Table Of ContentIon trap transducers for quantum electromechanical oscillators
W. K. Hensinger,1 D. W. Utami,2 H.-S. Goan,3 K. Schwab,4 C. Monroe,1 and G. J. Milburn2
1FOCUS Center and Department of Physics, University of Michigan, Ann Arbor, MI 48109-1120 USA
2Centre for Quantum Computer Technology and Department of Physics,
School of Physical Sciences, The University of Queensland, QLD 4072 Australia
3Department of Physics, National Taiwan University, Taipei 106 Taiwan
4Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park MD 20740 USA
(Dated: February 1, 2008)
Anenduringchallengeforcontemporaryphysicsistoexperimentallyobserveandcontrolquantum
behaviorinmacroscopicsystems. Weshowthatasingletrappedatomicioncouldbeusedtoprobe
thequantumnatureofamesoscopicmechanicaloscillatorprecooledto4K,andfurthermore,tocool
the oscillator with high efficiency to its quantum ground state. The proposed experiment could be
5 performed using currently available technology.
0
0 PACSnumbers: 62.25.+g,42.50.Vk,72.70.+m,73.23.-b,61.46.+w
2
n
Aquantumelectromechanicalsystem(QEMS)is ade- to posses sensitivity of QEMS energy at the level of a
a
vice where the quantum nature of either the electronic single quanta, a detection which is very difficult with a
J
or mechanical degrees of freedom becomes important in simple linear coupling to displacement. Finally, we show
9
the observable behavior [1, 2]. Potential applications of that the ion-QEMS system can be configured to provide
1 QEMS include single spin detection [3], single molecule a very effective cooling mechanism for the mechanical
v mass spectrometry [4], and readout for quantum infor- oscillators [9] and estimate that it should be possible to
7
mation devices [5]. An excellent example of such a sys- cool a QEMS cantilever, precooled only to 4 Kelvin, to
3
tem is a rf single electron-transistor integrated with a its ground state with very high efficiency.
0
1 lowloss,highfrequencynanomechanicalresonatorwhich Herewe describeaquantummodelofasingletrapped
0 demonstratedbothcontinuouspositiondetectionapprox- atomic ion which is electrostatically coupled to a very
5 imatelyafactoroffiveawayfromthe uncertaintyprinci- small doubly clamped cantilever, Fig. 1. This coupling
0
plelimit,anddirectobservationsofthemechanicalmode canbeswitchedonandoffusinganexternalbiasvoltage
/
h occupation factor as low as N 58 [6]. However, this de- at an electrode on the oscillator. The ion is held in a
≈
p vice suffers from both practical and fundamental limi- mesoscopic micro-fabricated quadrupole Paul trap [12],
-
t tations; picowatt levels of dissipation at the transistor and laser cooled using resolved sideband cooling [13].
n
is suspected to have blocked passive cooling of the me- Externallasers are used to couple the internal electronic
a
u chanics to lower temperatures and occupation factors, statesoftheiontoitsvibrationaldegreeoffreedom. The
q as well as the fundamental sensitivity limitation due to
: coupling to an intrinsically non QND (quantum non-
v
demolition) variable such as position. Other methods
i
X to perform measurements on quantum limited mechan-
r ical systems are clearly needed. Furthermore, coupling
a
atomic systems to nano-electronic and nano-mechanical
devices appearsto be a very promising and exciting new
frontier, where it is hoped one might combine the un-
matchedquantumcoherenceanddetectionefficienciesof
atomicphysics,withthe sophisticatedelectronicandmi-
FIG. 1: Schematic representation of a single trapped atomic
crostructureswhicharepossibleinthecondensedmatter
ion coupled to an nanoelectromecanical oscillator.
realm [7, 12].
