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‫גוריון בנגב‬-‫אוניברסיטת בן‬
Course: Atoms and Molecules
By Ron Folman
Lecture 10
Walks on the Bloch sphere:
Coherent manipulations of an atomic two-state system
Rabi Oscillations and Ramsey fringes
Thanks to Menachem Givon for preparing the lecture
Walks on the Bloch sphere
1. Introduction
Since the dawn of quantum mechanics in the 1920s, the atomic two-state system has
served as an important model of radiation-matter interaction. With the introduction (in
the 1960s) of lasers as a source of monochromatic and coherent radiation, detailed studies
of the coherent matter-radiation interactions became a major part of the research activities
and technological development. Magnetic Resonance Imaging (MRI), atomic clocks, the
Global Positioning System (GPS, based on cesium atomic clocks), are just few examples
of the many applications of the two-state system. Much of this work (SHO 1990 and
references therein) was based on atomic two-state models, sometimes modified to include
effects of additional states. The appearance of quantum computing concepts in the 1980s
(EIS 2004 and references therein) brought in a new possible role for the atomic two-state
system: that of the qubit, the quantum equivalent of the bit. The qubit is the basic
information storage and manipulation unit of the quantum computer, and realization of a
qubit is an important step towards the construction of a quantum computer.
The state Ψ of an atomic two-state system (and any other two-state system) can be
represented as a superposition of the two eigenstates of the Hamiltonian H0 describing the
system:
0
Ψ = C 0 0 + C1 1
Where C0,1 are complex numbers,
∑C C
i
(1.1)
*
i
Z
= 1, i = 0,1
θ
Ψ
i
and
H 0 0 = E 0 0 , H 0 1 = E1 1
(1.2)
ϕ
X
Y
Alternatively, we can write (1.1) as
Ψ = cos
θ
θ
0 + e iϕ sin 1
2
2
(1.3)
1
Figure 1.1 – The Bloch sphere
where we neglected the global phase, and 0 ≤ θ ≤ π , 0 ≤ ϕ ≤ 2π . Thus, the two angles
θ, ϕ define the normalized state Ψ up to a global phase. As long as our two-state
system does not interact coherently with the external world, this global phase does not
affect any observable, and can therefore be neglected.
We can geometrically demonstrate any such state as a point on a three-dimensional unit
sphere, called the “Bloch Sphere” (Fig 1.1). The “north pole” represents the pure state
0 while the “south pole” represents the pure state 1 . Any other point represent a
superposition state per (1.3), where θ, ϕ are the spherical polar coordinates of that point.
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Once we established the correspondence between states and the Bloch sphere, we can use
the Bloch sphere to visualize and demonstrate the system and its evolution (both θ and ϕ
may vary with time).
Side note: Much attention is being devoted today towards developing two-state systems
as qubits of the future quantum computer. If time allows we will discuss the uniqueness
of quantum computing and the application of two-state systems for the quantum
computer. Several two-state systems are currently utilized, or studied, as qubits,
including: Superconducting Quantum Interference Devices (SQUID), where the direction
of the current in the superconducting loop is the binary observable (ORL 2004); NMRbased devices, that were used to build a quantum computer capable of factoring the
number 15 (VAN 2001); systems based on the electron spin, that can be manufactured
utilizing existing chip fabrication technologies (CAR 2005); cold trapped ions (KIN
1999), (PEA 2006); and cold trapped neutral atoms (SCH 2004), (KHU 2005), (YAV
2006), (TRE 2006), (LAC 2006), (CHA 2006).
In 2000, DiVincenzo (DIV 2000) defined several requirements for the implementation of
quantum computation:
1.1 Scalable physical system with well characterized qubits. Scalability is important,
since any practical computer needs a large number of qubits. The meaning of the
term “well characterized” is harder to define exactly. It includes:
•
•
•
•
Stability, or at least meta-stability of two states that serve the qubit;
Accuracy of the physical parameters and the internal Hamiltonian;
Weak coupling to external fields (to avoid dephasing), but good coupling to
specific external fields which will be used to manipulate the qubit;
Weak coupling to other internal states and to other qubits (to prevent cross-talk).
1.2 Ability to initialize the qubits’ state. The main factor here is the initialization time –
if it is not considerably shorter then the quantum gate time (the actual
“computation”), one will need to replace the qubits after every gate operation.
1.3 Very long decoherence time (compared to gate time). This is necessary for the
implementation of “error correction” procedures.
1.4 A “universal” set of quantum gates. This is the heart of the quantum computer. In
principle, it is just a set of unitary transformations, applied sequentially, each to a
finite set of qubits. This implies that we will be able to turn some interaction “ON”,
direct it to a specific set of qubits, and then turn if “OFF”, without disturbing any
other qubits. In practice, no such system have bean realized yet, but even partly
realized system may be good enough for quantum computation. The details vary
considerably with the choice of qubit, but we may translate this requirement to the
following: the ability to address a specific qubit, to manipulate its state with
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“enough” precision, to control its interaction with other qubits, and to do so during a
time interval “much shorter” than the decoherence time.
1.5 Qubit-specific measurement capability. Theoretically, we should be able to measure
the state of each qubit independently of any other parameters of the system,
including the state of the nearby qubits, without disturbing any other qubit, and to do
so in a time interval “much shorter” then the decoherence time. In practice, we may
be able to relax this requirement by repeating the measurement several times and
averaging the results. This can be done either by repeating the computation several
times or by performing parallel computations simultaneously.
Based on these criteria, we can compare specific systems and methods and assess their
relative value as “components” of the future quantum computer. In this lecture, we will
focus on experimental procedures for rotating qubits (on the Bloch sphere) that are based
on ultracold neutral atoms. As an exercise, if time allows, we will use the above criteria
to estimate the suitability of the procedure as part of a future quantum computer.
In section 2 we will present some of the relevant theory. We will start with the
Schrödinger equation for a two-state atom and its interaction with external excitation.
Then we proceed to describe the density matrix, the Bloch sphere and the Bloch vector.
We will describe Rabi oscillations and manipulating two-state systems with π pulses,
chirped pulses, Raman pulses and Stimulated Raman Adiabatic Passage (STIRAP). In
section 3 we will review the state of the art in the field of manipulating two-state systems
based on ultracold atoms.
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2. Theory
In this section we review some of the theory relevant to the atomic two-state system. The
purpose here is to review the relevant theoretical background and establish notation.
Based on this background we will review the state of the art, and define the intended
research work included in this proposal. In this section we will mainly follow Shore
(SHO 1990), unless otherwise noted.
