0,0,2其. In the neighborhood of the crossing the effective Hamiltonian 共with adiabatic elimination of the state 3兲 is

冢冣 −

Heff 1 =

u2 uei␪ 2

ue

−i␪

u2 2

.

共65兲

.

共66兲

The immersion of the control manifold is then

冢冣 u cos ␪

f 1共u, ␪兲 =

u sin ␪ u2 − 2

The first immersed manifold is then diffeomorphic to an elliptic paraboloid. The second example is the Hamiltonian

冢

0

H2共␪, ␾兲 = 2共e−i␾ − 1兲 cos ␪

2共ei␾ − 1兲 cos ␪ 1 e

i␾

e−i␾ 1

冣

.

共67兲

This Hamiltonian corresponds to the RWA description of a three level atom interacting with four lasers such that the first and the second laser are quasiresonant with the transition 兩1典 to 兩2典, the third laser is quasiresonant with the transition 兩1典 to 兩3典 and the fourth laser is quasiresonant with the transition 兩2典 to 兩3典. The second laser is constant with an amplitude equal to 2 in reduced units, the first laser presents a phase modulation with a constant amplitude equal to 2, the third laser


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presents a polarization modulation with a constant amplitude equal to 1, and the fourth laser presents a phase modulation synchronized with the phase modulation of the second laser. The control parameters are ␾ 共the phase of the second and of the fourth lasers兲 and ␪ 共the angle between the polarization of the third laser and the atom electric dipole moment兲. For 共␪ , ␾兲 = 共␲ / 2 , 0兲,

H2

冢冣

冉冊

0 0 0 ␲ ,0 = 0 1 1 2 0 1 1

and in the neighborhood of the crossing, the effective Hamiltonian 共with adiabatic elimination of the state 3兲 is

Heff 2 共␪, ␾兲 =

冢

− 2共e

−i␾

sin2 ␪ 2

2共ei␾ − 1兲 − cos ␪ei␾

− 1兲 − cos ␪e

sin2 ␪ 2

−i␾

冣

.

共68兲


f 2共 ␪ , ␾ 兲 =

冢

共2 − cos ␪兲cos ␾ − 2 共2 − cos ␪兲sin ␾ sin2 ␪ − 2

冣

共69兲

.

The second immersed manifold is then diffeormorphic to a half torus. The third example is the Hamiltonian

H3共u, ␪兲 =

冢

2共ei␪ − 1兲 u cos

0 2共e−i␪ − 1兲 u cos

␪ 2

␪ 2

0

ue−i␪

uei␪

1

冣

.

共70兲

This Hamiltonian corresponds to the RWA description of a three level atom interacting with four lasers such that the first and the second laser are quasiresonant with the transition 兩1典 to 兩2典, the third laser is quasiresonant with the transition 兩1典 to 兩3典, and the fourth laser is quasiresonant with the transition 兩2典 to 兩3典. The second laser is constant with an amplitude equal to 2 in reduced units, the first laser presents a phase modulation with a constant amplitude equal to 2, the third laser presents a polarization modulation synchronized with the phase modulation of the first laser and it presents an amplitude modulation; finally, the fourth laser presents a phase modulation synchronized with the phase modulation of the second laser. The control parameters are u 共the amplitude of the third laser兲 and ␪ 共the phase of the second and of the fourth lasers, ␪ / 2 being the angle between the synchronized polarization of the third laser and the atom electric dipole moment兲. For 共u , ␪兲 = 共0 , 0兲,

冢冣 0 0 0

H3共0,0兲 = 0 0 0 , 0 0 1 and in the neighborhood of the crossing, the effective Hamiltonian 共with adiabatic elimination of the state 3兲 is


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David Viennot

Heff 3 共u, ␪兲 =

冢

u2 sin2

␪ 2

␪ 2共ei␪ − 1兲 − u2 cos ei␪ 2

2

␪ 2共e−i␪ − 1兲 − u2 cos e−i␪ 2

u2 sin2 −

2


f 3共u, ␪兲 =

冢

冉

2 − u2 cos

冉

冊

␪ cos ␪ − 2 2

冊

␪ sin ␪ 2 ␪ u2 sin2 2 2

2 − u2 cos

冣

.

␪ 2

冣

.

