# Born-Oppenheimer and the Geometry of Ray Space:
an application to cold atoms

Joseph Samuel  International Centre for Theoretical Sciences, Bangalore, India
560089
and
Raman Research Institute, Bangalore, India 560080

###### Abstract

It is known that, within the Born-Oppenheimer approximation, the slow modes of
the nuclear motion are altered by three effects that emerge from integrating
out the fast modes of the electronic motion. The first is an effective scalar
potential $V_{\mathrm{d}yn}$ coming from the eigenvalue of the electronic
state, the second is an effective magnetic field coming from the Berry phase
vector potential $A$. The third term is an additional potential
$V_{\mathrm{g}eom}$ originating in the geometry of ray space and the Fubini-
Study metric. In this article, we illustrate these effects and their geometric
origin in the context of a simple toy model of a slow neutron interacting with
a strong, spatially varying magnetic field. Regarding the neutron spin as a
fast degree of freedom, we work out the slow dynamics of the motion of the
neutron. Our treatment is geometrical and brings out the effects originating
in the Kähler geometry of ray space and the Fubini-Study metric. We then give
examples of magnetic field configurations which isolate these three separate
effects. Finally we apply these ideas to the trapping of cold atoms. Our main
result is that the geometric electric potential $V_{\mathrm{g}eom}$ dominates
for smaller traps and can be used to confine cold atoms in static traps. This
observation could result in better and smaller atomic clocks. This paper is
dedicated to Michael Berry on his 80th Birthday.

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##  1 Introduction

Berry’s work[1] has illuminated the geometry underlying the ray space of
quantum mechanics. The seminal ideas in [1] led to numerous experiments (for
example, [2, 3, 4]), applications and generalisations[5, 6]. It was also
realised that the idea had been anticipated several times[7, 8] in different
contexts. Michael Berry traces the seed of the idea as far back as William
Rowan Hamilton (1832) in his theoretical studies on conical double refraction
and their experimental confirmation by Herbert Lloyd. One of the anticipations
was due to Longuet-Higgins[9] in the context of the Born-Oppenheimer
approximation which is widely used in the quantum theory of lattice dynamics
and molecular physics. The Born-Oppenheimer approximation separates the fast
electronic motion from the slow nuclear motion. Initially the motion of the
nucleii is neglected and the frozen nuclear coordinates are treated as
parameters in the electronic problem. Invoking the adiabatic theorem to allow
for slow motion of the nuclei, one then uses a reduced Schrödinger equation to
solve for the nuclear motion. The purpose of this article is to elucidate the
derivation of the slow dynamics in a simple toy model. While this topic has
been studied before in Refs.[10, 11, 12], our treatment may perhaps appeal to
a geometrically minded reader. We go on to apply these geometric ideas to cold
atom physics and make suggestions which could potentially result in better
atomic clocks.

Let us motivate our discussion with a simple classical example . Consider a
well oiled bicycle wheel111The precession and nutation of a rapidly spinning
top can also be viewed as slow modes subject to a monopole magnetic field in
exactly the same way. mounted on an axle rod so that it is free to rotate
about an axis pointing along the rod. If the direction of the rod is held
fixed, and the wheel spun at angular velocity $\omega$ the wheel rotates
through an angle $\omega t$ in time $t$ (neglecting friction). If the
direction of the rod is slowly changed to traverse a closed circuit $C$ in the
space of directions, we may naively expect that the total angle of rotation is
still $\omega t$. However, this naive expectation is wrong. The total angle of
rotation includes an additional piece given by the solid angle traversed by
the circuit $C$. The explanation is that the wheel rotates about different
axes during the circuit and such rotations do not commute. This effect is the
Hannay angle[13], a classical analogue of the Berry phase. To perform this
experiment, it helps to mark the rim of the bicycle wheel and use a
stroboscope or camera to track the motion of the wheel.

So far we have regarded the direction of the rod as a parameter within the
control of the experimenter. However, parameters are also dynamical variables,
subject to the laws of motion. To realise this, let us suspend the wheel and
axle from the ceiling and repeat the experiment by spinning the wheel and
setting the axle pendulum in motion. We find that the trajectory of the axle
resembles that of a charged particle in a magnetic field. With more work one
can arrange for the axle to trace its path in a sand pit and see precessing
ellipses. Reversing the sense of the spin reverses the sense of the
precession. What we are seeing here, in the spirit of the Born-Oppenheimer
approximation, is the back reaction of the fast mode (the spin of the bicycle
wheel) on the slow modes (the oscillation of the pendulum) of the system. A
less elaborate experiment, less convincing, but more in tune with the sloppy
habits of theoretical physicists, is to twist a wet tea bag so its string is
wound up and let it untwist while oscillating, suspended from its string.
Michael Berry has been subjected to both these demonstrations on one of his
visits to the Raman Research Institute.

In this article, we consider a simple quantum mechanical toy model222 Another
toy model has been discussed in [14] to study the Born-Oppenheimer
approximation. This quantum model is described in section II. In section III,
we derive the dynamics of the slow modes taking into account the back reaction
of the fast mode. In section IV we describe some geometrical aspects of ray
space to bring out the geometric nature of the extra scalar potential. Section
V describes magnetic field configurations which bring out the effect of
different terms in the slow Hamiltonian. Section VI shows how the geometrical
ideas of the toy model carry over to the more realistic model of molecular
dynamics. In section VII, we suggest that geometric electromagnetic fields may
find application in developing better atomic clocks. Section VIII is a
concluding discussion.

