When RF amplitude is zero, the system reverts to the DC effect. This is most readily accomplished by irradiating the junction with low-intensity microwave radiation in the range of 10—20 GHz. The junction current then given by. As the voltage is swept, those points at which the radiation frequency obeys the resonance condition exhibit a series of stepwise increases in current, equal to. These steps were first observed by S. Shapiro in and can be considered a definitive validation of the AC Josephson effect.

The width of each voltage step is very precisely equal to. All plots are shown without numerical axes labels, since these magnitudes depend on the specific individual characteristics of instruments and materials. Thus, a qualitative descriptions of these phenomena suffices. Josephson junctions have important applications in SQUIDs superconducting quantum interference devices , which can be very sensitive magnetometers, and for RSFQ rapid single flux quantum digital electronic devices.

They have been suggested in several proposed designs for quantum computers. The Josephson effect provides a highly accurate frequency to voltage conversion, as expressed by the Josephson constant,. Since frequency can be very precisely defined by the cesium atomic clock, the Josephson effect is now used as the basis of a practical high-precision definition of the volt.

Snapshot 3: oscilloscope trace for typical AC Josephson effect, showing Shapiro steps.

## Synopsis: Putting a Spin on the Josephson Effect

Feynman, R. Leighton, and M. Download free CDF Player. The preceding can be solved for the two fundamental equations for the Josephson effect: and , where is the critical current, above which superconductivity is lost. The junction current then given by , for , , again neglecting sinusoidal voltages of unobservably high frequencies. Snapshot 1: DC Josephson effect, with zero external voltage Snapshot 2: no tunneling when barrier is sufficiently opaque Snapshot 3: oscilloscope trace for typical AC Josephson effect, showing Shapiro steps References [1] R.

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Share Demonstration. Embed Code Copy. Blinder Aharonov-Bohm Effect S. Blinder Einstein-de Haas Effect S. Ice Cube Melting in Water S. Blinder Absorption Spectroscopy S. Blinder Multipurpose Tool S. By now, good agreement between experiment and theory has been established. First, we shall discuss the cotunneling regime in detail, where one can employ perturbation theory in the dot—lead couplings. Second, in order to go beyond the cotunneling regime, we have performed a mean-field analysis similar in spirit to earlier work for S—QD—S devices [57,58].

The QD is modeled as an Anderson impurity [36] , i. The spinful nanowire proposal for TS wires [2,3] also requires a sufficiently strong bulk Zeeman field oriented along the wire in order to realize the topologically nontrivial phase, but for concreteness, we here imagine the field B as independent local field coupled only to the QD spin. One could use, e. We show below that unless B is inadvertently aligned with the MBS spin polarization axis, spin mixing will indeed generate a supercurrent.

These tunnel amplitudes contain density-of-states factors for the respective leads. The operator expression for the current flowing through the system is then given by. Later on we will address how our results change when the TS wire is modeled as spinful nanowire [2,3] , where the corresponding bGF has been specified in [43].

In that case, low-energy states with energy well below U are restricted to the single occupancy sector,. Moreover, is the anticommutator of the composite boundary fields. The leading contribution is then of second order in H int ,. We mention in passing that previous work has also pointed out that experiments employing QDs between N instead of S leads and TS wires can probe nonlocal effects due to MBSs [12,16,]. In our case, e. For the corresponding S—QD—S case, see [58,79].

We note that a full solution of this interacting many-body problem requires a detailed numerical analysis using, e. We start by defining the GF of the QD,. Note that this notation introduces double counting, which implies that only half of the levels are physically independent. Of course, the results below take this issue into account. In addition, we get finite-energy subgap poles for. From blue to yellow, the color code indicates increasing values of the spectral density. We next turn to self-consistent mean-field results for the phase-dependent ABS spectrum at finite U. For moderate interaction strength, e.

On the other hand, for stronger interactions, e. We notice, however, that changing the sign of B x would result in zero junction behavior. In fact, we will first describe the Josephson current for the elementary case of an S—TS junction using the spinful nanowire model. Surprisingly, to the best of our knowledge, this case has not yet been addressed in the literature.

In spatially discretized form, the spinful nanowire model for TS wires reads [2,3,43]. Under the condition. As we discuss below, the physics of the S—QD—TS junction sensitively depends on both the bulk Zeeman field V x and on the local magnetic field B acting on the QD, where one can either identify both magnetic fields or treat B as independent field.

Below we study these expressions in the zero-temperature limit. The Josephson current can be computed using the bGF expression for tunnel junctions in [40] , which is a simplified version of the above expressions for the S—QD—TS case. Close to the topological transition, the transparency is well approximated by. First, we observe that I c is strongly suppressed in the topological phase in comparison to the topologically trivial phase.

