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Quantum coherence in a single-Cooper-pair box: experiments in the frequency and time domains
Physica B 280 (2000) 405}409
Quantum coherence in a single-Cooper-pair box: experiments in the frequency and time domains Y. Nakamura *, Yu. A. Pashkin, J.S. Tsai NEC Fundamental Research Laboratories, 34 Miyukigaoka, Tsukuba 305-8501, Japan CREST, Japan Science and Technology Corporation (JST), Kawaguchi, Saitama, 332-0012, Japan
Abstract In a small superconducting electrode connected to a reservoir via a Josephson junction, we can realize an arti"cial two-level system in which two charge-number states of the electrode are coupled by Cooper-pair tunneling. We have investigated quantum coherence in the two-level system through two complementary experiments: (i) photon-assisted Cooper-pair-tunneling spectroscopy of the energy-level splitting in the frequency domain, and (ii) pulsed-gate control of the quantum state to observe the coherent oscillations in the time domain. Our results demonstrate the existence and the persistence of the coherence for at least &30 cycles of the oscillation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Coherent oscillations; Energy-level splitting; Josephson junction; Two-level system
1. Introduction In the microscopic world, we "nd many quantum two-level systems such as nuclear and electron spins, atomic states, and diatomic molecular orbitals. Coherence in such two-level systems has been studied in detail both in the frequency domain and in the time domain. Radio-frequency, microwave, and optical spectroscopy have been applied to observe the energy-level splitting between two eigenstates. Also, Rabi oscillations in the time domain have been observed in many systems. We have conducted analogous experiments on an arti"cial two-level system in a small-Josephson-junction circuit called a single-Cooper-pair box [1] (enclosed part of Fig. 1(a)). This circuit consists of a small superconducting electrode (a &box') coupled to a superconducting reservoir via a Josephson junction. The relevant states of the two-level system are two charge states that di!er by 2e, for example, "02 and "22, where the charge state "n2 is a superconducting ground state with a "xed excess-electron number n in the box. This system is unique in the
* Corresponding author. Tel.: #81-298-50-1148; fax: #81298-56-6139 . E-mail address: [email protected] (Y. Nakamura)
sense that it is a single two-level system consisting of two distinct macroscopic quantum states involving a large number of conduction electrons in the box, not an ensemble of a macroscopic number of microscopic twolevel systems. (We would not say macroscopically distinct in the sense de"ned in Ref. [2], since the essential di!erence between the two states in our case is just a single Cooper pair transferred across the junction.) Also, in contrast to the microscopic systems, the two-level system can be #exibly designed by modifying the macroscopic energy parameters, the Josephson energy E and the ( charging energy E , to which all the electrons in the box ! collectively contribute. Here, we report our observation of clear evidence of quantum coherence in the two-level system: energy-level splitting due to coherent superposition of the two charge states and coherent oscillations between the two charge states.
2. Experimental setup and measurement schemes The single-Cooper-pair box and the additional probe junction were fabricated by electron-beam lithography and shadow evaporation of Al "lms. The box electrode had dimensions of &700;50;15 nm and contained &10 conduction electrons. The Josephson junction
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 7 9 0 - 1
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Fig. 1. (a) Single-Cooper-pair-box circuit (enclosed by a dashed line). A superconducting voltage-biased probe electrode is attached to the box via a tunnel junction. (b) Schematic energy diagram of the device for Q /e"1.
