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arXiv:quant-ph/0309180v3 28 Aug 2005
Decoherence-free dynamical and geometrical entangling phase gates
Jiannis K. Pachos
∗
and Almut Beige
Blackett Laboratory, Imperial College London, Prince Consort Road, London, SW7 2BW, UK
(February 1, 2008)
It is shown that entangling two-qubit phase gates for quantum computation with atoms inside
a resonant optical cavity can be generated via common laser addressing, essentially, within one
step. The obtained dynamical or geometrical phases are produced by an evolution that is robust
against dissipation in form of spontaneous emission from the atoms and the cavity and demonstrates
resilience against fluctuations of control parameters. This is achieved by using the setup introduced
by Pachos and Walther [Phys. Rev. Lett. 89, 187903 (2002)] and employing entangling Raman- or
STIRAP-like transitions that restrict the time evolution of the sy stem onto stable ground states.
03.67.Pp, 42.50.Pq
I. INTRODUCTION
The phase of a state vector is undoubtedly one of the
central curiosities that differentiate quantum from clas-
sical mechanics. Especially, the presence of quantum
phases in non-intuitive interference expe riments has been
the focus of intellectual excitement in many studies (see
for example [1–5]). The presence of a phase is e ven more
striking in a state that exhibits entanglement. In par -
ticular, the research on highly correlated subsystems has
contributed to the boosting of innovative technological
applications [6–8]. Such an example is quantum compu-
tation where the entangling phase gate often provides the
basic two-qubit gate – an essential ingredient for the real-
ization of arbitrary quantum algorithms. This gate cor-
responds to a unitary oper ation that changes the phase
of one subsystem depending on the state of another one.
In the last years, considerable effort has been made to
find efficient ways for its experimental implementation.
For a proposed gate implementa tion to be feasible with
present technology, it is importa nt that the scheme is
widely independent from various control parameters like
the operation time. Minimizing the control errors will
augment the efforts for engineering scalable quantum in-
formation processing with high accuracy. Another prob-
lem arises from the fact that it is very difficult to isolate
a quantum mechanical system c ompletely from its en-
vironment without loosing the possibility to manipulate
its state. Uncontrollable enviro nmental couplings lead
in general to dissipation and the loss of information. In
addition, the phase of a state vector is very fragile with
respect to decoherence. As a result, phase factors are
rather hard to generate and to store in engineered sys-
tems.
This paper analyzes in detail entangling phase oper-
ations that are especially designed to bypass the dissi-
pation problem and guar antee very high fidelities for a
wide range of experimental parameters. In particular, we
consider the quantum behaviour of atoms inside an op-
tical cavity which has been observed experimentally by
Hennrich et al. [9] and, more recently, by J. McKeever et
al. [10] and by Sauer et al. [11 ]. The scheme presented
here involves two atoms trapped at fixed positions inside
an optical cavity and can be implemente d using the tech-
nology of the recent calcium ion experiment [12]. Alter-
natively, neutral atoms can be tr app e d with a standing
laser field [13], as in the exp e riment by Fischer et al. [14],
or in an optical lattice [11 ,15]. In the last decade, several
atom-cavity quantum computing schemes have been pro-
posed [16–24], e ach of them having its respec tive merits.
To implement quantum phase gates we utilize adia-
batic processes that result in the generation of dynam-
ical or geometrical phases [25]. Employing geometrical
phases is an intriguing way of manipulating quantum me-
chanical systems. They exhibit independence from the
operation time of the control evolution and are robust
against per tur bations of system parameters as long as
certain requirements, concerning the geometrical charac-
teristics of the evolution, a re satisfied. To avoid dissipa-
tion, we exploit the existence of dec oherence-free states
[26–29]. As in [20], only ground states with no pho-
ton in the cavity mode and the atoms in a stable state
become populated. Population transfers between those
ground states are achieved with the help of Raman or
STIRAP-like processes [30,31]. Their control pr ocedures
are ex actly the same a s for the usual Raman and STI-
RAP transitions, but now the c oupling of the atoms to
the same cavity mode allows for the creation of entan-
glement. Another advantage of the proposed scheme is
that it can be realized via common laser addressing and,
essentially, within one step.
In the following, each qubit is obtained from two
ground states of the same atom. High fidelities of the
final state are achieved even for moderate values o f the
∗
jiannis.pachos@imperial.ac.uk
1
atom-cavity c oupling co nstant g compared to the spon-
taneous cavity decay rate κ and the atom decay rate
Γ. Different fro m other atom-cavity quantum comput-
ing schemes [16–19,21], we assume the c onstant g
2
to be
about 100 κΓ which is close to the state of the art technol-
ogy [11] and should be within the range of experiments
in the nearer future [15].
