Project Report for Quantum Mechanics II (Spring 2023)
Charged particles in electromagnetic fields and
Aharanov Bohm effect
Shahmir Altamash1 Qasim Ali Chawla2
PHY-312, Department of Physics, SBASSE, LUMS.
E-mail: 24100069@lums.edu.pk, 25100093@lums.edu.pk
Abstract: The study of charged particles in electromagnetic fields is a fundamental concept in physics with a wide range of applications in various fields.In this paper, we will
explore this concept in more detail and focus specifically on the Aharonov-Bohm effect.
This effect is a fascinating quantum mechanical phenomenon that occurs when charged
particles are confined to a region where the electromagnetic field is zero, but there is a
non-zero vector potential. Our aim is to provide a detailed analysis of the Aharonov-Bohm
effect, including an explanation of the underlying physics and the experimental evidence
for the effect. This report is divided into five sections. The first three sections refer to the
theoretical understanding of the charged particles in electromagnetic fields and how the
Aharanov Bohm effect arises. The fourth section is about the experiments that were conducted to validate this theory. The last section deals with the applications and potential
development in this field.
Keywords: Charged Particles, Aharanov Bohm effect
Contents
1 Hamiltonian of charged particles in Electromagnetic Fields
2
2 Adiabatic Evolution
2
3 Berry’s phase
3.1 Aharonov-Bohm effect and the Berry phase
5
7
4 Experimental Techniques for validation of Aharanov Bohm effect
4.1 Observing the Aharonov-Bohm effect using electron holography
4.2 Observation of the Aharonov-Bohm Effect in a Carbon Nanotube Quantum
Ring
9
9
10
5 Applications for Aharanov Bohm effect
5.1 Spin Aharonov-Bohm effect and topological spin transistor
5.2 High-sensitivity rotation sensing with atom interferometers using AharonovBohm effect
10
10
6 Appendix A: Berry Curvature
13
–1–
12
1
Hamiltonian of charged particles in Electromagnetic Fields
The Hamiltonian of a charged particle in an electromagnetic field is given by:
H=
1
(p − qA)2 + qϕ,
2m
The resulting Schrodinger equation when will incorporate this Hamiltonian becomes:
1
∂
(p − qA)2 + qϕ Ψ(r, t),
iℏ Ψ(r, t) =
∂t
2m
Where Ψ(r, t) is the wave function of the particle, r is its position, t is time, p is
the momentum operator, q is its charge, m is its mass, A is the vector potential of the
electromagnetic field, ϕ is its scalar potential.
The commutator between position operator x̂ and momentum operator p̂ − qA is given
by:
[x̂i , pˆj − qA] = [x̂i , pˆj ] − [x̂i , Â]
ˆ
= [x̂i , pˆj ] − [x̂i , [Ax + Aˆy + Âz ]]
= [x̂i , pˆj ] − [[x̂i , Aˆx ] + [x̂i , Aˆy ] + [x̂i , Âz ]]
The entire second term reduces to 0 since the magnetic vector potential is a scalar and
not an operator, so the commutation relation with the position operator yields 0, and the
overall commutator then becomes:
[x̂i , pˆj − qA] = [x̂i , pˆj ]
= iℏδij
We can see how even though our Hamiltonian changed for a charged particle in an Electromagnetic field, the commutation relation of position and momentum operator remains
unaffected.
2
Adiabatic Evolution
The Adiabatic evolution is a result of the adiabatic theorem which states that the system
remains in instantaneous eigenstate of the Hamiltonian if the perturbation on it is acting
slowly enough to create a gap between the eigen value and the rest of the Hamiltonian’s
spectrum. More generally, suppose that we have a ψ(t) such that
H(t)|ψ(t)⟩ = E(t)|ψ(t)⟩
we’ll call |ψ(t)⟩, an instantaneous eigenstate. Here it is important to note an instantaneous eigenstate is not a solution of the time dependent Schrödinger equation. As it turns
out, it is a useful tool to construct approximate solutions to the Schrödinger equation.
