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Ultrafast geometric control of a single qubit using chirped pulses
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2012 Phys. Scr. 2012 014013
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IOP PUBLISHING
PHYSICA SCRIPTA
Phys. Scr. T147 (2012) 014013 (4pp)
doi:10.1088/0031-8949/2012/T147/014013
Ultrafast geometric control of a single
qubit using chirped pulses
Patrick E Hawkins1 , Svetlana A Malinovskaya1 and
Vladimir S Malinovsky1,2
1
Department of Physics and Engineering Physics, Stevens Institute of Technology, Hoboken,
NJ 07030, USA
2
ARL, 2800 Powder Mill Road, Adelphi, MD 20783, USA
E-mail: [email protected]
Received 13 August 2011
Accepted for publication 6 September 2011
Published 17 February 2012
Online at stacks.iop.org/PhysScr/T147/014013
Abstract
We propose a control strategy to perform arbitrary unitary operations on a single qubit based
solely on the geometrical phase that the qubit state acquires after cyclic evolution in the
parameter space. The scheme uses ultrafast linearly chirped pulses and provides the possibility
of reducing the duration of a single-qubit operation to a few picoseconds.
PACS numbers: 03.67.−a, 42.50.Ex, 42.50.Hz
In this paper, we address a single-qubit manipulation
solely based on the geometrical effect. We demonstrate how
to use the geometric phase, which is controllable by the
relative phase between external fields, to perform ultrafast
arbitrary unitary operations on a single qubit. Moreover, we
show that femtosecond laser systems can be used for quantum
computing, allowing us to perform unitary operations on a
single qubit in a few picoseconds. The proposed method
combines adiabaticity and pulse area control. In more detail,
we demonstrate that by controlling the pulse area and the
relative phase of the chirped pulses, we can prepare the
resonant qubits in an arbitrary coherent superposition, while
adiabaticity of the excitation guarantees to keep off-resonant
qubits unexcited. Keeping in mind that the decoherence rate
of the qubit transitions in a quantum register should be small
enough to allow many quantum operations [6, 19], we restrict
our consideration by choosing the two lowest levels in the
three-level system as the qubit states. External addressing
of the qubit is done by using the Raman excitation scheme
through a third ancillary state. We assume that the ancillary
state is far off-resonance with external fields to ensure that
decoherence on the ancilla–qubit transitions can be neglected.
1. Introduction
In the last few decades, many universal sets of quantum
gates have been proposed for quantum computation [1–3].
Various combinations of quantum gates have been intensively
discussed in the literature related to the universality
in quantum computation [1, 4–6]. To perform quantum
computation, one must have two major building blocks at
their disposal: (i) any arbitrary unitary operations on a single
qubit and (ii) a controlled-NOT operation on two qubits.
The most common method for implementing single-qubit
quantum gates is based on the Rabi solution for a two-level
system excited by an external field [1]. There is a very simple
reason for this choice. To construct a quantum gate, one
needs to know the exact form of the evolution operator of
the qubit under the external field excitation. The evolution
operator has an easily interpreted form in the case of the Rabi
solution [7]. Essentially, the whole dynamics of the qubit is
governed by the pulse area. There have been several proposals
to design quantum gates in a robust fashion, i.e. to use
adiabatic methods for a coherent superposition preparation
[8, 9]. The ingenious idea of using the geometric
phase [10–12] to manipulate the qubit state was recently
developed in a new direction called geometric quantum
computing [13–16]. The adiabatic manipulation of a quantum
system using only geometric phase has some advantages
since it reduces the requirements for perfect tuning of control
field parameters, and also the geometric operations can be
significantly more robust against noise [13, 17, 18].
