Download Observation of qubit state with a dc-SQUID and dissipation effect... Hideaki Takayanagi, Hirotaka Tanaka, Shiro Saito and Hayato Nakano

Observation of qubit state with a dc-SQUID and dissipation effect in the SQUID
Hideaki Takayanagi, Hirotaka Tanaka, Shiro Saito and Hayato Nakano
NTT Basic Research Laboratories, NTT Corporation
3-1 Morinosato-Wakamiya, Atsugi-shi, Kanagawa 243-0198 Japan
[email protected]
PACS Ref : 73.23.-b, 74.50.+r, 85.25.Dq
_
Two states of a flux qubit with three Josephson junctions were shown in a single measurement
with a dc-SQUID. The qubit is an aluminum superconductor loop surrounded by a dc-SQUID
for readout. It has two states, which have persistent currents flowing in opposite directions. The
readout data for three samples with different junction sizes suggest that the probability
distribution for the double-well potential depends on the ratio of E J / E C , where E J is the
Josephson energy and E C is the charging energy. The probability distribution was estimated by
calculating the wavefunctions and energy levels for the measured samples.
We measured the retrapping current of a dc-SQUID without the qubit as a function of the
magnetic field. It was found that the relative phase relation for the magnetic field dependence of
the retrapping current jumped from 0 to π at E J ∪ E C . This jump can be explained
phenomenologically by taking into account the two kinds of dissipation arising from ac currents
flowing through the lead and along the SQUID loop. The fact that the jump occurred at
E J ∪ E C strongly suggests that the quantum nature of the electromagnetic environment needs to
be taken into account in order to understand the origin of the dissipation.
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1. Introduction
Quantum computers have a novel architecture that makes
extremely fast
computation possible. The basic element of a quantum computer is a qubit, which is
expressed as a quantum two-level system. In principle, any quantum two-level system
has the potential to be a qubit. However each system has its own advantages and
disadvantages. Important criteria for qubit feasibility are tolerance to decoherence,
efficient qubit interaction, and scalability. Although the coherence of solid-state quantum
computers must be improved, qubits in a solid state, such as a semiconductor or
superconductor, have appropriate scalability when we utilize the well-established
nanometer-scale fabrication technology now widely used in the semiconductor industry.
The short coherence time in a solid-state quantum bit is due mainly to the existence of
many degrees of freedom. The gate operations in a solid-state quantum computer have
not yet been demonstrated experimentally. In contrast, the gate operations have already
been demonstrated in NMR [1] and ion trap quantum computers, for example, the
Controlled NOT gate. NMR has even been used to perform practical algorithms, such as
Deutsch-Jozsa, quantum search, and Shor s factorization [2], with several qubits. But
experimental schemes in NMR are not directly applicable to quantum computers with
over 10 qubits because of the short coherence time and initialization problem.
With solid-state quantum computers, we would expect a superconductor to have a
longer coherence time because of the superconducting energy gap which appears
between the condensed Cooper-pair state and the excited quasiparticle state at a
temperature below the critical one. This energy gap protects our qubits from unexpected
excitations that probably originate from the environment and destroy qubit coherence,
leading to errors or information loss during computation.
Both charge and flux qubits have already been confirmed experimentally [3-5]. In addition the
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charge qubit [3] has achieved full one-qubit operation, but it has a large charge noise in its
Josephson junctions. This charge noise, which sometimes shows 1/f characteristics [6], is
unavoidable when we use an evaporated metal superconductor with the Dolan [7] bridge
technique, which is the only method currently available for fabricating Josephson junctions
smaller than 100 nm. The charge noise exists over a wide range of time scales and degrades the
quality not only in the qubit but also in other applications, such as the current standard with a
single electron transistor_In contrast, flux noise, which is more important in a flux qubit [8],
also exists in metal superconductors. This is caused by magnetic traps around or inside the qubit.
The external electromagnetic environment also influences the flux qubit as well as the charge
qubit. But there is relatively less flux noise than charge noise in the small junctions of
evaporated metal superconductors and the flux qubit is insensitive to the charge noise. That
means the flux qubit should have a longer coherence time than the charge qubit.
