Spin Dephasing as a Probe of Mode Temperature, Motional State Distributions, and Heating Rates in a 2D Ion Crystal

1.6 MB·

by  Brian C. Sawyer

Spin Dephasing as a Probe of Mode Temperature, Motional State Distributions, and
Heating Rates in a 2D Ion Crystal
Brian C. Sawyer,1,∗ Joseph W. Britton,1 and John J. Bollinger1
1Time and Frequency Division, National Institute of Standards and Technology, Boulder, CO 80305
We employ spin-dependent optical dipole forces to characterize the transverse center-of-mass
(COM)motionalmodeofatwo-dimensionalWignercrystalofhundredsof9Be+. Bycomparingthe
measuredspindephasingproducedbythespin-dependentforcewiththepredictionsofasemiclassical
dephasing model, we obtain absolute mode temperatures in excellent agreement with both the
Doppler laser cooling limit and measurements obtained from a previously published technique (B.
C.Sawyeretal. Phys. Rev. Lett. 108,213003(2012)). Furthermore,thestructureofthedephasing
histogramsallowsfordiscriminationbetweeninitialthermalandcoherentstatesofmotion. Wealso
apply the techniques discussed here to measure, for the first time, the ambient heating rate of the
COM mode of a 2D Coulomb crystal in a Penning trap. This measurement places an upper limit
4 on the anomalous single-ion heating rate due to electric field noise from the trap electrode surfaces
1 of dn¯ ∼5 s−1 for our trap at a frequency of 795 kHz, where n¯ is the mean occupation of quantized
dt
0 COM motion in the axial harmonic well.
2
PACSnumbers: 52.27.Jt,52.27.Aj,03.65.Ud,03.67.Bg
n
a
J
I. INTRODUCTION ion. Thisgeneraltechniqueshouldbeapplicabletoother
3
systems such as neutral atoms in optical lattices [14] or

opto-mechanical systems [15] where spin degrees of free-
] Laser-cooled ions stored in radiofrequency (RF) or
h dom can be controlled and coupled to motional degrees
Penning traps readily form crystalline arrays. Sensitive
p of freedom.
measurements of the motion of ions in these arrays are
-
t importantforavarietyofstudiesinatomicphysics,quan- Spin-dependent forces are a key tool in trapped-ion
n
tum information science, and plasma physics. In atomic quantum simulation and quantum computing work. Ap-
a
u physics the small residual motion of trapped ions pro- plication of a spin-dependent force to a superposition
q duces systematic errors that can limit the performance of different spin states can generate entanglement be-
[ of atomic clocks and precision measurements [2, 3]. In tween the spins, while the concomitant coupling of the

quantuminformationscience,wheretrapped-ioncrystals spin and motional degrees of freedom typically produces
1
v provide a promising platform for quantum computation infidelitiesthatmustbemitigated[1,16–21]. Herewefo-
2 and simulation [4–8], the residual motion of the trapped cus on the coupling and potential entanglement between
7 ions produces infidelities that require careful evaluation. the spin and motional degrees of freedom produced by
6 In plasma physics, trapped-ion crystals provide a conve- a spin-dependent force, and work in a regime where the
0 nient laboratory platform for studies of strongly coupled induced spin-spin entanglement is negligible. However,
.
1 plasmas, which model dense astrophysical matter [9–12]. the discussion and measurements presented here provide
0 Careful measurements of ion motion are used to deter- insight into the size and nature of trapped-ion quantum
4 mine the ion energy and the plasma coupling. Energy gate errors produced by coupling of the spins to ther-
1
transport studies require detailed measurements of the mal fluctuations of the motional modes [22]. Spin-echo
:
v ion motion, typically as a function of time and resolved as well as other dynamical decoupling techniques can re-
i spatially or between different modes [13]. move the coupling of the spin and motional degrees of
X
freedom [16, 23, 24], but their efficacy depends on the
Here we discuss a new technique for measuring the
r
size of the error and the coherence of the motional state
a temperatureand, moregenerally,theenergystatedistri-
throughout an experiment, which can be evaluated with
bution of a trapped-ion crystal. The technique is mode
the techniques discussed here.
specific in that it can resolve the energy distribution of
differentmodesofthecrystal. Wedemonstratethetech- This study extends the results of Ref. [1], where we
niquebypresentingmeasurementsoftheenergydistribu- measured the decrease in the composite Bloch vector
tion of the axial center-of-mass (COM) mode of a single- length produced by the application of a homogeneous
plane array of several hundred Be+ ions stored in a Pen- spin-dependentopticaldipoleforce. Weshowedthatthis
ning trap. The technique requires isolating and control- decrease (or decoherence) of the Bloch vector depended
lingatwo-levelsystem–aneffectivespin-1/2ineachion on the average energy or temperature of the initial mo-
– and employs a weak, global spin-dependent force that tionalstate. Hereweshowthatthedephasingresponsible
couplesthespinandmotionaldegreesoffreedomofeach for this decoherence may be directly measured, reveal-
ing more detailed information about the motional state.
In addition to the average energy or temperature of a
mode, information on the energy distribution can also
∗Electronicaddress: [email protected] be obtained. Spin-dephasing produced through the ap-
2
plication of a spin-dependent force provides an alterna- precession is much larger than the quantum projection
tive to the well known Raman sideband technique for noise[26],andcanbemeasuredwithgoodsignal-to-noise
determiningtheenergydistributionofmotionalstatesof in one experimental shot.
trapped-ion crystals [25]. The spin-dephasing technique The rest of the manuscript is structured as follows. In
is particularly well-suited for many-ion crystals, and for Section II we describe the Penning trap setup where we
somesetups–inparticularforhigherfrequencytwo-level implement the spin dephasing technique to characterize
systemssuchasthe124GHzspin-fliptransitiondiscussed the energy distribution of the axial COM mode of a 2D
here (see Sec. II) – can be simpler to implement. trapped-ion crystal of hundreds of 9Be+ ions. In Section
Toillustratethebasicideaofspindephasingproduced III we discuss a more detailed dephasing model assum-
through the application of a spin-dependent force, we ing a thermal distribution of coherent states. In Sec-
consider the simple case of a single trapped ion whose tion IV we discuss dephasing measurements of the COM
motional degree of freedom along the z-axis (trap fre- mode energy distribution for both thermal states and
quencyω )iscoupledtotwointernalspinstatesthrough coherently-excited thermal states. We also present mea-
z
a sinusoidally time-varying spin-dependent force. The surements of the heating rate of the axial COM mode.
interaction Hamiltonian for this system is In Section V we summarize and conclude.
