Coherently displaced oscillator quantum states of a single trapped atom

Abstract

Coherently displaced harmonic oscillator number states of a harmonically bound ion can be coupled to two internal states of the ion by a laser-induced motional sideband interaction. The internal states can subsequently be read out in a projective measurement via state-dependent fluorescence, with near-unit fidelity. This leads to a rich set of line shapes when recording the internal-state excitation probability after a sideband excitation, as a function of the frequency detuning of the displacement drive with respect to the ion's motional frequency. We precisely characterize the coherent displacement based on the resulting line shapes, which exhibit sharp features that are useful for oscillator frequency determination from the single quantum regime up to very large coherent states with average occupation numbers of several hundred. We also introduce a technique based on multiple coherent displacements and free precession for characterizing noise on the trapping potential in the frequency range of 500 Hz to 400 kHz. Signals from the ion are directly used to find and eliminate sources of technical noise in this typically unaccessed part of the spectrum.

Full text

Coherently displaced oscillator quantum states of a
single trapped atom
Katherine C. McCormick1,2, Jonas Keller1,2, David J.
Wineland1,2,3, Andrew C. Wilson1, Dietrich Leibfried1
1National Institute of Standards and Technology, Boulder, CO 80305, USA
2University of Colorado, Department of Physics, Boulder, CO 80305, USA
3University of Oregon, Department of Physics, Eugene, OR 97403, USA
Abstract. Coherently displaced harmonic oscillator number states of a harmonically
bound ion can be coupled to two internal states of the ion by a laser-induced motional
sideband interaction. The internal states can subsequently be read out in a projective
measurement via state-dependent fluorescence, with near-unit fidelity. This leads to a
rich set of line shapes when recording the internal-state excitation probability after a
sideband excitation, as a function of the frequency detuning of the displacement drive
with respect to the ion’s motional frequency. We precisely characterize the coherent
displacement based on the resulting line shapes, which exhibit sharp features that are
useful for oscillator frequency determination from the single quantum regime up to very
large coherent states with average occupation numbers of several hundred. We also
introduce a technique based on multiple coherent displacements and free precession for
characterizing noise on the trapping potential in the frequency range of 500 Hz to 400
kHz. Signals from the ion are directly used to find and eliminate sources of technical
noise in this typically unaccessed part of the spectrum.
arXiv:1811.00668v1 [physics.atom-ph] 1 Nov 2018

Coherently displaced oscillator quantum states of a single trapped atom 2
1. Introduction
As this focus issue on Quantum Optomechanics showcases, the manipulation of quan-
tum states of a harmonic oscillator (HO) is a theme of current interest across a wide
range of experimental platforms. Often the methods developed on one platform, in
our case a single, harmonically bound atomic ion, can be adapted to many other plat-
forms, after suitable modifications of the original procedures. An important example
is resolved sideband-cooling of micro-fabricated oscillators, theoretically described in
[1, 2], that bears strong analogies to the methods that were first developed for single-ion
mechanical oscillator systems [3, 4, 5]. In this contribution to the focus issue, ground-
state cooling is combined with another basic idea that is widely applicable across HO
platforms, namely coherent displacements that can be conveniently implemented with
a classical force that is near resonant with the HO [6]. The final ingredient here is a
suitable two-level system, in our case two internal electronic states of a single ion, that
can be coupled to the HO motion and read out with a projective measurement to gain
information about the state of the HO. A superconducting qubit is just one example
of an effective two-level system that has been coupled to a co-located micro-fabricated
HO [7]. Therefore, the methods described in this contribution might also be adaptable
and useful to the rapidly growing community that studies HO systems in the quantum
regime.
Exciting the HO motion of trapped charged particles with a weak oscillating electric
field, a method often called a “tickle,” has long been used to determine motional fre-
quencies and subsequently the charge-to-mass ratio in various ion trap based devices, for
example in ion-trap mass spectrometers [8, 9]. The response of the ions can be detected
by counting resonantly ejected particles, by resonance absorption of the driving field
[10], or through image currents in the trap electrodes [11]. For certain atomic ions, it
is possible to detect the ion motion through changes of scattered light that depends on
velocity via the Doppler effect [12]. An important practical advantage of a tickle is that
it only interacts with the charge of the ion, therefore it is immune to magnetic field or
AC-Stark shifts that may restrict how well the HO frequency can be determined spec-
troscopically, for example by resolving the motional sidebands of internal transitions
of ions [13]. Many other implementations of HO systems have analogous mechanisms
available, for example excitation of a micro-fabricated resonator by driving it with a
piezo-electric element or with a capacitively coupled electric circuit.
The tickle method can be further refined with atomic ions that are cooled close to
the ground state of their motion and can be coupled to a two-level system through
resolved sideband transitions [4, 13]. Near the ground state of motion, the probability
of driving a “red sideband” transition, where the internal state change of the ion is
accompanied by reducing the number of quanta in the motional state |ni → |n−1i,
is strongly suppressed and can be used to determine the average harmonic oscillator

