Measuring the Momentum of a Nanomechanical Oscillator through the Use of Two Tunnel Junctions

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by  C. B. Doiron

Measuring the momentum of a nanomechanical oscillator through the use of two
tunnel junctions
C.B. Doiron,1 B. Trauzettel,2,1 and C. Bruder1
1Department of Physics, University of Basel, CH-4056 Basel, Switzerland
2Institute of Theoretical Physics and Astrophysics,
University of Wu¨rzburg, D-97074 Wu¨rzburg, Germany
(Dated: November2007)
8 Weproposeawaytomeasurethemomentumpofananomechanicaloscillator. The p-detectoris
0 based on two tunnel junctions in an Aharonov-Bohm-typesetup. One of the tunneling amplitudes
0 depends on the motion of the oscillator, the other one not. Although the coupling between the
2
detector and the oscillator is assumed to be linear in the position x of the oscillator, it turns out

n that the finite-frequency noise output of the detector will in general contain a term proportional
a to the momentum spectrum of the oscillator. This is a true quantum phenomenon, which can be
J realizedinpracticeifthephaseofthetunnelingamplitudeofthedetectoristunedbytheAharonov-

6 Bohm fluxΦ to a p-sensitivevalue.
1
PACSnumbers: 72.70.+m,73.23.-b,85.85.+j

]
l
l
a In nano-electromechanical (NEM) systems, it is the po- tor which has been analyzed theoretically in greatdetail
h
sition of the oscillator that typical measurement devices [9, 10, 11, 12, 13, 14, 15] and experimentally realized
-
s (like tunnel junctions or single electron transistors) are [16, 17] is depicted in Fig. 1a. It shows a single tun-
e
coupled to. Using these detectors, position measure- nel junction coupled to a NEM oscillator. A thorough
m
ments with sensitivities close to the standard quantum analysis of the coupled quantum system leads to the re-
.
t limit have alreadybeen observed[1, 2, 3]. Froma funda- sult that the output signal of the detector is sensitive to
a
mental point of view, it is desirable to go further, i.e. to the position spectrum S (ω) = dteiωth{xˆ(t),xˆ(0)}i of
m x
prepareandmanipulateNEMoscillatorsinthequantum the oscillator. The modificationoRfthe detector shownin
-
d regime. AquantumNEMsystemwouldallowustostudy Fig. 1b instead allows for a measurement of Sp(ω).
n anidealrealizationofacontinuousvariablequantumsys- TheHamiltonianofthecoupledsystemH =H +H +
osc B
o tem [4]. The exploration of such systems has to be seen H is the sum of the Hamiltonian of the (quantum)
tun
c
ascomplementarytothewidestudy oftwo-levelsystems harmonicoscillatorH (withmassM andfrequencyΩ),
[ osc
done in the context of quantum computing. the bath Hamiltonian H (describing the leads of the
B
2 detector), and the tunneling Hamiltonian H (which
v In order to be able to fully characterize a continuous tun
couples the dynamics of the electrons that tunnel across
9 variable quantum system that is described by two non-
the junction to the motion of the oscillator):
0 commutingoperatorsxˆandpˆ,weneedtobeabletomea-
7
sure expectation values of moments of both of them [5].
2 Onlythisallows,forinstance,todetecttheentanglement H =~Ω(a†a+ 1)= pˆ2 + MΩ2xˆ2 , (1)
. osc 2 2M 2
7 between two (or more) NEM devices [6]. The literature
0 H = ε c†c + ε c†c , (2)
already contains proposals regarding quantum measure- B k k k q q q
7 mentsofthemomentumofmacroscopicobjectslikethose Xk Xq
0
v: used for gravity-wave detection [7]. However, none of Htun =T(xˆ)Y† c†kcq+T†(xˆ)Y c†qck . (3)
i theseproposalshavebeenrealizedinpractice. InthisLet- Xk,q Xk,q
X ter,weproposearealisticandfeasiblewaytomeasurethe
r momentumofananometer-sized resonator. Thisisanon- Here,k (q)isawave-vectorintheright(left)lead,c(†) is
a trivial task since the coupling between the detector and the electron annihilation (creation) operator, and Y(†)
the oscillator is naturally described by an x-dependence is an operator that decreases (increases) m, the num-
but not a p-dependence. Nevertheless, the proposed ber of electrons that have tunneled through the system,
setup (shown in Fig. 1b) allows for a measurement of by one. It allows one to keep track of the transport
the momentum spectrum S (ω) = dteiωth{pˆ(t),pˆ(0)}i processes during the evolution of the system. For typ-
p
of the oscillator. This can be donRe because we have ical nanoresonators, the mass M varies between 10−18
found a way to tune the phase of the tunnel coupling and 10−15 kg, and the resonance frequency is usually
term that is sensitive to the position of the oscillator 1MHz<Ω/2π<1GHz [18].