共71兲

共72兲

The third immersed manifold is then diffeomorphic to a half Moebius strip. ជ 兩d 兩1 , Rជ 典, which is equal to f *A, and we For these three examples we have computed 具1 , R M have drawn the magnetic field of the control manifold dM f *A. We have moreover drawn the flux density of a monopole magnetic field through the immersed surfaces. The results are represented in Fig. 2. We see with these figures that the immersion can be used to interpret the magnetic field chart, particularly for the third example, where the sign inversion of the field is explained by the nonorientability 共the twist兲 of the Moebius strip, and for the second example, where the existence of the two lobes of the field is explained.

V. THE APPARENT CHARGE OF A TWO LEVEL CROSSING MAGNETIC MONOPOLE IN THE CONTROL MANIFOLD

In the universal model the monopole has a charge equal to 21 . Experimentally, it is the magnetic field on M that we see, and the apparent charge of the monopole in M can be different from 1 25 2 . Leboeuf and Mouchet proposed a physical realization of nonelementary magnetic monopoles in quantum adiabatic dynamics by introducing constraint parameters. They explored the properties of their nonelementary monopoles by an analysis of Dirac strings. In this paper we want to exhibit nonelementary monopoles in a different way, using the immersion of the control manifold in the generalized universal space. Although our analysis is similar to the method followed by Leboeuf and Mouchet, we present it briefly, without repeating the discussion about Dirac strings. The interested reader can see Ref. 25

A. The geometry of a level crossing

In the universal model the single monopole has a magnetic charge equal to 21 . For the Hamiltonian 关Eq. 共31兲兴 that corresponds to the conical crossing of the eigenvalues ±冑共x1兲2 + 共x2兲2 + 共x3兲2, a conical crossing being a zero-order contact between the energy surfaces. We recall that a contact at Rជ 0 between two surfaces defined by the equations z = E+共R1 , R2兲 and z = E−共R1 , R2兲 in the space ជ 兲 − E 共Rជ 兲 and each of its 共R1 , R2 , z兲 is said to be of order r if and only if the function E+共R − ជ . We know that the concept of contact order is invariant under derivatives of order 艋r vanish at R 0 diffeomorphism, so this notion is well defined even without endowing M with a metric 共Fig. 2兲. To have a complete exposition of contact manifold theory, the reader can see Ref. 26.


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ជ 兩d 兩1 , Rជ 典, Right: the immersed manifold and the flux FIG. 2. Left: the magnetic field of the control manifold dM具1 , R M density of a central field centered on 共0, 0, 0兲, from top to bottom: the first example 关Eq. 共64兲兴, the second example 关Eq. 共67兲兴, and the third example 关Eq. 共70兲兴. The colors of the fields density are such that a strong positive field 共or a strong positive flux兲 is black, a strong negative field 共or a strong negative flux兲 is white, a vanishing field is gray. For the first example: q = u cos ␪ and r = u sin ␪.

For the generic Hamiltonian 关Eq. 共36兲兴, the contact order between the energy surfaces 共the ជ 兲f j共Rជ 兲␦ be the eigenvalues order of the intersection兲 depends of the immersion f. Let E± = ± f i共R ij ជ 兲 = 关E 共Rជ 兲 − E 共Rជ 兲兴 / 2. By definition if Rជ is the position of a of the effective Hamiltonian. Let ⑀共R + − 0 ជ 兲 = 0 共i.e., "i, f i共Rជ 兲 = 0兲. We have: level crossing then ⑀共R

冑

0

0

⳵⑀ = ⳵R␮

=

⳵f1 1 f ⳵R␮

⳵fi j f ␦ij ⳵R␮

共73兲

冑 f i f j␦ij

⳵f2 2 f ⳵R␮

⳵f3 3 f ⳵R␮

冑共f 1兲2 + 共f 2兲2 + 共f 3兲2 + 冑共f 1兲2 + 共f 2兲2 + 共f 3兲2 + 冑共f 1兲2 + 共f 2兲2 + 共f 3兲2 .

共74兲

It is then clear that


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David Viennot

冋册 ⳵fi ⳵R␮

"i,

Rជ =Rជ 0

=0Þ

冋册 ⳵⑀ ⳵R␮

Rជ =Rជ 0

共75兲

= 0.

