PHYSICAL REVIEW B 84, 205316 (2011)

Mechanically probing coherent tunneling in a double quantum dot
J. Gardner,1 S. D. Bennett,2 and A. A. Clerk1
1

Department of Physics, McGill University, Montréal, Québec, Canada H3A 2T8
Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA(Received 7 September 2011; revised manuscript received 31 October 2011; published 16 November 2011)2

We study theoretically the interaction between the charge dynamics of a few-electron double quantum dot anda capacitively coupled atomic force microscopy cantilever, a setup realized in several recent experiments. We
demonstrate that the dot-induced frequency shift and damping of the cantilever can be used as a sensitive probe
of coherent interdot tunneling, and that these effects can be used to quantitatively extract both the magnitudeof the coherent interdot tunneling and (in some cases) the value of the double-dot T1 time. We also show how
the adiabatic modulation of the dot eigenstates by the cantilever motion leads to substantial effects which arecompletely absent in the more studied single-dot case.
DOI: 10.1103/PhysRevB.84.205316

PACS number(s): 73.21.La, 85.85.+j, 85.35.Gv

I. INTRODUCTION

Quantum dots have received significant attention both fortheir applications to quantum information and as laboratories
for studies of fundamental physics. Self-assembled, epitaxial
quantum dots (QDs) offer advantages over lithographicallydefined dots in terms of size, confinement potential, andscalability.1 Their small size, however, makes direct electricalcharacterization (e.g., via transport) extremely difficult. An
alternate approach uses an oscillating atomic force microscope(AFM) tip, which is only capacitively coupled to the QD
charge.2–6 The dot charge acts as a force on the cantilever,and hence its dynamics can alter the cantilever frequency anddamping rate. These effects provide detailed information on
the dot, similar to that revealed by transport measurements ordirect charge-sensing techniques (i.e., using a nearby coupledelectrometer). It can even reveal subtle effects involving theinterplay between orbital degeneracy and Coulomb-blockade
physics that are difficult to obtain by other means.5,7In this work, we focus theoretically on additional, physically distinct effects that arise when a low-frequency cantileveris coupled to a double quantum dot (DQD) (i.e., two QDs
which are coupled capacitively and via coherent tunneling).8,9
Unlike the single-dot case, the cantilever is now sensitive
to variations in the distribution of charge between the two
dots. We find that this sensitivity leads to new mechanisms
for a dot-induced cantilever damping and frequency shift.
These effects are not solely the consequence of incoherent
tunneling to a reservoir (as is the case for a single dot),
but instead depend sensitively on the strength of coherent
tunneling between the quantum dots. Our results are derived
using a linear-response, quantum master-equation calculation;
this extends the semiclassical Fokker-Planck treatments used
so often in quantum electromechanics5,10–15 to now include
coherent interdot tunneling.
The most prominent new effects emerge in the vicinity ofthe so-called charge-transfer line, where two DQD charge configurations having the same total charge are almost degenerate.
In this regime, we find a new mechanism for DQD-induced
cantilever damping that is enhanced by the relatively longtime scale for charge relaxation. We show that this effect canbe used to directly measure the T1 time of the DQD. We alsofind a new mechanism for a dot-induced cantilever frequency
1098-0121/2011/84(20)/205316(6)shift near the charge-transfer line. In this regime, the presenceof coherent tunneling implies that the DQD energy eigenstates
are superpositions of charge eigenstates. The oscillator motion
can adiabatically modulate the corresponding wave functions,which gives rise to the new frequency-shift mechanism. Not
surprisingly, as this effect is a direct consequence of having
superpositions of charge states, it can be used to probe thestrength of coherent interdot tunneling.
II. MODEL

Motivated by experiments,5,7 we consider a setup wherea self-assembled DQD sitting on a surface is capacitively
coupled to an oscillating metallized cantilever; the dots
are also tunnel coupled to a two-dimensional electron gas(2DEG) sitting below the surface. The DQD is describedby a standard Coulomb-blockade Hamiltonian, plus a term
describing coherent interdot tunneling (strength tc = |tc |). We
will be interested in the few-electron regime, where each dot
has at most one electron, and thus retain only a single orbitalin each dot; for simplicity, we also treat the case of spinlesselectrons, as including spin does not significantly change ourresults.
For a fixed cantilever position, we have††