by an Aharonov-Bohm (AB) flux Φ, see Fig. 1b. Re- We first discuss the model in the standard configuration
lated setups have been investigated recently in the con- shown in Fig. 1a and later on describe the new setup
text of dephasing due to the coupling of an AB ring in Fig. 1b. For small displacements with respect to the
structure to a NEM device [8]. A typical position detec- tunneling length (which is the relevant regime in typi-
2
FIG.1: (coloronline)aPositiondetector. ThefigureshowsschematicallyapositiondetectorofthemotionofaNEMoscillator
(red). The detector is based on a tunnel junction with a tunnel matrix element T(xˆ) which depends on the position of the
oscillator. The shaded regions (yellow) are assumed to be conducting. b Position and/or momentum detector. The figure
illustratesadetectorwhichcontainstwotunneljunctionsthatformaloopthreadedbyamagneticfluxΦ. Tuningthefluxwill
change theperformance of thedetectorfrom beingable todetect thepower spectrum of theposition operator of theoscillator
to being able to detect the power spectrum of the momentum operator of the oscillator. For clarity, the two insets show a
simplified illustration of thedetectors in a and b.
cal experiments on NEM devices), the tunneling ampli- T(xˆ,Φ)=(τ (Φ)+eiη(Φ)τ xˆ)/(2πΛ) with
0 1
tude T(xˆ) can be taken as a linear function of xˆ, namely
T(xˆ) = (eiϕ0/2πΛ) τ +eiητ xˆ , where τ and τ are Φ
0 1 0 1 τ2(Φ) = τ2 +τ2 +2τ τ cos 2π +ϕ −ϕ ,
real, and Λ is the de(cid:0)nsity of stat(cid:1)es. The phases ϕ0 and 0 0,d 0,u 0,d 0,u (cid:16) Φ0 0,d 0,u(cid:17)
η describe details of the detector-oscillator coupling [19]. Φ
η(Φ) = 2π +ϕ −ϕ
Itcanbe shownthatourdetectorisquantum-limitedfor Φ 1,d 0,u
0
danetyecηt,oarcicnorRdienf.g[t1o4]t.heWdietfihninititohneosfinaglqeu-jaunntcutmio-nlimseittuepd − Arg τ0,u+ei(2πΦΦ0+ϕ0,d−ϕ0,u)τ0,d , (4)
(cid:16) (cid:17)
(Fig. 1a), the relative phase η is sample-dependent and
cannot be tuned experimentally. In a typical device, where we have defined Tu ≡ eiϕ0,uτ0,u, Td ≡ eiϕ0,dτ0,d+
the x-dependence of the phase of T(xˆ) is much weaker eiϕ1,dτ1,dxˆ, τ1 ≡ τ1,d, and Φ0 = h/e [24]. The position-
than the x-dependence of the amplitude. Then, we can independent part of the tunneling amplitude τ0(Φ) dis-
set η ≃ 0, and the tunnel junction acts as a position- plays the standard AB oscillations as a function of flux.
to-current amplifier where the frequency-dependent cur- Likewise, the relative phase η(Φ) shows a distinct de-
rent noise SI(ω) of the detector contains a term propor- pendence on the flux. Importantly, for τ0,u > τ0,d,
tional to the position spectrum S (ω) of the oscillator, the phase η(Φ) can be tuned continuously in the whole
x
i.e. ∆SI(ω)=SI(ω)−2ehIi≈λ2xSx(ω)withλx thegain range [−π,π]. In the limit, where τ0,u ≫ τ0,d, η(Φ) ∼
of the amplifier [13, 20]. 2π Φ + η(Φ = 0) varies linearly with the applied flux.