Notation: We use Dirac’s braket notation for the fast mode but not for the
slow modes. Projectors onto one dimensional subspaces (proportional to the
vector $\ket{e}$) of the fast Hilbert space are denoted $P^{e}$ and satisfy
$P^{e}P^{e}=P^{e}$. The orthogonal projection is $Q^{e}=\mathbb{I}-P^{e}$. We
will use a global space independent frame $\\{\ket{+},\ket{-}\\}$ as well as a
local frame $\\{\ket{e_{+}(\vec{R})},\ket{e_{-}(\vec{R})}\\}$, which is space
dependent. Spatial derivatives, $\frac{\partial}{\partial R^{i}}$ will be
denoted $\nabla_{i}$. The projectors $P^{+}$ and
$Q^{+}=P^{-}=\mathbb{I}-P^{+}$, commute with $\nabla_{i}$, since the basis
$\\{\ket{+},\ket{-}\\}$ is independent of space. Let us also clarify here that
there are two kinds of electromagnetic effects in play here. Those
electromagnetic fields that are produced by charges and currents are called
laboratory electromagnetic fields. Those electromagnetic fields that emerge
from the geometry of ray space are referred to as geometric electromagnetic
fields (also referred to in [15] as “synthetic gauge fields”).

##  2 A Toy Model

We consider a slow neutron in a strong external magnetic field
${\vec{B}}({{\vec{R}}})$. Let us regard the spin as the fast variable and the
spatial position ${\vec{R}}$ of the neutron as the slow variable. The total
Hamiltonian of our system is the sum

$H_{T}=H_{f}+H_{s}\hskip 14.22636pt.$ (1)

The fast Hamiltonian $H_{f}$ is that of a spin half in a magnetic field

$H_{f}=-\mu{\vec{B}}({\vec{R}}).{\vec{\sigma}}\hskip 14.22636pt,$ (2)

where ${\vec{B}}({\vec{R}})$ is a spatially varying magnetic field and $\mu$
is the magnetic dipole moment of the neutron. We will assume that
$|\vec{B}(\vec{R})|$ does not vanish in the region of interest. In the spirit
of the Born-Oppenheimer approximation, we will assume that the magnetic field
$\vec{B}(\vec{R})$ varies slowly with ${\vec{R}}$. (More precisely, we would
like
$\frac{d\vec{R}}{dt}.\vec{\nabla}\big{(}\hat{B}(\vec{R})\big{)}<<\frac{\mu|\vec{B}(\vec{R})|}{\hbar}$,
where $\hat{B}$ is a unit vector and $\frac{d\vec{R}}{dt}$ a typical velocity
of the slow neutron .)

The slow Hamiltonian $H_{s}$ is given by the kinetic energy of the neutron

$H_{s}=-\frac{\hbar^{2}}{2m}{\vec{\nabla}}.{\vec{\nabla}}\hskip 14.22636pt.$
(3)

We now regard ${\vec{R}}$ as a parameter in the fast Hamiltonian
$H_{f}(\vec{R})$ and diagonalise the fast Hamiltonian. (In molecular physics,
this corresponds to holding the nuclear configuration fixed and solving for
the electronic eigenstates.) Let the total wave function of the system be

$\Psi({\vec{R}})=\begin{bmatrix}\psi_{+}(\vec{R})\\\
\psi_{-}(\vec{R})\end{bmatrix}=\psi_{+}({\vec{R}})\ket{+}+\psi_{2}({\vec{R}})\ket{-},$
(4)

where

$\displaystyle\ket{+}$ $\displaystyle=\begin{bmatrix}1\\\
0\end{bmatrix},\,\,\,\ket{-}$ $\displaystyle=\begin{bmatrix}0\\\
1\end{bmatrix}$ (5)

We will refer to the orthonormal basis $\\{\ket{+},\ket{-}\\}$ as the global
frame. We will also need a local frame adapted to the magnetic field. To do
this, let us solve the eigenvalue equation

$H_{f}(\vec{R})\ket{e({\vec{R}})}=\lambda\ket{e(\vec{R})}$ (6)

for the fast spin degree of freedom.

This equation has two independent solutions
$\\{\ket{e_{+}(\vec{R})},\ket{e_{-}(\vec{R})}\\}$ (satisfying
$\braket{e_{+}(\vec{R})}{e_{-}(\vec{R})}=0$) corresponding to the two distinct
eigenvalues $\\{\lambda_{+}=-\mu|B|\\},\lambda_{-}=\mu|B|\\}$. For normalised
eigenstates $\ket{e_{\pm}(\vec{R})}$, equations (6) determine
$\ket{e_{a}(\vec{R})},a=\\{+,-\\}$ up to a phase factor
$w_{a}(\vec{R}),|w_{a}(\vec{R})|=1$ a complex number of modulus unity, which
depends on $a$ as well as $\vec{R}$. Let us make an arbitrary smooth choice of
these phases333This is always possible locally.. We now have
$\\{\ket{e_{+}(\vec{R})},\ket{e_{-}(\vec{R})}\\}$ , which defines a local
orthonormal basis which is adapted to the magnetic field and provides us with
a moving frame. We refer to this as the local basis.

