A Single Trapped Ion as a Time-Dependent Harmonic Oscillator
A Single Trapped Ion as a Time-Dependent Harmonic Oscillator
Nicolas C. Menicucci1, 2, ∗ and G. J. Milburn2
Department of Physics, Princeton University, Princeton, NJ 08544, USA
School of Physical Sciences, The University of Queensland, Brisbane, Queensland 4072, Australia
(Dated: November 4, 2018)
We show how a single trapped ion may be used to test a variety of important physical mod-
els realized as time-dependent harmonic oscillators. The ion itself functions as its own motional
detector through laser-induced electronic transitions. Alsing et al. [Phys. Rev. Lett. 94, 220401
(2005)] proposed that an exponentially decaying trap frequency could be used to simulate (thermal)
Gibbons-Hawking radiation in an expanding universe, but the Hamiltonian used was incorrect. We
apply our general solution to this experimental proposal, correcting the result for a single ion and
showing that while the actual spectrum is different from the Gibbons-Hawking case, it nevertheless
shares an important experimental signature with this result.
PACS numbers: 03.65.-w, 32.80.Pj
I. INTRODUCTION
The time-dependent quantum harmonic oscillator has
long served as a paradigm for nonadiabatic time-
dependent Hamiltonian systems and has been applied to
a wide range of physical problems by choosing the mass,
the frequency, or both, to be time-dependent. The ear-
liest application is to squeezed state generation in quan-
tum optics [1, 2, 3], in which the effect of a second-order
optical nonlinearity on a single-mode field can be mod-
eled by a harmonic oscillator with a frequency that is
harmonically modulated at twice the bare oscillator fre-
quency. It was subsequently shown that any modulation
of the frequency could produce squeezing [4], and thus the
same model could be used to approximately describe the
generation of photons in a cavity with a time-dependent
boundary [5, 6].
The model has been used in a number of quantum
cosmological models. In Ref. [7], a time-dependent fre-
quency has been used to explain entropy production in a
quantum mini-superspace model. The model, with both
mass and frequency time-dependent, has been particu-
larly important in developing an understanding of how
quantum fluctuations in a scalar field can drive classical
metric fluctuation during inflation [8, 9]. In a cosmo-
logical setting the time-dependence is not harmonic and
is usually exponential. In all physical applications, of
course, the model is only an approximation to the true
physics, and its validity can be tested only with consid-
erable difficulty, especially in the cosmological setting.
Here we propose a realistic experimental context in which
the time-dependent quantum harmonic oscillator can be
studied directly.
Many decades of effort to refine spectroscopic measure-
ments for time standards now enable a single ion to be
confined in three dimensions, its vibrational motion re-
stricted effectively to one dimension, and the ion cooled
∗Electronic address: nmen@princeton.edu
to the vibrational ground state with a probability greater
than 99% [10]. Laser cooling is based on the ability to
couple an internal electronic transition to the vibrational
motion of the ion [11]. These methods can easily be ex-
tended to more than one ion and their collective normal
modes of vibration [12]. Indeed so carefully can the cou-
pling between the electronic and vibrational states be
engineered that is is possible to realise simple quantum
information processing tasks [13, 14]. We use the control
of trapping potential afforded by ion traps, together with
the ability to reach quantum limited motion, to propose
a simple experimental test of quantum harmonic oscilla-
tors with time-dependent frequencies. We also make use
of the ability to make highly efficient quantum measure-
ments, based on fluorescent shelving [10], to propose a
practical means to test our predictions.
In this paper, we calculate the excitation probability
of a trapped ion in a general time-dependent potential.
When beginning in the vibrational ground state of the
unchirped trap and starting the chirping process adia-
batically, the excitation probability is simply related to
the Fourier transform of the solution of the Heisenberg
equations of motion (which is also the same as the trajec-
tory of the equivalent classical oscillator). We compare
our result with that of Ref. [15] for the case of a single
ion undergoing an exponential frequency chirp. The cited
work attempts to use this experimental setup to model
a massless scalar field during an inflating (i.e., de Sitter)
universe, which would give a thermal excitation spectrum
as a function of the detector response frequency [16]. The
analysis is incorrect, however, because the wrong Hamil-
tonian was used. Nevertheless, the corrected calculation
presented here also gives an excitation spectrum with
a thermal signature, although the particular functional
form is different.