0031-8949/12/014013+04$33.00
2. The qubit evolution operator in the adiabatic
approximation
Let us consider the dynamics of a single qubit presented by
the two lowest levels in the three-level λ-system (see figure 1):
the two lowest quantum states |0i and |1i have energies 0 and
1
© 2012 The Royal Swedish Academy of Sciences
Printed in the UK
Phys. Scr. T147 (2012) 014013
P E Hawkins et al
|e >
Δ
ωS
ωP
δ
|1 >
|0 >
Figure 2. Rotation trajectory of the Bloch vector representing qubit
states |0i, |ii in panel (a) and states |1i, | − ii in panel (b) resulting
from the application of the sequence of the resonant π and π/2
pulses with relative phase ϕ + π .
Figure 1. Schematic diagram of the qubit states as the two lowest
states in the three-level system.
1 , respectively, and the excited ancillary state |ei has energy
e . To manipulate in time by the total wave function of the
quantum system |9(t)i = a0 (t)|0i + a1 (t)|1i + b(t)|ei, where
a0,1 (t) and b(t) are the probability amplitudes to be in state
|0i, |1i or |ei, we use the external fields, which in general can
(0)
be described as E P,S (t) = E (0)
P,S (t) cos φ P,S (t), where E P,S (t)
are the pulse envelopes and φ P,S (t) are the time-dependent
phases. We consider the case of linearly chirped pulses so that
φ P,S (t) = φ P,S + ω P,S t + α P,S t 2 /2, where φ P,S are the initial
phases, ω P,S are the center frequencies and α P,S are the chirps
of the pulses [20, 21].
Using the rotating wave approximation (RWA) and
assuming large detunings of the pump and Stokes’ field
frequencies from the transition frequencies to the ancillary
state, 1 = ω P − (e − 0 )/h̄ = ω S − (e − 1 )/h̄, we utilize
the adiabatic elimination of the ancillary state and obtain the
following form of the Hamiltonian in the field interaction
representation:
H = −δ(t)σ z h̄/2 − (ei1φ σ + + e−i1φ σ − )e (t)h̄/2 ,
qubit, δ(t) = δ = 0, we have θ (t) = θ (0) = π/4 and the
unitary evolution operator for the wave function of the
resonant qubit takes the form
U (t) = cos(S(t)/2) I + i sin(S(t)/2) (ei1φ σ + + e−i1φ σ − ),
(3)
Rt
where S(t) = 0 dt 0 e (t 0 ) is the effective pulse area. Note
that this solution of the Schrödinger equation in the adiabatic
approximation is the exact solution, since the nonadiabatic
coupling is exactly zero for the resonant qubit.
3. Geometric manipulation of the qubit states
To demonstrate a pure geometric operation on a single
qubit, we employ the Bloch vector representation by
using the Pauli matrix decomposition, % = (I + B · σ)/2,
of the qubit density matrix, % = |ΨihΨ|, where B =
(u, v, w) = (%01 + %10 , i(%01 −%10 ), %00 −%11 ), %i j = ai (t)a ∗j (t)
and i, j = {0, 1}. In this representation, the Bloch
equation Ḃ = Ω × B describes a precession of the
Bloch vector B about the pseudo-field vector, Ω =
(−e (t) cos 1φ, e (t) sin 1φ, −δ), with components determined by the effective Rabi frequency two-photon detuning,
and the relative phase between the pump and Stokes pulses.
Arbitrary unitary operations on a single qubit are
equivalent to the rotation of the Bloch vector B. It is
proven that any unitary rotation of the Bloch vector can
be decomposed as U = eiα0 Rz (α1 )R y (α2 )Rz (α3 ), where
Ri = eiασi (i = y, z) are the rotation operators [1, 3].
Therefore, to prepare an arbitrary state of a single qubit
we need to demonstrate rotations of the qubit Bloch vector
about the z- and y-axes by applying various sequences of the
external pulses. Below, we demonstrate how this can be done
by controlling the parameters of the external pulses that are
defined by the explicit form of the evolution operator (see
equation (3)).
To implement the rotation of the Bloch vector about the
z-axis (the phase gate) based on the geometrical phase, we can
use the evolution operator of the resonant qubit, equation (3).
