Stability of atomic clocks based on entangled atoms.

Abstract

We analyze the effect of realistic noise sources for an atomic clock consisting of a local oscillator that is actively locked to a spin-squeezed (entangled) ensemble of N atoms. We show that the use of entangled states can lead to an improvement of the long-term stability of the clock when the measurement is limited by decoherence associated with instability of the local oscillator combined with fluctuations in the atomic ensemble's Bloch vector. Atomic states with a moderate degree of entanglement yield the maximal clock stability, resulting in an improvement that scales as N(1/6) compared to the atomic shot noise level.

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arXiv:quant-ph/0401130v1 21 Jan 2004
Stability of atomic clocks based on entangled atoms
A. Andr´ e, A. S. Sørensen, and M. D. Lukin
Physics Department and Institute for Theoretical Atomic and Molecular Physics,
Harvard University, Cambridge, Massachusetts 02138
(Dated: February 1, 2008)
We analyze the effect of realistic noise sources for an atomic clock consisting of a local oscillator
that is actively locked to a spin-squeezed (entangled) ensemble of N atoms. We show that the use
of entangled states can lead to an improvement of the long-term stability of the clock when the
measurement is limited by decoherence associated with instability of the local oscillator combined
with fluctuations in the atomic ensemble’s Bloch vector. Atomic states with a moderate degree of
entanglement yield the maximal clock stability, resulting in an improvement that scales as N1/6
compared to the atomic shot noise level.
Quantum entanglement is the basis for many of the
proposed applications of quantum information science
[1]. The experimental implementation of these ideas is
challenging since entangled states are easily destroyed by
decoherence. To evaluate the potential usefulness of en-
tanglement it is therefore essential to include a realistic
description of noise in experiments of interest. Although
decoherence is commonly analyzed in the context of sim-
ple models [2], practical sources of noise often possess a
non-trivial frequency spectrum, and enter through a va-
riety of different physical processes. In this Letter, we
analyze the effect of realistic decoherence processes and
noise sources in an atomic clock that is actively locked
to a spin-squeezed (entangled) ensemble of atoms.
The performance of an atomic clock can be charac-
terized by its frequency accuracy and stability. Accu-
racy refers to the frequency offset from the ideal value,
whereas stability describes the fluctuations around, and
drift away from the average frequency. To improve the
long-term clock stability, it has been suggested to use
entangled atomic ensembles [3, 4, 5], and in this let-
ter we analyze such proposals in the presence of real-
istic decoherence and noise. In practice, an atomic clock
operates by locking the frequency of a local oscillator
(L.O.) to the transition frequency between two levels in
an atom. This locking is achieved by a spectroscopic mea-
surement determining the L.O. frequency offset δω from
the atomic resonance, followed by a feedback mechanism
which steers the L.O. frequency so as to null the mean
frequency offset. The problem of frequency control thus
combines elements of quantum parameter estimation the-
ory and control of stochastic systems via feedback [6, 7].
The spectroscopic measurement of the atomic tran-
sition frequency is typically achieved through Ramsey
spectroscopy [8], in which the atoms are illuminated by
two short, near-resonant pulses from the local oscillator,
separated by a long period of free evolution, referred to
as the Ramsey time T. During the free evolution the
atomic state and the L.O. acquire a relative phase differ-
ence δφ = δωT, which is subsequently determined by a
projection measurement. If a long time T is used, then
Ramsey spectroscopy provides a very sensitive measure-
ment of the L.O. frequency offset δω [9]. Here, we inves-
tigate the situation relevant to trapped particles, such
as atoms in an optical lattice [11] or trapped ions [12].
In this situation, the optimal value of T is determined
by atomic decoherence (caused by imperfections in the
experimental setup) which therefore determines the ulti-
mate performance of the clock.
We consider an ensemble of N two-level particles with
lower (upper) state | ↓? (| ↑?). Adopting the nomencla-
ture of spin-1/2 particles, we introduce the total angu-
lar momentum (i.e., Bloch vector)?J =?N
of the atoms has mean ??J? along the z direction and
?Jx? = ?Jy? = 0. Unavoidable fluctuations in the x and
y components ?J2
atomic projection noise. These fluctuations give rise to
an uncertainty in the Ramsey phase δφR ≃ ∆Jy/|?ˆJz?|
as indicated geometrically in Fig. 1 [3, 10]. For uncor-
related spins aligned along the z axis, the uncertainty
from independent spins are added in quadrature, result-
ing in the projection noise ∆Jy =
the measurement error it has been proposed [5, 13, 14]
and demonstrated [15] to use entangled atomic states (so
called spin squeezed states), which have reduced noise
in one of the transverse spin components (e.g., Jy) and
non-zero noise ∆Jz in the mean spin direction. Ideally
this gives an improvement by a factor ξ =√N∆Jy/?Jz?,
which can be as low as ξ = 1/√N for maximally entan-
gled states [4].
