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Superconducting single-Cooper-pair box as quantum bit
Physica C 357±360 (2001) 1±6
www.elsevier.com/locate/physc
Superconducting single-Cooper-pair box as quantum bit J.S. Tsai a,*, Y. Nakamura a, Yu. Pashkin b a
NEC Fundamental Research Laboratories, 34 Miyukigaoka, Tsukuba 305-8501, Japan b CREST, Japan Science and Technology Corporation, Kawaguchi 332-0012, Japan Received 16 October 2000; accepted 12 January 2001
Abstract We demonstrated the ®rst electronic control of 1-qubit achieved in a solid-state device by using a submicron electron device called a single-Cooper-pair box. The number of electrons in the box (typically 108 ) is quantized and they form a single macroscopic quantum charge-number state, corresponding to the number of excess electrons in the box. By making all the electrodes superconducting, we can couple two neighboring charge-number states coherently. In this way one can create an arti®cial two-level system. We attached an additional tunneling probe to the box to monitor the probability of one of the two states involved in the coherence. We applied a suciently fast voltage pulse to the gate to create a degenerated charge-number state, so that to force the two states to undergo a quantum oscillation. As a result, we indeed observed the coherent oscillation. This was the ®rst time that the quantum coherent oscillation was observed in a solid-state device whose quantum states involved a macroscopic number of quantum particles. Multiple-pulse experiments were also carried out. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 03.67.Lx; 74.90.n; 85.25.Cp; 85.30.Wx Keywords: Quantum bit; Quantum computing; Single-electron box; Single-Cooper-pair box; Aluminum
1. Introduction ± quantum information processing The proposal of quantum information processing has shown us the ®rst serious possibility, though still limited in the range of applications, to break away from the traditional spell that has been bounding the integration scale of the electron device and its information processing power for a long time. Qubit (quantum bit), the basic element of the quantum information processing, consists of
* Corresponding author. Tel.: +81-298-50-1161; fax: +81298-56-6139. E-mail address: [email protected] (J.S. Tsai).
a quantum two-level system that can be brought to a superposition by an external mean. Among the numerous possible two-level systems, various typical microscopic two-level systems such as polarized photon states, atomic states and nuclear spin states have been utilized ®rst to demonstrate the realization of the quantum calculation in a limited scale. However, to realize a scaled-up version of the quantum computing system consisting of hundreds of qubits, its solid-state electronics version is considered to be perhaps indispensable. We demonstrated the ®rst electronic control of 1-qubit achieved in a solid-state device. The general scalability of such a solid state device is considered to be a prerequisite for a practical quantum computer of the future.
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 1 7 2 - 1
2
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
2. Single-Cooper-pair box We utilized a submicron electron device called a single-Cooper-pair box, to demonstrate the operation of 1-qubit control in solid-state electron device. It is a small superconductive box connected to the outside by a Josephson tunnel junction and a gate capacitor. The Cooper pairs can tunnel into the box through the tunnel junction. However, due to the small capacitance C of the box, the transfer of a Cooper pair would be accompanied by a large increase in charging energy, making the transfer of the Cooper pairs energetically dicult at low bias voltage. When the charging energy Ec associated with the transfer of one electron charge across the tunnel junction, Ec e2 =2C, is smaller than gap energy of the superconductor D, the transfer of the quasi-particle can be ignored during the gate operation. The large resistance associated with the tunnel junction suppresses the quantum ¯uctuation of the charge of the box through the junction, thus the number of Cooper pairs in the box is quantized under a certain gate bias voltage. There are many (typically 108 in our experiment) Cooper pairs in the box, but they form a single macroscopic quantum charge-number state, corresponding to the number of excess electrons in the box. Fig. 1 shows the energy diagram of the system. The energy dierence between the upper and lower p band is DE EJ2 dE2 , where dE 4Ec Q=e 1 and Q Cg Vg . The Josephson coupling enables a dissipation-less transfer of an electron pair, thus two neighboring charge-number states, designated as a state j0i, and a state with one more Cooperpair j1i, can be coherently coupled. This was
Fig. 1. The band diagram of the single-Cooper-pair box.
achieved by biasing the box at Q e, the resonant condition, by gate, or by using photon-assisted coupling of two states at the o-resonant condition where Q 6 e (Fig. 1). In this way one can create an arti®cial two-level system. The result of previous experiments showing that the macroscopic quantum state of superconductivity is extraordinary well isolated from the internal and external dissipation [1] gave us a con®dence in such experiment that requires extremely low dissipation. For the detection of the quantum state, we attached an additional tunneling probe to the box, our original method, to study this twolevel system. With this probe, we could simply measure the current it carried to monitor the probability of one of the two states that was involved in the coherence.
