Download Geometric manipulation of the quantum states of two

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PHYSICAL REVIEW A 69, 050301(R) (2004)
Geometric manipulation of the quantum states of two-level atoms
Mingzhen Tian, Zeb W. Barber, Joe A. Fischer, and Wm. Randall Babbitt
Department of Physics, Montana State University, Bozeman, Montana 59717, USA
(Received 5 December 2003; published 6 May 2004)
Manipulation of the quantum states of two-level atoms has been investigated using laser-controlled geometric phase change, which has the potential to build robust quantum logic gates for quantum computing. For a
qubit based on two electronic transition levels of an atom, two basic quantum operations that can make any
universal single qubit gate have been designed employing resonant laser pulses. An operation equivalent to a
phase gate has been demonstrated using Tm3+ doped in a yttrium aluminum garnet crystal.
DOI: 10.1103/PhysRevA.69.050301
PACS number(s): 03.67.Lx, 42.50.Md, 03.65.Vf
Two-level systems, such as atoms or spins, have been
extensively used as prototypes for the quantum information
carrier—qubit, in quantum computing. The building blocks
of quantum computation consist of two types of accurately
controllable universal operations on a set of two-level systems: (i) any arbitrary unitary operations on a single qubit
and (ii) a controlled-NOT operation involving two qubits
[1,2]. These operations have been sought through dynamic
evolutions of various two-level systems driven by interaction
Hamiltonians from external sources, such as laser pulses and
magnetic fields. It was also realized in recent years that the
whole set of universal quantum operations can be accomplished purely through geometric effects on the wave functions of the qubits that are imparted by the external controls.
The geometric operations originate from the geometric
phase, a fascinating result of quantum theory, which shows
that a quantum system evolving through a cyclic path in the
wave function space gains a geometric phase in addition to
the Hamiltonian-dependent dynamic phase [3]. One of the
attractive features of geometric operation is the potential robustness. The geometric phase solely depends on the amount
of the solid angle enclosed by the evolution path. It does not
depend on the details of the path, the time spent, the driving
Hamiltonian, or the initial and final states of the evolution.
Therefore, it is expected that the geometric operations are
relatively robust compared to the dynamic ones and are resilient to errors caused by certain types of noises in the driving Hamiltonian. Take, for example, the noises that cause the
jitter on the evolution path, but keep the amount of solid
angle associated with the path unchanged. Quite a few allgeometric or hybrid (with dynamic operations) quantum operations have been proposed for various physical systems,
such as nuclear magnetic resonance (NMR) [4], iontraps [5],
neutral atoms [6], optical systems [7], and superconducting
nanocircuits [8]. However, the exploration of geometric
quantum computation is still in its early stage; experimental
demonstrations are rare [8–10] and the theoretical analyses
on robustness are still far from complete and conclusive.
In this paper, we propose a scheme for universal single
qubit operations on two-level atoms composed purely by
laser-controlled geometric phase changes. These geometric
operations are equivalent to the arbitrary rotations of the
Bloch vector of a two-level atom on the Bloch sphere. To
eliminate the infidelity of the operation associated with the
Hamiltonian-dependent dynamic phase, special paths have
1050-2947/2004/69(5)/050301(4)/$22.50
been designed to eliminate the dynamic phase change. We
also present the experimental demonstration of the laserpulse-controlled Bloch vector rotation of rare earth ions
doped in a crystal, which is one of the basic Bloch vector
rotations equivalent to the single qubit phase gate. The proposed single qubit operations are applicable to two-level atoms. Controlled geometric operations can also be designed
in the presence of some auxiliary levels in the system [11]. A
universal controlled-NOT gate involving two qubits can be
developed based on these single qubit operations for specified interactions that couple qubits, such as the dipole-dipole
interaction [12] or the coupling through photons in a cavity
[13].