The sequence of two π pulses with relative phase 1φ = ϕ + π
gives U π;ϕ+π U π ;0 = cos ϕI + i sin ϕσ z ≡ Rz (ϕ), where the
first subindex of U indicates the pulse area S(T ) and the
second one indicates the relative phase 1φ.
Figure 2 shows the Bloch vector trajectories: initially
the qubit is in the state |0i (|1i), which corresponds to the
(1)
where δ(t) = δ + (α S − α P )t + 1S(t), 1S(t) = (2P0 (t) −
2S0 (t))/(41) is the effective ac-Stark shift, e (t) =
 P0 (t) S0 (t)/(21) is the effective two-photon Rabi
frequency,  P0,S0 (t) = µ0e,e1 E (0)
are the Rabi
P,S (t)/h̄
frequencies, µ0e,e1 are the dipole moments, δ is the
two-photon detuning, 1φ = φ P − φ S , σ ± = (σ x ± iσ y )/2
are the Pauli raising and lowering operators and σ x,y,z are the
Pauli matrices.
The Hamiltonian in equation (1) controls the dynamics
of the qubit wave function as long as the approximation
of the adiabatic elimination is valid. Here we consider the
adiabatic excitation of the qubit. In this case, the solution of
the Schrödinger equation with the Hamiltonian in equation (1)
has the form
1φ
2 σz
3(t)
1φ
e−iθ(t)σ y ei 2 σ z eiθ(0)σ y e−i 2 σ z |9(0)i, (2)
p
Rt
where 3(t) = 0 dt 0 λ(t 0 ), λ(t) = δ 2 (t) + 2e (t) and
tan[2θ (t)] = e (t)/δ(t). Note that the general form of the
evolution operator in equation (2) is obtained in the adiabatic
approximation when the the nonadiabatic coupling has been
neglected. This approximation is well justified if the condition
˙ e (t)δ(t)| λ3 (t) is valid.
|e (t)δ̇(t) − 
In the case of completely overlapped pulses,  P0 (t) =
 S0 (t), with identical chirp rates, α P = α S , for the resonant
|9(t)i = ei
2
Phys. Scr. T147 (2012) 014013
P E Hawkins et al
Bloch vector pointing in the z (−z)-direction, while the vector
Ω1 = (−e , 0, 0) points in the −x-direction (for simplicity
we chose 1φ = 0 for the first π-pulse). The first π-pulse flips
the population to the state |1i (|0i), correspondingly the Bloch
vector turns about the effective field vector Ω1 (about the
x-axis); it stays in the y, z-plane all the time and points in
the −z (z)-direction at the end of the pulse. Due to the second
π-pulse, the population is transferred back to the initial state
|0i (|1i); therefore the Bloch vector returns to its original
position pointing along the z (−z)-axis. However, since we
chose 1φ = ϕ + π for the second π-pulse, the pseudo-field
vector is rotated counterclockwise by the angle ϕ + π in
the x, y-plane, Ω2 = (e cos ϕ, −e sin ϕ, 0), and the Bloch
vector moves in the plane perpendicular to the x, y-plane and
has the angle π/2 − α (−π/2 − α) with the x, z-plane.
The Bloch vectors representing a pair of orthogonal basis
states |0i and |1i follow a path that encloses, respectively,
solid angles of 2ϕ and −2ϕ. The geometrical phase is equal
to one half of the solid angle, which means the basis states |0i
and |1i gain phases ϕ and −ϕ and the evolution operator takes
the form of the Rz (ϕ) rotation. Therefore, by applying the
sequence of two π-pulses with the relative phase 1φ = ϕ + π,
we implement the rotation operator about the z-axis with the
relative phase controlling the rotation angle.