In this paper we investigate the use of a laser-cooled, bias gate on the oscillator carries a charge Q = C V (t)
o o
trapped atomic ion to both monitor and manipulate the where C is the capacitance of the gate. We allow for
o
number-stateofaQEMSoscillator,followingearliersug- the possibility for the bias gate voltage V (t) to be time
o
gestions by Wineland and others [9, 10]. As the tem- dependent so that it may be set to zero to turn off the
perature of an ion’s vibrational degree of freedom can electrostaticcouplingtothe trappedion. Theioncarries
be < 10−3 K, it is hoped that such a transducer could charge +e. In the geometry of Fig. 1, the interaction
generate far less thermal power than low temperature energy between the ion and the oscillator is given by
single-electrondevices. Furthermore,duetotheexcellent V =keV C /d+Xˆ(t) xˆ(t) wheredisthe equilibrium
c o o
| − |
optical readout achieved in ion trap systems via fluores- separation of the oscillator centre of mass position and
cence shelving [11], the ion trap transducer is expected theion. Wesetthe equilibriumpositionoftheionatthe
Typeset by REVTEX
2
originforsimplicity. InthatcaseXˆ(t)andxˆ(t)represent tonian for the three coupled systems may be written as
the small oscillations of the QEMS oscillator and the H =h¯ωa†a+h¯νb†b ¯hκ(a+a†)(b+b†)+h¯ωAσ +H where
− 2 z e
ion around their equilibrium positions, respectively. H describes the interaction between the external laser
e
We now assume that the deviations from equilibrium andtheelectronicstatesoftheion(seebelow). Thecou-
are small compared to the equilibrium separation plingconstantκisgivenbyκ=(mMνω)−1/2keV C /d3.
o o
d and expand to second order in (Xˆ(t) xˆ(t))/d. The electronic transition frequency is ω and σ =
A z
−
The interaction energy is then given by V = e e g g is a Paulimatrix. The interactionbetween
c
| ih |−| ih |
(keV C /d) 1 (Xˆ(t) xˆ(t))/d+(Xˆ(t) xˆ(t))2/d2 . the control laser and the electronic transition is given in
o o h i
− − − the dipole approximation by H = 2h¯Ωσ sin(k xˆ ω t)
The linear term may be absorbed into the definition of e x l − l
whereω andk arethefrequencyandwavevectorofthe
the equilibrium positions. The quadratic term includes l l
laser, Ω is the effective Rabi frequency for the transition
a coupling of the oscillators and a renormalisation
andσ = e g + g e. WewillworkintheLamb-Dicke
of the oscillation frequency for both the ion and the x | ih | | ih |
limit in which the amplitude of the ion’s motion in the
oscillator. This frequency shift is a small perturbation
direction of radiation is much less than the wave length
of the bare frequencies and is neglected. We then see
ofthelaser√n +1η =√n +1k (h¯/(2mν))1/2 <<1in
that the interaction Hamiltonian coupling the ion to the b b l
oscillator is H = χXˆ(t)xˆ(t) where χ = 2keV C /d3. which case we may take He = 2h¯ηΩσx(b+b†)sin(ωlt).
c o o
− We now move to the interaction picture and make the
A similar ion-oscillator system is described in [14].
rotating wave approximation for both the QEMS-ion
Iniontraptechnologya lasercanbe directedontothe
coupling and the electronic coupling, and assume that
ion to induce transitions between the internal electronic
ω = ω ν or ω = ω +ν so that we are resolving
states [13]. The electric field seen by the ion depends l A − l A
the first red or blue sideband. The total Hamiltonian is
on its motion in the trap, thus the external laser can
H = h¯∆a†a ¯hκ(a†b + ab†) + H with ∆ = ω ν
couple the internalelectronicstate to the vibrationalde- I − sb −
gree of freedom. In this way information on the vibra- and Hsb = h¯g(bσ+ + b†σ+†) for the red sideband and
tional motion canbe transferredto the electronic degree Hsb =h¯g(b†σ++bσ+†)forthebluesidebandwhereg =ηΩ
offreedomwhereitcanbeefficientlymeasuredusingthe andσ+ = e g . Inourmodelbothκandgcanbeturned
| ih |
on and off. The measurement protocol has two stages.
