Could two degenerate energy states be observed for a superconducting ring at $\Phi_{0}$/2?

A.A. Burlakov; A.V. Nikulov; S.V. Dubonos; V.A. Tulin; V.L. Gurtovoi

REVIEW

Could two degenerate energy states be observed for a superconducting ring at $\Phi_{0}$/2?

A.A. Burlakov, V.L. Gurtovoi, S.V. Dubonos, A.V. Nikulov, V.A. Tulin

Superconductivity · Submitted 1 Mar 2006 · 23 references

Pith's verdict: A worthwhile experimental report of a genuine anomaly, but the 'violation of macroscopic realism' framing rests on an unexamined assumption that phase slips do not shift the critical-current oscillations.

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CitationA.A. Burlakov, A.V. Nikulov, S.V. Dubonos, V.A. Tulin, V.L. Gurtovoi. "Could two degenerate energy states be observed for a superconducting ring at $\Phi_{0}$/2?." arXiv:cond-mat/0603005 (2006). https://pith.science/paper/NJFWKSMW

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Editor's note

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The thing to know about this paper is that the headline result is not the two-state interpretation but the unresolved critical-current anomaly. On asymmetric Al rings the authors find that the critical current measured in the two directions has minima shifted by half a flux quantum relative to each other, while the Little-Parks resistance and rectified-voltage oscillations on the same rings look like the usual ones. They read this as a contradiction that forces a choice between incorrect measurements and a violation of macroscopic realism. On the evidence in front of them, 'incorrect measurements' remains the more likely option, and their own last sentence admits it.

What is genuinely new here is the critical-current data: Ic+ and Ic− on the same asymmetric rings, measured over many periods, with the same half-period shift for different widths and temperatures. Putting R, Vdc, and Ic on the same ring is also a useful step. The resistance and rectified voltage data alone would be a re-application of standard flux quantization, but the combination is worth having. The authors cite their earlier rectification work and the relevant Leggett-Garg literature; there is no citation problem.

The soft spot is the load-bearing assumption in the critical-current model: that the winding number n stays fixed during the current ramp until the normal-state jump, and that Ic± is set by the pair velocity reaching vc in one half-ring. The I-V curves are strongly irreversible near Tc, which is exactly the regime where thermally activated phase slips can change n during the ramp, and in an asymmetric ring their rate can depend on current direction. The authors' argument that many pairs remain up to vc only says the order parameter is not zero; it does not rule out phase slips. Without sweep-rate or sweep-direction tests, the half-period shift may be a measurement artifact. The same shift across widths and temperatures is suggestive but not decisive, because a geometry-dependent phase-slip mechanism could be equally universal.

I would also be more careful about the strength of the two-state claim. Vdc=0 at half-integer flux plus R maximum is consistent with two degenerate states, but so is a classical mixture of the two persistent-current directions; it does not demonstrate macroscopic quantum superposition. The paper's title says 'two degenerate energy states', which is fair, but the concluding reference to realism enters only through the contradiction, so the weight lands on that contradiction.

Who is this for? Experimentalists working on Little-Parks oscillations and persistent currents in rings; people interested in Leggett-Garg tests will find it a cautionary tale rather than evidence. It deserves a serious referee, but I would expect heavy revision: the authors need to address the phase-slip artifact path explicitly, show sweep-rate controls, and soften the either/or conclusion unless they can rule out the artifact. Recommend: send to peer review rather than desk reject.

Referee report

Significance

If the two-state interpretation is upheld, the resistance and rectified-voltage results would provide a clean confirmation of the standard flux-quantization picture at half-integer flux, and the half-period shift in the critical current would be a genuinely puzzling anomaly. The manuscript has several strengths: measurements were made on multiple rings with different asymmetric widths, the shift was tracked over many periods with a stated precision of +/-0.02Phi0, and the comparison with symmetric rings is a useful check. However, the central claim is not uniquely determined by the data, and the contradiction embedded in the critical-current measurements is left unresolved. In its present form the paper does not establish that the data require two degenerate states, nor does it establish a failure of macroscopic realism; the value of the work lies mainly in posing a sharp experimental puzzle that needs additional controls.