We are studying a flux qubit with three Josephson junctions that was first proposed by
Mooij et al. [9] The qubit state can be readout by measuring the magnetic field using a dcSQUID which magnetically couples with the qubit. We showed the possibility of a nonensemble readout , namely single-shot measurement ofn a flux qubit and dc-SQUID
combination [10]. The single-shot measurements were performed for qubits with different
junction sizes. The change in junction size changes the ratio between the Josephson coupling
energy EJ and the charging energy EC.
In this paper, we first describe the flux qubit that we are now working on. Then we
report the results of single-shot measurements with a changing E J/EC ratio. The wave
function amplitude in the double-well potential is discussed in light of the results obtained.
We also studied the effect of dissipation on the SQUID, because dissipation in the
SQUID affects coherence in the qubit. We measured the retrapping current of the SQUID
with no qubit and found that the relative phase relation for the magnetic field dependence
3
of the retrapping current jumped from 0 to π at E J ∪ EC . This jump is discussed
phenomenologically taking into account the two kinds of dissipation arising from ac
currents flowing through the lead and along the SQUID loop.
2. Readout of the qubit state
2.1 Flux qubit
Figure 1 shows the qubit, located in a dc-SQUID, which is a highly sensitive device for
magnetic field measurement. Both the qubit and the dc-SQUID consist of an aluminum
micrometer-sized loop with two or three Josephson junctions. We fabricated the samples
using electron-beam lithography in a double-layer resist and the standard shadow
evaporation technique [7]. Two successive aluminum layers were deposited (50 and 50
nm) on a SiO2 substrate. The qubit square loop inside has a persistent circulating current.
This current generates flux through the ring; the flux orientation is determined by the
current direction. The energy profile of this qubit has already been calculated in detail and
there are two energy minima in the energy landscape. According to Mooij et al. [9], the
total energy U of the qubit is given by
U / E J = 2 + α − cos γ 1 − cos γ 2 − α cos(2πf + γ 1 − γ 2 ) ,
(1)
where γ i (i = 1,2,3) is the phase difference across the three junctions. Junctions 1 and 2 are
identical with regard to EJ and the capacitance C, but junction 3 with γ 3 has a small E J
and C ( αE J , αC (0 < α < 1)) . The filling f is the external flux Φ EXT through the qubit loop
normalized by the flux quantum Φ 0 . It is assumed that the inductance of the qubit loop is
negligibly small. Figure 2 shows the total energy U as functions of ϕ m and ϕ p , where
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ϕ m = (γ 1 − γ 2 ) / 2 and ϕ p = (γ 1 + γ 2 ) / 2 . Two energy minima are clearly seen. These two
states correspond to clockwise and anti-clockwise current circulation. These current states
generate magnetic flux and have opposite orientations.
Flux qubit states can be read out by measuring the magnetic field with an under-damped
dc-SQUID, namely without a shunt resistor. Although a standard dc-SQUID has shunt
resistors to prevent hystereses in the current-voltage characteristics, we used an un-shunted
SQUID to minimize the coupling between the qubit and the dissipative environment
introduced by the shunt resistors. To perform accurate measurement, the qubit state must
not be disturbed during operation. However, this is somewhat contradictory. Accurate
measurement is performed by increasing the mutual inductance between the qubit and the
SQUID. We can alter the mutual inductance by changing the distance between the qubit
and the SQUID. Large inductance provides good sensitivity but leads to large
decoherence.
It should also be noted that the two current states in a qubit behave as a single
wavefunction, because this current is carried by a supercurrent and the ground state of the
electrons in the superconductor is condensed in a single quantum state. A physical twolevel system is constructed using this superconductor and the double-well potential that is
provided by the three Josephson junctions. The coherent behavior in this system is of
interest and is called macroscopic quantum coherence (MQC).
We showed the possibility of a non-ensemble readout , namely single-shot measurement
in a flux qubit and DC-SQUID combination [10]. We believe this non-ensemble readout is
very effective in terms of providing a quantum algorithm for such algorithms as Shor s
factorization, although a more detailed mathematical approach is also required to prove the
relationship between the power of a QTM (quantum Turing machine) and the complexity of
the problem.
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2.2 Measurement setup
The readout of our qubit is the measurement of the switching current of the dc-SQUID.