Coherently displaced oscillator quantum states of a single trapped atom 3
occupation number ¯n[5, 13]. Starting near the ground state, a resonant tickle can add
quanta of motion such that the red sideband can be driven again, as discussed qualita-
tively in [14]. For weak excitation, ¯n≤1, we observe responses close to the Fourier limit
of the tickle pulse, as we will describe in more detail and experimentally demonstrate.
If the tickle excitation acts longer or with a larger amplitude, an ion in the ground state
can be displaced to coherent states with an average HO occupation number ¯n1.
The Rabi frequency of sideband transitions depends non-linearly on ¯n, which leads to
collapse and revival of internal state changes that are one of the hallmarks of the Jaynes-
Cummings model [15, 16, 17]. Here we examine the probability of changing the internal
state theoretically and experimentally, as a function of tickle detuning relative to the
frequency of a HO motional mode of the ion. When probing the red sideband transition
after displacing to ¯n1, we observe rich sets of features with steep and narrow side
lobes around the resonance center. Such nonlinear responses can in principle be used to
find the frequency of the motion with better signal-to-noise ratio than what the Fourier
limit implies for smaller coherent states where ¯n≤1 and the response of the ion is
essentially linear.
A sequence of coherent displacements alternating with free evolution of the motion,
inspired by spin-echos [18] and dynamical decoupling [19, 20], can be used to obtain a
frequency-filtered response of the ion. We implement and characterize such sequences
by observing and modeling the ion response to deliberately applied, monochromatic
modulations to the trapping potential curvature. Similar sequences can then be used
without applied modulations. In this case, the response of the ion can be attributed to
HO frequency noise that is intrinsic to our system, allowing us to characterize noise on
the trap potential in a frequency range of 500 Hz to 400 kHz, a wide frequency range
that has not been studied in detail in previous work [21]. With this method, several
narrow band technical noise components (spurs) in our setup could be identified through
the direct response of the ion. The noise was traced back to digital-to-analog converters
(DACs) used in our setup and eliminated by replacing them with analog bench power
supplies.
2. States and ion fluorescence signals from coherent displacements
We consider a single ion with charge qand mass mconfined in a harmonic trapping
potential with minimum position at r0, such that the motion of the ion can be described
by three normal HO modes with frequencies ωx≤ωy≤ωz. By using a coordinate system
where the axis directions coincide with the normal mode directions, we can write the ion
position as r=r0+δr. The interaction of the ion with an additional uniform electric
field Ecan be described as
HE=q(E·δr).(1)

Coherently displaced oscillator quantum states of a single trapped atom 4
For the HO in the x-direction we introduce ladder operators ˆaand ˆa†to write the energy
as
H0= ¯hωxˆa†ˆa. (2)
We have suppressed the ground state energy since it is a constant term that does not
change the dynamics. We replace δx by its equivalent quantum mechanical operator
δˆx=x0(ˆa†+ ˆa) with x0=q¯h/(2mωx) the ground state extent of the oscillator. For
the normal mode in this direction, and in the interaction picture relative to H0the
interaction with an oscillating electric field Ex(t) = E0cos(ωt +φ) becomes
HI= ¯hΩx(ˆa†eiωxt+ ˆae−iωxt)(ei(ωt+φ)+e−i(ω t+φ))
= ¯hΩx(ˆa†e−i(δt+φ)+ ˆaei(δt+φ)+ ˆa†ei(σ t+φ)+ ˆae−i(σt+φ)),(3)
with the coupling Ωx=qE0x0/(2¯h) and δ=ω−ωx,σ=ωx+ω. If the oscillating field is
close to resonance with the normal mode, |δ|  σ, the faster-rotating terms containing
σcan be neglected to a good approximation and the interaction takes the form of a
coherent drive detuned by δ
HI'¯hΩx(ˆa†e−i(δt+φ)+ ˆaei(δt+φ)).(4)
We can formally integrate the equation of motion for HI[6, 22, 23] to connect an initial
state |Ψ(0)iat t= 0 when the electric field is switched on to the coherently displaced
state after evolution for duration t,|Ψ(t)i
|Ψ(t)i=ˆ
D(α(t))eiΦ(t)|Ψ(0)i,(5)
where
ˆ
D(α) = exp(αˆa†+α∗ˆa),
α(t) = Ωxeiφ Zt
0eiδτ dτ =iΩxeiφ 1−eiδt
δ,
Φ(t) = Im Zt
0α(τ){∂τα∗(τ)}dτ=Ωx
δ2
[sin(δt)−δt].(6)
The phase Φ(t) can play an important role, for example, in two-qubit gates [23, 24] or
interferometric experiments that combine internal degrees of freedom of the ion with
motional states [25, 26]. Here, we will be interested only in the average occupation
number ¯n=hΨ(t)|ˆa†ˆa|Ψ(t)i, which does not depend on Φ(t). If the initial state is the
harmonic oscillator ground state, |Ψ(0)i=|0ithe average occupation is
¯n(t) = |α(t)|2= 2 Ωx
δ2
[1 −cos(δt)].(7)
On resonance (δ= 0) the coherent state amplitude grows linearly in tas α(t) = eiφΩxt
and the energy of the oscillator quadratically as ¯n(t)=Ω2
xt2. For a coherent state, the
probability distribution over number states |miis a Poisson distribution with average ¯n
P(0)
m=¯nme−¯n
m!.(8)