The two orthonormal bases $\ket{a}$ and $\ket{e_{a}(\vec{R})}$ are related by
a Unitary transformation $U(\vec{R})$

$\ket{e_{a}(\vec{R})}=U(\vec{R})\ket{a}$ (7)

We can transform our original eigenvalue problem (6) by defining
$\Psi(\vec{R})=U(\vec{R})\Phi(\vec{R})$. The new eigenvalue equation is

${\cal H}_{T}\Phi=E\Phi,$ (8)

where ${\cal H}_{T}=U^{-1}H_{T}U$ is given by

${\cal
H}_{T}=\lambda_{+}(\vec{R})\sigma_{3}-\frac{\hbar^{2}}{2m}U^{\dagger}(\vec{R})\nabla_{i}\nabla_{i}U(\vec{R})$
(9)

Let us now work out the dynamics governing the slow modes of the system.

##  3 The Reduced Hamiltonian

Let us assume that the spin is in the ground state $\ket{+}$ of the fast
Hamiltonian. The adiabatic theorem assures us that (with an exponentially
small chance of transition) the spin remains in the ground state. We can now
pass to the reduced space by projecting the wave function and the Hamiltonian
into the $\ket{+}$ subspace. $\phi(\vec{R})=\braket{+}{\Phi(\vec{R})}$. The
spin state can be described by the density matrix
$\rho_{f}=(1+\sigma_{3})/2=\ket{+}\bra{+}=P^{+}$, where $P^{+}$ is the one
dimensional projector on to the ground state. The reduced Hamiltonian for the
slow degrees of freedom is ${\cal H}_{red}=Tr[{\cal
H}_{T}\rho_{f}]=\bra{+}{\cal H}_{T}\ket{+}$.

Writing out the reduced Hamiltonian, we have for any wave function
$\xi(\vec{R})$

${\cal
H}_{red}\xi(\vec{R})=\lambda_{+}(\vec{R})\braket{+}{\sigma_{3}}{+}\xi(\vec{R})-\frac{\hbar^{2}}{2m}\bra{+}U^{\dagger}(\vec{R})\nabla_{i}\nabla_{i}U(\vec{R})\ket{+}\xi(\vec{R}),$
(10)

where it is understood that the derivative operators act on everything to
their right unless explicitly bracketed.

###  3.1 Dynamical Scalar Potential

The first term in ${\cal H}_{red}$, gives us the fast eigenvalue which gives
an effective space dependent scalar potential
$V_{dyn}(\vec{R})=-\mu|\vec{B}(\vec{R})|$ for the slow modes. We call this the
dynamical scalar potential as it is related to the eigenvalue of the fast
Hamiltonian. This term is well known in molecular physics: the electronic
eigenvalue serves as a scalar potential in which the slow nucleii move.
Adiabatically moving a neutron to a region of lower magnetic field requires
the mover to do some work.

###  3.2 Geometric Vector Potential

The second term in ${\cal H}_{red}\xi(\vec{R})$ can be rewritten as

$-\frac{\hbar^{2}}{2m}\braket{+}{U^{\dagger}(\vec{R})\nabla_{i}U(\vec{R})U^{\dagger}(\vec{R})\nabla_{i}U(\vec{R})}{+}\xi(\vec{R})$
(11)

by inserting the identity $UU^{\dagger}=1$ between the derivatives. It is
elementary to verify that for any function $\xi(\vec{R})$

$U^{\dagger}(\vec{R})\nabla_{i}U(\vec{R})\,\,\xi(\vec{R})=\nabla_{i}\xi(\vec{R})+\omega_{i}(\vec{R})\xi(\vec{R})$
(12)

where $\omega_{i}(\vec{R})=U^{\dagger}(\vec{R})\nabla_{i}U(\vec{R})$ is an
effective $U(2)$ gauge field (with vanishing $U(2)$ field strength). Writing
${\cal D}_{i}=\nabla_{i}+\omega_{i}$ as the covariant derivative in the $U(2)$
gauge field, we have

$Tr[P^{+}{\cal D}_{i}{\cal D}_{i}]=\braket{+}{{\cal D}_{i}{\cal
D}_{i}}{+}=\braket{+}{{\cal D}_{i}}{+}\braket{+}{{\cal
D}_{i}}{+}+\braket{+}{{\cal D}_{i}P^{-}{\cal D}_{i}}{+},$ (13)

where $P^{-}=(\mathbb{1}-P^{+})$. For any function $\xi(\vec{R})$,
$\braket{+}{{\cal D}_{i}}{+}\xi(\vec{R})=$

$\displaystyle\nabla_{i}\xi(\vec{R})+\braket{+}{\omega_{i}}{+}\xi(\vec{R})$
$\displaystyle=\nabla_{i}\xi(\vec{R})+[\braket{+}{U^{\dagger}(\vec{R})\nabla_{i}U(\vec{R})}{+}]\xi(\vec{R})=$
(14)
$\displaystyle\nabla_{i}\xi(\vec{R})+[\braket{e_{+}(\vec{R})}{\nabla_{i}}{e_{+}(\vec{R})}]\xi(\vec{R})$
$\displaystyle=\nabla_{i}\xi(\vec{R})+A_{i}\xi(\vec{R})$ (15)

where $A_{i}=\braket{e_{+}(\vec{R})}{\nabla_{i}}{e_{+}(\vec{R})}$ is the Berry
potential, which we also refer to as $A_{geom}$ the geometric vector
potential. As a result the first term in (13) gives us

$(\nabla_{i}+A_{i})(\nabla_{i}+A_{i})\xi(\vec{R})$ (16)

This term shows that the slow modes are subject to an effective magnetic field
represented by Berry potential $A_{i}$. This is a well known result, which has
been discussed in the molecular physics literature[16]. This is the quantum
analogue of the classical effect described in the introduction of this paper.

