We consider one source of decoherence for a single trapped ion due to
intensity and phase fluctuations in the exciting laser pulses. For simplicity
we assume that the stochastic processes involved are white noise processes,
which enables us to give a simple master equation description of this source of
decoherence. This master equation is averaged over the noise, and is sufficient
to describe the results of experiments that probe the oscillations in the
electronic populations as energy is exchanged between the internal and
electronic motion. Our results are in good qualitative agreement with recent
experiments and predict that the decoherence rate will depend on vibrational
quantum number in different ways depending on which vibrational excitation
sideband is used.Comment: 2 figures, submitted to PR

B. W. Shore

C. Monroe

C.W. Gardiner

D. Leibfried

D. M. Meekhof

G. J. Milburn

H. Moya-Cessa

J. I. Cirac

K. Kraus

L.-M. Kuang

S. Dyrting

S. Schneider

English

arXiv

We consider one source of decoherence for a single trapped ion due to intensity and phase fluctuations in the exciting laser pulses. For simplicity we assume that the stochastic processes involved are white noise processes, which enables us to give a simple master equation description of this source of decoherence. This master equation is averaged over the noise, and is sufficient to describe the results of experiments that probe the oscillations in the electronic populations as energy is exchanged between the internal and electronic motion. Our results are in good qualitative agreement with recent experiments and predict that the decoherence rate will depend on vibrational quantum number in different ways depending on which vibrational excitation sideband is used

Schneider, S.

Milburn, Gerard J.

