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Physica C 367 (2002) 197–203 www.elsevier.com/locate/physc
Implementation of a combined charge-phase quantum bit in a superconducting circuit A. Cottet, D. Vion, A. Aassime, P. Joyez, D. Esteve *, M.H. Devoret Service de Physique de l’Etat Condens e, CEA-Saclay, Quantronics group, F-91191 Gif-sur-Yvette, France
Abstract We discuss a qubit circuit based on the single Cooper-pair transistor (which consists of two ultrasmall Josephson junctions in series) connected in parallel with a large Josephson junction. The switching of this junction out of its zerovoltage state is used to readout the qubit. We report measurements of the discriminating power of the readout process, and we discuss its back-action on the qubit. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.Lx; 73.23.Hk; 85.25.Na Keywords: Qubits; Quantum computing; Charging effects; Quantum coherence
1. Introduction The recently proposed quantum computing schemes are based on the controlled evolution of a set of two-level systems, called qubits (see Ref. [1] for a review). Suitable qubits should have a quantum coherence time much longer than the duration of an elementary operation, and be measurable when necessary. Among the various systems proposed for implementing qubits, nanofabricated solid-state circuits are particularly attractive because they are more easily scalable. The presently most investigated solid-state qubits are the ‘‘single Cooper-pair box’’ [2], in which Rabi precession of a coherent superposition of the qubit states has been demonstrated [3], and the ‘‘flux box’’ [4]. In the Cooper-pair box, the qubit states
* Corresponding author. Tel.: +33-016-908-5529; fax: +33016-908-7442. E-mail address:
[email protected] (D. Esteve).
are spanned by discrete charge states of a superconducting small metallic island, and are controlled by the gate charge coupled to the island. In the flux box, the qubit states are spanned by flux states of a small superconducting loop, and are controlled by the flux threading the loop. Whereas some simple qubit manipulations have already been realised in these systems, no readout of a single qubit has yet been achieved. In the case of the Cooper-pair box, the measurement of the island potential either by means of a radio-frequency single electron transistor [5] or of a single Cooper-pair transistor [6], is nevertheless believed to reach a sensitivity sufficient for such a ‘‘singleshot’’ readout. In this paper, we discuss a qubit circuit which is controlled both by a charge and by a phase. In this combined charge-phase qubit, qubit control is performed by acting on the gate charge like in the Cooper-pair box, but the measured quantity is a supercurrent, like in the flux box. The main interest of this Q–d design is to provide (i) a good immunity respectively to the
0921-4534/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 1 0 1 4 - 0
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offset charge noise, which is presently considered as the major source of decoherence in charging devices, (ii) a good decoupling from the measuring system when readout is not active, and (iii) a fast pulsed readout. We explain below the operating principle of this Q–d qubit, and we report first measurements of the readout system resolution.
2. Operating principle of a combined Q–d qubit based on the single Cooper-pair transistor The qubit circuit we consider (see Fig. 1a) is based on the single Cooper-pair transistor (SCPT) [7], which consists of two nominally identical ultrasmall junctions in series. When a phase difference d is imposed across the transistor by means of
Fig. 1. (a) Schematic circuit of a Q–d qubit based on a superconducting SCPT in a flux-biased loop; (b) full circuit with a readout Josephson junction, and a bias-current source with admittance Y ðxÞ: Readout is based on the dependence of the critical current of the system on the qubit state. (c) Effective admittance implemented in the present experiment. An on-chip capacitor C 1 pF is connected in parallel with an 500 X–10 pF RC-series circuit made with surface mounted components.
a small superconducting loop threaded by an applied flux, this device is equivalent to a single Cooper-pair box with an effective Josephson coupling EJ ¼ 2EJ0 cos ðd=2Þ, where EJ0 is the Josephson energy of each junction with critical current iC ¼ EJ0 =u0 (u0 ¼ h=2eÞ. The hamiltonian of the transistor depends both on the phase d and on the gate charge ng ¼ Cg U =2e, and can be written as: 2 H ¼ 2EJ0 cos ðd=2Þ cos b h þ EC ðb n ng Þ :
ð1Þ
Here, the conjugated variables b h and b n are respectively the difference between the phases across each junction of the transistor, and the number of excess Cooper pairs in this island; EC ¼ ð2eÞ2 = ð2CR Þ is the charging energy of the island with total capacitance CR , and ng the reduced gate charge in units of 2e. This hamiltonian is easily diagonalised in the eigenbasis of the charge operator b n . The eigenstates jii, with energy Ei ðd; ng Þ, sustain a supercurrent Ii ¼ u1 0 ðoEi =odÞ and correspond to an island voltage Vi ¼ ð2eÞ1 ðoEi =ong Þ. When EJ0 is not much larger than EC , and ng close to 1=2; the two lowest energy states, labelled 0 and 1 are well separated from the other levels and can be used to implement a qubit. In this circuit, the qubit manipulation can be performed by applying microwave pulses at the qubit transition frequency X=2p ¼ ðE1 E0 Þ=h on the gate electrode. When EJ0 is not large compared to EC , the energies of the qubit levels are well approximated by: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi 2 E0;1 ¼ ðEJ0 cos ðd=2ÞÞ þ EC ð1 2ng Þ : ð2Þ Although the qubit states can be used over a wide parameter range, some biasing points are more attractive. In particular, at the electrostatic energy degeneracy point ng ¼ 1=2; one has oEi =ong ¼ 0, and, at d ¼ 0; oEi =od ¼ 0. At the biasing point ðng ¼ 1=2; d ¼ 0Þ, the qubit frequency is set by the Josephson energy of the junctions, and is only affected in second order by fluctuations of the control parameters. Dephasing of the qubit should thus be minimized at this biasing point. In particular, the random offset charge noise [9] should less affect this circuit than the Cooper-pair box [8]. In the proposed set-up, the qubit is maintained at the biasing point ðng ¼ 1=2; d ¼ 0Þ during its ma-
A. Cottet et al. / Physica C 367 (2002) 197–203
nipulation, and the phase d is varied only during the readout. The qubit circuit is shown in Fig. 1b: another Josephson junction, with a Josephson energy EJ ¼ u0 IC EJ0 , is inserted in the loop in order to readout the qubit. The principle of the readout is based on the dependence on the qubit state i of the critical current ICi of the parallel combination of the transistor and of the large junction. This critical current is the maximum value of ICi ðcÞ ¼ IC sin c þ Ii ð/ þ cÞ, where c is the phase across the readout junction, and ð/ þ cÞ the phase across the SCPT. Because the large junction imposes to a good approximation the phase across the transistor, the critical current almost corresponds to c ¼ p=2. We demonstrate in the following that the critical current can be determined with sufficient accuracy to discriminate both qubit states.