Coherently displaced oscillator quantum states of a single trapped atom 5
An initial number state with |Ψ(0)i=|ni, displaced by ˆ
D(αd), results in a more involved
probability distribution [6]
P(n)
m= ¯n|n−m|e−¯nn<!
n>!(L|n−m|
n<(¯n))2,(9)
where ¯n=|αd|2,n<(n>) is the lesser (greater) of the integers nand mand La
n(x) is a
generalized Laguerre polynomial.
In our experiments, the HO motion is coupled by laser fields to a two-level system
with states labelled |↓i and |↑i with energy difference E↑−E↓= ¯hω0>0. The internal
state is initialized in |↓i by optical pumping. After the state of motion is prepared,
the state |↓i |Ψ(t)ican be driven on a red sideband, resulting in population transfer
|↓i |mi ↔ |↑i |m−1ifor all m > 0 while the state |↓i |0iis unaffected [13]. The Rabi
frequencies depend on m > 0 as
Ωm,m−1= Ω0e−η2/2ηs1
mL1
m−1(η2),(10)
where Ω0is the Rabi frequency of a carrier transition |↓i ↔ |↑i of an atom at
rest, η=kxx0is the Lamb-Dicke parameter, with kxthe component of the effective
wavevector along the direction of oscillation and Lβ
m(x) is a generalized Laguerre
polynomial [13]. After driving the red sideband of state |↓i |Ψ(t)ifor duration τ, the
probability of having flipped the internal state to |↑i is
P↑(τ) = 1
2"1−P(n)
0−
∞
X
m=1
P(n)
mcos(2Ωm.m−1τ)#.(11)
We calibrate the red sideband drive time to be equivalent to a π-pulse on the |↓i |1i ↔
|↑i |0itransition, which implies 2Ω1,0τ=π. For an arbitrary displaced number state
the probability of the ion to be in |↑i becomes
Pπ
↑=1
2"1−P(n)
0−
∞
X
m=1
P(n)
mcos(πΩm.m−1
Ω1,0
)#.(12)
This probability is not a monotonic function of ¯nand exhibits maxima and minima
as the displacement changes. Experimental observations of this behavior for displaced
number states and comparisons to the predictions of Eq. (12) will be discussed in section
4.2. When the detuning in Eq. (7) is δ6= 0, the coherent drive displaces |Ψ(0)ialong
circular trajectories in phase space that can turn back onto themselves. Every time
δ t =mπ with ma non-zero integer, |α(t)|will reach a maximum of 2Ωx/δ for modd
and return to zero for meven. The non-monotonic behavior of Pπ
↑with respect to
¯ncreates feature-rich lineshapes when this probability is probed as a function of the
displacement detuning δrelative to the HO frequency.
3. Noise sensing with motion-echo sequences
The motion displacements discussed above enable sensitive tests of the ion’s motional
coherence. Intrinsic noise at the motional frequencies heats the ion out of the ground

Coherently displaced oscillator quantum states of a single trapped atom 6
state, and is observed in all traps at a level that often exceeds resistive heating by
orders of magnitude. This “anomalous heating,” is well documented [27, 28], but the
sources are not well understood. On much longer time scales than the ion-oscillation
period, motional frequencies are known to drift over minutes and hours due to various
causes, for example slow changes in stray electric fields and drifts of the sources that
provide the potentials applied to the trap electrodes. Much less work has been done to
characterize noise in the frequency range in between the HO frequency and slow drift
[21]. Here, we construct sequences of coherent displacements that alternate with periods
of free evolution and suppress the sensitivity to slow drifts of the harmonic oscillator
frequency. This allows us to isolate noise in a specific frequency band, in analogy to an
AC-coupled electronic spectrum analyzer. Coherent displacements can be implemented
on time scales of order of a few µs, making this method suitable for detecting noise at
frequencies in a range of 500 Hz to 400 kHz in the experiments described here.
3.1. Basic principle
The sequences of coherent displacements discussed here are closely related to spin-echo
experiments and dynamical decoupling in two-level systems [18, 19, 20]. In analogy to
the classic
(π/2-pulse)-τa-(π-pulse)-τa-(π/2-pulse)
spin-echo sequence [18] with τathe duration of a free-precession period, the ideal
“motion-echo”pulse sequence consists of
ˆ
D(Ωxτd/2)-τa-ˆ
D(−Ωxτd)-τa-ˆ
D(Ωxτd/2),
where the minus sign in the argument of the second displacement indicates that the
phase φof the displacement drive has changed by πrelative to the other displacement
operations. To simplify this initial discussion, we assume that all displacements are
instantaneous and not affected by fluctuations in the oscillator frequency. This condition
is similar to the “hard-pulse” limit for spin-echo sequences. If the frequency of the
oscillating electric field in Eq. (1), which we call the “local oscillator frequency” in this
context, is on resonance with the HO frequency, the displacements in the sequence add
up to zero, so any initial state is displaced back onto itself at the end of the sequence (see
Fig. 1 (a)). In analogy to a spin-echo sequence, if the local oscillator differs from the HO
frequency by a small, constant detuning δ2π/τd, the sequence will still result in a
final state that is very close to the initial state (see Fig. 1 (b)). However, if the detuning
changes sign between free-precession periods (see Fig. 1 (c)), the final state will not
return to the initial position and in general information about the oscillator frequency
fluctuations can be gained from the final displacement. This basic echo sequence can
be expanded by including additional blocks of the form
τa-ˆ
D(Ωxτd)-τa-ˆ
D(−Ωxτd)
after the first displacement ˆ
D(−Ωxτd) in analogy to dynamical decoupling sequences in
two-level systems. Ideally, this increases the number of free-precession sampling windows
which leads to a longer sampling time and narrower filter bandwidth of the extended