2.1 The Schrödinger equation for a two-state atom.
Let’s look at a bound atom which can be described by a stationary Hamiltonian H0 and a
complete set of orthonormal eigenfunctions ψ n such that:
and
H0 ψn = En ψn
(2.1)
ψ n ψ m = δ n,m
(2.2)
Any state Ψ(t) of the system can then be described as a linear combination of the
Ψ(t) = ∑ C n (t) ψ n
(2.3)
C n (t) = ψ n Ψ(t) ≡ n Ψ
(2.4)
ψ n s:
n
where
The Cn(t)s are complex numbers representing probability amplitudes. Pn(t), the
probability of finding the system in the state ψ n at the time t is:
2
Pn (t) = C n (t) = n Ψ
and, since ψ n is a complete set,
∑ P (t) = 1
n
2
(2.5)
(2.6)
n
Obviously, when there are no interactions between the bound atom and its environment
the system is stationary and all the Pn(t)s are constants. It follows that only the phase of
the Cn(t)s change with time for an isolated system:
C n (t) = e
i
− En t
h
C n (0)
(2.7)
We are interested in situations where we manipulate the atom with controllable, timedependent external interactions. Typically, those interactions are small relative to the
internal forces that define the stationary states ψ n . (If this is not the case, the external
forces will completely alter the internal atomic structure, the ψ n s will become
meaningless, and the perturbative approach we are following will collapse). Under this
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assumption we can define the Hamiltonian H, describing both the bound atom and its
interaction with the applied external forces as
Then the Schrödinger equation
H(t) = H 0 + V(t) .
(2.8)
∂
Ψ(t) = H(t)Ψ(t)
∂t
(2.9)
ih
describes the time evolution of the atomic state Ψ(t) under the influence of the external
interaction V(t), that may also be represented by its matrix elements Vnm(t):
Vnm (t) = ψ n V(t) ψ m
(2.10)
Vnm=Vmn*
(2.11)
with, since V(t) is Hermitian,
Let us now focus our attention at an atomic two-state system. Such a system is, by
definition, fully described by the two states: 0 ≡ ψ 0 and 1 ≡ ψ1 . Using (2.3) we can
expand any wave function Ψ(t)
in 0 and 1 . Substituting this expansion in the
Schrödinger equation (2.9) yield the following equation for the expansion coefficients:
ih
0
V01 (t) ⎤ ⎡C 0 (t)⎤
d ⎡C 0 (t)⎤ ⎡E 0 + V 00 (t)
=
⎢
⎥⎢
⎢
⎥
⎥
0
dt ⎣ C1 (t) ⎦ ⎣ V10 (t)
E1 + V11 (t)⎦ ⎣ C1 (t) ⎦
(2.12)
Once V(t) and the initial conditions are specified, Eq. (2.12) above provides the time
evolution of the two-state system subject to the external interaction.
2.2 Coherent interaction – Majorana spin flips
A bound two-state atom with no interaction with its environment will remain indefinitely
in its initial condition, as far as observables are considered. To illustrate some of the
phenomena that occur when such an interaction is present, let us subject a bound twostate atom to a very simple external time-dependent potential: zero at t < 0 and constant
afterwards.
t ≤ 0 : Vij (t) = 0, ( i, j = 0,1)
h
(2ϖ − ω 0 ) − E 00 , V11 (t) = h (2ϖ + ω 0 ) − E10 ,
2
2
1
V01 (t) = V10 (t) = const ≡ 2 hΩ
t > 0 : V00 (t) =
(2.13)
where hω 0 is the energy difference between the two states, and hϖ is the average of the
energies of the two states. Substituting (2.13) into (2.12) and setting ϖ = 0 (the zero
point of the energy is arbitrary) we get:
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i
which leads to
d ⎡C 0 (t)⎤ 1 ⎡− ω 0
=
dt ⎢⎣ C1 (t) ⎥⎦ 2 ⎢⎣ Ω
Ω ⎤ ⎡C 0 (t)⎤
+ ω 0 ⎥⎦ ⎢⎣ C1 (t) ⎥⎦
(2.14)
~
~
&& (t) = − 1 Ω
C
Ci (t), with Ω = Ω 2 + ω02 , i = 0,1
i
4
(2.15)
i
i
Solving (2.15) with initial conditions: C0(0)=1, C1(0)=0 C& 0 (0) = ω 0 , C&1 (0) = − Ω and
2
2
calculating the probability P1(t) of finding the system in the upper state we get:
(
)(
( ))
~ 2
~
P1 (t) = 12 Ω Ω 1 − cos Ωt
(2.16)
So we see that by adding this simple interaction, we caused periodic oscillations in the
two-state probability. When the interaction is weak ( V01 << hω 0 , or Ω << ω 0 ) the
oscillation amplitude is small, and its frequency is close to ω 0 . When the interaction is
strong ( Ω >> ω 0 ) the entire population moves periodically between each of the two
states, and its oscillation frequency is close to Ω . Similar oscillations of the two-state
atom are produced by almost any external excitation.
2.3 Pure and mixed states, coherence, density matrix, Bloch sphere and Bloch vector
The periodic probability oscillations described in the previous section can never be
observed by one measurement of a single two-state atom. To notice these oscillations we
need an ensemble of such atoms (and several other conditions that we will summarize in
the end of this section). In this section we will describe the basic tools to deal with
ensemble of quantum states, following Hideo Mabuchi (MAB 2001). (Note that many
observations of a single system constitute an ensemble)
Lets assume that at time t = 0 we can define an ensemble of two-state atoms through a set
of wave functions Ψ i (0) , each representing a fraction pi of the atoms in that ensemble.
We can then define the density matrix, or the density operator,
as:
ρ̂(0) = ∑ p i Ψ i (0) Ψ i (0) , with
i
∑p
i
=1
(2.17)
i
It can be shown that the equation of motion of ρ̂(t) is given by:
ih
∂
ρ̂(t) = [H, ρ̂(t)]
∂t
(2.18)
and that the expectation value of any operator  is given by
 = Tr(ρ̂Â)
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If one of the p i equals 1, say p1 = 1, then Ψ1 represents all the atoms in the ensemble,
and we have what is usually called a “pure state”. Otherwise, we have a “mixed state”.
(“Pure ensemble” and “mixed ensemble” seem more suitable names, but we will stick
with tradition).
r
We now introduce the spin operator S
r hr h
S = σ = (σ̂ x , σ̂ y , σ̂ z ) =
2
2
h ⎛⎛0 1⎞ ⎛0 − i⎞ ⎛1 0 ⎞⎞
⎜⎜
⎟, ⎜
⎟, ⎜
⎟⎟
2 ⎜⎝ ⎜⎝ 1 0 ⎟⎠ ⎜⎝ i 0 ⎟⎠ ⎜⎝ 0 − 1⎟⎠ ⎟⎠
(2.20)
r
and the Bloch vector ν , (which is the unit vector connecting the origin to a point θ, ϕ on
r
r
the Bloch sphere). S and ν will enable us to link together Bloch sphere, the density
operator and the pure and mixed states concepts.