3. Readout resolution The determination of the critical current can be performed by detecting the transition to a finite voltage state when the bias-current approaches it. This switching transition is easily detected by monitoring the voltage across the system. The switching out of the zero-voltage state of a current-biased Josephson junction has been thoroughly investigated. The current-biased junction is indeed a model system for the dynamics of a single degree of freedom whose coupling to an environment can be controlled [10,11]. The dynamics of the phase c is that of a particle with mass Cu20 , placed in a tilted washboard potential U ¼ EJ ðcos c þ scÞ, with s ¼ I=IC , and subject to retarded friction determined by the admittance Y ðxÞ of the circuit in parallel with the junction. Below the critical value s ¼ 1; the potential has metastable equilibrium positions. The phase dynamics is characterised by the frequency xP =2p and the quality factor Q of the oscillations at the bottom of the wells. For s close to 1; the oscillation frequency is xP =2p ’
pffiffiffi !1=2 2I C 1=4 ð1 sÞ : u0 Ceff
ð3Þ
199
In this expression, we have assumed that the capacitance Ceff includes the eventual contribution of the environment. The quality factor is then Q ¼ CxP =ReY ðxP Þ. For underdamped or moderately damped junctions ðQ P 1Þ, escape out one of these wells triggers the run-away of the phase down the washboard potential, and a finite voltage develops. At high temperature, the phase across the junction is an almost classical variable and switching occurs by thermal activation over the potential energy barrier. The rate of this process follows Kramers’ law: xP DU Ccl ¼ a exp ; ð4Þ kB T 2p pffiffiffi 3=2 where DU ’ ð4 2=3ÞEJ ð1 sÞ , and a is a prefactor of order 1 in the regime Q 1: Below the cross-over temperature defined by kB TCO ¼ hxP = 2p, escape occurs mainly by quantum tunneling ‘‘through’’ the potential energy barrier [11]. For Q P 1, the quantum rate does not depend on Q and is: sffiffiffiffiffiffiffiffi DU xP DU exp 7:2 Cq ¼ 52 : ð5Þ hxP 2p hxP It is useful to define the escape temperature Tesc through the relation: xP C¼ expðDU =kB Tesc Þ: ð6Þ 2p The theoretical predictions for a junction in the Q P 1 regime are Tesc ’ T above the cross-over temperature TCO , and Tesc ’ hxP =7:2kB below. Quantum corrections in the thermal activation regime, and thermal corrections in the quantum regime slightly increase the escape temperature above these values. When a bias-current pulse I is applied during a time s, the switching probability for each qubit state i is PSi ¼ 1 expðCi sÞ. If these probabilities are different enough, switching provides a singleshot measurement of the qubit. We define the discriminating power of the readout system as a ¼ jPS1 PS0 j. Single-shot readout is achieved when a is close to 1. The relations (4) and (5) show that the discriminating power is determined by thermal and quantum fluctuations, which limit the slope dC=dI.
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In order to increase this slope, one should decrease the escape temperature till it reaches the temperature of the experiment, and decrease IC . Since the quantum escape temperature is set by the readout junction plasma frequency, which is imposed if all junctions are fabricated in the same pump-down, the only way to decrease it significantly is to add an extra capacitance to the readout junction capacitance. Decreasing the critical current IC is in conflict with the requirement of a good phase bias of the SCPT, and a compromise has to be found, as discussed below. In the present design, the ratio between the readout junction critical current IC 1 lA and the SCPT supercurrent is about 100. Finally, the time resolution is limited by the readout pulse duration, provided that the readout junction stays at thermal equilibration in its potential well when the bias-current is raised. For that purpose, its Q factor should not be too large.
4. Back-action of the environment and of the readout system on the qubit In fine, the sensitivity of a readout system has to be weighted by its back-action on the qubit during manipulation and readout. As already discussed in the case of the single Cooper-pair box measured by an electrometer [5,12,13], different processes contribute to this back-action. First, the qubit can undergo a transition from the upper state to the ground state by transferring the energy hX to the degrees of freedom it is coupled. This process defines the relaxation time TR . The reverse excitation process is also possible, and is characterised by an excitation time TE . Both TR and TE are expected to differ when readout is off or on. In particular, TE is expected to be much longer in the off state, when no energy is available to excite the qubit, than in the on state. When a coherent superposition of qubit states is prepared, it is furthermore randomly dephased by the low frequency fluctuations of the qubit hamiltonian due to both the environment and the measuring system. Dephasing is expected to occur much faster when readout is on because the qubit is no longer decoupled from phase fluctuations.