3. Device The typical device picture is shown in Fig. 2 (the photograph and the schematic diagram). The whole structure was made of aluminum thin ®lm t 20 nm. The box has a dimension of 700 50 nm2 , a size quite readily reproducible by an electron-beam lithography. The coherence junction connecting the island and reservoir where quantum coherence was prepared was split into two junctions, forming a SQUID loop. In this way we could control the Josephson energy of the system
Fig. 2. The device picture and its schematic diagram.
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
by an external magnetic ®eld, an extra control that was proved to be quite useful as discussed below. The tunnel junction was prepared by the anglecontrolled evaporations of aluminum through a suspended stencil mask. The thickness of the tunnel barrier, natural oxide of aluminum, was controlled by regulating the partial pressure of the oxygen of the evaporation chamber. The probe junction had a thicker tunnel oxide barrier prepared by an additional oxidation process, and typically resulted with a resistance roughly three orders of magnitude larger compared to the coherence junctions. To couple a wide bandwidth electric signal to the box, a high-speed gate was made employing a coplanar wave-guide design 50 X. The entire device was made on top of a copper ground plane, with silicon nitride insulating layer in between. The typical Josephson energy of the coherent junction was 40 leV, the charging energy of the box was typically 100 leV. The experiments were carried out in an environment with low thermal energy, typically 30 mK or 3 leV, low enough compared with all relevant energies associated with the box. 4. Energy-domain experiment We studied the two-level system by two complimentary experiments, an energy-domain spectroscopic experiment [2] and a time-domain quantum oscillation experiment [3]. First, we performed the energy-domain experiment, utilizing a photon-assisted tunneling method. We attached an additional tunneling probe to the box to study this two-level system. Under the resonant condition, there was a resonant current ¯owing into the probe junction within the Coulomb blockade region, when the probe junction was appropriately biased. This was the current carried by the so-called Josephson-quasiparticle peak (JQP) resonant cycle [4]. The JQP current signi®es the cyclic production of a coherent state between states j0i and j1i, followed by decoherence of the coherent state due to transfer of two quasiparticles to the probe electrode. The aim of this experiment was to trace out the upper energy band as shown schematically in
3
Fig. 3. The energy band traced by the photon assisted JQP gap experiment.
Fig. 1. To carry out this feat, we utilized a photonassisted tunneling method. Even under the oresonant bias condition, if there are photons with appropriate energy, the states j0i and j1i can also be coupled. This photo-assisted coupling resulted with appearance of a photon-assisted JQP peak current at the corresponding o-resonant gate bias. By sweeping the photon frequency while varying the gate bias, we obtained a series of photonassisted JQP structures whose peak positions traced out the upper energy band of the system. When the resistance of the probe junction was suciently high (suciently small tunnel rate C giving providing Ch EJ ), the experiment showed a clear band gap structure in the eigenstates as shown in Fig. 3 [2]. The existence of the gap near the degeneracy point of states j0i and j1i signi®es the existence of quantum coherence between these two states. This was the ®rst time such gap was directly observed spectroscopically, although some indirect evidences of such gap were reported previously [5,6]. 5. Time-domain experiment ± quantum bit To further study the coherence, we carried out a bolder experiment designed to observe the dynamics of the coherence, namely, the coherent
4
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
oscillation. First, as the initial state, we prepared a pure state j0i by biasing the system far from the resonant condition. Next, we applied a suciently fast voltage pulse to the gate (a non-adiabatic gate operation) to create a degenerated charge-number state at the resonant condition. The voltage rise time had to be fast enough to initiate a nonadiabatic transition [7]. At the degenerate point, two neighboring charge states j0i and j1i were coherently coupled by the Josephson energy, so that the two states were forced to undergo a quantum oscillation during the time Dt that corresponds to the gate pulse width. At the end of pulse, the system was brought back to the original bias point. At this moment, the state of the system retained a coherent superposition of states j0i and j1i. At this point, a sampling measurement of state j1i by the attached probe junction was carried out. This was done by measuring the current carried by the probe junction under the repeated pulsing ex-