The Bloch vector components, in Cartesian coordinates, are:
r
ν = (ν x , ν y , ν x ) = (sinθ cosϕ , sinθ sinϕ , cosθ )
(2.21)
r
Calculating the expectation value S for the pure state represented by θ, ϕ (Eq. 1.3) we
r
h
get (see annex 1 for details): S = (sinθ cosϕ , sinθ sinϕ , cosθ )
(2.22)
2
So by combining (2.19), (2.21) and (2.22) we see that for a pure state we have:
r
r hr
r
S = Tr(ρ̂S) = ν, with ν = 1
2
(2.23)
Eq. (2.23) motivates the definition of the generalized Bloch vector (both for pure and for
mixed states) as:
r
hr
ν B = Tr(ρ̂S)
2
(2.23a)
Studying the properties of the density matrix and the generalized Bloch vector, we can
learn the following:
(1 + νr B ⋅ σr )
•
The density matrix can be written as ρ̂ =
•
The diagonal matrix elements of ρ̂ are real; they give the relative populations of the
two basis states 0 and 1 for the ensemble represented by ρ̂ . Therefore, their sum
1
2
(2.24)
is 1, or Tr (ρ̂ ) = 1 .
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•
•
•
The off-diagonal elements of ρ̂ , ρ12 = ρ ∗21 , represent the coherence between the
atoms in the ensemble. For example, if we have a completely incoherent ensemble of
r
two-state atoms, the off-diagonal elements, (as well as ν B ), vanish.
r
r r
dν B 1 r
= Tr[σ[H, (1 + ν B ⋅ σ )]]
(2.25)
dt
2
Example: when a static magnetic field Bz is applied, the solution of this equation
shows that the Bloch vector precesses about the z axis at the Larmor frequency, given
by ω L = γB z , where γ is the gyromagnetic ratio of the atom. For simplicity, the
r
dynamics of ν B may be calculated in a reference frame that rotates about the z axis at
the Larmor frequency.
r
The equation of motion for ν B is:
ih
Finally, the Bloch vector can represent dissipation and dephasing. For that purpose,
we define 3 additional parameters: longitudinal relaxation time T1, transverse
relaxation time T2 and ν 0z , which is the z component of the Bloch vector at thermal
r
equilibrium. Adding these parameters to the equation of motion for ν B , and solving
in the rotating frame with no field except Bz, we obtain the following results for the
components of the Bloch vector:
ν x (t) = ν x (0)e
ν y (t) = ν y (0)e
(
−
t
T2
−
t
T2
(2.26)
)
ν z (t) = ν z (0) − ν e
0
z
−
t
T1
+ ν 0z
This result is valid only in a very simple case, but it does present the general picture. We
see that if we start at t=0 at some pure state on the surface of the Bloch sphere, the
projection of the Bloch vector on the XY plane will start to shrink to zero with the T2
time constant, representing dephasing, or decoherence process. In parallel, but at a
different rate defined by T1, the Z component of the Bloch vector will decay towards its
thermal equilibrium value ν 0z .
Let us briefly return to the Majorana spin flips described in section 2.2. We now see that
there are several requirements if we wish to actually observe the probability oscillations:
•
We need an ensemble of measurements. It can be realized by repeated measurement
on a single two-state atom, or by a single measurement of a group of such atoms, or
some combination of these two methods.
•
We have to prepare the ensemble so that its initial state is as close as possible to a
pure state. Otherwise, the atoms in the ensemble will have random phases relative to
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each other. This will mask any observation of the probability oscillations. Typically,
the ensemble is prepared by some type of optical pumping (YUA 2004).
•
All the operations, including final measurements, must be completed well before the
smaller of the two relaxation times T1 and T2. Failing to do this will again lead to
averaging out the probability oscillations.
•
As we will see in section (2.5), there is one more condition: the initial state of the
system at t=0 should not be an eigenstate of the full Hamiltonian. If it is, the system
will remain in that state indefinitely.
All of these conditions are required if we wish to observe the Majorana spin flips, or any
other coherent phenomena.
2.4 Rabi oscillations, Rabi frequency, Rabi flopping pulses, Ramsey fringes
The two-state atom is described by the Hamiltonian H0, the eigenvalues E0, E1 and the
eigenstates 0 , 1 . When we subject the atom to external periodic excitation (such as
laser light), the atom is now described by the full Hamiltonian Ĥ(t) = H 0 + V̂(t) . For
circular polarized light we may have (SHO 1990, §3.7 – annex 2)
V00 (t) = V11 (t) = 0
V01 (t) = V10* (t) = 12 hΩ e-i (ωt + φ )
(2.27)
where ω is the laser light frequency and Ω is the Rabi frequency. For laser beam of
intensity I (W/cm2), the Rabi frequency is given by:
Ω=
r r
d 10 ⋅ ε
ea 0
8π(ea 0 )
I
h 2c 2
2
(2.28)
r
r
r
d10 is the dipole matrix element 1 d 0 and ε is the unit vector in the direction of the
electric field of the laser light. (Note: this dipole moment, and Eq. (2.28), are valid only if
0 and 1 are non-degenerate states)
Substituting (2.27) into the time dependent Schrödinger equation, transforming to the
rotating wave picture (reference frame that rotates at ω) and manipulating we get the
following equations for the probability amplitudes:
i
d ⎡C 0 (t)⎤ 1 ⎡− (ω 0 − ω)
= ⎢
Ω
dt ⎢⎣ C1 (t) ⎥⎦ 2 ⎣
Ω ⎤ ⎡C 0 (t)⎤
(ω 0 − ω)⎥⎦ ⎢⎣ C1 (t) ⎥⎦
(2.29)
where hω 0 is the energy difference between the two states. Solving this equation with the
initial condition C0(0)=1, and calculating the probabilities we get:
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2
[
( )]
Ω
1
~
~
P1 (t) = 1 − P0 (t) =
1 − cos Ωt , Ω =
2
2
2 (ω 0 − ω) + Ω
Ω + (ω 0 − ω )
2
2
(2.30)
We see that the probability of finding the atom in the state 1 oscillates indefinitely
(Rabi oscillations) with a frequency that is the RMS value of the Rabi frequency and the
detuning δ = ω 0 − ω of the laser light from the frequency of the atomic transition.
This behavior is similar to the Majorana spin flips presented in section 2.2. However
while the amplitude of the Majorana spin flips depend mainly on Ω , the amplitude of the
Rabi oscillations depends only on the detuning. At δ = 0 the amplitude is maximum and
it drops as the excitation field frequency moves away from this resonance point. Figure
2.1 below present the Rabi probability oscillations for Ω = 2rad/sec, and several values
of the detuning.