###  3.3 Geometric Scalar Potential

Like the Berry potential, the second term in (13) also has a simple geometric
interpretation. Expanding the expression for the covariant derivative, we have

$\braket{+}{(\nabla_{i}+\omega_{i})P^{-}(\nabla_{i}+\omega_{i})}{+}$ (17)

Since $\nabla_{i}$ commutes with $P^{-}$, all the derivative terms drop out in
(17) leaving us with
$\braket{+}{\omega_{i}P^{-}\omega_{i}}{+}=-\braket{+}{\omega_{i}^{\dagger}P^{-}\omega_{i}}{+}$
which equals

$\displaystyle-\braket{+}{\big{(}\nabla_{i}U^{\dagger}(\vec{R})\big{)}U(\vec{R})P^{-}U^{\dagger}(\vec{R})\big{(}\nabla_{i}U(\vec{R})\big{)}}{+}=-\big{(}\nabla_{i}\braket{e_{+}(\vec{R})}{\big{)}U(\vec{R})P^{-}U^{\dagger}(\vec{R})\big{(}\nabla_{i}}{e_{+}(\vec{R})}\big{)}$
(18)
$\displaystyle=-\big{(}\nabla_{i}\bra{e_{+}(\vec{R})}\big{)}(\mathbb{I}-\ket{e_{+}(\vec{R})}\bra{e_{+}(\vec{R}))\big{(}\nabla_{i}|e_{+}(\vec{R})}\big{)}=-\big{(}\nabla_{i}\braket{e_{+}(\vec{R})}{\big{)}Q^{e_{+}}\big{(}\nabla_{i}}{e_{+}(\vec{R})}\big{)}$
(19)

where $Q^{e_{+}}=P^{e-}=\mathbb{I}-P^{e+}$ is the projector orthogonal to the
state $\ket{e_{+}(\vec{R})}$. Supplying the necessary constants, we find the
final expression for the geometric scalar potential

$V_{geom}(\vec{R})=\frac{\hbar^{2}}{2m}{\huge\\{}\big{(}\nabla_{i}\braket{e_{+}(\vec{R})}{\big{)}\big{(}\nabla_{i}}{e_{+}(\vec{R})}\big{)}-\big{(}\nabla_{i}\braket{e_{+}(\vec{R})}{\big{)}\ket{e_{+}(\vec{R}}\bra{e_{+}(\vec{R}}\big{(}\nabla_{i}}{e_{+}(\vec{R})}\big{)}{\huge\\}},$
(20)

This potential is non-negative and vanishes only when the direction of
$\vec{B}$ is constant[11]. In the next section we will bring out the geometry
behind these terms in the slow Hamiltonian.

##  4 Geometry of Ray Space

The ray space of quantum mechanics ${\cal R}$ is defined as the set of
normalised states modulo multiplication by a unit complex number, a phase. For
finite dimensional Hilbert spaces, this is just complex projective space
$\mathbb{CP}^{n}$. In the toy model discussed here $n=1$ and we have
$\mathbb{CP}^{1}=S^{2}$. ${\cal R}$ has three intertwining structures on it: a
Riemannian structure, a symplectic structure and a complex structure. These
three structures combine to form a Kähler structure.

The Riemannian structure is physically accessible as the overlap between
quantum states: $|\braket{\phi}{\psi}|^{2}$ represents a transition
probability. Geometrically, the same quantity gives us the distance $d$
between rays $|\braket{\phi}{\psi}|=\cos{d/2}$. Passing to infinitesimally
separated rays $\ket{\phi}=\ket{\psi}+\delta\ket{\psi}$, we arrive at the
Fubini-Study metric [17, 12] for normalised states.

$ds^{2}=\braket{\delta\psi}{\delta\psi}-\braket{\delta\psi}{\psi}\braket{\psi}{\delta\psi}$
(21)

where we choose representative states from each ray. Given a normalised
$\ket{\psi}$ and two vectors $\delta_{1}\ket{\psi}$ and
$\delta_{2}\ket{\psi}$, the Fubini-Study metric is

$\gamma(\delta_{1}\ket{\psi},\delta_{2}\ket{\psi})=\braket{\delta_{1}\psi}{\delta_{2}\psi}-\braket{\delta_{1}\psi}{\psi}\braket{\psi}{\delta_{2}\psi}$
(22)

The geometric scalar potential (20) is exactly of this form with
$\delta\ket{\psi}$ replaced by $\nabla_{i}\ket{e_{+}(\vec{R})}$.