Major objections (3)

[Paragraph beginning 'The measurements [13] and of our numerous measurements...' and Fig. 4] The inference that Vdc=0 and Delta_R maximum at Phi=(n+0.5)Phi0 prove the existence of two degenerate states is underdetermined. A zero rectified voltage at half-integer flux is consistent with any model in which the current-voltage characteristic is symmetric at that flux, including equal thermal occupation of two states as one possibility but also other symmetry mechanisms. The manuscript does not provide an independent control, such as a fixed-flux measurement of Vdc as a function of ac amplitude, or measurements that distinguish statistical mixture from coherent superposition. At most, the data are consistent with the standard quantization condition; they do not uniquely establish the two-state coexistence claim.
[Paragraphs 'The expected magnetic dependencies of the critical current...' and 'The observations of the minimums...'…] The claimed contradiction with the Little-Parks and rectified-voltage data relies entirely on the expectation that Ic+ and Ic- minima should lie at Phi=(n+0.5)Phi0. That expectation assumes that the quantum number n is fixed during the current ramp up to the normal-state jump and that the critical current is set when the pair velocity in one half-ring reaches vc. The manuscript's argument that the number of pairs remains large up to vc does not exclude thermally activated phase slips, which can change n by one while the order parameter remains finite. In an asymmetric ring the phase-slip rate can be current-direction dependent, producing apparent minima displaced by half a period without any violation of realism. Since no sweep-rate, sweep-direction, or temperature-rate controls are reported, the measured Delta_phi=0.5 cannot yet be used either to support or to refute the two-state interpretation.
[Concluding paragraph] The concluding statement that the results are 'either incorrect or experimental evidence of violation of the principle of realism' is a false dichotomy. The unresolved possibility of a phase-slip artifact, an incomplete model of the critical-current condition, flux redistribution during measurement, or current-direction-dependent dissipation provides a third class of explanations. A revised manuscript should either rule out these alternatives with additional measurements or explicitly present the discrepancy as unresolved rather than as evidence for or against macroscopic realism.

Minor objections (5)

[Title and abstract] There is a typographical error in the displayed title: 'superco nducting' should read 'superconducting'.
[Section 'Measurements of the critical current...' and Fig. 1] The notation is confusing because sn and sw are used for cross-section areas while wn and ww are used for semi-ring widths; these symbols should be defined explicitly and used consistently throughout.
[Fig. 3] The theoretical branches for Ic+ and Ic- are not individually labelled in Fig. 3; please add labels or a legend indicating which curve corresponds to which current direction and state the values of Ic and Ip used for the calculation.
[Text near 'The additional flux Phi_I =...'] The inductance is given as 'L approximately 2 10^-11 G'; the unit should be H (henry), not G (gauss).
[Figs. 4-6] The figures do not show experimental error bars; the manuscript should state the uncertainties in the resistance, rectified voltage, and critical-current data, and display them where they are larger than the symbol size.

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The paper

The paper claims

The paper argues that Little-Parks resistance oscillations and rectified-voltage oscillations on asymmetric superconducting aluminum rings give evidence of two degenerate energy states at Φ0/2, while also reporting a half-period shift in…

The reading

The paper claims that Little-Parks resistance oscillations and quantum oscillations of the rectified voltage, measured on the same asymmetric superconducting aluminum rings, can only be explained if at magnetic flux Φ0/2 the ring has two degenerate energy states with equal probability—one carrying a persistent current in each direction. On average the persistent current vanishes, but its square stays at a maximum, which is why the resistance peaks while the rectified voltage crosses zero. The same samples, however, show critical-current oscillations in opposite directions shifted by half a period relative to each other, contradicting the expected relation between critical-current minima and resistance maxima. The authors leave this contradiction open, stating that the results are either incorrect or evidence of a violation of macroscopic realism at the mesoscopic level.

What carries the argument

The argument rests on the quantization of the superconducting pair velocity circulation, ∮ dl v = (2πℏ/m)(n − Φ/Φ0), which for a ring of unequal half-widths produces a persistent current Ip ∝ n−Φ/Φ0 and an anisotropy of the critical current Ic,an = Ic+−Ic− ∝ Ip. The two-state interpretation at Φ0/2 follows from comparing two observables on the same ring: resistance ΔR ∝ v² (same for n and n+1) and rectified voltage Vdc ∝ ⟨v⟩ (which averages to zero over the two states). The critical-current shift is characterized by a single parameter Δφ, the relative shift of the Ic+ and Ic− oscillations.