technique of fluorescence shelving [11]. Such measure-
In stage I, the oscillator and cantilever are coupled for a
ments are very nearly perfect projective measurements
time τ by setting κ = 0,g = 0. In stage II, κ is turned
onto one of two electronic states, g , e . If the laser
6
{| i | i} off and the electronic and vibrationaldegrees of freedom
is detuned to the first red (blue) motional sideband, the
of the ion are coupled by red and blue sideband excita-
ioncanmakeatransitionfromthe groundtotheexcited
tion. At the end of this stage, the ionic electronic state
state absorbing a laser photon and a single quanta of vi-
is measured.
brational energy is simultaneously added (deducted) in
the process [13]. The probability of excitation depends
Using the ion as an ultrasensitive sensor for the can-
on the phonon distribution in the trap, so subsequent
tilever motion is experimentally realistic. Assuming a
readout of the electronic state then reveals information
19.7 MHz cantilever, as reported recently by LaHaye et
on the phonon number in the trap.
al. [6], a radio-frequency microfabricated ion trap [12]
Theoverallschemeforusingthetrappedionasatrans-
could be used to confine the ion at a distance β of 50µm
ducerofQEMSmotionisasfollows. Theioniscooledto
from the cantilever at a secular frequency of 19.7 MHz.
thevibrationalgroundstateandpreparedinsomeappro-
Applying a static voltage of 7.5V one would obtain a
priate electronic state. Next, a voltage is applied to the
coupling frequency Ω 2π 52.5 kHz for a cadmium
d
oscillator gate, coupling the motion of the QEMS to the ≈ ×
ion. Assuming the cantilever at a temperature at 4 K
vibrationalstate ofthe ion. This transfersphonons from
(liquid Helium), the cantilever contains on the order of
the oscillator to the ion. In the next step an external
1000 quanta of motion. Therefore one would only re-
laser couples the trap phonons onto the electronic de-
quire an interaction time of 5 µs for the ion to undergo,
gree of freedom which is then readout using fluorescence
onaverage,asinglephononexcitation. Anharmonicityin
shelving.
the cantilevermotionisincluded inthe measuredQ(too
We define dimensionless annihilation and creation op- small to observe at 4K) and will not measurably distort
erators for the ionic vibration and the QEMS vibration the cantilever lineshape for any reasonable temperature
using the ground state standard deviation of position of [15]. An upper bound for the width of the secular mo-
the oscillator as a convenient length scale in each case, tion frequency due to the anharmonicity of the ion mo-
Xˆ =(h¯/(2Mω))1/2(a+a†)andxˆ=(h¯/(2mν))1/2(b+b†) tion at n¯ = 4000 is found to be ∆ν ν(z/β)2 where
b
whereν isthevibrationalfrequencyoftheioninthetrap z =(h¯n¯ /(2mν))1/2istheextentofth≈eatomicwa∼vefunc-
b
(also called the secular frequency), ω is the resonantfre- tion and β is the dimension of the ion trap. For the
quencyoftheQEMSoscillator,andM,marethemasses parameters considered here, ∆ν κ, so ion trap anhar-
≪
of the QEMS and the ion respectively. The total Hamil- monicitiesarenotexpectedto degradetheion-cantilever
3
coupling. The QEMS oscillator is always coupled to a stage one the QEMS oscillator and ionic vibrational
thermalising reservoirandbefore the coupling to the ion mode are in thermal states with mean phonon num-
is turned on we assume that the oscillator is in a ther- ber n¯a0 and n¯b0, respectively. For γa κ one ob-
malequilibriumwithmeanvibrationalquantumnumber tains n¯b(τ) = n¯a0 (n¯a0 n¯b0)e−γaτ/2co≪s2(κτ). The
n¯a0 =(eh¯ω/kBT 1)−1 kBT/(h¯ω). mean oscillator pho−non num−ber can also be found to be
When the cou−pling b≈etween the oscillator and the ion n¯a(τ) = n¯a0 −(n¯a0 −n¯b0)e−γaτ/2κ2sin2(Ωγτ)/Ω2γ. In
is turned on, the ion becomes indirectly coupled to the Fig. 2, we plot n (τ) (obtained from Eq. (2)) using re-
b
phonon reservoir of the QEMS oscillator. This may be alistic parameters corresponding to the mechanical can-
modelledusingaquantumopticsmasterequationforthe tileverofRef. [6]. Assumingn¯a0 =4000,n¯b0 =0,∆=0,
oscillator (if the frequency of the oscillators is not too κ=2π 52.5 kHz and assuming a Q factor for the can-
×
low)thatdescribesthejointdensitymatrixoperatorR(t) tilever of 30000 we find that the effect of the interaction
for the QEMS oscillator and the ionic vibration motion of the cantilever with the thermal bath is approximately
during stage I, negligible during the first quarter coupling period, how-
ever, becomes significant on longer timescales. In the
dR
dt = −i∆[a†a,R]+iκ[a†b+ab†,R] limit γaτ <<1 we find that n¯b(τ)≈n¯a0sin2(κτ).