between the reservoir and the box had normal resistance of &10 k), while the probe junction, which had a thicker tunnel barrier, had resistance R of &30 M). This large di!erence in the resistances is a key point in our experiments, as we will discuss below. Although it is not shown in Fig. 1(a), the reservoir junction was divided into two parallel junctions to form a SQUID loop through which we can control the e!ective Josephson coupling energy E by an externally applied magnetic ( #ux penetrating the loop. There were also two gate electrodes capacitively coupled to the box. One was for a DC gate voltage < (the DC gate) and the other was for microwave and high-speed pulse signals (the pulse gate; not shown in Fig. 1(a)). The superconducting gap of the electrodes D was 230$10 leV, and the charging energy of the box, E ,e/2CR , was 117$3 leV, where CR is ! the total capacitance of the box. The sample chip was mounted in a shielded copper package and placed at the base temperature of the dilution refrigerator (&30 mK; k ¹&3 leV). Special care was taken concerning the design of the microwave- (or pulse-) transmission line that went from a room-temperature environment to the sample. We used a low-loss silver-plated Be}Cu coaxial cable (above 4.2 K) and a Nb coaxial cable (below 4.2 K). At the end of the Nb cable, the central conductor of a V connector (Wiltron) was connected to the on-chip coplanar line which was terminated with the pulse gate of the device that acted as an open end. We measured the DC current through the probe junction under a "nite bias voltage < to detect the probabil ity density of the charge states of the box. In an energy diagram (Fig. 1(b)) we have drawn the addition energies
of the nth electron in the box for Cooper-pair tunneling dE (n)/2,(E(n)!E(n!2))/2 (normalized to the energy per electron; solid levels) and for single-electron tunneling dE (n),E(n)!E(n!1) (dashed levels) where E(n)"E (n!Q /e) is the electrostatic energy of the ! state "n2 (Q ,C < #const.). Because of the charging e!ect, those energies have discrete levels. The duplication of the levels for each n is due to the di!erence of the charge between a Cooper pair and a single electron. The ladder of the levels can be vertically shifted in proportion to the gate voltage, or the gate-induced charge Q . For the reservoir and the probe, we depict the Fermi level and the superconducting pair-breaking energy 2D for the quasiparticle tunneling. A Cooper pair resonantly tunnels through the reservoir junction when E " $ dE (n"2)/2"0 is ful"lled; in other words, when "02 and "22 are degenerate. This works as an e!ective two-level system. Other charge states with the same parity as n have higher energy, and the states with a di!erent parity are e!ectively decoupled because of the superconducting gap. No quasiparticle tunneling takes place at the reservoir junction under the relevant conditions. On the other hand, Cooper-pair tunneling through the probe junction can be ignored because of the negligibly small Josephson energy. Instead, because of the large bias voltage (< '(2D#E )/e), two sequential quasiparticle ! tunnelings can occur with the rates of C and C (C JR\), bringing the state from "22 to "02. We can use these processes to detect the charge states: "22 emits two electrons to the probe, while "02 does nothing. Although we cannot measure the single two-electron tunneling event, the cyclic process consisting of the resonant Cooper-pair tunneling and the two sequential quasiparticle tunnelings is repeated automatically to produce an easily measurable DC current [3,4]. A typical Q -dependence of the DC current I is shown as dashed lines in Figs. 2(a) and (b) that have a broad peak at Q /e"1; that is, a resonant tunneling peak. The abrupt drop of the current at Q /e&1.13 is because of the suppression of C due to the superconducting gap. In the spectroscopy of the energy-level splitting, we used photon-assisted Cooper-pair tunneling under microwave irradiation with a frequency f through the gate [5]. Fig. 2(c) illustrates the relative electrostatic energies of the two charge states as a function of Q (dashed lines). Coherent Josephson coupling of the two charge states results in two eigenenergy levels with an energy-level splitting of E at Q /e"1 (solid lines). When ( the photon energy hf matches the energy gap between the two eigenstates, a photon-assisted transition takes place and increases the population of "22, resulting in a photoinduced current peak as in Fig. 2(a). On the left side of the resonance, photon absorption dominates, while on the right side, photon emission does (not shown), since "02 is more populated than "22 due to the quasiparticle tunneling at the probe junction. The quasiparticle tunneling
Y. Nakamura et al. / Physica B 280 (2000) 405}409
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pulse height at the device. Thus, we swept the baseline of the pulse, that is, Q , and measured the I}Q curve as in Fig. 2(b) for each *t. The oscillation in the time domain was observed in the cross-section of the series of the I}Q curves [4].
3. Frequency-domain experiment: energy-level splitting
Fig. 2. Current through the probe junction as a function of Q with 33.8 GHz microwave irradiation (a) and with a pulse array (*t"160 ps, ¹ "16 ns) (b). The dashed curves are the data taken without microwave or the pulse array (< "650 lV and / "0.31). (c), (d) Schematic energy diagrams illustrating the ways of changing the quantum states of the box by microwave irradiation (c) and by a gate-voltage pulse (d). In (c) the solid lines represent eigenenergies of the box and the dashed lines show the electrostatic energy of each charge state (vice versa in (d)).