University of Queensland eSpace

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Decoherence in ion traps due to laser intensity and phase
fluctuations
S. Schneider and G. J. Milburn
Department of Physics, The University of Queensland, QLD 4072 Australia.
Abstract
We consider one source of decoherence for a single trapped ion due to in-
tensity and phase fluctuations in the exciting laser pulses. For simplicity we
assume that the stochastic processes involved are white noise processes, which
enables us to give a simple master equation description of this source of deco-
herence. This master equation is averaged over the noise, and is sufficient to
describe the results of experiments that probe the oscillations in the electronic
populations as energy is exchanged between the internal and electronic mo-
tion. Our results are in good qualitative agreement with recent experiments
and predict that the decoherence rate will depend on vibrational quantum
number in different ways depending on which vibrational excitation sideband
is used.
03.65.Bz,05.45.+b, 42.50.Lc, 89.70.+c
Typeset using REVTEX
1
I. INTRODUCTION
Recent advances in laser cooling now enable a single trapped ion to be prepared in a
chosen quantum state of the center-of-mass vibrational motion [1]. In some cases, the state
is highly nonclassical, such as the recent preparation of a superposition of two oscillator
coherent states [2]. Quantum dynamical features, such as collapse and revival oscillations,
have also been observed [3]. The key innovation in such experiments is the ability to tailor
the effective potential experienced by the ion by coupling the center-of-mass motion to the
electronic states by external laser pulses. If more than a single ion is trapped, individual
ions may be addressed by different laser pulses leading to an entanglement of the collective
vibrational motion of the ions and their electronic states. This is the principle behind the
suggestion of Cirac and Zoller [4] for implementing quantum computation in ion traps. So
far however only a single controlled-NOT gate, a key component in a quantum computer,
has been implemented experimentally [5].
Despite these heroic experimental achievements, the quantum motion of a single trapped
ion is obviously limited by sources of decoherence. Decoherence arises from random and
unknown perturbations of the Hamiltonian. If these perturbations cannot be followed ex-
actly, experiments must average over them. This leads to an effective irreversible evolution
of the trapped ion and a suppression of coherent quantum features through the decay of
off-diagonal matrix elements of the density operator in some basis. Complementary to the
decay of off-diagonal matrix elements, noise is added to conjugate variables. This can appear
as a heating of the ion if noise is added to the momentum variable.
In this paper we consider one source of decoherence for a single trapped ion due to
intensity and phase fluctuations in the exciting laser pulses. For simplicity we assume
that the stochastic processes involved are white noise processes, which enables us to give
a simple master equation description of this source of decoherence. Section II contains a
general overview of the kind of system we investigate. In the first subsection we concentrate
on intensity fluctuation in the laser, whereas the second subsection is devoted to phase
2
fluctuations. We conclude with a discussion on the experimental relevance of our results.
II. LASER FLUCTUATIONS.
A single two-level ion, with mass m, tightly bound in a harmonic trap, and laser cooled
to the Lamb-Dicke limit, can be prepared in a variety of states by carefully controlling
the effective detuning of external laser fields which couple the vibrational motion and the
internal electronic states. For simplicity we will assume the ion is constrained to move in a
single dimension at harmonic frequency ν. A reference frequency is provided by the atomic
transition frequency ωA. If the effective laser frequency ωL is tuned below or above this
frequency by multiples of the harmonic trap frequency ν, a variety of effective potentials
may be obtained. In the NIST experiments [1], two laser fields are used to excite two-photon
stimulated Raman transitions. However in this paper we will consider the simplest case of a
single classical laser, with wave vector kL and frequency ωL, where the field is propagating
in the same direction in which the ion is constrained to vibrate.
In the interaction picture with the dipole and rotating wave approximation the interac-
tion Hamiltonian is [1]
HI = h¯Ω(t)(σ+ + σ−)
(
eiη(a
−iνt+a†eiνt)−iδt+iφ(t)
)
+ h.c , (1)
where Ω(t) is the effective Rabi frequency for this transition, written as a function of time
to account for fluctuations resulting from laser intensity fluctuations, φ(t) represents fluc-
