PHYSICAL REVIEW B 71, 193304 共2005兲

Single-electron transistor backaction on the single-electron box
B. A. Turek,1 K. W. Lehnert,1 A. Clerk,1 D. Gunnarsson,2 K. Bladh,2 P. Delsing,2 and R. J. Schoelkopf1
1Department of Applied Physics and Department of Physics, Yale University, New Haven, Connecticut 06511, USA
2Microtechnology Center at Chalmers MC2, Department of Microelectronics and Nanoscience,

Chalmers University of Technology and Goteborg University, SE-412 96 Goteborg, Sweden
共Received 21 February 2005; published 26 May 2005兲
We report an experimental observation of the backaction of a single-electron transistor 共SET兲 measuring the
Coulomb staircase of a single-electron box. As current flows through the SET, the charge state of the SET
island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width,and asymmetry of the Coulomb staircase. A sequential tunneling model accurately recreates these effects,
confirming this mechanism of the backaction of a SET. This is a first step toward understanding the effects ofquantum measurement on solid-state qubits.
DOI: 10.1103/PhysRevB.71.193304

PACS number共s兲: 73.23.Hk, 72.70.⫹m, 85.35.Gv

In the recent work toward the goal of quantum computing,and in the study of single quantum systems in general, thesingle-electron transistor 共SET兲 is often used as a measurement device. It has been proposed as a readout device formechanical,1 spin,2 and charge3 quantum systems, and has
been successfully used to measure superconducting charge
qubits.4 As with any amplifier, the SET must produce electrical noise on its input, perturbing the measured system andcausing the unavoidable backaction of a quantum measurement.
SET backaction on a two-level system has been studied
extensively in the theoretical literature. It has been determined that the SET should be able to approach the quantum
limit of backaction, where it dephases a qubit as rapidly as itis reads the qubit state.5 Spectral components of the SET
backaction at the two-level system transition frequency canalso contribute to transitions between two qubit states.6,7 A
qubit could thus form a spectrum analyzer capable of probing previously inaccessible frequencies.8 These theoreticalanalyses presume SET backaction results from fluctuationsin the charge state of the SET island caused by the drainsource current, but no experimental measurements exist confirming that this is the dominant or the sole mechanism of the
SET’s backaction. Indeed, it often appears that the SET canpoison the Cooper-pair box, inducing nonequilibrium quasiparticles through other mechanisms.9,10
As a first quantitative test of SET backaction, we consider
the SET and box operated in the normal 共nonsuperconducting兲 state, created with the application of a 1-T magnetic
field. Analysis of the normal box is simpler than in the superconducting state because the box is no longer sensitive toparity and quasiparticle generation. The normal SET can also
be simply described by a sequential tunneling model, which
avoids the complication of the many possible quasiparticlepair tunneling cycles11 in the superconducting SET. Nevertheless, the primary mechanism of SET backaction is still thecapacitive electromagnetic coupling between the box and
SET, and the box remains a mesoscopic device that is sensitive to this backaction. Just as with the SSET-Cooper-pairbox system, sensitive measurements of the Coulomb staircase of the normal box can reveal the dynamics of thecoupled system, and probe the nature of SET backaction.
1098-0121/2005/71共19兲/193304共4兲/$23.00

The possibility of SET backaction on a single-electron
box was proposed with experiments in the field,12 but has
proven difficult to quantify. The signature of SET electricalbackaction is difficult to separate from simple heating of thesample.13,14 The backaction has been measured with verystrong coupling between the SET and the box,15 but few
measurements exist in systems that are as weakly coupled as
the proposed Cooper pair box-SET experiments. In this Brief
Report, we present an experimental analysis of a SET weakly
coupled to a single-electron box. We vary the operating pointof the SET, measure the Coulomb staircase of the box, andfind the variations in the shift, width, and asymmetry of thestaircases to be in agreement with a model that includes
backaction caused by the charge-state fluctuations of the SET
island. These variations in the measured staircases allow us
to measure average properties of the noise of the SET.
The SET 关Fig. 1共a兲兴 consists of a aluminum island connected through tunnel junctions to two leads 共the drain andthe source兲 and capacitively coupled to a third 共the gate兲. A
SET is described by its charging energy 共Ec = e2 / 2C⌺, theenergy to add an additional electron to the island兲, by thetunneling resistance of the junctions on the drain and thesource leads 共R j兲, and by the size of the capacitors coupling
it to the external control voltage 共Cge兲 and to the measured

FIG. 1. 共a兲 Circuit diagram of the single-electron box 共dashedbox兲 capacitively coupled to the SET 共dotted box兲. The normal-state
tunnel junctions are represented by boxes with a single line throughthem. 共b兲 Plot of the charge state on the SET island vs time. The
dotted line shows the mean value of the charge on the SET island.

