A realistic experiment to demonstrate macroscopic ... - CiteSeerX

Physica C 350 (2001) 171±176

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A realistic experiment to demonstrate macroscopic quantum coherence q Marc J. Feldman *, Mark F. Bocko Department of Electrical and Computer Engineering, Hopeman Hall-ECE, University of Rochester, Rochester, NY 14627, USA Received 17 April 2000; accepted 5 June 2000

Abstract According to quantum theory, microscopic objects exist in a superposition of distinct states until they are ``observed''. Nobody knows whether such quantum coherence can actually exist in a macroscopic system. In the experiment described here, a superconducting quantum interference device is extremely well isolated from any interaction with internal or external modes, and a superposition state can be demonstrated even if it lasts for less than 1 ns. This is accomplished by using superconducting digital electronic circuitry as the experimental apparatus. If successful, it is the ®rst step toward a future full-scale quantum computer fabricated using integrated circuit manufacturing techniques. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Macroscopic quantum coherence; Rapid single ¯ux quantum; Josephson junction; Quantum computing

During the 1980s there was a very active worldwide e€ort toward establishing and detecting the (now still unobserved) phenomenon of ``macroscopic quantum coherence'' (MQC) in the rf superconducting quantum interference device (SQUID). The intention was to use a macroscopic system to demonstrate one of the strangest implications of the theory of quantum mechanics, that an object can exist in a superposition of distinct states until a ``measurement'' forces it to choose one or another of the states. This superposition would be a true macroscopic ``Schr odingerÕs cat'', a coherent state with microamps of current circuq This research was supported in part by the US Army Research Oce. * Corresponding author. E-mail address: [email protected] (M.J. Feldman).

lating in opposite directions around the SQUID ring simultaneously, over a length scale of order one tenth millimeter [1]. Superconductivity is itself an example of quantum mechanics acting on a macroscopic scale, of course, and there are numerous other instances in which the ground state of a macroscopic system is a coherent superposition of a large number of particle states. In addition there are instances in which transitions between energy states can be observed spectroscopically in a macroscopic system. MQC, however, refers to the superposition of states each with macroscopically di€erent behavior, which evolves under the laws of quantum mechanics until an ``observation'' causes the ``collapse of the wave function'' into a state of de®nite macroscopic properties. In this sense MQC has never been observed in any physical system. These distinction are

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 0 ) 0 1 6 0 0 - 2

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quite clearly delineated in Ref. [1]. As Schr odingerÕs parable illustrated, a successful demonstration of MQC would be dicult to reconcile with our notions of the ``reality'' of the macroscopic world [2]. The SQUID MQC experiment deserves another look, for two reasons. First, there is a much better prospect of success today, as we will describe. Second, if this two-state coherence time is long enough, then the rf SQUID can be used as a quantum coherent bit (qubit) [3] to provide the basis for quantum computing [4]. Here we present a scheme to establish and to verify MQC in a rf SQUID. This experiment should achieve a very high degree of isolation from environmental electromagnetic modes because only DC leads are required, which implies a much longer decoherence time compared with earlier experiments. On the other hand, the time resolution of this experiment can be as short as 100 ps, which implies that MQC can be unambiguously demonstrated even if the decoherence time is very short, less than 1 ns. The earlier experiments required wideband coupling, to MHz frequencies, to attain a time resolution of order microseconds. Our scheme avoids this compromise by using superconducting digital circuitry to provide the interface between the rf SQUID and laboratory instrumentation. The theory of MQC in the rf SQUID was well developed, e.g., in Refs. [5,6]. A generic rf SQUID is sketched in Fig. 1. The Hamiltonian of this system, assuming no dissipation …R ˆ 1†, is H …U; Uex † ˆ

2 o2 h ‡ U …U; Uex †; 2C oU2

…1†

where C is the capacitance across the Josephson junction and the potential energy U is given by

Fig. 1. The rf SQUID consists of a Josephson junction with critical current Ic and shunt capacitor C and resistor R in an inductive loop L, biased with externally applied magnetic ¯ux Uex .

Fig. 2. rf SQUID potential energy versus internal ¯ux, at onehalf external ¯ux bias.