also leads to a "nite lifetime and, as a result, a level broadening C to the state "22 (the solid-line states in Fig. 2(c) are not eigenstates any longer), that gives an intrinsic energy resolution of the spectroscopy. Thus, we need to have a large resistance di!erence between the reservoir and the probe junctions to ful"l the condition
C ;E in order to see the splitting as evidence of the ( coherent superposition. To do a time-resolved measurement, we applied short DC-voltage pulses to the gate [6]. Fig. 2(d) shows the scheme. When Q is kept far to the left of the resonance, the system relaxes to the ground state ("lled circle). Then, a pulse with an appropriate height brings the system nonadiabatically to the resonance where coherent oscillation between the two charge states starts. The rise and fall times of the pulse must be short compared to the coherent oscillation period. Otherwise, the state of the two-level system just follows the ground state (dashed line), and no oscillation occurs. After a pulse length *t, the gate voltage comes back to the initial value with a superposition of the two charge states corresponding to the phase of the oscillation during *t. Then, "nally, the "22 component of the wave function decays to the ground state with two quasiparticle tunnelings at the probe junction. Hence, if we repeat the pulse with a repetition time ¹ longer than the relaxation time (C\ #C\ ), we have a measurable DC probe current (of an order of 2e/¹ ) that re#ects the weight of the state "22 in the superposition. In practice, we could not know the exact
Fig. 3 shows I}Q curves under microwave irradiation with frequencies from 2.2 to 40 GHz. Compared with the curve in Fig. 2(a), the data were taken at larger < . Thus, the cuto! of the probe current was shifted toward larger Q (&1.45e). On each side of the resonance peak at Q /e"1, we observed peaks corresponding to the photon-assisted Cooper-pair tunneling with either photon absorption or emission process. The reason why the emission peak is much weaker than the absorption peak is not clear yet [5]. We can see that the peak height of the photon-assisted peaks changes smoothly as a function of frequency; this means the coupling strength of the microwave signal to the sample was rather #at for the frequency range we used. Since the microwave power A ("!33 dB) was relatively high in this measurement, we observed higher-order peaks involving multiplephoton transitions. From another experiment on the microwave-power dependence (not shown; similar to the one in Ref. [7]), at 10 GHz, we estimated dE /hf as ! &0.77 for Fig. 3, where dE is the oscillation amplitude ! of the relative energy between the two relevant charge states as determined by the microwave "eld. To obtain the energy dispersion, we plot the positions of peaks in the I}Q curves as a function of frequency
Fig. 3. I}Q curves for various microwave frequencies from 2.2 (bottom) to 40 GHz (top) at < "700 lV and / & . Each curve is shifted by 0.5 pA for clarity.
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Y. Nakamura et al. / Physica B 280 (2000) 405}409
Fig. 4. Spectroscopy of the energy-level splitting. The black dots represent positions of the peaks in the I}Q curves as a function of microwave frequency for / "0 (a), (b), (c) and (d). The solid curves are "ts to the energy dispersion curve for E "83, ( 73, 42 and 0 leV, respectively for (a)}(d). The dashed lines are the dispersion for E "0. (
under a lower microwave power (A"!50 dB). Fig. 4 shows such plots for di!erent external magnetic "elds. The photon-absorption peak positions "t well with the expected dispersion curves with the gap frequency corresponding to E /h. The -dependence of the gap will be ( discussed in the next section. This observation of the energy-level splitting is evidence of coherent superposition of the two charge states. Such spectroscopic determination of the Josephson energy is unique in this two-level system and is not possible in principle in the classical (large) Josephson junction. However, we could not directly determine the decoherence time, nor actively control the superposition of the wave functions in this experiment. Although, in principle, we can obtain information on the decoherence from the peak-width analysis, in practice this has been di$cult with our current setup since we cannot clearly distinguish intrinsic broadening from that due to experimental noise at low energies less than a few leV.