The SQUID is an un-shunted type and it therefore has a large hysteresis. A triangular wave
oscillator and relatively high resistances in series with the dc-SQUID generate a current
sweep. The resistance in series is 100 k or 1 MΩ. The typical resistance of the dc-SQUID
was between 100 and 1 kΩ. When the ratio is at least more than 100, the dc-SQUID is
sufficiently current biased. Each measurement corresponds to one cycle of the current
sweep generated by the oscillator. We simultaneously measured the voltage across the dcSQUID using the four-terminal method of measurement. The current value when the
SQUID jumps to the voltage state is the switching current ISW.
We measured the sample at a base temperature of about 40 mK with an He3-He4 dilution
refrigerator. Because this refrigerator has no temperature-control system, the entire lowtemperature measurement was performed at the base temperature, which was very stable
with a fluctuation of less than 0.2 mK from one batch run to another. The sample was
mounted using a copper holder that provided a thermal connection between the mixing
chamber and chip holder.
To eliminate unexpected noise, we inserted RC low pass filters in each sample
measurement line. The refrigerator was located in an electromagnetic shielding room. A
superconducting solenoid was used to apply the external magnetic field to the dc-SQUID
and qubit. The shape of the double-well potential of the qubit can be completely controlled
by the external magnetic field. The measured data were logged into a PC by an AD
converter and stored in an HD. The current sweep speed was varied from 0.1 Hz to 75 Hz
depending on the required accuracy and time allowed. Multiple measurements in different
external magnetic fields required a lot of sweeps.
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2.3 Readout with a dc-SQUID
Figure 3 shows the current voltage characteristics of the dc-SQUID. This shows a
hysteretic response. The SQUID switching appeared as a horizontal line which jumped
from a zero voltage state to a finite voltage state of about 400_V. This voltage is almost the
same as the superconductor gap in bulk aluminum. After jumping to the finite voltage state,
the I-V curve follows a resistive line, the resistance of which is almost the same as the
tunnel resistance measured at room temperature. The jumping position of the current
depends on the magnetic flux penetrating the dc-SQUID loop. The supercurrent keeps its
maximum value when the external magnetic flux has an integer number of flux quanta. In
contrast, when the external flux is a half integer, the supercurrent is suppressed almost to
zero. When the value of the external flux is in between these two, the supercurrent changes
continuously between integer and half integer numbers. The switching current is
determined as follows:
I SW = 2 I C cos(πΦ / Φ 0 ) ,
(2)
where I SW is the switching current and I C is the critical current of a single Josephson
junction, and Φ is the flux through the SQUID loop. The experimentally observed
maximum switching current reflects the quality of the dc-SQUID and measurement setup.
It reached more than 800 nA with the dc-SQUID in Fig. 3. Comparing the maximum
switching current with the normal resistance using the Ambegaokar-Baratof relation [11],
we consider this sample to have an ideal SIS tunnel barrier. The measurement is also well
current-biased and has relatively little noise. We can also evaluate the junction quality from
the vertical supercurrent branch in the I-V characteristic.
Figure 4 shows the switching current of a dc-SQUID when the external magnetic field is
changed. We can see the periodic behavior as described before. The horizontal axis in Fig.
4 is the external magnetic field. There is a small discontinuous region where the ground
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state of the inner qubit changes and the orientation of the generated flux switches. These
regions appear slightly thicker and are placed at —0.5 and +0.5 in thefilling f = Φ EXT / Φ 0 .
The global periodic dependence in Fig. 4 is caused by the external magnetic field and the
discontinuous step is caused by the inside qubit.
The relationship between the readout signal and the sample geometry is interesting. To
investigate it, we first studied the effect of junction size on the readout signal. Figures 5(a),
(b) and (c) are enlargements of the area around +0.5 in the filling (indicated by an arrow in
Fig. 4) for three different samples. Each sample has the same dc-SQUID, but the size of the
junction in the qubit is different. The size of the two identical junctions in the qubit is
2
0.2 ↔W µm and the three samples have a W of 1, 0.5 and 0.3 µm. Hereafter, we call them
sample 1, 2, and 3 in this order; α is fixed at 0.8 for all three samples. Sample 1, which has
the largest junction, showed two clearly distinguishable parallel lines (Fig. 5 (a)). It should
be noted that each point in the figure corresponds to a single-shot measurement. Each
figure in Fig. 5 corresponds to around 5000 measurements. The two lines correspond to
clockwise and anti-clockwise qubit current. Although these two states might both be in the
ground state, this separation means that the dc-SQUID can detect changes in qubit ground
states with a single-shot measurement [10].