We consider two electronic levels of an atom as the qubit
[as shown in Fig. 1(a)] whose state is usually described with
a wave function as
冢 冣
cos
兩 ␺ 典 = e i␥
e
−i␾
␤
2
␤
sin
2
,
where the two basis states, 兩0典 = 共 01 兲 and 兩1典 = 共 10 兲, represent the
ground and the excited levels of the electronic transition,
respectively. The wave function can also be mapped into a
unitary vector 共Bloch vector兲 rជ on the Bloch sphere as shown
in Fig. 1共b兲. This mapping is accurate up to an arbitrary
FIG. 1. (a) Schematic of an electronic transition of a two-level
atom with the wave function 兩␺典 driven by a laser field
⍀0 cos共␻Lt + ␪兲 (b) Bloch vector rជ representing the wave function
ជ on the Bloch sphere.
兩␺典 and its precession about the Rabi vector ⍀
69 050301-1
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PHYSICAL REVIEW A 69, 050301(R) (2004)
TIAN et al.
global phase, ␥, which is insignificant for it does not affect
any observables in the closed two-level system. Any pair of
orthogonal bases of the system is a pair of vectors pointing in
opposite directions, for example, 兩0典 and 兩1典 are represented
by the vectors pointing down and up along axis 3̂, respectively. If a Hamiltonian drives a Bloch vector going through
a cyclic path, the tip of the Bloch vector traces out a circuit
on the Bloch sphere and returns to its initial position. This
evolution makes the wave function associated with the Bloch
vector gain a geometric phase besides the dynamic phase.
The geometric phase ␾ solely depends on the solid angle ␣
enclosed by the circuit at the center of the unitary globe, as
␸ = −␣ / 2. It does not depend on the shape of the circuit, the
driving Hamiltonian, or the initial and the final positions of
the Bloch vector. The geometric nature of this phase change,
independent of the state of the system and the interaction
Hamiltonian, makes it attractive for making robust qubit operations and quantum gates.
For an electronic transition of an atom, a laser pulse can
drive the evolution of the Bloch vector. The motion of the
Bloch vector is governed by the optical Bloch equation, as
ជ ⫻ rជ. The Rabi vector ⍀
ជ consists of three compodrជ / dt = ⍀
nents along the three axes of the Bloch sphere as
共−⍀0 cos ␪ , −⍀0 sin ␪ , ⌬ 兲 where the electric field of the laser
pulse is denoted as E共t兲 = ⍀0 cos共␻Lt + ␪兲 with frequency ␻L,
phase ␪, and amplitude ⍀0, in Rabi frequency. The detuning
⌬ = ␻a − ␻L is the frequency difference between the atomic
transition and the laser. The motion of the Bloch vector is a
ជ at Rabi frequency ⍀
precession about the torque vector ⍀
= 冑⍀02 + ⌬2 as shown in Fig. 1(b). The amount of rotation is
determined by the pulse area, defined as the product of the
Rabi frequency and the driving pulse duration. In general,
any 2␲ pulse makes the Bloch vector complete one cycle of
rotation about the Rabi vector. The solid angle enclosed by
the circuit is determined by the angle between the two vectors. However, one cannot use a single pulse to make an
arbitrary pure geometric phase operation, since a cyclic evolution is usually accompanied by a dynamic phase change as
well. Only when the Bloch and the Rabi vectors are perpendicular to each other throughout the entire circuit will the
dynamic phase of the system remain unchanged [14]. Thus,
more than one pulse is required to make up a circuit to enclose an arbitrary geometric phase for a qubit operation and
at the same time eliminate the dynamic phase change.
A universal single qubit operation is equivalent to a rotation of the Bloch vector, which can be accomplished as
U = U3共␦3兲U2共␦2兲U3共␦1兲. The two types of basic rotations, U3
and U2, are about axes 3 and 2 of the Bloch sphere, respectively [1]. ␦i 共i = 1, 2 , and 3兲 denotes the corresponding rotation angle. In matrix notation, the two basic rotations can be
written as
U 3共 ␦ 兲 =
冉
ei␦/2
0
0
e
−i␦/2
冊
and U2共␦兲 =
冉
cos共␦/2兲
sin共␦/2兲
− sin共␦/2兲 cos共␦/2兲
冊
FIG. 2. Bloch vector rotation about axis 3̂ driven by two resonant ␲ pulses, the corresponding Rabi vectors with relative phase
ជ and ⍀
ជ , respectively. (a) The path of Bloch
␦ / 2, are labeled as ⍀
1
2
vector ជr−3 representing wave function 兩0典 enclosing a solid angle,
2共␲ − ␦ / 2兲. (b) The path of the Bloch vector rជ3 representing 兩1典
enclosing solid angle, −2共␲ − ␦ / 2兲.