pulse-induced probe current *I is shown as a function of the pulse length *t in Fig. 5. Since we do not know the exact pulse height at the sample, we select an appropriate Q (&0.51e) as the baseline of the pulse to obtain the largest period of the oscillation. The observed period ¹ should correspond to h/E . Otherwise, the two ( charge states are not exactly degenerate during the pulse, and the observed oscillation has a smaller period [6]. We controlled E through the magnetic "eld, and plot the (
-dependence of h/¹ in the inset of Fig. 5. The values of E , determined independently from the spectroscopy in ( the previous section, are also plotted in the same graph. Both agreed well with the expected cosine -dependence of E , con"rming that the oscillation is due to the Joseph( son coupling between the two charge states. The decoherence time of the two-level system is the next thing we would like to know. Fig. 6 shows the oscillation up to *t"2 ns, and we see nearly 30 cycles of the oscillation. In the measurement, we expected to see a decaying envelope of the oscillation and to obtain a decoherence time. However, the noisy signal, which was mainly due to the background charge #uctuation near the device, prevented us from doing so. Since we measured the DC current with a time constant of 20 ms, low-frequency (1/f ) #uctuation of the background charge could degrade the signal by creating a #uctuating o!set in Q . Whether such a #uctuation a!ects the intrinsic decoherence time is not clear yet, since the #uctuation spectrum in the gigahertz range is not known. On the other hand, we can roughly estimate the intrinsic decoherence time of a single-Cooper-pair box embedded in a low-ohmic electromagnetic environment [1]. In the present sample, it is of an order of 100 ns assuming
4. Time-domain experiment: coherent oscillations Following the scheme in Fig. 2(d), we observed coherent oscillations between the two charge states in the time domain. The observed oscillating signal in the
Fig. 5. Pulse-induced current versus pulse length. Inset: Magnetic-"eld dependence of E determined by the two independent ( experiments in the time domain and the frequency domain. The solid curve is a cosine "tting with E ( "0)"84 leV. (
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operation is necessary to enable multiple-step operation of multiple qubits. In the present experiment, the shape of the pulse supplied by room-temperature electronics was far from rectanglar (the rise and fall times were &30}40 ps at the top of the cryostat), and there was jittering of an order of 10 ps. For 100-ps-scale qubit control, we need much faster and more precise electronics, such as a single-#ux-quanta circuit [11] located on the same chip. Though we have relied on averaged measurements over repeated pulse operations in the present work, singleshot quantum measurement of the charge states [12] is also indispensable. Fig. 6. Pulse-induced current versus pulse length up to *t"2 ns.
100 ) environmental impedance on the probe electrode. That is much larger than our experimentally con"rmed value of 2 ns. However, in our measurement scheme, quasiparticle tunneling at the probe junction also disturbs the coherence, and even worse, the probe is always connected to the two-level system even during the pulse operation. Thus, the decoherence time in the current setup is probably limited by 1/C &6 ns at most. The large, almost linear, background current in Fig. 6 is another side e!ect of the quasiparticle tunneling. As *t increases, the probability of the unwanted quasiparticle relaxation during the pulse increases.
5. Concluding remarks The method of observing the oscillation can also be considered a method of coherent control of the quantum states in the two-level system: With an appropriate pulse length and pulse height, in principle we can create arbitrary superposition of the charge states. Use of this twolevel system as a quantum bit (qubit), an elemental unit of quantum computing, has been proposed [8}10]. In that context, the decoherence would be the most serious problem and needs to be further studied in detail. From a technical point of view, more accuracy in the pulse
Acknowledgements This work was supported by the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST).
References [1] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M.H. Devoret, Phys. Scripta T 76 (1998) 165. [2] A.J. Leggett, Prog. Theor. Phys. Suppl. 69 (1980) 80; in: J. Souletie, J. Vannimenus, R. Stora (Eds.), Chance and Matter, North-Holland, Amsterdam, 1987, p. 395. [3] T.A. Fulton, P.L. Gammel, D.J. Bishop, L.N. Dunkleberger, G.J. Dolan, Phys. Rev. Lett. 63 (1989) 1307. [4] D.V. Averin, V.Ya. Aleshkin, JETP Lett. 50 (1989) 367. [5] Y. Nakamura, C.D. Chen, J.S. Tsai, Phys. Rev. Lett. 78 (1997) 2328. [6] Y. Nakamura, Yu.A. Pashkin, J.S. Tsai, Nature 398 (1999) 786. [7] Y. Nakamura, C.D. Chen, J.S. Tsai, Czech. J. Phys. 46 (1996) 2301. [8] A. Shnirman, G. SchoK n, Z. Hermon, Phys. Rev. Lett. 79 (1997) 2371. [9] D.V. Averin, Solid State Commun. 105 (1998) 659. [10] Yu. Makhlin, G. SchoK n, A. Shnirman, Nature 398 (1999) 305. [11] K.K. Likharev, V.K. Semenov, IEEE Trans. Appl. Supercond. 1 (1991) 3. [12] A. Shnirman, G. SchoK n, Phys. Rev. B 57 (1997) 15 400.