In fact, I c slowly decreases as one moves into the deep topological phase by increasing V x.

## Josephson effect in junctions of conventional and topological superconductors

However, a residual finite supercurrent can be observed even for rather large values of V x. This observation suggests that continuum contributions in this setup mainly originate from s -wave pairing correlations which are not particularly sensitive to the topological transition.

All other parameters are specified in the main text. In the topologically trivial phase, small deviations from the sinusoidal law can be detected, but once one enters the topological phase, these deviations become extremely small. Apart from providing a direct link to experimental control parameters, another advantage of using the spinful nanowire model of [2,3] for modeling the TS wire is that the angle between the local Zeeman field B and the MBS spin polarization does not have to be introduced as phenomenological parameter but instead results from the calculation [43].

Below we present self-consistent mean-field results obtained from this scheme. In view of the huge parameter space of this problem, we here only discuss a few key observations. A full discussion of the phase diagram and the corresponding physics will be given elsewhere.

This behavior is very similar to what happens in S—TS junctions with large transparency , cf. In what follows, we study the superficially similar yet rather different case of an S—TS—S junction. Over the past few years, impressive experimental progress [] has demonstrated that the ABS level system in a SAC [81] can be accurately probed and manipulated by coherent or incoherent microwave spectroscopy techniques.

We show below that an additional TS wire, cf. As we have discussed above, the supercurrent flowing directly between a given S lead and the TS wire is expected to be strongly suppressed. However, through the hybridization with the MBS, Andreev level configurations with even and odd fermion parity are connected.

This effect has profound and potentially useful consequences for Andreev spectroscopy. The p - and s -wave pairing correlations in the spinful TS wire are thereby spatially separated. Since the s - and p -wave bands represent normal modes, they are not directly coupled to each other in this scenario, i. We discuss this analogy in more detail later on. In practice, the SAC can be embedded into a superconducting ring for magnetic flux tuning of. However, we will also hint at how one can treat the general case.

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The dimensionless parameters b 1,2 describe the Zeeman field component along the MBS spin polarization axis, see below. After some algebra, we obtain the effective action. Moreover, we have defined an effective GF in Nambu space with frequency components. As before, the Josephson current through S lead no. In this case, we can approximate. After the rescaling. Moreover, we define the auxiliary parameters. We emphasize that H at neglects TS continuum quasiparticles as well as all types of quasiparticle poisoning processes.

### References

Let us briefly pause in order to make two remarks. First, we note that the Majorana field. The corresponding Hilbert subspace is spanned by four states,. Such a subgap state may be represented, e.

However, the MBS will automatically be located at zero energy and thus represents a generic situation. We close this paper by summarizing our main findings. We have studied the Josephson effect in different setups involving both conventional s -wave BCS superconductors S leads and topologically nontrivial 1D p -wave superconductors TS leads with Majorana end states. The TS wires have been described either by a spinless theory applicable in the deep topological regime, which has the advantage of allowing for analytical progress but makes it difficult to establish contact to experimental control parameters, or by a spinful nanowire model as suggested in [2,3].

For S—TS tunnel junctions, we find that in the topological phase of the TS wire, the supercurrent is mainly carried by above-gap continuum contributions. We confirm the expected supercurrent blockade [31] in the deep topological regime where the spinless theory is fully valid and thus no residual s -wave pairing exists , while for realistic parameters, a small but finite critical current is found. The dependence of the critical current on the bulk Zeeman field driving the TS wire through the topological phase transition shows a kink-like feature at the critical value, which is caused by a sudden drop of the Andreev state contribution.

The supercurrent blockade in the deep topological phase could be lifted by adding a magnetic impurity to the junction, also allowing for the presence of a local magnetic field B. Such a magnetic impurity arises from a spin-degenerate quantum dot QD , and we have studied the corresponding S—QD—TS problem for both the spinless and the spinful TS wire model.

As a final example for devices combining conventional and topological superconductors, we have shown that S—TS—S devices allow for a Majorana-induced parity switch between Andreev state sectors with different parity in a superconducting atomic contact. This observation could be useful for future microwave spectroscopy experiments of Andreev qubits in such contacts. We acknowledge funding by the Deutsche Forschungsgemeinschaft Grant No. Twitter: BeilsteinInst. Beilstein J. Toggle navigation. Please enable Javascript and Cookies to allow this site to work correctly!

Alex Zazunov. Albert Iks. Miguel Alvarado. Alfredo Levy Yeyati. Reinhold Egger. Jump to Figure 1. Results and Discussion. Cotunneling regime We now address the CPR in the elastic cotunneling regime, 15 where perturbation theory in H int is justified. We start by defining the GF of the QD, 24 Note that this notation introduces double counting, which implies that only half of the levels are physically independent.

Jump to Figure 2. Jump to Figure 3.