Sample 2, which has a medium size junction, showed two lines linked by points (Fig. 5
(b)), and finally in sample 3, which has the smallest junction, the two lines merged with
each other (Fig. 5 (c)). Van der Wal et al. reported very similar ISW-f characteristics [5] but
they applied averaging to the multiple measurement readout. We have to note again that the
data points in the line in Fig. 5 (c) were observed in a single-shot measurement. Therefore,
the results obtained strongly suggest that the shape of the probability distributions P of the
particle with a mass C in the double-well potential for three samples are as shown in Fig. 6.
The P for sample 1 seems to be strongly localized in two potential wells. On the other hand,
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the P for sample 2 seems to be widely distributed, and the P for sample 3 seems to have a
sharp peak around the center of the potential energy.
2.4 Discussion
To test the suggestion made in Fig. 6, we calculated P using the evaluated EJ and EC. The
barrier height in the double-well potential is determined by EJ and the eigenvalue of each energy
level in the well depends on EJ/EC. Neither the critical current nor the junction capacitance in
the qubit can be measured directly. However, EJ is evaluated from the ISW (Φ = 0) of the junction
in the dc-SQUID, which was fabricated simultaneously and could be measured directly. The
junction capacitance was evaluated from the junction area and the typical specific capacitance of
60 fF/µm2. The evaluated EJ and EJ / EC are 500 GHz and 320 for sample 1, 250 GHz and 70 for
sample 2, and 100 GHz and 20 for sample 3. Using these values, we calculated the energy
eigenvalues and wavefunctions of each level in the double-well potential taking the tunneling
effect through the barrier into account. We define Ψi (ϕ ) as a wavefunction of the ith energy
level. Ψ0 (ϕ )
and Ψ1 (ϕ ) indicate the ground and the first exited states, respectively. We
performed multiple measurements of the single quantum state one-by-one. When T = 0, we
always find the qubit state in the ground one. However, the actual qubit is excited from the
ground state as a result of interaction with the environmental thermal bath. This effect gives us a
possible way of finding out the qubit state in the excited states. The probability of finding the
state in the nth energy level is given by
Pn = exp(− β En ) /
exp(− β Ei ) ,
(3)
i
where Ei is the eigenvalue of the ith level and β = 1 / k BT . Then multiple measurements
give the probability distribution P(ϕ) as
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2
P(ϕ ) =
Pi Ψi (ϕ ) .
(4)
i
Figure 7 plots the calculated P for the three samples at T= 40 and 400 mK. As would be
expected from the data in Fig. 5 (a), the P for sample 1 is well localized in two wells even
at a high temperature of 400 mK (Fig. 7(a)). However, the P is still localized even in the
case of sample 3 at a high temperature (Fig. 7(b)). At present, we cannot explain the
difference between the results suggested in Fig. 6 and the calculated ones given in Fig. 7.
However, there is a possible explanation.
In the previous discussion (Eq. (1)), we assumed that the inductance of the qubit loop L
was negligibly small. When L is considered again, the magnetic coupling between the qubit
and the dc-SQUID is given by the interaction Hamiltonian,
H int = − M /( LLS ) Φ 20 /(4π 2 ) (2ϕ m + γ 3 − 2πf ) (γ S − − 2πΦ / Φ 0 ).
(5)
Here, LS is the inductance of the SQUID loop and Φ is the external flux trapped in the SQUID
loop. γ S− is defined by γ S1 − γ S 2 , where γ Si is the phase difference across the junction in the
SQUID. This shows that the SQUID measures the qubit state via the quantity (ϕ m + γ 3 / 2 − πf ) ,
which is different from ϕ m discussed above.
The potential energy U (ϕ p = 0, ϕ m , γ 3 ) for sample 3 is plotted in Fig. 8. The horizontal
axis (ϕ m − γ 3 / 2 − πf ) coincides with the limit of L ♦ 0 to ϕ m in the previous figure of the
potential (Fig. 2). The vertical axis (ϕ m + γ 3 / 2 − πf ) corresponds to the quantity observed
by the SQUID. The distance ∆ϕ between two minimum positions is proportional to the
product of the Josephson energy and the inductance of the qubit, EJL. Although the width of
2
the wells in the horizontal direction is independent of the EJL product, when E J L(2e / h ) is
small, the ratio of the width in the vertical direction to ∆ϕ grows rapidly. This means that
the wavefunction of the qubit in each well is well localized in the horizontal direction.