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BRIEF REPORTS

system 共Cc兲. A high tunneling resistance 共R j ⬎ h / e2兲 andlarge charging energy 共Ec ⬎ kBT兲 suppress the addition of
charge to the island by quantum or thermal fluctuations, so
the island may be considered confined to a discrete set of
charge states. A bias voltage 共Vds兲 provides the energy necessary for the system to switch between charge states, allowing current to flow from the drain to the source. The amountof current is controlled by the rate of transition between accessible charge states, which is a function of the potential of
the island. Thus the SET forms a very sensitive electrometer,where changes in the total charge capacitively coupled to theisland modulate the current flowing through the transistor.
The SET is operated by fixing the values of the externallyapplied Vds and Vge, and observing variations in the conductance as the charge coupled to the SET from the measured
system changes. The point at which Vds and Vge are fixed istermed the operating point; the same measurement can be
performed by observing conductance variations about manydifferent operating points.
The box 关Fig. 1共a兲兴 consists of another island capacitively
gated by an external lead 共Vgb兲 and connected through atunnel junction to ground. As with the SET, the gate leadcontrols the potential of the box and changes the relativeelectrostatic energies of the available charge states. We express the gate voltages for both the box and the electrometerin terms of the number of electrons on the corresponding
gate capacitors: ngb = CgbVgb / e and nge = CgeVge / e. When ngb
is raised by one electron, the island charge state of minimum
energy changes, and a single electron tunnels on to the island
to keep it in its ground state. Plotting the time-averaged number of additional electrons on the island as a function of ngb
gives the familiar “Coulomb staircase” 关Fig. 2共b兲兴.12 The
width of this staircase is normally a function only of thetemperature of the sample. In this Brief Report we quantify
SET backaction by observing additional variations in the
Coulomb staircase that are systematic with the SET operating point.
The coupling capacitor 关Cc in Fig. 1共a兲兴 couples together
the potential on the two islands, allowing the SET to measure
the box and also allowing the potential on the SET island toaffect the box. The strength of this coupling is expressed
either as the fraction of the electrometer charge coupled tothe box 共␬ = Cc / C⌺SET兲 or as the temperature necessary tocause changes in a Coulomb staircase comparable to thosecaused by backaction 共T␬ = ␬Ecbox / kB兲. As the polarization
charge on Cc changes, the total charge coupled to the SETchanges, changing the tunneling rates in the SET and modulating the current that flows from the drain to the source. The
charge on the box is then inferred from the change in current
through the SET. Cc also couples the charge on the SET
island to the box, and in doing so creates the effects that we
see as the SET’s backaction.
The discrete nature of charge causes two kinds of noise in
the SET. The drain-source current flows not as a continuous
fluid, but as individual charges, causing an uncertainty in the
SET’s measurement due to shot noise. In addition to shotnoise on the output 共the drain-source current兲, there is also
charge noise on the SET input 共the gate capacitor兲 that affects the measured system. Electrons tunneling on and off the

FIG. 2. 共a兲 Plot of the reflected power from the SET as a function of gate 共nge兲 and drain-source 共Vds兲 voltage. 共b兲 Coulomb staircases measured at the operating points marked in 共a兲 as a functionof the box gate voltage ngb. The time-averaged number of electrons
on the box is measured with a precision of ±1 ⫻ 10−3 and an accuracy of ±2 ⫻ 10−3. 共c兲 Derivatives of these Coulomb staircases andof the corresponding Coulomb staircases generated with a sequential tunneling model with ECSET / kB = 2.3 K, ECbox / kB = 1.6 K, R jSET
= 47 k⍀, R jbox = 15.4 k⍀, and Cc / C⌺SET = 0.048. The derivative of
the Coulomb staircase is reported with an accuracy of ±0.4.