U …U; Uex † ˆ

…U

Uex † 2L

2

Ic U 0 U cos2p : U0 2p

…2†

U is the magnetic ¯ux through the SQUID loop, and U0 ˆ h=2e is the quantum of magnetic ¯ux in a superconductor. (Note in these equations that U is analogous to the coordinate x of a particle whose mass is C.) If Uex is exactly U0 =2 then U (U) has two symmetric minima, as shown in Fig. 2 for LIc ˆ 2U0 . In equilibrium a current close to U0 =2L circulates around the SQUID loop either adding to or canceling the applied ¯ux; call these the ``1'' and the ``0'' states, respectively. As the SQUID parameters L and Ic are made smaller the potential barrier decreases and the minima move closer together. Then the ¯ux in the SQUID loop can tunnel between the two minima. This implies that the degenerate ground state energy of the SQUID is split by an amount DE related to the tunneling matrix element, and the ``0'' and ``1'' ¯ux states are energy superposition states. Therefore if the coherence of this superposition can be maintained long enough, the wave function of a ¯ux initially localized in the ``0'' state will oscillate back and forth to the ``1'' state at frequency DE=h. This oscillation is a clear signature of a coherent superposition state. The ``standard'' MQC experiment [7] follows this protocol to demonstrate the oscillation of the wave function: the rf SQUID biased at one-half ¯ux is prepared in a distinct ¯ux state, allowed to freely evolve for some time t, after which its state is projected by a classical measurement. Repeating this sequence the probability distribution of ®nal ¯ux state P is plotted as a

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function of t. With full coherence P(t) is a sinusoid of frequency DE=h. Without coherence P(t) monotonically tends to 0.5 with relaxation time given by thermal noise and tunneling. With incomplete coherence P(t) is a damped sinusoid with relaxation given by the decoherence time. Thus an unambiguous con®rmation of MQC is possible. For this experiment to succeed, it is essential that the SQUID evolution be isolated from all other degrees of freedom. Coherent superposition is extremely fragile and is rapidly destroyed by any interaction with its environment [8,9]. Therefore, the diculty of this experiment is evident. On one hand, the experiment should proceed as rapidly as possible so that an oscillation cycle may be observed before decoherence occurs. This means that DE=h must be large. On the other hand, the control leads should be heavily ®ltered to isolate the rf SQUID from external degrees of freedom which will cause decoherence. This precludes high-frequency measurements, and so DE=h must be small. The solution to this conundrum is to place the SQUID in an entirely superconducting environment. Unlike most other macroscopic objects, superconductors at very low temperature have very few internal degrees of freedom, and in fact their speci®c heat vanishes at zero temperature. For the MQC experiment to succeed, it is essential that the rf SQUIDÕs environment consist exclusively of superconductors far below their transition temperature. Note that Nakamura et al. recently demonstrated the coherent superposition of energy states di€ering by a single Cooper pair in a mesoscopic superconducting system [10]. Only three electronic functions are required to perform the ``standard'' MQC experiment ± initialization, start/stop with programmed delay interval, and read-out. All three can be accomplished using superconducting circuitry, as sketched in Fig. 3. Much of the circuitry shown is derived from the rapid single ¯ux quantum (RSFQ) logic family [11]. RSFQ logic is a new but fairly well developed integrated circuit technology. Relatively complex circuits consisting of roughly 100 gates clocked at frequencies of about 10 GHz have been demonstrated by several groups. For a recent review and comparison to semiconductor integrated circuitry see Ref. [12]. RSFQ circuits perform digital com-

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Fig. 3. An all-superconductor circuit to investigate the coherent evolution of the macroscopic wave function of an rf SQUID. The quantum state of the SQUID can be projected by a classical measurement at any increment of 100 ps up to 12.8 ns after the coherent evolution begins. The purely superconducting environment of the SQUID should allow a very long decoherence time, but macroscopic quantum coherence can be demonstrated even if the decoherence time is substantially less than 1 ns.

putation by manipulating magnetic ¯ux quanta and the single ¯ux quantum (SFQ) voltage pulses which accompany moving ¯ux quanta. RSFQ circuits consist almost entirely of interconnected SQUID loops, each including two or more relatively shunted Josephson junctions and an inductor. An SFQ can be stored in a SQUID loop and can be transferred between loops. It is important to note that, between operations, an RSFQ circuit has no voltages and there is no dissipation. The MQC experiment sketched in Fig. 3 includes an rf SQUID, a circuit to bias the rf SQUID at exactly one-half ¯ux, a SET signal to prepare the rf SQUID in a distinct ¯ux state, a circuit that can be preprogrammed to respond after a chosen time delay, and a START signal which both allows the rf SQUID to begin its coherent evolution and which also generates the END signal after the chosen delay. The END signal performs a classical measurement of the state of the rf SQUID. Each ``'' in Fig. 3 represents an unshunted Josephson junction. Everything else in Fig. 3 is composed of resistively shunted Josephson junctions and superconducting inductors, exclusively. The rf SQUID Josephson junction is denoted by `` '' has a critical current of several microamps. All of the others have critical currents of hundreds of microamps. The half-¯ux bias for the rf SQUID is provided by a single very weakly coupled inductor, which is placed in a hysteretic SQUID