two components in an arbitrary wave function. This rotation
can be done with two resonant pulses, with the pulse area of
␲, but with a relative phase difference of ␦ / 2. As shown in
Fig. 2共a兲, the two Rabi vectors associated with the two control pulses are all in the 1-2 plane, since the pulses have zero
frequency detuning. We set the phase of the first pulse to
ជ , assozero and the second to ␦ / 2 so that the Rabi vector ⍀
1
ciated with the first pulse, points to the −1̂ direction, and
ជ 兩 = 兩⍀
ជ 兩, but ⍀
ជ is rotated ␦ / 2 counterclockwise in the 1-2
兩⍀
2
1
2
plane. We track the motion of a vector rជ = rជ−3 pointing down
along −3̂ axis, which corresponds to the ground-state wave
ជ by ␲ to
function 兩0典. Driven by pulse 1, rជ rotates about ⍀
1
position rជ3. The tip of the vector traces out a path, which is
half of the great circle of the Bloch sphere in the 2-3 plane.
ជ by ␲ back
Then, pulse 2 drives this vector rotating about ⍀
2
to its initial position rជ−3 through a path on another great
circle of the globe in a plane including axis 3̂ and angling
from axis 2̂ by ␦ / 2. We denote the operation of these two
pulses, O3共␦ / 2兲. This operation drives rជ through a cyclic
path of two halves of the great circles back to is initial state.
The solid angle enclosed by the path is 2共␲ − ␦ / 2兲. Since the
Bloch vector is perpendicular to the driving Rabi vectors on
the whole path, the dynamic phase is unchanged. The result
of O3共␦ / 2兲 acting on rជ−3 imparts a pure geometric phase
−共␲ − ␦ / 2兲 to the corresponding wave function as
O3共␦ / 2兲兩0典 = e−i共␲−␦/2兲兩0典. Similarly, O3共␦ / 2兲 drives a Bloch
vector starting from position rជ3 through a different cyclic
path, as shown in Fig. 2共b兲 enclosing a solid angle of −2共␲
− ␦ / 2兲, while the Bloch vector stays perpendicular to the
Rabi vectors over the entire path. This results in a pure geometric phase ␲ − ␦ / 2 added to the associated wave function
兩1典 as O3共␦ / 2兲兩1典 = ei共␲−␦/2兲兩1典. Therefore, we can write
O3共␦/2兲 = e−i␲
.
These basic rotations can be realized purely by controlled
geometric phase change. We discuss U3共␦兲 first. This rotation matrix results in a relative phase change ␦ between the
冉
ei␦/2
0
0
e
−i␦/2
冊
as the operation matrix on any arbitrary wave function. Ignoring the global phase factor in front of the matrix we have
the two-pulse-controlled operation O3共␦ / 2兲, which is equiva-
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GEOMETRIC MANIPULATION OF THE QUANTUM…
FIG. 3. Bloch vector rotation about axis 2̂ driven by three resoជ ,⍀
ជ ,
nant pulses, the corresponding Rabi vectors are labeled as ⍀
1
2
ជ
ជ
ជ
and ⍀3, respectively. ⍀1 and ⍀3 represent two ␲ / 2 pulses with the
ជ is a ␲ pulse with a relative phase of ␲ + ␦ / 2.
same phase while ⍀
2
The path of Bloch vector ជr2 representing wave function, 兩i典
= 共1 / 冑2兲共兩0典 + 兩i典兲 encloses solid angle, ␦ / 2.
lent to the state-independent rotation U3共␦兲. The rotation
angle ␦ is controlled by the relative phase between the two
control ␲ pulses.