ĤDQD = Ĥc + tc d̂L d̂R + tc d̂R d̂L + Ĥres ,(1)

where d̂α† is the electron addition operator for dot α = L,R.
The charging Hamiltonian Ĥc takes the standard form8Ĥc =ECα (n̂α − Nα )2 + ECm (n̂L − NL ) (n̂R − NR ) .
α=L,R

(2)
ECα is the charging energy of dot α, ECm describes theircapacitive coupling, and n̂α = d̂α† d̂α is the dot α electron
number operator. We focus exclusively on the Coulombblockade regime where ECα  kB T . Finally, Ĥres describes
the 2DEG as a free-electron gas, and dot-2DEG tunneling.
We take the 2DEG to be in equilibrium at temperature T , andassume for simplicity that the 2DEG-dot tunnel matrix element
is the same for both dots. We use  to denote the maximum
Fermi-golden-rule tunnel rate from a given dot to the 2DEG.

205316-1©2011 American Physical Society

J. GARDNER, S. D. BENNETT, AND A. A. CLERKPHYSICAL REVIEW B 84, 205316 (2011)

The only parameters in ĤDQD depending on the cantilever position rtip are the dimensionless gate voltages Nα =−VB Ctip,α (|rtip − rα |)/e, where Ctip,α is the cantilever-dot
capacitance, rα denotes the position of dot α, and VB is thevoltage applied between the cantilever and the 2DEG. As
demonstrated in Refs. 4–6, by varying the tip position at a fixedheight above the DQD sample plane, one effectively varies
NL ,NR and thus maps out the well-known DQD “stability
diagram”8 (i.e., realizes different ground-state charge configurations). One can thus effectively view NL ,NR as independent
parameters, similar to conventional gated devices.
We now allow the cantilever height zm to oscillate, letting
zm = 0 denote its equilibrium position. The coordinate zm iswell-described as a harmonic oscillator having a frequency
ωm and mass m. The coupling between the oscillations andthe DQD electrons arises solely through the dependence ofthe tip-sample capacitance (and hence Nα ) on zm . For thesmall oscillations of interest, we can linearize this dependence.
Equation (2) then yields the DQD-cantilever interaction
HamiltonianĤint = −ẑm
(3)
Aα n̂α ≡ −ẑm F̂ ,α=L,R

Aα = 2ECα

∂Nα
∂Nᾱ
∂E+α
+ ECm
=−
,
∂zm
∂zm
∂zm

(4)

where α = L,R and ᾱ = R,L are complementary indices. In
the last equality, we have introduced the electron-additionenergies Eα+ = ECα (1 − 2Nα ) − ECm Nᾱ associated withadding a single electron to dot α to an initially empty DQD
state (i.e., zero electrons in either dot).
We are most interested in the regime where the total DQD
charge is fixed at one, and where the electrostatic energy
detuning δ = (E+L − E+R )/2 of the two relevant charge states|10 (electron on left) and |01 (electron on right) is small.
In this regime, we can safely neglect charge states where
the total DQD charge is larger than 2. Further, we will
focus on DQDs where the coherent tunneling tc is muchlarger than both the DQD-2DEG tunneling rate  and themechanical frequency ωm ; this condition is readily achievedin self-assembled QDs (cf. Ref. 16). It is thus useful to workin the basis of adiabatic eigenstates: the eigenstates of ĤDQD
determined by the instantaneous value of ẑm . Two of thefour eigenstates are simply charge eigenstates: |1[zm ] = |11,|4[zm ] = |00. For the remaining eigenstates, note that Eq. (3)implies that the detuning δ will vary linearly with zm . We thus
define




tc
tc
= arctan.
θ [zm ] = arctan
δ[zm ]
δ − zm (AL − AR )/2(5)
The remaining adiabatic eigenstates are thus