The paper is organized as follows. Section II intro-
duces the decoherence-free subspace of the sy stem with
respect to leakage of photons through the cavity mir-
rors. An effective Hamiltonian describing the possible
time evolutions of the system within that subspace is
derived. In order to avoid spontaneous emission from
the atoms, we present in Section III different parame-
ter regimes that minimize the population in the excited
atomic levels and result in entangling Raman and entan-
gling STIRAP transitions. In Section I V we employ the
correspo nding time evolutions to generate dynamical and
geometrical phases for the rea lization of two-qubit phase
gates for quantum computation. Finally, we s ummarize
our results in Section V.
II. ELIMINATION OF CAVITY DECAY
In the following we consider two ato ms (or ions), each
of them comprising a four-level system. A possible level
configuration, which is suited for the implementation of
quantum computing with calcium ions, is shown in Fig-
ure 1. Each qubit is obtained from the stable ground
states |0i and |1i of the same atom. In addition, each
atom posses ses an e xcited state |2i, that becomes only
virtually populated during gate operations, and a third
auxiliary ground state |σi. In order to realize an entan-
gling opera tion, the two atoms involved should be posi-
tioned in the cavity, where both see the same coupling
constant g, and laser fields should be applied as shown.
As we see later, the detuning δ plays an important role
in regulating the phase factor of the final state of the
system.
atom 2:
σ
+
σ
−
qubit 1
qubit 2
2
σ
0
1
g
S
1/2
3/2
D
1/2
P2
σ
0
1
g
S
1/2
3/2
D
1/2
P
atom 1:
∆−δ
∆
∆−δ
∆
+
σ
−
σ
FIG. 1. Level configuration for atom-cavity quantum com-
puting with calcium ions showing the d riv ing laser fields. The
2-σ and the 2-1 transitions are activated by the laser radia-
tions with amplitudes and polarizations given by Ω
σ
, σ
+
and
Ω
1
, σ
−
respectively. Each qubit is obtained from two degen-
erate S
1/2
ground states of the same ion.
Let us denote the cavity photon a nnihilatio n and cre-
ation operator by b and b
†
while ∆ is the detuning of the
cavity with respect to the 2-1 transition of each atom.
The occupation of the cavity is then given by n = hb
†
bi.
One laser drives the 2-1 transition of atom i with Rabi
frequency Ω
(i)
1
and detuning ∆; another one excites the
2-σ transition with Rabi frequency Ω
(i)
σ
and detuning
∆ − δ. Going over to the interaction picture with re-
sp e c t to H
0
+ ¯h
P
i
[ δ|σi
ii
hσ| + ∆ |2i
ii
h2|] where H
0
is
the interaction-free Hamiltonian, one finds
H
I
=
2
X
i=1
¯hg
|2i
ii
h1|b + H.c.
+
1
2
¯h
Ω
(i)
1
|1i
ii
h2| + Ω
(i)
σ
|σi
ii
h2| + H.c.
−¯hδ |σi
ii
hσ| − ¯h∆ |2i
ii
h2| . (1)
The generatio n of a non-trivial time evolution requires
in gener al a non-homoge neity in the Rabi frequencies of
the laser fields with respect to the two atoms. Without
loss of generality, we assume here that the Rabi frequen-
cies of laser fields coupling to the same transition differ
only in phase. In the fo llowing, e ntangling operations are
realized by choosing
Ω
(1)
σ
= − Ω
(2)
σ
≡ −Ω
σ
,
Ω
(1)
1
= − Ω
(2)
1
≡ −Ω
1
/
√
2 . (2)
These parameters can be implemented via common laser
addressing by driving each transition with the same laser
field and from an angle that produces a π phase difference
between atom 1 and 2.
One of the main sources for decoherence in atom-cavity
setups is the leakage of photons through the cavity mir-
rors. However, the presence of high spontaneous de-
cay rates does not necessar ily lead to dissipation. As it
has been shown in the past, the presence of a relatively
strong atom-cavity co upling constant g [32] c an have
the same effect as continuous measurements whether the
cavity mode is empty or not. As a consequence, the
time evolution of the system becomes restricted onto the
decoherence-free states of the system with respect to cav-
ity decay. In this Section, we exploit this effect and its
consequences for the time evolution to significantly sim-
plify the requirements for atom-cavity quantum compu-
tation.