Φ0
In the opposite regime τ ≪ τ , η no longer depends
0,u 0,d
We now demonstrate that a tunnel junction with a on Φ. Therefore, it is crucial to put the tunneling am-
phase η = π/2modπ acts as a momentum detector plitudes in the regime where η(Φ) can be tuned to π/2.
and ∆S (ω) ≈ λ2S (ω), where λ is the gain of the We will show below that a feasible way to calibrate η(Φ)
I p p p
momentum-to-current linear amplifier. The critical re- to the p-sensitive point π/2 is a measurementof the flux
quirement to build a momentum detector is to be able dependence of the current through the AB detector.
to vary η experimentally. This can be done using the Similarly to Refs. [13, 20], we study the coupled system
AB-type setup shown in Fig. 1b: a metallic ring where using the quantum equation of motion for the charge-
one arm is a standard tunnel junction position detector resolveddensity matrix within the Born-Markovapprox-
with tunneling amplitude T (xˆ), and the other arm is a imation, assuming that eV ≫~Ω.
d
position-independent tunnel junction with tunneling am- It has been derived previously that, under the assump-
plitudeT [21]. ThetotaltransmissionamplitudeT(xˆ,Φ) tion that the tunneling amplitude depends linearly on xˆ,
u
ofthedeviceisthesumofbothtunnelingamplitudes[23]. the equation of motion for the reduced density matrix
Since only one arm shows a position-dependence, the in- of the oscillator is of Caldeira-Leggett form [12, 13, 15].
duced phase difference between the two arms affects the Thus, it contains both a damping and a diffusion term.
position-independent and the position-dependent parts Whenthe electrontemperatureismuchsmallerthanthe
of the tunneling amplitudes τ and τ in a different way. applied bias (and taking V > 0), the detector-induced
0 1
Explicit calculation shows that the tunneling amplitude dampingcoefficientisγ =~τ2/(4πM)andthediffusion
+ 1
isgiven(uptoaglobalgauge-dependentphasefactor)by coefficient is D =2Mγ k T with T =eV/2k .
+ + B eff eff B
3
In general, the oscillator is not only coupled to the de- trum is therefore ∆S =∆S +∆S with
1 2
tector but also to the environment. The coupling to this
additional bath is controlled via γ0 = Ω/Q0 (related to ∆S (ω)= λ2 1− ~Ω ∆x20 (7)
thefinitequalityfactorQ0 ofthemode,whichincurrent 1 h x(cid:18) 2eV hhx2ii(cid:19)
experiments varies from 103 to 106 [18]) and the associ- MΩ eV
−λ λ τ2∆x2 S (ω)
ated diffusion constant D0 = 2Mγ0kBTenv that must be x p(cid:18) 2π 1 0MΩ2hhx2ii(cid:19)i x
addedtothedetector-induceddampinganddiffusioncon-
MeV hhp2ii 4(Ω2−ω2)
stantstofindthetotaldampingcoefficientγ =γ +γ −λ λ 1−
tot + 0 x p(cid:18) hhp2ii(cid:19) M 4γ2 ω2+(ω2−Ω2)2
and the total diffusion coefficient D =D +D . T tot
tot 0 + env
denotes the temperature of the environment. In typical ∆S (ω)=λ2 1− MeV S (ω) (8)
experiments, it varies from 30mK to 10K. Within our 2 p(cid:18) hhp2ii(cid:19) p
model, all the these system parameters are independent ~Ω ∆x2
+ λ λ 1− 0
of the applied flux. p x(cid:18) 2eV hhx2ii(cid:19)
h
It is now straightforward to calculate the current and MΩ eV 4MΩ2hhx2ii(Ω2−ω2)
−λ2 τ2∆x2 ,
the current noise of the detector. We skip the details p(cid:18) 2π 1 0MΩ2hhx2ii(cid:19)i 4γt2otω2+(ω2−Ω2)2
here (see Ref. [20] for η = 0) and directly turn to the
results. The average current of the detector is given by where the position and the momentum gain are given
by λ = 2eτ τ (eV/h)cosη and λ = (e/2πM)τ τ sinη,
x 0 1 p 0 1
respectively. We now discuss several limits of the cur-
I = e2V τ2+2cosητ τ hxi+τ2hx2i rent noise SI(ω) of the detector in the case of a gen-
h 0 0 1 1 eral phase η. For η = 0modπ, we recover Eq. (30) of
(cid:0) (cid:1)
2eγ+τ0 Ref. [20] – the position detector result. More interest-
− sinηhpi−eγ . (5)
~τ1 + ingly,forη =π/2modπ, λx =0andthedetectoroutput
contains only two terms: The firstone is proportionalto
S (ω)andthereforepeakedaroundΩ. Thesecondoneis
Forη 6=0modπ,theaveragecurrentcontainsatermpro- p
proportionalto(Ω2−ω2)andcontributesnegligiblynear
portionaltotheaveragemomentumoftheoscillatorthat
resonance ω ≈Ω. Hence, for η =π/2, we obtain
does not vary with the applied bias [15]. However, since
hpi=0 in the steady-state, the average current contains
MeV
no information about the momentum of the oscillator. ∆S(ω ≈Ω)≈λ2p(cid:18)1− hhp2ii(cid:19)Sp(ω). (9)
Therefore,the currentofthe detector cannotbe usedas
a p-detector in the steady-state. Nevertheless, the cur- Thus, the addednoise is directly proportionalto the mo-
rent is important to calibrate η to the p-sensitive value mentumspectrumoftheoscillator. Thisisthekeyresult
π/2. AcarefulanalysisofthecurrentI asafunctionofΦ of our Letter.