For the low temperatures that we focus on, the DQD will
primarily occupy the states |2[zm ] and |3[zm ], and thus willapproximate the physics of a two-level system (TLS).
III. CALCULATIONAs a result of the coupling in Eq. (3), the average force
from the dot F̂  will respond with a delay to the motion of theoscillator, resulting in both a spring-constant shift kdot and extra
damping γdot ; for the weak couplings of interest, this is fully
described within linear response theory.17 To quantitatively
describe these effects in the regime h̄ωm  kB T , we derive a
Lindblad master equation describing the DQD and cantilever.
For a single-QD system, this approach yields a classical masterequation with incoherent tunneling rates set by the instantaneous oscillator position.5,10–15 To extend this approach toinclude coherent tunneling, note that since we will calculate
kdot and γdot using linear response theory, we can without lossof generality replace the cantilever position ẑm by its averagevalue: ẑm → zm (t) = z0 cos(ωm t). The understanding of how
the DQD responds to this time-dependent classical field willthen yield kdot and γdot . By defining Û [x] via Û [x]|j [0] =|j [x], we define ρ̂rot (t) = Û [zm (t)]† ρ̂(t) Û [zm (t)], where ρ̂(t)is the Schrödinger-picture reduced density matrix of the DQD.
ρ̂rot is simply the DQD density matrix in the adiabatic basis.
By using the experimentally relevant conditions |tc |  h̄and (ĀL,R z0h̄ωm )  (kB T )2 , and making Born-Markov andsecular approximations, we obtain4∂ ρ̂rot1= [Ĥeff ,ρ̂rot ] +j k D[Ŝj k ]ρ̂rot .
∂t
ih̄
j,k=1

The first term on the right-hand side (RHS) describes coherent
evolution. Using Pauli matrices to describe the states |2 ≡|2[0], |3 ≡ |3[0] as a two-level system in the naturalway, e.g., σ̂z = |33| − |22|, σ̂x = |32| + |23|, σ̂y =i(|23| − |32|), we findĤeff = ¯ n̂tot + Em |44| +with corresponding adiabatic eigenenergies 
E+L + E+R − (AL + AR )zm∓ (δ[zm ])2 + tc2
Eg,e [zm ] =2≡ ¯ [zm ]∓ [zm ].
(7)

σ̂z −

h̄ ∂zm ∂θσ̂y .
2 ∂t ∂zm(9)

The last term here describes an effective state precession; its
origin is the rotation of the adiabatic eigenstates brought on byzm (t). Note that similar adiabatic approaches have been used
to study periodically modulated, dissipative two-level systems,with the dissipation being treated phenomenologically,18–22 or
as a bosonic bath.23 In contrast, our system is effectively afour-state system, and the dominant dissipation due to 2DEG
tunneling is treated microscopically.
The remaining terms on the RHS of Eq. (8) have thestandard form of Lindblad dissipation. By letting Ŝj k = |j k|,the superoperators D[Ŝj k ] are defined as†|2[zm ] ≡ |g[zm ] = − sin (θ/2) |10 + cos (θ/2) |01, (6a)|3[zm ] ≡ |e[zm ] = cos (θ/2) |10 + sin (θ/2) |01,
(6b)

(8)

†

†

D[Ŝj k ]ρ̂rot = Ŝj k ρ̂rot Ŝj k − 12 (Ŝj k Ŝj k ρ̂rot + ρ̂rot Ŝj k Ŝj k ).

(10)

In the case where the total charge in states |j  and |k differs
by 1, j k is simply a Fermi-golden-rule rate for DQD-2DEG
tunneling determined by the instantaneous eigenstate energies
Ej [zm (t)], Ek [zm (t)]. In contrast, the rates 23 ,32 describe theintrinsic relaxation of the DQD (e.g., due to electron-phonon
interactions), with 1/T1,int ≡ 23 + 32 . We will treat suchprocesses phenomenologically by taking 1/T1,int to be a205316-2

MECHANICALLY PROBING COHERENT TUNNELING IN A . . .

