Macroscopic Quantum Phenomena in Josephson Elements

J. Clarke*, Gif-sur-Yvette and G. Schön, Jülich (CEN, Saclay and Institut für Festkörperforschung, KFA) Josephson's original paper on the quantum effects was initiated by the applied current and the Johnson effect named after him has long been a Leggett 1) who questioned the validity noise current T (t) generated by the re­ classic. Until recently, the experimental of quantum mechanics on a macrosco­ sistance, one finds: manifestations of Josephson tunnelling pic level. A central issue is the fact that have also been classical in the sense as one begins to couple a macroscopic ϬU(φ)/Ϭ(hφ)/2e) = T'(t) (1) that the phase difference across the system to external or unspecified inter­ The first term on the left-hand side ari­ junction could be regarded as a classical nal degrees of freedom, an unavoidable ses from charging of the capacitance, variable. In the past few years, however, dissipation sets in. The major challenge the second from the dissipative current progress in microfabrication and the to the theory has been the need to ac­ that flows through the resistor, and the understanding of the importance of ex­ count for this dissipation within a rigo­ third from the supercurrent Is = Ic sin φ ternal noise have enabled us to build rous quantum mechanical framework and the applied current lx. For the mo­ Josephson circuits that no longer be­ and to determine the influence of dissi­ ment, we have ignored the normal cur­ have classically but instead, although pation on various quantum phenomena rent due to tunnelling of quasiparticles. macroscopic in size, display various 2). One gains considerable insight into the quantum mechanical effects. For exam­ In Fig. 1 we summarize the essential predictions of Eq. (1) by realizing that it is ple, macroscopic quantum tunnelling features of the classical Josephson completely analogous to the equation and quantization of energy have been effect. The single degree of freedom for a particle moving in a tilted wash­ demonstrated experimentally. Other ef­ that characterizes the effect is the phase board potential. There is a one-to-one fects such as the existence of the super­ difference φ across the junction. By correspondence between the capaci­ position of wave functions on a macro­ equating the currents flowing through tance, phase difference, and charge for scopic level and of Bloch oscillations the three components of the resistively the junction and the mass, position, and have been predicted. The study of these shunted junction (Fig. 1b) to the sum of momentum of the particle. The potential describing a current biased junction with lx < Ic is illustrated in Fig. 1c. In the stationary state V=φ= 0, the junction (or particle in the local minimum) may also undergo oscillations with a frequen­ cy ωp = ωpo [1-(Ix/Ic)2]1/4 where the plasma frequency is ωp0 = 2e Ic/hC. The classical description is valid provided kBT >> hωpo', where kB is Boltzmann's constant and T is the absolute tempera­ ture. It is obvious that Eq. (1) contains Planck's constant, h, the presence of which underscores the quantum me­ chanical origin of the Josephson effect. However, each factor of h is divided by the charge of a Cooper pair, 2e, and the quantity h/2e = Φo/2π is not small; Eq. (1) is a classical equation of motion. Provided one can ignore the dissipa­ Fig. 1 — (a) A Josephson tunnel junction consists of two superconducting electrodes tive term, the general quantum mecha­ separated by an insulating barrier. The Cooper pairs in each electrode are described by a nical description is a straightforward ex­ macroscopic wave function of the form Δexp (iΦ). A supercurrent Is, involving the tunneling tension of Eq. (1). The charge Q on the of Cooper pairs through the barrier, flows if there exists a phase difference φ = φR- φL capacitor and (periodic functions of) the across the barrier according to the relation ls = lc sin φ; lc is the critical current of the junc­ phase difference hφ/2e are treated as tion, the maximum supercurrent it can sustain. In the presence of a voltage V across the bar­ non-commuting variables, so that rier, the phase difference evolves asφ= 2eV/h. The self-capacitance C of the junction allows Q = (h/i) ∂/∂/hφ/2e) (2) a charge Q to accumulate. In (b), an external shunt resistor R has been added to the junction. If we impose a current lx, The junction is then described by the Ha­ current conservation leads to Eq. (1), where the potential is U(φ) = - (hlc/2 e)cos tp - miltonian lxhφ/2e. This potential is plotted in (c) for Ix < Ic. The equation of motion, Eq. (1), is H0 = Q2/2C + U(φ) (3) analogous to that for the motion of a particle in this tilted washboard potential. In (d), we This description can be completely justi­ show a superconducting loop interrupted by a Josephson tunnel junction. The potential is fied on the basis of microscopic theory U(φ) = - (hlc/2e)cosφ - (Φ- ΦX) 2/2L, where Φ is the total magnetic flux threading the loop and ΦX is the externally applied flux. Flux quantization leads to the constraint φ/2π = * on leave from the Univ. of California and the 0/0o, where Φ0 = h/2e is the flux quantum. Lawrence Berkeley, Laboratory. 