We first discuss the back-action when readout is off, i.e. at zero-bias-current. In this case, phase fluctuations are small, and the whole circuit connected to the SCPT can be treated as an effective impedance Zeff ðxÞ ¼ ðY ðxÞ þ jCeff x jIC = u0 xÞ1 . Symmetry arguments show that the qubit is not coupled to its environment when the SCPT junctions are balanced, as already mentioned. When SCPT junction asymmetry d ¼ jEJ01 EJ02 j= ðEJ01 þ EJ02 Þ is taken into account, the relaxation time TR is: TR
4 u0 ; d 2 Re½Zeff ðXÞiC
ð7Þ
in which we have used Eq. (2). Excitation is negligible ðTE TR Þ, and no dephasing still occurs. Taking into account practical limitations on the junction asymmetry ðd > 0:1Þ, we estimate TR > 20 ls for the admittance shown in Fig. 1c, and for the parameters EJ0 ¼ 2EC ¼ 0:4kB K. Assuming that the gate charge can be tuned at better than 0:02e, we estimate that the phase coherence time is limited by the offset charge noise [9] at Tu > 1 ls. Note that a duration of 1 ls allows to perform about 100 qubit manipulations. When readout is on, the phase excursions are not bounded, and only approximate expressions have been obtained. The main result is that the phase coherence time is extremely short Tu ’ 50 ns. This simply means that the readout system has dephased the qubit before the ‘‘pointer’’ has moved. We find that TE is still longer than TR > 10 ls, which we have calculated by averaging over the phase dynamics, assuming that the readout junction plasma frequency is lower than the qubit transition frequency. Circuit design thus results from a trade-off between efficient readout, long relaxation and dephasing times when readout is off, and weak relaxation/excitation when readout is on.
5. Sample fabrication and experimental set-up Samples were fabricated by electron-beam lithography in a two-step process. First, an interdigitated 1.0 pF capacitor and small metallic
A. Cottet et al. / Physica C 367 (2002) 197–203
islands, made out of gold, were deposited. The role of the capacitor is to decrease the readout junction resonance frequency in order to improve the discriminating power, while that of the metallic islands is to provide a normal metal sink for spurious quasiparticles in the SCPT electrodes [7]. It is indeed essential to avoid a single quasiparticle entering the SCPT island, which would destroy the desired qubit states at ng ¼ 1=2. In a second step, the SCPT and the readout junction were fabricated by depositing aluminum layers at two angles through a suspended shadow-mask [14]. The first layer was oxidized in order to grow the tunnel junction barriers. Similar SCPT and readout junction are fabricated together with the qubit circuit for the sake of tunnel resistance control. The sample was wire-bonded onto a miniature circuit-board with surface mounted components. The board was fitted in a shielded copper box thermally anchored to the mixing chamber of a dilution refrigerator. All lines were carefully filtered [15] in order to reach thermal equilibrium at the base temperature 14 mK. The bandwidth of the current-biasing line was however kept large enough to pass submicrosecond readout square pulses. When switching occurs, the voltage on the measuring line with capacitance CM 0:5 nF rises at a rate IC =CM 1 mV=ls. The switching was detected by monitoring the voltage across the readout junction using a low-noise amplifier with a 1 MHz bandwidth.
201
Fig. 2. Switching probability PS of the readout junction when 0.5 ls long bias-current square pulses with variable amplitude are applied. The critical current is IC ¼ 1:17 lA. The steepest step, obtained at the lowest temperature, would correspond to a discriminating power of 0.6 in a single shot readout of the qubit.
6. Experimental results We report here switching experiments in a sample with the SCPT disconnected on purpose. We have measured the switching rate of the readout junction alone as a function of bias-current and temperature, using square pulses with adjustable height, or linear ramps [16] for currentbiasing the readout junction. The variations of the switching probability when the height of 0.5 ls readout pulses is varied are shown in Fig. 2 for different temperatures. The maximum slope, obtained at the lowest temperature, would correspond to a discriminating power of 0.6 for the readout. Switching rates obtained using both
Fig. 3. Switching rate of the readout junction as a function of bias-current IB ¼ sIC at different temperatures. Full symbols correspond to 0:5 ls long bias-current square pulses, and open symbols to linear bias-current ramps at dIB =dt ¼ 0:84 lA/ms.
measurement techniques are shown in Fig. 3 as a function of bias-current, for different temperatures. The escape temperature defined by Eq. (6), and determined following the procedure described in Ref. [17], is plotted in Fig. 4. The lowest measured escape temperature is about 90 mK, which is
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7. Conclusions We have discussed a new design for a combined charge-phase qubit, based on a SCPT connected in parallel with a large readout junction. Whereas the qubit is addressed using the gate charge coupled to the transistor island, the readout is performed using the supercurrent driven by the phase difference across the transistor, which controls the switching of the readout junction out of the zerovoltage state. From measurements of the switching probability of this junction when a square biascurrent pulse is applied, we have determined the discriminating power of the readout. We conclude that single-shot measurements should be possible in this system. Experiments on a full qubit circuit are in progress. Fig. 4. Escape temperature versus sample temperature. Full symbols correspond to bias-current square pulses, and open symbols to linear bias-current ramps. We attribute the difference found at low temperature between both methods to the larger heating of the damping circuit when linear ramps are used. The dashed line indicates the predicted escape temperature assuming that the on-chip capacitance fully adds to the junction capacitance, and that no spurious heating occurs.
higher than the design value of 60 mK predicted using Eq. (5), assuming that the on-chip capacitor fully contributes to the capacitance Ceff . We attribute this discrepancy to two effects. First, the SMC resistor in the bias-current line is slightly heated by the bias-current, as indicated by the increase observed when relatively slow bias-current ramps are used instead of fast pulses. Second, the on-chip added capacitance is not fully effective because of the residual inductance of the connecting lines and of the capacitance itself. Quantitatively, we have calculated that the quantum tunneling rate is the same as if about half of the onchip capacitance was effectively contributing [18]. These two effects explain well the value of the minimum escape temperature, and of the resulting discriminating power. These results, which already allow interesting measurements on the qubit to be performed, can be improved, and a discriminating power of 0.95 with 200 ns long readout pulses could be achieved if the predicted escape temperature of 60 mK is reached.