Although we have used representative kets instead of rays, this is actually a
metric on the ray space. This metric can also be expressed directly in terms
of density matrices. If $\rho$ is a density matrix and $\dot{\rho}$ and
$\rho^{\prime}$ tangent vectors at $\rho$, the Fubini-Study metric is given by

$\gamma(\dot{\rho},\rho^{\prime})=\frac{1}{2}\mathrm{Tr}[\dot{\rho}\rho^{\prime}]$
(23)

The symplectic structure is physically manifested in the geometric phase which
is detectable in interference phenomena. The symplectic form on ray space is
the curvature of the Berry connection and acts like a magnetic field. Given
any closed curve $C$ in ray space, the Berry phase is $\exp{i\phi}$, where
$\phi$ is the symplectic area of any disc that $C$ bounds. The Berry phase for
open curves can be similarly defined (provided the ends of the curve are not
on orthogonal rays) by closing the ends by a geodesic curve. The geodesic in
ray space is determined by the Fubini-Study metric. The work of
Pancharatnam[7] in optics drew attention to the importance of geodesics. These
ideas were carried over to the Hilbert space of quantum mechanics in [18].

The complex structure is intimately tied up with the symmetry of time
reversal. Time reversal is an anti-unitary operator in quantum mechanics.
Wigner’s theorem tells us that all ray space isometries are either unitary or
anti-unitary. To discriminate between these two possibilities we use the
Bargmann invariant [19, 20] which is an example of the Pancharatnam phase[7].
For Hamiltonians which are time reversal invariant, the geometric phase can
only take values $\pm 1$ and becomes a topological phase. This was the context
for Longuet-Higgins discovery of conical intersections in molecular physics.

Let us now understand the geometric scalar potential in terms of the Fubini-
Study metric. At every point $\vec{R}$, Eq. (6) determines a ray in the fast
Hilbert space. In the case of the two state sytem we consider here the ray
space is $S^{2}$, the Bloch sphere.

$f:\mathbb{R}^{3}\mapsto S^{2}$ (24)

The map $f$ can be used to pull back the Fubini-Study metric tensor to real
space. This gives us a second rank tensor in real space. Contracting this
tensor with the spatial metric gives us the geometric potential. Using the
unit vector $\hat{n}$, where $\rho(\vec{R})=\frac{1+\hat{n}(\vec{R})}{2}$, to
describe a point on the Bloch sphere, $f$ gives us $\hat{n}({\vec{R}})$ and we
have the pull back explicitly written as

$\gamma_{ij}=\frac{1}{2}\mathrm{Tr}[\big{(}\nabla_{i}\rho(\vec{R})\big{)}\big{(}\nabla_{j}\rho(\vec{R})\big{)}]=\frac{1}{4}(\nabla_{i}\hat{n}(\vec{R}))\cdot(\nabla_{j}\hat{n}(\vec{R})),$
(25)

The potential $V_{geom}$ is then given by
$g^{ij}(\vec{R})\gamma_{ij}(\vec{R})$ the contraction of $\gamma_{ij}$ with
the metric tensor of space. Although $g_{ij}=\delta_{ij}$ is flat in this
simple toy model, we include $\vec{R}$ dependence to leave open the
possibility of using generalised coordinates on the slow configuration space.
The Fubini-Study metric is a Riemannian metric on the ray space and so is
positive definite. However, the pull back $\gamma_{ij}$ of this tensor is not
a Riemannian metric on space since it may be degenerate. It is however, non-
negative (positive semi definite). Similarly, the pullback of the symplectic
form on the ray space, gives us a two-form on space, but this could be
degenerate (or indeed vanish entirely).

##  5 Isolating the separate effects

Putting together the work of the last sections, we find the Hamiltonian for
the slow modes:

${\cal
H}_{red}=V_{dyn}(\vec{R})-\frac{\hbar^{2}}{2m}(\nabla_{i}-A_{i}(\vec{R}))(\nabla_{i}-A_{i}(\vec{R}))+V_{geom}(\vec{R}),$
(26)

where $V_{dyn}(\vec{R})=\lambda_{+}(\vec{R})$, $A_{i}$ is the Berry potential
and $V_{geom}(\vec{R})$ is the geometric potential. We see that the slow modes
are subject to three effects by the fast modes.

  1. 1.

The first is a dynamic scalar potential $V_{dyn}(\vec{R})$ arising from the
eigenvalue of the fast mode.

  2. 2.

The second is modification of the kinetic energy term by the magnetic field
$F=dA$ of the Berry potential. Let us be clear that this is not the same as
the external $\vec{B}(\vec{R})$ we introduced; remember the neutron is neutral
and insensitive to the external magnetic field $\vec{B}(\vec{R})$. Only the
spin couples to $\vec{B}(\vec{R})$. What we are seeing here is geometric
magnetism, which is distinct from the magnetic effects created by currents.

  3. 3.

The third is the scalar geometric potential $V_{geom}$ that is of interest in
this paper. One could think of this as a “geometric electricity” in analogy
with the usual scalar potential in electrostatics.

Let us look at some examples to isolate these separate effects. By choosing a
magnetic field $\vec{B}(\vec{R})=b(\vec{R})\hat{i}$ which varies only in
magnitude, but not in direction, we eliminate effects 2 and 3 and isolate the
dynamic scalar potential. To isolate the effect of the geometric vector
potential, we consider the magnetic field
$\vec{B}(\vec{R})=B_{0}(\hat{i}\cos{y}+\hat{j}+\hat{k}\sin{y})$ which has
constant dynamic and geometric potentials. Here effects 1 and 3 are eliminated
and only effect 2 remains.