Quantum coherence in a single-Cooper-pair box: experiments in the frequency and time domains Y. Nakamura *, Yu. A. Pashkin, J.S. Tsai NEC Fundamental Research Laboratories, 34 Miyukigaoka, Tsukuba 305-8501, Japan CREST, Japan Science and Technology Corporation (JST), Kawaguchi, Saitama, 332-0012, Japan
Abstract In a small superconducting electrode connected to a reservoir via a Josephson junction, we can realize an arti"cial two-level system in which two charge-number states of the electrode are coupled by Cooper-pair tunneling. We have investigated quantum coherence in the two-level system through two complementary experiments: (i) photon-assisted Cooper-pair-tunneling spectroscopy of the energy-level splitting in the frequency domain, and (ii) pulsed-gate control of the quantum state to observe the coherent oscillations in the time domain. Our results demonstrate the existence and the persistence of the coherence for at least &30 cycles of the oscillation. 2000 Elsevier Science B.V. All rights reserved. Keywords: Coherent oscillations; Energy-level splitting; Josephson junction; Two-level system
1. Introduction In the microscopic world, we "nd many quantum two-level systems such as nuclear and electron spins, atomic states, and diatomic molecular orbitals. Coherence in such two-level systems has been studied in detail both in the frequency domain and in the time domain. Radio-frequency, microwave, and optical spectroscopy have been applied to observe the energy-level splitting between two eigenstates. Also, Rabi oscillations in the time domain have been observed in many systems. We have conducted analogous experiments on an arti"cial two-level system in a small-Josephson-junction circuit called a single-Cooper-pair box [1] (enclosed part of Fig. 1(a)). This circuit consists of a small superconducting electrode (a &box') coupled to a superconducting reservoir via a Josephson junction. The relevant states of the two-level system are two charge states that di!er by 2e, for example, "02 and "22, where the charge state "n2 is a superconducting ground state with a "xed excess-electron number n in the box. This system is unique in the
* Corresponding author. Tel.: #81-298-50-1148; fax: #81298-56-6139 . E-mail address: [email protected] (Y. Nakamura)
sense that it is a single two-level system consisting of two distinct macroscopic quantum states involving a large number of conduction electrons in the box, not an ensemble of a macroscopic number of microscopic twolevel systems. (We would not say macroscopically distinct in the sense de"ned in Ref. [2], since the essential di!erence between the two states in our case is just a single Cooper pair transferred across the junction.) Also, in contrast to the microscopic systems, the two-level system can be #exibly designed by modifying the macroscopic energy parameters, the Josephson energy E and the ( charging energy E , to which all the electrons in the box ! collectively contribute. Here, we report our observation of clear evidence of quantum coherence in the two-level system: energy-level splitting due to coherent superposition of the two charge states and coherent oscillations between the two charge states.
2. Experimental setup and measurement schemes The single-Cooper-pair box and the additional probe junction were fabricated by electron-beam lithography and shadow evaporation of Al "lms. The box electrode had dimensions of &700;50;15 nm and contained &10 conduction electrons. The Josephson junction
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 1 7 9 0 - 1
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Fig. 1. (a) Single-Cooper-pair-box circuit (enclosed by a dashed line). A superconducting voltage-biased probe electrode is attached to the box via a tunnel junction. (b) Schematic energy diagram of the device for Q /e"1.