Acknowledgements We gratefully acknowledge discussions with C. Urbina and Y. Nakamura, and the help of H. Pothier.
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REPORTS to ⬃150 GPa, consistent with the modulus values of large SWNT bundles (22). Although an individual SWNT has an elastic modulus of ⬃1 TPa, the value can decrease to ⬃100 GPa for nanotube bundles, owing to the internanotube defects (for example, imperfect lattice of nanotube bundles owing to different nanotube diameters) present along the bundles. The long nanotube strands created by our direct synthesis technique are an alternative to the fibers and filaments spun from nanotube slurries (4). The mechanical and electrical properties of these strands are superior to the latter fibers: The strands can be produced in high yield and continuously, and the thickness of the strands and their length may be further optimized by tuning the processing conditions to produce practically useful nanotube-based macroscale cables. 1. 2. 3. 4. 5. 6. 7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18. 19. 20. 21. 22.
23.
886
References and Notes
S. Iijima, Nature 354, 56 (1991). Z. Pan et al., Nature 394, 631 (1998). H. M. Cheng et al., Chem. Phys. Lett. 289, 602 (1998). B. Vigolo et al., Science 290, 1331 (2000). L. J. Ci et al., Carbon 38, 1933 (2000). M. Endo et al., J. Phys. Chem. Solids 54, 1841 (1993). R. Andrews et al., Chem. Phys. Lett. 303, 467 (1999). H. M. Cheng et al., Appl. Phys. Lett. 72, 3282 (1998). Thiophene and ferrocene were dissolved in the carbon source (liquid n-hexane), sprayed into the hydrogen stream, and fed from the top of a vertical heated furnace. The hydrogen flow rate was then adjusted to provide the optimum conditions for nanotube strand formation. The gas flow carried the strands downstream, and the nanotube strands were collected at the bottom of the furnace. A. Thess et al., Science 273, 483 (1996). P. Launois et al., J. Nanosci. Nanotechnol. 1, 125 (2001). Our data correspond to the larger tube numbers in the bundle (N ⫽ 37; N ⫽ 91), as displayed in figure 4 of S. Rols, R. Almairac, L. Henrard, E. Anglaret, J. L. Sauvajol [Eur. Phys. J. B 10, 263 (1999)]. J. E. Fischer et al., Phys. Rev. B 55, R4921 (1997). P. H. Zhang, P. E. Lammert, V. H. Crespi, Phys. Rev. Lett. 81, 5346 (1998). The true strain εT is defined as εT ⫽ ln(Lf/L0), where Lf is the real length of the sample and L0 is the original length of the sample. The true strain could also be described as εT ⫽ 2 ln(D0/Df), assuming that the sample volume is constant, where D0 and Df are the original diameter and the real diameter of the sample during the measurement, respectively. Therefore, the true stress can be described as T ⫽ P/A ⫽ 4P exp(εT)/D02, where P is the load and A is the real surface area supporting the load. Because ln(1⫹ε) ⬇ ε when ε ⬍ 0.1, the true stress versus true strain curve has almost the same slope as that of the load versus engineering strain curve in the elastic strain regime. R. H. Baughman et al., Science 284, 1340 (1999). M. M. J. Treacy, T. W. Ebbesen, J. M. Gibson, Nature 381, 678 (1996). E. W. Wong, P. E. Sheehan, C. M. Lieber, Science 277, 1971 (1997). M. F. Yu, B. S. Files, S. Arepalli, R. S. Ruoff, Phys. Rev. Lett. 84, 5552 (2000). J. P. Lu, Phys. Rev. Lett. 79, 1297 (1997). F. Li, H. M. Cheng, S. Bai, G. Gu, M. S. Dresselhaus, Appl. Phys. Lett. 77, 3161 (2000). L. Forro ´, C. Scho¨nenberger, in Carbon Nanotubes: Synthesis, Structure, Properties and Applications, M. S. Dresselhaus, G. Dresselhaus, Ph. Avouris, Eds. (Springer, New York, 2001), pp. 329 –390. H.W.Z., C.L.X., and D.H.W. acknowledge financial support from the State Key Project for Fundamen-
tal Research of the Ministry of Science and Technology, China, under grant G20000264-04. P.M.A. acknowledges financial support from the NSF through a CAREER grant. B.Q.W. acknowledges support from the NSF Nanoscale Science and Engineering Center, at Rensselaer Polytechnic Insti-
tute, for the directed assembly of nanostructures. We also acknowledge useful discussions with N. Koratkar and technical assistance from D. Vansteele and Y. Choi. 10 October 2001; accepted 22 March 2002
Manipulating the Quantum State of an Electrical Circuit D. Vion,* A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina,† D. Esteve, M. H. Devoret‡ We have designed and operated a superconducting tunnel junction circuit that behaves as a two-level atom: the “quantronium.” An arbitrary evolution of its quantum state can be programmed with a series of microwave pulses, and a projective measurement of the state can be performed by a pulsed readout subcircuit. The measured quality factor of quantum coherence Q ⬵ 25,000 is sufficiently high that a solid-state quantum processor based on this type of circuit can be envisioned. Can we build machines that actively exploit the fundamental properties of quantum mechanics, such as the superposition principle or the existence of entangled states? Applications such as the transistor or the laser, often quoted as developments based on quantum mechanics, do not actually answer this question. Quantum mechanics enters into these devices only at the level of material properties, but their state variables such as voltages and currents remain classical. Proposals for true quantum machines emerged in the last decades of the 20th century and are now being actively explored: quantum computers (1), quantum cryptography communication systems (2), and detectors operating below the standard quantum limit (3). The major difficulty facing the engineer of a quantum machine is decoherence (4). If a degree of freedom needs to be manipulated externally, as in the writing of information, its quantum coherence usually becomes very fragile. Although schemes that actively fight decoherence have recently been proposed (5, 6), they need very coherent quantum systems to start with. The quality of coherence for a two-level system can be quantitatively described by the quality factor of quantum coherence Q ⫽ 01T, where 01 is its transition frequency and T is the coherence time of a superposition of the states. It is generally accepted that for active decoherence compensation mechanisms, Q’s larger than 104 01 top are necQuantronics Group, Service de Physique de l’Etat Condense ´, Direction des Sciences de la Matie `re, Commissariat ` a l’Energie Atomique–Saclay, 91191 Gifsur-Yvette, France. *To whom correspondence should be addressed. Email:
[email protected] †Member of CNRS. ‡Present address: Applied Physics Department, Yale University, New Haven, CT 06520, USA.