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However, it spreads widely along the observed quantity ϕ m + γ 3 / 2 ; in other words, the
quantity ϕ m + γ 3 / 2 is uncertain. This fact might explain the switching current distribution
we observed. More theoretical investigation of this direction is needed. In particular, the
wavefunction should be calculated and considered.
3. Dissipation effect on the dc-SQUID
It is believed that the dissipation in the qubit and dc-SQUID af fects the coherence of the
qubit state. Since the retrapping current of the SQUID is related to the dissipation of the
SQUID and the qubit, it is useful to study the retrapping current of the dc-SQUID from the
viewpoint of dissipation.
The retrapping current observed in Nb/NbxOy/PbIn single junctions demonstrates that the
dissipation in the retrapping process is dominated by the tunnelling of thermally excited
quasiparticles [12, 13]. In this case, the subgap resistance of the junction and the dc current
flowing in the junction cause the dissipation. However, experimental work on small single
junctions has highlighted problems caused by the junction s environment [14]. In the latter
case, the retrapping current occurs at a relatively high subgap voltage, so that high
frequency components are important in determining the impedance. We studied the
retrapping current of the dc-SQUID without the qubit in order to characterize the dcSQUID itself. We found that two kinds of dissipation due to high frequency componets
play important roles in small-capacitance SQUIDs. These dissipations are closelyy related
to the SQUID geometry, which has two degrees of freedom, and are also related to the
quantum nature of the electric magnetic environment.
3.1 Observation of the retrapping current
We altered the SQUID loop inductance LS from 10 to 320 pH by changing the loop size,
11
and altered the junction capacitance C by changing the overlapping area of the junction so
that C ranged from 1.2 to 24 fF. These capacitances were also evaluated on the basis of
junction area and typical specific capacitance of 60 fF/µm2.
Figure 9 shows an enlargement of the square area in Fig. 3. The retrapping current Ir of the
SQUID depends on the magnetic field. In this work, we define the retrapping current as the
current value when the I-V curve crosses 350 µV. Figures 10 (a) and (b) show the magnetic field
dependence of the retrapping currents and the switching currents for the sample with
capacitance C = 1.2 fF (a), and the sample with capacitance C = 12 fF (b). The switching
currents of both samples show typical oscillatory magnetic field dependence. However the
retrapping current of the lower-capacitance sample shows cosine-like dependence; in contrast,
that of the higher-capacitance sample shows minus-cosine-like dependence. The transition
between the two kinds of magnetic field dependence occurs at around C = 2.4 fF. However, as
discussed below, this value is not intrinsic to the transition. The transition occurred when
E J / EC ∪1 . Another characteristic feature of the data is that both retrapping currents in Figs 10
(a) and (b) have an offset.
The retrapping current I r of a Josephson junction depends on a dissipation. When we apply
the resistively and capacitively shunted junction (RCSJ) model [15] to the junction, the
temperature dependence of the retrapping current I r (T ) exhibits the exponential temperature
dependence associated with quasi-particle resistance Rqp (T ) = RN exp(∆ 0 / k B T ) [164]. Here, RN
is the normal resistance of the junction and ∆0 is the gap energy. In a small and highly resistive
junction, however, I r (T ) abruptly stops decreasing at a certain temperature and then remains
constant down to the lowest temperatures [16-18]. Johnson et al. explained this phenomenon
with an energy balance argument [14]. They considered the current components at two
frequencies: dc and the Josephson frequency ω J = 2eV / h , where V is the dc average voltage
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across the junction. The high frequency current I SW 0 sin ω J t is generated at the junction
through the ac Josephson effect. Here, I SW 0 is the bare switching current of the junction which
is determined by the Ambegaokar-Baratoff relation [11]. The total impedance Z (ω J ) for the
high frequency current was modeled using the parallel contributions of the quasiparticle
resistance Rqp, the junction capacitance C, and the real impedance Z1 of the lead, which works
only for the high frequency. The lead typically shows a characteristic impedance
Z1 ∪ Z 0 / 2π = 60 Ω , apart from logarithmic corrections for geometric factors, where
