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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. 1 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 2 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 4 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. 5 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. 6 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 7 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, 8 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 9 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. 10 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 12 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 13 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 14 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 15 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 16 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 References [1] Chuang, I. L., Gershenfeld, N. and M. Kubinec, M., Phys. Rev. Lett. 80, 340 (1998). Chuang, I. L., Vandersypen, L. M. K., Zhou, X., Leung D.W. and Lloyd, S., Nature 393, 143 (1998). Jones, J.A., Mosca, M. and Hansen, R. H., Nature 393, 144 (1998). [2] Shor, P., Proceedings 35th Annual Symposium on Foundations of Computer Science 124 (1994). [3] Nakamura, Y., Pashkin, Yu. 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L., Raider, S.I. and Fisher, M. P. A., Phys. Rev. Lett. 61, 2372 (1988). [13] Cristiano, R., Frunzio, L., Nappi, C., Castellano, M. G., Torrioli, G. and Cosmelli, C., J. Appl. Phys. 81, 7418 (1997). 19 [14] Johnson, A. T., Lobb, C. J. and Tinkham, M., Phys. Rev. Lett. 65, 1263 (1990). [15] McCumber, D. E., J. Appl. Phys. 39, 3113 (1968). [16] Iansiti, M., Hu, Qing, Westervelt, R. M. and Tinkham, M., Phys. Rev. Lett. 55, 746 (1985). [17] Iansiti, M., Johnson, A. T., Smith, W. F., Rogalla, H., Lobb, C. J. and Tinkham, M., Phys. Rev. Lett. 59, 489 (1987). [18] Iansiti, M., Tinkham, M., Johnson, A. T., Smith, W. F. and Lobb, C. J., Phys. Rev. 39, 6465 (1989). 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