j=1?Sj, where
Initially the statee.g.Sj
z= (| ↑?j?↑ | − | ↓?j?↓ |)/2.
x? = ?J2
y? ?= 0, result in the so-called
√N/2. To reduce
Using a simple noise model it was shown in Ref. [16]
that entanglement provides little gain in spectroscopic
sensitivity in the presence of atomic decoherence.
essence, random fluctuations in the phase of the atomic
coherence cause a rapid smearing of the error contour
in Fig. 1a. For example, dephasing of individual parti-
cles results in an additional contribution (N/4)?δφ2? to
the noise, where ?δφ2? denotes the variance of the phase
fluctuations (increasing with T as ?δφ2? = γT for white
noise, where γ is the dephasing rate). In practice, the
stability of atomic clocks is often limited primarily by
fluctuations of the L.O. As we show below, the L.O. fluc-
In

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tuations result in the added noise ∆J2
is the initial variance in Jz. This added noise is due to
the error in the feedback loop, caused by the longitudi-
nal noise ∆Jz. For weakly entangled states, the added
noise is considerably smaller than in the case of atomic
dephasing and the use of entangled states can lead to a
significant improvement in clock stability.
In what follows we outline a model that incorporates
the effects of atomic noise and spin squeezing as well
as that of the feedback loop.
note that qualitative considerations along these lines were
noted in Ref. [17]. At the operating point, the error sig-
nal in Ramsey spectroscopy [5] measured at time tk is
determined by the operator
z?δφ2?, where ∆J2
z
Before proceeding, we
ˆE(tk) ≃
N
?
j=1
ˆSj
zsin[δφj(tk)] +ˆSj
ycos[δφj(tk)], (1)
where δφj(tk) is the phase acquired by the jth atom dur-
ing the interrogation time T and all operators refer to
the initial atomic state. We separate the phase into two
parts δφj= δφO+δφj
is the phase due to the frequency fluctuations δω(t) of
the L.O., and δφj
Eis a phase induced by the interaction
of the jth atom with the environment. In order to lock
the L.O. to the atomic frequency, the interrogation time
should be short enough that ?δφ(tk)2?<
in terms of δφ(tk), we find the measured error signal
E, where δφO(tk) =?T
0δω(tk−t)dt
∼1. Expanding
E(tk) ≃ ?ˆJz?
?
δφO(tk) −δφO(tk)3
3!
?
+
N
?
j=1
Sj
zδφj
E(tk)
+ [Jy(tk) + δJz(tk)δφO(tk)] + ···.
Here δJz(tk) = Jz(tk) − ?ˆJz?, where Jz(tk) and Jy(tk)
are random numbers with a distribution corresponding to
the initial atomic state (we consider here states for which
?ˆJyˆJz? = 0, so that we may treat Jz(tk) and Jy(tk) as
independent random variables). The term multiplying
?ˆJz? in (2) is used to estimate the frequency offset, while
the remaining terms represent measurement noise.
The feedback is started at t = 0 and, at the end of
each Ramsey cycle, at tk= kT (k = 1,2,...), the detec-
tion signal is used to steer the frequency of the oscillator
to correct for the fluctuations accumulated during the
last cycle δω(t+
refer to before and after the correction, and ∆ω(tk) is
the frequency correction. Assuming that negligible time
is spent performing the π/2 pulses and in preparing and
detecting the state of the atoms, the mean frequency off-
set after running for a period τ = nT is then
(2)
k) = δω(t−
k) + ∆ω(tk), where t−
kand t+
k
δ¯ ω(τ) =
1
τ
?τ
n
?
0
δω(t)dt
=
T
τ
k=1
?δφO(tk)
T
+ ∆ω(tk)
?
. (3)
We begin by analyzing the simplest case of linear feed-
back (in E(tk)) and later extend to the more optimal
nonlinear feedback case. With ∆ω(tk) =
(2) and substituting in (3), we find, ignoring for now the
δφO(tk)3term,
−E(tk)
?ˆ Jz?T, using
δ¯ ω(τ) =
−1
τ?ˆJz?