Hˆ =F cos(µt)zˆσˆz, (1)
0
II. EXPERIMENTAL SETUP
where zˆ is the position operator of the ion in the z-
direction, σˆz is the Pauli spin matrix associated with
As described in previous publications, we employ a
the two internal energy levels, and µ is the frequency
Penning trap to confine crystals of hundreds of 9Be+
of the applied spin-dependent force. We assume the
ions [1, 8]. Depicted in Fig. 1a, the trap consists of
ion spin state is initialized in an equal superposition
√
a static electric (E) quadrupole produced from a stack
{|↑(cid:105)+|↓(cid:105)}/ 2 of the |↑(cid:105),|↓(cid:105) internal levels. This spin
of cylindrical electrodes (inner radius of 2.0 cm) placed
state can be represented as pointing along the x-axis in
within the room-temperature bore of a ∼4.46 T super-
the rotating frame of the Bloch sphere. Suppose the
ion temperature is large compared to (cid:126)ω /k , where (cid:126) conductingmagnet. Theorientationofthisuniformmag-
z B netic (B) field defines the z-axis in our system. Har-
and k are the Planck and Boltzmann constants, and
B monic axial (z-axis) ion confinement with a frequency
we may treat the ion motion as classical. The ini-
of ω = 2π × 795 kHz is obtained by applying -1 kV
tial motional state of the ion can then be written as z
to the central ring electrodes relative to grounded up-
z(t) = Z cos(ω t+φ), where Z and φ fluctuate from
A z A per and lower endcap electrodes. The cylindrical axis
one shot (or realization) of the experiment to the next,
of the trap electrodes is aligned with the uniform mag-
consistent with a thermal distribution. Application of
netic field. Radial ion confinement results from E(cid:126) ×B(cid:126)
Eq.1producesanadditionalspin-dependentmotion,but
inducedrotationthroughthemagneticfield. Weapplya
we assume this driven spin-dependent motion is small
weak quadrupolar “rotating wall” potential to precisely
comparedwiththeinitialthermalfluctuation(Z )(valid
A
forourworkwithhundredsoftrappedBe+ ions),andwe controltherotationfrequency(ωr)andhence,radialcon-
can approximate the Hamiltonian, Hˆ, as fining force, of the ion cloud [27]. Neglecting the weak
azimuthal dependence of the rotating wall potential, the
Hˆ ≈ F cos(µt)Z cos(ω t+φ)σˆz following trap potential describes ion confinement in the
0 A z
= F02ZA {cos[(µ−ωz)t−φ]+cos[(µ+ωz)t+φ]}σˆz. Penning trap as seen in a frame rotating at ωr [28]:
(2) qΦ (r,z) = 1Mω2(cid:0)z2+β r2(cid:1), (3)
For µ = ωz this Hamiltonian is simply a constant shift trap 2 z r
F|↑0(cid:105)Z,A|↓(cid:105)colsev(φel)s/,(cid:126)pliunstaherafpriedqluyevnacryyidnigffetreernmcethbaettwaeveenratghees βr ≡ ωr(Ωωc2−ωr) − 12, (4)
to zero for time intervals long compared to π/ω . If z
z
the spin-dependent force is applied for a time interval where M (q) is the mass (charge) of a single 9Be+,
τ, then the Bloch vector undergoes a precession by an Ω = 2π × 7.597 MHz is the cyclotron frequency, and
c
angle Φ = (F Z cos(φ)/(cid:126))τ. Fluctuations in Z and z (r) is the axial (radial) distance from the trap center.
p 0 A A
φ from one shot (or realization) of the experiment to the For rotation frequencies where the radial confinement is
next produce spin dephasing when averaged over many weakrelativetotransverseconfinement(β (cid:28)1),weob-
r
experimental realizations. By measuring this dephasing tain a single ion plane. The rotation frequency at which
directly we show that it is possible to acquire informa- the ion configuration transitions from two planes to one
tionontheinitialmotionalstate(forexample,theenergy plane depends sensitively on the ion number [29]. For
distribution)ofthetrapped-ionharmonicoscillator. The mostexperiments,weoperatewith100to300ions,which
sensitivity to motion of this technique is very high. For necessitatesω (cid:46)2π×48 kHzforsingle-planeconditions
r
themodestparametersusedinthemeasurementsofSec. with ω =2π×795 kHz.
z
III and IV, F = 10−23 N and τ = 1 ms give Φ = 30◦ We use Doppler laser cooling along both the axial (z-
0 P
for Z (cid:39) 6 nm. With N (cid:38) 100 trapped ions, a 30◦ axis, out-of-plane) and radial (in-plane) trap dimensions
A
3
a) b) mJ a) π/2 |y τ π |x τ π/2 |∆φ
+3/2
B
2P3/2 +1/2 N
~79 GHz
-1/2
-3/2
b) z z
z
2P1/2 +1/2
-1 kV y y
-1/2 y
ωr Cooling(313 nm)~~ ~~Repump ~~~~ptical Dipole Force ∆xφ = 0 ∆xφ = π/2 ∆xφ = π
O
2S1/2 ~124 GHz +1/2
z -1/2
x y 2 cm
FIG. 1: (color online) (a) Simplified illustration of the Pen-
ning trap electrode structure in cross section (not to scale).
Photon Counts (arb. u.)
Staticvoltagesareappliedtotheelectrodesintheforeground
(orange), while three of the six rotating wall electrodes (red)
are shown in the background. A simulated single-plane ion FIG. 2: (color online) (a) An example spin echo pulse se-
configuration is shown at the trap center and magnified by quence used for qubit state manipulation. Spins are initial-
∼100 for visibility. The 313-nm intersecting optical dipole izedto|↑(cid:105) ,andthe124GHzispulsedforthegivenduration
N
forcebeamsarealsoillustrated. (b)Low-lyingelectroniclev- ataspecificphase,where∆φisdefinedrelativetothey-axis
els of 9Be+ in the 4.46 T B-field of the Penning trap. Pro- and is varied depending on the intended final spin state. (b)
jection of total electronic angular momentum (m ) is given Detected histograms for three collective spin orientations of
J
totherightofeachlevel. Nuclearspinprojectionshavebeen the qubit ensemble consisting of 174(10) spins. The collec-
excluded for clarity. Relevant transitions near 313 nm are tive state at the end of the spin-echo sequence is represented
labeled. on a Bloch sphere above the corresponding histogram. The
horizontal axis is scaled such that the mean photon counts
detected when in state |↑(cid:105) corresponds to unity. Each his-
N
togram is the result of 1000 state preparation and detection
to produce the Coulomb crystal. As shown in Fig. 1b,
sequences, and bin widths have been adjusted between the
the hyperfine structure of 9Be+ exhibits a strong Zee-
three histograms to place them on the same vertical scale.