periments. Under an appropriate voltage bias condition of the probe junction, the j1i state would decay to j0i state by emitting two quasi-particles (electrons) to the probe electrode with an observation rate C (the tunneling rate of the probe junction). It was important to keep the observation ``weak'', namely, hC < EJ , so that the quantum coherence would not be destroyed easily by the existence of the probe junction. The time delay between the pulses was set to be longer than the measurement time 1=C, so each measurement could be considered as independent of the previous measurement. As a result, the current we monitored re¯ected the averaged occupation probability of the j1i state after sampling for many events (about one million events in the actual experiment). The samplings were carried out at dierent Dt conditions that resulted with dierent linear combinations of states j0i and j1i. By sweeping Dt with an increment of 1 ps or so, we succeeded in
Fig. 4. The quantum oscillation observed in the non-adiabatic gate operation experiment.
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
tracing out the time evolution of the quantum oscillation (Fig. 4). The period of the oscillation in this case was expected to be h=EJ . We could control the eective Josephson energy EJ by applying an external magnetic ®eld to the DC SQUID as shown in Fig. 2. As a result, the period of the oscillation was modulated as anticipated. The Josephson energies obtained from the oscillation periods and the photon-assisted spectroscopy agreed extremely well under all the magnetic ®eld values tested. They were also modulated by magnetic ®eld following exactly the expected cosine function as shown in the inset of Fig. 4. This was the ®rst time that the quantum coherent oscillation was observed in a solid-state electron device whose quantum states involved a macroscopic number of quantum particles. By adjusting Dt of the pulse, as well as the strength of the magnetic ®eld, one can achieve the generation of any desired linear superposition of the two states, or the 1-qubit control operation. 6. Decoherence In quantum information processing, computation that requires many gate operations has to be performed before the system loses quantum coherence. Decoherence happens when the quantum system interacts with the external environment that accompanying energy dissipation. For example, the decoherence time due to the coupling to the vacuum (spontaneous emission) via the probe junction can be estimated to be roughly 100 ns at the resonant condition. However, since the probe junction attached to the box had a sampling time of typically 6±9 ns for the most of the experimental devices we used, this probing time restricted the upper limit of the decoherence time in our experiment. The experimentally con®rmed coherent oscillation lasted for about 5 ns, when biased at the resonant condition. The oscillation period was typically 100 ps, so this corresponds to about 50 gate operations. Considering the sampling rate of the probe junction, the observed 5 ns oscillation was perhaps not so unreasonable. To study the decoherence under the o-resonant condition, we carried out a multi-pulse experi-
5
ment [8]. Here, two consecutive voltage pulses were applied to the gate before the probe junction executed the observation (quasiparticles tunneling). The ®rst pulse would bring the system into a coherent state. After this pulse, the system would be biased at the o-resonant condition but still maintaining coherence. Then, during the inter-pulse period td , the quantum state will acquire an extra quantum phase p dependent factor exp i DE=htd , where DE EJ2 dE2 is the energy dierence between the upper and lower band. If one makes a measurement at this moment, the resultant current (or the probability of the j1i state) would not be much in¯uenced by the time td when dE EJ . However, when we applied the second pulse with time delay td before the measurement, the probability of the j1i state would be modi®ed by the length of td . A spin representation of the system as shown in Fig. 5 illustrates the above physics geometrically [8]. Here, a surface of a sphere de®ned by a ``spin'' vector represents the linearly superposed state of the j0i and the j1i states. The Josephson energy represents the ``magnetic ®eld'' in x direction, and energy dierence dE 4Ec Q=e 1 represents the ``magnetic ®eld'' in z direction. When the system is put into the quantum coherence, the spin vector would undergo a precession under the p in¯uence of the resultant ``magnetic ®eld'' DE EJ2 dE2 . Thus the axis of rotation can be controlled by selecting the operating point Q that provides a desired dE Q.