island cause both the charge state and the potential of theSET island to fluctuate between two values 关Fig. 1共b兲兴. The
fluctuating potential on the SET island coupled through Cc isfound to be the source of the SET’s backaction. Three averaged properties of the fluctuating potential have effects visible on the Coulomb staircase and can be varied with theoperating point of the SET. The mean charge on the SET
island varies by as much as one electron, and leads to shifts
in the position of the Coulomb staircase by as much as ␬e.
The rms magnitude of the charge fluctuations on the SET
island broaden the measured Coulomb staircase by anamount that varies with nge. Finally, the telegraph-noise nature of the charge-state fluctuations on the SET island causes
the staircases to be asymmetric; the magnitude and directionof that asymmetry varies with the SET’s operating point.
A sequential tunneling model for the full SET-box systemaccurately recreates both the measurement and the backaction. The tunneling rates between any two box and SET
charge states are calculated as a function of nge, ngb, and Vds
共for details, see Ref. 16兲. The time-averaged charge state of
the SET-box system corresponds to the steady state of these
coupled rates. The current through the transistor is calculated
as the product of the time-averaged charge on the SET islandand the rate at which charge tunnels off the island. Thismodel allows us to replicate the Coulomb staircases taken atvarious operating points with only the electron temperature

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as a free parameter. The elevated temperature of the best-fit
model steps 共T = 27± 1 mK in a fridge at T = 13 mK兲 reflected
the broadening of the measured steps due to quantum fluctuations of charge,17 and is well understood. Theoreticalcurves also correctly account for higher-order effects in thebox-SET system. At certain operating points 共e.g., nge = 21 ,Vds = 0, ngb = 21 兲, the SET’s backaction is a sensitive functionof the state of the box. Changes in the Coulomb staircase
measured at such operating points can only be understood bya sequential tunneling model for the full coupled box-SET
system.
Coulomb staircases were measured in a dilution refrigerator at 13 mK, where the available thermal energy was far lessthan the charging energy of either the SET or the box island.
The SET was operated as a rf-SET,18 with a LC resonant
circuit reflecting an amount of microwave power that varied
as the oscillator was damped by the varying conductance of
the SET. Staircases were measured by sweeping ngb over arange corresponding to 1 / 4e. While the box gate was swept,
the SET gate was swept in the opposite direction to cancel
the parasitic capacitance of the box lead to the electrometer’s
island. Before each Coulomb staircase was measured, nge
was swept to find the reflected microwave power as a function of charge coupled to the SET island. Variations in reflected power with ngb were then converted 共via this lookup
table兲 to charge on Cc 共for a more detailed description, see
Ref. 17兲. The measured charge on the box is thus reported
from the amount of charge on Cge necessary to cause anequivalent electrometer response.
Backaction effects were found to be very sensitive tovariations in ngb and nge and our experiment therefore required that these voltages be set with high precision. Drifts
were removed by referencing the steps to a fiducial step every 20 min. First, nge was swept at Vds = 0 and the value ofnge that maximized SET conductance was determined asnge = 21 关see Fig. 2共a兲兴. Next, a Coulomb staircase was measured with the SET operated at nge = 0.44, Vds = 0. The valueof ngb at the center of this step was determined. Charge offset
noise and 1 / f noise drifts add constant offsets to either nge or
ngb; measuring the fiducial step as described here allows us
to quantify the change in these offsets on both the box and
the SET. Measurements found to contain large charge jumps
in nge or ngb were discarded. This procedure allowed measurement of Coulomb staircases with an uncertainty of 1
⫻ 10−3e in the charge and an uncertainty of 5.5⫻ 10−4e in thehorizontal position of the steps. The uncertainty in the applied nge was found to be 5 ⫻ 10−3e.
The differences in Coulomb staircases measured at different operating points allow us to measure average propertiesof the fluctuating potential of the SET island. Staircases measured at different operating points are shifted in ngb 关Fig.
2共b兲兴. The shift of each staircase is proportional to the mean
charge on the SET island. The mean charge on the SET
island varies by as much as one electron with SET operating
point, and the corresponding charge that couples to the box
and adds to ngb varies by as much as ␬e. We measure staircase shift by reporting the value of ngb at each step’s midpoint, measured relative to the center of a fiducial step 关Fig.
3共a兲兴. The sequential tunneling model accurately recreates
these variations in the step position.

