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Rabi oscillations
lectures in Quantum Informatics 5-6th of September 2011
Thilo Bauch and Göran Johansson
In this lecture I will go through atom-field interactions and Rabi oscillations.
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Performing single qubit gates
In the previous lectures we showed how to map any qubit hamiltonian onto
the hamiltonian of a spin 1/2 particle in a magnetic field. For the Cooper-pair
box qubit, the Josephson energy denotes the strength of Cooper-pair tunneling
between the island and the reservoir. In the charge basis, this corresponds to
a constant magnetic field in the x-direction. Since the charging energy is large
EC EJ , the direction of the effective magnetic field can be tuned in the
xz-plane by changing the gate voltage Vg .
As we discussed in connection to the Bloch sphere, any single qubit gate
can be performed by three consecutive rotations along two non-parallel axes.
Thus, arbitrary single qubit gates can be performed by quickly changing the
gate voltage between two different values, e.g. from the degeneracy point ng =
Cg Vg /2e = 1/2, where the effective field is in the x-direction to a position away
from the degeneracy point, where the field is almost purely in the z-direction.
This technique was used by Nakamura et al. in their original work showing
coherent oscillations in a Cooper-pair box.
Later developments has shown that charge noise is the dominant noise source
for superconducting qubits. At the charge degeneracy point the two qubit eigenstates have equal charge, and thus the qubit is insensitive to charge noise (to
first order in the coupling to the noise). Thus, there was a motivation to develop
schemes for making gates, where the qubit never leaves the degeneracy point.
The natural method is to use a small amplitude harmonic perturbation of the
gate voltage ng = 1/2 + δng cos ωt, which is on resonance with the qubit energy
splitting h̄ω = EJ . This method was first discussed by Rabi (Phys. Rev. 51, 652
(1937))in the context of nuclear magnetic resonance (NMR), where e.g. spins in
molecules can be probed. It is from this field we have the Bloch sphere, and also
the Bloch equations describing the dissipative dynamics, which we will introduce
later. Another field where this method of manipulating two-level systems is used
extensively is quantum optics (see e.g. ”Introductory Quantum Optics” by C. C.
Gerry and P. L. Knight). Specific transitions between electronic states in atoms
can be induced by applying laser pulses, with frequency on resonance with the
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corresponding energy difference. In fact, this is a good way to selectively choose
eigenstates, and thus reduce the dynamics to a two-level system.
2
Atom-field interaction
We start from the Hamiltonian for an electron in the potential from the nucleus
Ĥ0 =
P̂2
+ V (r),
2m
(1)
where P̂ = −ih̄∇. In the presence of an external electromagnetic field the
Hamiltonian changes into
h
i2
P̂ + eA(r, t)
− eΦ(r, t) + V (r),
(2)
Ĥ =
2m
where A(r, t) is the vector potential and Φ(r, t) the scalar potential of the field.
Since the electromagnetic field is invariant under so-called gauge transformations,
∂χ
A → A + ∇χ, Φ → Φ −
,
∂t
where χ(r, t) is an arbitrary scalar field. We simplify this Hamiltonian by using
the Coulomb gauge (also known as transverse or radiation gauge), which is given
by the constraint
∇ · A = 0.
The scalar potential in the Coulomb gauge is given by the Poisson equation
−∇2 Φ =
ρ
,
ε0
where ρ is the charge density, and ε0 is the permittivity of free space. If there
are no additional charges close to the atom, the scalar potential reduces to
the nucleus potential V (r) already accounted for in the solutions without an
external field. In this case, the vector potential satisfies the wave equation
∇2 A −
1 ∂2A
= 0,
c2 ∂t2
with plane-wave solutions on the form
A(r, t) = A0 ei(k·r−ωt) + c.c.,
where |k| = 2π/λ is the wave vector of the radiation and λ is the wavelength.
The Coulomb gauge is not relativistically invariant, which is clear since the
scalar potential is given by the instantaneous charge distribution. But for quantum optics is most often non-relativistic, so the simplification that the radiation
field is determined completely by the vector potential is more important.
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We can now write down the Hamiltonian in the Coulomb gauge,
h
i2
P̂ + eA(r, t)
Ĥ =
2m
+ V (r),
(3)
In order to simplify the atom-field interaction further we first note that the
spatial extension of the electron cloud in an atom is a few Ångström, while
the wavelength λ of the light is typically a few hundred nanometers. Putting
the origin r = 0 of the coordinate system in the atom nucleus, we find that
k · r 1 in our problem. Thus we consider a spatially uniform vector potential
A(r, t) ≈ A(t), which is known as the dipole approximation. Finally we perform
a gauge transformation using χ(r, t) = −A(t) · r, giving
∇χ(r, t) = −A(t),
∂A
∂χ
(r, t) = −r ·
= r · E(t),
∂t
∂t
where E(t) is the external electric field at the atom. Finally we arrive at the
Hamiltonian for the field-atom interaction in the dipole approximation
Ĥ =
P̂2
+ V (r) + er · E(t) = Ĥ0 − d̂ · E(t),
2m
(4)
where d̂ is the dipole moment operator. Fortunately the derivation of this
Hamiltonian is valid both for classical and quantized radiation fields. First we
consider the classical case.