PHYSICAL REVIEW B 84, 205316 (2011)

parameter. We also assume that the bath responsible for the
intrinsic relaxation has the same temperature as the 2DEG;
as such, the zm = 0 stationary solution of Eq. (8) is simply athermal occupation of the states |j . To find the dot-induced
damping and spring-constant shift of the cantilever, we use
Eq. (8) to find the first-order-in-zm correction to ρ̂rot , and use
the corresponding change in F̂ (t) to get γdot and kdot in thestandard manner (see, e.g., Ref. 17).

nonadiabatic Landau-Zener transitions will have negligible
probability and can be safely neglected. Such nonadiabatic
transitions were recently studied in a two-mode optomechanical system;25 unlike our work, the focus was on the regime
where ωm was much larger than the effective tunneling tc . We
stress that kdot,ad is a direct consequence of having coherent
interdot tunneling, and vanishes in the limit tc → 0. It is mostpronounced at low temperatures (kB T < tc ), where it gives rise
to a feature near the charge-transfer line whose width (in δ) is∼tc . Further, at such low temperatures, this effect dominates
all other contributions to kdot near the charge-transfer line. It
thus provides a direct means for both detecting the presence ofcoherent interdot tunneling, and for measuring its magnitude.
Shown in Fig. 1 is a full calculation of the DQD-induced
frequency shift ω = kdot /(2mωm ) obtained using Eq. (8) andlinear response theory, keeping all contributions. We haveused experimentally relevant DQD and cantilever parameters;
see caption for details. The lower inset shows results for asmall value of tc ; similar to a single-QD system, the only
appreciable frequency shift occurs at charge-addition lines
where lead tunneling is strong and the total dot charge canfluctuate. In contrast, for a larger value of tc , one obtains
a large frequency shift along the charge-transfer line, inagreement with Eq. (13). Again, seeing this effect provides
a direct probe of coherent interdot tunneling. Note that
there exists a somewhat similar adiabatic contribution to theIV. BASIC MECHANISMS

In the low-frequency limit, the linear-response results takethe form
mγdot = τ∂F̂ 
∂F̂ 
, kdot = −
,
∂zm
∂zm

(11)

where τ is an effective response time.17 In the single-dot
case, the static susceptibility ∂F̂ /∂zm is just proportional
to the charge susceptibility ∂n̂tot /∂N . γdot and kdot arethus only significant when the QD is tuned to a point
where its total charge can fluctuate via 2DEG-QD tunneling;correspondingly, τ ∼ 1/ .7 In contrast, these fluctuations of
total charge are exponentially suppressed in a DQD near the
charge-transfer line. As we now show, γdot and kdot are instead
determined by the response and dynamics of the DQD charge
distribution.
In the vicinity of the charge-transfer line, the DQD-induced
force operator F̂ takes the form [Aδ = (AL − AR )/2]:F̂ −AL + AR2

1

γ (Hz)20 nm

Aδ (n̂L − n̂R ) = Aδ (cos θ σ̂z − sin θ σ̂x ) .

1.1

(11)
(10)
(01)
(00)

(12)

(00)

0

0

(11)

(01)

Nr

It follows that both γdot and kdot will have contributions fromthree distinct mechanisms, corresponding to the susceptibilities ∂zm σ̂x , ∂zm σ̂z , and ∂zm θ . The first of these involves theoscillator motion modifying the coherence between the |eand |g DQD eigenstates. This corresponds to the well-known
resonant-damping mechanism of acoustic vibrations by a
two-level system,19,20,24 and is strongly suppressed in oursystem as h̄ωm  tc ; we thus do not discuss it further. The
remaining two mechanisms are important for our system, andwe discuss their effects in turn.