N
?
n
?
k=1
[Jy(tk) + δJz(tk)δφO(tk)
+
j=1
Sj
zδφj
E(tk)](4)
Note that the acquired offsets δφO(tk)/T (k = 1,...,n)
due to L.O. frequency fluctuations are corrected by the
feedback loop and do not appear in (4), while measure-
ment noise is added at the detection times tk. The first
two terms in (4) are uncorrelated for different tk since
the atomic noise for different detection events is uncorre-
lated. If the dephasing noise is uncorrelated for different
tk, then the fractional frequency fluctuation (Allan devi-
ation) [19] σy(τ) = ?(δ¯ ω(τ)/ω)2?1/2, is
σy(τ) =
?∆J2
y+ ∆J2
z?δφ2
ω√τT?ˆJz?
O? + (λ?J2
z?)?δφ2
E??1/2
. (5)
Here λ accounts for the possibility of collective decoher-
ence, so that for atoms dephasing collectively (indepen-
dently) λ → 1 (λ → (N/4)/?ˆJ2
the atoms in a similar fashion than collective dephasing.
Note, however, the significant difference between collec-
tive environmental dephasing, which enters expression
(5) as ?ˆJ2
is the relevant expression. The feedback loop results in
a large cancellation of the effect of the L.O. noise on the
stability; the uncanceled part of the noise is now propor-
tional to ∆J2
z?.
When decoherence is negligible, ?δφ2
the long term frequency stability is given by σy(τ) =
∆Jy/ω√τT?ˆJz? as shown in Refs. [3, 5]. For an uncorre-
lated atomic state, the stability improves with increasing
number of atoms as N−1/2[10]. The maximum possi-
ble improvement using spin-squeezed states is a factor of
N−1/2, yielding a stability σy(τ) ∝ N−1[4].
The best long term stability is obtained with the
longest possible interrogation time T. When the inter-
rogation time is limited by environmental decoherence,
the latter cannot be ignored. This corresponds to the
situation considered in Refs. [16, 20], in which case no
substantial improvement is possible. In the practically
relevant case where the main source of noise is from the
L.O. [12, 21, 22] the situation is quite different. In this
case it is undesirable to use a very highly squeezed state
with ∆Jy∼ 1 because it has a very large uncertainty in
the z-component of the spin ∆Jz ∼ N, which accord-
ing to Eq. (5) has a large contribution to the noise. A
z?). The L.O. noise affects
z??δφ2
E?, and L.O. noise, for which ∆J2
z?δφ2
O?
z≪ ?ˆJ2
O? = ?δφ2
E? = 0,

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10
−2
10
0
10
2
10
−2
10
−1
10
0
0.0010.01 0.11
1
10
100
y
Jz
f/γ δϕR
σy
1/
ξ = N−1/6
ξ = 1
c)
x
J
J
a) b)
L.O.
noise spectrum
?δφ2? = γT
FIG. 1: a) Representation of the probability distribution on
the Bloch sphere for a spin squeezed state |ψ(κ)?, with κ =
N1/4corresponding to the squeezing parameter ξ = N−1/4
(N = 10, both the initial state and the state just before de-
tection are shown for clarity). Thick lines indicate initial un-
certainties in?J. b) Noise spectra due to L.O. frequency fluc-
tuations when free running, when stabilized to unsqueezed
atoms (ξ = 1), and when stabilized to spin squeezed atoms
(ξ = N−1/6), N = 103and γT = 10−2.
tional frequency stability 1/σy (arbitrary units) vs.
sey time for white L.O. noise, N = 105, with linear feed-
back to uncorrelated atoms (◦); linear feedback to correlated
atoms (⋄, ξ = N−1/4); nonlinear feedback to uncorrelated
atoms (•); and nonlinear feedback to correlated atoms (filled
⋄, ξ = N−1/4). Points: numerical simulations, lines: analyti-
cal results.
c) Inverse frac-
Ram-
moderately squeezed state can, however, lead to a con-
siderable improvement in the stability. This observation
is the main result of the present Letter.
To find the optimal stability, we first optimize (5) with
respect to the interrogation time. Considering uncorre-
lated atoms first, we have ∆Jy =
Eq. (5) then predicts that σy(τ) decreases indefinitely as
1/√T. To derive Eq. (5), however, we have linearized
the expression in Eq. (1), and this linearization breaks
down when the (neglected) cubic term in (2) is com-
parable to the noise term that we retained, i.e., when
δφ(tk)3∼ ∆Jy/?ˆJz?.
based on Eq. (2), including perturbatively the nonlin-
ear terms in a stochastic differential equation, we find
the optimal time γT = (2∆J2
√N/2 and ∆Jz = 0;
In a more careful analysis [18]
y/?ˆJz?2)1/3. At this point
the stability is given by σy(τ) = ζN−1/3γ/ω√γτ where
ζ =
3
24/3N1/3


?