II. GENERAL SOLUTION
The quantum Hamiltonian for a single ion in a time-
dependent harmonic trap can be well-approximated in
http://arxiv.org/abs/0704.0135v2
mailto:nmen@princeton.edu
one dimension by
ν(t)2q2 , (1)
where ν(t) is time-dependent but always assumed to
be much slower than the timescale of the micromo-
tion [10]. For emphasis, we have indicated the explicit
time-dependence of the frequency ν; we will often omit
this from now on. Working in the Heisenberg picture, we
get the following equations of motion for q and p:
, (2)
ṗ = −Mν2q . (3)
Dots indicate total derivatives with respect to time. Dif-
ferentiating again and plugging in these results gives
0 = q̈ + ν2q , (4)
0 = p̈− 2
ṗ+ ν2p . (5)
As we shall see, only Eq. (4) is necessary for calculating
excitation probabilities, so we will focus only on it. These
equations are operator equations, but they are identical
to the classical equations of motion for the analogous
classical system. Interpreting them as such, we will la-
bel the two linearly independent c-number solutions as
h(t) and g(t), where the following initial conditions are
satisfied:
h(0) = ġ(0) = 1 and ḣ(0) = g(0) = 0 , (6)
Writing q(0) = q0 and p(0) = p0, the unique solution for
q to the initial value problem above is
q(t) = q0h(t) +
g(t) . (7)
By differentiating and using the relations above, we know
also that
p(t) = Mq0ḣ(t) + p0ġ(t) . (8)
To check our math, we can verify that [q(t), p(t)] = i~,
which is fulfilled if and only if the Wronskian W (h, g) of
the two solutions is one for all times—specifically,
hġ − ḣg = 1 , (9)
where we have assumed that [q0, p0] = i~.
Moreover, if the initial state at t = 0 is symmetric with
respect to phase-space rotations, then we have additional
rotational freedom in choosing the initial quadratures.
(This would be the case, for instance, if we start in the
instantaneous ground state.) Notice that Eq. (7) can be
written as the inner product of two vectors:
q(t) =
h(t), ν0g(t)
(and similarly for Eq. (8)), where we have normalized
the quadrature operators to have the same units. As an
inner product, this expression is invariant under simulta-
neous rotations of both vectors. Thus, if the initial state
possesses rotational symmetry in the phase plane, then
the rotated quadratures are equally as valid as the orig-
inal ones for representing the initial state, which means
that an arbitrary rotation can be applied to the second
vector above without changing any measurable property
of the system. This freedom can be used, for instance, to
define new functions h′(t) and g′(t) that are more con-
venient for calculations, where the linear transformation
between them and the original ones (with prefactors as
in Eq. (10)) is a rotation. We will use this freedom in the
next section.
One reason why ion traps have become a leading im-
plementation for quantum information processing is the
ability to efficiently read out the internal electronic state
using a fluorescence shelving scheme [10]. As the internal
state can become correlated with the vibrational motion
of the ion, this scheme can be configured as a way to
measure the vibrational state directly [17]. To correlate
the internal electronic state with the motion of the ion,
an external laser can be used to drive an electronic tran-
sition between two levels |g〉 and |e〉, separated in energy
by ~ωA. The interaction between an external classical
laser field and the ion is described, in the dipole and
rotating-wave approximation, by the interaction-picture
Hamiltonian [10]
HL = −i~Ω0
σ+(t)e
ik cos θq(t) − σ−(t)e
−ik cos θq(t)
where Ω0 is the Rabi frequency for the laser-atom inter-
action, ωL is the laser frequency, k is the magnitude of
the wave vector ~k, which makes an angle θ with the trap
axis, q(t) is given in Eq. (7), and
σ±(t) = e
±i∆tσ± . (12)
The electronic-state raising and lowering operators are
defined as σ+ = |e〉〈g| and σ− = |g〉〈e|, respectively, and
∆ = ωA − ωL (13)
is the detuning of the laser below the atomic transi-
tion. We can construct a meaningful quantity that char-
acterizes the “size” of q(t) based on the width of the
ground-state wave packet for an oscillator with frequency
ν(t), namely
~/2Mν(t). As long as this quantity is
much smaller than k cos θ throughout the chirping pro-
cess, then we can expand the exponentials in Eq. (11) to
first order and define the interaction Hamiltonian HI be-
tween the electronic states and vibrational motion (still
in the interaction picture) by
HI = ~Ω0k cos θq(t)
e−i∆tσ− + e
+i∆tσ+
. (14)
where we have assumed that ωL is far off-resonance, and
thus ∆ 6≃ 0.