1
δ=0
P1(t)
0.75
δ=Ω
0.5
0.25
0
0
1
2
3
4
5
6
δ=2Ω
time
Figure 2.1 – Rabi oscillations
2.4.1 Rabi pulses
Focusing our attention on the resonance case, and following Kuhr (KUH 2003), we can
r
represent the effect of the Rabi pulse of duration time t on the Bloch vector ν B (0)
(section 2.3) by the following 3x3 matrix:
0
0 ⎞
⎛1
⎜
⎟ r
r
ˆ
ν B (t ) = ⎜ 0 cos φ (t ) sin φ (t ) ⎟ ⋅ν B (0) = Θ
φ ( t ) ⋅ν B (0)
⎜ 0 − sin φ (t ) cos φ (t ) ⎟
⎝
⎠
r
(2.31)
t
with
φ (t ) = ∫ Ω(t ' )dt '
(2.32)
0
In other words, a Rabi pulse rotates the Bloch vector about the x axis.
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For φ (t) = π and φ (t) = π/2 and the matrix becomes very simple:
Θπ
2
⎛1 0 0⎞
⎛1 0 0 ⎞
⎜
⎟
⎜
⎟
= ⎜ 0 0 1 ⎟ , Θπ = ⎜ 0 − 1 0 ⎟
⎜ 0 −1 0⎟
⎜ 0 0 − 1⎟
⎝
⎠
⎝
⎠
(2.33)
Θ π is called a π pulse, or Rabi flopping pulse, and Θ π / 2 is called a π 2 pulse.
The matrix representation for the free precession is:
⎛ cos φ (t ) sin φ (t ) 0 ⎞
t
⎜
⎟
Θ FREE (t ) = ⎜ − sin φ (t ) cos φ (t ) 0 ⎟, φ (t) = ∫ δ (t' )dt'
0
⎜
0
0
1 ⎟⎠
⎝
(2.34)
Note that the frequency of the free precession of the Bloch vector is δ = ω 0 − ω , since our
results were obtained in a rotating wave frame that rotates at ω .
a.
b.
0
0
c.
0
φ
1
0
d.
1
1
e.
0
1
1
f.
0
1
Figure 2.2: some examples of Rabi pulses.
is the Bloch vector before the
pulse,
is the vector after the pulse.
Figure 2.2 above presents some examples of Rabi pulses: (a) presents a general Rabi
pulse. The x component of the Bloch vector is saved, while its projection on the yz plane
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is rotated by φ . (b) Presents the operation of a π pulse on 0 . It flips the population
from 0 to 1 , which explains its name - Rabi flopping pulse. In (c) a π/2 pulse takes the
population from 0 to a point on the equator representing coherent equal mixture of the
two base states with no phase difference between them (ϕ = 0) . In (d) another π/2 pulse
applied immediately takes the population from that point to 1 . (e) shows free precession
of the Bloch vector in the xy plane. Finally, (f) shows how a π/2 pulse effect the Bloch
vector after it was allowed to precess for some time.
The Rabi pulses may be utilized directly to manipulate our two-state atom system.
However, the “two-state atom” is usually just an approximate model, where we pick two
out of its many states and set them to be the states that will realize a qubit or any other
useful device. As it turns out, typically those two states are separated by microwave scale
energies (∼10GHz), so that the Rabi pulses will need to have microwave frequency.
Microwave pulses are much harder to manipulate than optical frequency laser beams, and
their spatial resolution is measured in centimeters, compared with less than a micrometer
for the optical range. Moreover, in some cases the direct transition between the two states
may be forbidden or limited by selection rules, making direct Rabi flopping impossible.
Therefore, Rabi pulses are rarely utilized for direct manipulation of two-state ensembles.
2.4.2 Ramsey fringes
Consider the following sequence of operations, performed on an ensemble of two-state
atoms:
•
Prepare the ensemble in the pure state Ψ (0) = 0 .
•
Apply a π/2 Rabi pulse by a coherent, monochromatic laser beam at ω ≈ ω 0
•
•
Block the laser beam for time T
Apply a second π/2 Rabi pulse by same laser
•
Measure the population p1 (t) of level 1
In terms of the Bloch vectors and the matrices defined in (2.33) and (2.34), we can
r
summarize the sequence described above as measuring the z component of ν RAMSEY (T )
defined below:
r
r
ν RAMSEY (T ) = Θπ 2 ⋅ Θ FREE (T ) ⋅ Θ π 2 ⋅ν B0
(2.35)
Based on the examples presented in Figure 2.2, it is clear that p1 (t) is a periodic function
of the accumulated phase φ (see 2.34). Actually, we have:
T
ν Z , RAMSEY (T ) = − cos φ = − cos( ∫ δ (t )dt )
(2.36)
0
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To observe the oscillatory behavior of (2.36), we need δ to be small compared to the
Rabi frequency and the spectral pulse width, such that the pulse can be approximated as
near resonant one, and complete population transfer can occur.
The “Ramsey fringes” created by (2.36) are a very delicate interferometeric tool. The
interferometeric nature can be revealed if we look at a π/2 pulse as a 50/50 optical
beamsplitter. The first pulse splits the atomic wavefunction into a superposition of the
two states. During the free propagation time, the relative phase of the two states evolves
at a rate proportional to the energy difference between the states, and the coupling driving
field accumulates a phase ωT . The second pulse recombines the two states,
interferometrically comparing the accumulated relative phase.
Below are some examples to demonstrate how this tool works. To simplify, let’s define
the following set of units:
~
δ =δ Ω
(2.36a)
~
T = TRAMSEY /2πΩ
(2.36b)
where Ω is given by (2.28). In the following example, we will set the Rabi frequency to a
convenient number: Ω =1000*2π.
First let review the effect that T has when the detuning is small and independent of time:
~
δ = 0 ÷ 0.5 . For very small detuning, 2.36 is a good approximation. However, for larger
detuning we need to replace Θ π 2 in (2.35) with (2.31). Based on that we can calculate the
~
~
population of 1 : P1 = (1 − ν Z,RAMSEY (T))/2 . (Note that T measures time in units of Rabi
cycles). We can see below that as the detuning goes up, the cycling of the population is
more rapid, while the maximum population decreases.
1
0.8
Detuning =0
P1
0.6
0.4
Detuning =0.1
Detuning =0.5
`
0.2
0
0
5
10
T
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In the graph in this page we
see the effect of the detuning
on the population of 1 , for
1
0.8
two values of T. We can see
that as T is larger, the
population cycling is more
rapid. Thus, using the Ramsey
fringes with large T, enables us
to
determine
the
Rabi
frequency
with
higher
accuracy.