between the reservoir and the box had normal resistance of &10 k), while the probe junction, which had a thicker tunnel barrier, had resistance R of &30 M). This large di!erence in the resistances is a key point in our experiments, as we will discuss below. Although it is not shown in Fig. 1(a), the reservoir junction was divided into two parallel junctions to form a SQUID loop through which we can control the e!ective Josephson coupling energy E by an externally applied magnetic ( #ux penetrating the loop. There were also two gate electrodes capacitively coupled to the box. One was for a DC gate voltage < (the DC gate) and the other was for microwave and high-speed pulse signals (the pulse gate; not shown in Fig. 1(a)). The superconducting gap of the electrodes D was 230$10 leV, and the charging energy of the box, E ,e/2CR , was 117$3 leV, where CR is ! the total capacitance of the box. The sample chip was mounted in a shielded copper package and placed at the base temperature of the dilution refrigerator (&30 mK; k ¹&3 leV). Special care was taken concerning the design of the microwave- (or pulse-) transmission line that went from a room-temperature environment to the sample. We used a low-loss silver-plated Be}Cu coaxial cable (above 4.2 K) and a Nb coaxial cable (below 4.2 K). At the end of the Nb cable, the central conductor of a V connector (Wiltron) was connected to the on-chip coplanar line which was terminated with the pulse gate of the device that acted as an open end. We measured the DC current through the probe junction under a "nite bias voltage < to detect the probabil ity density of the charge states of the box. In an energy diagram (Fig. 1(b)) we have drawn the addition energies
of the nth electron in the box for Cooper-pair tunneling dE (n)/2,(E(n)!E(n!2))/2 (normalized to the energy per electron; solid levels) and for single-electron tunneling dE (n),E(n)!E(n!1) (dashed levels) where E(n)"E (n!Q /e) is the electrostatic energy of the ! state "n2 (Q ,C < #const.). Because of the charging e!ect, those energies have discrete levels. The duplication of the levels for each n is due to the di!erence of the charge between a Cooper pair and a single electron. The ladder of the levels can be vertically shifted in proportion to the gate voltage, or the gate-induced charge Q . For the reservoir and the probe, we depict the Fermi level and the superconducting pair-breaking energy 2D for the quasiparticle tunneling. A Cooper pair resonantly tunnels through the reservoir junction when E " $ dE (n"2)/2"0 is ful"lled; in other words, when "02 and "22 are degenerate. This works as an e!ective two-level system. Other charge states with the same parity as n have higher energy, and the states with a di!erent parity are e!ectively decoupled because of the superconducting gap. No quasiparticle tunneling takes place at the reservoir junction under the relevant conditions. On the other hand, Cooper-pair tunneling through the probe junction can be ignored because of the negligibly small Josephson energy. Instead, because of the large bias voltage (< '(2D#E )/e), two sequential quasiparticle ! tunnelings can occur with the rates of C and C (C JR\), bringing the state from "22 to "02. We can use these processes to detect the charge states: "22 emits two electrons to the probe, while "02 does nothing. Although we cannot measure the single two-electron tunneling event, the cyclic process consisting of the resonant Cooper-pair tunneling and the two sequential quasiparticle tunnelings is repeated automatically to produce an easily measurable DC current [3,4]. A typical Q -dependence of the DC current I is shown as dashed lines in Figs. 2(a) and (b) that have a broad peak at Q /e"1; that is, a resonant tunneling peak. The abrupt drop of the current at Q /e&1.13 is because of the suppression of C due to the superconducting gap. In the spectroscopy of the energy-level splitting, we used photon-assisted Cooper-pair tunneling under microwave irradiation with a frequency f through the gate [5]. Fig. 2(c) illustrates the relative electrostatic energies of the two charge states as a function of Q (dashed lines). Coherent Josephson coupling of the two charge states results in two eigenenergy levels with an energy-level splitting of E at Q /e"1 (solid lines). When ( the photon energy hf matches the energy gap between the two eigenstates, a photon-assisted transition takes place and increases the population of "22, resulting in a photoinduced current peak as in Fig. 2(a). On the left side of the resonance, photon absorption dominates, while on the right side, photon emission does (not shown), since "02 is more populated than "22 due to the quasiparticle tunneling at the probe junction. The quasiparticle tunneling
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pulse height at the device. Thus, we swept the baseline of the pulse, that is, Q , and measured the I}Q curve as in Fig. 2(b) for each *t. The oscillation in the time domain was observed in the cross-section of the series of the I}Q curves [4].
3. Frequency-domain experiment: energy-level splitting
Fig. 2. Current through the probe junction as a function of Q with 33.8 GHz microwave irradiation (a) and with a pulse array (*t"160 ps, ¹ "16 ns) (b). The dashed curves are the data taken without microwave or the pulse array (< "650 lV and / "0.31). (c), (d) Schematic energy diagrams illustrating the ways of changing the quantum states of the box by microwave irradiation (c) and by a gate-voltage pulse (d). In (c) the solid lines represent eigenenergies of the box and the dashed lines show the electrostatic energy of each charge state (vice versa in (d)).