Extended reading notes

Core claim

The central claim is that a superconducting ring pierced by half a flux quantum, Φ=(n+0.5)Φ0, has two permitted states with the same minimum energy, corresponding to quantum numbers n and n+1, i.e., persistent current Ip ∝ n−Φ/Φ0 = ±1/2. The observed simultaneous maximum of resistance (∝ v²) and zero of rectified voltage (∝ average v) on the same asymmetric Al ring constitute evidence for these two degenerate states, because v² = ((1/2)²+(−1/2)²)/2 = 1/4 is maximal while average v = 0. The paper further reports that the critical current measured in opposite directions, Ic+ and Ic−, is shifted by Δφ = 0.5 relative to each other on asymmetric rings, with minima at Φ = (n+0.5±0.25)Φ0, contradicting the expectation that both minima should occur at Φ = (n+0.5)Φ0; this contradiction is presented as unresolved.

Load-bearing premise

The paper assumes that the critical current in each direction is set by the condition that the superconducting pair velocity reaches the same critical value vc in the narrow or wide half-ring, so that both Ic+ and Ic− should have minima at half-integer flux; if this depairing condition is wrong, the observed half-period shift has a mundane explanation and the claimed contradiction evaporates.

Editorial extensions

If this is right

If the two degenerate states exist, a superconducting ring at Φ0/2 is a mesoscopic system poised between two macroscopic persistent-current states, making it a candidate for studying quantum superpositions on a scale visible to classical probes.
The measured rectified-voltage oscillation Vdc(Φ/Φ0) is confirmed to be a consequence of the periodic asymmetry of the current-voltage curves, directly linking it to the persistent current.
The observed universal shift Δφ = 0.5 for all asymmetric rings examined (width ratios from 1.25 to 2) means that existing depairing models based solely on velocity quantization do not describe the critical current in these rings.
If the contradiction is resolved in favor of the critical-current data, the Little-Parks-based degeneracy evidence would have to be reinterpreted; if resolved in favor of the Little-Parks data, some aspect of the critical-current measurement violates simple symmetry expectations.
The result, if correct, implies that the principle of macroscopic realism—that a macroscopic system always occupies one definite state—could fail for superconducting rings at half-integer flux.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A time-resolved measurement of the persistent-current sign in a single ring at Φ0/2, for example with a nano-SQUID or a weak-link interferometer, could distinguish genuine two-state behavior from a static configuration with zero net current; the paper does not report such a measurement.
The half-period shift might be an artifact of the critical-current measurement procedure itself, such as asymmetric Joule heating or flux redistribution during the current sweep, which would dissolve the contradiction without invoking macroscopic realism.
The fact that near-symmetric rings show small shifts (0.05–0.07) suggests a continuous dependence of Δφ on width ratio; mapping Δφ(ww/wn) across the range 1 to 1.25 would test whether the half-period shift is intrinsic to the asymmetry.
If the two-state picture is correct, the ring at Φ0/2 should also exhibit switching or telegraph-like fluctuations in its persistent current as the system moves between the n and n+1 states; looking for such fluctuations in the voltage or magnetic susceptibility is a testable extension.

Original abstract

The Little-Parks oscillations of the resistance and the quantum oscillations of the rectified voltage observed for asymmetric superconducting Al rings give experimental evidence of two degenerate energy states at the magnetic flux $\Phi_{0}$/2. The quantum oscillations of the critical current as a function of magnetic field have also been measured. On the one hand, these oscillations confirm that the quantum oscillations of the rectified voltage are a consequence of periodical dependence of the asymmetry of the current-voltage curves and, on the other hand, comparison of the oscillations with Little-Parks measurements results in contradiction.

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The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Little-Parks oscillations of the resistance and the quantum oscillations of the rectified voltage observed for asymmetric superconducting Al rings give experimental evidence of two degenerate energy states at the magnetic flux $\Phi_{0}$/2. The quantum oscillations of the critical current as a function of magnetic field have also been measured. On the one hand, these oscillations confirm that the quantum oscillations of the rectified voltage are a consequence of periodical dependence of the asymmetry of the current-voltage curves and, on the other hand, comparison of the oscillations with Little-Parks measurements results in contradiction.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Little-Parks oscillations of the resistance and the quantum oscillations of the rectified voltage observed for asymmetric superconducting Al rings give experimental evidence of two degenerate energy states at the magnetic flux $\Phi_{0}$/2.	result 0.96	0.83	0.92	asserted 0.99
s1	The quantum oscillations of the critical current as a function of magnetic field have also been measured.	method 0.58	0.88	0.03	asserted 1.00
s2	On the one hand, these oscillations confirm that the quantum oscillations of the rectified voltage are a consequence of periodical dependence of the asymmetry of the current-voltage curves and, on the other hand, comparison of the oscillations with Little-Parks measurements results in contradiction.	interpretation 0.87	0.88	0.09	asserted 1.00

What kind of paper the classifier reports this is: empirical study 1.00 · evidence from observation or experiment 1.00.

