We report the first observation of resonant tunneling of a system between two macroscopically distinct states: energy levels in different Auxoid wells of a weakly damped superconducting quantum interference device which differ in mean current by approximately 6 p, A. Near 50 mK, the tunneling rate I (4,) from the metastable well vs applied Ilux &9, is found to exhibit a series of local maxima where the levels (spaced by =1.9 K) in the two wells cross. The positions of these maxima agree well with the level crossings calculated using independently determined system parameters. PACS numbers: 74.50.+r, 73.40.Gk Throughout the history of quantum mechanics, there have been a series of paradoxes resulting from attempts to reconcile the description of nature at the microscopic level with everyday observations of macroscopic objects. While many explanations have been proposed, the dilemma remains in the minds of many [1,2]. It is of some interest then to try to observe quantum effects, familiar in the microscopic world, in variables describing macroscopically distinct states. Josephson junctions, along with the superconducting quantum interference device (SQUID), have proven to be excellent systems for such investigations. The IIux 4 linking the SQUID (or the phase difference across a current biased junction) describes the collective motion of a macroscopic number of particles and yet manifests quantum behavior at accessible temperatures. Further, it is possible to characterize these systems very well, permitting quantitative comparison with theory. Tunneling of these macroscopic variables to the continuum (MQT) [3 -7] has been widely studied and generally found to be in excellent agreement with theory. Level quantization within a well has been observed [8], again agreeing with theory [9,10]. However, answers to more fundamental questions (including macroscopic quantum coherence [1]) require measurements of the behavior of systems involving transitions between macroscopically distinct, discrete final and initial states. Extensive theoretical analysis of such transitions has been done using the two state approximation to the SQUID [11,12].These predict that, under conditions of low damping and temperature, 4 should display the quantum interference behavior familiar for microscopic systems. The limited experimental work on such systems has tended to confirm these predictions in the higher damping regime where evidence for discrete final states is indirect [13]. Recently reported results on magnetic systems have been interpreted in terms of macroscopic quantum behavior [14]. However, there is currently substantial controversy as to the interpretation of the results that, it is argued, differ markedly from theory [15]. In this Letter, we report the observation of resonances in the tunneling rate of the fiux between two macroscopically distinct Iluxoid wells of a SQUID when the ground state of the upper well is aligned with an excited state in the lower well. We refer to this phenomenon, in which the quantized final and in...
The flux change ␦ ⌽ through a bistable superconducting quantum interference device has been measured in the presence of thermally induced switching ͑with rate ⌫͒ versus ␦ ⌽ x , the change in the applied flux. For small ␦ ⌽ x , ␦ ⌽ is proportional to ␦ ⌽ x with a measured flux gain g, depending on the temperature, barrier height, and frequency ⍀, with a maximum of about 16. In agreement with theories of periodically driven stochastic bistable systems, g͑⍀͒ is nearly frequency independent up to ⌫ and is proportional to ⍀ Ϫ1 for ⍀ӷ⌫. For larger amplitude signals, harmonic generation has been measured in the adiabatic limit ͑⍀Ӷ⌫͒ and found to be in good agreement with theory. Possible applications of this system for flux measurement are discussed. © 1995 American Institute of Physics.The superconducting quantum interference device ͑SQUID͒, a superconducting loop interrupted by one or two Josephson junctions, has proven to be an extraordinarily sensitive magnetometer ͑see, e.g., Ref. 1 for a review͒. Essentially, the operation of these devices involves switching the loop between fluxoid states using either a rf flux ͑single junction rf SQUID͒ or the Josephson oscillations resulting from a dc voltage across the junctions ͑two junction dc SQUID͒. In general, the noise per unit bandwidth is reduced as this switching frequency is increased. Recent theories of periodically driven stochastic bistable systems [2][3][4] show that, in principle, high sensitivity can also be achieved if the intrinsic thermal fluctuations of the device are the source of this switching. In this letter, we apply these results to a SQUID system and demonstrate that substantial flux gain can be achieved in both passive and active versions of the system.The system being studied here, which we refer to as a stochastic ͑or S-͒ SQUID, uses the rf SQUID configuration, i.e., a superconducting loop of inductance L interrupted by a Josephson junction of critical current I c . 5 The equation of motion for the flux ⌽ through this S-SQUID is homologous to that of a particle of mass C ͑junction's shunt capacitance͒, moving in a potential U͑⌽͒ and having a friction coefficient ␥ϭ1/R, where R is the junction's shunt resistance. This potential, shown in Fig. 1 ͑insert͒, is given by: ͑1͒Here, ⌽ x is the externally applied magnetic flux through the SQUID. ⌽ 0 is the flux quantum, and E J ϭ⌽ 0 I c /2 is the maximum Josephson coupling energy of the junction. In Eq. ͑1͒ the origins of ⌽ and ⌽ x have been chosen at ⌽ 0 /2 so that at ⌽ x ϭ0 the potential is symmetric in ⌽ with two minima at Ϯ⌽ m separated by a barrier ⌬U ͓cf. Fig. 1 ͑insert͔͒. For ⌽ x 0, the symmetry of the potential is broken and an energy difference 2⑀ between the left and right wells appears with ⑀ϭ⌽ m ⌽ x /L for small ⌽ x .Thermally induced switching between the two wells occurs with a rate given ͑for ⌬U/k B Tӷ1͒ by the Kramers relation as