–2–
We build a relation between |ψ(t)⟩ and |Ψ(t)⟩ to construct the solution to the Schrödinger
equation
iℏ∂t |Ψ(t)⟩ = H(t)|Ψ(t)⟩
An ansatz to building the solution |Ψ(t)⟩ using the instantaneous eigenstate:
|Ψ(t)⟩ = c(t) exp
1
iℏ
t
Z
E t′ dt′ |ψ(t)⟩
(2.1)
0
with c(t) a function of time to be determined and where we have included a familiar
time dependent phase.
The left hand side of the Schrödinger equation then looks like
iℏ∂t |Ψ(t)⟩ = ċ(t) exp
1
iℏ
Z
t
′
′
E t dt
|ψ(t)⟩+E(t)|Ψ(t)⟩+c(t) exp
0
1
iℏ
Z
t
′
′
E t dt
|ψ̇(t)⟩
0
For the right hand side, using the instantaneous eigenstate equation, we have
H(t)|Ψ(t)⟩ = c(t) exp
1
iℏ
t
Z
′
′
E t dt
H(t)|ψ(t)⟩ = E(t)|Ψ(t)⟩
0
Equating the two sides, we get
ċ(t) exp
1
iℏ
Z
t
′
′
E t dt
|ψ(t)⟩ + c(t) exp
0
1
iℏ
Z
t
′
′
E t dt
|ψ̇(t)⟩ = 0
0
and canceling the two exponentials we have
ċ(t)|ψ(t)⟩ = −c(t)|ψ̇(t)⟩
(2.2)
Multiply by ⟨ψ(t)| we get a differential equation for c(t) :
ċ(t) = −c(t)⟨ψ(t) | ψ̇(t)⟩,
which we can solve. Letting c(t = 0) = 1 we can write
Z tD
E ′
′
′
c(t) = exp −
ψ t | ψ̇ t dt .
0
The above exponential is a phase because the bracket in the integrand is actually
purely imaginary:
Z
Z
Z
d
dψ ∗
∗ dψ
∗
⟨ψ(t) | ψ̇(t)⟩ = dxψ
= dx (ψ ψ) − dx
ψ
dt
dt
dt
Z
d
dxψ ∗ ψ − ⟨ψ̇(t) | ψ(t)⟩
=
dt
Since the wavefunction is normalized we have
–3–
⟨ψ(t) | ψ̇(t)⟩ = −⟨ψ̇(t) | ψ(t)⟩ = −⟨ψ(t) | ψ̇(t)⟩∗
showing that indeed ⟨ψ(t) | ψ̇(t)⟩ is purely imaginary. To emphasize this fact we write
Z t D
E ′
′
′
c(t) = exp i
i ψ t | ψ̇ t dt
0
Having apparently solved for c(t) we now return to our ansatz (2.1), we get
Z t
Z t D
E ′
′
1
′
′
′
|Ψ(t)⟩ ≃ c(0) exp i
i ψ t | ψ̇ t dt exp
E t dt |ψ(t)⟩·
iℏ 0
0
(2.3)
The equation we had to solve, (2.2), is a vector equation, and forming the inner product
with ⟨ψ(t)| gives a necessary condition for the solution, but not a sufficient one. We must
check the equation forming the overlap with a full basis set of states. Indeed since |ψ(t)⟩
is known, the equation can only have a solution if the two vectors |ψ̇(t)⟩ and |ψ(t)⟩ are
parallel. This does not happen in general. So we really did not solve equation (2.2). The
conclusion is that, ultimately, the ansatz in (2.1) is not good enough.
We can see the complication more formally. At t = 0 equation (2.2) reads
ċ(0)|ψ(0)⟩ = −c(0)|ψ̇(0)⟩
(2.4)
Using Gram-Schmidt we can construct an orthonormal basis B for the state space with
the choice |ψ(0)⟩ for the first basis vector:
B = {|1⟩ = |ψ(0)⟩, |2⟩, |3⟩, . . .}
Equation (2.4) requires
⟨n | ψ̇(0)⟩ = 0 n = 2, 3, . . .
