Download Reading out Charge Qubits with a Radio Frequency Single Electron... K. Bladh, D. Gunnarsson, G. Johansson, A. K¨

Reading out Charge Qubits with a Radio Frequency Single Electron Transistor
K. Bladh, D. Gunnarsson, G. Johansson, A. Käck, G. Wendin, P. Delsing
Microtechnology Center at Chalmers MC2, Department of Microelectronics and Nanoscience,
Chalmers University of Technology and Göteborg University, S-412 96, Göteborg, Sweden
A. Aassime
Service de Physique de l’Etat Condensé, CEA-Saclay, F-91191 Gif-sur-Yvette,France
M. Taslakov
Swedish National Testing and Research Institute, SP Box 857, SE-501 15, Borås, Sweden and
Institute of Electronics Bulgarian Academy of Sciences, Sofia, Bulgaria
(Dated: April 19, 2002)
We describe fabrication and measurements of Single Electron Transistors operated in the Radio
Frequency mode (RF-SET). We demonstrate very high sensitivity for RF-SETs and we evaluate the
back-action from an RF-SET, when used as a read-out device for charge qubits. We conclude that
single-shot read-out is possible with a signal to noise ratio greater than one, and that niobium qubits
would substantially improve the signal to noise ratio. Furthermore we describe how the RF-SET
could be used to read out a differential qubit in a gradiometer coupling.
PACS numbers: 03.67.Lx, 73.23.Hk, 85.25.Na
INTRODUCTION
Quantum computers are predicted to perform certain
computational tasks much faster than ordinary classical
computers [1, 2]. Even though the difficulties in realizing
a quantum computer are tremendous, the distant goal
of quantum computers has focused a lot of attention on
quantum state engineering and quantum measurement,
areas where substantial progress has been made. A large
number of qubit systems have been suggested, but so far
only relatively primitive systems with a limited number
of qubits have been tested. The most advanced demonstration so far is a seven qubit implementation [3] of
Shor’s algorithm [1]. The different types of qubit systems
can be divided into two categories: i) those which are
based on microscopic systems such as molecules, trapped
atoms, or photons [3–8], and ii) those which are based
on solid state systems [9–13]. Both categories have advantages and disadvantages, especially the microscopic
systems have long coherence times, but are hard to integrate into systems with a larger number of qubits. For
the solid state systems the situation is the opposite.
The system that we will discuss in this paper is the
Single Cooper-pair Box (SCB)[14]. It is a solid state system, which is based on a small superconducting island
connected to a charge reservoir via a Josephson junction. The Karlsruhe group has, in a series of papers,
described theoretically how this system can be used as
a qubit [9, 15, 16]. Quantum coherence in a SCB was
first demonstrated by Nakamura et al. [17, 18], however
the coherence times were quite short, of the order of 10
ns. In a very recent experiment [19] coherence times of
the order of 1 µs have been observed. The read-out of a
SCB with a single electron transistor (SET) [20, 21] has
been discussed in a number of papers [16, 22–24]. The
radio-frequency version of the SET (the RF-SET) [24–27]
is especially interesting since it combines very high sensitivity with high speed. An alternative read-out scheme
which uses the switching current of a large Josephson
junction has been developed by the Saclay group [19]. In
both cases it seems that single-shot read-out should be
possible, even if it has not yet been demonstrated.
In this paper we discuss the read-out of a SCB qubit
with a Radio Frequency Single Electron Transistor. We
also analyze the back-action of the RF-SET on the qubit
based on measurements on two different SETs. In section
2 we describe the fabrication of the SETs and the qubits.
We then describe the measurement setup, and in section
4 we discuss the measured data. In section 5 we discuss
the back-action, and in section 6 we suggest a new type of
differential qubit which can be read out in a gradiometer
configuration.
SAMPLE FABRICATION
Choosing the right parameters and chemicals is essential to have sample fabrication working well. For shadow
evaporation of metals, different systems have been developed to have certain characteristics in combination
with the materials being used. P. Dubos et al. [28]
have demonstrated the use of a trilayer resist based on
polyethersolfone(PES)/germanium/PMMA for use with
niobium evaporation. The thermostable PES can withstand the elevated temperatures during niobium evaporation so that the typical problems with out-gassing from
the resist does not degrade the superconducting properties of the strips being fabricated. A calixarene derivative
2
FIG. 1: An example of one of the structures fabricated with
the method described in the text. It shows an SET with a gate
and some nearby ground planes fabricated by evaporation of
250+400 Å aluminum at two different angles. The smallest
line width is 70 nm.
e-beam resist has been developed by J. Fujita et al. [29]
to have extremely high resolution, higher than PMMA,
so that sub 10 nm lines can be fabricated. Some of the
samples presented in this paper were made using a bilayer resist consisting of PMGI in the bottom layer and
ZEP520A in the top layer. However, this bottom layer
created a number of problems, and we are now fabricating samples with a bilayer resist system that has very
high yield and quality.