∆Jy
?ˆJz?
?4/3
+24/3
3
?
∆Jz
?ˆJz?
?2

1/2
. (6)
To evaluate the potential improvement in stability
by using squeezed states (i.e., the scaling with increas-
ing number of atoms N, in the limit N ≫ 1), it is
convenient to use a family of states parametrized by
a small number of parameters. A one-parameter fam-
ily of states that includes the uncorrelated state as well
as spin squeezed states is given by the Gaussian states
|ψ(κ)? = N(κ)?
total angular momentum quantum number is J = N/2,
and N(κ) is a normalization factor. The transverse noise
for these states is given by ∆Jy = κ/2.
number of atoms N ≫ 1, the uncorrelated state is well
approximated by |ψ(κ =
states are obtained when κ → 1.
ily of states the optimal value is ζ ≃ 1.42N−1/6for
κ ≃ 21/16N1/4(ξ ∼ N−1/4) giving a stability scaling
as N−1/2.This represents an improvement by a fac-
tor of N1/6compared to uncorrelated states, for which
ζ = 3/24/3and the stability scales as N−1/3. We em-
phasize that these results are derived assuming a linear
feedback loop.
To confirm these predictions, we have made exten-
sive numerical simulations of the frequency control loop,
along the lines of Ref. [23]. The noise spectrum of the
free-running oscillator is defined by S(f)δ(f + f′) =
?δω(f)δω(f′)?, where δω(f) is the Fourier transform of
the stochastic process δω(t).
sponding time-series and at the detection times tk= kT,
the accumulated phase δφO(tk) is calculated and the
atomic noise is generated from the probability distribu-
tions of Jy and Jz. The error signal E(tk) is found and
a frequency correction ∆ω(tk) is generated. The noise
spectrum of the slaved oscillator, see Fig. 1b, clearly
shows that while for short time scales (<
quencies) the noise is given by that of the free-running
oscillator, at longer time scales (lower frequencies) the os-
cillator is locked to the atoms and the remaining (white)
noise is determined by the atomic fluctuations. The low-
frequency white noise floor determines the long-term sta-
bility of the clock and is the quantity we seek to optimize.
In Fig. 1(c) we compare our analytical results with the
results of the numerical simulations as a function of Ram-
sey time T, and in Fig. 2(a) we show the scaling with
the number of atoms. The analytical and numerical ap-
proaches are in excellent agreement.
So far we have assumed linear feedback and white
noise; we now relax these assumptions.
limit identified above is mainly determined by the break-
down of the assumption of small (i.e., linear) phase fluc-
m(−1)me−(m/κ)2|m?, where |m? are
eigenstates of the Jyoperator with eigenvalue m and the
For a large
√N)?, while highly-squeezed
Within this fam-
We generate the corre-
∼T, high fre-
The stability

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100 100010000
1e+05
10
100
1000
100 100010000
10
100
y
1/σy
NN
a)
1/
b)
σ
FIG. 2: Inverse fractional frequency stability 1/σy (arbitrary
units) vs. number of atoms N, with Ramsey time optimized
for a) white noise and b) 1/f noise. Points: numerical simu-
lations, lines: analytical results. Uncorrelated atoms (◦) and
optimal spin squeezed atoms (⋄), both for linear feedback (full
lines, open symbols) and nonlinear feedback (dashed lines,
filled symbols).
tuations. In fact, the stability can be improved consider-
ably by using a feedback ∆ω which is a nonlinear func-
tion of the error signal E. To investigate this we have
included a nonlinear feedback ∆ω(tk) ∝ arcsin[E(tk)/J]
in our numerical simulations. In Fig. 1(c) it is seen that
nonlinear feedback performs better, and that it extends
the validity of Eq. (5) all the way to γT ∼ 0.1. For larger
γT, the feedback loop fails, resulting in a rapid decrease
in stability. If we optimize the Allan deviation in Eq. (5)
for nonlinear feedback, under the condition γT ≤ 0.1, we
find that the optimally squeezed states have ∆Jy∼ N1/3
(ξ ∼ N−1/6) resulting in a stability scaling as N−2/3.