In the next Section, Raman and STIRAP-like pro-
cesses [30,31] are introduced in order to minimize the
population in the excited atomic state |2i. Assuming
that the population in level 2 is negligible, the only con-
dition that has to be fulfilled to avoid cavity decay is
that the Rabi freq uencies Ω
1
and Ω
σ
are sufficiently weak
compared to the atom-cavity coupling constant g,
|Ω
1
|, |Ω
σ
| ≪ g (3)
This relation induces two different time scales in the sys-
tem and allows for the calculation of the time evolution
with the help of an adiabatic elimination. Figure 2 shows
2
the most r e le va nt transitions if the sy stem is initially pre-
pared in a qubit state with an empty cavity (n = 0) and
uses the abbreviations
|αi ≡
|12i − |21i
/
√
2 ,
|Ai ≡
|σ1i + |1σi
/
√
2 ,
|˜αi ≡
|σ2i − |2σi
/
√
2 ,
|
˜
Ai ≡
|σ1i − |1σi
/
√
2 . (4)
Setting the amplitudes of all fast varying s tates equal to
zero and using the Hamiltonian (1), one finds that the
system evolves effectively according to the Hamiltonian
H
eff
=
1
2
¯h
Ω
1
|αih11| + Ω
σ
|αihA| + H.c.
−¯hδ |AihA| − ¯h∆ |αihα|
⊗ |0
cav
ih0
cav
| . (5)
As expected for adiabatic processes, this Hamiltonian re-
stricts the time evolution onto a subspace of slowly vary-
ing states, which is why the states |˜αi and |
˜
Ai are not
present in Equation (5). The subspace of the slowly vary-
ing states includes, in addition to the ground sta tes , only
one state w ith population in the excited state |2i. This
state, |αi|0
cav
i, is a zero eigenstate of the atom-cavity
interaction.
∆
...
g
...
g
laser excitation
to other levels via
01,n=0
02,n=0
10,n=0
20,n=001,n=1
...
ΩΩ
1
1
10,n=1
11,n=0
α,n=0
σσ,
n=0
α,n=0
~
A,n=0
...
g
1
Ω Ω Ω Ω
1
σ
laser excitation
to other levels via
σ
A,n=1
~
∆
∆−δ
∆−δ
∆
∆
FIG. 2. Level configuration showing the most relevant
transitions for the initial qub it states |11i, |01i and |10i in
the presence of mechanisms (see Section III) that minimize
the population in level 2. The presence of a strong cavity cou-
pling constant g prohibits a transfer of population between
|Ai and |σσi. This does not effect t he t ransition between |11i
and |Ai. The atomic state |00i does not see any laser fields
and does not change its amplitude in time.
Note that the Hamiltonian (5) indeed restricts the time
evolution of the sy stem onto its deco herence-free sub-
space with respec t to cavity decay. A state belongs to
this subspace if the resonator mode is in its vacuum state
|0
cav
i and the atom-cavity interaction cannot transfer ex-
citation from the atoms into the resonator. Hence, the
decoherence-free s ubspace includes all gr ound states and
the state |αi|0
cav
i. They are exactly the slowly varying
states of the system. Furthermore, the effective Ha mil-
tonian can be written as
H
eff
= IP
DFS
H
laser
IP
DFS
, (6)
where IP
DFS
is the projector onto the dec oherence-free
subspace
IP
DFS
=
h
|αihα| +
X
i,j=0,1,σ
|ijihij|
i
⊗ |0
cav
ih0
cav
| , (7)
and H
laser
is the laser term in (1). One way to interpret
this is to state that condition (3) effectively induces con-
tinuous measurements whether the cavity field is empty
or not [33]. As a consequence, entanglement can be gen-
erated between qubits via excitation of the ma ximally
entangled state |αi.
In the event that condition (3) is not fulfilled, as in all
realistic experiments, the ab ove argumentation does not
hold and corrections to the effective time evolution (5)
have to be taken into acc ount. To identify the evolution
of the system more accurately, we use the quantum jump
approach [34–36] which predicts the no-photon time evo-
lution of the sy stem with the help of the conditional, non-
Hermitian Hamiltonian H
cond
. Given the initial state
|ψi, the state of the system equals U
cond
(T, 0) |ψi/k·k at
time T under the condition of no emission. For conve-
nience, H
cond
has been defined such that
P
0
(T ) = kU
cond
(T, 0) |ψik
2
(8)
is the probability for no photon in (0, T ). If photo ns
are emitted, then the computation failed and has to be
repeated. To some extent, this can be compensated by
monitoring photon emissions with good detectors. Alter-
natively, quantum teleportation can be employed to per-
form a whole algorithm by selecting the successful gates
[37].
