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LC11258
Current to frequency conversion in a Josephson circuit
F. Nguyen, N. Boulant, G. Ithier,∗ P. Bertet, H. Pothier, D. Vion, and D. Esteve†
Quantronics Group, Service de Physique de l’État Condensé (CNRS URA 2464),
DSM/DRECAM/SPEC, CEA-Saclay, 91191 Gif-sur-Yvette, France
Abstract
The voltage oscillations which occur in an ideally current-biased Josephson junction, were pro-
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posed to make a current standard for metrology by Likharev and Zorin [1]. We demonstrate similar
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oscillations in a more complex Josephson circuit derived from the Cooper pair box, the Quantron-
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ium. When a constant current I is injected in the gate capacitor of this device, oscillations develop
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at the frequency fB = I/2e, with e the electron charge. We detect these oscillations through the
sidebands induced at multiples of fB in the spectrum of a microwave signal reflected on the circuit,
TR
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conversion experiment for metrology.
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up to currents I exceeding 100 pA. We discuss the potential interest of this current to frequency
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Keywords:
IS
IE
PACS numbers: 74.50.+r, 74.25.Fy, 74.45.+c, 74.78.Na, 73.63.-b
1
(March 14, 2007)
Exploiting the quantum properties of a current-biased Josephson junction to make a
current standard suitable for metrology was proposed by Likharev and Zorin [1]. This system
has a simple mechanical analog: the phase difference ϕ across the junction is equivalent to
the position of a particle moving in the Josephson potential −EJ cos ϕ, the voltage across the
junction to the particle velocity, and the bias current I to an applied force. The dynamics
of such a particle is well explained within the framework of the Bloch energy bands [2]
ǫi (p) formed by the eigenstates of the particle in the periodic potential, where p is the
quasi-momentum of the particle. It was predicted in particular that the voltage across the
junction oscillates periodically at the Bloch frequency fB = I/2e [1]. Such an experiment,
which would provide a direct link between time and current units, would be of fundamental
interest for electrical metrology. However, current-biasing a junction over a large frequency
range by embedding it in high impedance circuits [3, 4] is utmost difficult. On the other
hand, it is easy to impose the quasi-momentum, that is the total bias charge Q delivered by
the source to the junction [1], by connecting it to a small gate capacitor Cg in series with a
voltage source Vg , so that Q = Cg Vg . Despite this Cooper pair box [5] biasing scheme cannot
impose a constant DC current, we report experiments on a modified setup, shown in Fig 1a,
with which we observe oscillations at the Bloch frequency fB = I/2e when an AC current
taking two values dQ/dt = ±I is injected in the box through a gate capacitor. We discuss
the interest of this new current-to-frequency conversion method for current metrology, for
which electron pumping [6, 7] and electron counting [8] have also been proposed.
The Quantronium device [9, 10] is a box with a split Josephson junction forming a
loop including also a Josephson resonator used to probe the box [11]. The box includes
two small junctions with Josephson energies EJ (1 ± d)/2 (d is an asymmetry coefficient)
which define a small island with total capacitance C. The superconducting phase θb of this
b of extra Cooper pairs inside, forms the single degree
island, conjugated to the number N
of freedom of the box [5, 12]. The resonator, formed by a third larger junction in parallel
with an added on-chip capacitor Ca , has a low enough plasma frequency fp (tuned in the
1 − 2 GHz range), and a low enough quality factor Q to ensure that the phase difference γ
across it is almost a classical variable, and thus also the phase difference δ = γ + φ/ϕ0 across
the two box junctions, φ being the magnetic flux applied through the loop, and ϕ0 = ~/2e.
The control parameters of the split-box are thus the phase difference δ and the reduced gate
charge Ng = Cg Vg /2e, with Cg the island gate capacitance, and Vg the gate voltage. The
2
(March 14, 2007)
i (Ng,δ)
I
Cg
γ
δ
Ca
Φ
Vg(t)
Measure
a)
U(t)
L(t)
b)
c)
Ng=CgVg / 2e
N
g
∆N
N
off
1
L
t
1/f
e)
d) L
Power
f = | dNg / dt |
B
t
f
0
frequency
FIG. 1: Operating principle of the Quantronium circuit for the production and detection of Blochlike oscillations. The circuit(a) is a split Cooper-pair box with a probe junction for the detection
of the oscillations which develop when the gate charge is swept linearly. When the linear sweep
is replaced by a triangular sweep (b) with extrema corresponding to symmetry points of the inductance modulation pattern (c), the time variations of the inductance (d) are the same as for a
continuously increasing linear sweep. This modulation manifests itself as sidebands in the spectrum
(e) of a microwave signal reflected onto the circuit.