Fig. 5. The spin representation of the single-Cooper-pair box.
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J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
The precession around x- and z-axes could be traced by sweeping the inter-pulse delay time td and sampling the probe current (probability of j1i state). The result of the experiment indeed showed a quantum oscillation with oscillation period of DE= h as expected [8]. The decoherence time at the o-resonant condition was found to be about 1 ns, much shorter than the decoherence time found at the resonant condition. Probably the shorter decoherence could be attributed to the ¯uctuation in Q of the system due to the infamous background charge ¯uctuation. Such ¯uctuation would generate a large ¯uctuation in DE at the o-resonant condition, but not so much at the resonant condition. By projecting the observed low-frequency 1=f background charge noise up to the relevant frequency of the experiment, we indeed obtained a roughly similar o-resonant decoherence time estimation [8]. In addition, the control of the phase of the quantum precession (by selecting the time delay td ) in the two-pulse experiment also corresponded to the operation of the quantum phase gate. The energy relaxation time in this two-level system was also carried out by the similar doublepulse experiment, and it was found that there was no other fast energy relaxation process that faster than the probe measuring time (9 ns) existed [8]. 7. Summary Utilizing small Josephson junction device having large charging energy, we have performed the energy-domain and the time-domain experiments showing the existence of quantum coherence between neighboring charge states. The time-domain experiment corresponded to the realization of the 1-qubit operation, the ®rst of such demonstration in a solid-state electron device. Both rotation gate (controlling the amplitude of state j1i) and phase gate (controlling of the phase of the state) were demonstrated. The experimentally con®rmed decoherence time at the resonant condition was more than 5 ns, and was probably limited by the probe measuring time. The decoherence time at the oresonant condition was about 1 ns. The much faster decoherence found at this bias condition
could be attributed to the background charge ¯uctuation in the device. Such ¯uctuation would induce the decoherence at the o-resonant condition, but would not do so at the resonant condition where relevant factor associated with this eect, dDE=dQ, is much smaller. The qubit operation at the resonant point is a unique approach compared with the other more common Rabioscillation qubits. It is unique in the sense that during the operation, the system would not acquire quantum phase, thus less susceptible to the ¯uctuations that aect the phase [9]. Some dynamical noise-suppression procedure [10] should be eective in reducing the decoherence originated from charge ¯uctuation. No energy relaxation channel that faster than the probe measurement was found that would relax the j1i state to the j0i state. Acknowledgements This work has been supported by the CREST project of Japan Science and Technology Corporation (JST). References [1] J.S. Tsai, A.K. Jain, J.E. Lukens, Phys. Rev. Lett. 51 (1983) 316. [2] Y. Nakamura, C.D. Chen, J.S. Tsai, Phys. Rev. Lett. 79 (1997) 2328. [3] Y. Nakamura, Yu.A. Pashkin, J.S. Tsai, Nature 398 (1999) 786. [4] T.A. Fulton, P.L. Gammel, D.J. Bishop, L.N. Dunckleberger, G.J. Dolan, Phys. Rev. Lett. 63 (1989) 1307. [5] V. Bouchiat, D. Vion, D. Esteve, M.H. Devoret, in: K. Ismail, S. Bandyopadhyay, J.P. Leburton (Eds.), Quantum Devices and Circuits, Imperial Collage Press, London, 1997, p. 204. [6] L.S. Kuzmin, D.B. Haviland, Phys. Rev. Lett. 67 (1991) 2890. [7] J.R. Rubbmark, M.M. Kash, M.G. Littman, D. Kleppner, Phys. Rev. A 23 (1981) 3107. [8] Y. Nakamura, Yu.A. Pashkin, J.S. Tsai, Proceeding of the International Workshop on Macroscopic Quantum Coherence and Computing (MQC2 ), in press. [9] A. Ekert, M. Erivsson, P. Hayden, H. Inamori, J. Jones, D. Oi, V. Vedral, arXiv: quantum-ph/0004015. [10] L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 82 (1999) 2417.