Using first-order time-dependent perturbation theory,
the probability to find the ion in the excited state is
P (1) =
dt2 〈HI(t1)PeHI(t2)〉
= Ω20k
2 cos2 θ
dt2 e
−i∆(t1−t2) 〈q(t1)q(t2)〉 ,
where Pe = 1vib ⊗ |e〉〈e| is the projector onto the ex-
cited electronic state (and the identity on the vibrational
subspace). We always assume that the ion begins in the
electronic ground state. If the ion also starts out in the
instantaneous vibrational ground state for a static trap
of frequency ν0 = ν(0) at t = 0 (which is most useful
when the chirping begins in the adiabatic regime), then
we can evaluate the two-time correlation function as
〈q(t1)q(t2)〉ground =
h(t1)h(t2) +
g(t1)g(t2)
〈q0p0〉
h(t1)g(t2)− h(t2)g(t1)
h(t1)− iν0g(t1)
h(t2) + iν0g(t2)
f(t1)f
∗(t2) , (16)
where we have used the facts that for the vibrational
ground state,
(p0/Mν0)
= ~/2Mν0 and
〈q0p0〉 =
〈{q0, p0}+ [q0, p0]〉 = i~/2, and we have de-
fined the complex function
f(t) = h(t)− iν0g(t) , (17)
which is the solution to Eq. (4) with initial the conditions,
f(0) = 1 and ḟ(0) = −iν0. Plugging this into Eq. (15)
gives, quite simply,
P (1) → (Ω0η0)
2 |F̧|
, (18)
where
dt e−i∆tf(t) , (19)
and we have defined the unitless, time-dependent Lamb-
Dicke parameter [10] as
η(t) =
~k2 cos2 θ
2Mν(t)
, (20)
and η0 = η(0). Recalling that f(t) can be considered a
complex c-number solution to the equations of motion for
the equivalent classical Hamiltonian, Eq. (18) shows that
the excitation probability is simply related to the Fourier
transform of the classical trajectories when beginning in
the vibrational ground state.
III. EXPONENTIAL CHIRPING
Recent work [15] has suggested that an exponen-
tially decaying trap frequency has the same effect on
the phonon modes of a string of ions as an expand-
ing (i.e., de Sitter) spacetime does on a one-dimensional
scalar field [18]. An inertial detector that responds to
such an expanding scalar field would register a thermal
bath of particles, called Gibbons-Hawking radiation [16].
Ref. [15] suggests that the acoustic analog [19] of this
radiation could be seen in an ion trap, causing each ion
to be excited with a thermal spectrum with temperature
~κ/2πkB, as a function of the detuning ∆, where κ is
the trap-frequency decay rate. The analysis used an in-
correct Hamiltonian that neglected squeezing and source
terms that have no analog in the expanding scalar field
model but which are present when considering trapped
ions in this way, and the results are incorrect. In this sec-
tion, we revisit this problem and calculate the excitation
probability for a single ion in an exponentially decaying
harmonic potential, as a function of the detuning ∆.
We write the time-dependent frequency as [20]
ν(t) = ν0e
−κt . (21)
This results in
q̈ + ν20e
−2κtq = 0 . (22)
Solutions with initial conditions (6) are
h(t) =
, (23)
g(t) =
where the time dependence is carried in ν = ν(t) from
Eq. (21), and Jn and Yn are Bessel functions. We could
plug these directly into the formulas from the last section,
but we will simplify the calculations by considering the
limits of slow and long-time frequency decay, represented
ν0 ≫ κ and ν0e
−κT ≪ κ , (25)
respectively. This allows us to do several things. First, it
allows us to use the usual ground state of the unchirped
trap at frequency ν0 as a good approximation to the
ground state of the expanding trap at t = 0, since at that
time the system is being chirped adiabatically. This is
important because it allows the experiment to begin with
a static potential, which is useful for cooling. Second, it
allows us to simplify h(t) and g(t) using the phase-space
rotation freedom discussed above. Using asymptotic ap-
proximations for the Bessel functions in the coefficients,
≃ −Y1
, (26)
, (27)
we get
h(t) ≃
sinϕY0
+ cosϕJ0
, (28)
ν0g(t) ≃
− cosϕY0
+ sinϕJ0
. (29)
where ϕ = ν0/κ − π/4. Since we are taking the initial
state to be the ground state, which is symmetric with
respect to phase-space rotations, we can use the freedom
discussed in the previous section to undo the rotation
represented by Eqs. (28) and (29) and define the simpler
functions
h(t) → h′(t) =
, (30)
g(t) → g′(t) =
. (31)
The primes are unnecessary due to the symmetry of the
initial state, so we drop them from now on and plug di-
rectly into Eq. (17):
f(t) =
− iJ0
, (32)
where H
n is a Hankel function of the first kind. The
integral in Eq. (19) can be evaluated in the limits (25)
using techniques similar to those used in Ref. [15]. First,
define
, τ = α− κt , u = eτ , and x = ∆/κ .