Coherently displaced oscillator quantum states of a single trapped atom 7
1
3
5
1
3
5
1
3
5
a) b) c)
4
2
4
2
4
2
Figure 1. Schematic phase-space sketch of the displacements in the simplest
motion-echo sequence. Here, all displacements are assumed to act instantaneously
(hard-pulse limit), such that the effect of oscillator detuning during displacements
can be neglected. (a) Without fluctuations of the oscillator frequency, the ground-
state minimum uncertainty disk (green) is coherently displaced by Ωxτd/2 (step 1),
then remains stationary during a free-precession period (step 2), it is then displaced
symmetrically through the origin by −Ωxτd(step 3), followed by another free-
precession period (step 4). The final displacement by Ωxτd/2 (step 5) returns the
state to the origin. (b) With a small, constant detuning, the state drifts perpendicular
to the direction of the first displacement in step 2. However, it drifts an equal amount
in the opposite direction during step 4, to still end up in the ground state after step 5.
This immunity to constant detuning can be thought of as a HO analogy to a spin-echo
sequence in a spin-1/2 system. (c) If the detuning changes sign between steps 2 and
4, the state does not return to the origin. In all three cases (a)-(c), the final state
reflects the sum of additional displacements during the operation of the sequence that
are caused by time-dependent changes in HO detuning.
sequence, while still producing no total displacement from the initial state if the HO is
stable, even if the local oscillator is slightly detuned from the HO resonance.
3.2. Effects of oscillator frequency fluctuations
If the local oscillator frequency is not on resonance with the HO, or if the detuning is
not constant in time, a realistic coherent drive (not assuming the hard-pulse limit) will
not always displace the state of motion along a straight line. The differential equation
describing the coherent displacement α(t) as a function of time in that generalized case

Coherently displaced oscillator quantum states of a single trapped atom 8
is (from Eq. (6))
˙α(t) = αiδ(t)+Ωxeiφ,(13)
where δ(t) is the instantaneous detuning between the HO and the local oscillator at
time t. In the special case where δdoes not depend on time and α(0) = 0, the solution
is α(t) from Eq. (6). If there is noise on the trap frequency, δ(t) will fluctuate randomly
as a function of t. A general solution of Eq. (13), at time t0+τas it evolves from the
initial state α(t0) at time t0, can be formally written as
α(t0, τ ) = exp[iI1(t0, τ)][α(t0)+ΩxeiφI2(t0, τ )],(14)
with
I1(t0, τ ) = Zt0+τ
t0
δ(τ1)dτ1,
I2(t0, τ ) = Zt0+τ
t0
exp −iZτ2
t0
δ(τ1)dτ1dτ2.(15)
This form is useful for numerical calculations and can be explicitly solved for special
cases of δ(t). Motion-echo sequences are most useful if the accumulation of phase during
τis small, I1(t0, τ )2π. In such cases, we can expand the exponential functions in
Eqs. (14) and (15) to linear order and approximate
α(t0, τ )' {α(t0)+Ωxτeiφ}+i(α(t0)+Ωxτ eiφ)I1(t0, τ )
−iΩxeiφI3(t+ 0, τ ),
I3(t0, τ ) = Zt0+τ
t0Zτ2
t0
δ(τ1)dτ1dτ2.(16)
The different terms in Eq. (16) have straightforward interpretations: the term in curly
braces characterizes the displaced coherent state for no detuning, δ(t) = 0. Finite de-
tuning rotates this state around the origin in phase space and to lowest order this effect
is captured by the term proportional to I1(t0, τ ). The final term reflects the effect of the
detuning while the state is displaced by a coherent drive, which results in a correction
proportional to I3(t0, τ ). For a free-precession period, Ωx= 0, during displacement by
a coherent drive, Ωx6= 0.
In this linear approximation, it is straightforward to keep track of the displacements
and the corrections from δ(t)6= 0 when periods of driving and free precession are con-
catenated. Because corrections on earlier corrections are higher order than linear, the
correction from each period only acts on the zero-order displacement of any previous
period. This implies that the zero-order terms in curly brackets and the corrections
can be summed up separately for a sequence. In this way, we can calculate the total
zero-order displacement αnand first order correction ∆αnof a sequence with nsteps
starting at time t= 0 in state |α(0)i. For the k-th step starting at tk, the displacement
drive Rabi frequency is Ωx,k, the drive duration τk, and the phase φk. In the linear
approximation with α(0) = α0the sums are
αn=α0+
n
X
k=1
Ωx,kτkeiφk,