3
(March 14, 2007)
Hamiltonian of the box alone writes:
b − Ng )2 − EJ cos( δ ) cos(θ)
b + dEJ sin( δ ) sin(θ)
b ,
b = EC (N
H
2
2
(1)
with EC = (2e)2 /2C.
The eigenenergies ǫi (Ng , δ) vary periodically with Ng (period 1) and δ (period 2π) [10].
The experiment consists in imposing a linear variation of the reduced quasi-momentum
Ng , in order to induce a periodic evolution of the quantum state along the first Bloch
band ǫ0 (Ng , δ), at the Bloch frequency fB = dNg /dt = I/2e. Therefore, the current
i(Ng , δ) = ϕ−1
0 ∂ǫ0 (Ng , δ)/∂δ through the two small junctions, as well as the associated
effective inductance
L(Ng , δ)−1 =
for
small
phase
excursions,
and
1 ∂ 2 ǫ0 (Ng , δ)
ϕ20
∂δ 2
hence
the
(2)
admittance
Y (ω)
=
j [Cω − I0 /ϕ0 ω − 1/L(Ng , δ)ω] as seen from the measuring line, vary periodically with
time. We measure this admittance Y (ω) by microwave reflectometry, as achieved for the
rf-SET [13]. In our experiment, we apply a triangular modulation of the gate voltage with
frequency fg corresponding to dNg /dt = ±I/2e. Due to the symmetry properties of the
quantum states with respect to Ng , the inductance varies as for a linear sweep, as shown in
Fig. 1, provided that the extremal values of Ng are integer or half-integer. The corresponding Bloch frequency is then fB = 2∆Nfg with ∆N the amplitude of the gate modulation.
In order to obtain the largest gate charge modulation of the inductance, a static flux is
applied to the loop in order to adjust the phase at δ ≈ π. When a small microwave signal at
frequency f0 is sent on the measuring line, the periodic modulation of the reflection factor
yields sidebands in the spectrum of the reflected signal [13], which we call Bloch lines [14],
shifted from the carrier by multiples of fB . Due to the periodic excitation, the stationary
outgoing amplitude can be written as a sum of discrete frequencies:
vout (t) =
X
vk exp[2iπ(f0 + kfg )t].
(3)
k
The circuit equations and the loop-current expression [9, 10] allow to calculate all sideband
amplitudes vk . When the sideband frequencies depart from the resonance, the amplitudes
get smaller and become asymmetric (v−k 6= vk ).
The sample was fabricated using electron-beam lithography [9, 10] and aluminum deposition. In order to avoid quasiparticle poisoning, the island was made thinner than the leads
4
(March 14, 2007)
N= 80 N
f = 50 Hz
g
off
40
= 0
g
-0.2
30
N
demodulator output (V)
0.0
-0.4
-0.6
20
3
time (ms)
4
5
FIG. 2: (Color online) Demodulated output signal (left scale) recorded with a 300 Hz-30 kHz
bandwidth during a 3 ms time-window, when a triangular wave voltage corresponding to a Bloch
frequency fB = 8 kHz is applied to the gate (right scale). Each period corresponds to the injection
of an extra Cooper pair.
(13 nm and 42 nm respectively), and gold quasiparticle traps were used [15]. In the present
experiment, a sizeable asymmetry d was introduced on purpose in order to maintain a large
gap G0 = ǫ1 − ǫ0 at (Ng = 1/2, δ = π) and avoid microwave driven transitions towards excited bands. The sample was placed in a sample-holder fitted with microwave transmission
lines, at a temperature T = 30 mK. The gate was connected to a 250 MHz-bandwidth rf
line. The microwave signal after reflecting on the probe junction, went through 3 circulators
before being amplified by a cryogenic amplifier with noise temperature TN = 2.2 K, and a
room temperature amplifier. The signal was then either demodulated with the CW input
signal in order to observe oscillations in the time domain, or sent to a spectrum analyser.
In the latter case, the applied power was ≈ −132 dBm, corresponding to phase excursions
smaller than ±0.1 rad, and the total measurement gain was ≈ 88 dB. As a function Ng and
δ, the plasma resonance varied in the range 1.11 − 1.21 GHz, slightly below the circulator
bandwidth, which yielded an extra attenuation of the signal due to a spurious interference
with the leakage signal through circulator [16]. The variations of the resonance were used to
obtain the best fit sample parameters : fp = 1.19 GHz, with Q ≈ 17, EC = 1.42 ± 0.2 kB K,
EJ = 2.88 ± 0.2 kB K and d = 0.15 ± 0.03 yielding G0 ≈ 7hf0 .