The integral in question then becomes (neglecting the
prefactor)
dt e−i∆tH
dt e−i∆tH
α−κt)
dτ e−ix(α−τ)H
e−ixα
dτ eixτH
e−ixα
du uix−1H
0 (u) . (34)
Inserting a convergence factor with x → x− iǫ, and then
taking the limit ǫ → 0+, we can use the formula
du uix−1H
0 (u) = −2
ix Γ(ix/2)
(eπx − 1)Γ(1− ix/2)
to evaluate
Γ(ix/2)
Γ(1− ix/2)
(eπx − 1)2
(eπx − 1)2
. (36)
When plugging in for the dummy variables (33), this
gives
P (1) = (Ω0η0)
2 2πν0
(eπ∆/κ − 1)2
. (37)
The calculated result from Ref. [15] for a single ion is
GH = (Ω0η0)
e2π∆/κ − 1
, (38)
which contains a Planck factor with Gibbons-
Hawking [16] temperature T = ~κ/2πkB but is
different from the actual result for a single ion, given by
Eq. (37).
Several things should be noted about these functions.
First, they both break down as ∆ → 0 because of the ap-
proximation made in obtaining Eq. (14). They also fail
if the time-dependent Lamb-Dicke parameter (20) ever
becomes too large throughout the chirping process. Fur-
thermore, most cases of interest will be ∆ ≃ ν0 (the first
red sideband) and near ∆ ≃ −ν0 (the first blue side-
band), which means that |∆| ≫ κ, since ν0 ≫ κ. The
first red sideband represents a detector that requires the
absorption of one phonon (plus one laser photon) in order
to excite the atom—the usual thing we mean by “particle
detector” when the particles are phonons. The first blue
sideband, on the other hand, represents a detector that
emits a phonon in order to excite the atom (along with
absorbing one laser photon).
There are a couple of ways to compare these functions.
First, we can take the ratio of the two for both the red-
and blue-sideband cases. In both cases, we obtain
P (1)
(1 + 2e−π|∆|/κ) (39)
plus terms of order O(e−2π|∆|/κ). Since |∆| ≃ ν0, the
prefactor is close to one, and the second term is very
small (since ν0 ≫ κ). Furthermore, it is cumbersome
to directly compare the measured probability to the full
function (with all the prefactors). It is often easier in-
stead to make measurements on both the first red side-
band and the first blue sideband and then take the ratio
of the two. The constant prefactors disappear in this
calculation, and both functions then have the same ex-
perimental signature:
P (1)(∆)
P (1)(−∆)
GH(∆)
GH(−∆)
= e−2π∆/κ , (40)
which is that of a thermal distribution with tempera-
ture T = ~κ/2πkB, which is of the Gibbons-Hawking
form [16] with the expansion rate given by κ. There-
fore, although the Hamiltonian used in the calculations
in Ref. [15] was missing terms, the intuition (at least for
a single ion) was correct in that the actual experimental
signature in this case matches that of an ion undergoing
thermal motion in a static trap, where the temperature
is proportional to κ.
To see whether this experiment is feasible, we must ex-
amine the validity of our approximations. For a typical
trap, we expect that ν0 ≃ 1 MHz, and thus if we take
κ ≃ 1 kHZ, we easily satisfy the first of conditions (25),
namely ν0 ≫ κ. The second of these conditions gives a
constraint on the modulation time T . For these param-
eters we expect that T ≃ a few msec. This is compat-
ible with typical cooling and readout time scales and is
less than those for heating due to fluctuating patch po-
tentials [10]. Thus, this is a realizable experiment with
current technology.
IV. CONCLUSION
We have shown that a single trapped ion in a modu-
lated trapping potential can serve as an experimentally
accessible implementation of a quantum harmonic oscilla-
tor with time-dependent frequency, including robust con-
trol over state preparation, manipulation, and measure-
ment. The ion itself serves both as the oscillating particle
and as the local detector of vibrational motion via cou-
pling to internal electronic states by an external laser.
For the case of a general time-dependent trap frequency,
we calculated the first-order excitation probability for the
ion in terms of the solution to the classical equations of
motion for the equivalent classical oscillator. We applied
this general result to the case of exponential chirping and
corrected the calculation in Ref. [15] for a single ion. We
found that while the results from the two calculations dif-
fer, the experimental signature in both cases is the same
and equivalent to that of a thermal ion in a static trap.
We thank Dave Kielpinski for invaluable help with the
experimental details. We also thank Paul Alsing, Bill
Unruh, John Preskill, Jeff Kimble, Greg Ver Steeg, and
Michael Nielsen for useful discussions and suggestions.
NCM extends much appreciation to the faculty and staff
of the Caltech Institute for Quantum Information for
their hospitality during his visit, which helped bring this
work to fruition. NCM was supported by the United
States Department of Defense, and GJM acknowledges
support from the Australian Research Council.
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