To isolate the effect of the geometric scalar potential, we present an example
in which the first two effects are absent and the third is present. This will
convince the reader that the third effect is independent of the first two. It
may also help an experimental detection of the third effect if the first two
effects are absent. We will now describe a magnetic field configuration
$\vec{B}(\vec{R})$, (with $\vec{\nabla}.\vec{B}=0$) with the property that

  * •

$|\vec{B}(\vec{R})|$ is constant and so $V_{dyn}(\vec{R})$ is independent of
$\vec{R}$.

  * •

the “magnetic field” $F=dA$ due to the Berry potential vanishes.

With $\vec{R}=(x,y,z)$, consider
$\vec{B}(\vec{R})=B_{0}(\cos{a(y)}\hat{i}+\sin{a(y)}\hat{k})$ where $a(y)$ is
any function depending only on $y$. This makes the fast Hamiltonian
$H_{f}=-\mu\vec{B}(\vec{R}).\vec{\sigma}$ real and proportional to
$\cos{a(y)}\sigma_{3}+\sin{a(y)}\sigma_{1}$, which is

$\begin{bmatrix}\cos{a(y)}&\sin{a(y)}\\\ \sin{a(y})&-\cos{a(y)}\\\
\end{bmatrix}$ (27)

The ground state of this Hamiltonian $H_{f}$ is

$\ket{e_{+}(y)}=\begin{bmatrix}\sin{a(y)/2}\\\ -\cos{a(y)/2}\\\ \end{bmatrix}$
(28)

It is easily checked that 1) $\lambda_{+}(\vec{R})=-\mu B_{0}$ is actually
independent of $\vec{R}$, so this term has no effect on the slow modes. 2) The
Berry potential vanishes identically $A_{i}=0$ and so there are no “geometric
magnetic” effects and 3) the geometrical scalar potential $V_{geom}$ does not
vanish and is given by $\frac{\hbar^{2}}{8m}(\frac{da(y)}{dy})^{2}$. Note that
$V_{geom}$ is non-negative.

##  6 More Realistic Models

Let us carry these ideas over to the more realistic situation of molecular
dynamics. The total Hamiltonian is

$H_{T}=T_{N}+V_{NN}+T_{e}+V_{ee}+V_{Ne}$ (29)

where the terms are, in order, $T_{N}$ the nuclear kinetic energy, $V_{NN}$
the nuclear-nuclear interaction, $T_{e}$ the electronic kinetic energy,
$V_{ee}$ the electron-electron interaction and $V_{Ne}$ the nuclear-electronic
interaction. The fast Hamiltonian consists of all those terms that depend on
electronic coordinates,

$H_{f}=T_{e}+V_{ee}+V_{Ne}(R),$ (30)

where it is understood that the dependence of $V_{Ne}$ on the nuclear
coordinates $R$ is parametric. We use the symbol $R$ to represent a set of
generalised coordinates in the nuclear configuration space. The slow
Hamiltonian consists of the remaining terms

$H_{s}=T_{N}+V_{NN}$ (31)

Following the Born-Oppenheimer approach, we need to solve eq.(6) for the
eigenstates $\ket{e_{n}(R)}$ and eigenvalues $\lambda_{n}(R)$. While this
problem is considerably more difficult than our simple toy model, it is within
the reach of physical chemists aided by modern computers. We will assume that
there are no degeneracies $\lambda_{n}(R)\neq\lambda_{m}(R)$ for $m,n$
distinct. The general ideas we used for the toy model carry over to these more
realistic and difficult models. After performing a unitary transformation to
the basis $\\{\ket{e_{n}(R)}\\}$, the slow mode Hamiltonian, including the
back reaction of the fast modes on the slow ones is given by

${\cal H}_{red}=\tilde{T}_{N}+V_{NN}+V_{dyn}+V_{geom},$ (32)

where $\tilde{T}_{N}$ is the nuclear kinetic energy modified by the Berry
potential $A$, and $V_{dyn}$ and $V_{geom}$ are the two scalar potentials of
dynamical and geometric origin. Assuming that the fast system is in the state
$\ket{e_{n}(R)}$ with eigenvalue $\lambda_{n}(R)$, here are explicit
expressions for these

$V_{dyn}(R)=\lambda_{n}(R).$ (33)

The effect of the geometric vector potential and the geometric scalar
potential can be expressed in terms of the second rank tensor ${\cal
V}_{ij}=\overline{{\cal V}}_{ji}$ on the nuclear configuration space

${\cal V}_{ij}(R)=\sum_{m\neq
n}\frac{\braket{e_{n}(R)}{\partial_{i}H_{f}(R)}{e_{m}(R)}\braket{e_{m}(R)}{\partial_{j}H_{f}(R)}{e_{n}(R)}}{(\lambda_{n}(R)-\lambda_{m}(R))^{2}}$
(34)

The effective magnetic field strength due to the Berry potential is given by
the imaginary part of the tensor ${\cal V}$:

$F_{ij}(R)=\frac{{\cal V}_{ij}-{\cal V}_{ji}}{2i}.$ (35)

This agrees with eq.(10) of Ref. [1]. The pull-back of Fubini-Study metric on
the fast ray space to the nuclear configuration space is given by the real
part:

$\gamma_{ij}(R)=\frac{{\cal V}_{ij}+{\cal V}_{ji}}{2}$ (36)

The geometric scalar potential is given by

$V_{geom}(R)=g^{ij}(R)\gamma_{ij}(R),$ (37)

where the metric $g$ on the nuclear coordinate space has been contracted with
$\gamma$. Note also that the metric on the nuclear configuration space is
determined by the kinetic energy term, which weights the nucleii according to
their masses. It may be advantageous to leave open the choice of coordinates
on this space. For instance nuclear separations may be better coordinates than
nuclear positions.