The rotation operator about the y-axis controlled by
the relative phase and determined by the geometrical phase,
which the Bloch vector gains along the cyclic path, can be
constructed by using three pulses. The first and third pulses
are π/2-pulses with 1φ = 0, while the second pulse is the
π-pulse with the relative phase π + ϕ. It is easy to show
using equation (3) that this three-pulse sequence results in
the rotation operator about the y-axis, U π2 ;0 U π ;π +ϕ U π2 ;0 =
eiϕσ y ≡ R y (ϕ), and the relative phase controls the rotation
angle.
To demonstrate the geometrical nature of the operation
R y (ϕ), we note that this operation creates the relative
phase
√
between two basis states | ± ii = (|0i ± i|1i)/ 2. In the
Bloch representation, these states have the form | ± ii =
cos π4 |0i + e±iπ/2 sin π4 |1i, which are two vectors pointing in
the y- and −y-directions (see figure 2). The trajectory of
the Bloch vector representing the states | ± ii is shown in
figure 2. The pseudo-field vectors Ω1 and Ω3 are defined by
the effective Rabi frequencies of the first and third pulses
and point in the −x-direction since we chose 1φ = 0. The
second pseudo-field vector Ω2 is rotated counterclockwise by
the angle ϕ + π in the x, y-plane, the same as in the case
above. The initial Bloch vector points in the y (−y)-direction.
The first π/2-pulse rotates the Bloch vector about Ω1 to the
position pointing in the −z (z)-direction. The second pulse
flips the direction of the Bloch vector. The third π/2-pulse
returns the Bloch vector to its original position. During
the whole evolution, the Bloch vector and the pseudo-field
vector are orthogonal. As we observe, similar to the previous
case, the basis states |ii and | − ii follow a path enclosing,
respectively, solid angles of 2ϕ and −2ϕ. Therefore, they
gain the relative phase 2ϕ, which is the geometrical phase
controllable by the relative phase between the external pulses.
This phase gate in the | ± ii basis is equivalent to the R y (ϕ)
rotation operator in the |0i, |1i basis.
4. Conclusions
We have demonstrated ultrafast arbitrary operations on a
single qubit, which are solely based on the geometrical phase.
The operations are robust with respect to external noise,
since the final states of the qubit do not depend on the
dynamical phase [13, 17, 18]. Our proposal combines the
pulse area control with adiabaticity by using chirped pulses.
An analytic expression for the evolution operator for the
resonant qubit, which utilizes the Raman excitation of the
three-level λ-system with a large single-photon detuning,
provides a clear geometrical interpretation of the qubit
dynamics.
Note that the general solution, equation (2), of the
Schrödinger equation in the adiabatic approximation also
provides a description of the off-resonant qubit dynamics. It
can be easily shown that for the off-resonant qubit, δ 6= 0,
the evolution operator (in the adiabatic limit) resulting from
the adiabatic return at the end of the pulse excitation,
T , has the form U (T ) = cos[3(T )/2]I + sin[3(T )/2]σ z .
Therefore, the off-resonant qubits in the quantum register
acquire only the additional phase determined by the value of
the dynamical phase, 3(T ). This phase can be accounted for
and corrected later at the end of quantum computation.
To estimate the time scale of the proposed geometric
operations, we can use 100 fs pulses with intensity of the
order of 1012 W cm−2 . Pulse duration increases to a few
picoseconds by applying a linear chirp, α P,S , of the order
of 10−5 fs−2 . This amount of chirping is sufficient to provide
the adiabatic excitation of the three-level system [20–22] and
can be readily produced experimentally using commercially
available pulse-shaping systems. Note that the pump and
Stokes Rabi frequencies,  P0,S0 (t), depend on the chirp
rate [20, 21] and this gives us a way to control the value
of the nonadiabatic coupling to satisfy the adiabaticity
conditions. At the same time, the effective pulse area, S(T ) =
RT 0
0
0 dt e (t ), is independent of the chirp rate [22], which
provides additional flexibility for a potential experimental
realization.
Acknowledgments
The authors acknowledge partial financial support from
DARPA (HR0011-09-1-0008) and NSF (PHY-0855391).
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