Let us now consider a concrete example to see how
well the elimination of cavity decay works. For the two
four-level atoms in Figure 1 it is [20]
H
cond
= H
I
−
i
2
¯hκ b
†
b −
i
2
¯hΓ
2
X
i=1
|2i
ii
h2| . (9)
Suppose that the sy stem is initially prepared in |01i|0
cav
i
and a laser field is applied with Ω
1
6= 0 that excites the
2-1 transition of both atoms for a significant time T .
Equation (5) then implies the inhibition of any time evo-
lution since |01i|0
cav
i is a zero eigenstate of H
eff
. Indeed,
the numerical solution of the time evolution given by the
Hamiltonian (9) reveals that the final state coincides with
the initial state of the system with fidelity F ≡ 1 under
the condition of no photon emission in (0, T ). The suc-
cess rate P
0
(T ) as a function of ∆ and Ω
1
is shown in
Figure 3. As expected, P
0
is the closer to one the smaller
3
the Rabi fr e quency Ω
1
and its size is widely independent
from the detuning ∆.
FIG. 3. The success rate P
0
(T ) as a function of the
Rabi frequency Ω
1
and the detuning ∆ for the initial state
|01i|0
cav
i, κ = Γ = 0.1 g, δ = 0 and T = 2000/g. If one al-
lows for a short transition time at the end of the operation for
unwanted states to decay, then the fin al state of the system
equals exactly the initial one.
III. ELIMINATION OF ATOM DECAY AND
ENTANGLING OPERATIONS
For optical cavities, the atom decay rate Γ is in gen-
eral of about the s ame size as the parameters g and κ.
We therefore still need to overcome the problem of spon-
taneous emission from the atoms in order to realize a
coherent time evolution. This can be achieved by mini-
mizing the population in level 2 and keeping the system
effectively in the ground states |11i|0
cav
i and |Ai|0
cav
i.
To realize this we employ in the following Raman or
STIRAP-like transitions [30,31]. These are directly ap-
plicable since the relevant states comprise a Λ-type three-
level configuration. Entanglement between the atoms is
created by coupling |11i to the maximally entangled state
|Ai.
A. Entangling Raman transitions
One way to implement transitions without populating
the excited state |αi is to choose the detuning ∆ much
larger than the Rabi frequencies Ω
1
and Ω
σ
,
|Ω
1
|, |Ω
σ
| ≪ ∆ (10)
This choice of parameters results in the realization of a
Raman-like transition since the la rge detuning ∆ intro-
duces an additional time scale in the time e volution (5).
Adiabatically eliminating the excited state |αi reveals
that the system is e ffectively g overned by the Hamilto-
nian
H
eff
=
1
2
¯hΩ
|11ihA| + H.c.
−¯h∆
11
|11ih11| − ¯h∆
A
|AihA| (11)
with the abbreviations
Ω ≡
Ω
1
Ω
∗
σ
2∆
, ∆
11
≡ −
|Ω
1
|
2
4∆
, ∆
A
≡ δ −
|Ω
σ
|
2
4∆
. (12)
Solving the time evolution for the case where the lasers
are turned on for a period T with constant amplitude
yields
U
eff
(T, 0) = exp
i
2
(∆
11
+ ∆
A
)T
h
cos
KT
2
+
i(∆
11
−∆
A
)
K
sin
KT
2
|11ih11|
−
Ω
K
sin
KT
2
|11ihA| + |Aih11|
+
cos
KT
2
−
i(∆
11
−∆
A
)
K
sin
KT
2
|AihA|
i
, (13)
where
K ≡
|Ω|
2
+ (∆
11
− ∆
A
)
2
1/2
(14)
plays the role of a Ra bi frequency. As the evolution (13)
can be used to create entanglement, we call it an entan-
gling Raman (E-Raman) transition.
In order to see how well the elimination of the dissi-
pative states works, let us consider as an example the
preparation of the maximally entangled s tate |Ai. If the
atoms are initially in |11i, this can be achieved by apply-
ing a laser pulses of length T = π/K = 2π∆|Ω
1
|
2
with
δ = 0 and Ω
σ
= Ω
1
. Figures 4 and 5 present the fidelity
F and success rate P
0
, respec tively, of this entangling
operation as a function of the Rabi frequency Ω
1
and the
detuning ∆. They result from a numerical integration of
the time evolution with the full Hamiltonian (9).