Z 0 = 377 Ω is the free space impedance. From this model, Z (ω J ) is given by
1 Z (ω J ) = i ω J C + 1 Rqp (T ) + 1 Z1 .
(6)
The input power is balanced by the sum of the dc and ac dissipations. This gives
IV =
2
V 2 I sw
0 Re( Z ) .
+
Rqp
2
(7)
The minimum current which satisfies Eq.(7) is I r and the corresponding voltage is the
retrapping voltage Vr . As the temperature decreases, I r decreases and Vr increases. At
sufficiently low temperatures, Vr is pinned at the gap voltage Vg ∪2∆ 0 / e and R qp (T ) can be
disregarded because Rqp (T ) >> Z 1 . The substitution of Eq. (6) into Eq. (7) gives
Ir ∪
2
2
I sw
I sw
0 Re( Z )
0 Z1
,
∪
2Vg
2Vg [1 + (4∆ 0 CZ1 h ) 2 ]
(8)
which has no temperature dependence at low temperatures. Thus, in small and highly resistive
junctions, ac dissipation dominates at low temperatures, whereas dc dissipation dominates at
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higher temperatures.
Figure 11 shows the temperature dependence of the measured retrapping current of the dcSQUID. The retrapping current clearly shows saturation at low temperatures. Hence, at T < 0.4
K, we can expect the ac dissipation to be dominant and we can ignore the dc dissipation. Thus
Eq. (8) can be used at the lowest temperature of 40 mK. If we consider the SQUID as a single
junction by replacing C with 2C and I SW 0 with 2 I SW 0 cos(π Φ / Φ 0 ) in Eq. (8), we obtain the
cosine dependence of I r . However, this model does not provide the minus-cosine dependence
for a high capacitance SQUID. Furthermore, when the filling Φ / Φ 0 is half integers, the
retrapping current vanishes and there is no offset. Therefore, the single junction model cannot be
applied to our experimental results.
3.2 Discussion
Here we propose a SQUID model instead of a single junction one. We consider the two
components of the ac current: I lead and I loop . I lead is the ac current flowing through the lead
and I loop is the ac current circulating around the SQUID loop. Under a magnetic field Φ / Φ 0 ,
and by using fluxoid quantization γ S1 − γ S 2 = 2π (n + Φ / Φ 0 ) , I lead and I loop are written as
I lead = I1 + I 2 = 2 I sw0 cos( π Φ / Φ 0 ) sin( γ S1 + π Φ / Φ 0 ) ,
(8)
I loop = ( I1 − I 2 ) / 2 = − I sw0 sin(πΦ / Φ 0 ) cos(γ S1 + πΦ / Φ 0 ) .
(9)
Here I1 and I 2 are ac currents flowing through junctions 1 and 2, γ S1 and γ S 2 are the phase
differences across junctions 1 and 2, and n is an integer. We assume that the input energy I rV g
is balanced by the sum of the dissipations due to I lead and I loop . Then we obtain
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2
2
I rVg = I lead
Re( Z lead ) + I loop
Re( Z loop )
[
]
[
]
2
2
I sw
I sw
0
=
4 Re( Z lead ) + Re(Z loop ) + 0 4 Re(Z lead ) − Re( Z loop ) cos( 2π Φ / Φ 0 )
4
4
. (11)
Here Z lead and Z loop are the impedances for the lead and loop currents. The first term on the
right-hand side of Eq. (11) shows that the retrapping current has an offset. The second term
shows that the magnetic field dependence of the retrapping current can be cosine- or minuscosine-dependent. If the lead impedance is dominant (i.e., 4 Re( Z lead ) > Re( Z loop ) ), the
retrapping current shows cosine-dependence, which corresponds to the result for the lowercapacitance samples (see Fig. 10 (a)). If the loop impedance is dominant, the retrapping current
shows minus-cosine-dependence, which corresponds to the results for the higher-capacitance
sample (see Fig. 10 (b)). This model can thus explain the experimental results qualitatively. We
can estimate Re( Z lead ) and Re( Z loop ) from the retrapping currents at Φ / Φ 0 = 0 and Φ / Φ 0 =
1/2. In this estimation, we use I sw0 which is deduced from Vg and the normal resistance of the
junction R N using the Ambegaokar-Baratoff relation. The value of V g = 420 µV is extracted
from the I-V curve of the sample, and R N is estimated from the high bias part of the I-V curve.