94 down the tilted washbord. In the WKB gy, producing distinct resonances, approximation, the tunnelling rate at thereby enabling one to evade the cor­ T = 0 is respondence limit. If the potential were Γ q ~ aq(wp/2π) exp(-7.2 ΔU/hωp). harmonic, all the energy level splittings Here, aq is a known prefactor and AU is would be identical; one would observe the height of the barrier, which like ωp is only a single resonance, and one would controlled by the bias current lx. At a not be able to distinguish between clas­ non-zero temperature, the junction may sical and quantum behaviour. also escape by thermal activation over Several other macroscopic quantum the barrier, a process which, in the classi­ effects have been suggested. One is the cal limit and for moderate damping, existence of the superposition principle occurs at a rate on a macroscopic level in a system con­ Γt = (ωp/2π) exp(-ΔU/kBT). sisting of a superconducting loop inter­ 1 Thus to observe quantum tunnelling we rupted by a single Josephson junction require hωp > 7.2 kBT. For the junction 1). For an applied flux of Φo/2, the two Fig. 2 — Observation of MQT. To observe parameters C = 5 pF and lc = 10 µA, absolute minima occur at the same MQT, the Josephson tunnel junction is cool­ this constraint implies T < 6 0 mK; this potential energy, as indicated in Fig. 4. ed to millikelvin temperatures in a dilution temperature range is readily accessible in The degeneracy of their ground states is refrigerator. Extensive filtering against room a dilution refrigerator. lifted by the tunnelling between the temperature noise is essential. The junction wells. The two lowest eigenstates, one is current-biased in the zero voltage state It should be emphasized that in the with an appropriate tilt of the washboard in process of macroscopic quantum tun­ symmetric the other antisymmetric, are Fig. 1c so that the lifetime against tunnelling nelling the state of the entire junction — separated in energy by an amount δ pro­ into the free running state has a reasonable characterized by φ — changes spon­ portional to the tunnelling rate. If we value, say 1µs to 10 ms. The elapsed time taneously as it makes a transition prepare the system in the left-hand between the application of the current bias through a region of φ-space that is clas­ minimum, that is, in a superposition of and the appearance of a voltage across the sically inacessible. This effect has been two eigenstates, it should perform junction is measured. This process is repea­ observed experimentally by several coherent oscillations between the two ted a large number of times (104 to 105) to groups 4·5). A description of one such wells at a frequency δ/h. Unfortunately, obtain a distribution of lifetimes from which experiment in which the effects of dam­ the range of parameters for which the a mean value for the tunnelling rate is effect is observable is more restricted calculated. In one of the experiments 5), the ping were negligible appears in Fig. 2. tunnelling rate was expressed as an escape The results are in quantitative agree­ temperature, Tesc, defined as F = ωp/2π) ment with theoretical predictions. exp(-ΔU/kBTesc) in both the classical and Another quantum mechanical effect, quantum regimes. In the classical limit, Tesc namely the quantization of the energy in is equal to the bath temperature T, while in the quantum limit Tesc is a constant, in­ a bound state, has also been observed in dependent of temperature (see the expres­ a Josephson junction. Given the Hamil­ sions forTq and Tt). The measured values of tonian of Eq. (3), we naturally expect the Tesc are plotted vs. T; the solid line is Tesc = eigenstates of a particle confined to a T. As the temperature is lowered from 1 K, well to have discrete energies. Strictly the measured values of Tesc are initially speaking, the potential U(φ) shown in equal to T, eventually flattening off to a con­ Fig. 1c is not bounded below, and the stant value when MQT takes over from ther­ eigenstates are not well defined. How­ Fig.3 — Observation of quantized energy mal activation. Using junction parameters ever, provided the period of oscillation in levels 6). The energy levels in a particular determined in situ in the classical regime, well of the tilted washboard potential are the authors found excellent agreement bet­ one of the minima, 2π/ωp, is short com­ shown in the inset. Note that the spacings of ween the measured low temperature value pared with the lifetime 1/Γ against the levels decrease with increasing energy. of Tesc and the T = 0 prediction indicated to macroscopic quantum tunnelling out of At an