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loop including a Josephson junction with large critical current. When Ibias is increased, magnetic ¯ux quanta enter the loop one by one. Then Ibias is turned o€ and a current, close to the desired value, circulates in the bias loop; it will circulate forever if not disturbed. Then ibias is adjusted to get precisely the half-¯ux bias. The DC/SFQ is an input device which generates an SFQ pulse in response to a slow current ramp. The SET pulse proceeds to a splitter ``S'' and feeds two DRO cells. A destructive read-out (DRO) cell stores an SFQ that enters from the left (as drawn in Fig. 3), and then released it as an SFQ pulse to the right when a second SFQ enters from the top. The stored ¯ux in the lower DRO is very weakly coupled to the bias loop, just enough to change the exact half-¯ux bias so that the rf SQUID is initially in the desired ``1'' or ``0'' state. The START pulse clears the DRO to reestablish the exact half-¯ux bias. The START pulse also starts the CLOCK to produce the END pulse after the preprogrammed delay. The CLOCK is an SFQ transmission line ring identical to that used in Ref. [13]. An injected SFQ pulse circulates around the ring with a transit time of, say 100 ps, and emits a stream of SFQ pulses at a frequency of 10 GHz. These enter the DELAY device composed of a chain of divide-bytwo T-¯ip-¯op cells. A convenient number is seven T-¯ip-¯ops, and so the DELAY circuit emits a single pulse every 27 clock periods, in this case 12.8 ns. The T-¯ip-¯op chain can be initialized (while monitoring the READ TIME output) to choose the ®rst output (the END pulse) at any increment of 100 ps up to 12.8 ns. The ®rst END pulse proceeds to the comparator formed by the two large Josephson junctions connected to the rf SQUID, and then the comparator sends an SFQ pulse to the SFQ/DC output device if and only if the current in the rf SQUID is clockwise. Thus the quantum state of the rf SQUID can be projected at any time up to 12.8 ns after initialization. This measurement can be repeated many thousands of times per hour, with a variety of preprogrammed durations. The timing determines the desired size of the ground state energy splitting DE: we are interested in DE=k between 3 mK which gives an oscillation period of 16 ns, and 100 mK which gives an oscillation period of 0.5 ns.

The physical parameters of the rf SQUID required for this experiment are not especially formidable. A Schr odingerÕs equation analysis of Eqs. (1) and (2) considering the many experimental trade-o€s involved, suggests that a good choice for the three circuit parameters L  100 pH, Ic  4 lA, and C  50 fF. The disparity between this Ic and that of the other Josephson junctions in the circuit, hundreds of microamps, is crucial to the success of this experiment. On one hand, it means that the other junctions will be far below their critical current during the coherent evolution of the SQUID, and so behave like bulk superconductors. On the other hand it means that the duration of an SFQ pulse, about 5 ps, is far shorter than any response time of the rf SQUID, and so the comparator performs an essentially instantaneous projection of the state of the rf SQUID. Ic must be adjustable during an experiment so that a range of DE can be examined to unambiguously demonstrate that any oscillation observed is due to MQC. This can be accomplished by replacing the rf SQUID junction by two such junctions in parallel in a small inductance loop, subject to an applied magnetic ®eld. This stratagem was ®rst employed in Ref. [14] and has since become standard in a variety of experiments, including Ref. [10]. The circuit in Fig. 3 includes on the order of 100 Josephson junctions and generates about 20 lW power dissipation. This is suitable for laboratory cryocoolers, but if desired this power could be much decreased with minimal e€ort. Thus far we have stressed the advantages of our scheme to demonstrate MQC and hardly mentioned the diculties inherent in this experiment. A major practical problem is parameter control. As emphasized in Ref. [7], the rf SQUID bias must be maintained quite precisely at one-half ¯ux, with fractional stability better than DE=Ic U0 ; the Fig. 3. circuit is designed to accomplish this. Also the adjustable Ic must be ®xed without drift or ¯uctuation; roughly, a 1% change in Ic produces a factor of 2 change in DE. The more daunting and more fundamental problem is decoherence, which must be in fact the central issue of any MQC experiment. Successful macroscopic tunneling experiments [15] and recent spectroscopic measurements of avoided crossings