Both Re( Z lead ) and Re( Z loop ) are of the order of 100 Ω. Using these values, the SQUID model
reproduces the magnetic field dependence of the retrapping current very well, as shown in Fig.
10.
Now let us consider the physical origin of Z lead and Z loop . It was experimentally found that
Re( Z lead ) was strongly dependent on the junction capacitance but not on the loop inductance.
The capacitance dependence of Re( Z lead ) can be reproduced using the single junction model
with Z1 of the values from 50 to 300 Ω (not shown). In this model, we ignore the loop
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inductance and consider the SQUID as a single junction which is shunted by the impedance Z1
for a high frequency current. The values used for Z1 (50 to 300 Ω) are somewhat distributed but
are of the order of the Z1 ∪ Z 0 / 2π = 60 Ω that was given in the single junction model. Thus,
Z lead can be modeled by the impedance of the single junction model.
The radiation resistance of the SQUID loop is 1↔
10 −6 Ω for a loop size of 100 µm2 and at
the typical frequency of 2eVg / h ∪ 200 GHz. This value is too small to account for our data (of
the order of 100 Ω). Therefore we cannot explain Z loop by considering the effect of the loop
only. Experimental results showed that Re( Z loop ) also exhibited junction capacitance
dependence (however, it was weaker than that of Re( Z lead ) ) and that Re( Z loop ) was not
dependent on the loop inductance. Judging from these results, we think that current is induced
in the lead by the circulating current and that this effect should be taken into account in
explaining the physical origin of Z loop .
Figure 12 shows A …4 Re( Z lead ) − Re( Z loop ) as a function of E J / EC . Here, E J = hI SW 0 2e
is the Josephson coupling energy of the junction and EC = (2e )2 2C is the charging energy of
the junction for a Cooper pair. A positive value of A means that the retrapping current shows
cosine-like dependence. A negative value of A means that it shows minus-cosine-like
dependence. It can be clearly seen that the magnetic field dependence of the retrapping current
changes at E J ∪ EC . We therefore consider that competition between the charge and the phase
causes these changes in the retrapping current behavior described above. It should be noted that
the temperature dependence of the retrapping current in a small Josephson junction does not
change at E J ∪ EC . The superconducting loop is essential to observing the change in the
behavior of the retrapping. However, the loop size is irrelevant. It is not the loop characteristics
but the junction characteristics (EJ and E C ) that determine the behavior. This means that a
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junction with a superconducting loop is more sensitive to the quantum nature of an
electromagnetic environment. In other words, the SQUID is a very effective tool for investigate
dissipation effects caused by the environment.
4. Conclusions
We described the use of a dc-SQUID to readout the ground state of a qubit with three
Josephson junctions. Single-shot measurements were performed for the qubits with different
junction sizes; in other words, the ratio of E J / EC varied. We also calculated the energy
levels and wave functions for the measured samples, and discussed the probability amplitude
in the double-well potential. The calculated results could not explain the probability
amplitude that the experimental results suggested for the sample with the smallest junction.
However, we put forward a simple idea that might explain the the results of the experiment.
Although more detailed theoretical and experimental work is required, the clear
experimental result obtained for a non-ensemble readout of fers excellent prospects for the
future.
We measured the retrapping current of a dc-SQUID without a qubit and found that the
dependence of the retrapping currents on the magnetic field undergoes a π-phase shift when the
charging energy of a Cooper pair becomes comparable to the Josephson energy. The magnetic
field dependence of the retrapping current can be explained in terms of the impedance at the
lead and at the loop. However, the value of the loop impedance estimated from the geometrical
configuration is too small to account for the observed π-phase shift. The fact that the shift occurs
at E J ∪ E C strongly suggests that the quantum nature of the electromagnetic environment plays
an important role here. Uunderstanding the dissipation at the SQUID is particularly important
because it is used as a readout device for flux qubits. We therefore believe that the findings
reported in this paper merit further experimental and theoretical investigation.