Now we discuss the rotation about axis 2̂, U2共␦兲. This
operation changes the relative phase of another pair of the
orthogonal basis states,
兩 + i典 =
冉冊
冉 冊
1 1
1 1
and 兩− i典 =
冑2 − i ,
i
冑2
as U2共␦兲兩 ± i典 = e⫿i␦/2兩 ± i典. The two basis states 兩 + i典 and 兩−i典
correspond to Bloch vectors rជ2 and rជ−2, pointing along axes 2̂
and −2̂, respectively. Therefore, we need to design a pulse
sequence to drive the two vectors through their corresponding cyclical paths and to gain the opposite phase of ␦ / 2.
These can be done with three resonant pulses; the correជ ,⍀
ជ , and ⍀
ជ in Fig.
sponding Rabi vectors are labeled as ⍀
1
2
3
3. The three pulses have pulse areas of ␲ / 2, ␲, and ␲ / 2,
respectively. The first and third pulses have a zero phase and
the second pulse has a relative phase of ␲ + ␦ / 2. This makes
ជ and ⍀
ជ aligned with −1̂ axis, while ⍀
ជ points in a direc⍀
1
3
2
tion angled ␦ / 2 from axis 1̂ in the 1-2 plane. We denote the
operation of this pulse sequence with O2共␦ / 2兲. Under this
operation, a Bloch vector rជ = rជ2, representing wave function
ជ by ␲ / 2 to a position pointing
兩 + i典. rotates first about ⍀
1
ជ by ␲ to point up
down along −3̂ axis, then rotates about ⍀
2
ជ by ␲ / 2 back to the
along axis 3̂, and finally rotates about ⍀
3
initial position. The first and the third segments of the cyclic
path are on the great circle in plane 2-3 and the second segment on another great circle in a plane including axis 3̂ rotated counterclockwise from axis 2̂ by ␦ / 2. The solid angle
enclosed by the path is ␦. Thus, the wave function 兩 + i典 gains
a geometric phase of −␦ / 2 as O2共␦ / 2兲兩 + i典 = e−i␦/2兩 + i典. Under
the same operation, a Bloch vector rជ = rជ−2, pointing in the
FIG. 4. Regular pulse sequence of a photon echo process. (b)
Pulse sequence with two control pulses to perform rotation, U3共␦兲,
where the rotation angle ␦ is controlled by the relative phase ␦ / 2
between the two control pulses. The rotation results in the echo field
phase change by ␦. (c) Schematic of the experimental setup.
opposite direction of rជ2, goes through a cyclic path enclosing
a solid angle of −␦ and the corresponding wave function
gains a geometric phase as, O2共␦ / 2兲兩−i典 = ei␦/2兩−i典. Therefore,
O2共␦ / 2兲 is equivalent to the rotation U2共␦兲 about axis 2. The
rotation angle is controlled by the relative phase between the
driving pulses. If it is desired to make the basic rotation
about axis 1̂, instead of U2, a similar approach can be used to
design a resonant three-pulse-driven operation U1 in which
the rotation angle, again, is controlled by the relative phase
between the driving pulses.
In a proof-of-concept experiment we demonstrated the
laser-pulse-controlled Bloch vector rotation, U3共␦兲 on Tm3+
doped in yttrium aluminum gamet (YAG) crystal. The resulting quantum states of the ions were measured using photon
echoes [15] as the analogue of spin echoes in NMR systems
[16]. The rephasing process of the photon echo cancels the
dynamic phase differences for atoms at different frequencies.