FIG. 3. 共a兲 The horizontal position of the center of the Coulomb
staircase for various operating points of the SET. The model is asolid line, and circles are experimental measurements. No measure1
ments exist near nge = 2 where the electrometer had no gain. Representative error bars are shown in the bottom right-hand side of theplot. Horizontal uncertainty reflects the measured instability of theSET operating point due to charge noise. 共b兲 The maximum slope ofthe staircases measured with the SET at the same series of operating
points. Confidence bands show the model curve for 27± 1 mK.

The measured Coulomb staircases also exhibit variations
in width that change with operating point 关Fig. 3共b兲兴. Threedifferent mechanisms broaden the Coulomb staircase: quantum fluctuations, thermal excitation, and SET backaction.
Quantum fluctuations of charge on the box cause broadening,but only away from the center of the step.17 Our measurement, which quantifies broadening as the maximum slope atthe center of each Coulomb step, is therefore insensitive toquantum broadening. Thermal excitations of the box alsobroaden the Coulomb staircase. SET heating varies with operating point, and, for large values of Vds, can produce atrend in staircase width similar to the effects of backaction.
All of our data were taken, however, at Vds = 0, where heatingfrom the SET was negligible. Finally, SET backaction broadens the Coulomb staircase when the charge-state fluctuationsof the SET island cause the box to switch between charge
states. SET backaction broadens staircases by as much as ␬e,and broadens staircases most at operating points where the

FIG. 4. 共a兲 Derivatives of steps measured at the operating points
in Fig. 2共a兲, offset in ngb to eliminate the shift in position of thesteps. Note that the tails of the two steps are asymmetric. 共b兲 Steps
measured at the same operating points with the sample at 100 mK.
The asymmetry is no longer visible. The inset demonstrates thethermal broadening by showing a 1 / 4 scale curve from the top
graph plotted on the x axis for the bottom graph.

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rms magnitude of the SET charge-state fluctuations is largest.
The observed variations in staircase broadening with operating point 关Fig. 3共b兲兴 are fully accounted for with our sequential tunneling model.
The staircases are also asymmetric in a manner that varies
predictably with operating point. Each staircase was found tohave a longer tail in the direction away from which the staircase was shifted. The asymmetry of the Coulomb staircase isbest viewed in the derivative of the steps 关Fig. 2共b兲, or withthe curves shifted to overlay in Fig. 4共a兲兴, where it clearly
follows the same trend as the model produces. Unfortunately,
differentiating our data increased the noise and made it difficult to quantify the asymmetry; qualitatively, however, themodel reproduces the experimentally observed trends. The
staircase asymmetry is caused by the nature of the chargestate fluctuations on the SET island. The potential of the SET
island lies preferentially to one side of the mean potential,with infrequent fluctuations far to the other side 关Fig. 1共b兲兴.
The staircases are thus broadened asymmetrically in the +ngb
and −ngb directions. The preferred charge state, and thus theasymmetry of the measured staircase, is found to switch atnge = 21 .
The model also shows good agreement with our data at
higher temperatures, where the various effects of the back-action change predictably. At higher temperatures, the meanpotential of the SET island still changes with nge, and thus
step shifts are still visible. For T ⬎ T␬, however, the range 共inngb兲 of thermal broadening is greater than the range of thebackaction broadening or the asymmetry, and neither ofthese effects are therefore visible 关Fig. 4共b兲兴.
In these experiments we confirm that charge-state fluctuations of the SET island are the primary source of SET backaction. We observe the differences in Coulomb staircases
measured with the SET biased at a variety of different operating points, and note changes in the shift, width, and asymmetry of the steps that are accurately recreated by a sequential tunneling model. This confirms that electromagnetic
coupling to the fluctuating SET island potential can provide
the ultimate lower bound on SET backaction.

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Bladh, D. Gunnarsson, T. Duty, P. Delsing, and R. J. Schoe-This work was supported by the National Security Agency
共NSA兲 and Advanced Research and Development Activity
共ARDA兲 under Army Research Office 共ARO兲 Contracts No.
DAAD-19-02-1-0045 and DAAD-19–01–1–0611, the NSF
ITR program under Grant No. DMR-0325580, the NSF under Grant No. DMR-0342157, the W. M. Keck Foundation,and the David and Lucile Packard Foundation.

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