essary, top being the duration of an elementary operation (7). Among all the practical realizations of quantum machines, those involving integrated electrical circuits are particularly attractive. However, unlike the electric dipoles of isolated atoms or ions, the state variables of a circuit, like voltages and currents, usually undergo rapid quantum decoherence because they are strongly coupled to an environment with a large number of uncontrolled degrees of freedom (8). Nevertheless, superconducting tunnel junction circuits (9–13) have displayed Q’s up to several hundred (14), and temporal coherent evolution of the quantum state has been observed on the nanosecond time scale (10, 15) in the case of the single Cooper pair box (16). We report here a new circuit built around the Cooper pair box with Q in excess of 104, whose main feature is the separation of the write and readout ports (17, 18). This circuit, which behaves as a tunable artificial atom, has been nicknamed a “quantronium.” The basic Cooper pair box consists of a low-capacitance superconducting electrode, the “island,” connected to a superconducting reservoir by a Josephson tunnel junction with capacitance Cj and Josephson energy EJ. The junction is biased by a voltage source U in series with a gate capacitance Cg. In addition to EJ, the box has a second energy scale, the Cooper pair Coulomb energy ECP ⫽ (2e)2/ 2(Cg ⫹ Cj) . When the temperature T and the superconducting gap ⌬ satisfy kBT ⬍⬍ ⌬/lnN and ECP ⬍⬍ ⌬, where N is the total number of paired electrons in the island, the number of excess electrons is even (19, 20). The Hamiltonian of the box is then ˆ ⫽ E CP共Nˆ ⫺ N g兲 2 ⫺ E J cosˆ H
(1)
where Ng ⫽ CgU/2e is the dimensionless gate
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REPORTS charge and ˆ the phase of the superconducting order parameter in the island, conjugate to the number Nˆ of excess Cooper pairs in it (16). In our experiment, EJ ⬵ ECP and neither Nˆ nor ˆ is a good quantum number. The box thus has discrete quantum states that are quantum superpositions of several charge states with different N. Because the system is sufficiently nonharmonic, the ground ⱍ0典 and first excited ⱍ1典 energy eigenstates form a two-level system. This system corresponds to an effective spin one-half sជ, whose Zeeman energy h01 goes to a minimal value close to EJ when Ng ⫽ 1/2. At this particular bias point, both states ⱍ0典 (sz ⫽ ⫹1/2) and ⱍ1 典 (sz ⫽ ⫺1/2) have the same average charge 具Nˆ典 ⫽ 1/2, and consequently the system is immune to first-order fluctuations of the gate charge. Manipulation of the quantum state is performed by applying microwave pulses u(t) with frequency ⬵ 01 to the gate, and any superposition ⱍ⌿典 ⫽ ␣ ⱍ0典 ⫹  ⱍ1典 can be prepared. A novel type of readout has been implemented in this work. The single junction of the basic Cooper pair box has been split into two nominally identical junctions in order to form a superconducting loop (Fig. 1). The Josephson energy EJ in Eq. 1 becomes EJ cos(␦ˆ /2) (21), where ␦ˆ is an additional degree of freedom: the superconducting phase difference across the series combination of the two junctions (22). The two states are discriminated not through the charge 具Nˆ典 on the island (10, 23), but through the supercurrent in the loop 具Iˆ典 ⫽ (2e/ប) 具H/␦典. This is achieved by entangling sជ with the phase ␥ˆ of a large Josephson junction with Josephson energy EJ0 ⬇ 20 EJ inserted in the loop (17, 24). The phases are related by ␦ˆ ⫽ ␥ˆ ⫹ , where ⫽ 2e⌽/ប, ⌽ being the external flux imposed through the loop. The junction is shunted by a capacitor C to reduce phase fluctuations. A trapezoidal readout pulse Ib(t), with a peak value slightly below the critical current I0 ⫽ 2eEJ0/ប, is applied to the parallel combination of the large junction and the small junctions (Fig. 1C). When starting from 具␦ˆ 典 ⬇ 0, the phases 具␥ˆ 典 and 具␦ˆ 典 grow during the current pulse, and consequently an sជ-dependent supercurrent develops in the loop. This current adds to the bias current in the large junction, and by precisely adjusting the amplitude and duration of the Ib(t) pulse, the large junction switches during the pulse to a finite voltage state with a large probability p1 for state ⱍ1典 and with a small probability p0 for state ⱍ0典 (17). This readout scheme is similar to the spin readout of Ag atoms in a Stern and Gerlach apparatus, in which the spin is entangled with the atom position. For the parameters of the experiment, the efficiency of this projective measurement should be ⫽ p1 ⫺ p0 ⫽ 0.95 for optimum readout condi-