www.elsevier.com/locate/physc
Superconducting single-Cooper-pair box as quantum bit J.S. Tsai a,*, Y. Nakamura a, Yu. Pashkin b a
NEC Fundamental Research Laboratories, 34 Miyukigaoka, Tsukuba 305-8501, Japan b CREST, Japan Science and Technology Corporation, Kawaguchi 332-0012, Japan Received 16 October 2000; accepted 12 January 2001
Abstract We demonstrated the ®rst electronic control of 1-qubit achieved in a solid-state device by using a submicron electron device called a single-Cooper-pair box. The number of electrons in the box (typically 108 ) is quantized and they form a single macroscopic quantum charge-number state, corresponding to the number of excess electrons in the box. By making all the electrodes superconducting, we can couple two neighboring charge-number states coherently. In this way one can create an arti®cial two-level system. We attached an additional tunneling probe to the box to monitor the probability of one of the two states involved in the coherence. We applied a suciently fast voltage pulse to the gate to create a degenerated charge-number state, so that to force the two states to undergo a quantum oscillation. As a result, we indeed observed the coherent oscillation. This was the ®rst time that the quantum coherent oscillation was observed in a solid-state device whose quantum states involved a macroscopic number of quantum particles. Multiple-pulse experiments were also carried out. Ó 2001 Elsevier Science B.V. All rights reserved. PACS: 03.67.Lx; 74.90.n; 85.25.Cp; 85.30.Wx Keywords: Quantum bit; Quantum computing; Single-electron box; Single-Cooper-pair box; Aluminum
1. Introduction ± quantum information processing The proposal of quantum information processing has shown us the ®rst serious possibility, though still limited in the range of applications, to break away from the traditional spell that has been bounding the integration scale of the electron device and its information processing power for a long time. Qubit (quantum bit), the basic element of the quantum information processing, consists of
* Corresponding author. Tel.: +81-298-50-1161; fax: +81298-56-6139. E-mail address: [email protected] (J.S. Tsai).
a quantum two-level system that can be brought to a superposition by an external mean. Among the numerous possible two-level systems, various typical microscopic two-level systems such as polarized photon states, atomic states and nuclear spin states have been utilized ®rst to demonstrate the realization of the quantum calculation in a limited scale. However, to realize a scaled-up version of the quantum computing system consisting of hundreds of qubits, its solid-state electronics version is considered to be perhaps indispensable. We demonstrated the ®rst electronic control of 1-qubit achieved in a solid-state device. The general scalability of such a solid state device is considered to be a prerequisite for a practical quantum computer of the future.
0921-4534/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 ( 0 1 ) 0 0 1 7 2 - 1
2
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
2. Single-Cooper-pair box We utilized a submicron electron device called a single-Cooper-pair box, to demonstrate the operation of 1-qubit control in solid-state electron device. It is a small superconductive box connected to the outside by a Josephson tunnel junction and a gate capacitor. The Cooper pairs can tunnel into the box through the tunnel junction. However, due to the small capacitance C of the box, the transfer of a Cooper pair would be accompanied by a large increase in charging energy, making the transfer of the Cooper pairs energetically dicult at low bias voltage. When the charging energy Ec associated with the transfer of one electron charge across the tunnel junction, Ec e2 =2C, is smaller than gap energy of the superconductor D, the transfer of the quasi-particle can be ignored during the gate operation. The large resistance associated with the tunnel junction suppresses the quantum ¯uctuation of the charge of the box through the junction, thus the number of Cooper pairs in the box is quantized under a certain gate bias voltage. There are many (typically 108 in our experiment) Cooper pairs in the box, but they form a single macroscopic quantum charge-number state, corresponding to the number of excess electrons in the box. Fig. 1 shows the energy diagram of the system. The energy dierence between the upper and lower p band is DE EJ2 dE2 , where dE 4Ec Q=e 1 and Q Cg Vg . The Josephson coupling enables a dissipation-less transfer of an electron pair, thus two neighboring charge-number states, designated as a state j0i, and a state with one more Cooperpair j1i, can be coherently coupled. This was
Fig. 1. The band diagram of the single-Cooper-pair box.
achieved by biasing the box at Q e, the resonant condition, by gate, or by using photon-assisted coupling of two states at the o-resonant condition where Q 6 e (Fig. 1). In this way one can create an arti®cial two-level system. The result of previous experiments showing that the macroscopic quantum state of superconductivity is extraordinary well isolated from the internal and external dissipation [1] gave us a con®dence in such experiment that requires extremely low dissipation. For the detection of the quantum state, we attached an additional tunneling probe to the box, our original method, to study this twolevel system. With this probe, we could simply measure the current it carried to monitor the probability of one of the two states that was involved in the coherence.