tuations in the laser phase, η = kL(h¯/2mν)
1/2 is the Lamb-Dicke parameter, δ = ωA − ωL
is the detuning between the laser and the electronic states, and σ± are the usual two-level
atom transition operators. In order to excite particular transitions, |g, n〉 ↔ |e, n′〉, of the
coupled electronic/vibrational spectrum we choose the detuning
δ = ν(n− n′) , (2)
where n, n′ are integers. In this paper we will assume that the amplitude of the ions motion
in the direction of the laser field is much less than a wavelength. In this limit we can expand
3
the interaction Hamiltonian to lowest order in η [1]. Furthermore, to illustrate the effect
of laser fluctuations it will suffice to consider four cases: (i) carrier excitation, n = n′, (ii)
first red sideband excitation n′ = n− 1, (iii) first blue sideband, n′ = n+ 1, (iv) second red
sideband, n′ = n+ 2. The interaction Hamiltonians for these four cases are:
HI(t) = h¯Ω(t)(1 + η
2a†a)
[
σ+e
iφ(t) + h.c
]
carrier , (3)
HI(t) = h¯Ω(t)η
[
σ+ae
iφ(t) + h.c
]
red sideband , (4)
HI(t) = h¯Ω(t)η
[
σ+a
†eiφ(t) + h.c
]
blue sideband , (5)
HI(t) = h¯Ω(t)η
[
σ+a
2eiφ(t) + h.c
]
second red sideband . (6)
The red sideband Hamiltonian corresponds to the familiar Jaynes-Cummings Hamiltonian
[7] of quantum optics.
We will specify the noise by defining a stochastic process for Ω(t) and φ(t). In the case
of laser amplitude fluctuations we take
Ω(t)dt = Ω0(dt+
√
ΓdW (t)) , (7)
where dW (t) is the increment of a real Wiener process [6], and Ω0 is the non-fluctuating
component of the Rabi frequency. The parameter Γ scales the noise. The interpretation of
Γ is given by integrating Eq.(7) to obtain the pulse area, A(T ), which is also a stochastic
variable,
A(T ) = Ω0T +∆A(T ) , (8)
where T is the pulse duration. The last term in this equation is a Gaussian random variable
with mean zero and variance E(∆A(T )2) = Ω20ΓT . If we then consider the ratio of the r.m.s.
fluctuations in the pulse area to the deterministic pulse area we find
E(∆A(T )2)1/2
Ω0T
=
√
Γ
T
. (9)
For phase fluctuations we take a simple diffusion,
φ(t) =
√
γW (t) , (10)
where W (t) is the Wiener process.
4
A. Intensity fluctuations
The Hamiltonians in Eqs. (6) are stochastic. We first consider the effect of laser intensity
fluctuations and ignore phase fluctuations. We thus set φ(t) = 0 (constant). To obtain the
corresponding Schro¨dinger equation requires some care, as the white noise process is quite
singular. We can however define a stochastic Schro¨dinger equation in the Ito formalism [8],
or more appropriately a stochastic Liouville-von Neumann equation,
dρ(t) = −i[G, ρ]dt− i
√
Γ[G, ρ]dW (t)− Γ
2
[G, [G, ρ]]dt , (11)
where G takes one of the four forms,
G = Ω0σx(1 + η
2a†a) carrier , (12)
G = ηΩ0(aσ+ + a
†σ−) red sideband , (13)
G = ηΩ0(a
†σ+ + aσ−) blue sideband , (14)
G = ηΩ0(a
2σ+ + (a
†)2σ−) second red sideband . (15)
This equation gives the evolution of the system density operator conditioned on a particular
noise history. In an experiment involving a number of pulses, with data from each pulse
combined, the noise is effectively averaged and we obtain the following master equation
describing the system
dρ(t) = −i[G, ρ]dt− Γ
2
[G, [G, ρ]]dt . (16)
This equation has a similar form to that considered in reference [9] for a model of intrinsic
decoherence. Indeed, that model of intrinsic decoherence has been explicitly solved for the
case of the Jaynes-Cummings Hamiltonian (the red side-band case) by Moya-Cessa et al.
[10] (see also [11]).
The last term in Eq.(16) is responsible for decoherence and, complementary to deco-
herence, it leads to diffusive growth in observables that do not commute with G, which is
proportional to the interaction Hamiltonian of the system. In each case, the eigenstates of
5
G are discrete and labeled by an index n corresponding to a vibrational quantum number,
and a sign index ± arising from the two dimensional Hilbert space of the electronic motion.
We will designate these states as {|e±n 〉}. Thus
∂
∂t
〈e±n |ρ(t)|e±m〉 =
[
−i(e±n − e±m)−
Γ
2
(e±n − e±m)2
]
〈e±n |ρ(t)|e±m〉 . (17)
In this form the decay of off-diagonal coherence is quite explicit. Notice however that
decoherence takes place in a joint basis of both the electronic and vibrational Hilbert spaces.
This is in contrast to many similar proposals for decoherence, such as Brownian motion,
which only involve a single Hilbert space. Furthermore, the form of the decoherence ensures
that total energy is conserved even in the presence of noise, as expected for a stochastic
Hamiltonian.
The eigenstates and eigenvalues for each of the operators in Eqs.(11-13) are as follows.
For the case of carrier excitation:
|e±n 〉 =
1√
2
(|g, n〉 ± |e, n〉) (18)
e±n = ±
h¯Ω0
2
(1 + nη2) . (19)
For the case of red side-band excitation (Jaynes-Cummings):
|e±n 〉 =
1√
2