This will not hold in general. The key insight, however, is that (2.3) is a fairly accurate
solution if the Hamiltonian is slowly varying. Making the definitions:
Z
1 t
E t′ dt′ , ν(t) ≡ i⟨ψ(t) | ψ̇(t)⟩,
ℏ 0
so that θ, ν and γ are all real, the state reads
θ(t) ≡ −
Z
γ(t) ≡
t
ν t′ dt′
0
|Ψ(t)⟩ ≃ c(0) exp(iγ(t)) exp(iθ(t))|ψ(t)⟩
The phase θ(t) is called the dynamical phase and the phase γ(t) is called the geometric
phase.
Instantaneous eigenstates are rather ambiguous. If one has the |ψ(t)⟩, they can be
modified by the multiplication by an arbitrary phase. This is also the gauge transformation
applied to the state.
|ψ(t)⟩ → ψ ′ (t) = e−iχ(t) |ψ(t)⟩
–4–
3
Berry’s phase
Adiabatic theorem: |ψ(t = 0)⟩ = |ψn (0)⟩ and with instantaneous eigenstates |ψn (t)⟩ we
find
|ψ(t)⟩ ≃ eiθn (t) eiγn (t) |ψn (t)⟩
and the phases can be expressed as following:
Z
1 t
θn (t) = −
En t′ dt′
ℏ
D 0
E
νn (t) = i ψn (t) | ψ̇n (t)
Z t
γn (t) =
νn t′ dt′
0
We now understand the relevance of γn (t), the geometrical phase: Assume the hamiltonian H depends on a set of coordinates
R(t) = (R1 (t), R2 (t), . . . , RN (t))
several parameters that are time dependent.
Therefore, the instantaneous states
H⟨(R) |ψn (R)⟩ = E⟨(R) |ψn (R)⟩
In order to evaluate the geometric phase, we would start by computing
d
νn (t) = i ψn (R(t))
ψn (R(t))
dt
we need
d
dt
|ψn (R(t))⟩
N
X d
d
dRi
⃗ R |ψn (R(t))⟩ · dR(t)
|ψn (R(t))⟩ =
=∇
|ψn (R(t))⟩
dt
dRi
dt
dt
i=1
so that
νn (t) = i
D
E
⃗ R ψn (R(t)) · dR(t)
ψn (R(t)) ∇
dt
and
Z
γn (τ ) ≡
τ
Z
τ
νn (t)dt =
0
0
D
E
⃗ R ψn (R(t)) · dR(t) dt
i ψn (R(t)) ∇
dt
hence the geometrical phase γn (tf ), also known as Berry phase, is
Z
Rf
γn (tf ) =
E
D
⃗ R ψn (R) · dR
i ψn (R) ∇
Ri
–5–
The integral depends on the path, but does not depend on time so regardless of the
time taken
to transition through
the path the geometric phase remains unchanged.
D
E
⃗
i ψn (R) ∇R ψn (R) is an N -component object that belongs to the parameter space.
It is called the Berry connection An (R), associated with |ψn (t)⟩
D
E
⃗ R ψn (R)
An (R) = i ψn (R) ∇
In this way we can rewrite the Berry phase as
Z
Rf
An (R) · dR
γn =
Ri
If we redefine the instantaneous states by an overall phase
E
fn (R) = e−iβ(R) |ψn (R)⟩
|ψn (R)⟩ → ψ
where β(R) is an arbitrary real function. Calculating geometric phase for n-dimensional
parameter space:
D
E
fn (R) ∇
fn (R)
fn (R) = i ψ
⃗R ψ
A
D
E
⃗ R e−iβ(R) ψn (R)
= i ψn (R) eiβ(R) ∇
⃗ R β(R) ⟨ψn (R) | ψn (R)⟩ +An (R)
= i −i∇
|
{z
}
1
fn (R) = An (R) + ∇
⃗ R β(R)
A
(3.1)
Over here we can notice that this equation (3.1) is analogous to the vector potential
in
⃗′ → A
⃗ ′ + ∇E .
EM A
To see whether the berry phase has the same form we will integrate it over a closed
path:
Z
γ
fn (Rf ) =
Rf
Z
Rf
fn (R) · dR = γn (Rf ) +
A
Ri
⃗ R β(R) · dR =⇒
∇
Ri
γ
fn (Rf ) = γn (Rf ) + β (Rf ) − β (Ri )
The geometrical phase is completely well defined for closed paths in parameter space.