+γa(n¯a0+1) [a]R+γan¯a0 [a†]R
D D
+µ1 [b]R+µ2 [b†]R, (1)
D D
where [ ]ρ= ρ † ( † ρ+ρ † )/2 is defined for
DO O O − O O O O
arbitrary operators and ρ, and the damping rate γ
a
O
is related to the quality factor Q of the QEMS oscilla-
tor by the expression γ =ω/Q. Note that choosing the
a
damping rates µ1 =γb(n¯b0+1), µ2 =γbn¯b0 corresponds
to the dynamicsforthe ionicvibrationalmotioninduced
byinteractingwithathermalbath, asinthe caseforthe
QEMSoscillator. However,if the mainnoise sourcecou- FIG.2: Themeanphononnumberoftheionn¯b(τ)asafunc-
tion of time τ obtained from Eq. (2) is plotted (solid line)
pleddirectly to the vibrationalmodes ofthe trappedion
and compared to the case where the coupling of cantilever
isthe fluctuatingelectricfieldgeneratedinthe trapelec-
and thermal reservoir is neglected during the cantilever-ion
trodes[16], wecanmodelitasaclassicalstochasticelec- interaction (dashed line). The mean phonon number of the
tric field. In this case, we take µ1 = µ2 = (1/τ1), where cantilever n¯a(τ) is plotted as dotted line. The inset shows
τ1 is the characteristic heating time [17]. Extrapolated quadraticshort time behaviour of theion.
from recent experiments and associated theory for char-
acteristic heating times [16, 18], we obtain an expected From Fig. 2 we see that eventually the ionic vibra-
approximateheatingrateontheorderof0.06quantaper tional degree of freedom comes into thermal equilibrium
msfortheabovegeometryforthecadmiumion. Compar- withthe cantilever. Howeverdue tothe verystrongcou-
ing this heating rate with the coupling constant κ from pling between the two, the dynamics of this process is
ourexampleaboveitisreasonabletoneglectionheating certainly not exponential. The oscillations in Fig. 2
during the cantilever-ion interaction. We are interested are thus evidence of non-exponential relaxation. The
in the mean phonon number n¯ (τ) = b†(τ)b(τ) of the experiment proposed here provides a convenient way to
b
h i
thermal state of the ionic vibrational modes at the end study the transition from exponential relaxation to non-
of stage one. Using Eq. (1) the equations of motion for exponential short time behavior by varying the coupling
quadratic moments form a closed set and do not involve strength κ with respect to the damping rate of the can-
higher-order moments. tilever, γ .
a
Neglecting heating due to a classical stochastic field The readoutprocessofstageII issimilarto the exper-
during the ion-cantilever interaction (as justified above) iments described in Ref. [13]. In stage II we couple the
andassumingthesecularionfrequencyequaltothereso- electronic state of the ion to its vibrational motion for a
nancefrequencyofthe cantileveronecanderiveasimple time T using the first red and blue sideband transitions.