The new resist system consists of a top layer of
ZEP520A:Anisole (supplied by Nippon Zeon Co Ltd) diluted 1:2, spun to a suitable thickness and baked 10 min
at 170 ◦ C. The bottom layer is 8%wt P(MMA-MAA)
copolymer in ethyl lactate baked 5 min at 170 ◦ C. After
e-beam exposure the top layer is developed 40 sec in oxylene, rinsed 30 sec in IPA and dried in N2 . The bottom
layer is developed in IPA:H2 O 5:1 until enough undercut
is produced, rinsed 30 sec in IPA and dried in N2 . Rinsing of the samples is important to get the best results.
After evaporation of 250+400 Å aluminum, excess metal
is lifted off in n-methyl pyrrolidone. An example of a
device fabricated with this new resist is shown in Fig. 1
The main advantage of this bilayer is the reproducibility and the high sample yield. In total 46 SETs (92
junctions) in different geometries have been fabricated
resulting in 45 usable devices with resistances in the kΩ
range and only one SET not working. The spread in resistance of the sample was such that 80% of the devices
were within ±15 kΩ of the target resistance of 40 kΩ. We
think that this spread in resistance is not due to differences in the bilayer but rather the irreproducibility of the
conditions in the vacuum chamber during evaporation.
With an evaporation system working in a controlled cli-
FIG. 2: The layout for the integrated qubit and SET, The
Qubit on the left is made in a SQUID geometry such that
the Josephson coupling energy can be tuned with a magnetic
field.
mate we think this bilayer could produce tunnel junctions
with near 100% yield and reduce the current spread in
resistance so that most fabricated circuits have the same
resistance within a few kΩ. Compared to numerous other
resist systems we have tested, none came close to having
the same performance. The system also shows excellent
stability against changes in external parameters such as
humidity and ageing, which is good for the long-term
reproducibility of these results.
In Fig. 2 we show the layout for a qubit-SET sample.
The qubit on the left is made in a SQUID geometry such
that the Josephson coupling energy can be tuned with
an external magnetic field, and the charge on the box
can be controlled via the gate lead which couples to the
qubit from the left via the gate capacitance CgQ . The
SET is coupled to the qubit via the coupling capacitance
CC , and the working point of the SET can be adjusted
with the SET-gate which is coupled via the capacitance
CgS . The sum capacitance of the SET is CΣ = CSET +
CC + CgS .
MEASUREMENT SET-UP
The measurements were performed in a dilution refrigerator with a base temperature of about 20 mK. A block
TABLE I:
SET-Sample
1
2
3
R [kΩ]
44.1
41.0
48.0
CΣ [aF]
370
270
600
√
δQ [µe/ Hz]
6.3
3.2
60
δ[h̄]
13
4.8
730
3
30
P (nW)
20
10
0
250
300
350
400
Frequency (MHz)
FIG. 4: The power spectrum of the shot noise of the SET
when a current of 1µA is sent through the SET (Sample #1)
and after the amplifier noise has been subtracted. The noise
is coupled out to the amplifiers via the tank circuit, and thus
gives important information about the tank circuit. The dotted line is a fit to the data using a simple LRC-model.
FIG. 3: A block scheme of the RF-SET measurement system.
scheme of the measurement set-up is shown in Fig. 3. A
weak RF-signal (the carrier) is launched towards the tank
circuit via a directional coupler and a bias tee. The reflected signal is amplified by a cold amplifier and two
warm amplifiers. We have used carrier frequencies in
the range of 300-500 MHz. The reflected signal can be
analyzed either in the frequency domain or in the time
domain.
The tank circuit consists of a capacitance which is just
the stray capacitance of the contact pad of the sample
substrate and an inductor. We have used two different
types of inductors, a commercial chip inductor, which
was flip chipped onto the substrate, and a circuit board
inductor which was bonded to the substrate [30]. Both
types of inductors give similar results, but the circuit
board inductor is easier to connect, and has a higher self
resonance frequency.