0.6
P1
T=5
T=15
0.4
0.2
0
-1.5
-1
-0.5
0
0.5
1
1.5
Detuning
2.5 Dressed states, adiabaticity, chirped pulses, adiabatic population transfer
Our two-state atom is described by the Hamiltonian H0, the eigenvalues E0, E1 and the
eigenstates 0 , 1 . When we add external excitation, the atom is now described by the
full Hamiltonian Ĥ(t) . It is of interest to study the full Hamiltonian eigenstates. Let the
interaction with the external excitation (such as monochromatic laser beam) be:
h⎡ 0
V̂(t) = ⎢ − iωt
2 ⎣Ωe
Ω*eiωt ⎤
⎥
0 ⎦
(2.37)
and the rotating wave frame state:
Ψ(t) = C0 (t) 0 eiωt 2 + C1 (t) 1 e-iωt 2
(2.38)
(The energy zero is midway between the states). Substituting (2.37) and (2.38) in the time
dependent Schrödinger equation lead to:
ih
∂ ⎛ C 0 (t) ⎞
⎜
⎟=
∂t ⎜⎝ C1 (t) ⎟⎠
h ⎛ - δ Ω * ⎞⎛ C 0 (t) ⎞
⎜
⎟⎜
⎟
2 ⎜⎝ Ω δ ⎟⎠⎜⎝ C1 (t) ⎟⎠
(2.39)
where δ = ω 0 − ω, and hω 0 = E1 − E 0 . As we can see, in the rotating wave frame the
matrix representation of Ĥ(t) is time independent, and so are its eigenvalues, given by:
~
h
hΩ
2
2
E± = ±
Ω +δ =±
2
2
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Calculating the eigenvectors we get (up to a global phase):
Φ + (t) = e iωt 2
Φ − (t) = e
i ωt 2
1
δ
1
δ
(1 + ~ ) 0 + e -iωt 2
(1 − ~ ) 1
2
2
Ω
Ω
1
δ
1
δ
(1 − ~ ) 0 + e -iωt 2
(1 + ~ ) 1
2
2
Ω
Ω
(2.41)
the states Φ ± (t) are called “dressed states”, as opposed to the “bare states 0 , 1 . It is as
if the full Hamiltonian “dresses” the bare states.
We can utilize the dressed states to manipulate our system of two-state atoms.
•
First, from (2.41) we can see that at far red detuning, Φ − (t) is mainly 0 , while at far
blue detuning, Φ - (t) is mainly 1 .
•
Second, we note that if a system is represented at some time by either one of the
dressed states, it will remain in this state indefinitely, as this state is an eigenstates of
the full Hamiltonian.
•
Third, it turns out that if we change the frequency of the exciting field “slowly”, a
system that is in a pure dressed state will remain in this dressed state. Such a change
is an adiabatic change. The adiabaticity can be measured by (RUB 1981)
γ=
Ω
2
dδ dt
(2.42)
The probability that an atom will make a transition to another dressed state is P = e −2πγ so
that with large enough γ (see 2.24), we can assume that no atom will make a transition to
another dressed state.
There are several procedures that utilize adiabaticity to manipulate atomic ensembles.
Chirped pulses
To demonstrate this procedure, let’s start with an ensemble that has all the atoms in the
ground state 0 . (This may be the thermal equilibrium at the ensemble’s temperature, or
a result of some other procedure). Then we slowly ramp up the intensity of a far-red
detuned laser field. Since the Φ − (t) dressed state is then mainly 0 , all the atoms will be
in this state. Then we sweep the frequency adiabatically up to a far-blue detuned region.
There, the Φ − (t) dressed state is then mainly 1 . At this point we adiabatically ramp the
intensity down to zero, leaving almost all the atoms in 1 . Thus we have a “chirped
pulse” method to manipulate the state of an ensemble of atoms.
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2.5
Adding other states: Raman transitions, STIRAP
Typically, our two-state atom has many other states that we ignore. Sometimes, however,
it is advantageous to bring more states into the game. It adds additional parameters that
may be used to manipulate our two-state system, and may facilitate otherwise forbidden
transitions. We will review two schemes here: Raman transitions and STIRAP.
2.5.1 Raman transitions
Figure 2.3 presents a scheme to excite Rabi probability oscillation between our two states
through a third state. This is known as a Raman transition.
The scheme shown in figure 2.3 is
typically called a “Λ configuration”. Two
coherent light beams are applied
simultaneously to the system. (By
“coherent” we mean that the phase of the
beat-note of the two light fields is
independent of time.) We define here two
Rabi frequencies per (2.28): Ω P for the
transition, and Ω S for the
1 ⇔ 2
ω0
2 ⇔ 3 transition. The energy zero is at
Figure 2.3: Raman transition scheme
level 1 .
Following Dotsenko (DOT 2002), this system can be described by the state:
Ψ (t ) = C1 (t ) 1 + C 2 (t ) 2 + C3 (t ) 3
(2.43)
and its Hamiltonian Ĥ is given by (using the rotating wave approximation) :
⎛ 0
⎜
h
Hˆ = ⎜ Ω P
2⎜
⎝ 0
ΩP
2Δ
ΩS
0 ⎞
⎟
ΩS ⎟
2δ ⎟⎠
(2.44)
Substituting (2.43) and (2.44) into the time dependent Schrödinger equation, we get
equations for the Ci(t)s. For C2(t) we have:
iC& 2 (t ) = 12 (Ω P C1 (t ) + Ω S C3 (t )) + ΔC 2 (t )
(2.45)
As typically Δ >> Ω P , Ω S , C2(t) oscillates at high frequency. Therefore we may replace
its time derivative in (2.45) with its average over large number of cycles, namely zero
(see DOT 2002, pp5). Using the modified (2.45) to calculate C2(t) as a function of the
other Ci(t)s, we get an effective Hamiltonian:
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2
Ω P ΩS Δ ⎞
1⎛ Ω Δ
⎟
Ĥ EFF = ⎜⎜ P
4 ⎝ Ω P Ω S Δ Ω S2 Δ − 4δ ⎟⎠
(2.46)
The off diagonal elements give the coupling of the state 1 and 3 by the Raman beams,
while the diagonal elements represent the light shift of the energy levels due to the
interaction with the light fields. By solving the time dependent Schrödinger equation with
the effective Hamiltonian, and calculating the population probabilities, we obtain:
P3 (t) = 1 − P1 (t) =
Ω 2R
sin 2 (Ω 0 t 2)
Ω 02
(2.47)
where at t=0 all the population is in 1 , and we have defined Ω 0 = Ω 2R + δ 2 and
Ω R = Ω P Ω S 2Δ .
So operating with two light fields we may to induce Rabi oscillations between our two
states as if we were coupling them with a single δ-detuned field.