(10)

Nl

1

0

V. ADIABATIC FREQUENCY SHIFT

Equation (12) implies that as F̂ has an explicit dependenceon θ , the intrinsic zm dependence of θ [cf. Eq. (5)] willcause a modulation of F̂ . Physically, this corresponds to theadiabatic modulation of the DQD eigenstates by the cantilever
oscillation (via the cantilever’s modulation of the electrostatic
detuning δ). The corresponding oscillation in n̂L − n̂R  causes
a force oscillation, which is in phase with zm (t); it thus
contributes to the DQD-induced spring-constant shift kdot . One
finds simply
kdot,ad = −Aδ σ̂z ∂ cos θ−A2δ sin2 θ tanh( /kB T )
,
=
∂zm
(13)

where the RHS should be evaluated at zm = 0. Note that as we
focus on a small cantilever frequency [h̄ωm  tc2 /(z0 Aδ )],FIG. 1. (Color online) Calculated DQD-induced frequency shiftω = kdot /(2mωm ), using parameter values similar to the experimentof Ref. 5. Main plot: Frequency shift vs dimensionless gate voltage forωm = 160 kHz, k0 = 7 N/m,  = 10 kHz, T = 4.2 K, and tc = 1, ECL =20, ECR = 25, ECm = 12 meV. Most interesting is the feature along thecharge-transfer line (indicated with a dashed line), which results fromthe adiabatic modulation of the DQD eigenstates by the cantilever.
The width of this feature is a measure of tc . For simplicity, we haveused fixed couplings AL ≈ 8 and AR ≈ 6 meV/nm. Lower inset:
Same as main plot, but now tc = 0.001 meV  kB T . The adiabaticfeature is absent. Upper inset: Simulated AFM data of frequency shift
vs lateral tip position, for a tip height of 20 nm and a bias voltage
VB = −7.1 V; white dots indicate the centers of the two dots. The
parameters and coherent tunneling are the same as the main plot, but
the couplings AL ,AR now vary with tip position. See Appendix A for
more details.

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J. GARDNER, S. D. BENNETT, AND A. A. CLERKPHYSICAL REVIEW B 84, 205316 (2011)

TLS–acoustic-wave interaction in glasses,21 but it is neglectedin the standard early treatments.18–20

1(11)(10) (01)Equation (12) indicates a second mechanism that contributes to kdot and γdot near the charge-transfer line: thecantilever’s modulation of σz , which is the population
asymmetry of the two low-energy DQD eigenstates. This
corresponds directly to the well-known mechanism of nonresonant damping by a TLS, studied in the context of acoustic
damping in glasses.18–23,26 On a heuristic level, the cantilever
oscillations cause the DQD splitting to oscillate [cf. Eq. (7)],which in turn causes the occupancy of the states |2, |3 tooscillate. The corresponding oscillations in σ̂z  (and henceF̂ ) are phase shifted with respect to zm (t) due to the finite
DQD T1 time; the mechanism thus contributes both to γdot andkdot . This mechanism for damping relies on the DQD being
coupled to a bath (e.g., the reservoir electrons), which allows
its populations to reequilibrate in response to changes in the
splitting energy ; hence, the energy dissipated is ultimatelytransferred to this bath. Note also that this mechanism issuppressed at low temperatures kB T  , as in this case the
DQD is always in its ground state.
We find that the DQD-induced damping due to this process
is given by
A2δ cos2 θT1mγdot =,
(14)
2T2
1 + ωmkB T cosh2 ( /kB T )1
in agreement with previous works.18–20 Unlike previous works,
in this system one knows the precise microscopic nature of the
TLS (i.e., an electron in the DQD) and at least some of theprocesses contributing to its relaxation time T1 . For our modelof spinless electrons, we find

Nr

VI. EFFECTIVE TLS DAMPING−1
T1−1 =  cosh(β )(e−β|¯ | + eβ(|¯ |−Em ) ) + T1,int,

Δω (Hz)

20 nm

(11)

(00)

(01)
(00)

(10)

tc

kB T
(01)
(00)

0

0

(11)

(10)