##  7 Better Clocks from Geometric Fields

Many laboratories over the globe cool and trap atoms and ions. This field has
overlap with condensed matter physics, quantum simulation and quantum
computation. An important motivation for cold atom studies is to achieve more
accurate and stable time-keeping. The best clocks available today are atomic
clocks, which lose less than a second in the age of the Universe. Better
clocks lead to a host of other applications, including global positioning
systems, space exploration, metrology and in fundamental physics. Atomic
clocks used to be confined in cells. But this results in a disturbance to the
atoms when there are collisions with the cell walls. This problem is
circumvented by electromagnetic trapping, where the walls are replaced by a
confining potential. An ideal clock would be an atom or ion at rest in space
confined by static electromagnetic (and gravitational) fields. An obstacle to
such confinement is Earnshaw’s theorem[21]: a static electric (or magnetic )
field satisfying the source-free Maxwell’s equations cannot trap ions or
atoms. Static electric fields are divergence and curl free and so satisfy the
Laplace equation. The electrostatic potential is a harmonic function and the
maximum principle then tells us that ions cannot be confined in a static
electric field trap. Since static equilibrium is impossible, one uses dynamic
electromagnetic fields as in the Paul trap, where an ion is kept in place by
an electric field configuration rotating at a radio frequency.

Ions are charged therefore susceptible to disturbance from stray
electromagnetic fields. A neutral atom would be less susceptible to such
fields but conversely, not as easy to manipulate. If the atom has spin, it can
be manipulated using magnetic fields: a spinning atom in a magnetic field
$\vec{B}$ has a potential energy $V_{dyn}(R)=-\mu|\vec{B}|$. However Wing’s
theorem tells us that this potential is also unable to confine atoms. It can
be shown [22, 23] that $-\nabla^{2}V_{dyn}\geq 0$ and again the maximum
principle forbids local minima of the potential. Static confinement is
impossible and so experimenters use the magneto-optical trap in which
electromagnetic fields at optical frequencies are used to confine the atom.

These general results are a consequence of the fact that laboratory
electromagnetic fields which are static have to satisfy Laplace’s equation.
However geometric potentials can evade the theorems as they do not have to
satisfy the source-free Maxwell’s equations. The geometric electric field is
curl-free (in regions where $|\vec{B}|$ is non-vanishing) but it need not be
divergence free. We have explicitly constructed examples of static trapping
potentials using geometric fields. These examples clearly prove that theorems
analogous to Earnshaw’s and Wing’s theorem (and its generalisations[23]) do
not apply when one includes geometric electromagnetic fields.

We describe two cases: static atom traps which confine neutral spinning atoms
in three dimensions and atom wave-guides which confine them in two dimensions
(say $x$ and $y$) and allow free motion in the $z$ direction. Neither traps
nor wave-guides can be realised using static laboratory electromagnetic fields
because of Wing’s theorem. Both these can be achieved by using geometric
electric fields.
Geometric atom trap: The laboratory magnetic field that is needed has two
parts: a constant magnetic field $\vec{B}_{0}$ which is created by a solenoid
and an inhomogeneous magnetic field $\vec{B}_{1}(\vec{x})$ which is created by
twelve wires located along the infinite line got by extending the edges of a
cube of side $a$. A unit current is passed along these wires in the direction
of the positive $x$, $y$ and $z$ directions. The total magnetic field is given
by

$\vec{B}(\vec{x})=\vec{B}_{0}+\vec{B}_{1}(\vec{x})$ (38)

The potential $V(\vec{x})$ seen by the atom is the sum

$V(\vec{x})=V_{dyn}(\vec{x})+V_{geom}(\vec{x})$ (39)

of the dynamical potential $V_{dyn}=-\mu|\vec{B}|$ and the geometric potential
$V_{geom}=1/4\nabla_{i}\hat{B}^{a}\nabla_{i}\hat{B}^{a}$. We have used
mathematica to compute the equipotentials of $V(\vec{x})$ and explicitly
verified that it admits local minima. The equipotentials in the neighbourhood
of the local minima have spherical topology, rather than the hyperbolic
surfaces one gets from saddle points. This proves that static magnetic fields
can trap neutral atoms if the geometric electric field is taken into account.
Wave-guides: To construct a geometric wave-guide we use $N$ straight wires
along the $z$ direction equally spaced on the circumference of a circle of
radius $a$ in the $x-y$ plane. These $N$ wires generate the field
$B_{1}(\vec{x})$, which is in the $x-y$ plane. In addition a solenoid
generates a constant field $B_{0}$ in the $z$ direction. The resulting
potential $V(\vec{x})$ (39) is plotted in Figure 1 for the case $N=10$. The
equipotentials of $V(\vec{x})$ are topological circles clearly showing two
dimensional confinement.