The contact order between the energy surfaces is 艌1 if "i, "␮关⳵ f i / ⳵R␮兴Rជ =Rជ 0 = 0. More generally, we can prove that n−1

1 1 ⳵ n⑀ ⳵n−p f i ⳵p f j = ␦ij 兺兺 ⳵R␮1 . . . ⳵R␮n 2⑀ p=0 ␴苸Sn 共n − p兲!p! ⳵R␮␴共1兲 . . . ⳵R␮␴共n−p兲 ⳵R␮␴共n−p+1兲 . . . ⳵R␮␴共n兲 n−1

−

1 1 ⳵n−p⑀ ⳵ p⑀ , 兺兺 ␮␴共1兲 ␮␴共n−p兲 ␮␴共n−p+1兲 2⑀ p=1 ␴苸Sn 共n − p兲!p! ⳵R . . . ⳵R ⳵R . . . ⳵R␮␴共n兲

共76兲

where Sn is the nth group of permutations. Then, the order of the crossing is greater than or equal to r if "n 艋 r, "i, "␮1 , . . . , ␮n关⳵n f i / ⳵R␮1 . . . ⳵R␮n兴Rជ =Rជ 0 = 0. B. Perturbative analysis

We will see that the apparent charge of the magnetic monopole in M is related to the order of ជ . We the eigenvalue crossing and thereby is related to the number of zero derivatives of f at R 0 3 3 ជ consider a neighborhood U of R0. We suppose that f共U兲 is tangent to the plane x = 0 in R , i.e., that the Dirac strings are orthogonal to f共U兲. If this is not the case, we can always use a gauge transformation such that the Dirac strings rotate to be orthogonal to f共U兲; in other words we change the coordinates system such that the x3 axis coincides with the Dirac strings. In agreement ជ 苸 Uf 3共Rជ 兲 ⯝ 0. In the with this hypothesis, and with U sufficiently small, we can consider that "R 3 plane x = 0 we have the gauge potential A=

i x2dx1 − x1dx2 i xidx j⑀ij = . 2 共x1兲2 + 共x2兲2 2 xix j␦ij

共77兲

Note that in this expression the Kronecker symbol ␦ij plays the role of the metric tensor of the Euclidian plane and the Levi-Civita symbol ⑀ij plays the role of the vector cross product in the Euclidian space. They are then metric dependent tensors 共contrary to the previous parts of this paper where the Levi-Civita symbol has been used to represent the sum over the permutations兲. ជ is The gauge potential in M in the neighborhood of R 0 j ជ 兲 ⳵ f ⑀ dR␮ f i共R ij ⳵R␮ i . f *A = 2 f i共R ជ 兲f j共Rជ 兲␦

共78兲

ij

But in the neighborhood of Rជ 0 we can use a Taylor expansion:

冋册冋

i ជ 兲 = f i共Rជ 兲 + 共R␮ − R␮兲 ⳵ f f i共R 0 0 ⳵R␮

+ ¯ +

Rជ =Rជ 0

+

共R␮ − R0␮兲共R␯ − R0␯兲 ⳵2 f i 2 ⳵ R ␮⳵ R ␯

冋

共R␮1 − R0␮1兲 . . . 共R␮n − R0␮n兲 ⳵n f i ␮1 n! ⳵ R . . . ⳵ R ␮n

册

Rជ 0

册

Rជ 0

+ O共共共R␮ − R0␮兲共R␯ − R0␯兲␦␮␯兲n/2兲. 共79兲

Now we suppose that "r 艋 n − 1, "i, "␮1 , . . . , ␮r关⳵r f i / ⳵R␮1 . . . ⳵R␮r兴Rជ =Rជ 0 = 0; the contact order of the level crossing is then equal to n − 1. It is clear that any Taylor expansion must be of an order ជ we have equal to n. "Rជ in the neighborhood of R 0


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ជ兲 ⯝ f i共R

冋

共R␮1 − R0␮1兲 . . . 共R␮n − R0␮n兲 ⳵n f i ␮1 n! ⳵ R . . . ⳵ R ␮n

册

Rជ 0

冋

共R␮1 − R0␮1兲 . . . 共R␮n−1 − R0␮n−1兲 ⳵fi ⳵n f i ␯ ⯝ ␮1 ⳵R 共n − 1兲! ⳵R . . . ⳵R␮n−1⳵R␯

ជ兲 = g␮␯共R

⯝

冋

⯝

Rជ 0

共81兲

,

共82兲

共R␮1 − R0␮1兲 . . . 共R␮n−1 − R0␮n−1兲 ⳵n f i 共n − 1兲! ⳵R␮1 . . . ⳵R␮n−1⳵R␮

冋

册册

共R␯1 − R0␯1兲 . . . 共R␯n−1 − R0␯n−1兲 ⳵n f i 共n − 1兲! ⳵R␯1 . . . ⳵R␯n−1⳵R␯

Rជ 0

Rជ 0

␦ij ,

共83兲

冋

共R␮1 − R0␮1兲 . . . 共R␮n − R0␮n兲共R␯1 − R0␯1兲 . . . 共R␯n − R0␯n兲 ⳵n f i 共n!兲2 ⳵ R ␮1 . . . ⳵ R ␮n ⫻