Coherently displaced oscillator quantum states of a single trapped atom 9
∆αn=i
n
X
k=1 nαk−1+ Ωx,kτkeiφkI1(tk, τk)−Ωx,k eiφkI3(tk, τk)o.(17)
3.3. Motion-echo sequences
We restrict ourselves to sequences with Nsteps acting on an initial ground state, α0= 0,
where the sum over all unperturbed displacements of a sequence is αN= 0. In this way,
the final state is equal to |∆αNiand directly reflects the effects of non-zero detuning.
Moreover, we can construct the displacements in such a way, that a constant detuning
δ6= 0, results in ∆αN= 0. This mimics the feature of spin-echo sequences that
small constant detunings have no effect on their final state. The motion-echo sequences
preserve this feature, if the linear approximation is valid, even when taking the effect
of the detuning onto the displacement operations into account. For constant δ, the
integrals I1(tk, τk) = δτkand I3(tk, τk) = 1/2δτ 2
kare independent of tkand the total
displacement simplifies to
∆αN=iδ
N
X
k=1
τk(αk−1+ 1/2 Ωx,keiφkτk).(18)
For the motion-echo sequences, Ωx,k = Ωxis the same for all displacements and the
coherent state parameter before each of the ˆ
D(±Ωxτd) operations is ∓Ωxτd/2. In this
case, the second contribution in the (...) braces is ±1/2Ωxτd, equal and opposite to the
initial state parameter, so all displacement terms in the sum Eq. (18) are equal to zero
individually, except for the first and last displacement which is ˆ
D(Ωxτd/2). However,
since α0= 0, these terms sum to iδτd/2(1/2Ωxτd/2−Ωxτd/2 + 1/2Ωxτd/2) = 0, which
leaves only the free precession terms to be considered. All sequences contain an even
number 2naof free-precession periods (na>0, integer), with half of them contributing
iδτaΩxτd/2 each and the other half −iδτaΩxτd/2, so, as previously noted (see Fig. 1)
these terms also sum to zero and ∆αN= 0 for a constant detuning.
3.4. Response to a monochromatic modulation
Next, we can determine the response to a monochromatic modulation at frequency
ωnof the form δn(t) = Ancos(ωnt+φn). On the one hand, the HO frequency can
be deliberately modulated in this way, which enables us to compare the response of
the motion-echo sequence to the theoretical expectation. On the other hand, some of
the frequency noise acting on the oscillator can be characterized as a noise spectrum
consisting of a sum of such modulation terms with distinct frequencies ωn,j , possibly
varying amplitudes An,j and random phases φn,j. In addition, the harmonic oscillator
may be affected by noise with a continuous spectrum, but we will restrict ourselves to
discrete, narrow-band noise spurs here. The noise spectrum can be characterized with
motion-echo sequences, if the response to a monochromatic modulation at ωnallows for
determination of that frequency within a band that depends on the resolution of the
sequence. The amplitude of the response ∆αNis proportional to the noise amplitude
and is zero when averaged over the random noise phase φn(denoted by h...i), but because

Coherently displaced oscillator quantum states of a single trapped atom 10
the final occupation ¯nfin is proportional to |αN|2, after integrating over φn, we get an
average final occupation
h¯nfini=1
2πZ2π
0|∆αN|2dφn.(19)
This is proportional to the noise power inside the filter bandwidth of the motion-echo.
For the monochromatic modulation, the integrals I1and I3have analytic solutions:
I1(t0, τ ) = An
ωn
[sin(ωn(t0+τ) + φn)−sin(ωnt0+φn)]
I3(t0, τ ) = An
ω2
n
[cos(ωnt0+φn)−cos(ωn(t0+τ) + φn)−
−ωnτsin(ωnt0+φn)].(20)
Now, the integrals depend on t0and τ, therefore the sum over a motion-echo sequence
is non-zero in general. Inserting the integrals into Eq. (17) and summing over the
motion-echo sequences is straightforward but tedious, and yields closed expressions for
the final displacement ∆αNand the corresponding average occupation number of the
motion ¯nfin =|∆αN|2. Taking the average over the random phase φnyields
h¯nfini=8A2
nΩ2
x
ω4
n
sin2[ωnτd/4]{cos[ωnτd/4] −cos[ωn(τa+ 3τd/4)]}2×
×sin2[naωn(τd+τa)]
sin2[ωn(τd+τa)] .(21)
If the free-evolution time τais varied in the motion-echo sequence, the expression in
the upper line produces an envelope that is oscillating at frequency ωnwith phase
shifts proportional to τd.‡The first main peak appears when ωn(τd+τa)'πand the
spacing between adjacent main peaks is exactly ∆τa= 2π/ωn, which allows for deter-
mination of ωnfrom this interference pattern. The width of the narrow main peaks
can be characterized by the distance δτaof the two zeros of the response closest to a
peak, which are spaced by δτa= 2π/(naωn). It is possible to resolve a pair of main
peaks produced by modulations at ωnand ωn+δωnrespectively, as separate maxima if
|δωn| ≥ π/[na(τa+τd)]. If a continuous noise power spectral density a2
n(ωn) is sampled
in this way, δωndetermines the bandwidth of the sample filter that relates the noise
power density to the actual noise power detected in this band.
To have h¯nfiniapproximately reflected in the ion state population Pπ
↑, the average mode
occupation should be kept below h¯nfin i ≤ 1, which is possible by reasonable choices for
the displacement Ωxτdand the number of free-precession periods 2na. Choosing quan-
tities that are too large has the same effect as over-driving the mixer in an electronic
spectrum analyzer, which leads to a response that is not linear in the input signal,
resulting in a distorted output.
‡The expression in the lower line is equivalent to the intensity far-field pattern of a transmission
grating with naslits [29] and describes a na-times sharper response that produces a more narrowly
peaked interference pattern with nearly symmetric side lobes.