man shift in the large uniform B-field of the Penning
trap. We laser cool along the ∼313 nm |J = 1/2,m =
J
+1/2(cid:105) → |3/2,+3/2(cid:105) cycling transition between the
2S and2P manifolds,whereJ andm arethetotal shown includes both π/2- and π-pulses about the given
1/2 3/2 J
electronic angular momentum and its projection along Bloch sphere axes. The phase of the final pulse (∆φ) is
the B-field axis, respectively. The linewidth of this cool- defined relative to that of the first π/2-pulse, which we
ing transition is Γ ∼ 2π × 17.97 MHz [30], yielding a definetobearotationaboutthey-axis,andisvariedde-
Doppler cooling limit of (cid:126)Γ/2k ∼ 0.43 mK. For all ex- pending on the intended final spin state. We use a spin
B
perimentsdescribedhere,the9Be+ areopticallypumped echo sequence with free evolution periods of τ ∼0.1 to 1
to the |I =3/2,m =+3/2(cid:105) nuclear spin state [31]. ms to mitigate the deleterious effects of radial B-field
I
The two qubit states for our experiments are the inhomogeneity over the ∼400 µm ion plane diameter,
|↑(cid:105) ≡ |1/2,+1/2(cid:105) and |↓(cid:105) ≡ |1/2,−1/2(cid:105) valence electron and to cancel B-field fluctuations at frequencies below
spinprojectionsofthe2S electronicgroundstate. The τ−1 [32, 33]. After the pulse sequence, we measure the
1/2
cooling and repump transitions illustrated in Fig. 1b al- population of spins in state |↑(cid:105) (P↑) by switching on the
low for efficient preparation of all N trapped ions to the Doppler cooling beams and counting scattered photons
state |↑(cid:105) ≡ |↑↑...↑(cid:105). The splitting between qubit lev- collected by an ultraviolet-sensitive photomultiplier. An
N
els is ∼124 GHz, and we perform global qubit rotations f/5 objective imaging the side of the ion plane collects
via direct application of resonant millimeter-wave radi- thescatteredcoolingphotons,andatypicalphotoncount
ation to the ions. We typically achieve π-pulse times rate per ion is 103 s−1.
(tπ) of ∼70 µs. As discussed below, for the experiments The histograms of Fig. 2b give experimental results
described here we perform global readouts of the qubit for three different values of ∆φ as measured in a system
state through state-dependent resonance fluorescence on of N = 174(10) spins. Each color-coded histogram is
the Doppler cooling transition. the result of 1000 pulse sequences (see Fig. 2a) and sub-
Figure 2a illustrates a typical pulse sequence for qubit sequent qubit state readouts. Bin widths for the three
manipulation. We first prepare |↑(cid:105) using the Doppler histograms are adjusted for clear presentation on a sin-
N
cooling and repump lasers. The spin echo sequence gle vertical scale, and the horizontal axis is scaled to the
4
photoncountscollected for thestate|↑(cid:105) . Thestandard tionalmodesplayanimportantroleinquantuminforma-
N
deviation of the ∆φ = π histogram suggests technical tion experiments with trapped ions. If all trapped ions
noisethatiscomparabletoshotnoisewith3510photons possess the same charge-to-mass ratio, the COM mode
collected. The increase in the standard deviation of the frequencyisindependentofionnumberand,inthecaseof
∆φ = π data is due to quantum spin projection noise transverse modes, constitutes the highest-frequency and
2
(∝N−1/2) [26]. longest-wavelength oscillation. In traps whose electrode
dimensions are much larger than those of the ion crystal
We generate a spin-dependent optical dipole force
(i.e. Penning traps), the COM mode is the transverse
(ODF) by interfering two ∼313 nm laser beams at the
mode most susceptible to noise from fluctuating poten-
ion plane to create an optical lattice (see Fig. 1a) [1,
tials on trap electrodes.
8]. The resulting lattice wavelength (λ ) is determined
l
The interaction Hamiltonian for the spins and the
by the crossing angle of the ODF beams as λ =
l
λODF (cid:2)2sin(θ2R)(cid:3)−1, where θR is the full beam crossing COM degree of freedom is
wanogrlke,aθnRd=λO4D.2F(2)is◦,thwehiOchDFresluasltesriwnaλvlel∼eng3t.7h.µFmo.rTthhies HˆODF =F0cos(µt+ϕ)√z0 (cid:0)aˆe−iωzt+aˆ†eiωzt(cid:1)(cid:88)N σˆiz,
ODF laser frequency is detuned by ∼20 GHz from the N
i=1
nearest resonances for the |↑(cid:105) and |↓(cid:105) states (see Fig. 1b) (5)
and the linear polarization of each beam is chosen so as where the sum is over the N spins, z = (cid:112)(cid:126)/(2Mω )
0 z
to produce a polarization gradient at the ion plane that is the ground state wavefunction size of a single trapped
impartsequal-magnitude,opposite-signforcestothetwo ion, aˆ (aˆ†) are the lowering (raising) operators for the
qubitstateswithamagnitudeof∼10−23 Nperspinat1 COM mode, and ϕ is the ODF phase. In general the
−→
W cm−2 per beam. The optical lattice wavevector (∆k) timeevolutionoperatorfortheaboveHamiltoniancanbe
is oriented along the z-axis of the trap to preferentially written as the product of a spin-dependent displacement
excite motion transverse to the crystal plane. The two operator, exp(cid:16)(cid:2)αaˆ†−α∗aˆ(cid:3)(cid:80)N σˆz(cid:17), and an evolution
i=1 i
ODF laser beams are produced from a single beam us-
operatorforageneralIsinginteractionthatinvolvesonly
ing a 50/50 beamsplitter, and their relative frequency is
pairwise spin interactions [1, 17]. For resonant drives
adjustedfromzeroto∼10MHzusingacousto-opticmod-
(µ ≈ ω ), the effect of the spin-dependent displacement
ulators, enabling production of a standing- or running- z
typically dominates, and we neglect the induced Ising
wave spin-dependent optical lattice.
interaction throughout this manuscript. In this case the
evolution operator for Hˆ separates into a product of
ODF
N individual spin-dependent displacement operators,
III. THERMAL DEPHASING MODEL
N
Dˆ (α)=(cid:89)exp(cid:0)(cid:2)αaˆ†−α∗aˆ(cid:3)σˆz(cid:1). (6)
SD i
In Ref. [1] we excited arbitrary drumhead modes of a
i=1
2D trapped-ion crystal through application of a homo-
geneous spin-dependent force. The spin-dependent force The displacement amplitude, α, for resonant (µ = ωz)
coupled the 9Be+ ground-state valence electron spin and spin-dependent excitation of the COM mode for a time,
transverse motional degrees of freedom. We measured τ, and phase, ϕ, is
the decrease in the composite Bloch vector of the spins
F z τ
due to this coupling, and showed that this decrease (or α(τ,ϕ)=−i 0√0 eiϕ. (7)
decoherence)dependedontheaverageenergyortemper- 2(cid:126) N
ature of the motional state. We sketch the calculation
For pulse sequences involving separated periods of ODF
of Ref. [1] in Appendix A. In the Fock state basis ther-
excitation (e.g. spin echo), the final motional displace-
mal motional states are described by a diagonal density
ment is simply a sum of individual displacements of the
matrix. The calculation proceeds by assuming an ini-
form of Eq. 7 with appropriate phases (ϕ) and times
tial Fock state |n(cid:105) for a mode. Application of a spin-
(τ) for each of the ODF excitations within the sequence.