tions. The readout is also designed so as to minimize the ⱍ1典 3 ⱍ0典 relaxation rate using a Wheatstone bridge–like symmetry. Large ratios EJ0/EJ and C/Cj provide further protection from the environment. Just as the system is immune to charge noise at Ng ⫽ 1/2, it is immune to flux and bias current noise at ⫽ 0 and Ib ⫽ 0, where Iˆ ⫽ 0. The preparation of the quantum state and its manipulation are therefore performed at this optimal working point. A quantronium sample is shown in Fig. 1B. It was fabricated with standard e-beam lithography and aluminum evaporation. The sample was cooled down to 15 mK in a dilution refrigerator. The switching of the large junction (25) to the finite voltage state is detected by measuring the voltage across it with a room-temperature preamplifier, followed by a discriminator. By repeating the experiment, the switching probability, and hence the occupation probabilities of the ⱍ0典 and ⱍ1典 states, can be determined. The readout part of the circuit was tested
by measuring the switching probability p at thermal equilibrium as a function of the pulse height Ip, for a readout pulse duration of R ⫽ 100 ns. The discrimination between the estimated currents for the ⱍ0典 and ⱍ1典 states was found to have an efficiency of ⫽ 0.6, which is lower than the expected ⫽ 0.95. Measurements of the switching probability as a function of temperature and repetition rate indicate that the discrepancy between the theoretical and experimental readout efficiency could be due to an incomplete thermalization of our last filtering stage in the bias current line. Spectroscopic measurements of 01were performed by applying to the gate a weak continuous microwave irradiation suppressed just before the readout pulse. The variations of the switching probability as a function of the irradiation frequency display a resonance whose center frequency evolves with dc gate voltage and flux as the Hamiltonian predicts, reaching 01 ⬵ 16.5 GHz at the optimal working point (Fig. 2). The small discrepancy
Fig. 1. (A) Idealized circuit diagram of the quantronium, a quantum-coherent circuit with its tuning, preparation, and readout blocks. The circuit consists of a Cooper pair box island (black node) delimited by two small Josephson junctions (crossed boxes) in a superconducting loop. The loop also includes a third, much larger Josephson junction shunted by a capacitance C. The Josephson energies of the box and the large junction are EJ and EJ0. The Cooper pair number N and the phases ␦ and ␥ are the degrees of freedom of the circuit. A dc voltage U applied to the gate capacitance Cg and a dc current I applied to a coil producing a flux ⌽ in the circuit loop tune the quantum energy levels. Microwave pulses u(t ) applied to the gate prepare arbitrary quantum states of the circuit. The states are read out by applying a current pulse Ib(t ) to the large junction and by monitoring the voltage V(t ) across it. (B) Scanning electron micrograph of a sample. (C) Signals involved in quantum state manipulation and measurement. Top: Microwave voltage pulses u(t ) are applied to the gate for state manipulation. Middle: A readout current pulse Ib(t ) with amplitude Ip is applied to the large junction td after the last microwave pulse. Bottom: Voltage V(t) across the junction. The occurrence of a pulse depends on the occupation probabilities of the energy eigenstates. A discriminator with threshold Vth converts V(t) into a boolean output for statistical analysis.
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REPORTS Fig. 2. (A) Calculated transition frequency 01 as a function of and Ng for EJ ⫽ 0.865 kBK and EJ/ECP ⫽ 1.27. The saddle point at the intersection of the blue and red lines is an ideal working point where the transition frequency is independent, to first order, of the bias parameters. (B) Measured center transition frequency (symbols) as a function of reduced gate charge Ng for reduced flux ⫽ 0 [right panel, blue line in (A)] and as a function of for Ng ⫽ 0.5 [left panel, red line in (A)], at 15 mK. Spectroscopy is performed by measuring the switching probability p (105 events) when a continuous microwave irradiation of variable frequency is applied to the gate before readout (td ⬍ 100 ns). Continuous line: Theoretical best fit leading to EJ and EJ/ECP values indicated above. Inset: Lineshape measured at the optimal working point ⫽ 0 and Ng ⫽ 0.5 (dots). Lorentzian fit with a FWHM ⌬01 ⫽ 0.8 MHz and a center frequency 01 ⫽ 16463.5 MHz (solid line). Fig. 3. (A) Left: Rabi oscillations of the switching probability p (5 ⫻ 104 events) measured just after a resonant microwave pulse of duration . Data were taken at 15 mK for a nominal pulse amplitude Uw ⫽ 22 V ( joined dots). The Rabi frequency is extracted from an exponentially damped sinusoidal fit (continuous line). Right: Measured Rabi frequency (dots) varies linearly with Uw, as expected. (B) Ramsey fringes of the switching probability p (5 ⫻ 104 events) after two phasecoherent microwave pulses separated by ⌬t . Joined dots: Data at 15 mK; the total acquisition time was 5 mn. Continuous line: Fit by exponentially damped sinusoid with time constant T ⫽ 0.50 s. The oscillation corresponds to the beating of the free evolution of the spin with the external microwave field. Its period indeed coincides with the inverse of the detuning frequency (here ⫺ 01 ⫽ 20.6 MHz).