Coherently displaced oscillator quantum states of a single trapped atom 11
4. Experimental implementation and results
4.1. Experimental setup
Experiments were performed with a single 9Be+ion trapped 40 µm above a cryogenic
('4 K) linear surface-electrode trap described elsewhere [30, 31]. The coherent dis-
placements are performed on the lowest frequency mode (axial) of the three orthogonal
harmonic oscillator modes of the ion, with frequency ωx'2π×8 MHz. We use two lev-
els within the electronic 2S1/2ground-state hyperfine manifold, |F= 1, mF=−1i=|↑i,
|F= 2, mF=−2i=|↓i, where Fis the total angular momentum and mFis the com-
ponent along the quantization axis, defined by a 1.43mT static magnetic field. Direct
“carrier”-transitions between the states |↓i |niand |↑i |niare driven by microwave fields
induced by a ω0'2π×1.281 GHz current through one of the surface trap electrodes.
The ion is prepared in |↓i |0iwith a fidelity exceeding 0.99 by Doppler laser cool-
ing, followed by ground-state cooling [23] and optical pumping. Sideband transitions
|↓i |ni ↔ |↑i |n±1iare implemented with stimulated Raman transitions driven by two
laser fields that are detuned from the 2S1/2→2P1/2transition (λ'313 nm) by ap-
proximately 40 GHz [32]. This allows us to prepare nearly pure number states of the
motion as described in more detail in [33]. We implement the tickle by applying a
square-envelope pulse with oscillation frequency near the axial mode frequency to the
same electrode that is used for the microwave-driven hyperfine transitions, which pro-
duces an electric field at the position of the ion with a component along the direction
of the motional mode.
We distinguish measurements of the |↑i and |↓i states with state-dependent fluo-
rescence [23]. When scattering light from a laser beam resonant with the |↓i ↔
P3/2, F = 3, mF=−3Ecycling transition, 11 to 13 photons are detected on average
over 400 µs with a photo-multiplier if the ion is in |↓i, while only 0.2 to 0.5 photons
(dark counts and stray light) are detected on average if the ion is projected into |↑i.
In the experiments detailed below, the signal indicates the deviation of the final mo-
tional state from |n= 0i. Population in the ground state of motion is discriminated from
that in excited states of motion by performing the RSB pulse theoretically described in
Eq. (12), connecting population in |↓i |n > 0ito |↑i |n−1iwhile leaving population in
|↓i |n= 0iunchanged. For average excitation ¯n≤1/2 the probability Pπ
↑of changing
the internal state is approximately equal to ¯n. A subsequent microwave carrier π-pulse
exchanges population in |↓i and |↑i. The latter state has a low average count rate, which
minimizes shot noise in the photomultiplier signal. This is helpful when determining
small deviations from |n= 0iwith high signal-to-noise ratio.

Coherently displaced oscillator quantum states of a single trapped atom 12
-2-1 1 2
0.1
0.2
0.3
0.4
0.5
detuning
Figure 2. Spin-flip probability Pπ
↓(See Sec. 4.2) of ion after 13 µs tickle excitation
on |↓i versus detuning from ion oscillation frequency. The average occupation ¯nof
the ion motion in response to tickle excitation is mapped onto the spin state by
performing a RSB pulse, which connects levels |↓i |nito |↑i |n−1ifor n > 0, while
leaving population in |↓i |n= 0iunchanged. A subsequent microwave carrier π-pulse
exchanges populations in |↑i and |↓i to reduce measurement projection noise. The
solid line is a fit using Eq. (7) and free parameter ¯nand an experimentally determined
vertical offset of 0.05(1) to account for background counts and imperfect ground state
cooling. The fit yields an on resonance average occupation of ¯n= 0.61(1).
4.2. Displaced number states
As briefly described in [14], tickling an ion that has been cooled to near the motional
ground state to determine the ion oscillation frequency is a practical calibration tool.
The experiment is performed here as follows: the ion is cooled to near the ground state
and prepared in |↓i, then a tickle tone with a fixed amplitude and detuning δis applied
for a fixed duration τd= 13 µs. The resulting coherent state is characterized by apply-
ing a RSB π-pulse for the |↓i |1i → |↑i |0itransition followed by a microwave carrier
π-pulse, then detected via state-selective fluorescence as described above.
The symbols with error bars (1-σstatistical error, from shot noise in the photo-
multiplier count rate averaged over 600 experiments per detuning value) in Fig. 2
show the measured Pπ
↓as a function of tickle detuning for low on-resonance occupation
(¯n= 0.61(1)). The line shape is well described by Eqs. (7) and (12). The solid line is
a fit to these equations with ¯n= 0.61(1) as the only free parameter after subtracting
an offset of 0.05(1) due to stray light background and imperfect ground state cooling
that was determined independently. In this case, Pπ
↓is roughly linear in ¯nand reflects
the sinc2-shape of the Fourier transform of the square-envelope tickle pulse. Keeping