Additionally, if the ψn (t) are real, then the Berry phase vanishes:
Z
E
d
νn = i ψn (t) | ψ̇n (t) = i dxψn∗ (t, x) ψn (t, x)
dt
Z
Z
d
i
d
= i dxψn (t, x) ψn (t, x) =
dx (ψn (t, x))2 =
dt
2
dt
Z
i d
=
dx |ψn (t, x)|2 = 0
2 dt
D
–6–
D
E
In general grounds it is understood that ψn (t) | ψ̇n (t) is purely imaginary. But if
ψn is real this overlap cannot produce a complex number, so it can only be zero.
If there is just one coordinate, i.e. R = R, Berry’s phase vanishes for a loop
I
An (R)dR = 0
In here the integral from Ri to Rf is cancelled by the integral back from Rf to Ri .
- In 3D we can make use of Stoke’s theorem to simplify the calculation of γn [C]
ZZ ZZ
I
⃗
⃗
⃗
Dn · dS
An (R) · dR =
∇ × An (R) · dS ≡
γn [C] =
S
C
S
where S is the surface with boundary C and Dn is the Berry curvature
⃗ × An (R)
Dn ≡ ∇
If we think of the Berry connection as the vector potential, the Berry curvature is the
associated magnetic field. Note that the curvature is invariant under phase redefinition
fn (R) = ∇
⃗ ×A
⃗ × An (R) + ∇β(R)
⃗
Dn → D′n = ∇
⃗ × An (R) + ∇
⃗ × ∇β(R)
⃗
⃗ × An (R) = Dn .
=∇
=∇
3.1
Aharonov-Bohm effect and the Berry phase
Figure 1. a diagram of how to observe the aharonov-bohm effect
Classically we see that a charged particle remains unaffected in a region where there is
no magnetic field but on the quantum scale things behave in a peculiar way, even though
the magnetic field is 0, the path of the particle still gets altered all due to the non-zero
magnetic vector potential. The phase that the wavefunction picks up due to this change
in the path is the geometric phase. This same geometric phase that the particle picks up
when it is moved around a solenoid in a closed loop is known as the Berry phase. To
compute it we first need to see what the Hamiltonian for this system is:
1
(p − qA(r)2 + V (r − R),
2m
The Hamiltonian is according to the distances marked in the diagram. Now in order
to get a potential free equation we do a gauge transformation.
H=
–7–
ψ = eiκ ψ ′
where κ =
q
ℏ
R
⃗ · d⃗l. After gauge transformation the hamiltonian becomes:
A
−ℏ2 2
⃗
H=
∇ + V (⃗r − R)
2m
⃗ the energy eigenfunctions also become funcSince hamiltonian is now a function of (⃗r − R)
⃗
tions of (⃗r − R).
To find the geometric phase we need to calculate the follwing: ⟨ψn′ |∇R | ψn′ ⟩ so that we
H
⃗ and the answer
can take the integral over the entire loop γn (t) = i ⟨ψn | ∇R | ψn ⟩ · dR
will be our berry phase.
Evaluating the action of the del operator on the ket vector:
E
⃗
∇R | ψn′ ⟩ = ∇R eiκ φ′n (⃗r − R)
h
i
⃗
⃗ + eiκ ∇R ψ ′ (⃗r − R)
= ∇R eiκ ψn′ (⃗r − R)
n
h
i
⃗ + eiκ ∇R ψn′ (⃗r − R)
⃗
= i∇R (κ)eiκ φ′n (⃗r − R)
h
i
qr
⃗
⃗ + eiκ ∇R ψn′ (⃗r − R)
(3.2)
= −i A(R)eiκ ψn′ (⃗r − R)
ℏ
Now taking the inner product, we get integration over the entire space[completeness
theorem]
For the first term in eq 3.2 the integral will be:
Z
qr
= −i A(R) d3⃗re−iκ ψn′∗ eiκ ψn′
ℏ
qr
= −i A(R)
ℏ
The wavefunctions are normalized hence their integration gives 1, whereas the exponentials cancel out.