1

Nl

0

FIG. 2. (Color online) Calculated DQD-induced damping γdot .
DQD parameters are identical to Fig. 1, except for tc = 0.3 meV, T =8.4 K, and  = 10 MHz. We have also included an intrinsic relaxation−1∼ 100 kHz), which imposes a lower bound on themechanism (T1,int
TLS relaxation rate. The AFM tip parameters are ωm = 75 kHz andk0 = 3 N/m. The main plot shows the damping as a function of thedimensionless gate voltages, while the inset shows a simulated AFM
damping-vs-position image. The damping features near the chargetransfer line correspond to the TLS damping mechanism discussed in
the text.

to the cantilever damping, γdot . It is interesting to note thatthe presence of coherent tunneling causes the effect to vanishat δ = 0, as
has no linear dependence on zm here. One
can thus use the suppression of this damping effect on the
charge-transfer line as a direct probe of coherent interdot
tunneling.

(15)

where β = 1/(kB T ). The first term in the relaxation rate
corresponds to relaxation processes involving 2DEG-DQD
tunneling and an intermediate state where the total DQD
charge is either 2 or 0; note that this term depends explicitly on
both and ¯ , and will thus vary as one moves (in gate voltage
space) along and away from the charge-transfer line. The formof this term corresponds to the simple case of spinless electronsand equal dot-2DEG tunnel couplings; the more general form
is given in Appendix B. The second term in Eq. (15) describes
intrinsic relaxation processes in the DQD (e.g., due to acoupling to phonons). Note that for this mechanism (i.e., theσ̂z contribution to F̂ ), T1 plays the role of the response time τin Eq. (11).
One might naı̈vely think that significant dot-induced
damping could only occur near charge-addition lines where
DQD-2DEG tunneling is strong. However, for a low-frequency
cantilever, we see that near the charge-transfer line, γdot scales
as T1 ; in contrast, the more conventional γdot mechanism neara charge-addition line scales as 1/ . Thus, if T1  > 1, this“TLS damping” mechanism can be equal to or even greater inmagnitude than the more conventional damping peaks foundnear charge-addition lines. This behavior is shown in Fig. 2,
where we use our full calculation to plot the DQD contribution-2VII. MEASURING T1

In the simple case of a low-frequency cantilever and a single
mechanism contributing to both kdot and γdot , Eq. (11) suggeststhat one can simply measure the relevant response time τ bytaking the ratio of the two effects, without having to precisely
know the strength of the dot-cantilever coupling. A similar
approach can be used to extract the DQD T1 time near the
charge-transfer line, though more care is needed, as there are
two mechanisms contributing to kdot . First, note that the γdot
as given by Eq. (14) is only appreciable for kB T  . For
kB T  , one finds that the two spring-constant mechanisms
combine in a particularly simple manner, and that the damping
versus spring-constant shift ratio takes the simple form
mγdot /kdot− cos2 θ T1 .

(16)

By fitting the experimentally measured γdot and kdot to thisformula near the charge-transfer line, one can thus get a direct
measure of the DQD T1 time. This shows an advantage ofthis technique over conventional charge sensing: as one ismeasuring dynamic phenomena (as opposed to simply theaverage value of the charge in the two dots), it is possible
to directly extract important DQD time scales.

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MECHANICALLY PROBING COHERENT TUNNELING IN A . . .

PHYSICAL REVIEW B 84, 205316 (2011)arm asVIII. CONCLUSIONS

Using a somewhat different master-equation approach inconjunction with linear response theory, we have studied
theoretically how charge dynamics in a DQD can cause
damping and frequency shifts of a low-frequency mechanicalresonator (such as an AFM tip). Qualitatively new effects arise
compared to the case of a single dot due to the cantilever’s
sensitivity to charge distribution, and due to the presenceof coherent interdot tunneling. We demonstrated that theseeffects could be used to detect and measure the magnitude ofcoherent tunneling, as well as extract the DQD T1 time near
the charge-transfer line.
ACKNOWLEDGMENTS

We thank Y. Miyahara and L. Cockins for useful conversations, and acknowledge research support from CIFAR,NSERC, and the McGill Centre for the Physics of Materials.

α(ρ,h)1+

α0(h/dh )2 + (ρ/dρ )2

.