2.5.2 STIRAP - Stimulated Raman Adiabatic Passage
STIRAP is a method for the complete transfer of an ensemble of atoms from one state to
another, with the indirect “help” of a third state. In this subsection we will follow
Bergmann (BER 1998) and use his notation.
In Fig 2.4 we see the three levels: Level 1 is the
ground state while level 3 is the excited state. The
idea is to transfer the population from 1 to 3 ,
without populating level 2 . (Atoms excited to level
2 will undergo spontaneous emission to levels other
than 1 and 3 and would be lost).
Fig 2.4: STIRAP scheme
The STIRAP method utilize two laser fields: the Pump beam, linking 1 and 2 , with
detuning Δ P , and the Stokes beam, linking 2 and 3 , with detuning Δ S . Clearly, this
system is very similar to the Raman scheme (Figure 2.3), and we will use a similar
Hamiltonian as a starting point:
0
Ω P (t )
⎞
⎛ 0
⎟
⎜
h
Hˆ (t ) = ⎜ Ω P (t ) 2Δ P
Ω S (t ) ⎟
2⎜
Ω S (t ) 2(Δ P − Δ S ) ⎟⎠
⎝ 0
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Here, however, we intend to modify the Rabi frequencies during the process, so we
explicitly indicated their time dependence.
The next step is to define the following 3 eigenstates through the mixing angle Θ:
TanΘ =
Ω P (t)
Ω S (t)
a + = sin Θ sin Φ 1 + cos Φ 2 + cos Θ sin Φ 3
a
0
(2.49)
= cos Θ 1 − sin Θ 3
a − = sin Θ cos Φ 1 − sin Φ 2 + cos Θ cos Φ 3
(The angle Φ is a function of the Rabi frequencies and detuning and is not relevant here.)
The three eigenstates above have the following eigenvalues (dressed states):
ω + = Δ P + Δ 2P + Ω 2P + Ω S2
ω0 = 0
(2.50)
ω − = Δ P − Δ 2P + Ω 2P + Ω S2
We will now describe the population transfer
procedure, with the help of figure 2.5.
At t=0, both laser intensities and the mixing
angle are zero, all the population is in state 1 ,
and the three dressed states are degenerate.
In the first step we ramp the Stokes laser up
adiabatically (a). We notice (c) a split of the
dressed states to three distinct levels. The
population (d) is still at 1 since the Stokes
laser couples two empty states. However, as the
mixing angle (b) is still zero all the population
is at a 0 also, as a 0 = 1 at this point.
At the next step we ramp up (a) the Pump laser,
while ramping down the Stokes laser. This is
done in order to smoothly change the mixing
angle (b) from zero to 90°. During this step, all
the population remains in a 0 , since it is an
Figure 2.5: STIPAP procedure
eigenstate of the full Hamiltonian.
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However, as Θ changes smoothly from 0 to 90°, a 0 , which is a superposition of 1 and
3 , changes smoothly from 1 to 3 (see (2.49)). We now ramp the Pump laser
intensity down smoothly to finish the procedure. Using this sequence, we are able to
transfer all the population from state 1 to 3 without populating 2 , from which
population may be lost via spontaneous emission. This method can be modified to
include more than one additional level, and may be used for arbitrary manipulation of a
two-state atom.
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3. State of the art
During the last decade, a large amount of work was aimed at the manipulation of twostate atoms, also known as qubit rotations. Several methods were used, such as single
pulses, chirped pulses, several variations of the STIRAP methods and Raman beams. The
two-state neutral atoms manipulated thereby were held in magnetic traps on atomchips, in
optical traps (such as optical lattice) and in magneto-optical traps. In other experiments
the atoms were part of a moving beam, or confined to a vapor cell. To picture the state of
the art in this field, we present below several examples of these efforts that have been
performed in the last three years.
3.1
STIRAP based method
We will start our review with a method based on STIRAP, performed in Kaiserslautern
by Vewinger et al (VEW 2003). They used a STIRAP procedure to achieve a coherent
superposition of two degenerate states in a moving beam of 20Ne atoms.
The relevant level diagram is presented in Figure
3.1. At the start of the sequence, the atoms are at
the 3P0 state. The target is to transfer them to the
states 3P2(m = +1), 3P2(m = -1), and control the
relative phase. Two laser beams are used: The
linearly polarized Pump laser P, coupling 3P0 with
the intermediate level 3P1(m = 0), and another
linearly polarized laser S, which can be viewed as
a superposition of two σ+ and σ− beams. These
beams, marked S+, S-, couple the intermediate level
Figure 3.1 – Level diagram (see text)
with the target states.
The Hamiltonian of the combined system (atom and radiation) has four adiabatic states,
two of which are “dark” states, which do not have components of the 3P1 state. These
states are used for the adiabatic transfer process:
Φ1 (t) = cosϑ (t) 0,0 − sinϑ (t)cosϕ (t)e −iχ 2,−1 − sinϑ (t)sinϕ (t)e + iχ 2,+1
(3.1)
Φ 2 (t) = sinϕ (t)e −iχ 2,−1 − cosϕ (t)e +iχ 2,+1
where the states are marked by J, m notation, the mixing angles ϑ and ϕ given by:
tanϑ (t) =
Ω P (t)
Ω S (t)
Ω (t)
tanϕ (t) = s+ , Ω s (t) = Ω S2+ + Ω S2−
Ω S− (t)
(3.2)
and the angle χ measures the inclination of the polarization plane of the S beam, as seen
in figure 3.2 below.
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Generally, Φ1
and Φ 2
are coupled. In our case they are not coupled, since ϕ is
constant. Therefore, in the adiabatic limit the system reaction to detuning changes is
given by Φ1 , as Φ 2 does not include the initial state.
The 20Ne atoms interact first with the Stokes beam (see Figure 3.2 below). As they move
on, they interact with both the stokes and the pump beams, with the stokes beam intensity
fading and that of the pump beam rising. Then pump beam fade out and the atoms move
out of the interaction region. (see section 2.5.2 for details). With this sequence, we have
(see 3.2) ϑ (−∞) = 0 at the beginning of the sequence and ϑ (+∞) = π / 2 at its end. Since
we know that the system will evolve per Φ1 (in the adiabatic limit) we can substitute
these values in Eq. 3.1, and see that the system will evolve from the initial state 0,0 to
the desired final state:
Ψ =
[
1
2,−1 e −i ( χ +φ ) + 2,+1 e +i ( χ +φ )
2
]
(3.3)
where the relative phase can be controlled by rotating the polarization plane of the S
Beam.
The paper then describes the method to measure the actual resulting superposition, which
we will not follow here.