dependent force produces spin-dependent displacements
Wedefinespin-independentdisplacements,Dˆ(α ),inthe
oftheFockstateanddecoherenceofthespinsisnaturally 0
usual way:
described in terms of spin-motion entanglement and the
increasingdisplacementsensitivityofFockstateswithn. Dˆ(α )=exp(cid:0)α aˆ†−α∗aˆ(cid:1). (8)
0 0 0
Hereweuseamodelmotivatedbythedephasingpicture
of the Introduction that does not require quantum en- In contrast to Ref. [1], we consider the initial state
tanglement of the spin and motional degrees of freedom of the COM mode for each experiment to be a coherent
for thermal excitations large compared with the ground state|α (cid:105). Wedenotetheexpectationvalueofaquantum
0
(cid:68) (cid:69)
state size. operator Oˆ at the end of an experiment as Oˆ . We
For simplicity, we describe the dephasing model for
then perform an average over a thermal distribution of
the axial COM mode, although a generalization to other (cid:68)(cid:68) (cid:69)(cid:69)
drumheadmodesisstraightforward. Center-of-massmo- expectation values which we denote as Oˆ . More
th
5
precisely, we calculate thermal averages of a function, π/2 |y π/2 |∆φ
A(ξ), of the continuous variable ξ ≡|α |2 as τ
0
(cid:90) ∞
(cid:104)A(ξ)(cid:105) ≡β A(ξ)e−βξdξ, (9)
th 124 GHz
0
where β = (cid:126)ω (k T)−1 for COM mode temperature T.
z B
For completeness, the Fock state calculations in the Ap- ODF
pendix involve the corresponding thermal average over
discrete Fock state expectation values, An, as ψ ψ ψ
1 2 f
∞
(cid:104)A (cid:105) ≡(cid:0)1−e−β(cid:1)(cid:88)A e−βn. (10)
n th n FIG. 3: (color online) Ramsey pulse sequence for global 124-
n=0 GHz qubit rotations (upper) and ODF excitation (lower) as
used for the derivations of Section III. The quantum states
|ψ (cid:105), |ψ (cid:105), and |ψ (cid:105) are labeled at appropriate points in the
A. Bloch Vector Length (∆φ=0) 1 2 f
sequence as described in the text.
Here we are interested in calculating the expectation
value of a component of the composite Bloch vector,
(cid:16)Sˆx,Sˆy,Sˆz(cid:17)=(cid:16)(cid:80)Ni=1 σˆ2ix,(cid:80)Ni=1 σˆ2iy,(cid:80)Ni=1 σˆ2iz(cid:17), forinitial where θ0 = Im{αα0∗}. It is useful to pause at Eq. 14
before applying the final microwave pulse and evaluate
spin states which are product states. This reduces to
(cid:104)Sˆ (cid:105) and (cid:104)Sˆ (cid:105). We find
calculating the expectation value of a component of an x y
individual spin (cid:126)σ . For the evolution operator of Eq. 6,
i
whichisaproductofcommutingdisplacementoperators
N N
involving individual spins, only the displacement opera- (cid:104)Sˆ (cid:105) = (cid:104)σˆx(cid:105)= cos(4Im{α∗α })e−2|α|2 (15)
tor involving σˆz non-trivially enters into the calculation. x 2 2 0
i N N
We assume that each experiment begins with the state, (cid:104)Sˆ (cid:105) = (cid:104)σˆy(cid:105)= sin(4Im{α∗α })e−2|α|2. (16)
|↑(cid:105)|α (cid:105) = |↑(cid:105)Dˆ(α )|0(cid:105), where |α (cid:105) is a coherent state of y 2 2 0
0 0 0
COMmotion. InSec.IIIAandIIIB,weconsideraRam-
seypulsesequenceconsistingoftwoπ/2pulsesseparated
From Eqs. 15 and 16, we see that the ODF has caused
byatimeτ asshowninFig.3. ForthisRamseysequence
a coherent rotation of the composite Bloch vector about
consisting of a single ODF excitation period, Eq. 7 is
the z-axis by
used to calculate spin-dependent displacements, α. The
measurements of Section IV involve spin echo sequences,
but all of the theory results of Sections IIIA and IIIB (cid:16) (cid:17)
θ =arctan (cid:104)Sˆ (cid:105)/(cid:104)Sˆ (cid:105) =4Im{α∗α }. (17)
apply with small modifications for calculating the spin- coh y x 0
dependent displacement, α.
The first π/2 pulse of the sequence of Fig. 3 yields the
qubit rotation Theeffectofspin-motionentanglementisreflectedinthe
π term e−2|α|2 of Eqs. 15 and 16, which deviates negligibly
|ψ1(cid:105) = Rˆ(2,0)|↑(cid:105)|α0(cid:105) from unity for the dephasing measurements discussed in
Sec. IV. Thermal averages may be performed over the
1
= √ (|↑(cid:105)+|↓(cid:105))Dˆ(α0)|0(cid:105), (11) continuous variable ξ ≡ |α0|2, where the magnitude ξ is
2 now weighted according to Boltzmann statistics and the
phase of α ∈ C is evenly distributed over 2π radians.
where we define the following qubit rotation matrix 0
(cid:18) cos(θ) −e−iφsin(θ)(cid:19) Defining α=|α|eiφ(cid:48) and α0 =|α0|eiφ0, we calculate:
Rˆ(θ,φ)= 2 2 . (12)
eiφsin(θ) cos(θ)
2 2
We now consider the effect of the spin-dependent ODF β (cid:90) 2π(cid:90) ∞
(cid:104)θ2 (cid:105) = 16|α|2ξsin2(φ(cid:48)−φ )e−βξdξdφ
actingforthefreeevolutiontime,τ. Thisyieldsthestate coh th 2π 0 0
0 0
|ψ (cid:105) = √1 |↑(cid:105)Dˆ(α)Dˆ(α )|0(cid:105) = 8|α|2β−1. (18)
2 0
2
1
+√ |↓(cid:105)Dˆ(−α)Dˆ(α )|0(cid:105) (13)
2 0 In Sec. IV, a typical |α| for a 30-yN ODF driving at ωz
1 for 100 µs is ∼0.05, while β−1 ∼12 at the Be+ Doppler
= √ e−iθ0|↑(cid:105)|α+α0(cid:105) cooling limit, producing a non-negligible rotation angle
2
standard deviation of ∼30◦.