(A2)

Representative parameters are α0 ∼ 0.1, dh ∼ 10, and dρ ∼20 nm. Using this simple form and assuming moderate dotspecific parameter variation, we model the addition energies
and coupling strengths for both dots at all oscillator positions.
APPENDIX B: EFFECTIVE T1 TIMESDUE TO ELECTRON TUNNELINGThe effective TLS relaxation rate 1/T1 given in Eq. (14)describes relaxation due to electron tunneling in the simplest
case of spinless electrons, and where the 2DEG is symmetrically coupled to the two dots of the DQD. Relaxing bothof these assumptions in our master-equation treatment results
in a modified form for the first (tunneling-induced) term in
Eq. (14),
T1−1 tunnel = ¯ 1 − cos2 θ δ2 cosh(β )(e−β|¯ | + 2eβ(|¯ |−Em ) ).

APPENDIX A: CAPACITANCE AND
COUPLING MODELING(B1)

We briefly describe in this appendix the modeling ofparameters used to generate Figs. 1 and 2; we stress that themain results and equations of the paper are independent of thismodeling. To generate plots of AFM damping and frequency
shift as a function of tip position rtip , one needs to understandthe dependence of the dimensionless gate voltages Nβ =−VB Ctip,β (|rtip − rβ |)/e on rtip . This dependence determines
both the addition energies [E+β = ECβ (1 − 2Nβ ) − ECm Nβ̄ ]and the coupling strengths (Aβ = ∂E+β /∂zm ). We describe thedependence of Ctip on tip position using a simple functional
form [see Eq. (A2) below] derived from the experimental
results of Ref. 5; we also take parameter values in this form
that correspond to typical values in those experiments.
Following standard convention, we define the “lever arm”
αβ between the 2DEG and dot β as
αβ =Ctip,β,C,βT1−1 tunnel =  cosh(β )[cos2 θ e−β|¯ |+ (1 + cos2 θ )eβ(|¯ |−Em ) ],(A1)

where C,β = e2 /(2ECβ ) is the total capacitance of the dot.
In general, C,β  Ctip,β , and thus one can safely neglect
the position dependence of C,β . Recent work reported
in Refs. 5 and 27 has addressed the variation of α withtip position, both experimentally and using finite-difference
modeling; the results can be approximated with a simple
analytic form. Considering a single, isolated dot and defining ρand h such that (rtip − r) = ρ + hẑ, we approximate the lever

1Here, tunneling between the left (right) dot to the 2DEG is
described by the total rate L (R ), and we have defined ¯ =(L + R )/2, δ = (L − R )/(L + R ). The factor of two
in the last term of the above expression reflects the presenceof spin degeneracy.
The above result [as well as Eq. (15)] neglects the possibilityof coherence in the 2DEG-DQD tunneling (i.e., the possibilityof interference between tunneling into a given DQD eigenstate
via the left dot or via the right dot). In the limit where theinterdot spacing is much larger than the Fermi wavelength of
the 2DEG, such interference terms are strongly suppressed.
In contrast, for a small interdot spacing, this interference willcontribute. One thus finds a different expression for the 2DEG
tunneling. Taking δ = 0, the tunneling contribution to theeffective TLS relaxation rate 1/T1 becomes

H. Drexler, D. Leonard, W. Hansen, J. P. Kotthaus, and P. M. Petroff,
Phys. Rev. Lett. 73, 2252 (1994).
2
M. Woodside and P. L. McEuen, Science 296, 1098
(2002).
3
J. Zhu, M. Brink, and P. L. McEuen, Appl. Phys. Lett. 87, 242102(2005).

(B2)

where the factors of cos θ [cf. Eq. (5) in the text] arise
from electron interference. This result suggests that the leadmediated T1 time is particularly long near the charge-transfer
line (cos θ ∼ 0) due to destructive interference. Because T1,int
is likely to dominate the relaxation rate in this region, suchan effect will have minimal influence on TLS damping in
the one-electron DQD. Nevertheless, this interference effect isclosely intertwined with coherent tunneling in the DQD, andcould offer an interesting topic for future investigations.

4

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