The reflected signal demodulated with the carrier is shown in Fig. 2 for a triangular wave
voltage applied on the gate, corresponding to a Bloch frequency fB = 8 kHz. Due to noise,
5
(March 14, 2007)
harmonic k
-70 -60 -50 -40 -30 -20 -10
-50
0
10 20 30 40 50 60 70
N= 4.64
f = 200 Hz
g
N
off
= -0.16
power (dBm)
-100
-50
*
*
N= 5
N
off
= 0.02
-100
-50
*
*
N= 5
N
off
= 0
-100
f =
1.13999
1.14000
1.14001 GHz
FIG. 3: Spectrum of a reflected CW signal at 1.14 GHz for different triangular wave gate modulation
patterns at frequency fg = 200 Hz. The spectrum consists of sidebands shifted by kfg from the
carrier. Progressive tuning of the amplitude ∆N and of the offset Noff yields a spectrum consisting
only of Bloch lines shifted from the carrier by a multiple of fB = 2∆N fg . The Bloch lines of order
1 (k = ±2∆N ) are marked by an asterisk.
such time-domain measurements could only be performed with a bandwidth narrower than
100 kHz [17]. In the following, we present measurements of the reflected signal using a
spectrum analyzer with a 1 Hz bandwidth.
A series of spectra recorded at f0 = 1.14 GHz with fg = 200 Hz, and taken with progressively tuned gate sweep signal amplitude and offset is shown in Fig. 3. When the gate voltage
is tuned as shown in Fig. 1, the spectrum consists only of Bloch lines, as predicted. All lines
have a linewidth (1 Hz) limited by the spectrum analyser. A wide range of amplitudes ∆N
and frequencies fg were explored.
6
(March 14, 2007)
The comparison between the measured and predicted sideband amplitudes when the offset
Noff or the amplitude ∆N of the gate modulation is varied is shown in Fig. 4 for fg = 1 kHz.
The measured sideband amplitudes are well accounted for by the solution of Eq. 2, but for the
carrier, which suffers from the spurious interference effect already mentioned [16]. Although
the overall agreement for the sidebands, and in particular the cancelation of some of them at
particular offsets and amplitudes, demonstrate the phase-coherence of the measured signal,
the coherent quantum dynamics of the Quantronium is not proved. Indeed, because of the
opposed inductance modulation in the the excited state, the excitation from the ground
state to the first excited state during short times would only reduce the amplitude of the
Bloch lines in proportion of the time spent in the excited level. Spectra obtained at larger
frequencies fg and larger amplitudes ∆N, corresponding respectively to currents and Bloch
frequencies I = 32 pA, fB = 100 MHz, and I = 130 pA, fB = 408 MHz, are shown in Fig. 5.
These results demonstrate that Bloch oscillations persist at Bloch frequencies larger than
the resonator bandwidth fp /Q ≈ 70 MHz, even though Bloch lines become weaker. The
successful current to frequency conversion performed up to currents I > 100 pA using Bloch
oscillations is the main result of this work. However, the amplitudes of the Bloch lines at
these high frequencies are smaller than predicted by the model, and additional sidebands
are present. These imperfections are not explained by Zener transitions to the first band
ǫ1 (Ng , δ) whose rate for each Cooper pair passed is γZ = exp(−fZ /fB ), with fz ≈ 13 GHz
in the present experiment. We rather attribute them to the rounding of the gate triangular
wave signal at its turning points, and to drifts of the gate-charge due to background charge
noise.
In addition to sidebands in a reflected signal, we have also observed that the gate modulation yields microwave emission at the measuring port in absence of any incident microwave
signal. More precisely, the resonator gets excited when a given multiple nfB of the Bloch
frequency falls within the bandwidth of the resonance. Two phenomena can be invoked to
explain this behavior. First, the circuit acts as an AC current source periodically modulated
at the Bloch frequency, but with a vanishing amplitude at δ = π. Second, the inductance
modulation yields a parametric excitation of the probe oscillator (Eq. 2), which is expected
to yield spontaneous oscillations, in particular at δ = π where the inductance modulation
is the largest. However, numerical simulations of our circuit at zero temperature fail to
account quantitatively for the emitted power.