also leads to a "nite lifetime and, as a result, a level broadening C to the state "22 (the solid-line states in Fig. 2(c) are not eigenstates any longer), that gives an intrinsic energy resolution of the spectroscopy. Thus, we need to have a large resistance di!erence between the reservoir and the probe junctions to ful"l the condition
C ;E in order to see the splitting as evidence of the ( coherent superposition. To do a time-resolved measurement, we applied short DC-voltage pulses to the gate [6]. Fig. 2(d) shows the scheme. When Q is kept far to the left of the resonance, the system relaxes to the ground state ("lled circle). Then, a pulse with an appropriate height brings the system nonadiabatically to the resonance where coherent oscillation between the two charge states starts. The rise and fall times of the pulse must be short compared to the coherent oscillation period. Otherwise, the state of the two-level system just follows the ground state (dashed line), and no oscillation occurs. After a pulse length *t, the gate voltage comes back to the initial value with a superposition of the two charge states corresponding to the phase of the oscillation during *t. Then, "nally, the "22 component of the wave function decays to the ground state with two quasiparticle tunnelings at the probe junction. Hence, if we repeat the pulse with a repetition time ¹ longer than the relaxation time (C\ #C\ ), we have a measurable DC probe current (of an order of 2e/¹ ) that re#ects the weight of the state "22 in the superposition. In practice, we could not know the exact
Fig. 3 shows I}Q curves under microwave irradiation with frequencies from 2.2 to 40 GHz. Compared with the curve in Fig. 2(a), the data were taken at larger < . Thus, the cuto! of the probe current was shifted toward larger Q (&1.45e). On each side of the resonance peak at Q /e"1, we observed peaks corresponding to the photon-assisted Cooper-pair tunneling with either photon absorption or emission process. The reason why the emission peak is much weaker than the absorption peak is not clear yet [5]. We can see that the peak height of the photon-assisted peaks changes smoothly as a function of frequency; this means the coupling strength of the microwave signal to the sample was rather #at for the frequency range we used. Since the microwave power A ("!33 dB) was relatively high in this measurement, we observed higher-order peaks involving multiplephoton transitions. From another experiment on the microwave-power dependence (not shown; similar to the one in Ref. [7]), at 10 GHz, we estimated dE /hf as ! &0.77 for Fig. 3, where dE is the oscillation amplitude ! of the relative energy between the two relevant charge states as determined by the microwave "eld. To obtain the energy dispersion, we plot the positions of peaks in the I}Q curves as a function of frequency
Fig. 3. I}Q curves for various microwave frequencies from 2.2 (bottom) to 40 GHz (top) at < "700 lV and / & . Each curve is shifted by 0.5 pA for clarity.
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Fig. 4. Spectroscopy of the energy-level splitting. The black dots represent positions of the peaks in the I}Q curves as a function of microwave frequency for / "0 (a), (b), (c) and (d). The solid curves are "ts to the energy dispersion curve for E "83, ( 73, 42 and 0 leV, respectively for (a)}(d). The dashed lines are the dispersion for E "0. (
under a lower microwave power (A"!50 dB). Fig. 4 shows such plots for di!erent external magnetic "elds. The photon-absorption peak positions "t well with the expected dispersion curves with the gap frequency corresponding to E /h. The -dependence of the gap will be ( discussed in the next section. This observation of the energy-level splitting is evidence of coherent superposition of the two charge states. Such spectroscopic determination of the Josephson energy is unique in this two-level system and is not possible in principle in the classical (large) Josephson junction. However, we could not directly determine the decoherence time, nor actively control the superposition of the wave functions in this experiment. Although, in principle, we can obtain information on the decoherence from the peak-width analysis, in practice this has been di$cult with our current setup since we cannot clearly distinguish intrinsic broadening from that due to experimental noise at low energies less than a few leV.