1
+√2eiθ0|↓(cid:105)|−α+α0(cid:105), (14) Forcompleteness, wepresent|ψf(cid:105), P↑ and(cid:104)Sˆz(cid:105)results
6
following the final π/2-pulse, Rˆ(π,0), Eq. 26 since, for COM excitation, all spins feel a force
2
of equal magnitude and there is no differentiation be-
|ψ (cid:105) = 1|↑(cid:105)(cid:0)e−iθ0|α+α (cid:105)−eiθ0|−α+α (cid:105)(cid:1) tweenspinpairs. Inevaluatingthetwo-spinexpectation,
f 2 0 0 (cid:104)σˆzσˆz(cid:105), only the displacement operators in Eq. 6 involv-
1 2
+1|↓(cid:105)(cid:0)e−iθ0|α+α (cid:105)+eiθ0|−α+α (cid:105)(cid:1)(19) ing σˆ1z and σˆ2z non-trivially enter into the calculation.
2 0 0 We then compute the thermal average of this expecta-
1(cid:16) (cid:17) tion value and obtain:
P(α0) ≡ 1−cos(4Im{α∗α })e−2|α|2 (20)
↑ 2 0
N N(N −1)
P↑ = (cid:104)P↑(α0)(cid:105)th (cid:104)(cid:104)Sˆz2(cid:105)(cid:105)th = 4 + 4 σ2 (27)
= 1(cid:16)1−e−2|α|2(2β−1+1)(cid:17) (21) σ2 ≡ (cid:104)(cid:104)σˆ1zσˆ2z(cid:105)(cid:105)th (28)
2 1(cid:16) (cid:17)
(cid:104)Sˆ (cid:105) = −N cos(4Im{α∗α })e−2|α|2 (22) = 2 1−(cid:104)cos(8Im{α∗α0})(cid:105)the−8|α|2 (29)
z 2 0 1(cid:16) (cid:17)
N = 1−e−8|α|2(2β−1+1) . (30)
(cid:104)Sˆ (cid:105) = − e−2|α|2(2β−1+1). (23) 2
z th 2
The first term of Eq. 27 is the contribution of spin pro-
We note that β−1 =(cid:104)|α0|2(cid:105)th. Also, for n¯ (cid:29)1, β−1 ∼n¯ jection noise while the second is due to dephasing. The
and the P↑ of Eq. 21 agrees with the treatment that quadraticscalingofthedephasingwithN relativetothe
assumes a thermal distribution of Fock states (Eq. A5) linear scaling of projection noise indicates that thermal
given in Appendix A, but offers a more classical descrip- dephasing can be more accurately measured with larger
tion. That is, for each possible coherent state amplitude ion numbers. We may recast the normalized thermal de-
andphase,theresultoftheexperimentalsequenceisthat phasing portion (σ2) in terms of the COM mean phonon
the composite Bloch vector undergoes coherent rotation number (n¯ ∼β−1):
by some angle θ . Over many such experimental se-
coh
quences, wemeasureadephasingofthecompositeBloch 1 (cid:20) 1 (cid:21) 1
vector associated with the angular variance, (cid:104)θc2oh(cid:105)th, of n¯ ∼ 16|α|2 ln 1−2σ2 − 2. (31)
Eq. 18.
Using Eq. 31, we can now extract the mean phonon oc-
cupation of the COM mode from measurements of spin
B. Dephasing (∆φ= π) dephasing. To determine σ2 we subtract the calculable
2
spin variance (N) from the measured (cid:104)(cid:104)Sˆ2(cid:105)(cid:105) and nor-
4 z th
A final qubit rotation of Rˆ(π,π) (∆φ = π/2) trans- malizebythesquaredlengthofthecompositeBlochvec-
forms rotations and dephasing2in2 the xy-plane of the tor (|S(cid:126)|2 = N2). For N > 100, the difference between
4
Bloch sphere to the detection (z) basis. Below we cal- N2 and N(N−1) is below 1 %. In Appendix B we derive
culate the thermal average of the expectation value (cid:104)Sˆ2(cid:105) a4n analogou4s expression to Eq. 30 using Fock states that
z
from which a temperature determination can be ob- agrees in the limit β−1 ∼n¯.
tained. In addition, we discuss the implementation of AsdetailedinSec.IV,thisapproachtocalculatingspin
a Monte Carlo analysis (see Sec. IV) that is in excellent dephasing using coherent states motivates a straightfor-
agreementwithmeasurementsofthermalaswellasnon- wardMonteCarloanalysisinwhich,foreachexperiment,
thermal motional state distributions with n¯ (cid:29)1. we choose a random initial coherent state of motion and
For the ∆φ= π pulse sequence, we obtain the follow- subsequentlyapplytheexperimentalpulsesequencesand
2
ing expression for (cid:104)Sˆ (cid:105) after the final π/2 pulse: state readout. The initial coherent state magnitudes are
z
weighted according to a thermal distribution while the
(cid:104)Sˆ (cid:105)= N sin(4Im{α∗α })e−2|α|2. (24) phase is random and unweighted in the range [0,2π).
z 2 0 Each simulation includes a Bloch vector rotation of θ
coh
determined by the randomized initial coherent state and
NotethatthethermalaverageoftheexpressioninEq.24
fixed spin-dependent displacement α. After many such
vanishes, so we instead calculate dephasing through the
Monte Carlo runs, we bin the outcomes into simulated
second moment of Sˆ :
z histograms for direct comparison with experimental his-
tograms.
N
(cid:104)Sˆ2(cid:105) = 1(cid:88)(cid:104)σˆzσˆz(cid:105)+ 1(cid:88)(cid:104)σˆzσˆz(cid:105) (25)
z 4 i i 4 i j
i=1 i(cid:54)=j
IV. CENTER-OF-MASS MEASUREMENTS
N N(N −1)
= + (cid:104)σˆzσˆz(cid:105) (26)
4 4 1 2
In this Section, we describe measurements of spin de-
where (cid:104)σˆzσˆz(cid:105) is an expectation value involving any two phasing for different initial states of COM motion. We
1 2
non-identical spins within the ensemble. We simplify to show experimentally that such measurements reveal not
7
only the effective temperature of the COM mode (n¯) [1], whose temperature is 0.4(1) mK, the Doppler cooling
but allow for a more detailed characterization of the ini- limit. The frequency width of the spectral feature is ap-
tial motional state (e.g. thermal, coherent, or a mixture proximatelygivenbytheFourierwidthoftheODFpulse
of the two). duration of 2τ = 1 ms, and the degree of decoherence
1
Figure 4a shows the experimental pulse sequences for measured near δ = 0 is determined by the ODF magni-
ODFlaserbeamsandqubitrotationsusedtomeasurethe tudeandn¯accordingtoEqs.21and32[1]. Thedetuning-
Bloch vector length (∆φ = 0, Sec. IIIA) as in Ref. [1], independent background decoherence level of P ∼ 0.07
↑
alongwithaplotofexperimentaldataandcorresponding in Fig. 4a is due to spontaneous emission from the off-
theory fit. The collective spin of the trapped ions is first resonant ODF laser beams, and is fully characterized for
prepared in the state, |↑(cid:105) , and the pulses at ∼124 GHz this system [1, 36].