In particular for κ = Γ = 0.1 g, the fidelity of the final
state can be as high as 0.993 if one chooses Ω
1
= 0.01 g
and ∆ = 1.357 g and allows the success probability to
be as high as P
0
= 0.857. Higher fidelities can only be
obtained in the presence of sma ller decay rates. In gen-
eral one can improve the success rate P
0
by a few percent
by increasing the detuning ∆ and by sacrificing an e rror
from the maximal fidelity of the order of 10
−3
. While
the fidelity exhibits a maximum as a function of ∆, the
success probability increases monotonically with ∆. For
large detunings, the adiabatic elimination of the cavity
mode is no longer efficient (see Figure 2) resulting in the
reduction of F . Varying the Rabi freq uency gives larger
fidelities for smaller Ω
1
since the adiabatic elimination
4
of cavity excitation as well as the elimination of |αi be-
comes mo re efficient. However, the smaller Ω
1
the la rger
the operation time T of the entangling evo lution.
FIG. 4. The fidelity F for the preparation of the maximally
entangled state |Ai given the initial state |11i as functions of
the Rabi frequency Ω
1
and the detuning ∆ for Ω
σ
= Ω
1
,
Γ = κ = 0.1 g and δ = 0.
FIG. 5. The success rate P
0
for the preparation of the
maximally entangled state |Ai for the same parameters as
in Figure 4.
B. Entangling STIRAP processes
Another way to transfer population between the
ground states |11i and |Ai without ever populating the
excited state |αi is to employ a STIRAP-like process
[30,31]. As this is an entangling process we can call it
an entangling STIRAP (E- STIRAP ) transition. To see
how the s uppression of spontaneous emission from the
atoms works in this case let us consider again the ef-
fective Hamiltonian (5) and note that its eigenvalues for
δ ≪ ∆ are
E
0
= 0 , E
±
=
1
2
(−∆ ±
p
|Ω
σ
|
2
+ |Ω
1
|
2
+ ∆
2
) . (1 5)
For convenience, we now introduce the interaction pic-
ture with respect to the Hamiltonian H
0
= −¯hδ|AihA|.
Within that picture, the eigenvectors of the effective
Hamiltonian become time dependent and the dark state
of the system equals
|E
0
i ≡
1
p
|Ω
σ
|
2
+ |Ω
1
|
2
Ω
σ
|11i − e
−iδt
Ω
1
|Ai
. (16)
The other two eigenstates |E
±
i occupy the antisymmet-
ric state |αi and hence possess population in the excited
atomic state |2i.
Spontaneous emission fr om the atoms can therefore be
avoided if the s ystem remains constantly in the dark state
|E
0
i. Nevertheless, a nontrivial entangling time evolution
between the gr ound states |11i and |Ai can be induced,
by va rying the laser amplitudes Ω
1
and Ω
σ
in time. If
this happens slow c ompared to the time scale given by
the eigenvalues E
±
, no population is transferred into the
eigenstates |E
±
i. Assuming adiabaticity, the system fol-
lows the changing parameters and remains in the dark
state. The fact that |E
0
i incorporates a time dependent
phase factor can be used to induce a certain phase on
the final state |Ai by choosing the small laser detuning δ
appropriately.
In the following we assume ∆ = 0 since this maxi-
mizes the energy distance between |E
0
i and its nearest
eigenstate. For convenience, we also consider the case
where Ω
1
and Ω
σ
are real and define tan θ ≡ Ω
1
/Ω
σ
and
φ ≡ −δt. The eigenstate corresponding to the zero eigen-
value then take s the familiar for m
|E
0
i = co s θ |11i− e
iφ
sin θ |Ai . (17)
The adiabatic condition is satisfied if
|
˙
θ|, |
˙
φ| ≪
p
Ω
2
σ
+ Ω
2
1
(18)
If the time dependence of θ and φ is indeed such that
the initial state of the system coincides with the dark
state |E
0
(0)i, the effective evolution along a path C in
the (θ, φ) parameter space can easily be calculated. As a
consequence of the adiabatic theorem, the effective evo-
lution is diagonal with respect to the basis given by the
eigenvectors |E
0
i and |E
±
i. Especially for the case where
θ(0) = 0 and with respect to the basis states |E
0
(0)i,
|E
+
(0)i and |E
−
(0)i, the effective time evolution of the
system can be wr itten as
U
eff
(T, 0) = R(θ, φ)
e
iϕ
0
0 0
0 e
iϕ
+
0
0 0 e
iϕ
−
(19)
with θ = θ(T ), φ = φ(T ) and
R(θ, φ) =
1
2
×
2 cos θ
√
2 sin θ
√
2 sin θ
−
√
2e
iφ
sin θ (1 + e
iφ
cos θ) −(1 − e
iφ
cos θ)
√
2e
iφ
sin θ (1 − e
iφ
cos θ) −(1 + e
iφ
cos θ)
. (20)
5
The phases ϕ
i
≡ ϕ
d
i
+ ϕ
g
i
(i = 0, ±) are in ge neral the
sum of a dynamical and a geometrical phase with
ϕ
d
0
= 0 , ϕ
d
+
= −ϕ
d
−
= −
1
2
Z
T
0
q
Ω
2
1
+ Ω
2
σ
dt (21)
while the geometrical phases equal
ϕ
g
0
=
I
C
sin
2
θ dφ , ϕ
g
+
= ϕ
g
−
=
1
2
I
C
cos
2
θ dφ (22)
for a closed circular loop C.