appropriate temperature there is a the left of the ordinate axis. the metastable state, one would expect significant thermal population of the lower- to observe quasi-eigenstates. In other lying excited states. The escape rate Γ(O) 3). From this treatment, hφl2e and Q words, the wave packet centred at the from the zero voltage state is first measured emerge naturally as quantum mechani­ minimum has only a tiny penetration as described in Fig. 2. One then applies 2 cal variables, provided one is justified in under the barrier, and one neglects this GHz microwaves to the junction and remeasures the escape rate Γ(P), where P is ignoring the dynamics of all the other leakage in the calculation of the energy the microwave power, as a function of the degrees of freedom, for example, the levels. If the junction, prepared in a bias current. As the current is decreased, magnitude of the order parameter. metastable state, is exposed to micro- the spacings of the energy levels increase. Given the Hamiltonian of Eq. (3), we wave radiation, one observes an enhan­ Whenever a particular spacing corresponds expect to observe all the quantum ef­ cement in the escape rate from the zero to the microwave frequency, the escape rate fects predicted by elementary quantum voltage whenever the energy difference is resonantly enhanced. For example, the mechanics. For example, the junction between two eigenstates coincides peak in the escape rate at the right-hand side prepared in a metastable minimum of with the microwave frequency. Fig. 3 of the figure results from the microwave ex­ the potential (Fig. 1c) can escape by a describes the experiment 6) and shows citation of the particle from the ground state process that has become known as one of the results; the agreement bet­ to the first excited state, in which its lifetime is substantially reduced. The second and macroscopic quantum tunnelling (MQT) ween the observed and predicted reso­ third peaks correspond to transitions from into the non-zero voltage state: In the nance is very good. It should be empha­ the first excited state to the second and from parlance of the mechanical analogue, sized that the anharmonicity of the po­ the second to the third, respectively. The po­ the particle has tunnelled through the tential well causes the energy level split­ sitions of the resonances are in excellent local potential maximum and runs freely tings to decrease with increasing ener­ agreement with theoretical predïctions. 95 than for the examples described above charge transfer process leads to shot 7). In particular, the effect of dissipation noise rather than to Gaussian noise. (see below) is much stronger and may In many respects, the effect of dissi­ even change the picture qualitatively 8). pation is to make the system behave in a Furthermore, one must take care to more classical way, even though it is ful­ minimize the perturbation due to the ly quantum mechanical. For example, measurement process 7). dissipation reduces the spread of the A second predicted macroscopic wave function describing a particle in a quantum effect is concerned with Bloch well. As a result, the MQT rate is lowered oscillations 9) in junctions where the by a factor that depends exponentially charging energy of a Cooper pair on the strength of the dissipation. For (2e)2/2C is larger than or comparable to weak dissipation, the coherent oscilla­ Fig. 4 — Potential of a superconducting the Josephson coupling energy, which, loop containing a Josephson tunnel junction tions in the double well potential of Fig. 4 with an external flux Φo/2. The two lowest in turn, must exceed kBT. Thus for T = 1 decay exponentially with time, so that eigenstates are split in energy by δ. K, one requires C < 2e2/kBJ = 3fF. For the system ends up in one or other of the zero bias current, the Josephson po­ minima with equal probability 12). At the external perturbations. Nonetheless, it is tential is periodic in the phase difference, same time, the splitting of the energy to be hoped that some of these effects, and the wave function is thus also levels in Fig. 4 is reduced, as are the at least, will be observed in the course of periodic, apart from a phase factor, widths of the energy bands in the perio­ the next few years. Ψn,Qx(Φ + 2π) = exp(2πiQx/2e) ψnQx(Φ)· dic potential. For strong dissipation, a Physically, Qx represents an external phase transition 8, 10, 13) is predicted to charge introduced onto the electrodes occur in a double well or periodic poten­ REREFENCES through (weakly coupled) external leads tial. Above a critical strength of the 1. Leggett A.J., Progr. Theor. Phys. 69 (1980) 10). From our experience with periodic, dissipation, the system becomes localiz­ 80. potentials and Bloch states in solid state 2. Caldeira A.O. and Leggett A.J., Ann. Phys. ed: if prepared in one of the potential (N.Y.) 149 (1983) 347. physics, we are not surprised to learn wells, the particle will remain there, and 3. Eckern U., Schön G. and Ambegaokar V., that coupling of the potential wells turns there will be no coherent oscillations. In Phys. Rev. B 30 (1985) 6419; each discrete energy level into an energy the case of very strong dissipation, it is Larkin A.I. and Ovchinnikov Yu.N., Phys. Rev. B band. By increasing Qx by means of an often sufficient to use a quantum Lange- 28 (1983) 6281; for a review see also: Schön external current we can sweep through vin equation in which the particle is des­ G. in SQUID '85, eds. H.