Figure 3.2: Geometry of the experiment. The direction of polarization (indicated by the small
arrows) for the pump laser (λ = 616nm) is chosen to be the z-axis, its direction of propagation
defines the x-axis, while the neon beam propagates in y-direction. The two circularly polarized
Stokes lasers are generated by a linearly polarized laser (λ = 588 nm) propagating in zdirection, whose direction of polarization forms an angle χ with the x-axis. The propagation of
the so called filter laser (λ = 588 nm) is parallel to the Stokes laser and its direction of
polarization forms an angle α with the x-axis. The detection laser (λ= 633 nm) is unpolarized.
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3.2 Spin flips with π pulses and chirped pulses
In 2004, Schrader et al. (SCH 2004), (Bonn
University, Meschede’s group) developed a
quantum register based on trapping a string
of single cesium atoms in an optical dipole
trap (Figure 3.3). Cesium atoms were
transferred from a MOT to a standing wave
dipole trap created by a far-red detuned laser
beam (λ=1064nm). The light from the
cesium atoms was collected by an Avalanche
PhotoDiode (APD) and Intensified ChargeFigure 3.3: setup for quantum register (see text)
Coupled Device camera (ICCD).
They chose the levels: F = 4, m F = 4 ≡ 0 , F = 3, m F = 3 ≡ 1
as the qubit states,
separated by a ~9.2GHz transition. When they operated the system, cesium atoms filled
some of the optical dipole traps created by the standing wave light radiation. (Figure 3.4,
a). To address each atom separately, they added a magnetic field with a gradient of 15
G/cm along the trap axis. Due to the Zeeman shift of the energy levels, this magnetic
field created a shift of ~9kHz in the resonance frequency for atoms separated by 2.5 μm.
Each atom has its own resonance
frequency as a function of its position.
The position of each cesium atoms is
located by the ICCD camera, and its
resonant frequency is calculated and store
in the system’s computer (Fig. 3.4 – a).
The next step is initialization (by optical
pumping) of all atoms (5 in our case) to
the state 00000 (Figure 3.4 b).
Then two microwave π pulses, one with
the resonanse frequency of the second
atom and one with that of the fourth atom,
are sent to the register via open ended
wave guide. Now our register reads
01010 (Figure 3.4 c). To verify that this
is really the case, a push-out laser beam
removes atoms in the 0 state from the
trap. (Figure 3.4 d). This is confirmed by
the camera picture (e).
Figure 3.4: Cesium atoms register (see text)
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Schrader et al measured decoherence time of 0.6ms, and crosstalk (effect on neighboring
atoms) of about 1%.
Using the same system, Khudaverdyan et al (KHU 2005) manipulate the state of the
cesium atoms with chirped pulses (otherwise known as adiabatic passage). They used two
microwave signal generators: one provides a 10.2 GHz signal with fast AM capabilities,
and the other provides 1 GHz signal with FM capabilities. Using a mixer to subtract the
signals, they got the required 9.2 pulse with computer controlled AM and FM
modulation. Thus, they were able to produce pulses with arbitrary shape. The sequence
they used was: Trapping cesium atoms in the optical dipole trap, initializing them to 0
with optical pumping, then transmitting the microwave pulse and finally measuring the
resulting population.
They showed a robust transition efficiency of up to 90%- 95% for variation of up to ±40
kHz in the signal’s central frequency. In addition, they demonstrated an ability to use a
fixed frequency microwave radiation to flip the atoms while the atoms were in motion.
They moved the atoms by an optical conveyor belt across a region where their resonance
frequency was a function of position (due to magnetic field gradient). As the atoms
reached the point where their resonance frequency matches the fixed microwave field,
their state was flipped.
3.3 Raman transitions in optical tweezers
M. P. A. Jones et al, (Jon 2006) in Cedex, France (Grangier’s Group) used a tightly
focused far-off resonant (λ=810nm) laser beam to create a dipole trap for 87Rb atoms.
Their trap had the property that the number of atoms in the trap to be either zero or one,
and was loaded with cold rubidium atoms (90μK) from optical molasses. As seen in
Figure 3.5 below, they chose as qubit states
and
F = 1, m F = 0 ≡ 0
F = 2, mF = 0 ≡ 1 . The also utilized the laser beam that created the trap as one of the
Raman beams. The other beam was produced by injection locking a sideband of a
microwave modulated bridge laser to the master (FORT) laser, and using the other
sideband to lock a slave laser. Thus the slave laser’s beam frequency is 6.8 GHz away
from the master’s, while maintaining the coherence between the beams. The Raman
beams have orthogonal linear polarization in the y-z plane, to drive ΔmF = 0 transitions.
The experiment sequence was as follows: The system monitors the fluorescence from the
molasses cooling light via the avalanche photodiode (APD). As an atom populate the
trap, the fluorescence goes drastically up, triggering the shut down of the molasses light
and the experiment sequence start.
First the atom is optically pumped to 0 . Then a Raman pulse is send to the atom,
followed by a “push out” laser beam that remove atoms in the 1 state from the trap.
Finally, the molasses light are turned on again and the status of the trap is detected.
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Rabi frequency of 2π x 6.7MHz was demonstrated, providing π/2 rotation time of 37ns.
Coherence time was 370μs. Using spin echo method, coherence time of 34 ms was
reached, 6 orders of magnitude higher then the qubit rotation time.
Figure 3.5: System setup and level diagram (see text)
3.4
Raman transitions on atom chip
Treutlein at el (TRE 2006) review the work in Max-Planck institute for quantum optics.
They realized single qubit rotation based on microwave and radio frequency Raman
transition. They chose the following levels F = 1, m F = −1 ≡ 0 , F = 2, m F = +1 ≡ 1
as the qubit states. The motivations for this choice are:
• Both states can be trapped by magnetic potential (“low field seekers”)
• The magnetic moments and the corresponding static Zeeman shifts of the two states
are approximately equal, leading to a strong common mode suppression of magnetic
field induced decoherence
•
Both states have nearly identical trapping potentials in magnetic traps. This will
reduce undesirable entanglement between the internal and external states.
In addition, at a magnetic field of 3.23G, both states experience the same first-order
Zeeman shift. Therefore, the atoms were held in magnetic traps with B0=3.32G, leading
to Zeeman shift of few MHz. Figure 3.6 (top) presents the level diagram of the 87Rb
atom, and the two Raman beams used in this experiment. The microwave Raman beam
was produced by a microwave signal generator phase locked to a 10 MHz reference
oscillator, and detuned 1.2 MHz above the intermediate 2,0 state. An RF signal
generator, phase locked to the same reference oscillator, generate the second (RF) Raman
beam.
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An ensemble of about 104 0 atoms, at 0.6μK, trapped in a Ioffe-type microtrap was
prepared, and then Raman pulse was applied.
After The Raman pulse, the number N1 of atoms transferred to 1 was measured.