⯝

册

⳵fi ⳵f j ␦ij ⳵R␮ ⳵R␯

⫻

f i f j␦ij ⯝

共80兲

,

⳵n f j ⳵ R ␯1 . . . ⳵ R ␯n

册

Rជ 0

册冋 Rជ 0

␦ij

共84兲

共R␮n − R0␮n兲共R␯n − R0␯n兲 g ␮n␯n n2 N n2

共85兲

共86兲

.

We then have f *A ⯝

in2 2N ⫻

冉冋

册冋册冊

共R␮1 − R0␮1兲 . . . 共R␮n − R0␮n兲 ⳵n f i n! ⳵ R ␮1 . . . ⳵ R ␮n

共R␯1 − R0␯1兲 . . . 共R␯n−1 − R0␯n−1兲 ⳵n f j 共n − 1兲! ⳵ R ␯1 . . . ⳵ R ␯n

Rជ 0

Rជ 0

⑀ij .

共87兲

However, the cross product in M induced by the Euclidian cross product is defined by 共R␮ − R0␮兲dR␯⑀␮g ␯ = 共R␮ − R0␮兲dR␯

⳵fi ⳵f j ⑀ij ⳵R␮ ⳵R␯

⯝共R␮ − R0␮兲dR␯ ⫻

共88兲

冋

共R␮1 − R0␮1兲 . . . 共R␮n−1 − R0␮n−1兲 ⳵n f i ␮1 共n − 1兲! ⳵R . . . ⳵R␮n−1⳵R␮

冋

共R␯1 − R0␯1兲 . . . 共R␯n−1 − R0␯n−1兲 ⳵n f j 共n − 1兲! ⳵R␯1 . . . ⳵R␯n−1⳵R␯

册

Rជ 0

⑀ij .

册

Rជ 0

共89兲

In summary, we have the result


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FIG. 3. 共Color online兲 The energy surfaces drown with respect to the coordinates of M for 共from left to right兲: the ជ 兲 = 共R1 , R2 , 0兲共a monopole with charge 1 兲, the immersion f共Rជ 兲 = 共共R1兲2 , 共R2兲2 , 0兲共a monopole with charge 1兲, immersion f共R 2 ជ 兲 = 共共R1兲3 , 共R2兲3 , 0兲共a monopole with charge 3 兲. and the immersion f共R 2

f *A ⯝

ni 共R␮ − R0␮兲dR␯⑀␮g ␯ . 2 共R␭ − R␭0 兲共R␳ − R0␳兲g␭␳

共90兲

By comparing this expression with Eq. 共77兲, we see that f *A is the gauge potential of a monopole with magnetic charge equal to n / 2 in the curved space M with metric g. We conclude that if the magnetic monopole in M has a charge equal to n then the contact order of the energy surface crossing is equal to n − 1 共Fig. 3兲.

C. Topological analysis

The previous analysis has the advantage that it exhibits the monopole gauge potential in M, but a shorter analysis is possible using a topological result. Let S1 be a circle in the plane x3 = 0 centered over 0. We have 共see Ref. 27, p. 152兲 −i 2␲

冖

1 4␲

A=

S1

冖

S1

x2dx1 − x1dx2 1 = 共x1兲2 + 共x2兲2 2

共91兲

more generally for any loop C in the plane M 共a loop is a closed path兲 we have 1 2␲

冖

C

x2dx1 − x1dx2 = n, 共x1兲2 + 共x2兲2

共92兲

where n is the number of oriented turns of C around 0. This number is a topological character of the loop in R2 \ 兵0其: it is invariant by homotopy and is called the winding number. We can use this ជ . f共M兲 is invariant to characterize the monopole charge. Let S1 be a circle on M centered on R 0 1 not a plane, but if the radius of S is sufficiently small we can consider that f共M兲 is approximately flat in the neighborhood of f共S1兲. We use the complex coordinate of M, z = R1 + iR2,