For the second term in the eq 3.2 the integral will be:
Z
⃗
d3⃗re−iκ ψn′∗ eiκ ∇R ψn′ (⃗r − R)
Z
⃗
= − d3⃗rψn′∗ ∇r ψn′ (⃗r − R)
Z
i
ℏ
⃗
=−
d3⃗rψn′∗ ∇r ψn′ (⃗r − R)
ℏ
i
=
(3.3)
Equation 3.3 is the average value of the momentum and it can also be verified through
the ehrenfest theorem that this integral boils down to a 0. Hence the only contribution to
the inner product is from the first term.
–8–
ψn′ |∇R | ψn′ = −i
qr
A(R)
ℏ
Integrating this over the closed loop we will finally get the berry phase:
I
γn (t) = i
−i
qr
⃗
A(R)dR
ℏ
Applying stokes theorem to get:
Z
q
γn (t) =
(∇ × A).dA
ℏ
Z
q
B.dA
=
ℏ
qϕ
=
ℏ
ϕ here is the magnetic flux, as seen the berry phase is simply the flux times charge
over ℏ.
4
Experimental Techniques for validation of Aharanov Bohm effect
The presence of Aharanov Bohm effect was widely questioned and was thought of as pure
mathemtical construct. To conter this many experiments were performed to validate this
theory. We have listed few experiments that were used to observe Aharanov Bohm effect.
4.1
Observing the Aharonov-Bohm effect using electron holography
Electron holography makes it possible to measure the phase shift directly of a high-energy
electron wave that has passed through a specimen in a transmission electron microscope.
Similarly this technique used to validate the Aharanov Bohm effect. The experiment used
tiny toroidal magnets or solenoids to create complete flux circuits and a new method of
holographic interference microscopy was employed to obtain electron phase contour maps
and quantify any potential leakage. Permalloy thin films were used to make toroids of
various sizes, and the magnetization was observed to be closed within the magnet. The
experiment confirmed the Aharonov-Bohm effect and demonstrated the utility of electron
holography for studying electromagnetic phenomena.
The photographs reveal that a phase difference really exists between two electron
beams that have passed through the inner and outer spaces of a toroidal magnet where
there were no magnetic fields in those spaces.[1]
–9–
Figure 2. (a) Contour map of electron phase. (b) Interferogram of electron phase
4.2
Observation of the Aharonov-Bohm Effect in a Carbon Nanotube Quantum Ring
In this section we have discussed the experimental observation of the Aharonov-Bohm effect
in a carbon nanotube quantum ring.
The current flowing through the nanotube ring in the presence of a magnetic field and
observed oscillations in the current as a function of the magnetic flux passing through the
ring. These oscillations were attributed to the Aharonov-Bohm effect, where the electrons
in the nanotube experience a phase shift due to the magnetic flux, even in the absence of
any magnetic field within the ring.
The experimental setup consisted of a semiconductor nanotube connected to two metallic electrodes, which were used to measure the conductance of the nanotube.
To observe the Aharonov-Bohm effect, the researchers applied a magnetic field perpendicular to the nanotube and varied the magnetic field strength by changing the current
through a nearby coil. The conductance of the nanotube was then measured as a function
of the magnetic field strength. The researchers observed oscillations in the conductance as
a function of the magnetic field, which were attributed to the Aharonov-Bohm effect.
To confirm that the observed oscillations were due to the Aharonov-Bohm effect, the
researchers performed additional measurements. They varied the temperature and bias
voltage across the nanotube and observed that the oscillations persisted, indicating that
they were not due to other effects.[2]
5
5.1
Applications for Aharanov Bohm effect
Spin Aharonov-Bohm effect and topological spin transistor
One potential use application of Aharanov Bohm effect is in Spin transistors. In this the
spin of the electron is managed by a magnetic flux, without any electromagnetic field having
an impact on the electron. The Quantum Spin Hall (QSH) insulator plays a crucial role in
this non-local spin manipulation, which is accomplished in an Aharonov-Bohm ring. The
– 10 –
topological property of the quantum spin Hall edge states allows to take advantage of the
direct link between spin polarization and direction of propagation. This idea is can be
utilized to develop a novel spintronics device called the topological spin transistor. This
transistor allows complete control over spin rotation by a magnetic flux of hc/2e.