888
between theoretical and experimental values of the transition frequency at nonzero magnetic flux is attributed to flux penetration in the small junctions not taken into account in the model. These spectroscopic data have been used to precisely determine the relevant circuit parameters, EJ ⫽ 0.865 kBK and EJ/ ECP ⫽ 1.27. At the optimal working point, the linewidth was found to be minimal, with a 0.8-MHz full width at half-maximum (FWHM). When varying the delay between the end of the irradiation and the readout pulse, the resonance peak height decays with a time constant T1 ⫽ 1.8 s. Supposing that the energy relaxation of the system is only due to the bias circuitry, a calculation similar to that in (26) predicts that T1 ⬃ 10 s for a crude discrete element model. This result shows that no detrimental sources of dissipation have been seriously overlooked in our circuit design. Controlled rotations of sជ around an axis x perpendicular to the quantization axis z have been performed. Before readout, a single pulse at the transition frequency with variable amplitude Uw and duration was applied. The resulting change in switching probability is an oscillatory function of the product Uw (Fig. 3A), which is in agreement with the theory of Rabi oscillations (27), proving that the resonance indeed arises from a two-level system. The proportionality ratio between the Rabi period and Uw was used to calibrate microwave pulses for the application of controlled rotations of sជ. Rabi oscillations correspond to a driven coherent evolution but do not give direct access to the intrinsic coherence time T during a free evolution of sជ. This T was obtained by performing a Ramsey fringe experiment (28), on which atomic clocks are based. One applies to the gate two phase-coherent microwave pulses, each corresponding to a /2 rotation around x (29) and separated by a delay ⌬t, during which the spin precesses freely around z. For a given detuning of the microwave frequency, the observed decaying oscillations of the switching probability as a function of ⌬t (Fig. 3B) correspond to the “beating” of the spin precession with the external microwave field (30). The oscillation period agrees exactly with the inverse of the detuning, allowing a measurement of the transition frequency with a relative accuracy of 6 ⫻ 10⫺6. The envelope of the oscillations yields the decoherence time T ⬵ 0.50 s. Given the transition period 1/01 ⬵ 60 ps, this means that sជ can perform on average 8000 coherent free precession turns. In all the time domain experiments on the quantronium, the oscillation period of the switching probability agrees closely with theory, which proves controlled manipulation of sជ. However, the amplitude of the oscillations is smaller than expected by a factor of 3 to 4.
3 MAY 2002 VOL 296 SCIENCE www.sciencemag.org
REPORTS This loss of contrast is likely to be due to a relaxation of the level population during the measurement itself. In order to understand what limits the coherence time of the circuit, measurements of the linewidth ⌬01 of the resonant peak as a function of U and ⌽ have been performed. The linewidth increases linearly when departing from the optimal point (Ng ⫽ 1/2, ⫽ 0, Ib ⫽ 0). This dependence is well accounted for by charge and phase noises with root mean square deviations ⌬Ng ⫽ 0.004 and ⌬(␦/2) ⫽ 0.002 during the time needed to record the resonance. The residual linewidth at the optimal working point is well explained by the second-order contribution of these noises. The amplitude of the charge noise is in agreement with measurements of 1/f charge noise (31), and its effect could be minimized by increasing the EJ/ECP ratio. The amplitude of the flux noise is unusually large (32) and should be significantly reduced by improved magnetic shielding. An improvement of Q by an order of magnitude thus seems possible. Experiments on quantum gates based on the controlled entanglement of several capacitively coupled quantronium circuits could already be performed with the level of quantum coherence achieved in the present experiment.
21. D. V. Averin, K. K. Likharev, in Mesoscopic Phenomena in Solids, B. L. Altshuler, P. A. Lee, R. A. Webb, Eds. (Elsevier, Amsterdam, 1991). 22. J. R. Friedman, D. V. Averin, Phys. Rev. Lett. 88, 50403 (2002). 23. A. Aassime, G. Johansson, G. Wendin, R. J. Schoelkopf, P. Delsing, Phys. Rev. Lett. 86, 3376 (2001). 24. A different Cooper pair box readout scheme using a large Josephson junction is discussed by F. W. J. Hekking, O. Buisson, F. Balestro, and M. G. Vergniory, in Electronic Correlations: From Meso- to Nanophysics, T. Martin, G. Montambaux, J. Traˆn Thanh Vaˆn, Eds. (Editions De Physique, Les Ulis, France, 2001), pp. 515–520. 25. For C ⫽ 1 pF and I0 ⫽ 0.77 A, the bare plasma frequency of the large junction is p/2 ⬵ 8 GHz, well below 01. 26. A. Cottet et al., in Macroscopic Quantum Coherence and Quantum Computing, D. V. Averin, B. Ruggiero, P. Silvestrini, Eds. (Kluwer Academic, Plenum, New York, 2001), pp. 111–125. 27. I. I. Rabi, Phys. Rev. 51, 652 (1937).
26 December 2001; accepted 20 March 2002
Coherent Temporal Oscillations of Macroscopic Quantum States in a Josephson Junction Yang Yu,1 Siyuan Han,1* Xi Chu,2† Shih-I Chu,2 Zhen Wang3 We report the generation and observation of coherent temporal oscillations between the macroscopic quantum states of a Josephson tunnel junction by applying microwaves with frequencies close to the level separation. Coherent temporal oscillations of excited state populations were observed by monitoring the junction’s tunneling probability as a function of time. From the data, the lower limit of phase decoherence time was estimated to be about 5 microseconds.