冉

f共z兲 = 冑共f 1共z兲兲2 + 共f 2共z兲兲2 exp i arctan

冉

= ⑀共z兲exp i arctan

f 2共z兲 f 1共z兲

冊

冊

f 2共z兲 . f 1共z兲

共93兲

共94兲

If the contact order is equal to n − 1, then ⑀ has a zero of order n at z0. We can then write f共z兲 = 共z − z0兲nh共z兲, where h共z兲 is a holomorphic function and where 共z − z0兲n = rnein␪. We have: arg f共z兲 = arg 共z − z0兲n + arg h共z兲 = n␪ + arg h共z兲,

共95兲


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1 2␲

冖

d arg f =

S1

1 2␲

冖

S1

1 d arg h共z兲 2␲

nd␪ +

共96兲

共97兲共98兲

=n. We also have

d arg f = d arctan

f 2 f 1df 2 − f 2df 1 = = − i2f *A f 1 共f 1兲1 + 共f 2兲2

共99兲

and so conclude that

−i 2␲

冖

f *A =

S1

−i 2␲

冖

f共S1兲

n A= . 2

共100兲

The monopole in M thus has an apparent charge equal to n / 2.

VI. NON-ABELIAN MONOPOLE ASSOCIATED WITH A MULTILEVEL CROSSING

In the previous parts we have considered only two-level crossings between nondegenerate states. Now we consider multilevel crossings, for example the crossing of three nondegenerate states or the crossing of a doubly degenerate state with a nondegenerate state.

A. Immersion of the control manifold

By elimination of the other states, we suppose that the Hamiltonian can be written in the neighborhood of the multilevel crossing as

ជ 兲 = f i共Rជ 兲J , Heff共R i

共101兲

where 兵Ji其 is a set of generators of a real Lie algebra g associated with a Lie group G 共the Hamiltonian symmetry group兲. Usually we have G = U共m兲, where m is the number of levels involved in the crossing 共or the sum of the degeneracy degrees of the levels involved in the crossing兲. But we know that the physics of an m-level system can be described simply by G = SU共m兲共see Refs. 28 and 29兲. Let d be the number of generators of G 共d = dim g兲. The immersion ជ 兲 = 共f 1共Rជ 兲 , . . . , f d共Rជ 兲兲 is then a map from M to Rd. Rd plays the role of the f defined by f共R generalized universal space in place of R3. Let 兵Hi , E␣其i,␣ be the Cartan basis of g. The Cartan subalgebra 共the algebra generated by 兵Hi其i兲 is the Lie algebra of the maximal torus T of G. Let ជ 兲 be the diagonal matrix of the eigenvalues of H共Rជ 兲共the matrix of levels involved in the E共R ជ 兲 = bi共Rជ 兲H . Then the diagonalcrossing兲. As the diagonal matrix group is Abelian, we have E共R i eff ization of H can be written


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David Viennot

ជ 兲 = ei␹␣共Rជ 兲E␣E共Rជ 兲e−i␹␣共Rជ 兲E␣ Heff共R

冉

共102兲

冊

ជ 兲E + ¯z␣共Rជ 兲E 兲 E共Rជ 兲 =exp i 兺共z␣共R ␣ −␣ ␣⬎0

冉

冊

ជ 兲E + z␣共Rជ 兲E 兲 , ¯␣共R ⫻exp − i 兺共z ␣ −␣ ␣⬎0

共103兲

ជ 兲 = ei␹␣共Rជ 兲E␣ is the diagonalizing matrix for Heff共Rជ 兲. The Berry phase depends only on where T共R ជ 兲 or equivalently on ␹␣共Rជ 兲. Moreover it is clear that 兵␹␣其 and 兵z␣其 are, respectively, real and z␣共R ␣ ␣ complex coordinates systems for the manifold G / T. This manifold is called a flag manifold, Mostafazadeh30 has shown that it is also the universal manifold for the system with symmetry ជ 兲 = 共␹␣共Rជ 兲兲 is the universal map. Clearly the flag group G, and ␹ : M → G / T defined by ␹共R ␣ manifold G / T is a complex manifold which has a Riemannian structure induced by the embedding G / T CPn−1 共see Ref. 30兲, where n is the dimension of the original Hilbert space. The Kählerian structure of CPn−1 induces the Riemannian structure of G / T considered as a real manifold. Let r be the rank of g 共the dimension of the Cartan subalgebra兲. Rd is foliated with leafs which are r d ជ 兲其 diffeomorphic to G / T and with the foliation parameters 兵bi共R i=1,. . .,r, 共G / T , R 兲 ⯝ R . Considering a particular leaf G / T, we have the following commutative diagram: M