Figure 1: Figure shows the equipotential lines (arbitrary units) of the cross
section of a geometric wave-guide. The inner most contours are of circular
topology and contain a minimum of the potential $V(\vec{x})$. The potential
rises from a minimum at the centre to a maximum in the white region and then
falls away in the radial direction. This shows that geometric fields can be
used to evade Wing’s theorem[22].

This proves that geometric fields can be used to evade the general no-go
theorem of [22] and [23] preventing static trapping of neutral atoms.

Magneto-optical traps in current use are about a millimetre in size. Assuming
that the Zeeman splitting due to the magnetic field is $10^{10}{\textrm{Hz}}$,
it is possible to roughly estimate $V_{dyn}$ as about $10^{-5}\rm{ev}$.
$V_{geom}$ is about four orders smaller than this and so entirely negligible
in many current experiments. However, the situation changes dramatically if
one reduces the size of the trap to micron sizes. There is an ongoing effort
to miniaturise atomic clocks [24], putting them on a chip. If one reduces the
size of the trap by a factor $f<1$, $V_{dyn}$ is unchanged (if the magnetic
field remains the same), while $V_{geom}$ increases by a factor of $1/f^{2}$.
In smaller traps $V_{geom}$ is comparable in magnitude to $V_{dyn}$ and this
would permit static trapping of neutral atoms. Reducing the magnetic field
decreases $V_{dyn}$, while leaving $V_{geom}$ unchanged. Thus, in going to
smaller traps one must consider geometric electric fields and in fact, one can
use them to advantage to achieve static trapping of neutral atoms444The
adiabaticity condition is comfortably satisfied in the regime of interest..
The potential $V_{geom}$ describing the geometric electric field can trap
neutral atoms without an optical field.

##  8 Conclusion

We have described a geometric effect that arises in the Born-Oppenheimer
approximation in a toy model . This is a scalar potential which has its origin
in the geometry of ray space, namely the Fubini-Study metric. We have given a
magnetic field configuration in which this geometric potential can be isolated
from the other effects. One difference between $V_{geom}$ and the effects due
to $V_{dyn}$ and $A_{geom}$ is that $V_{geom}$ has the same effect on both
spin states. But the other two act oppositely on the $\ket{+}$ and $\ket{-}$
states. The effect of $V_{geom}$ on neutrons is to favour regions where the
magnetic field direction is slowly varying and avoid regions of rapid
variation[11], where the conditions of the adiabatic theorem would be
violated.

While the more realistic models of molecular physics are not easy to solve,
the same geometric ideas apply there. If one solves the fast eigenvalue
problem on a computer, one can extract the slow dynamics in terms of
eigenvalues and eigenstates in an entirely geometric manner. The metric on the
nuclear configuration space is the one derived from the kinetic energy, which
weights different nucleii according to their mass. The non-negativity of the
geometric scalar potential remains true here as it follows from general
arguments. The magnitude of the geometric scalar potential is smaller than the
dynamical scalar potential by the ratio of the electronic mass to the nuclear
mass. The effect of the geometric potential, while small, is measureable in
the rotational and vibrational spectra of molecules.

In our toy model, we have treated the specific example of a neutron in a
strong magnetic field. However, the mathematics we presented here has wider
applications. One example is neutral cold atoms trapped using magnetic
fields[25]. Such configurations are used in cold atom laboratories to achieve
Bose-Einstein condensation as well as Fermi-degeneracy in dilute atomic gases.
Magnetic traps only confine atoms in the weak field-seeking states. For weak
magnetic field gradients and slow atoms, spin-flip transitions are suppressed
by the adiabatic theorem. In practice, these transitions do occur in regions
of very small magnetic fields which creates a region of trap loss in the
magnetic trap due to spin flips to untrapped states and are referred to as
Majorana losses. The background field ${\vec{B}}_{0}$ we used in our examples
is designed to avoid situations where the magnetic field is vanishingly small.
We have also emphasised that the geometric field will become more relevant as
the size of traps becomes smaller, since it depends on gradients of the
direction of the magnetic field. This geometric field can also be used to
advantage to develop better clocks.

Another application, unrelated to quantum mechanics, is polarised waves. When
polarised light propagates through a birefringent medium, the medium acts like
a magnetic field. In general one could have linear, circular and even
elliptical birefringence, in which two orthogonal polarisations traverse the
medium at different speeds. Mathematically, this problem of polarised wave
optics maps on all fours to the toy model of this paper. If one restricts to
linear birefringence and arranges for a helical configuration of molecules,
one can even reproduce the example of section V in which the waves are
affected only by the geometric scalar potential. Such helical configurations
can be generated as supramolecular structures using liquid crystals. The ideas
described here in the quantum context could be experimentally verified in the
classical wave context of optics. The effect of geometric potential can be
seen in an interference experiment. This experiment could also be done in a
scaled up setup by replacing light by radio waves and replacing the molecular
helical variation with centimetre size dielectric rods. One possible
application is to make achromatic lenses using the geometric scalar potential
to retard the beam.

We hope to interest geometers as well as experimenters in understanding the
uses of this geometric scalar potential.

Acknowledgements: This paper was motivated by a talk given by Berry at ICTS,
Bangalore, titled: The geometric phase and the separation of the world, during
a program titled: Geometric Phases in Optics and Topological Matter. I thank
Sanjukta Roy for clearing up my ideas on atom traps and Supurna Sinha for
reading through the paper and suggesting improvements. This work was supported
by a grant from the Simons Foundation (677895, R. G.).

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