(|g, n+ 1〉 ± |e, n〉) (20)
e±n = ±ηΩ0
√
n + 1 . (21)
For the blue side-band case:
|e±n 〉 =
1√
2
(|g, n− 1〉 ± |e, n〉) (22)
e±n = ±ηΩ0
√
n . (23)
For the second red sideband case:
|e±n 〉 =
1√
2
(|g, n+ 2〉 ± |e, n〉) (24)
e±n = ±ηΩ0
√
n(n + 1) . (25)
6
The general solution to Eq. (16) may be written explicitly as
ρ(t) = e−iGt−
Γt
2
G2
(
eJ tρ(0)
)
eiGt−
Γt
2
G2 , (26)
where the superoperator in the middle of this expression is defined by the power series
expansion for the exponential with
Jmρ = ΓmGmρGm . (27)
In the energy eigenstate basis we can solve Eq.(17) explicitly to give
〈e±n |ρ(t)|e±m〉 = exp
[
− i
h¯
t(e±n − e±m)−
Γt
2h¯2
(e±n − e±m)2
]
〈e±n |ρ(0)|e±m〉 . (28)
However it is of rather more use to exhibit the solution explicitly for particular initial
conditions of relevance to the experiments. With this in mind we will assume that the initial
state is prepared to be a particular vibrational energy eigenstate with the ion prepared in
the electronic ground state,
|ψ(0)〉 = |g, n〉 . (29)
In the experiments done so far, it is possible to probe directly which electronic state
the ion occupies. In the work of Meekhof et al. [3] the internal state |g〉 is the 2s 2S1/2
(F = 2,MF = 2) state of
9Be+, and the state |e〉 corresponds to the 2s 2S1/2 (F = 1,MF =
1) state as shown in figure 1. The state |g〉 is detected by applying a nearly resonant σ+
polarized laser probe field to drive a strong transition between the state |g〉 and another state,
2P3/2 (F = 3,MF = 3). As this other state can only decay back to |g〉, any fluorescence
observed on this transition is evidence that the atom was in the state |g〉 at the start of this
probe pulse. As the intensity of the fluorescence on this transition is strong, it is almost
certain to detect a photon and thus the quantum efficiency of this state determination is
near unity. In other words, this measurement scheme realizes an almost perfect projection
valued measurement onto the electronic state |g〉. A sequence of probe pulses delayed a time
τ after the initial state preparation can thus be used to build up the probability Pg(t) to find
7
the ion in the ground state. Of course to do this many repeats of the experiment must be
performed and it is not possible to track laser fluctuations exactly one each run. The final
result for Pg(t) must then represent an ensemble average over these fluctuations, and the ion
dynamics is then described by Eq.(16). Given Pg(t), one easily sees that Pe(t) = 1− Pg(t).
With ψ(0) = |g, n〉, the solution for Pg(t) in each case is as follows. For carrier excitation:
Pg(t) =
1
2
[
1 + exp
{
−ΓtΩ
2
0
2
(1 + 2nη2)
}
cos
(
Ω0t(1 + nη
2)
)]
. (30)
For red side-band excitation:
Pg(t) =
1
2
[
1 + exp
(
−2Γη2Ω20nt
)
cos
(
2ηΩ0
√
nt
)]
. (31)
For blue sideband excitation;
Pg(t) =
1
2
[
1 + exp
(
−2Γη2Ω20(n+ 1)t
)
cos
(
2ηΩ0
√
n + 1t
)]
. (32)
For second red sideband excitation:
Pg(t) =
1
2
[
1 + exp
(
−2Γη2Ω20n(n− 1)t
)
cos
(
2ηΩ0
√
n(n− 1)t
)]
. (33)
So to test the dependence of decoherence on the excitation in the vibrational state experi-
mentally, the second red sideband (or any higher order sideband) is a better choice than just
the first order sidebands, since the dependence of the damping on n is quadratic (or of even
higher order for higher sidebands). In figure 2 we illustrate a typical result using Eq. (32)
and the parameters given in [3].
B. Phase fluctuations
Now we investigate phase fluctuation in the laser instead of intensity fluctuations, i.e.
we set Ω(t) in Eqs. (3)–(6) to Ω0 and introduce white phase noise φ(t) =
√
λW (t) instead.
The means that the Hamiltonian is a nonlinear function of the noise which will lead to
technical difficulties in deriving the corresponding Ito differential equation for the system
state. However for our purposes a simple transformation can simplify matters considerably.
8
We follow [8] and use a random canonical transformation to get rid of the nonlinearity in the
noise source. In effect this is an instantaneous rotation of the system through a fluctuating
angle, analogous to the standard method of transformation to an interaction picture.
Uˆ = exp (iφ(t)σˆ+σˆ−) . (34)
Thus
ρ −→ ρ˜ = exp (−iφ(t)σˆ+σˆ−) ρ exp (iφ(t)σˆ+σˆ−) (35)
H −→ H0 = G− φ(t)σˆ+σˆ− , (36)
where G is one of the operators defined by Eqs. (12)–(15), depending on which sideband
the laser is tuned to. The reason we can use this stochastic transformation is that we are
only interested in the population of the electronic levels. As the generator of the unitary
transformation commutes with the population operator, the populations in the instantaneous
transformed frame are the same as those in the original frame. However other moments will
not be the same, and we could not easily use the transformed state, ρ˜, after averaging, to
reconstruct moments in the original frame. This transformation can be described covering