␹␣

G/T

→

f i ↓ ⌽␣ Rd

→

↓ CPn−1

where ⌽ is defined by

ជ 兲 = Ad共ei␹␣共Rជ 兲E␣兲bi共Rជ 兲H = ei␹␣共Rជ 兲ad共E␣兲bi共Rជ 兲H = ⌽i共Rជ 兲H + ⌽␤共␹共Rជ 兲兲E ., Heff共R i i i ␤

共104兲

Here Ad is the adjoint action of the Lie group on its Lie algebra, and ad is the adjoint action of the Lie algebra on itself. We suppose that G is compact and semisimple; then the Killing form of its complex Lie algebra gC is positively defined. Following the commutativity structure of the previous diagram, the Riemannian structure of G / T is also induced by the Killing form of gC considered as a scalar product on Rd. If G = SU共N兲 then 2NK is the Euclidian metric of Rd 共where K is the Killing form of SU共N兲兲. In the case of a two-level crossing, G = SU共2兲, d = 3 and the maximal torus is U共1兲; the flag manifold is then SU共2兲 / U共1兲 ⯝ S2 in agreement with the discussion in previous sections. In the case of a three-level crossing, G = SU共3兲, the universal space is R8, the maximal torus is T2 = U共1兲 ⫻ U共1兲 and the flag manifold is SU共3兲 / 共U共1兲 ⫻ U共1兲兲. For the generic case of an N-level crossing, the symmetry group is SU共N兲, its dimension is d = N2 − 1, and the standard monopole 2 exists in the universal space RN −1. The flag manifold 共the universal manifold兲 is SU共N兲 / TN−1 ⯝ CPN−1 › CPN−2 › . . . › CP1 where TN−1 is the 共N − 1兲-torus:

and where › denotes a possible nontrivial topological product. The induced metric on M is g␮␯ = ␦ij

⳵fi ⳵f j ⳵R␮ ⳵R␯

共105兲

with i = 1 , . . . , N2 − 1 and ␮ = 1 , 2.


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B. An analogy with the field theory 2

The monopole gauge potential is A 苸 ⍀1共RN −1 , su共N兲兲. The monopole field is obtained by the 2 Cartan structure equation F = dA + 21 关A , A兴苸 ⍀2共RN −1 , su共N兲兲 and satisfies half of the Yang-Mills equations, i.e., the Bianchi identity dF + 关A , F兴 = 0. In M, f *A 苸 ⍀1共M , su共N兲兲 and f *F 苸 ⍀2共M , su共N兲兲 satisfy similar equations. Consider the Abelian case of a pure adiabatic transport. We suppose that the evolution of the system is a path C on M which passes in the neighborhood of a two-level crossing but is sufficiently far away or is traversed at such a speed that no popuជ 共t兲. Let 兵兩 + , Rជ 典N , 兩− , Rជ 典N其 be lation transfer occurs. The path C is parametrized by a function t 哫 R N ជ the two eigenvectors. If we suppose that ␺共0兲 = 兩 + , R共0兲典 then we have

冉冕

t

␺共t兲 = exp − iប−1

ជ 共t⬘兲兲dt⬘ − E+共R

0

冕

Rជ 共t兲

Rជ 共0兲

冊

ជ 兲兩 + ,Rជ 共t兲典N , A+N共R

共106兲

ជ 兩d 兩 + , Rជ 典N. The gauge group associated with this situation is U共1兲. We have a where A+N = N具+ , R M complete analogy with electrodynamics, where ␺ is the matter field of a charged particle which is moving on the path C in M and which is subject to the monopole magnetic field. Indeed, Ezawa31 notes that a single matter field ␾共x兲 cannot be used in the whole space-time in the presence of a magnetic monopole. He uses a path-dependent formalism with ␾共x , P兲 = exp共兰⬁x A兲␾共x兲 where P is a path from infinity to x. This new field is gauge invariant. In the same way, in the generalized universal space R3 we cannot use a single field 兩 + , xជ 典 as a matter field because 1

兩 + ,xជ 典N =

冑2r共r + x3兲

兩 + ,xជ 典S =

1

冉

x1 + ix2

冉

r − x3

r + x3

冊

is not defined for x3 艋 0 and

冑2r共r − x3兲

x1 − ix2

冊

is not defined for x3 艌 0. Following Ezawa, by elimination of the dynamical phase, we set ˜␺共xជ , C兲 = exp共−兰xជ A 兲兩 + , xជ 典 where C is a path from an arbitrary point xជ to xជ . Let U and U be the 0 S N xជ 0 + charts for which A+S and A+N, respectively, are well defined. Suppose that xជ 0 苸 US. Then

˜ 共xជ ,C兲 = ␾

冦

冉冕冊冉冕冊冉冕冊 xជ

exp −

xជ 0

A+S 兩 + ,xជ 典Sif xជ 苸 US

xជ 1

exp −

xជ 0

xជ

A+S exp −

xជ 1

AN 兩 + ,xជ 典N if xជ 苸 UN, with xជ 1 苸 UN 艚 US .