N
constitute a Hahn spin echo that, in the absence of the
ODF beams, leaves the spins in state |↓(cid:105) with >99 %
N
fidelity [34]. We apply the ODF laser beams during each
A. Thermal Distributions
freeevolutionperiodofthespinechoforadurationτ =
1
500µs. Therelativefrequencyofthetwobeamsisµ,and
The measurement of Fig. 4a is one of composite Bloch
therelativephaseoftheODFbeatbetweenthefirstand
vector length, and is only second-order sensitive to spin
second arms is given by φ . Note that, for φ =π
ODF ODF dephasing. Tomeasuredephasingmoredirectlyandgain
andµ=ω ,thenetresultoftheexperimentalsequenceis
z morecompleteknowledgeofthespinstatistics,weimple-
identicaltothatoftheRamseysequenceofFig.3withno
menttheexperimentalpulsesequenceofFig.4b. Forthis
intermediate π-pulse and a single free-precession period
experiment, we resonantly drive the COM mode (δ =0)
of 2τ . This is due to the condition that the ODF on
1 with the spin-dependent ODF during free-evolution pe-
state |↑(cid:105) (F ) is opposite in sign but equal in magnitude
↑ riods of τ = 100 µs. We set the phase of the final π/2-
tothaton|↓(cid:105)(F ),where(F −F )=2F (seeEq.5). In 2
↓ ↑ ↓ 0 pulse of the sequence to be the same as the intermediate
thiscasetheODFofthesecondarmreinforcesthatofthe
π-pulse(∆φ= π),therebyrotatinganydephasingwithin
first. Inotherwords,thephaseadvanceoftheODFbeat 2
the xy-plane to lie along the qubit axis. After each such
byπ radiansreversestheeffectoftheintermediatequbit
sequence,weperformaprojectivemeasurementofP and
π-pulse but retains the suppression of spin decoherence ↑
repeat for a total of 1000 experiments. The collection of
inherent in the spin echo. However, for the case µ(cid:54)=ω ,
z 1000P valuesarebinnedanddisplayedashistogramsin
the finite duration of the qubit π-pulse leads to a phase ↑
Fig. 4b. As an additional check, we choose φ to be
offset between the ion crystal oscillation and ODF drive ODF
either 0 or π. In the case of φ =0 (light gray bars),
at the start of the second arm given by δ(τ +t ), where ODF
1 π the spin echo effectively cancels the spin-dependent ex-
δ ≡ (µ−ω ), that must be included in the theoretical
z citationandconcomitantBlochvectorrotation, allowing
analysis. Assuming that we interact exclusively with the
for characterization of other sources of dephasing such
COM mode, the final position (α ) of the ion crystal in
SE as spin projection noise, photon shot noise, AC Stark
phase space after application of the two ODF pulses of
shift fluctuations from the ODF lasers between the two
Fig. 4a is given by:
spinechoarms,andexcessmagneticfieldfluctuationsnot
F z (cid:0)1−eiδτ(cid:1) fullycanceledbythespinecho. ForφODF =π(redbars),
αSE(τ) = 2(cid:126)0√N0 δ theprecessioninducedbythespin-dependentforceinthe
first arm is enhanced in the second and we observe that
(cid:16) (cid:17)
× eiϕ0 −ei[ϕ0+δ(τ+tπ)+φODF] (32) thedetectedspinvarianceisgreatlyincreasedrelativeto
the φ = 0 case. We extract a COM temperature of
ODF
whereτ isthedurationofeachevolutionperiod,ϕ isthe 0.42(7)mKfromthismeasurementbyapplyingEq.31to
0
ODF phase at the start of the first free evolution period, the excess variance of the φ = π experiments. This
ODF
and we have included the additional φ to denote the temperature is in excellent agreement with the theoret-
ODF
added phase advance in the second free evolution period ical Doppler cooling limit as well as the Bloch vector
of the spin echo sequence. The common phase, ϕ , does length measurement of Fig. 4a.
0
not contribute to any experimental observables and may To further compare the measurements of Fig. 4b with
be disregarded. The relative minus sign between phase the model of Sec. IIIB, we use a Monte Carlo algorithm
factorsinthefinaltermofEq.32isduetotheintermedi- toproducesimulatedhistogramsconsistingof50,000‘de-
ateπ-pulseofthespinechosequence,whichremovesdis- tection’ events. This is a factor of 50 more detections
placements common to both free evolution periods since thanfortheexperimentalmeasurements,andissochosen
F =−F . toreducenoiseintheMonteCarlohistogramsforclearer
↑ ↓
Previous experiments with ions confined within RF distinction between experiment and simulation. Impor-
Paul traps have shown that Doppler laser cooling pro- tantly, the simulations have no adjustable parameters –
duces thermal states of ion motion [35]. In Fig. 4a, we allinputstotheMonteCarloalgorithmareexperimental
show that measurements of the Bloch vector length un- parameters (e.g. α , τ, t ) or obtained from measure-
SE π
der application of a spin-dependent force with µ ∼ ω ment (e.g. n¯ from the measured σ). The simulated his-
z
are consistent with that of a thermal state of motion tograms are scaled vertically to match the experimental
8
a) π/2 |y τ π |x τ π/2 |y b) π/2 |y τ π |x τ π/2 |x
N 1 1 N 2 2
124 GHz 124 GHz
(µ, 0) (µ, π) (ωz, 0) (ωz, φODF)
ODF ODF
0.4(1) mK φODF = 0 0.42(7) mK
φODF = π
δ / 2π (kHz)
FIG. 4: (color online) Experimental pulse sequences for global 124-GHz qubit rotations (upper) and ODF excitation (lower)
alongwithcorrespondingdata. Relativephasesandrotationanglesaregivenforeachqubitrotation. TheODFdrivefrequencies
(µ) and phases (φ ) are given as (µ,φ ). (a) Measurement of COM temperature using collective Bloch vector length.
ODF ODF
The microwave spin echo sequence produces the state |↓(cid:105) (P = 0) in the absence of the ODF, and the ODF frequency is
N ↑
swept over the COM resonance (ω =2π×795 kHz) with τ =500 µs. Frequency-dependent deviation from P =0 is fit to a
z 1 ↑
theoretical expression derived in Ref. [1]. The resulting experimental data (black points with error bars) is fit (solid red line)
toextractamodetemperatureof0.4(1)mKgiventheappliedODF.(b)ExtractionofCOMmodetemperaturethroughdirect
measurementofspindephasing. TheCOMmodeisresonantlydrivenineacharmforτ =100µsandφ forthesecondarm
2 ODF
is chosen to either undo (φ =0) or enhance (φ =π) the Bloch vector rotation produced by the spin-dependent force.