The tank circuit is analyzed by applying a relatively
large current trough the SET and detecting the shot noise
which is coupled out to the amplifiers via the tank circuit. In Fig. 4, the noise spectrum of such a measurement
is shown. In this measurement the noise spectrum at zero
current was subtracted from the spectrum taken at 1 µA
to remove the noise contribution coming from the amplifier. The form of the noise spectrum can be easily fitted
to a simple circuit model from which we can extract the
parameters of the tank circuit. For the typical case shown
in Fig. 4 we get a resonant frequency of 331 MHz and a
Q value of 18. This corresponds to an inductor value of
710 nH and a capacitance of 330 fF.
To analyze the signal in the frequency domain we used
an Advantest spectrum analyzer. A typical frequency
spectrum can be seen in Fig. 5. Here a small gate signal
with a frequency of 2 MHz and an amplitude of 0.035 erms
was applied to the gate. This modulation generates the
side peaks which can be seen in Fig. 5.
Just as in an AM-radio, there are two different methods to retrieve the modulating signal and study it in the
time domain. Either one can rectify and low pass filter the amplitude modulated carrier, or one can mix the
reflected signal with another RF-source, which is phase
FIG. 5: The frequency domain response of the RF-SET (Sample # 2) when a 2 MHz signal with an amplitude corresponding to 0.035 erms is applied to the gate.
4
20
15
10
I (nA)
5
0
-5
Pure RF-Mode
-10
RF-Amplitude
-15
-20
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
V (mV)
30
FIG. 6: The time response from an RF-SET using the rectifying method. The input is a step function corresponding to a
0.2 e charge change on the gate capacitance. The bandwidth
of this measurement is limited to 1 MHz. A single trace is
shown with no averaging.
10
I (nA)
locked to the carrier. By adjusting the phase between
the two sources, the output amplitude from the mixer
can be optimized. The two methods give similar results,
but the mixing method normally gives somewhat higher
signal to noise ratio. A typical time domain response to
a 0.2e gate signal using the rectifying method is shown
in Fig. 6.
20
0
RF+DC-Mode
-10
RF-Amplitude
-20
DC-bias
-30
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
V (mV)
PERFORMANCE OF THE RF-SET
In the following we will discuss and compare the results obtained in three different SETs, samples #1 and
#2 were optimized for best sensitivity, whereas sample
#3 was integrated with a qubit. The parameters for
the different samples, are summarized in Table I. The
current voltage (IV) characteristics of Sample #1 and
#2 are shown in Fig. 7 a and b respectively. The IVcharacteristic of the SETs could be modulated with a
gate voltage as shown for sample #1 and #2 in Fig. 7.
The gate voltage period was ∆Vg =10 mV corresponding
to a gate capacitance CgS =16 aF for both samples. We
have worked with two different modes of RF-excitation,
as indicated in Fig. 7. Either we used a relatively large
RF-amplitude and no dc-voltage (pure RF-mode), or we
used a dc-voltage and a small RF-amplitude (RF+DCmode). Generally one can say that the RF+DC-mode
normally gave better sensitivity.
For a given gate voltage, the IV-characteristic is asymmetric due to the asymmetric bias. As the bias voltage
is increased an additional charge is induced on the gate
capacitance. For a negative bias this induced charge is
of the opposite sign, and thus the effective gate charge is
FIG. 7: Current voltage characteristics for a) sample #1 and
b) sample #2. Curves are given for four different gate voltages. The two different modes of operation, pure RF mode
and RF+DC-mode are also indicated in the a) and b) figures,
respectively.
different as can be seen in Fig. 7a.
To evaluate the sensitivity performance of an SET, we
measure the signal to noise ratio of one of the side peaks
in the frequency spectrum when a small ac signal is applied to the gate. One of the side peaks is shown in
detail In Fig. 8 for a gate amplitude corresponding to
0.0095 erms and a gate frequency of 2 MHz.
The charge sensitivity of the SET can be calculated as
δQ = √
∆Q
RBW × 10SNR[dB]/20
(1)
where ∆Q is the value of the gate signal measured in
electrons (rms), RBW is the resolution bandwidth used
for the spectrum analyzer, and SNR is the signal to noise
ratio for the side peak, measured in dB. From the data
√ in
Fig. 8 we can deduce a charge sensitivity of 3.2 µe/ Hz.
5
FIG. 8: One of the side peaks in the frequency spectrum for
sample 2, with the gate frequency of 2 MHz and a gate amplitude corresponding to 0.0095 erms . The resolution bandwidth
was 10 kHz.
FIG. 10: The signal to noise ratio as a function of RFamplitude for a fixed bias voltage of 0.72 mV i.e at the Josephson Quasi Particle (JQP) peak, cf Fig. 9. The resolution bandwidth was 10 kHz.