冧

共107兲

We see that the formalism used by Ezawa to define a matter field in the presence of a magnetic monopole is similar to the adiabatic transport formula. Then we can say that the quantum system driven along a path C in the control parameter manifold M is similar to, for example, an electron which is forced to follow C in presence of a monopole magnetic field. Now consider the case of a crossing of three nondegenerate states. The wave function is 3

冋冉

␺共t兲 = 兺 T exp − iប−1 b=1

冕

t

0

ជ 共t⬘兲兲dt⬘ − E共R

冕

Rជ 共t兲

Rជ 共0兲

ជ兲 A共R

冊册

ជ 共t兲典兩b,R

共108兲

ba

ជ 典 , 兩2 , Rជ 典 , 兩3 , Rជ 典其 and the gauge group is SU共3兲. SU共3兲 is represented on the space spanned by 兵兩1 , R 1,0 by the irreducible representation D 共see Refs. 32兲. The situation is completely analogous to that of a quark field ␺ in the presence of a colored monopole, see Ref. 5, 6, and 31,


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ជ 典 , 兩2 , Rជ 典 , 兩3 , Rជ 典其 being a color triplet. As is the case in chromodynamics, “since SU共3兲 is 兵兩1 , R unbroken symmetry, (¼) the specification of the colour of a quark field depends on an arbitray choice of basis for the three-dimensional space of colours at each point in space-time” 共see Ref. ជ 典 , 兩2 , Rជ 典 , 兩3 , Rជ 典其 defined the colors at each point Rជ of M, the color operators 33兲, the basis 兵兩1 , R being

冢

1

0

0

冣

J3 = 0 − 1 0 , 0 0 0

冑3J8 =

冢

1 0

0

0 1

0

0 0 −2

冣

共109兲

ជ 典 , 兩2 , Rជ 典 , 兩3 , Rជ 典其. The initial point of the the matrices being written in the Rជ -dependent basis 兵兩1 , R dynamics Rជ 共0兲 can be considered as being a privileged point and consitutes a symmetry breaking. For example, in the case of an atom or a molecule driven by a pulse chirped laser, the initial point ជ 共0兲典 , 兩2 , Rជ 共0兲典 , 兩3 , Rជ 共0兲典其 is the position in M for which the laser is off. In this case the basis 兵兩1 , R is privileged as the basis for the bare atom or molecule unperturbed by the field. The two possible ជ -dependent basis,” descriptions of the color using “the symmetry breaking basis” and using “the R are associated with the choice of the molecular time-dependent description of population transfers, using “the unperturbed basis” or using “the instantaneous dressed basis.” VII. FINAL REMARKS AND CONCLUSION A. Global description versus local description

All of the previous discussion has concerned the neighborhood of a crossing, and has thus given a local analysis. If we consider M globally these might be several crossings 共monopoles兲 on M. Let n be the dimension of the original Hilbert space and m be the dimension of the adiabatic subspace. We know that the universal manifold of the gobal description is the Grassmanian manifold Gm共Cn兲. The relation between global and local description is the following. Let f be the ជ be a point where there exists a level crossing. Let global universal map from M to Gm共Cn兲. Let R 0 G / T be the flag manifold of this level crossing 共the local universal manifold兲 and ␹ be the local universal map from M to G / T. Gm共Cn兲 = U共n兲 / 共U共n − m兲 ⫻ U共m兲兲 where U共n兲 is the group of unitary tranformations of the original space, U共m兲 is the group of unitary transformations of the adiabatic space and U共n − m兲 characterizes the adiabatic independence of the quantum system from the exterior of the adiabatic space. It is clear that G is a subgroup of U共n兲 and T is a subgroup of ជ 兲 = ␹共Rជ 兲. U共m兲, and then G / T is a submanifold of Gm共Cn兲 which is localized with the point f共R 0 0 The local description depends on the choice of the effective Hamiltonian technique. Indeed, Heff does not have the same expression in the different partitioning methods.20–22 The geometric ជ 兲 is good approximation of H共Rជ 兲 for the local description will be relevant if and only if Heff共R ជ兲 crossing, i.e., if the behaviors of the eigenvalues and of the overlaps of the eigenvectors of Heff共R ជ 共and their first derivatives兲 are close to the behaviors of the associated quantities of H共R兲. In general the problem of finding an efficient effective Hamiltonian associated with a crossing is not a simple one. B. The control manifold