Figure 3.6 (bottom) is a plot of N1 as a function of the Raman pulse length, Showing
320Hz Rabi oscillations. The reason for this low Rabi frequency is the low radiation
power. Both beams are transmitted form outside the vacuum chamber. Decoherence
time, measurement by Ramsey spectroscopy, is 2.8 seconds.
A possible improvement for this method is to include microwave wave guide and RF
wires on the atom chip. This will increase dramatically the Raman power that reaches the
atoms, and thus raise the Rabi frequency to 1.8 MHz.
Figure 3.6: See text
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3.5
Proposal for realization of a qubit on atom chip
As the last item in this chapter, we present the proposition of Charron et al (CHA 2006)
regarding the realization of qubits on an atom chip.
The central idea in this proposal is to use the external degrees of freedom of a trapped
atom for quantum “calculations”, and then use Raman beams to transfer the “results” to
the internal states for long term storage.
In Figure 3.7 (top) we see the
physical arrangement of the trap.
The trapping potential is calculated
for single ultracold 87Rb atoms, and
for realistic currents and magnetic
fields. The resulting double well
potential of the trap is presented in
figure 3.7 (bottom). It is presented
along x` axis, tilted by 14.8° to the
original x axis. The transverse (y`)
potential is neglected, as it is
assumed that the energy of the atom
is well below the quanta of the
transverse degree of freedom.
The height of the internal barrier
(marked ξ) can be controlled by a
proper change of the currents. When
the barrier is high (Figure 3.7, a) the
atoms in the two wells do not
interact. When the barrier is lowered
(b), they do interact, and the
combined wavefunction depend
upon the states of each atom prior
lowering of the barrier.
Figure 3.7: Double wall microtrap (see text)
The following interaction sequence is proposed:
•
•
At t = 0 the barrier is high.
During 0 < t < T0 the barrier is smoothly lowered.
•
During T0 < t < T0+T1 the barrier is kept at its low position.
•
During T0+T1 < t < 2T0+T1 the barrier is smoothly raised to its initial condition.
Assuming that the atoms have low enough energy so that they can be either in the ground
vibrational state g = 0 or in the first exited vibrational state e = 1 , we have a two
qubit gate, encoded in the vibrational states of the two atoms.
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Then Charron et al propose to use two-photon Raman transition to encode the result of
the gate operation to the internal state of the atoms.
Some 87Rb levels are presented in figure 3.8.
Out of the 8 ground state sub-levels only 3 are
low field seekers which can be trapped by static
magnetic potential: 1,−1
2,1 and 2,2 . The
motivation for the assignment of the qubit states
1,−1 ≡ 1 2,1 ≡ 0 was already discussed in
section 3.4.
To induce two-photon Raman transitions
between those two levels, one of Raman beams
has to be σ− polarized, and the other σ+. If we
detune the Raman beams below the 5P1/2 level,
we may disregard the 5P3/2 16 sublevel, as the
Raman beams are more then 7 THz detuned
from them. (Typically, Raman beams may be
detuned up to 300 GHz from the relevant level).
Out of the remaining 8 5P1/2 sublevels, selection
rules dictate that only 4 will participate in the
transition, marked A, B, C, D in Figure 3.8.
Figure 3.8:
87
Rb levels (see text)
Charron et al proceed to calculate the effective interaction Hamiltonian of the system for
r
two (σ±) Raman beams with ω± frequencies and k ± wavevector, and their result is:
~
~
~
~ ⎞⎤
σ ⎞ ⎛σ
σ
hΩ − Ω + iη ⎡⎛ σ
~
H int =
e ⎢⎜⎜ 01 − 01 ⎟⎟ + ⎜⎜ 01 − 01 ⎟⎟⎥ + Hermitian conjugate terms (3.4)
4 3
⎣⎝ Δ A1 Δ C1 ⎠ ⎝ Δ A0 Δ C0 ⎠⎦
(
) (
)
r r
r r
where η = k + ⋅ r − ω + t − k - ⋅ r − ω - t , Ω± are the Rabi frequencies of the σ± Raman
beams, The detunings Δ are given by:
Δ A0 ≡ ω A − ω 0 − ω - Δ A1 ≡ ω A − ω1 − ω +
Δ C0 ≡ ω C − ω 0 − ω - Δ C1 ≡ ω C − ω1 − ω +
(3.5)
and the operators ~
σ are given by:
~ = i j ; i, j = 0,1, A, B, C, D
σ
ij
(3.6)
When we wish to set the Raman beam frequencies ω±, we need to detune them far
enough from any 5S1/2⇔5P1/2 transition frequency, otherwise we will have non-negligible
population in the 5P1/2 levels during the Raman transition, with unwanted spontaneous
decay to the 5S1/2 levels. Typically, the Raman detuning is few orders of magnitude
bigger then the excited level natural line width. However, in this case we may have
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Δ A1 ≈ Δ C1 , Δ A0 ≈ Δ C0
and our effective interaction Hamiltonian (3.4) will vanish
identically. The physical meaning is that that the two Raman processes through the A and
C levels interfere destructively, so that the probability for a transition between the qubit
states is zero.
However, the very structure of the 5P1/2 sublevels offers a possible solution to this
problem. While the line width of the 5P1/2 sublevel is 5.75MHz, the hyperfine splitting
between its F=1 and F=2 levels (between A and C) is some 800MHz. So it may possible
to detune the Raman beams far enough (relative to 5.75MHz) below A to prevent
populating it during the Raman pulse, but close enough (relative to 800MHz), to prevent
the Hamiltonian (3.4) from vanishing.
3.6
Summary
Neutral atoms can behave, under favorable conditions, as a two-state system. The two
hyperfine ground levels of alkali atoms seem like most favorable candidates to serve as
qubits. As a result, considerable scientific effort is devoted to the theoretical and
experimental study these levels and their manipulation. We have presented recent
samples of these works.
In this summary, we would like to note that it seems that the larger part of works deal
with neutral ultracold atoms held by optical dipole traps (optical tweezers, optical lattice,
etc.) Less work is devoted to atoms held by magnetic microtrap, typically realized on
atomchips.
As for the status of the magnetic microtraps on atomchips, we would like to quote
Treutlein (TRE 2006):
“Atom chips combine many important features of a scalable architecture for
quantum information processing. The long coherence lifetimes of qubits
based on hyperfine states of neutral atoms, accurate control of the coherent
evolution of the atoms in tailored micropotentials, and scalability of the
technology through microfabrication – which allows the integration of many
qubits in parallel on the same device while maintaining individual
addressability. Furthermore, atom chips offer the exciting perspective of
creating interfaces between the atomic qubits and other QIP systems
integrated on the same chip, such as photons in optical fiber cavities or solidstate QIP systems located on the chip surface. However, the experimental
demonstration of a fundamental two-qubit quantum gate on an atom chip is
an important milestone which still has to be reached.”
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