3. Device The typical device picture is shown in Fig. 2 (the photograph and the schematic diagram). The whole structure was made of aluminum thin ®lm t 20 nm. The box has a dimension of 700 50 nm2 , a size quite readily reproducible by an electron-beam lithography. The coherence junction connecting the island and reservoir where quantum coherence was prepared was split into two junctions, forming a SQUID loop. In this way we could control the Josephson energy of the system
Fig. 2. The device picture and its schematic diagram.
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
by an external magnetic ®eld, an extra control that was proved to be quite useful as discussed below. The tunnel junction was prepared by the anglecontrolled evaporations of aluminum through a suspended stencil mask. The thickness of the tunnel barrier, natural oxide of aluminum, was controlled by regulating the partial pressure of the oxygen of the evaporation chamber. The probe junction had a thicker tunnel oxide barrier prepared by an additional oxidation process, and typically resulted with a resistance roughly three orders of magnitude larger compared to the coherence junctions. To couple a wide bandwidth electric signal to the box, a high-speed gate was made employing a coplanar wave-guide design 50 X. The entire device was made on top of a copper ground plane, with silicon nitride insulating layer in between. The typical Josephson energy of the coherent junction was 40 leV, the charging energy of the box was typically 100 leV. The experiments were carried out in an environment with low thermal energy, typically 30 mK or 3 leV, low enough compared with all relevant energies associated with the box. 4. Energy-domain experiment We studied the two-level system by two complimentary experiments, an energy-domain spectroscopic experiment [2] and a time-domain quantum oscillation experiment [3]. First, we performed the energy-domain experiment, utilizing a photon-assisted tunneling method. We attached an additional tunneling probe to the box to study this two-level system. Under the resonant condition, there was a resonant current ¯owing into the probe junction within the Coulomb blockade region, when the probe junction was appropriately biased. This was the current carried by the so-called Josephson-quasiparticle peak (JQP) resonant cycle [4]. The JQP current signi®es the cyclic production of a coherent state between states j0i and j1i, followed by decoherence of the coherent state due to transfer of two quasiparticles to the probe electrode. The aim of this experiment was to trace out the upper energy band as shown schematically in
3
Fig. 3. The energy band traced by the photon assisted JQP gap experiment.
Fig. 1. To carry out this feat, we utilized a photonassisted tunneling method. Even under the oresonant bias condition, if there are photons with appropriate energy, the states j0i and j1i can also be coupled. This photo-assisted coupling resulted with appearance of a photon-assisted JQP peak current at the corresponding o-resonant gate bias. By sweeping the photon frequency while varying the gate bias, we obtained a series of photonassisted JQP structures whose peak positions traced out the upper energy band of the system. When the resistance of the probe junction was suciently high (suciently small tunnel rate C giving providing Ch EJ ), the experiment showed a clear band gap structure in the eigenstates as shown in Fig. 3 [2]. The existence of the gap near the degeneracy point of states j0i and j1i signi®es the existence of quantum coherence between these two states. This was the ®rst time such gap was directly observed spectroscopically, although some indirect evidences of such gap were reported previously [5,6]. 5. Time-domain experiment ± quantum bit To further study the coherence, we carried out a bolder experiment designed to observe the dynamics of the coherence, namely, the coherent
4
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
oscillation. First, as the initial state, we prepared a pure state j0i by biasing the system far from the resonant condition. Next, we applied a suciently fast voltage pulse to the gate (a non-adiabatic gate operation) to create a degenerated charge-number state at the resonant condition. The voltage rise time had to be fast enough to initiate a nonadiabatic transition [7]. At the degenerate point, two neighboring charge states j0i and j1i were coherently coupled by the Josephson energy, so that the two states were forced to undergo a quantum oscillation during the time Dt that corresponds to the gate pulse width. At the end of pulse, the system was brought back to the original bias point. At this moment, the state of the system retained a coherent superposition of states j0i and j1i. At this point, a sampling measurement of state j1i by the attached probe junction was carried out. This was done by measuring the current carried by the probe junction under the repeated pulsing ex-
periments. Under an appropriate voltage bias condition of the probe junction, the j1i state would decay to j0i state by emitting two quasi-particles (electrons) to the probe electrode with an observation rate C (the tunneling rate of the probe junction). It was important to keep the observation ``weak'', namely, hC < EJ , so that the quantum coherence would not be destroyed easily by the existence of the probe junction. The time delay between the pulses was set to be longer than the measurement time 1=C, so each measurement could be considered as independent of the previous measurement. As a result, the current we monitored re¯ected the averaged occupation probability of the j1i state after sampling for many events (about one million events in the actual experiment). The samplings were carried out at dierent Dt conditions that resulted with dierent linear combinations of states j0i and j1i. By sweeping Dt with an increment of 1 ps or so, we succeeded in
Fig. 4. The quantum oscillation observed in the non-adiabatic gate operation experiment.