All of the discussion can be generalized to a control manifold with dimension greater than two, exept for the links between ⴱg f *B and the monopole magnetic flux, which is the most interesting aspect of the monopole magnetic field in the two-dimensional case. Note that for a two-level crossing, the map f : M → R3 cannot be an immersion if dim M ⬎ 2. We can remark that in a theoretical study, the choice of the control manifold can be arbitrary; in order to have a physical meaning for M and its virtual magnetic monopoles, we can choose M


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to be closely appropriate to the experimental situation. The two parameters of M must be two “control levers” of the experimentalist, in order that the monopole effects can appear naturally in the experimental results C. The charge of non-Abelian monopoles

The definition of the charge of a non-Abelian monopole, for example, for a three level crossing, is not clear. In the Abelian case the generalized universal space is R3 and we can apply the method followed in field theory. In the non-Abelian cases the generalized universal space is Rd with d ⬎ 3, which cannot be identified with the usual physical space. The topological methods usually followed must be adapted and more subtle studies are needed to find a relevant definition of the charge of the non-Abelian adiabatic monopoles. D. About nonadiabatic evolutions

The adiabatic assumption states that the quantum dynamical system can be described at each time by a little set of instantaneous eigenvectors. In the nonadiabatic cases, there exist some techniques to describe the quantum system with a time-dependent non-eigenbasis 共see, for example, the time-dependent wave operator theory兲.34 In this case, the dynamics can be represented by a finite dimensional effective Hamiltonian. We can suppose that these time-dependent vectors can be expressed as control parameter-dependent vectors 共in fact this is a strong assumption because in general these vectors depend not only on the instantaneous time but also on the past ជ 共t兲典 is not an evolution兲. In this case the wave function takes the form of Eq. 共3兲 but where 兩a , R ជ 共t兲兲 = 具a , Rជ 共t兲兩Heff共Rជ 共t兲兲兩b , Rជ 共t兲典 is not a diagonal matrix. Following eigenvector and where E共R ab the works of Mostafazadeh and Bohm35,36 we can suppose that there exists a smooth map ˜ eff共R ជ 兲 = Heff共F共Rជ 兲兲. Let f be the immerF : M → M such that 兵兩a , Rជ 共t兲典其a are the eigenvectors of H eff ˜ sion map associated with H , we can consider f ⴰ F as the immersion map of Heff, this map is well ˜ eff. associated with the properties of magnetic monopoles since 兵兩a , Rជ 典其a are the eigenvectors of H Nevertheless the physical significance of these monopoles is not clear because they are not associated with an eigenlevel crossing. Moreover the assumption of the existence of F and above all the assumption of the control parameters dependence of the basis are important limitations on the class of nonadiabatic quantum systems for which we can apply the theory presented in this paper. E. Conclusion

The knowledge of the magnetic field in the control manifold can be very important for the numerical simulations of quantum adiabatic dynamics. By computing this field, which is equal to the monopole field density of f共M兲, we can localize the level crossings. It is well known that the variations of the wave function are more important in the neighborhood of the level crossings. If we model the control manifold by using a discrete numerical lattice X, then we need more vertices in the neighborhood of the level crossings in order to have a minimal data storage requirement, together with a good description of the wave function. We must then employ a nonhomogeneous lattice with small cells in the neighborhood of the crossing and with larger cells elsewhere. We can use the field ⴱg f *B to obtain a criterion about the local choice of cell sizes. ACKNOWLEDGMENTS

I would like to thank Professor John P. Killingbeck and Professor Jacky Cresson for their help. M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 共1984兲. B. Simon, Phys. Rev. Lett. 51, 2167 共1983兲. 3 F. Wilczek and A. Zee, Phys. Rev. Lett. 52, 2111 共1984兲. 4 D. Viennot, J. Math. Phys. 46, 072102 共2005兲. 5 Y. M. Cho, Phys. Rev. Lett. 44, 1115 共1980兲. 6 A. Sinha, Phys. Rev. D 14, 2016 共1976兲. 7 J. Moody, A. Shapere, and F. Wilczek, Phys. Rev. Lett. 56, 893 共1986兲. 1 2


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