17
Acknowledgements
We thank J. E. Mooij, C. J. P. M. Harmans, and S. Ishihara for motivating this research, and
M. Ueda, Caspar H. van der Wal, Y. Nakamura, Y. Sekine, and E. Laffose for useful
discussions.
18
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19
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20
Figure captions
Figure 1
SEM micrograph of qubit (inside) and dc-SQUID (outside). The three constrictions in the qubit
and the two in the dc-SQUID are Josephson junctions. The size of the SQUID loop is about 10
µm ↔10 µm.
Figure 2
{
}
The total energy of the three-junction qubit in the phase space ϕ m ; ϕ p when f = 0.5 and α =
0.8. The two black areas around the center of the figure correspond to the two wells.
Figure 3
Current-voltage characteristics of dc-SQUID with different magnetic field Φ / Φ 0 . This SQUID
has no inside qubit for native dc-SQUID characterization. The SQUID parameters are LS = 20
pH, C = 6 fF, RN = 0.47 kΩ, T = 40 mK, where LS is the loop inductance of the SQUID and RN is
the normal resistance of the junction.
Figure 4
Measured switching current-magnetic field characteristics in the SQUID with a qubit. Arrows
indicate where a jump occurs due to a change in the current direction in the qubit (see Fig. 5).
Figure 5
Switching current-filling characteristics, a, b, and c are enlarged graphs of Fig. 4 for samples 1,
2, and 3, respectively.
Figure 6
21
Probability distribution P for the double-well potential suggested by the switching current
distribution in Fig. 5.
Figure 7
Calculated probability distribution P for the phase space for samples 1, 2, and 3 (from top to
bottom).
Figure 8
Potential energy U (ϕ p = 0, ϕ m , γ 3 ) for sample 3 with f = 0.5. When the loop inductance of the
qubit L = 0, the horizontal axis (ϕ m − γ 3 / 2 − πf ) coincides with ϕ m . The vertical axis
(ϕ m + γ 3 / 2 − πf ) corresponds to the quantity observed by the SQUID.
Figure 9
Current-voltage characteristics of the SQUID with changes in the magnetic field. This graph is
an enlargement of the square area in Fig. 3.
Figure 10
Switching and retrapping currents as a function of the magnetic field Φ / Φ 0 , where Φ is the
flux through the SQUID loop. Dotted curves are experimental data. Solid red lines represent
SQUID model. Retrapping currents have an offset for both samples. (a) LS = 20 pH, C = 1.2 fF,
RN = 8.4 kΩ , T = 40 mK. The retrapping current shows cosine-like dependence; i.e. the
retrapping current and switching current are in-phase. (b) LS = 20 pH, C = 12 fF, RN = 1.1 kΩ, T
= 40 mK. The retrapping current shows minus cosine-like dependence, in other words, the
retrapping current and switching current are out of phase.
22
Figure 11
Temperature dependence of retrapping currents. At T > 0.4 K, retrapping currents decrease with
decreasing temperature. At these temperatures, dc dissipation is dominant. When T < 0.4 K, the
retrapping currents become constant. At these temperatures, ac dissipation is dominant. (a) LS =
20 pH, C = 1.2 fF, RΝ = 11 k Ω. (b) LS = 20 pH, C = 12 fF, RN = 1.1 kΩ.
Figure 12
4 Re( Z lead ) − Re( Z loop ) as a function of E J EC . The upper half of the figure shows that the
retrapping current exhibits cosine-like dependence. The lower half shows that the retrapping
current exhibits minus-cosine-like dependence. The magnetic field dependence of the retrapping
current changes at E J ∪ EC .
23
U
ϕ
p
ϕm
2000
Current (nA)
Φ/Φ0
1000
0.5
0.3
0
Isw
0
-1000
-2000
-400
0
Voltage (µV)
400
(a)
(b)
(c)
P
U
ϕm
0
f=0.5
ϕm
0
(a)
40mK
400mK
(b)
40mK
400mK
(c)
40mK
400mK
100
Current (nA)
80
Φ/Φ0
60
40
0.5
0.3
0
Ir
20
0
300
350
400
Voltage (µV)
450
(a)
(b)
(a)
(b)
4Re(Zlead) - Re(Zloop) (Ω)
1000
800
600
400
200
0
-200
-400
2
0.1
4 6
2
4 6
1
2
10
EJ / EC
4 6
100