shows that the inflection points of I(Φ) correspond pre-
From the parameter dependence of each gain, we can es-
cisely to values of η =π/2modπ. Therefore, we can use
timate that the momentum signal at η = π/2 should be
a current measurement to tune η to a p-sensitive value.
typically smaller than the position signal at η = 0 by a
In the experimentally relevant regime, where τ2hx2i ≪ factor(eV/~Ω)2. Nevertheless,itisunambiguouslypossi-
1
τ2, and for ω ∼ Ω, the dominant contributions to the bletoidentifyapsignalinthecurrentnoise. Wenowde-
0
current power spectrum of the detector are scribethreedifferentwaystodothis. First,sinceλx ∝V
whileλ isindependentofV,thebiasvoltagedependence
p
of the noise spectrum can also be used to confirm that
∞
momentum fluctuations are measured. Secondly, for an
S (ω)=2ehIi+8e2ω dt sin(ωt)×
I
Z oscillatorundergoingBrownianmotion, the temperature
0
eV γ τ dependence of both signals differs qualitatively. Like in
+ 0
cosητ τ hhxmii− sinηhhpmii , (6)
(cid:20) h 0 1 ~τ (cid:21) the position detector case, the momentum signal is re-
1
ducedbyaquantumcorrection(thetermproportionalto
−MeV/hhp2ii in Eq.(9)) that arisesfromthe finite com-
where hhabii = habi−haihbi. We now further analyze mutatorofxˆ andpˆ. However,there is afundamental dif-
the added noise due to the presence of the oscillator, ferencebetweenthex-detectorresult(Eq.(7)ofRef.[13])
∆S = S (ω)−2ehIi. This noise spectrum is the sum and the p-detector result (Eq. (9)). In the former case,
I
of a contribution arising due to correlations between the the quantum corrections are always small compared to
transfered charge m and position (term ∼hhxmii in Eq. the leading terms and therefore the peak at resonance
(6)), which we call ∆S , and one due to correlations is always positive. In contrast, the two terms in Eq. (9)
1
between m and the momentum of the oscillator (term can be of equal magnitude and compete about the sign
∼ hhpmii in Eq. (6)), which we call ∆S . The full spec- of ∆S(ω ≈ Ω). The p-sensitive current noise in Eq. (9)
2
4
changes sign when the effective temperature of the oscil- We have demonstrated how the proposed detector can
latorisequalto(eV/k )(1−γ /2γ )/(1−γ /γ ). For be made sensitive to either displacement or momentum
B + tot + tot
a coldenvironmentT ≪eV, ∆S(ω)is negativeatthe fluctuations of the oscillator.
env
resonance, whereas, for a hot environment T > eV,
env
We would like to thank W.A. Coish, L.I. Glazman, O.
∆S(ω) is positive. This change of sign never appears
Gu¨hne, and I. Martin for interesting discussions. This
during a position measurement, so this pronounced dif-
workwasfinanciallysupportedbytheNSERCofCanada,
ference betweena x-dependent and a p-dependent signal
the FQRNT of Qu´ebec, the Swiss NSF, and the NCCR
canbeuseddistinguishthetwo. Weillustratethechange
Nanoscience.
of sign in the inset of Fig. 2, where the added current
noise for η =π/2 is plotted for different T [25].
env
0.4
1
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tunnel junctions in an AB-type setup, it is possible to
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5
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microns in metallic thin films at low temperatures) and [24] Strictly speaking, the analysis in Eq. (4) is only valid in
that it is large enough such that we do not have to take the single-channel case. For the multi-channel case with
into account the change of the area of the AB loop due N channels in the AB loop, the magnitude of the AB
tothefluctuatingpositionoftheoscillator. Furthermore, oscillations is reduced by a factor 1/N.
thesetupisdesignedsuchthattheelectronscannotmake [25] Theplotsaredoneforanoscillatorinthethermalregime:
roundtripsintheABloopbutleavetheringinsteadafter hhp2ii=MkBTeff with γtotTeff =Tenvγ0+γ+(eV/2kB).
going either through the upper or the lower arm. This
explains why (in general) |T(xˆ,Φ)|2 6= |T(xˆ,−Φ)|2 [see