What was examined, and what was not

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: An SEM image of the asymmetric Al ring with radius

**Figure 2.** FIG. 2: The current-voltage curve measured on an asymmet

**Figure 3.** FIG. 3: Magnetic dependencies of the permitted velocity

**Figure 4.** Fig.4. The results of the measurements of the rec

**Figure 6.** FIG. 6: The oscillations of the resistance

Showing 5 of the paper's 6 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:cond-mat/0603005 by the authors.

References

Works cited (23)

[1]

A. J. Leggett, J.Phys. Con. Mat. 14, R415 (2002); J. of Supercon. 12, 683 (1999)

View on Pith 2002
[2]

4 µm . This nano-structures were fabricated by ther- mally evaporated on oxidized Si substrates, e-beam lithography and lift-oﬀ process. Their sheet resis- tance was 0 . 2 ÷ 0. 5 Ω / ⋄ at 4.2 K, the resistance ratio R(300K)/R (4. 2K) = 2 . 5 ÷ 3. 5, and critical temperature was Tc = 1 . 24 ÷ 1. 27 K. The coherence length of the like aluminum structures ξ(...

View on Pith
[3]

4 µm , Tc = 1. 235 K at T = 1. 225 K (◦). Φ / Φ 0 = 1/ 2 or −1/ 2 at Φ = ( n + 0. 5)Φ 0, Fig.3. Whereas the average value obtained at a multiple measurement should be equal zero Ip ∝ n − Φ / Φ 0 ∝ 1/ 2 + (−1/ 2) = 0, Fig.3. We have found that the values of the critical current corresponding to the jump in the normal state change periodically in magnetic ﬁ...

View on Pith
[4]

A. Einstein, B. Podolsky, and N. Rosen Phys. Rev. 47, 777 (1935)

View on Pith 1935
[5]

3 µm and Tc = 1. 23 K at diﬀerent temperature T = 1. 271 K (1) and 1 . 243 K (2). voltage observed on asymmetric loop and its system [13]. But the experimental periodical dependencies Ic+(Φ / Φ 0), I c− (Φ / Φ 0), Fig.3,5, diﬀer in essence from the expected one. First of all we have found that the anisotropy dependencies Ic,an(Φ / Φ 0) correspond rather m...

View on Pith
[6]

5 + ∆ φ/ 2)Φ 0 and Φ = ( n + 0 . 5 − ∆ φ)Φ 0 but not at Φ = ( n + 0. 5)Φ 0 as it should be expected, Fig.3, and as it is observed for symmetrical ring [14]. The shift value ∆Φ can be measured enough precisely, up to ±0. 02Φ 0, because of the Ic+(Φ / Φ 0), I c− (Φ / Φ 0) periodicity and the possibility to observe many, up to 25, periods. Our measurements i...

View on Pith
[7]

99Tc have shown that with this precision the shift has the same value ∆ φ = 0 . 5 for all asymmetric ring with ww/w n ⁄= 1 which we investigated. The shift value does not change with both temperature and value of ring anisotropy in the investigated interval ww/w n = 2 ÷ 1. 25: ∆ φ = 0 . 5 ± 0. 02 for the rings with ww = 0 . 4 µm ww = 0 . 3 µm and ww = 0 ....

View on Pith
[8]

N.D. Mermin, Physics Today, 38, 38 (1985)

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[9]

M.A.Nielsen and I.L.Chuang, Quantum Computation and Quantum Information Cambridge University Press, 2000

View on Pith 2000
[10]

of resistance ∆ R(Φ / Φ 0) of superconducting loop [11] prove quantization of velocity circulation ∮ l dlv = 2π ℏ m (n − Φ Φ 0 ) (1) of superconducting pairs and that the permitted state with minimum energy has overwhelming probability even FIG. 1: An SEM image of the asymmetric Al ring with radius r = 2 µm and semi-ring width wn = 0 . 2 µm , ww = 0 . 4 µ...

View on Pith
[11]

A. M. Steane, Rept.Prog.Phys. 61, 117, 1998

View on Pith 1998
[12]

J. S. Bell, Physics 1, 195 (1964)

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[13]

as a consequence of the rectiﬁcation of the ac external current Iext(t): Vdc = Θ − 1 ∫ Θ dtV (Iext(t)). The quan- tum oscillations Vdc(Φ / Φ 0) appear when the amplitude I0 of the external current Iext(t) = I0 sin(2πf t) exceeds minimum from the Ic+, I c− values I0 > min(Ic+, I c− ), its amplitude VA mounts the maximum at max(Ic+, I c− ) > I0 > min (Ic+, ...