M.J. Feldman, M.F. Bocko / Physica C 350 (2001) 171±176

in the rf SQUID are convincing evidence that coherent overlap between localized-¯ux wave functions can exist for very short times, but it is unknown whether the physical laws of nature allow such coherence to be maintained over the subnanosecond time scale required here. This question can only be decided by the MQC experiment itself. In the theory of MQC in the rf SQUID developed in the 1980s, the rate at which decoherence occurs is a complicated function of many parameters, but with reasonable assumptions the predicted phase coherence time is given by [16]: tU ˆ

2 h2 R ; d2 kT

…3†

where d is the ¯ux separation of the minima U …U† (Eq. (2)) which is about U0 =4 for the parameters we have chosen, and R includes all of the dissipation coupled to the rf SQUID represented as a parallel ohmic resistor as in Fig. 1. (It must be noted that simple linear theories like that leading to Eq. (3) are likely to seriously over- or underestimate the e€ects of decoherence [17].) If the external environmental contribution to R can be suciently minimized, then R represents the internal dissipation of the Josephson junction. Real Josephson junctions always have some quasiparticle ``leakage current,'' which contributes to R, along with the pair supercurrent. An excellent choice for the MQC experiment will be Nb/ Al trilayer junctions (speci®c capacitance  50 fF lm 2 ); because of the relatively large energy gap of Nb, widespread experience with RSFQ circuits which use these junctions, and the possibility to achieve low leakage current. Then each of the paired junctions in the rf SQUID will have area 0.5 lm2 . We are not aware of any Nb/Al junctions this small that have demonstrated low enough leakage current for the MQC experiment, but that should be experimentally feasible. Even the ®rst small (10 lm2 ) Nb/Al Josephson junctions measured at low temperature [18] had R=RN  1000 (RN is the normal state resistance, which is 500 X for our chosen parameters). If this result can be scaled to 20 times smaller area while maintaining the same junction quality then Eq. (3) would predict tU  10 ns at T ˆ 300 mK.

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If the MQC experiment is successful one expects a major e€ort to develop quantum coherent logic gates and eventually a full scale quantum computer based on rf SQUID qubits, using RSFQ circuits to perform the control functions which are provided by laser pulses in other candidate technologies for quantum computation. In fact, it is likely that proponents of other solid state qubits such as [10] will also ®nd it advantageous to use superconducting integrated circuit technology as a cryogenic interface to laboratory instrumentation. This has the great advantage that the superconducting circuitry is amenable to the same integrated circuit manufacturing techniques used to fabricate 10-million transistor semiconductor chips. (Note the serious e€orts toward developing a peta¯ops-scale computer based on RSFQ logic, described in Ref. [19]). In addition, coding information in magnetic ¯ux quanta rather than electric charges as in Ref. [10] bene®cial because it allows double superconducting ground planes to be used to isolate the qubit from distant background modes [20] and to suppress unwanted long-range interactions of the qubits. The disadvantage shared with any macroscopic scheme is that no two qubits will be precisely identical. Before the rf SQUID can be considered a qubit, techniques must be developed to rotate its quantum state within its Hilbert space by external control, such as the p and p=2 pulses for ion-trap qubits. An attractive possibility is to manipulate the SQUID potential function (Eq. (2)) using the applied electromagnetic ®elds of SFQ pulses, generated on chip. For example, Schr odinger equation simulations demonstrate that a chain of 10±20 SFQ pulses inductively coupled to the SQUID junction causes rapid but predictable evolution of the qubit wave function and acts analogous to the p=2 pulse. Entanglement schemes should be straightforward in this technology, using either the magnetic ®elds or the currents of the SQUID qubits. Nobody knows whether quantum coherence can actually exist over useful time intervals in a macroscopic system. But if the standard mathematics of quantum mechanics leading to Eq. (3) applies, one may expect phase coherence times measured in seconds rather than nanoseconds

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using future Josephson junctions with R=RN ˆ 1010 at T ˆ 10 mK. (Ratios greater than 108 have been experimentally achieved [21].) This will facilitate the development of a full scale quantum computer.

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