This allows us to observe the pure geometric phase change
of an ensemble of inhomogeneously detuned two-level systems. Figure 4 shows the schematics of the pulse sequences
and the experimental setup. In an ordinary photon echo process [Fig. 4(a)], pulse 1 (usually a ␲ / 2 pulse) excites the
atoms from the ground state to their superposition states to
create a coherence, which dephases during the time interval ␶
between pulses 1 and 2 due to the different resonant frequencies. Pulse 2 (usually a ␲ pulse) causes the rephasing of the
coherence of all the atoms resulting in a macroscopic field as
an echo pulse delayed by ␶ after the second pulse. The electric field of the echo is proportional to e−i␾ sin ␤ summing
over all excited atoms, where ␤ and ␾ are the two angular
coordinates of the Bloch vector representing an atomic state
at the echo rephasing time. In the pulse sequence of the
photon echo process, ␤ usually changes when the atoms are
excited by a pulse, and ␾ changes between the pulses due to
the dephasing and rephasing of the atoms at different detunings. When the echo is generated, the Bloch vectors of all
atoms have the same ␾, which was set to zero in our experiment. When the two control pulses, as shown in Fig. 4(b),
are applied between the first and second pulses or the second
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TIAN et al.
and echo pulses to perform the U3共␦兲 operation, an extra
phase ␦ is introduced to the coherence between the ground
and excited states, which means the echo field gains the same
amount of extra phase. We measured this phase by heterodyning the echo with a reference pulse, whose frequency was
shifted by 20 MHz from the echo’s frequency. As shown in
Fig. 4(c), the laser source in the setup was a cw Ti:sapphire
laser, frequency stabilized to submegahertz using the spectral
hole burning technique [17]. All the input pulses including
the heterodyning reference (not shown in Fig. 4) were generated by an acoustic optical modulator driven by an arbitrary waveform generator. Pulses 1 and 2 were 500 ns long,
delayed by ␶ = 7 ␮s. Two control pulses were 200 ns each,
applied between pulses 1 and 2. An 11 ␮s long heterodyning
reference pulse was added to the sequence right after pulse 2.
The relative phase ␦ / 2 of the two control pulses was varied
by the arbitrary waveform generator from 0 to ␲ with 0.05␲
increments The phase change of the echo field was calculated from a 409.6-ns long segment of the heterodyned echo
signal using fast Fourier transform (FFT). The results are
plotted in Fig. 5 (depicted as dots), showing the echo phase
change as a function of the control phase. The expected values of the geometric phase change ␦ are also plotted in Fig.
5 as a solid line. The three insets are the heterodyning signal
segments at ␦ / 2 = 0 , 0.5␲, and ␲, respectively. The experimental signals are averaged over 32 shots. We can see a good
consistency between the experiment and the theory with a
standard deviation of 0.03␲. We also captured 50 single
shots of the echo signal under the same condition, but without the control pulses. The standard deviation of the measurement of the echo phase was calculated to be 0.15␲. For
an average over 32 shot, the standard deviation due to measurement error would be 0.027␲. Therefore, the errors on the
controlled geometric phase change are mainly measurement
limited.
In summary, we proposed the scheme of universal single
qubit quantum operations by pure geometric manipulation of
two-level atoms with laser pulses. A complete set of basic
Bloch vector rotations (about axes 3̂ and 2̂ in our case) was
designed using geometric phase changes, each controlled by
the relative phase of a series of resonant laser pulses (two
pulses for U3 and three for U2兲. The rotation about axis 3̂,
U3, was demonstrated in the experiment with an ensemble of
inhomogeneously detuned two-level Tm3+ doped in a YAG
crystal. The rotation angle controlled by the relative phase of
the two control pulses was measured using heterodyned photon echoes. The experimental results are in good agreement
with the theory within measurement-limited errors. The rotation accuracy can be improved by a more stable laser source.
The phase measurement technique demonstrated in this paper can be further used to evaluate the operation fidelity
when various noises are present on the control pulses, such
as frequency jitter, amplitude fluctuation, and pulse duration
inaccuracy. The other basic rotation, U2, can also be demonstrated in a similar way.
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FIG. 5. Phase change of the photon echoes vs the control phase.
Experimental results (dots) were calculated from 409.6-ns segments
of echo signals (see the three examples shown in the insets) using
FFT. The theoretical values, ␦, (solid line) are twice of the control
phase ␦ / 2.
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