This represents again a relative improvement in scaling
of N1/6compared to the uncorrelated state for which the
stability scales as N−1/2. Detailed derivation of these re-
sults will be presented elsewhere [18].
The assumption of white noise ?δφ2? = γT, is conve-
nient for theoretical calculations, but in practice very-
low-frequency noise is likely to have nontrivial spectrum
such as 1/f noise. To find the scaling with the number
of atoms in this situation, we replace ?δφ2? = γT with
the behaviour expected for 1/f noise: ?δφ2? ∼ (γT)2.
Repeating all the calculations above we again find an im-
provement by a factor of N1/6by using squeezed states
for the nonlinear feedback loop, and a factor of N5/24
for linear feedback. In Fig. 2b we compare these scal-
ing arguments to the numerical simulations and the two
approaches are seen to be in very good agreement.
To summarize, we have shown that entanglement can
provide a significant gain in the frequency stability of an
atomic clock when it is limited by the stability of the os-
cillator used to interrogate the atoms. The optimal sta-
bility is achieved by using moderately squeezed states,
with a relative improvement that scales approximately
as N1/6with the number of atoms. These results are
in contrast to previous studies using simplified decoher-
ence models, which found that no practical improvement
can be achieved with entangled states. Finally, we note
a few interesting questions raised by our work. First, it
would be interesting to see if there exists special quantum
states of atoms and feedback mechanisms which optimize
the performance of the clock. Second, the present results
highlight that it is essential to have a realistic model of
the noise (and possible stabilization mechanism) present
in specific realizations of quantum information protocols.
Although the protocol considered in this Letter exploits
entanglement to stabilize a classical system (the local os-
cillator), it would be interesting to study how similar
considerations (e.g.1/f noise and collective decoher-
ence) affect protocols such as quantum error correction
codes [1], which use entanglement to stabilize a quantum
system and protect it from decoherence.
We are grateful to David Phillips, Ron Walsworth, and
David Wineland for useful discussions and comments on
the manuscript. This work was supported by the NSF
through its CAREER award and the grant to ITAMP, by
the Packard and Sloan Foundations, and by the Danish
Natural Science Research Council.
[1] D. Bouwmeester, A. Ekert, and A. Zeilinger (eds.), The
Physics of Quantum Information, (Springer, New York,
2000).
[2] C. W. Gardiner and P. Zoller, Quantum Noise, (Springer,
New York, 2000).
[3] W. H. Itano et al., Phys. Rev. A 47, 3554 (1993).
[4] J. J. Bollinger et al., Phys. Rev. A 54, R4649 (1996).
[5] D. J. Wineland et al., Phys. Rev. A 50, 67 (1994).
[6] J. M. Geremia et al., Phys. Rev. Lett. 91, 250801 (2003).
[7] L. K. Thomsen, S. Mancini, and H. M. Wiseman, Phys.
Rev. A 65, 061801 (2002).
[8] N. F. Ramsey, Molecular Beams, (Oxford University
Press, London 1956).
[9] In practice, T may be limited by various mechanisms,
e.g., in atomic fountain clocks, T is limited by the avail-
able distance the atoms may fall under gravity [10].
[10] G. Santarelli et al., Phys. Rev. Lett. 82, 4619 (1999).
[11] M. Takamoto and H. Katori, Phys. Rev. Lett. 91, 223001
(2003).
[12] R. J. Rafac et al., Phys. Rev. Lett. 85, 2462 (2000).
[13] M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993).
[14] A. Sørensen, L.-M. Duan, J. I. Cirac, and P. Zoller, Na-
ture 409, 63 (2001).
[15] V. Meyer et al., Phys. Rev. Lett. 86, 5870 (2001).
[16] S. F. Huelga et al., Phys. Rev. Lett. 79, 3865 (1997).
[17] D. J. Wineland el al., J. Res. Natl. Inst. Stan. Technol.
103, 259 (1998).
[18] A. Andr´ e, A. S. Sørensen, and M. D. Lukin, (in prepara-
tion).
[19] J. A. Barnes et al., IEEE Trans. Instrum. Meas. 20, 105
(1971).
[20] D. Ulam-Orgikh and M. Kitagawa, Phys. Rev. A 64,
052106 (2001).
[21] D. J. Berkeland et al., Phys. Rev. Lett. 80, 2089 (1998).
[22] S. A. Diddams et al., Science 293, 825 (2001).
[23] C. Audoin et al., IEEE Trans. Ultrason., Ferroelect.,
Freq. Contr. 45, 877 (1998); C. A. Greenhall, ibid. 45,
895 (1998).