We present evidence for transitions between fluxoid wells of a SQUID due to cascaded, two-photon processes. Such transitions are evidenced by an anomalous dependence on the transition rate from the one-photon resonant level within the initial well, which cannot be explained by previously observed macroscopic resonant tunneling. These two-photon processes may be a significant source of decoherence in SQUID qubits subject to microwave radiation. PACS numbers: 74.50. + r, 85.25.Dq The degree to which macroscopic degrees of freedom (MDFs) obey quantum mechanics is a perennial source of paradoxes and debates [1][2][3][4]. Examples of these MDFs include the center of mass of a baseball, the magnetization vector of a solid, and (the focus here) the phase difference w of the superconducting wave function across a Josephson junction or (equivalently) the magnetic flux F enclosed by a SQUID. w and F typically represent the collective motion of a large number ͑ϳ10 10 ͒ of Cooper pairs, putting them well into the "macroscopic" regime. In recent years, it has been predicted theoretically and verified experimentally that under appropriate conditions, such as low temperature and weak damping, at least some aspects of the behavior of MDFs must be described quantum mechanically [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. However, the existence of a coherent superposition of states of MDFs remains largely untested. Some of the most remarkable macroscopic quantum effects, such as energy level quantization, resonant tunneling, and resonant photon assisted tunneling between macroscopically distinct levels, have been observed in Josephson junctions and SQUIDs [16,[20][21][22]. Results from SQUID experiments have shown that SQUIDs can, in many respects, be custom designed quantum elements, which are promising for applications in fundamental scientific research (e.g., tests of macroscopic quantum coherence and macrorealism) as well as potential technological applications (e.g., qubits for quantum computing). The interaction between SQUIDs and microwave fields in the quantum regime plays an important role in many of these potential applications [23] but has only just begun to be tested. In this Letter, we report the first observation of the effect of twophoton processes on the transitions of a SQUID between its macroscopically distinct fluxoid states.The details of the SQUID system have been reported elsewhere [24]. We summarize its key features here. Figure 1 shows the schematic and equivalent potential of a SQUID biased with an applied flux (in units of the fluxThe two wells of the potential represent the f 0 and 1 fluxoid state of the SQUID, which for the parameters of our experiment, have counter circulating currents with magnitudes greater than 2 mA. The energy levels are calculated, neglecting damping, by numerical solution of Schrödinger's equation. The effect of damping on the energy of a level is of second order [25,26] and should be negligible in the low damping limit appropriate here. For f x 1 2 , the potenti...
Recently, Cosmelli et al. (hereafter the authors) [1] measured the escape rate ͑G͒ vs normalized flux bias ͑X ϵ F x ͞F 0 ͒ of an rf SQUID from a metastable fluxoid state at a temperature below 50 mK. The data were compared to calculations from which an effective damping resistance R 4 MV was extracted. However, in the following discussion we show that, in Ref. [1], (i) the energy level structure used to calculate the escape rate was significantly incorrect, and (ii) treating system temperature as a free fitting parameter could not be justified. Therefore, the value of R inferred from the data is unreliable.In Ref.[1], the measured escape rate G͑X͒ was compared to the solution of the master equation. Using the SQUID parameters given in Ref.[1], we found the number of levels in the well is N Ӎ 11 to 13; that is in a stark contrast to the value of 20 to 30 estimated by the authors. Furthermore, we found the parameter h, which is completely set by Z 0 ϵ p L͞C, b L ϵ 2pLI c ͞F 0 , and X, varies smoothly from 660 to 700 for 20.505 , X , 20.485. Thus, the value h ഠ 900 obtained by the authors is ϳ30% ͑ϳ6s͒ greater than the independently determined value. The calculated barrier height DU, small oscillation frequency v 0 , DU͞hv 0 , and h ϵ 2pd͑DU͞hv 0 ͒͞dX vs X are shown in Fig. 1. Obviously, the result of rate calculation depends crucially on the level structure. Hence, the use of incorrect values of N and h is sufficient to raise question about the validity of the G͑X͒ calculations in Ref. [1].The observed oscillations in G͑X͒ have been attributed to a depletion of the highest active level, denoted as the nth excited level with energy E n , that contributes the most to escape. Roughly speaking, the amplitude of oscillations in G͑X͒ can be taken as a measure of how fast the level n is being repopulated from below. Since the nth level couples most strongly to its nearest neighbors, a good approximation on the rate of repopulating it is given bywhere W n21, n is the transition rate from the ͑n 2 1͒th to the nth level, and v is the level spacing [2,3]. The last equation is valid at T øhv͞k B . The rate is expressed explicitly in R to emphasize that W n21, n depends exponentially on T but only linearly on R. Thus, a very small overestimate of T could result in a huge increase in the extracted value of R. For this reason, in order to obtain R from the fit unambiguously, it is necessary to have T independently verified. It is well known that an effective system temperature significantly higher than the bath temperature T b often indicates serious problems in shielding the sample from FIG. 1. Several key system parameters vs the normalized external flux calculated using SQUID parameters given in Ref. [1]. Note, the number of levels in the well is N ഠ round͑DU͞hv 0 1 0.5͒. extrinsic electromagnetic noise. Therefore, treating T as an adjustable fitting parameter, especially in a range well above T b , requires justification. The fact that the system was observed to follow Kramers' thermal activation behavior down to T b ഠ 1 K ...
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