Diagram below illustrates the mechanism proposed for realizing the spin AB effect.
The system under consideration is a two-terminal device consisting of a confined region of
the Quantum Spin Hall (QSH) insulator with a hole threaded by a magnetic flux ϕ. Due to
time-reversal (TR) symmetry, edge electrons propagating clockwise with their spin pointing
out-of-plane along z (spin up | ↑⟩) must have electrons propagating counterclockwise with
the opposite spin along −z (spin down | ↓⟩). If electrons are injected from a ferromagnetic
(FM) lead on the left with spin polarized along the x direction | →⟩ = √12 (| ↑⟩ + | ↓⟩), the
electron beam is split coherently into a | ↑⟩ beam propagating along the top edge and a | ↓⟩
beam propagating along the bottom edge upon entering the QSH region. When the electron
beams are recombined on the right side of the ring, the electrons along the top and bottom
edges acquire a phase difference of φ = 2πϕ/ϕ0 due to the AB effect, where ϕ0 = hc/e is the
flux quantum. As a result, the output state is given by √12 | ↑⟩ + e−iφ | ↓⟩ , indicating that
the electron spin is rotated by an angle φ in the xy plane. The electromagnetic fields are
zero in the region where electrons propagate because the magnetic flux is confined to the
hole in the device, allowing for the spin to be rotated by a purely quantum-mechanical Berry
phase effect. For collinear FM leads ( θ = 0 ), the conductance is expected to be maximal
for ϕ = 0 (modϕ0 ) and minimal for ϕ = 21 ϕ0 (modϕ0 ), thus realizing a ”topological” spin
transistor. This effect is topological in the sense that the spin is always rotated by one
cycle for each period of flux ϕ0 , regardless of the details of the device, such as the size of
the system or the shape of the ring.
Figure 3. Schematic picture of the spin AB effect. A ring of QSH insulator threaded by a magnetic
flux is connected to two magnetic leads. Spin-polarized electrons injected from the left lead enter
the QSH region as a superposition of spin-up and spin-down states. The spin-up down state can
only propagate along the top bottom edge of the QSH ring, and the two spin states thus acquire
an AB phase difference proportional to ϕ. Consequently, upon exiting the QSH region the two
edge states recombine into a state with spin rotated with respect to the injected direction. The
magnetization direction of the right lead generally differs from that of the left lead by an angle θ.
The two-terminal conductance G=G(ϕ,θ) of the device depends on the relative angle between the
spin polarization of the outgoing state and that of the right lead
This study investigates the potential of utilizing the Quantum Spin Hall (QSH) insulator state as a means of manipulating electron spin through quantum-mechanical methods,
rather than through classical electromagnetic fields. This manipulation is made possible by
– 11 –
the unique topological and helical nature of the QSH edge states, combined with the Berry
phase effect.[3] The study proposes a new type of spin transistor, which differs from previous proposals in that it does not rely on classical forces or torques acting on the electron
spin. The proposed transistor would benefit from the dissipation-less edge transport in the
QSH regime, resulting in lower power consumption than previous spin transistor designs.
Overall, the findings have potential implications for spintronics research and demonstrate
the practical applications of topological states of quantum matter.
5.2
High-sensitivity rotation sensing with atom interferometers using AharonovBohm effect
Another application of the Aharonov Bohm effect is in atom interferometers surrounded
by a Faraday Cage. Before delving into the details of how this works it is important to
understand what an interferometer is. An interferometer is a scientific tool that is used to
precisely measure the relative position or the motion of objects.
Figure 4. Diagram of a sagnac interferometer. a) the interferometer isn’t rotating. b) the interferometer is rotating
The conventional interferometer (the sagnac interferometer) uses the concept of splitting light beams into two and making them propagate around a closed loop, one in each
direction. Along with this the interferometer is also made to rotate at an angular frequency
Ω (a pictorial representation is shown above). When the interferometer is not rotating,
both beams reach the detector at the same time but when it is rotating the wave counter
moving to the interferometer will reach the detector before the co-moving wave. This time
delay corresponds to a phase difference also known as the sagnac phase shift and it is given
by: ∆Φ = ω∆τ = 2πR2 ωΩ/c2 where ω is the frequency of the waves, R is the radius of the
loop and τ is the time delay in both waves reaching the detector. The phase shift depends
on the Area of the closed loop A=πR2 so it can be generalized to ∆Φ = 4πΩ·A
where Ω · A
λc
is the vectorial product of of the axis of the rotation and the plane of the interferometer.