J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
tracing out the time evolution of the quantum oscillation (Fig. 4). The period of the oscillation in this case was expected to be h=EJ . We could control the eective Josephson energy EJ by applying an external magnetic ®eld to the DC SQUID as shown in Fig. 2. As a result, the period of the oscillation was modulated as anticipated. The Josephson energies obtained from the oscillation periods and the photon-assisted spectroscopy agreed extremely well under all the magnetic ®eld values tested. They were also modulated by magnetic ®eld following exactly the expected cosine function as shown in the inset of Fig. 4. This was the ®rst time that the quantum coherent oscillation was observed in a solid-state electron device whose quantum states involved a macroscopic number of quantum particles. By adjusting Dt of the pulse, as well as the strength of the magnetic ®eld, one can achieve the generation of any desired linear superposition of the two states, or the 1-qubit control operation. 6. Decoherence In quantum information processing, computation that requires many gate operations has to be performed before the system loses quantum coherence. Decoherence happens when the quantum system interacts with the external environment that accompanying energy dissipation. For example, the decoherence time due to the coupling to the vacuum (spontaneous emission) via the probe junction can be estimated to be roughly 100 ns at the resonant condition. However, since the probe junction attached to the box had a sampling time of typically 6±9 ns for the most of the experimental devices we used, this probing time restricted the upper limit of the decoherence time in our experiment. The experimentally con®rmed coherent oscillation lasted for about 5 ns, when biased at the resonant condition. The oscillation period was typically 100 ps, so this corresponds to about 50 gate operations. Considering the sampling rate of the probe junction, the observed 5 ns oscillation was perhaps not so unreasonable. To study the decoherence under the o-resonant condition, we carried out a multi-pulse experi-
5
ment [8]. Here, two consecutive voltage pulses were applied to the gate before the probe junction executed the observation (quasiparticles tunneling). The ®rst pulse would bring the system into a coherent state. After this pulse, the system would be biased at the o-resonant condition but still maintaining coherence. Then, during the inter-pulse period td , the quantum state will acquire an extra quantum phase p dependent factor exp i DE=htd , where DE EJ2 dE2 is the energy dierence between the upper and lower band. If one makes a measurement at this moment, the resultant current (or the probability of the j1i state) would not be much in¯uenced by the time td when dE EJ . However, when we applied the second pulse with time delay td before the measurement, the probability of the j1i state would be modi®ed by the length of td . A spin representation of the system as shown in Fig. 5 illustrates the above physics geometrically [8]. Here, a surface of a sphere de®ned by a ``spin'' vector represents the linearly superposed state of the j0i and the j1i states. The Josephson energy represents the ``magnetic ®eld'' in x direction, and energy dierence dE 4Ec Q=e 1 represents the ``magnetic ®eld'' in z direction. When the system is put into the quantum coherence, the spin vector would undergo a precession under the p in¯uence of the resultant ``magnetic ®eld'' DE EJ2 dE2 . Thus the axis of rotation can be controlled by selecting the operating point Q that provides a desired dE Q.