pulse-induced probe current *I is shown as a function of the pulse length *t in Fig. 5. Since we do not know the exact pulse height at the sample, we select an appropriate Q (&0.51e) as the baseline of the pulse to obtain the largest period of the oscillation. The observed period ¹ should correspond to h/E . Otherwise, the two ( charge states are not exactly degenerate during the pulse, and the observed oscillation has a smaller period [6]. We controlled E through the magnetic "eld, and plot the (
-dependence of h/¹ in the inset of Fig. 5. The values of E , determined independently from the spectroscopy in ( the previous section, are also plotted in the same graph. Both agreed well with the expected cosine -dependence of E , con"rming that the oscillation is due to the Joseph( son coupling between the two charge states. The decoherence time of the two-level system is the next thing we would like to know. Fig. 6 shows the oscillation up to *t"2 ns, and we see nearly 30 cycles of the oscillation. In the measurement, we expected to see a decaying envelope of the oscillation and to obtain a decoherence time. However, the noisy signal, which was mainly due to the background charge #uctuation near the device, prevented us from doing so. Since we measured the DC current with a time constant of 20 ms, low-frequency (1/f ) #uctuation of the background charge could degrade the signal by creating a #uctuating o!set in Q . Whether such a #uctuation a!ects the intrinsic decoherence time is not clear yet, since the #uctuation spectrum in the gigahertz range is not known. On the other hand, we can roughly estimate the intrinsic decoherence time of a single-Cooper-pair box embedded in a low-ohmic electromagnetic environment [1]. In the present sample, it is of an order of 100 ns assuming
4. Time-domain experiment: coherent oscillations Following the scheme in Fig. 2(d), we observed coherent oscillations between the two charge states in the time domain. The observed oscillating signal in the
Fig. 5. Pulse-induced current versus pulse length. Inset: Magnetic-"eld dependence of E determined by the two independent ( experiments in the time domain and the frequency domain. The solid curve is a cosine "tting with E ( "0)"84 leV. (
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operation is necessary to enable multiple-step operation of multiple qubits. In the present experiment, the shape of the pulse supplied by room-temperature electronics was far from rectanglar (the rise and fall times were &30}40 ps at the top of the cryostat), and there was jittering of an order of 10 ps. For 100-ps-scale qubit control, we need much faster and more precise electronics, such as a single-#ux-quanta circuit [11] located on the same chip. Though we have relied on averaged measurements over repeated pulse operations in the present work, singleshot quantum measurement of the charge states [12] is also indispensable. Fig. 6. Pulse-induced current versus pulse length up to *t"2 ns.
100 ) environmental impedance on the probe electrode. That is much larger than our experimentally con"rmed value of 2 ns. However, in our measurement scheme, quasiparticle tunneling at the probe junction also disturbs the coherence, and even worse, the probe is always connected to the two-level system even during the pulse operation. Thus, the decoherence time in the current setup is probably limited by 1/C &6 ns at most. The large, almost linear, background current in Fig. 6 is another side e!ect of the quasiparticle tunneling. As *t increases, the probability of the unwanted quasiparticle relaxation during the pulse increases.
5. Concluding remarks The method of observing the oscillation can also be considered a method of coherent control of the quantum states in the two-level system: With an appropriate pulse length and pulse height, in principle we can create arbitrary superposition of the charge states. Use of this twolevel system as a quantum bit (qubit), an elemental unit of quantum computing, has been proposed [8}10]. In that context, the decoherence would be the most serious problem and needs to be further studied in detail. From a technical point of view, more accuracy in the pulse
Acknowledgements This work was supported by the Core Research for Evolutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST).
References [1] V. Bouchiat, D. Vion, P. Joyez, D. Esteve, M.H. Devoret, Phys. Scripta T 76 (1998) 165. [2] A.J. Leggett, Prog. Theor. Phys. Suppl. 69 (1980) 80; in: J. Souletie, J. Vannimenus, R. Stora (Eds.), Chance and Matter, North-Holland, Amsterdam, 1987, p. 395. [3] T.A. Fulton, P.L. Gammel, D.J. Bishop, L.N. Dunkleberger, G.J. Dolan, Phys. Rev. Lett. 63 (1989) 1307. [4] D.V. Averin, V.Ya. Aleshkin, JETP Lett. 50 (1989) 367. [5] Y. Nakamura, C.D. Chen, J.S. Tsai, Phys. Rev. Lett. 78 (1997) 2328. [6] Y. Nakamura, Yu.A. Pashkin, J.S. Tsai, Nature 398 (1999) 786. [7] Y. Nakamura, C.D. Chen, J.S. Tsai, Czech. J. Phys. 46 (1996) 2301. [8] A. Shnirman, G. SchoK n, Z. Hermon, Phys. Rev. Lett. 79 (1997) 2371. [9] D.V. Averin, Solid State Commun. 105 (1998) 659. [10] Yu. Makhlin, G. SchoK n, A. Shnirman, Nature 398 (1999) 305. [11] K.K. Likharev, V.K. Semenov, IEEE Trans. Appl. Supercond. 1 (1991) 3. [12] A. Shnirman, G. SchoK n, Phys. Rev. B 57 (1997) 15 400.