all four cases of G in one, since it only affects the internal state and the operators describing
that (σˆ+ and σˆ− appear in the same order and form in all the interactions we consider here).
The corresponding master equation after averaging out the noise reads
dρ˜
dt
= −i [G, ρ˜]− λ [σˆ+σˆ−, [σˆ+σˆ−, ρ˜]] . (37)
For the initial condition ρˆ = |g〉〈g| ⊗ |n〉〈n| we can solve this for all sidebands. However,
here we restrict ourselves to the red sideband since that gives an idea on how the effects
arising from phase fluctuations are different from intensity fluctuations. The probability for
the atom to be in the upper state in this case is given by
P˜g(t) =
1
2
[
1 + cos(ω˜n) exp(−(λ/2)t) + λ
2ω˜n
sin(ω˜n) exp(−(λ/2)t)
]
(red sideband), (38)
with
9
ω˜n =
√
4η2Ω20n− λ2/4 . (39)
So here the effective Rabi frequency depends on the coherence decay rate and not vice versa
as in the case of intensity fluctuations.
III. DISCUSSION
We have shown how to model fluctuations in the laser causing the interaction between
center-of-mass motion and internal states in ion traps. Intensity fluctuations lead to deco-
herence processes which depend on the kind of interaction the laser is causing.
What is the value for Γ for fluctuations in the laser intensity? A rough estimate from
the figures given in [3], using the quoted experimental values, leads to Γ ≈ 1.4 · 10−8s. But
this is really a very rough estimation. If we define the fractional error by the quotient of the
r.m.s. and the pulse area,
r.m.s.
A(T )
=
Ω0
√
ΓT
Ω0T
=
√
Γ
T
, (40)
a fractional error of one percent leads to Γ ≈ 10−10. With a fractional error of ten percent,
however, we get Γ ≈ 10−8 and we are in the range of the roughly estimated value for Γ.
In [3] they experimentally estimate the n-dependence of their damping which they fit
with
Pg(t) =
1
2
[
1 +
∑
n=0
Pn cos(2Ωn,n+1t)e
−γnt
]
(41)
to be of the form
γn = γo(n+ 1)
0.7 . (42)
We derive the coefficient to be 0.5 instead of 0.7 if the decoherence is just due to intensity
fluctuations alone. However there are other sources of decoherence due, for example, to
fluctuations in the trap potential itself. These fluctuations lead to, among other things, a
fluctuating trap center point and will be addressed in a future publication.
10
ACKNOWLEDGMENTS
S. S. gratefully acknowledges financial support from a “DAAD Doktorandenstipendium
im Rahmen des gemeinsamen Hochschulsonderprogramms III von Bund und La¨ndern” and
from the Center of Laser Science.
11
REFERENCES
[1] D.J.Wineland, C. Monroe, W.M. Itano, D. Liebfried, B.King, and D.M. Meekhof, ”Ex-
perimental issues in coherent quantum-state manipulation of trapped atomic ions”,
submitted to Rev. Mod. Phys. (1997).
[2] C. Monroe, D.M. Meekof, B.E. King, and D.J. Wineland, Science, 272, 1131 (1996).
[3] D.M. Meekof, C. Monroe, B.E. King, W.M. Itano, and D. J. Wineland, Phys. Rev. Lett.
76, 1796 (1996) and Phys. Rev. Lett. 77, 2346 (Erratum) (1996).
[4] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4094 (1995).
[5] C. Monroe, D.M. Meekhof, B.E. King, W.M. Itano, and D.J. Wineland, Phys. Rev.
Lett. 75, 4714 (1995).
[6] C.W. Gardiner,Handbook of Stochastic Processes for Physics, Chemistry and the Natural
Sciences, (Springer-Verlag, Berlin, 1985).
[7] B.W. Shore and P.L. Knight, J. Mod. Opt. 40, 1195 (1993).
[8] S. Dyrting and G.J. Milburn, Quantum Semiclass. Opt. 8, 541 (1996).
[9] G.J. Milburn, Phys. Rev. A 44, 5401 (1991).
[10] H. Moya-Cessa, V. Buzek, M.S. Kim, and P.L. Knight, Phys. Rev. A 48, 3900 (1993).
[11] L.-M. Kuang, X. Chen, G.-H. Chen, and M.-L. Ge, Phys. Rev. A 56, 3139 (1997).
12
FIGURES
 


 
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FIG. 1. Internal level scheme of 9Be+. The ground state |g〉 is the 2s 2S1/2 (F = 2,MF = 2)
state and the excited state |e〉 is the 2s 2S1/2 (F = 1,MF = 1) state. The state |g〉 is detected by
applying a nearly resonant σ+ polarized laser probe field to drive a strong transition between |g〉
and another state 2P3/2 (F = 3,MF = 3) as indicated. As this other state can only decay back to
|g〉, any fluorescence observed on this transition is evidence that the atom was in the state |g〉 at
the start of the probe pulse.
13
 

 
150 300
0.5
1




	


FIG. 2. Pg(t) for an initial |g, n = 0〉 state of the atom driven by the first blue sideband,
Eq. (32). The time parameter τ = Ω0t is scaled with the effective Rabi frequency Ω0 = 470 kHz.
The value for the scaled damping coefficient Γ′ = ΓΩ0 is 0.041 and the one for η is 0.2. All the
values are rough estimates from the experimental values given in [3].
14


Decoherence in ion traps due to laser intensity and phase fluctuations

Milburn, G. J.

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Decoherence in ion traps due to laser intensity and phase fluctuations

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