analytical expression (∆=µ1 =µ2 =0): If we write the probability for the atom to found in the
excited state after time T as PR(T) and PB(T) for red
n¯b(τ) = n¯a0− (n¯a08Ω−2n¯b0)e−γaτ/2 andbluesidebandexcitationresepectivelyitceanbeshown
γ [16, 19] that the mean phonon number n¯ (τ) is given by
b
[(4Ω2 2κ2+iγ Ω )e−i2Ωγτ +4κ2 n¯ (τ)/(1 +n¯ (τ)) = PR(T)/PB(T) R . As this re-
× γ − a γ b b e e ≡ e
+(4Ω2 2κ2 iγ Ω )e2iΩγτ], (2) sult is independent of the coupling time T, noise due to
γ − − a γ heating during stage two can be kept small by minimiz-
where Ω = κ2 (γ /4)2. The initial conditions ing T. Indeed such experiments are routinely performed
γ p − a
used to obtain Eq. (2) assume that at the start of [18, 19, 20] and a phonon number smaller than unity
4
can be determined. As the parameters κ,τ are assumed This scheme is not a single shot readout of phonon
to be known this gives n¯a0 directly. In particular, for numberintheoscillator,ratheritenablesastatisticalin-
the short time region of Fig. 2 (and γ τ << 1) we find ferenceofthemeanphononnumberoftheoscillator. Can
a
that PeR(T)/PeB(T) ≈ n¯a0κ2τ2. Thus measurement of it be used to measure a weak classicalforce ? The sensi-
the ratio of excitation probability on the first red and tivity to changes in n¯ (τ) is best for values of n¯ (τ) less
b b
bluesidebandyieldsn¯a0 directly. Notethatn¯b(τ)should theunity. Thiscanbeachievedbyensuringn¯a0κ2τ2 1.
≤
be on the order or smaller than n = 20 for reliable It should then be possible to detect a change in the in-
max
measurement [10, 13, 16]. ferredmeanphononnumberoftheoscillatorofthe order
From Fig. 2 we see that it is possible to almost com- of one quanta. If a weak classical force is continuously
pletely exchange the mean phonon number of the can- applied to the cantilever after it has been preparedclose
tileverandtheionicvibration. Sincetheionbeginsinthe to the ground state (eg. via sympathetic cooling via the
vibrational ground state, the result after one exchange ion), its equilibrium phonon distribution will shift by an
time is to transfer the thermal energy of the cantilever amountproportionaltothesquareoftheratioofapplied
rapidlyto the ionicvibration,leavingthe cantilevernear force to the energy damping rate. Thus we should be
its ground state. able to infer the size of a classical force so weak as to
A number of cooling procedures appear possible. One shift the mean phonon number of the oscillator by one
couldsimplydumptheresultinghotionoronecouldim- quanta. This corresponds to a displacement sensitivity
plement two ion traps that are located adjacent to the at the standard quantum limit ∆x =(h¯/2mω)1/2.
SQL
cantilever: one ion trap to be used to cool the mechan-
GJM acknowledges the support of the ARC Federa-
ics to the ground state, while the other ion trap is used
tion Fellowship Grant. This work was supported by FO-
to perform the quantum measurement of the cantilever.
CUSseedfundingandtheU.S.NationalSecurityAgency
Alternately, an iterative cooling mechanism could be ac-
andAdvancedResearchandDevelopmentActivityunder
complishedbydecouplingthe resultingionfromthe cold
Army Research Office contract DAAD19-01-1-0667 and
cantilever (instead of dumping it) by detuning the ion
theNationalScienceFoundationInformationTechnology
trap secular frequency and then laser cooling the ion.
Research Program.
One could also accomplish cooling of the cantilever and
obtain a reasonable temperature estimate using just a
single ion provided n¯a0πγa/κ<nmax (meaning that the
ion carries less than n phonons after one full cou-
max
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