Just as for SQUIDs one can convert this charge sensitivity
to an (uncoupled) energy sensitivity.
as a function of the bias voltage when a relatively small
RF-amplitude is applied. One can see that there are
a number of maxima which are due to modulation of
different features in the IV-curve. Note also that the
SNR curve is not symmetric, which is again due to the
gate charge induced due to the asymmetric bias voltage.
The signal to noise ratio also varies with rf amplitude,
as can be seen in Fig. 10. Here the RF+DC mode is used,
and it can be seen that the SNR does not vary strongly
with the RF-amplitude, the maximum being relatively
flat.
δ
=
δQ2
2CΣ
(2)
In this case we find an (uncoupled) energy sensitivity
δ
= 4.8 h̄, which is approaching the shot noise limit. The
shot noise in an ordinary SET should give a lower limit
around δ
≈ 0.7 h̄ [22], and for an RF-SET the noise
is predicted to be slightly higher [31] so that we could
expect to reach δ
≈ 1 h̄ at best.
The SNR is of course dependent on a large number
of parameters, such as carrier frequency and amplitude,
gate frequency, dc-bias etc. In Fig. 9 we show the SNR
BACK-ACTION
When performing a measurement on a qubit, the measurement necessarily dephases the qubit. However, there
can also be a mixing of the two qubit states which destroys the information that the read-out system tries to
measure. The mixing occurs due to voltage fluctuations
of the SET island SV (ω) and the characteristic time for
this process is the mixing time tmix , which can be expressed in a simple form [16, 22] as
Γ1 =
FIG. 9: The signal to noise ratio as a function of bias voltage. IV-characteristics for several values of gate charge are
also shown. Data is taken from Sample #3 with a resolution bandwidth of 3kHz, 0.1 erms applied at the gate and an
RF-amplitude of -97 dBm
1
tmix
=
e2 2 EJ2
κ
SV (∆E/h̄),
h̄2 ∆E 2
(3)
where we define the coupling coefficient as κ = Cc /Cqb
and where the Josephson energy of the box EJ is assumed
to be small compared to ∆E.
As can be seen in Eq.(3), the mixing caused by backaction depends strongly on ∆E, and on the spectral density of the voltage fluctuations at the frequency corresponding to the energy difference between the two qubit
states ∆E/h̄. These fluctuations can be thought of in
terms of two contributions, one from the shot noise,
6
/
FIG. 11: Comparison of the calculations of the spectral density of the voltage fluctuations of the SET island using the full
quantum mechanical calculation (full line), the classical shot
noise (dashed line), and the quantum fluctuations assuming
a linear SET impedance (dotted line). Parameters for sample
#1 are used in all calculations
and one from the quantum fluctuations in the SET. At
low frequency, the shot noise will dominate, while the
quantum fluctuations will dominate at high frequencies.
At high frequencies the impedance of the SET can be
thought of as a linear resistance in parallel with the junction capacitances and the quantum fluctuations of the
SET can be expressed as
SV (ω) = 2h̄ωRe [ZSET (ω)]
(4)
To obtain the spectrum of fluctuations in the intermediate regime it is necessary to solve the full quantum
problem, which was done in [23]. The result of this calculation and the comparison to the shot noise and quantum
fluctuation results in the two limits, are shown in Fig. 11
for sample # 1. As can be seen the result of the full calculation coincides with the shot noise and the quantum
noise in the low and high frequency limits respectively.
In addition to this spectrum, the RF excitation gives
a component at fRF , and the non-linearity of the IVcharacteristics gives an additional component at 3fRF .
However these frequencies are much lower than the relevant mixing frequency ∆E/h̄
To resolve the two states of the qubit, which differ by
two electron charges, we need a measuring time tm which
can be expressed as [16]:
FIG. 12: Calculated SV (f ) for samples #1(full line) and
#2(dashed line). The arrows show the energy separations
of the two qubit states for an aluminum and a niobium qubit,
respectively.
tmix /tm . The spectral densities SV (f) for the two samples at the optimum charge sensitivity, at a current of
6.7 and 8.0 nA for samples #1 and #2, respectively, are
displayed in Fig. 12. As can be seen both from Eq. 3
and in Fig. 12, the mixing time increases strongly with
increasing ∆E. In our case ∆E is limited by the superconducting energy gap ∆, which for aluminum films
corresponds to about 2.5 K. Using niobium as the qubit
material would substantially increase the mixing time. If
we assume that EC and thus also ∆E can be scaled with
∆ and that the coupling coefficient κ is kept constant,
mixing times of several ms can be reached. In that case,
other sources would most probably dominate the mixing.