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Rabi - a two-level atom interacting with weak
harmonic classical radiation
Consider two atomic levels with different parity, called ground state |gi and
excited state |ei, with transition frequency ω0 = (Ee − Eg )/h̄. Apply harmonic
radiation E(t) = E0 cos (ωt), with frequency close to resonance |ω − ω0 | ω0 .
The truncated two-level Hamiltonian reads
Ĥ = −
h̄ω0
σz − A cos (ωt)σx ,
2
(5)
where A = he|d̂ · E0 |gi. Writing an ansatz for the state
|Ψ(t)i = Cg (t)e−iEg t/h̄ |gi + Ce (t)e−iEe t/h̄ |ei,
(6)
with Eg = −h̄ω0 /2 and Ee = h̄ω0 /2 (see Eq. ??), we get from the Schrödinger
equation
i
Ċg = A cos (ωt)e−iω0 t Ce ,
h̄
3
1
hD/A=0
0.8
hD/A=1
e
P (t)
0.6
0.4
hD/A=3
0.2
0
0
0.5
1
t (s)
1.5
2
-8
x 10
Figure 1: Rabi oscillations. Probability of finding a (two level) atom, subject
to ”classical” harmonic radiation, in the excited state |ei as a function of time
t. The transition dipole moment he|d̂ · E0 |gi/2πh̄ was chosen to be 250 MHz.
The detuning between the harmonic radiation frequency and the atom transition
frequency ∆/2π = (ω − ω0 )/2π is varied between 0 MHz and 750 MHz.
i
A cos (ωt)eiω0 t Cg .
h̄
Expanding the term cos (ωt) = (eiωt + e−iωt )/2, we find the slowly rotating
terms e±i(ω−ω0 )t and the fast terms e±i(ω+ω0 )t . Since the time-evolution induced
by the applied field is much slower than ω0 , we can neglect the quickly rotating
terms, an approximation named the Rotating Wave Approximation (RWA),
leaving
i
Ċg =
Aei(ω−ω0 )t Ce ,
2h̄
i
Ċe =
Ae−i(ω−ω0 )t Cg .
2h̄
We can eliminate Cg to arrive at
Ċe =
C̈e + i(ω − ω0 )Ċe +
1 A2
Ce = 0.
4 h̄2
From the trial solution Ce (t) = eiλt we find two roots
q
1
2
2
2
λ± =
∆ ± ∆ + A /h̄ ,
2
4
where ∆ = ω0 − ω is the detuning. The general solution is
Ce (t) = C+ eiλ+ t + C− eiλ− t ,
and considering the initial conditions Cg (0) = 1 and Ce (0) = 0 we can find the
solution
A i∆t/2
Ce (t) = i
e
sin (ΩR t/2),
ΩR h̄
∆
−i∆t/2
Cg (t) = e
cos (ΩR t/2) + i
sin (ΩR t/2) ,
ΩR
q
where ΩR = ∆2 + A2 /h̄2 is the so-called Rabi frequency. The probability to
find the atom in state |ei is given by
Pe (t) = |Ce (t)|2 =
A2
2
sin (ΩR t/2) ,
Ω2R h̄2
which is shown in figure ?? for three different detuning values ∆.
4
Qubit gates in the rotating frame
Mapping the Rabi problem to the spin 1/2 Hamiltonian and the Bloch sphere,
the energy difference Ee − Eg = h̄ω0 corresponds to a strong static magnetic
field in the z-direction. Any qubit state will thus precess quickly around the
z-axis with angular frequency ω0 /2 (see chapter 7.1 in the lecture notes ”A
quantum bit – in theory and one realization”). We simplify the description of
the dynamics by using a coordinate system rotating along with the precession.
(The coefficients Cg (t) and Ce (t) in Eq. (??) are actually the coefficients for the
|gi and |ei in this rotating frame.)
Ĥrot = R̂Ĥ R̂† + ih̄
where R̂ = e−i
ω0
2
tσz
∂ R̂ †
R̂ ,
∂t
.
Ĥrot = −A cos (ωt − δ0 )
0
e−iω0 t
0
=
eiω0 t
A
A
0
e−iδ0 ei(ω−ω0 )t
0
eiδ0 e−i(ω+ω0 )t
=−
−
,
0
0
2 e−iδ0 ei(ω+ω0 )t
2 eiδ0 e−i(ω−ω0 )t
where we introduced phase δ0 , corresponding to a controllable time for switching
on the harmonic radiation.
At exact resonance ω = ω0 the Hamiltonian in the rotating frame and using
the rotating wave approximation (RWA) is
A
A
0
e−iδ0
Ĥrot = −
= − (cos δ0 σx + sin δ0 σy ) .
(7)
0
2 eiδ0
2
Thus we find that by controlling the phase of the harmonic signal we can perform
rotations around any axis in the xy-plane, in the rotating frame. Rotations
around the z-axis are performed by waiting.
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