Coherently displaced oscillator quantum states of a single trapped atom 13
the excitation small gives us the practical advantage that only one prominent peak in
Pπ
↓versus detuning exists, making fitting to find the resonance frequency straightfor-
ward. However, the precision with which we can determine the frequency of oscillation
is Fourier-limited by the time that we apply the pulse.
With the development of a theoretical understanding of line shapes for larger exci-
tations, where Pπ
↓is non-linear in ¯n, we have found that we can determine the resonant
frequency with a precision that increases approximately linearly with the size of the
excitation |α|in the range of 0 <|α|<17, which implies that we can improve on the
Fourier limit of the tickle pulse. We have measured Pπ
↓(see Eq. 12) versus detuning of
the tickle frequency for various displacement amplitudes up to |α| ≈ 17, corresponding
to a coherent state with an average occupation of ¯n≈300. Fig. 3 shows four such cases
with ¯nof 3.22(3) (Fig. 3a), 10.4(1) (Fig. 3b), 98.4(7) (Fig. 3c) and 299(1) (Fig. 3d).
The lines are fits with free parameters Ωxand HO resonance frequency ωx. An experi-
mentally determined vertical offset of 0.05(1) is added to the fit function to account for
background counts, as in the evaluation of the data presented in Fig. 2.
The steep slopes of some of the line-shape features imply a stronger response to
small changes in the detuning, as compared to cases where ¯n≤1 (dashed lines in Fig.
3). Moreover, the response is symmetric around δ= 0, so these steeper slopes can be
exploited to find δ= 0 without a detailed understanding of the line shapes beyond this
symmetry. This enables line-center determination with a signal-to-noise ratio beyond
the Fourier limit of the linear response (¯n≤1).
We can also scan the amplitude of the coherent displacement while the tickle frequency
is resonant with the ion’s oscillation. As the amplitude of the coherent state increases,
the Rabi frequency of the RSB interaction varies with nas predicted by Eq. (10), pro-
ducing the non-monotonic response of the ion’s fluorescence shown in Fig. 4 together
with fits based on Eq. (10) with ¯nas a free parameter. We perform this experiment by
preparing the ion in pure number states n= 0 (Fig. 4 a)), n= 2 (Fig. 4 b)), n= 4
(Fig. 4 c)) and n= 6 ((Fig. 4 d)) and applying a resonant tickle tone with fixed Ωxfor
times ranging from 0.05to12 µs, resulting in coherent displacements up to |αd| ≈ 17.
4.3. Motion-echo experiments
We first perform the motion-echo experiments with a 10 kHz or 100 kHz tone applied
to one of the trap electrodes. The tone modulates the potential curvature of the trap
at the position of the ion and therefore the ion’s oscillation frequency. The purpose of
this is two-fold: To experimentally characterize the response of the ion and compare
the results against theory for a known perturbation and to explore the range of noise
that is detectable with this method in our setup. With the tone applied continuously

Coherently displaced oscillator quantum states of a single trapped atom 14
detuning
-3-2-11 2 3
0.1
0.2
0.3
0.4
0.5
0.6
n≃3.2
-3-2-1 1 2 3
0.1
0.2
0.3
0.4
0.5
0.6
n≃10
-1.5 -1.0 -0.5 0.5 1.0 1.5
0.2
0.4
0.6
0.8
n≃98
-0.5 0.5
0.2
0.4
0.6
0.8
1.0
n≃300
a)
d)
c)
b)
Figure 3. Response of ion to tickle excitation versus detuning from ion oscillation
frequency for maximal excitations on resonance of a) ¯n= 3.22(3), b) ¯n= 10.4(1),
c) ¯n= 98.4(7) and d) ¯n= 299(1). As the average occupation of the coherent state
increases, the line shape becomes more sharply featured. Solid lines are fits with free
parameters Ωxand ωx. We determine the maximum excitation ¯nof each experiment
from the fitted values of Ωx. Dashed lines represent a Fourier-limited response resulting
from a weaker excitation (¯n= 0.64) for comparison.
and the phase φnchanging randomly from experiment to experiment (to mimic the
uncontrolled phase of noise), we perform a series of motion-echo sequences with various
numbers of free-precession periods 2na, scanning the wait time τa. We find that Pπ
↓
depends on the relationship between τaand the frequency of the applied tone in the
expected manner according to Eq. (21). This can be seen by comparing the measured
points in Fig. 5 to the solid lines that show fits with Anas a free parameter. All fitted
values of Anare consistent with each other to within 2 times the standard deviation
(see caption of Fig. 5) and indicate a relative modulation depth of An/ωx'1.4×10−4.
Similar experiments were performed with applied tones from 500 Hz to 400 kHz and
while the results qualitatively agreed with theory, attenuation and distortion of the tones
through various filters with uncharacterized parasitic capacitance and resistive loss at
4 K in our experimental setup prevented us from comparing quantitatively to the theory.
Finally, we perform motion-echo experiments without a purposely applied tone, to
sense and characterize intrinsic frequency noise in our setup. With na= 10 (20 free-
precession periods), τd= 4 µs, and coherent displacements Ωxτd/2=3.44(2), we find
several peaks in the time scan corresponding to a single narrow-band noise spur at