ODF ODF
Temperature is extracted from the excess width of the φ = π histogram (red bars) using Eq. 31, and the solid black-line
ODF
histogramistheresultofaMonteCarlosimulationof50,000experimentalrunswithnoadjustableparameters. Wedetermine
a COM temperature of 0.42(7) mK using this dephasing measurement.
data given bin widths and total detection events. direct measurements of spin dephasing are less sensitive
to COM frequency and ODF phase drift within a single
The simulation procedure is the following: for each experiment. The effects of spontaneous emission deco-
Monte Carlo run, k ∈ {1,...,5 × 104}, we choose a herence are also negligible in this parameter regime.
random initial coherent state of COM motion given by
|αk|eiφk. Theprobabilityofchoosingagivenmagnitude,
|α |,isgivenbyBoltzmannstatisticsforathermalstate.
k
B. Thermal Distributions with Large Coherent
As such, the chosen magnitudes |α | follow a probabil-
k
Displacements
ity distribution proportional to exp(−β|α |2). The ran-
k
dom value of φ is unweighted and assigned from the
k
set [0,2π), which assumes the phase of the initial state We now describe the spin dephasing signature of rela-
of COM motion is uncorrelated with that of the ODF tively large coherent motional displacements, αd, acting
andvariesforeachexperimentalsequence. Followingthe in addition to thermal fluctuations characterized by n¯.
pulse sequence of Fig. 4b with φ =π, we apply rele- Such analysis may be relevant when narrow-bandwidth
ODF
vant qubit rotations in sequence with coherent z-axis ro- electric field fluctuations exist on trap electrodes at fre-
tations due to the spin-dependent ODF given by Eqs. 15 quencies near that of the COM. As in the previous sec-
and 16, where α constitutes the initial coherent state tion,weassumethatthephasesofthermal,coherent,and
k
foreachrunandα isthespin-dependentdisplacement ODF displacements are all uncorrelated. It may seem
SE
calibrated as in Refs. [1, 8]. For completeness, we also that little can be inferred from dephasing in the pres-
include the smaller measured variance in the φ = 0 ence of coherent excitations with a random phase, but
ODF
casephenomenologicallybyaddinganother,uncorrelated we demonstrate that a constant magnitude is all that is
random rotation to the composite Bloch vector. The an- required to distinguish such noise sources.
gle of this additional rotation follows a Gaussian proba- We use resonant RF excitation of the COM mode to
bilitydistributionwiththemeasuredvariance(lightgray produce coherent states of motion whose square magni-
barsinFig.4b)[51]. Thesimulatedhistogram(blackline) tude, |α |2, is larger than the thermal magnitude given
d
ofFig.4bshowsgoodagreementwithexperimentalmea- by n¯. To this end, we apply an oscillating voltage with
surements, and further supports the absolute tempera- a frequency of ω to the upper endcap electrode of the
z
turemeasurementofFig.4a. Notethatthehighersensi- Penning trap for a period of 20 µs as in Ref. [37]. This
tivityofthedirectspin-dephasingmeasurementsenables homogeneous COM mode excitation is applied following
the use of a shorter free-evolution period. As a result, the initial Doppler cooling and state preparation pulses
9
a) 60 b)70 simulate the experimental histograms of Fig. 5a and 5b.
0.7(1) mK 0.7(1) mK
50 60 |α| = 8 We simulate 50,000 runs to minimize noise relative to
d
40 50 the experimental data. The simulation results given in
40 Fig. 5a (black-line histogram) use the same procedure as
30
30 described in Sec. IVA. To include the coherent excita-
20
20 tion for Fig. 5b, we simply redefine the initial coherent
100 100 state amplitude to be (cid:16)|αk|eiφk +|αd|eiφ(cid:48)k(cid:17). Only the
0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0
variables with the k subscript change with each simu-
c) pz d) α pz lation run, as |αd| = 8 is fixed for all pulse sequences.
SE Theamplitudes|α |2 onceagainfollowaBoltzmanndis-
α α k
SE d tribution, but here we use T = 0.7 mK to reflect the
α α elevatedCOMtemperatureforthissetofmeasurements.
k k
z z The phases φk and φ(cid:48)k are uncorrelated and chosen ran-
domly from the set [0,2π) for each simulation run. Ex-
FIG. 5: (color online) (a) Histogram of 1000 experimental ample paths of COM motion through phase space for
runs (red bars) under similar conditions as that shown in each Monte Carlo simulation run are shown schemati-
Fig. 4b with a measured n¯ = 18(3). These data were taken callyinFigs.5cand5dbothwithoutandwiththeinitial
without an initial coherent drive of the COM motion. Solid RF pulse, respectively. The effect of the spin-dependent
black histograms are obtained from Monte Carlo simulations ODF is depicted as two oppositely-oriented vectors lead-
of 50,000 experiment sequences with no adjustable parame- ingtoaseparationofthespinstatesby2α ∼0.1,while
SE
ters. (b) Measured and simulated histograms resulting from
all other excitations are common to both spins.
insertionofaspin-independentcoherentexcitation(|α |=8)
d
before the spin dephasing measurement. The effect of co-
herent excitation is visible despite the fact that the relative
phasesofα andtheODFα areuncontrolledfromoneex-
d SE C. Heating Rates
perimenttoanother. (c,d)Examplephasespacetrajectories
of the COM mode in a frame rotating at ω for each Monte
z
Carlosimulationstep,k,bothwithoutandwiththecoherent MotionalheatinginRFiontrapshasgainedincreased
RF drive. The initial displacement, α , for each simulation prominence in recent years due primarily to studies of
k
run is chosen randomly according to Boltzmann statistics to so-called “anomalous heating” from ion-surface proxim-
reflect thermal fluctuations of COM motion. ity [38–43]. Recent work suggests surface contamination
as the culprit [44, 45]. Additionally, micromotion in RF
trapslimitstheuseablesizeofCoulombcrystalsforquan-
but before the experimental pulse sequence of Fig. 4b. tum information and quantum simulation experiments.