View on Pith
[14]

A. Aspect, P. Grangier, and G. Roger, Phys.Rev. Lett 47 460, (1981); P. G. Kwiat et al., idid 75, 4337 (1995); G. Weihs et al., idid 81, 5039 (1998); W. Tittel et al., Phys. Rev. A 57, 3229 (1998); J. W. Pan et al., Nature 403, 515 (2000)

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A. J. Leggett and A. Garg Phys. Rev. Lett. 54, 857, (1985)

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M.Tinkham, Introduction to Superconductivity. McGraw- Hill Book Company (1975)

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W. A. Little and R. D. Parks, Phys.Rev.Lett. 9, 9 (1962)

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H. Vloeberghs et al., Phys. Rev.Lett. 69, 1268, 1992

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A.V. Nikulov, Phys. Rev. B 64, 012505 (2001)

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S. V .Dubonos et al., Pisma Zh.Eksp.Teor.Fiz. 77, 439 (2003) ( JETP Lett. 77, 371 (2003)); available also at cond-mat/0303538

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[21]

D. S. Golubovic and V. V. Moshchalkov, will be published in Appl. Phys. Lett. , available at cond-mat/0509332. 5

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T. A. Fulton, L. N. Dunkleberger, and R. C. Dynes, Phys. Rev. B 6, 855 (1972)

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How the literature uses it

6 arXiv papers cite this in 34 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 21 sentences without a confident label

Contests it 2 papers

But measurements of asymmetric ring (with the different section of the ring-halves, Fig.1) have discovered fundamental disagreements with the theoretical prediction, see the relation (2) and Fig.19 in [ref] and Fig.2 in [ref] .

Could quantum mechanics describe completely and consistently all superconducting and other quantum phenomena? · Experimental results contradicting to the theoretical predictions · contest 0.78
Some experimental results [ref] seem to contradict not only this naive description but even the demand of quantization (2).

Energy of magnetic moment of superconducting current in magnetic field · Conclusion · contest 0.74

Uses its data or values 1 paper

Authors of [ref] measured critical currents for opposite directions of the applied current $I_c+(Φ/Φ_0+ϕ_l(T))$ and $I_c-(Φ/Φ_0-ϕ_l(T))$ as functions of the normalized magnetic flux in superconducting single circularly-asymmetric aluminum rings and such rings in series with different parameters, including different values of asymmetry, at different temperatures.

Shift of the maxima of the critical currents of different polarity relative to the zero magnetic flux along the flux axis in a superconducting asymmetric aluminum ring · Phase shift of the maxima of the critical currents of different polarity relative to the zero flux in opposite direction · 0.97

Compares with it 1 paper

In addition, we compared the theoretical values of the phase shift with the experimental values [ref] .

Shift of the maxima of the critical currents of different polarity relative to the zero magnetic flux along the flux axis in a superconducting asymmetric aluminum ring · Introduction · 0.94

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

A. J. Leggett, J.Phys. Con. Mat. 14, R415 (2002); J. of Supercon. 12, 683 (1999)

View on Pith 2002

[2] [2]

4 µm . This nano-structures were fabricated by ther- mally evaporated on oxidized Si substrates, e-beam lithography and lift-oﬀ process. Their sheet resis- tance was 0 . 2 ÷ 0. 5 Ω / ⋄ at 4.2 K, the resistance ratio R(300K)/R (4. 2K) = 2 . 5 ÷ 3. 5, and critical temperature was Tc = 1 . 24 ÷ 1. 27 K. The coherence length of the like aluminum structures ξ(...

View on Pith

[3] [3]

4 µm , Tc = 1. 235 K at T = 1. 225 K (◦). Φ / Φ 0 = 1/ 2 or −1/ 2 at Φ = ( n + 0. 5)Φ 0, Fig.3. Whereas the average value obtained at a multiple measurement should be equal zero Ip ∝ n − Φ / Φ 0 ∝ 1/ 2 + (−1/ 2) = 0, Fig.3. We have found that the values of the critical current corresponding to the jump in the normal state change periodically in magnetic ﬁ...