As an example, let us consider the case where the ini-
tial state |11i is transferred into the maximally entangled
state |Ai. This can be achieved by varying the Rabi fre-
quencies Ω
1
and Ω
σ
slowly in a so-called counterintuitive
pulse sequence. Figure 6 and 7 show the result of a nu-
merical solution of the corresponding time evolution tak-
ing the full Hamiltonian (9) into account. In particula r,
we calculated the fidelity and the success of the scheme
as a function of the maximum laser amplitude Ω and the
frequency ω for ∆ = δ = 0, κ = Γ = 0.1 g and
Ω
σ
(t) =
(
Ω sin ωt , for t ∈
0,
2
3
T
0 , for t ∈
2
3
T, T
(23)
and
Ω
1
(t) =
(
0 , for t ∈
0,
1
3
T
Ω sin ω
t −
1
3
T
, for t ∈
1
3
T, T
(24)
with the total operation time T = 3π/(2ω).
FIG. 6. The fidelity F for the preparation of the maximally
entangled state |Ai as a function of the maximum Rabi fre-
quency Ω and frequency ω for ∆ = δ = 0 and κ = Γ = 0.1 g.
FIG. 7. The success rate P
0
for the proposed entangled
state preparation scheme for the same parameters as in Fig-
ure 6.
In contrast to E-Raman processes, the fidelity F of the
final state can, in principle, b e arbitrarily close to one in-
dependent of the size of the spontaneous decay rates κ
and Γ. For example, for Ω = 0.02 g a nd ω = 4 · 10
−5
g
the fidelity becomes F = 0.998 and corresponds to the
success rate P
0
= 0.876 which is still within acceptable
limits. In general, a large Ω leads to the population of
the excited state |2i which might result in the emiss ion of
a photo n w ith rate Γ or in the populatio n of the cavity
mode followed by the leakage of a photon through the
cavity mirror s. As a result, F and P
0
decrease for in-
creasing laser amplitudes Ω. On the other hand, for very
weak Rabi frequencies (Ω ≈ 0) the fidelity and the suc-
cess rate increase as ω decreases which results in a longer
operation time T . This is in agreement with condition
(18) which implies that the adiabatic evolution is more
successful for slower evolutions. Neve rtheless, the suc-
cess rate P
0
cannot be increase d arbitrarily. The reason
is that very long oper ation times unavoidably lead to the
population of the cavity mode in the presence of a finite
value for Ω and the elimination of cavity decay does no
longer hold.
IV. DECOHERENCE-FREE DYNAMICAL AND
GEOMETRICAL PHASE GATES
In this Section we employ the decoherence-free E-
Raman and E-STIRAP processes intro duced in the last
Section for the realization of two-qubit quantum gate op-
erations. In particular, we consider the implementation
of the controlled phase gate
CP = dia g
1, 1, 1, e
iϕ
, (25)
where the qubit state |11i collects the phase ϕ while the
other qubit s tates remain unaffected. This gate consti-
tutes, together with a general single-qubit rota tio n, a uni-
versal set of gates and allows for the implementation of
6
any desired unitary time evolution. Attention is paid
to their sta bility a gainst fluctuations of most system pa-
rameters and we analyze in detail the efficiency of various
processes.
A. Raman-based Dynamical Phase Gates
Let us begin with the descr iption of a po ssible imple-
mentation of (25) with the help of the E-Raman process
introduced in Section IIIA. From Equation (13) one sees
that an applied laser pulse of length T evolves the initial
state |11i according to
U
eff
(T, 0)|11 i = exp
i
2
(∆
11
+ ∆
A
)T
×
h
cos
KT
2
+ i
∆
11
−∆
A
K
sin
KT
2
|11i −
Ω
K
sin
KT
2
|Ai
i
,
(26)
while the amplitudes of the qubit states |00i, |01i and |10 i
do not change in time. The desired phase gate (25) can
therefore b e realized in the minimal time if one chooses
T = 2π/K . (27)
For this parameter choice, the sine term in (2 6) vanishes
and the cosine term equals −1 giving finally
ϕ = π +
1
2
(∆
11
+ ∆
A
)T . (28)
Again, the phase ϕ can be controlled by adjusting the
size of the detuning δ. Especially, for
δ =
1
4∆
|Ω
1
|
2
+ |Ω
σ
|
2
(29)
one obtains ϕ = π a nd the amplitude of the sta te |11i
accumulates a minus sign. This particular evo lution is
illustrated in Figure 8 using a Bloch’s sphere represen-
tation. Note that the condition of timing the evolution
such that the sine term beco mes zero, is robust against
first order timing errors in T due to the behaviour of the
sine function around π.