The amplitude of the applied voltage is 110 µV (1.2 mV Penning traps use static potentials for ion confinement
m−1 attheionposition),correspondingtoacoherentex- and therefore do not induce micromotion, enabling the
citation of |α | ∼ 8 (∼ 30 nm amplitude). Histograms formation of large ion crystals [46, 47]. Furthermore,
d
compiled from 1000 experimental runs without and with the large physical size of typical Penning traps means a
theresonantRFexcitationareshowninFigs.5aand5b, likely insensitivity to anomalous heating processes. De-
respectively. The absolute COM temperature is deter- spite these encouraging features, no measurements have
mined to be 0.7(1) mK for the sequence without coher- yet been reported for ambient heating of a resolved mo-
ent excitation. This elevated COM temperature is due tional mode of an ion crystal in a Penning trap. A pre-
totheconnectionofthedirectdigitalsynthesizerusedto vious study of global (not mode-resolved) ambient heat-
applyresonantRFtotheupperendcap. Nevertheless,we ing of large 3D crystals in our Penning trap estimates
achieve |α |2/n¯ ∼ 3.5 for these experiments. Figure 5b that background gas collisions are a primary contribu-
d
shows the effect of applying the coherent motional exci- tor [48]. Heating rates of ∼65 mK/s were measured for
tation,namelythatthespindistributionissplitintotwo background pressures of ∼4×10−9 Pa (3×10−11 Torr),
peaks whose separation is determined by the combined which translates to dn¯ ∼ 1.7×103 s−1 at our trap fre-
dt
coherentandODFdisplacements. Iftherelativephaseof quency of 795 kHz.
αd and αSE were fixed for every experimental sequence, We apply the thermometry techniques summarized in
then the mean of the histogram of Fig. 5a would simply Fig. 4 to obtain an initial measurement of the ambient
be shifted to a new position corresponding to a coherent heating rate of the axial COM mode of our 2D crystals.
Bloch vector rotation, with the standard deviation still WeincludeavariabledelaybetweeninitialDopplercool-
reflecting the COM temperature of 0.7(1) mK. However, ing/state preparation and application of the experimen-
the randomness of the relative phases of αd and αSE in tal pulse sequences. Any increase in n¯ over this initial
combination with their constant amplitudes leads to a delay period is measured in the subsequent decoherence
characteristic splitting of the spin distribution about the or dephasing measurement. Figure 6 shows measured
mean. absolute COM temperatures as a function of initial de-
As in Sec. IVA, we use a Monte Carlo algorithm to lay (points with error bars) along with linear fits to each
10
data set (solid lines). The data of panels 6a and 6b are a) b)
identical, but plotted on linear and logarithmic vertical
250 ions
axes, respectively, for clarity. The fitted slopes reflecting
dn¯ foreachcurvearedisplayedinFig6bandcolor-coded 185 ions
dt
to match the corresponding data set.
Wemeasurethelargestheatingrateof1.4(2)×104 s−1
(black points) when the trap endcap electrodes are held 240 ions
at 0 V using the high voltage power supplies responsi-
ble for initial ion loading and transport. Small voltage
fluctuations from these power supplies as well as elec-
tromagnetic interference along the connecting cables are FIG. 6: (color online) Measured COM heating rates (points
the likely cause of this COM heating. Upon grounding witherrorbars)andcorrespondinglinearfits(solidlines)plot-
ted with both (a) linear and (b) logarithmic vertical scales.
theendcapsdirectlytothevacuumchamberatthehigh-
Thehighestheatingrateof dn¯ =1.4(2)×104 s−1 isobtained
voltage feedthrough, we observe an order-of-magnitude dt
with the trap endcap electrodes grounded through the high
drop in the heating rate to between 4.7(8) × 102 s−1
voltage power supplies used for initial ion loading. The two
(blue points) and 1.2(3) × 103 s−1 (red points). This
lower heating rates are measured with the endcaps shorted
approximate factor-of-two variation in heating rates is directly to the trap vacuum ground at the high voltage vac-
representativeofourday-to-dayobservationswithdiffer- uumfeedthrough. Thevariationbetweenthetwolowestrates
ent ion samples, and we see no evidence of a correlation is representative of the range of heating rates measured thus
betweentotalionnumberandheatingratewiththeend- far under the given conditions.
cap electrodes grounded at the feedthrough. The mean
ofthetwolowestheatingratescorrespondsto∼30mK/s,
which is near previous collisional heating estimates [48].
Additional filtering of the central ring and rotating-wall tinction between coherent and thermal states of motion,
electrodes outside the vacuum envelope yielded no mea- despitetherandomnessoftheRFdrivephaserelativeto
surable improvement in COM heating [52]. our optical dipole force.
Because the distance from the trapped-ion arrays to
The methods and analysis presented here enable very
the trap electrode surfaces (≥2 cm) is large compared to
sensitive detection of coherently driven motion of a
the diameter of the planar array (<0.5 cm), electric field
trapped-ioncrystal,andmaybeusedtophase-sensitively
noise from trap electrode surfaces will be uniform across
detectweakforces[37,49]. SectionIIIcanbeusedtoes-
thearrayandpreferentiallyheattheCOMmode. Foran
timate and optimize the force detection sensitivity for
array with N ions this results in a linear dependence of
a given N and temperature of the trapped-ion crystal.
theCOMheatingrateonionnumberduetouniformelec-
Weestimatethatthespin-motioncouplingtechniquedis-
tricfieldnoise[5]. ThelackofanobservedN-dependence
cussed here could improve on the force sensitivity ob-
inthemeasuredambientheatingrateindicatesthesource
tained in Ref. [37] by more than an order-of-magnitude.
of the heating is likely not electric field noise. However,
any potential anomalous heating must be less than the We also present the first measurements of ambient
measured ∼103 s−1 ambient heating rate. Dividing this heating of a resolved mode of motion in a Penning trap.
limit by the number of trapped ions (N ∼ 200) gives a Future crystal heating measurements will include other
limit on the anomalous heating rate for a single trapped resolvedtransversemotionalmodeswiththegoalofmore
ion of ∼5 s−1 at a trap frequency of 795 kHz. clearly distinguishing between electric field fluctuations
(mode-specific, N-dependent) and background gas col-
lisions (mode- and N-independent). Nevertheless, we
demonstratethatlowheatingratesareindeedachievable
V. CONCLUSION
in Penning ion traps.
In summary, we have demonstrated a new technique
for analyzing the motional state of a resolved ion crystal
mode. The methods presented here do not rely on stim-
ulated Raman transitions or Doppler linewidth analysis,
and are in principle applicable to any resolved motional
Acknowledgments
mode at any temperature provided the Lamb-Dicke con-
finement criterion is satisfied for the given mode. The
sensitivityofourspindephasingmeasurementsallowsfor This work was supported by the DARPA-OLE pro-
a regime of operation with negligible spin-motion entan- gramandNIST.TheauthorsthankJ.P.Home,K.R.A.
glement and spontaneous emission decoherence. Monte Hazzard, M. Foss-Feig, and A. M. Rey for useful discus-
Carlo simulations based on the semiclassical description sionsaswellasS.KotlerandJ.P.Gaeblerforcomments
of Sec. III are in excellent agreement with spin dephas- on the manuscript. This manuscript is a contribution of
ing measurements. Furthermore, we observe a clear dis- NIST and not subject to U.S. copyright.