The results for the samples #1 and #2 are summarized
in Table II, where a coupling κ = 0.01 is assumed.
A DIFFERENTIAL QUBIT
Although the results above show that a Single Cooperpair Box may be measured in a single-shot measurement
with a reasonable signal-to-noise ratio, the present design
suffers from a number of problems. In an attempt to improve the present system we here suggest an alternative
qubit layout.
TABLE II:
tm =
δQ
κe
2
.
(5)
Now we can use the measured data for samples #1
and #2 to calculate both tm and tmix , and thus we can
also get the expected signal-to-noise ratio for a singleshot measurement, which is simply given by SN RSS =
SET
Qubit
δQ
√
Sample material e/ Hz
1
Al
6.3
1
Nb
6.3
Al
3.2
2
2
Nb
3.2
∆E SV (ω)
[K] [nV2 /Hz]
2.4
0.29
15.5
0.056
2.4
0.39
15.5
0.080
tm
[µs]
0.40
0.40
0.10
0.10
tmix SNR
[µs]
8.6 4.6
1860 68
6.4 8.0
1300 114
7
but with increased bandwidth.
Furthermore, the 1/f charge noise, which is possibly
the worst source of decoherence [33], can be divided into
two different parts: the noise coming from long and short
distances. The noise sources which are far away from the
qubit will couple similarly to both islands and thus the
effect of this noise will act as a common mode signal and
thus be reduced. However, this is probably not a major
improvement, since the noise sources located close to the
SET-island are believed to dominate.
L
C
CJ
CC
Cgs
CJ
CC
CQB
Cgs
VS1
VS2
CJ
Cg
Cg
CJ
CONCLUSIONS
-Vg/2
Vg/2
FIG. 13: A differential qubit read out by a gradiometer coupled RF-SET. The two SETs are tuned in anti-phase such
that when a charge moves from left to right on the qubit, the
current increases in both SETs.
A problem with the ordinary SCB is that it sometimes
shows e-periodicity instead of 2e-periodicity. The origin
of this problem is not yet fully understood, but one source
is that quasi particles, generated in the reservoir, diffuse
to the junction and tunnel onto the island. Using a twoisland box (or if you like, an unconnected junction) would
drastically decrease this problem since the islands are no
longer coupled to a large reservoir. In fact this was the
system that was originally suggested by Shnirman et al.
[15].
Apart from the parity improvement there are also a
number of other advantages with the two island box. One
advantage is that one can make the read-out system in
the form of a gradiometer, where two SETs read out one
island each, as shown in Fig. 13. In such a configuration
the two SETs can be tuned into opposite points of their
transfer functions, so that the current would increase in
both SETs when charge increases on the gate of the left
SET and decreases on the gate of the right SET. The
sensitivity and back-action from the SETs will be similar
to the case described above (i.e. in Fig. 3), however this
differential set-up will improve the system in two different ways. First, the noise from the cold amplifier which
is coupled to the qubit via the tank circuit and the two
SETs will be a common mode signal, and will thus not
be coupled into the box. The amount to which this noise
source can be reduced depends on how well the two SETs
and the coupling capacitances can be matched. Just as
in the case of the Saclay qubit [19] and the new Delft
qubit [32] a symmetric layout improves the system. Secondly, this configuration also lowers the impedance of the
two parallel coupled SETs, and thus one can use a tank
circuit with a lower Q-value with maintained sensitivity,
In summary we have demonstrated a very high charge
sensitivity for Radio Frequency Single Electron Transistors,√at best we achieve a charge sensitivity of δQ=3.2
µe/ Hz and an uncoupled energy sensitivity of δ
=4.8 h̄.
We have also calculated the back-action which we would
get in reading out a qubit, using the experimental results
for two of the samples. We conclude that a single-shot
measurement should be possible, and that the use of niobium qubits would substantially improve the signal to
noise ratio of such a measurement. Furthermore, we also
suggest a new type of differential qubit and show how
such a qubit could be read out with a differentially coupled RF-SET.
We would like to acknowledge fruitful discussions with
R. Shoelkopf, K. Lehnert, D. Esteve, M. Devoret, D.
Vion, J.E. Mooij, P. Wahlgren, T. Claeson, and V.
Shumeiko. Samples were made at the Swedish Nanometer Laboratory and at the Microtechnology Centre at
Chalmers. We were supported by the Swedish VR, the
Wallenberg and Göran Gustafsson foundations, as well as
the European Union under the IST, Growth, and TMR
programs.
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