|11
|11
C
E−Raman
−|11
|A
|A
E−STIRAP
FIG. 8. The E-Raman and E-S TIRAP evolut ions depicted
on t he Bloch sphere. In the E-Raman evolution, the genera-
tion of the phase factor in front of |11i is explicitly shown. I n
the E-STIRAP evolution, the acquired phase relates to the
solid angle spanned by the path C on the sphere.
To minimize the experimental effort for the realiza-
tion of the phase gate (2 5), one should notice that a
non-trivial E-Rama n transition can also be induced using
only one laser field coupling to the 1-2 transition of each
atom. Indeed, for ∆
A
= 0 and the laser pulse duration
T = π/K, Equation (26) yields
U
eff
(T, 0)|11 i = −exp
i∆
11
T
|11i . (30)
For this case, Figure 2 shows that the initial state |11 i
couples only to the excited state |αi via the applied heav-
ily detuned las e r field. To see that the pha se operation
(30) is robust against fluctuations of Ω
1
, we now consider
the Rabi frequency being time dependent. In this case
the phase accumulated by the initial state |11i becomes
ϕ = π +
R
T
0
∆
11
dt. This clearly shows that statistical
perturbations of Ω
1
average out due to dependence of
the acquired phase on the integral over ∆
11
along (0, T ).
B. STIRAP-based Geometrical Phase gate
Another possibility to implement the co ntrolled two-
quit phase gate (25) is provided by the E-STIRAP pro-
cess. Let us first consider the case of the g e neration of a
dynamical phase as in [20]. To do so we var y θ from zero
to π, then apply a 2π pulse to transform the state |σi to
−|σi for both atoms and then reduce θ back to zero. At
the end of this time evolution, the state |11i acquires an
overall minus sign, while the other qubit states |00i, |01i
and |10i rema in unchanged. The result is the conditional
phase gate with ϕ = π.
Alternatively, a geometrical phase can be generated by
continuously vary ing the parameters θ and φ in a cyclic
fashion [38,39]. Starting from θ = 0 one can perform a
cyclic adiabatic evolution on the (θ, φ) pla ne along a loop
C. As predicted by Equations (19)-(22), the qubit s tate
|11i then acquires the geometrical phase
ϕ = ϕ
g
0
=
I
C
sin
2
θ dφ (31)
while the dynamical phase ϕ
d
0
is identically zero. The
concrete size of ϕ, which characterizes the phase gate
(25), depends therefore only on the solid angle spanned
by the path C on the Bloch sphere (see Figure 8). Hence
it is very robust against statistical fluctuations of the
Rabi frequencies of the applied laser fields and it is inde-
pendent of the total gate operation time.
Let us now consider a concr e te laser configuration,
where the laser field Ω
σ
is turned on and kept constant.
To perform a non-trivial loop in the control parameter
space (θ, φ) starting from the (0, 0) point we proceed as
follows. As describe d in Section III B, a non-zero detun-
ing δ results in the accumulation of a pha se φ = −δt when
7
we introduce a non zero value for θ. Let us consider the
simple case where
θ = arctan (x) with x ≡ Ω
1
/Ω
σ
, (32)
changes from zero to a non-zero value. More concretely,
we increase and decrease Ω
1
by linear ramps such that
x(t) =
(
αt , for t ∈
0,
1
2
T
α(T − t) , for t ∈
1
2
T, T
(33)
or by a sine r amp given by
x(t) = x
max
sin βt (34)
with t ∈ (0, π/β). For those two cases one can e asily
derive the geometrical phases as functions of the exper-
imental control parameters T , α, x
max
and β and we
present the results in Figure 9.
ϕ/δ
(b)
β
x
max
(a)
ϕ/δ
T
α
FIG. 9. The resulting geometrical phase ϕ
g
/δ produced
(a) by a linear ramp and (b) by a sine ramp. The produ ced
geometrical phases are largely independent of the control pa-
rameters α or x
max
respectively. α and β are depicted in
inverse time units, while x
max
is dimensionless.