7
(March 14, 2007)
-50
k = 1
3
Power (dBm)
-60
5
k = 2
7
4
-70
6
8
-50
-60
-70
0
0.5
N
1
N = 2
off
2
N
3
N
off
= 0
FIG. 4: (Color online) Comparison of the measured (symbols) and calculated (lines) sideband
amplitudes as a function of the sweep offset Noff and amplitude ∆N , for fg = 1 kHz. Left panels:
Offset dependence for ∆N = 2; the observed sidebands correspond as predicted to odd multiples
of fg (top) and to the Bloch line and its harmonics (bottom). Right panel: amplitude dependence
for Noff = 0. The Bloch line corresponds to harmonic 4 at ∆N = 2, and to harmonic 6 at
∆N = 3. Calculated curves were shifted by 45 dB (estimated value was 44 dB) to best match the
experimental Bloch line of order 1.
An important application of these experimental results could be to establish a direct
link between a DC current and a frequency through the Bloch frequency fB = I/2e, in
order to close the triangle of quantum metrology [18]. Such an experiment would aim at
measuring the current IH passing through a Quantum Hall bar device in terms of a rate N˙H
of transferred Cooper pairs, in order to check the consistency of the Quantum Hall effect
(QHE) with the AC Josephson effect. The Hall voltage VH = (h/e2 )IH across a QHE bar
can indeed be related using the AC Josephson effect to a frequency fH through the relation
(h/e2 )IH = (h/2e)fH . If the description of both QHE and Josephson experiments is exact,
one predicts N˙H = fH /4. A consistency check of this relation at the 10−8 level is presently a
8
(March 14, 2007)
harmonic k
0
50
100
150
200
0
*
f = 1 MHz
N=50
power (dBm)
g
N
off
= 0
-50
f = 1.15
1.20
0
1.25
1.30
85
GHz
170
0
N=85
f = 2.4 MHz
g
N
off
= 0
*
-50
f =
1.2
1.3
1.4
1.5
GHz
FIG. 5: (Color online) Amplitude of the sidebands at positive harmonics (top scale) of the gate
frequency fg . Top panel: ∆N = 50,fg = 1 MHz(I = 32 pA,fB = 100 MHz) ; bottom panel:
∆N = 85, fg = 2.4 MHz (I = 130 pA, fB = 408 MHz). The Bloch line of order 1 is marked by
an asterisk. The continuous line is the noise level. We attribute the presence of non Bloch lines to
the imperfections in the gate signal, and to charge noise.
major goal in metrology because, in conjunction with a metrological realization of the mass
unit by a Watt-balance experiment [19], it would provide a serious basis for a redetermination
of the SI unit system in terms of electrical experiments involving only fundamental constants.
Due to the large current IH ≈ 1 µA needed for QHE experiments, a direct measurement of
the associated rate N˙H is however beyond reach, but a superconducting transformer with
a topologically defined transformation ratio [20] can be used to transpose IH to a more
convenient current range 0.1 − 1 nA. This current range, which is still beyond reach of
single electron pumps [6] or of direct electron counting experiments [8], could be accessed
with the sluice Cooper pair pump [7], or with the method demonstrated in this work. The
challenge here is to measure the Bloch frequency with a true DC current injected in the
transistor island by the means of a suitable high impedance current source. Two conditions
9
(March 14, 2007)
are necessary for that purpose. First, the bias impedance as seen from the box ’island’ has
to be large enough to preserve single Cooper pair effects, which implies an impedance of the
order of or larger than RQ = h/4e2 . Second, the temporal fluctuations of the injected current
have to be small in order to obtain Bloch lines narrow enough to allow for a precise frequency
measurement. Developing a suitable current source for charge injection is thus a challenging
prerequisite to metrology experiments based on Bloch physics in the Quantronium circuit.
In conclusion, we have demonstrated the accurate conversion of a current ±I to a frequency fB = I/2e in a Josephson device biased through a small capacitor, by means of the
sidebands in the spectrum of a reflected microwave signal. This new method is simple to
operate and works for currents I > 0.1nA. It would be extremely appealing for metrology
if it could be operated with a DC current.
We acknowledge discussions with M. Devoret, F. Piquemal, W. Poirier, N. Feltin, and
within the Quantronics group. This work has been supported by the European project
Eurosqip and by a C’nano grant.
∗
Present address: Department of Physics, Royal Holloway, University of London, Egham, Surrey, TW20 OEX, UK
†
Corresponding author: [email protected]
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10
(March 14, 2007)
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[11] A related setup was reported recently in J. Koenemann et al., cond-mat/0701144.
[12] In contrast to the case of a current-biased junction, the phase θb is defined on a circle due to
b.
the quantization of N
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[16] the circulator leakage requested to explain our data varied from −7 dB to −23 dB over the
plasma frequency excursion range, which is compatible with room temperature measurements.
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