Coherently displaced oscillator quantum states of a single trapped atom 15
displacement amplitude
a)
d)
c)
b)
2 4 6 8 10 12
0.2
0.4
0.6
0.8
2 4 6 8 10 12
0.2
0.4
0.6
0.8
2 4 6 8 10 12
0.2
0.4
0.6
0.8
2 4 6 8 10 12
0.2
0.4
0.6
0.8
Figure 4. Response of ion to tickle excitation versus displacement αfor initial
number states a) n= 0, b) n= 2, c) n= 4 and d) n= 6. As the displacement
of the ion’s motion increases, the Rabi frequency of the RSB interaction varies non-
monotonically. Lines are produced with theory in Eq. (12) using fit parameters of
the tickle strength Ωxand contrast of the final state readout, and an experimentally
determined vertical offset of 0.05(1) to account for background counts and imperfect
initial state preparation.
ωn'2π×260 kHz and with an amplitude An= 2π×2.4(2) kHz (see Fig. 6a), which
corresponds to a relative modulation depth An/ωx'3×10−4. In this run, the electrode
potentials are supplied from digital to analog converters (DACs), so the spurs are likely
caused by cross-talk of digital circuitry in the DACs to the outputs. After switching
the electrode potential sources to low-noise analog power supplies, we observed a nearly
uniform noise floor without the spurs (see Fig. 6b).
5. Summary and conclusions
We have theoretically described and experimentally demonstrated a number of features
of coherently displaced harmonic oscillator (HO) number states, including displaced
ground states. Coherent displacements are a universal concept that applies to all har-
monic oscillators and, because they correspond to a classical, near-resonant force on
the oscillator, they can often be implemented in a simple way in concrete experimental
settings. Here, we characterized the responses of a single, harmonically trapped atomic

Coherently displaced oscillator quantum states of a single trapped atom 16
10 kHz 100 kHz
2na= 12
2na= 8
2na= 6
2na= 4
2na= 2 2na= 4
2na= 6
2na= 8
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Figure 5. Characterization of motion-echo experiments with applied tones of 10 kHz
and 100 kHz for various numbers of free-precession periods 2na(see top of plots). Lines
are fits based on Eqs. (21) and (12) with free parameters An. For the experiments
with the 10 kHz tone applied, fitted values of Anwere 2π×1.3(4) kHz, 1.5(1) kHz,
1.4(1) kHz and 1.2(1) kHz for 2na= 2, 4, 6 and 8, respectively. Likewise for the 100
kHz tone, fitted values of Anwere 2π×1.0(4) kHz, 1.1(3) kHz, 1.1(2) kHz and 1.1(1)
kHz for 2na= 4, 6, 8 and 12, respectively.
ion to an electric field that oscillates close to resonance with the ion motion. The ion
response was then characterized by coupling the HO to an internal two-level system of
the ion. This was realized by driving a red sideband on an internal state transition that
sensitively depends on the state of the HO and subsequently detecting state-dependent
fluorescence from the atom. The ion fluorescence exhibits a non-linear response in ¯n
when driving the sideband for final states with average occupation number ¯n > 1. The
resulting line shapes of internal-state population versus detuning exhibit a complicated
but symmetric structure with steep and narrow side lobes to the central resonance fea-
ture which are accounted for by the theory.
By applying a sequence of coherent displacements, alternating with free evolution of the
state of motion, we obtained a frequency-filtered response of the HO. In this way, fluc-
tuations of the HO frequency in a certain frequency band can be isolated and sensitively
detected by the ion itself. We demonstrated the key features of this mechanism from
the response to deliberately applied monochromatic modulations on the trap frequency,

Coherently displaced oscillator quantum states of a single trapped atom 17
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Figure 6. Noise sensing with motion-echo experiments. The two sub-plots compare
noise on electrode potentials delivered from two different sources: a) Digital-to-
Analog converters (DACs) and b) linear bench power supply. Experiments were
performed with na= 10 (20 free-precession periods) and coherent displacements of
Ωxτd/2=3.44(2). The solid black line in a) is a fit to the data taken on the DACs
based on Eq. (21) with free parameters Anand ωn, and a vertical offset of 0.07 to
account for background counts and imperfect state preparation. From this fit, we
determine that the DACs introduce noise at ωn/(2π)'260 kHz with An= 2π×2.4(2)
kHz.
and then used the motion-echo sequence to detect and eliminate inherent technical HO
frequency noise in our ion trap system.
We anticipate that the basic concepts exhibited in this work can be transferred to
many other HO systems and foresee various extensions and refinements of the current
work. For example, it should be possible to find interesting modifications of the coherent
displacement sequences by utilizing different displacement patterns in phase space or by
displacing non-classical quantum states [33, 34].

Coherently displaced oscillator quantum states of a single trapped atom 18
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Acknowledgements
We thank David Allcock, Shaun Burd and Daniel Slichter for helpful discussions and
assistance with the experimental setup and Alejandra Collopy and Kevin Gilmore for
useful comments on the manuscript. This work was supported by IARPA, ARO, ONR
and the NIST quantum information program. K.C.M. acknowledges support by an ARO
QuaCGR fellowship through grant W911NF-14-1-0079. J.K. acknowledges support by
the Alexander von Humboldt Foundation.