References and Notes
1. M. A. Nielsen, I. L. Chuang, Quantum Computation and Quantum Information (Cambridge Univ. Press, Cambridge, 2000). 2. The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation, D. Bouwmeester, A. Ekert, A. Zeilinger, Eds. (Springer-Verlag, Berlin, 2000). 3. V. B. Braginsky, F. Ya. Khalili, Quantum Measurement (Cambridge Univ. Press, 1992). 4. W. H. Zurek, J. P. Paz, in Coherent Atomic Matter Waves, R. Kaiser, C. Westbrook, F. David, Eds. (Springer-Verlag, Heidelberg, Germany, 2000). 5. P. W. Shor, Phys. Rev. A 52, R2493 (1995). 6. A. M. Steane, Phys. Rev. Lett. 77, 793 (1996); Rep. Prog. Phys. 61, 117 (1998). 7. J. Preskill, J. Proc. R. Soc. London Ser. A 454, 385 (1998) 8. Y. Makhlin, G. Scho¨n, A. Shnirman, Rev. Mod. Phys. 73, 357 (2001). 9. M. H. Devoret et al., in Quantum Tunneling in Condensed Media, Y. Kagan, A. J. Leggett, Eds. (Elsevier Science, Amsterdam, 1992). 10. Y. Nakamura, Yu. A. Pashkin, J. S. Tsai, Nature 398, 786, (1999). 11. C. H. van der Wal et al., Science 290, 773 (2000). 12. S. Han, R. Rouse, J. E. Lukens, Phys. Rev. Lett. 84, 1300 (2000). 13. S. Han, Y. Yu, X. Chu, S. -I. Chu, Z. Wang, Science 293, 1457 (2001). 14. J. M. Martinis, S. Nan, J. Aumentado, and C. Urbina (unpublished data) have recently obtained Q’s reaching 1000 for a current-biased Josephson junction. 15. Y. Nakamura, Yu. A. Pashkin, T. Yamamoto, J. S. Tsai, Phys. Rev. Lett. 88, 047901 (2002). 16. V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M. H. Devoret, Phys. Scr. T76, 165 (1998). 17. A. Cottet et al., Physica C 367, 197 (2002). 18. Another two-port design has been proposed by A. B. Zorin [Physica C 368, 284 (2002)]. 19. M. T. Tuominen, J. M. Hergenrother, T. S. Tighe, M. Tinkham, Phys. Rev. Lett. 69, 1997 (1992). 20. P. Lafarge, P. Joyez, D. Esteve, C. Urbina, M. H. Devoret, Nature 365, 422 (1993).
28. N. F. Ramsey, Phys. Rev. 78, 695 (1950). 29. In practice, the rotation axis does not need to be x, but the rotation angle of the two pulses is always adjusted so as to bring a spin initially along z into the plane perpendicular to z. 30. At fixed ⌬t , the switching probability displays a decaying oscillation as a function of detuning. 31. H. Wolf et al,. IEEE Trans. Instrum. Meas. 46, 303 (1997). 32. F. C. Wellstood, C. Urbina, J. Clarke, Appl. Phys. Lett. 50, 772 (1987). 33. The indispensable technical work of P. Orfila is gratefully acknowledged. This work has greatly benefited from direct inputs from J. M. Martinis and Y. Nakamura. The authors acknowledge discussions with P. Delsing, G. Falci, D. Haviland, H. Mooij, R. Schoelkopf, G. Scho¨n, and G. Wendin. Partly supported by the European Union through contract IST-10673 SQUBIT and the Conseil Ge ´ne ´ral de l’Essone through the EQUM project.
The question of whether macroscopic variables obey quantum mechanics has stimulated extensive theoretical interests (1, 2). The experimental search for macroscopic quantum phenomena (MQP) did not start until the early 1980s, when theory showed that the experimental conditions for observing MQP in Josephson junction– based devices were achievable (3–5). Many MQP, such as macroscopic quantum tunneling (MQT) (6–10), energy level quantization (11, 12), quantum incoherent relaxation (13), resonant tunneling and photon-assisted tunneling (14), and photo-induced transition and population inversion between macroscopic quantum states (15, 16), have since been observed. Recent spectroscopy evidence of superposition of Department of Physics and Astronomy, 2Department of Chemistry, University of Kansas, Lawrence, KS 66045, USA. 3Kansai Advanced Research Center, Communication Research Laboratory, Ministry of Posts and Telecommunications, 588-2 Iwaoka, Iwaoka-cho, Nishi-ku, Kobe, 651-24 Japan. 1
*To whom correspondence and requests should be addressed. E-mail:
[email protected] †Present address: Institute for Theoretical Atomic and Molecular Physics, Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA.
fluxoid states and persistent-current states in superconducting quantum interference devices has also been reported (17, 18). However, time domain coherent oscillations between macroscopic quantum states (MQS), which is more direct evidence for the superposition of MQS, has thus far evaded experimental detection. One of the methods proposed to create coherent temporal oscillations between two MQS is via Rabi oscillation, an effect that is well established and understood in atomic and molecular systems (19). The principle of Rabi oscillations is that by applying a monochromatic electromagnetic (EM) field to a quantum two-level system, which interacts with the EM fields, the system will be in a superposition of the two energy eigenstates that results in oscillations between the lower and upper levels with Rabi frequency ⍀. The amplitude of the population oscillations is at a maximum when the frequency of the EM wave is in resonance with the level spacing ⌬E, i.e., ⫽ ⌬E/ប. Rabi oscillation is a coherent quantum phenomenon that provides the foundation to a wide variety of basic research and applications, ranging from coherent excitation of atoms and molecules by laser to quantum computation (20–22). Re-
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