– 12 –
Figure 5. Diagram of an atom interferometer enclosed in a faraday cage and connected to a
generator
Another type of interferometer is atom interferometer, instead of light beams interfering, particles interfere. This is where Aharonov Bohm effect really comes into action. If
we enclose the interferometer in a faraday cage and connect a generator that will provide
potential between the atomic cavity and cage it is possible to get results even more precise
as compared to the conventional interferometer (a pictorial representation is shown above).
The way this works is the particles are let into the interferometer and once all of them
are in, the generator is turned on and set at a potential Vo . Now the particles will move
and interfere under the potential Vo . Note that during all this the interferometer is also
rotating at an angular frequency of Ω. The resulting phase difference in this case will be
proportional to the flight time difference to the detector ∆τ The phase difference due to the
0 Ω·A
applied potential will then be ∆Φp = 2qVℏv
. To see how the results have changed we take
2
a
the ratio of the phase difference of the sagnac interferometer and the atom interferometer
∆Φ
qV0
enclosed in the faraday cage and we get ∆Φp = mv
2.
a
The ratio shows that the phase shift for a given rotation rate can be made larger and
this inturn leads to increase in the sensitivity of the interferometer by magnitudes, and its
enhanced ability to detect small changes and produce precise results.[4]
6
Appendix A: Berry Curvature
If Berry connection is thought of as the vector potential, the Berry curvature is the associated magnetic field. This curvature is invariant under phase redefinition
fn (R) = ∇
⃗ × An (R) + ∇β(R)
⃗
⃗ ×A
Dn → D′n = ∇
⃗ × An (R) + ∇
⃗ × ∇β(R)
⃗
⃗ × An (R) = Dn .
=∇
=∇
Berry phase for an electron in a slowly varying magnetic field. If we take a magnetic
field with fixed intensity and slowly vary the direction B(t) = B⃗n(t) and then if we put an
electron inside this magnetic field, so that we get an interaction described by the following
Hamiltonian
H(t) = −µ · B(t) = µB B⃗n(t) · ⃗σ
The magnetic field B direction covers a closed loop C on the surface of an imaginary
sphere of radius ρ in a time scale T ≫ µBℏB . We can think of the instantaneous eigenstates
|χ± (R(t))⟩ satisfying
– 13 –
H(t) |χ± (R(t))⟩ = ±µB B |χ± (R(t))⟩
R(t) = (
r(t) , θ(t), ϕ(t))
|{z}
=ρ( fixed )
Acknowledgments
We thank our hardworking Teaching Assistant Mr. Qasim Javed for his help and guidance
throughout this project.
References
[1] A. Tonomura, T. Matsuda, B. Suzuki, A. Fukuhara, N. Osakabe, H. Umezaki, J. Endo, K.
Shinagawa, Y. Sugita, and H. Fujiwara, ”Observation of Aharonov-Bohm Effect by Electron
Holography,” Vox. Umz. 48, 24 May 1982, Number 21.
[2] Bachtold, A., Strunk, C., Salvetat, J.-P., Bonard, J.-M., Forró, L., Nussbaumer,
T.,Schönenberger, C. (1999). Aharonov–Bohm oscillations in carbon nanotubes. Nature, 397
(6721), 673–675. doi:10.1038/17755.
[3] Maciejko, J., Kim, E.-A., Qi, X.-L, Spin Aharonov-Bohm effect and topological spin
transistor, Journal of Applied Physics 108(9)093712 .
[4] Meric, Ozcan, High-sensitivity rotation sensing with atom interferometers using
Aharonov-Bohm effect” Proc. SPIE 6127, Quantum Sensing and Nanophotonic Devices III,
61271E (2006).
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