From the plots we see that the accumulated geomet-
rical phase ϕ is, as expected, widely independent on the
size of α or x
max
giving the resilience of the proposed E-
STIRAP process from control errors, especially from the
exact value of the laser amplitudes or their time deriva-
tives. This independence of ϕ makes the proposed quan-
tum gate implementation an attractive candidate for ex-
periments where the exact control o f all laser parameters
is not possible. In addition, it is experimentally appealing
to construct a setup where the two employed laser fields
can be produced from the same source, separated in two
by a beam splitter and having the ratio Ω
1
/Ω
σ
modulated
e.g. by an Acousto-Optical Modulator (AOM). With this
setup, the ratio is also resilient to the fluctuations of the
amplitude of the initial laser beam.
V. CONCLUSIONS
In this paper, we discussed in detail concrete proposals
for the realization of universal two-qubit phase gates for
quantum c omputing in atom-cavity setups. The atoms
(or ions) are trapped at fixed positions inside an opti-
cal cavity and each qubit is obtained from two stable
ground states of the same atom. To ac tivate a time evo-
lution, two laser fields a re applied simultaneously induc-
ing transitions in an effectively Λ-like system. In order to
minimize the experimental effort, we seek quantum com-
puting schemes that are as simple as possible in terms of
resources . The quantum gate implementatio ns proposed
here do not require individual addressing of the atoms
and can be realized, essentially, within one step. In par-
ticular, we considered the case where the atom-c avity
coupling constant g is only one order of mag nitude larger
than each of the spontaneous decay rates of the system
and we as sumed g
2
= 100 κΓ.
To avoid dissipation, we populate only the ground state
of the system including all stable ground states o f the
atoms while the cavity is in its vacuum state. Never-
theless, an interaction between qubits can take place via
virtual po pulation of the cavity mode and excited atomic
levels. We first eliminated the possibility of leakage of
photons through the cavity mirrors. This was a chieved
by cho osing the parameters such that adiabaticity re-
stricts the time evolution of the system effectively onto
a deco herence-free subspace with respect to cavity decay
that includes highly entangled atomic states. Using these
states for the implementa tion of Raman and STIRAP-
like evolutions, entanglement can be created betwe e n the
ground states of the atoms.
Comparing the E-Raman and E-STIRAP procedures
we see that they give in general similar fidelities and suc-
cess rates, with E-STIRAP having a slight advantage.
Their control procedures are quite different in bo th c ases,
providing a flexibility to adapt the proposed scheme to
different experimental setups. The main conceptual as
well as practical differe nce appears when we produce the
entangling pha se ga tes . As it is seen in Figure 8 the phase
produced by the E- Raman tra nsition on the state |11i is
given explicitly by the evolution of this state from the
north to the south pole of the Bloch sphere. In contrast,
with the E-STIRAP procedure initial po pulation in the
|11i s tate circula tes along a closed path and returns ex -
actly to the initial point. Nevertheless, at the end the
state |11i acquires a geometrical phase.
The idea of using measurements on ancilla states, like
the measurements on the cavity mode considered in this
paper, leads to a realm of new possibilities for qua ntum
computing. A recent example is the linear optics scheme
by Knill et al. [37] where photons become entangled with-
out ever having to interact. As long as the measurements
on the ancilla state do not reveal any information about
the qubits, they induce a unitary time evolution between
them which can then be used to implement universal gate
operations [40]. In contrast to other schemes, the gate
success rate for atom-cavity quantum computing can, at
least in principle, be arbitrarily close to one since the
measurements are performed almost continuously. This
allows to utilize the quantum Z e no effect [18] which as-
sures that transitions in unwanted states are strongly in-
8
hibited.
The proposed schemes exhibit stability against fluctu-
ations of most system parameters. The presented dy-
namical and geometrical phases are obtained by a time
evolution that does not dependent on the exa c t value
of the e xternal contr ol parameters. Using E-Raman or
E-STIRAP processes, we enjoy the experimental advan-
tages, such as, independence of the final state on the ex-
act intermediate value of the laser field amplitude. Gates
are constructed dynamically as well as geometrically in
order to take advantage of the additional fault-tolerant
features of geometrical quantum computation [41]. Re-
cently, many attempts have been made in exploring the
decoherence cha racteristics of geometrical phases [42,43].
Whether they are advantageous over equivalent dynami-
cal evolutions depends on the employed physical system.
In contrast to this, we presented here evolutions that
are intrinsically decoher e nce -free and naturally allow the
generation of entangling dynamical as well as geometrical
phases with simple control pictures.
Acknowledgements. This work was supported in part
by the Europe an Union and the EPSRC. A.B. thanks
the Royal Society for a James Ellis University Research
Fellowship.
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