View on Pith

[4] [4]

A. Einstein, B. Podolsky, and N. Rosen Phys. Rev. 47, 777 (1935)

View on Pith 1935

[5] [5]

3 µm and Tc = 1. 23 K at diﬀerent temperature T = 1. 271 K (1) and 1 . 243 K (2). voltage observed on asymmetric loop and its system [13]. But the experimental periodical dependencies Ic+(Φ / Φ 0), I c− (Φ / Φ 0), Fig.3,5, diﬀer in essence from the expected one. First of all we have found that the anisotropy dependencies Ic,an(Φ / Φ 0) correspond rather m...

View on Pith

[6] [6]

5 + ∆ φ/ 2)Φ 0 and Φ = ( n + 0 . 5 − ∆ φ)Φ 0 but not at Φ = ( n + 0. 5)Φ 0 as it should be expected, Fig.3, and as it is observed for symmetrical ring [14]. The shift value ∆Φ can be measured enough precisely, up to ±0. 02Φ 0, because of the Ic+(Φ / Φ 0), I c− (Φ / Φ 0) periodicity and the possibility to observe many, up to 25, periods. Our measurements i...

View on Pith

[7] [7]

99Tc have shown that with this precision the shift has the same value ∆ φ = 0 . 5 for all asymmetric ring with ww/w n ⁄= 1 which we investigated. The shift value does not change with both temperature and value of ring anisotropy in the investigated interval ww/w n = 2 ÷ 1. 25: ∆ φ = 0 . 5 ± 0. 02 for the rings with ww = 0 . 4 µm ww = 0 . 3 µm and ww = 0 ....

View on Pith

[8] [8]

N.D. Mermin, Physics Today, 38, 38 (1985)

View on Pith 1985

[9] [9]

M.A.Nielsen and I.L.Chuang, Quantum Computation and Quantum Information Cambridge University Press, 2000

View on Pith 2000

[10] [10]

of resistance ∆ R(Φ / Φ 0) of superconducting loop [11] prove quantization of velocity circulation ∮ l dlv = 2π ℏ m (n − Φ Φ 0 ) (1) of superconducting pairs and that the permitted state with minimum energy has overwhelming probability even FIG. 1: An SEM image of the asymmetric Al ring with radius r = 2 µm and semi-ring width wn = 0 . 2 µm , ww = 0 . 4 µ...

View on Pith

[11] [11]

A. M. Steane, Rept.Prog.Phys. 61, 117, 1998

View on Pith 1998

[12] [12]

J. S. Bell, Physics 1, 195 (1964)

View on Pith 1964

[13] [13]

as a consequence of the rectiﬁcation of the ac external current Iext(t): Vdc = Θ − 1 ∫ Θ dtV (Iext(t)). The quan- tum oscillations Vdc(Φ / Φ 0) appear when the amplitude I0 of the external current Iext(t) = I0 sin(2πf t) exceeds minimum from the Ic+, I c− values I0 > min(Ic+, I c− ), its amplitude VA mounts the maximum at max(Ic+, I c− ) > I0 > min (Ic+, ...

View on Pith

[14] [14]

A. Aspect, P. Grangier, and G. Roger, Phys.Rev. Lett 47 460, (1981); P. G. Kwiat et al., idid 75, 4337 (1995); G. Weihs et al., idid 81, 5039 (1998); W. Tittel et al., Phys. Rev. A 57, 3229 (1998); J. W. Pan et al., Nature 403, 515 (2000)

View on Pith 1981

[15] [15]

A. J. Leggett and A. Garg Phys. Rev. Lett. 54, 857, (1985)

View on Pith 1985

[16] [16]

M.Tinkham, Introduction to Superconductivity. McGraw- Hill Book Company (1975)

View on Pith 1975

[17] [17]

W. A. Little and R. D. Parks, Phys.Rev.Lett. 9, 9 (1962)

View on Pith 1962

[18] [18]

H. Vloeberghs et al., Phys. Rev.Lett. 69, 1268, 1992

View on Pith 1992

[19] [19]

A.V. Nikulov, Phys. Rev. B 64, 012505 (2001)

View on Pith 2001

[20] [20]

S. V .Dubonos et al., Pisma Zh.Eksp.Teor.Fiz. 77, 439 (2003) ( JETP Lett. 77, 371 (2003)); available also at cond-mat/0303538

View on Pith Pith review arXiv 2003

[21] [21]

D. S. Golubovic and V. V. Moshchalkov, will be published in Appl. Phys. Lett. , available at cond-mat/0509332. 5

View on Pith Pith review arXiv

[22] [22]

T. A. Fulton, L. N. Dunkleberger, and R. C. Dynes, Phys. Rev. B 6, 855 (1972)

View on Pith 1972

[23] [23]

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