Fig. 5. The spin representation of the single-Cooper-pair box.
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J.S. Tsai et al. / Physica C 357±360 (2001) 1±6
The precession around x- and z-axes could be traced by sweeping the inter-pulse delay time td and sampling the probe current (probability of j1i state). The result of the experiment indeed showed a quantum oscillation with oscillation period of DE= h as expected [8]. The decoherence time at the o-resonant condition was found to be about 1 ns, much shorter than the decoherence time found at the resonant condition. Probably the shorter decoherence could be attributed to the ¯uctuation in Q of the system due to the infamous background charge ¯uctuation. Such ¯uctuation would generate a large ¯uctuation in DE at the o-resonant condition, but not so much at the resonant condition. By projecting the observed low-frequency 1=f background charge noise up to the relevant frequency of the experiment, we indeed obtained a roughly similar o-resonant decoherence time estimation [8]. In addition, the control of the phase of the quantum precession (by selecting the time delay td ) in the two-pulse experiment also corresponded to the operation of the quantum phase gate. The energy relaxation time in this two-level system was also carried out by the similar doublepulse experiment, and it was found that there was no other fast energy relaxation process that faster than the probe measuring time (9 ns) existed [8]. 7. Summary Utilizing small Josephson junction device having large charging energy, we have performed the energy-domain and the time-domain experiments showing the existence of quantum coherence between neighboring charge states. The time-domain experiment corresponded to the realization of the 1-qubit operation, the ®rst of such demonstration in a solid-state electron device. Both rotation gate (controlling the amplitude of state j1i) and phase gate (controlling of the phase of the state) were demonstrated. The experimentally con®rmed decoherence time at the resonant condition was more than 5 ns, and was probably limited by the probe measuring time. The decoherence time at the oresonant condition was about 1 ns. The much faster decoherence found at this bias condition
could be attributed to the background charge ¯uctuation in the device. Such ¯uctuation would induce the decoherence at the o-resonant condition, but would not do so at the resonant condition where relevant factor associated with this eect, dDE=dQ, is much smaller. The qubit operation at the resonant point is a unique approach compared with the other more common Rabioscillation qubits. It is unique in the sense that during the operation, the system would not acquire quantum phase, thus less susceptible to the ¯uctuations that aect the phase [9]. Some dynamical noise-suppression procedure [10] should be eective in reducing the decoherence originated from charge ¯uctuation. No energy relaxation channel that faster than the probe measurement was found that would relax the j1i state to the j0i state. Acknowledgements This work has been supported by the CREST project of Japan Science and Technology Corporation (JST). References [1] J.S. Tsai, A.K. Jain, J.E. Lukens, Phys. Rev. Lett. 51 (1983) 316. [2] Y. Nakamura, C.D. Chen, J.S. Tsai, Phys. Rev. Lett. 79 (1997) 2328. [3] Y. Nakamura, Yu.A. Pashkin, J.S. Tsai, Nature 398 (1999) 786. [4] T.A. Fulton, P.L. Gammel, D.J. Bishop, L.N. Dunckleberger, G.J. Dolan, Phys. Rev. Lett. 63 (1989) 1307. [5] V. Bouchiat, D. Vion, D. Esteve, M.H. Devoret, in: K. Ismail, S. Bandyopadhyay, J.P. Leburton (Eds.), Quantum Devices and Circuits, Imperial Collage Press, London, 1997, p. 204. [6] L.S. Kuzmin, D.B. Haviland, Phys. Rev. Lett. 67 (1991) 2890. [7] J.R. Rubbmark, M.M. Kash, M.G. Littman, D. Kleppner, Phys. Rev. A 23 (1981) 3107. [8] Y. Nakamura, Yu.A. Pashkin, J.S. Tsai, Proceeding of the International Workshop on Macroscopic Quantum Coherence and Computing (MQC2 ), in press. [9] A. Ekert, M. Erivsson, P. Hayden, H. Inamori, J. Jones, D. Oi, V. Vedral, arXiv: quantum-ph/0004015. [10] L. Viola, E. Knill, S. Lloyd, Phys. Rev. Lett. 82 (1999) 2417.