-D. Hahlbohm, H. Lüb- the energy bands, causing the energy cribed classically while the noise driving big (de Gruyter, Berlin) 1985. and hence the voltage across the junc­ term has a quantum spectral density. 4. Voss R.F. and Webb R.A., Phys. Rev. Lett. 47 tion to oscillate at a frequency f = 1/2e. Both Gaussian 14) and shot noise 3) have (1981) 647. This result is potentially of practical im­ been analyzed in this approach. 5. Devoret M.H., Martinis J.M. and Clarke J., portance because of the link it provides Phys. Rev. Lett. 55 (1985) 1908. Of the various predicted effects of dis­ 6. Martinis J.M., Devoret M.H. and Clarke J., between the Josephson voltage-fre­ sipation, the reduction in the rate of Phys. Rev. Lett. 55 (1985) 1543. quency relation 2eV = hf and the quan­ MQT due to an ohmic resistor or single 7. See, e.g. de Bruyn R. Ouboter, Physica 126 tum Hall relation V = (h/ne2)l. electron tunnelling has been observed B+C, (1984) 423. Finally, we briefly discuss the effects experimentally 15). The results are in 8. Chakravarty S, Phys. Rev. Lett. 49 (1982) of dissipation on the quantum pheno­ qualitative agreement with theory. For 681; Bray A.J. and Moore M.A., Phys. Rev. mena described above. The pioneering example, the factor representing dissi­ Lett. 49, (1982) 1545; Hakim V., Guinea F. and paper on this subject is that of Caldeira pation in the exponent of the MQT rate Muramatsu A., Phys. Rev. B 30 (1984) 464. and Leggett 2), who considered a model 9. Widom A., Megaloudis G. Clark T.D., has been shown to scale as T2, in accor­ Prance H. and Prance R.J., J. Phys. A 15, in which the single degree of freedom φ dance with predictions 11). Measure­ (1982) 3877; Likharev K.K. and Zorin A.B., describing the junction is coupled linear­ ments have also been made of the noise Low J., Temp. Phys. 59 (1985) 347; Prance ly to a bath of harmonic oscillators. If the generated in an overdamped Josephson R.J., Clarke T.D., Mutton J.E., Prance H., Spiller distribution of amplitudes and frequen­ junction current-biased at a voltage V so T.P. and Nest R., Phys. Rev. Lett. 107A (1985) cies of these oscillators and the coupling that hωj >> kBT, where ωj = 2eV/h is 233. strengths to them are chosen suitably, in the Josephson frequency 15). The re­ 10. Guinea F. and Schön G., Europhys. Lett. 1 the classical limit the model describes a sults were in good agreement with pre­ (1986) 585. system in which the damping is propor­ dictions of the quantum Langevin equa­ 11. Larkin A.I. and Ovchinnikov Yu.N., Zh. Eksp. tional to the velocity φ. Thus, the dissi­ Theor. Fiz. 86 (1984) 719 [Sov. Phys. JETP 59 tion. 420); for a review see: H. Grabert in SQUID pation is linear in the voltage across the At this stage, most of the theoretical '85 (cf. Ref. 3). junction, that is, it is ohmic, and the cur­ groundwork has been laid, and the tech­ 12. Chakravarty S and Leggett A.J., Phys. Rev. rent noise generated by the resistance niques for describing dissipation are well Lett. 52 (1984) 5; Grabert H. and Weiss U., has a Gaussian distribution. understood. Experimentally, the exis­ Phys. Rev. Lett. 54 (1985) 1605; F. Guinea, An alternative approach to the pro­ tence of MQT and of quantized energy Phys. Rev. B 32 (1985) 4486; Leggett A.J., blem of dissipation starts with the mi­ levels has been demonstrated unequivo­ Chakravarty S., Dorsey A.T., Fisher M.P.A., croscopic tunnelling Hamiltonian for a cally, while the effects of dissipation on Garg A. and Zwerger W., submitted to Review the MQT rate have been shown to be of Modern Physics. Josephson junction 3). In the absence of 14. Koch R.H., van Harlingen D.J. and Clarke dissipation, this model reduces to the qualitatively in accord with theory. The J., Phys. Rev. B. 26 (1982) 74; A. Schmid, J. Hamiltonian of Eq. (3). In general, how­ other effects described above await ex­ Low Temp. Phys. 49 (1982) 609. ever, the model contains a dissipative perimental demonstration. The experi­ 15. Schwartz D.B., Sen B., Archie C.N. and element due to the tunnelling of single ments are difficult, usually demanding Lukens J.E., Phys. Rev. Lett. 55 (1985) 1547; electrons. This element is non-linear very low temperatures, very small devi­ Washburn S., Webb R.A., Voss R.F. and Faris rather than ohmic, and the discrete ces, and extraordinary screening against S. Phys. Rev. Lett. 54 (1985) 2712. 96