Testing axion physics in a Josephson junction environment

Testing axion physics in a Josephson junction environment Christian Beck Queen Mary, University of London 1

Testing axion physics in a Josephson junction environment Christian Beck Queen Mary, University of London 1

Contents 1 Introduction: What is a Josephson junction? 2 Analogy between eqs. of motions of axions and Josphson junctions C. Beck, Physica C 473, 21 (2012) [arxiv:1008.2085] 3 Josephson junctions as detectors for axions: Phase synchronization effects C. Beck, Mod. Phys. Lett. A 26, 2841 (2011) [arxiv:1110.5871] 4 Axions tunneling a junction (ATJ): Complement of the Light shining through wall (LSW) experimental setup 5 An observed axion candidate signal in S/N/S Josephson junctions C. Beck, arxiv:1309.3790 6 Summary 2

1 Introduction: What is a Josephson junction? Josephson junction (JJ) consists of two superconductors separated by a weak-link region (yellow) weak link-region is an insulator for tunnel junctions and a normal metal for S/N/S junctions distance between superconducting plates: d 1nm for tunnel junctions, d 1µm for S/N/S junctions If voltage V is applied then JJ emits Josephson radiation of frequency hω J = 2eV 3

Important technical device: Two Josephson junctions can form a bounded state, a SQUID (Superconducting Quantum Interference Device) Used e.g. for high-precision magnetic flux measurements See any textbook on superconductivity (e.g. Introduction to Superconductivity by M. Tinkham) how this works 4

2 Analogy between equations of motions of axions and Josephson junctions Axion field a = f a θ. Classical eq. of motion of the axion misalignement angle θ: θ + Γ θ + m2 a c4 h 2 sin θ = g γ π 1 f 2 a c 3 e 2 E B (1) f a axion coupling, m a axion mass, g γ = 0.97 for KSVZ axions, or g γ = 0.36 for DFSZ axions. In the early universe, Γ = 3H, where H is the Hubble parameter. E, B: external electric and magnetic field. 5

2 Analogy between equations of motions of axions and Josephson junctions Axion field a = f a θ. Classical eq. of motion of the axion misalignement angle θ: θ + Γ θ + m2 a c4 h 2 sin θ = g γ π 1 f 2 a c 3 e 2 E B (1) f a axion coupling, m a axion mass, g γ = 0.97 for KSVZ axions, or g γ = 0.36 for DFSZ axions. In the early universe, Γ = 3H, where H is the Hubble parameter. E, B: external electric and magnetic field. Compare this with the eq. of motion of a Josephson junction (JJ). The phase difference δ of a JJ driven by a bias current I satisfies δ + 1 δ + 2eI c 2e sin δ = RC hc hc I (2) I c : critical current of the junction, R: normal resistance, C: capacity of the junction. 5

2 Analogy between equations of motions of axions and Josephson junctions Axion field a = f a θ. Classical eq. of motion of the axion misalignement angle θ: θ + Γ θ + m2 a c4 h 2 sin θ = g γ π 1 f 2 a c 3 e 2 E B (1) f a axion coupling, m a axion mass, g γ = 0.97 for KSVZ axions, or g γ = 0.36 for DFSZ axions. In the early universe, Γ = 3H, where H is the Hubble parameter. E, B: external electric and magnetic field. Compare this with the eq. of motion of a Josephson junction (JJ). The phase difference δ of a JJ driven by a bias current I satisfies δ + 1 δ + 2eI c 2e sin δ = RC hc hc I (2) I c : critical current of the junction, R: normal resistance, C: capacity of the junction. Equations of motion are basically the same. 5

2 Analogy between equations of motions of axions and Josephson junctions Axion field a = f a θ. Classical eq. of motion of the axion misalignement angle θ: θ + Γ θ + m2 a c4 h 2 sin θ = g γ π 1 f 2 a c 3 e 2 E B (1) f a axion coupling, m a axion mass, g γ = 0.97 for KSVZ axions, or g γ = 0.36 for DFSZ axions. In the early universe, Γ = 3H, where H is the Hubble parameter. E, B: external electric and magnetic field. Compare this with the eq. of motion of a Josephson junction (JJ). The phase difference δ of a JJ driven by a bias current I satisfies δ + 1 δ + 2eI c 2e sin δ = RC hc hc I (2) I c : critical current of the junction, R: normal resistance, C: capacity of the junction. Equations of motion are basically the same. The numerical values of the coefficients for typical QCD axion physics and typical JJ physics are also quite similar (see C. Beck, Mod. Phys. Lett. 26, 2841 (2011) for examples). 5

3 Josephson junctions as detectors for axions: phase synchronization effects Consider an axion with misalignment angle θ that enters the weak link region of a Josephson junction with phase difference δ: Joint axion Josephson wave function Ψ = Ψ e iϕ must be single-valued. This means that for a given closed integration curve (dashed line above) one has ϕ d s + δ + θ = 0 mod 2π (3) SC = δ, θ are no longer independent of each other but influence each other. 6

In the presence of a vector potential A define gauge-invariant phase differences γ i by γ 1 := δ 2π A d s (4) Φ 0 weak link 1 γ 2 := θ 2π A d s. (5) Φ 0 weak link 2 7

In the presence of a vector potential A define gauge-invariant phase differences γ i by γ 1 := δ 2π A d s (4) Φ 0 weak link 1 γ 2 := θ 2π A d s. (5) Φ 0 weak link 2 Standard formalism exploiting uniqueness of axion-josephson wave function then yields ˆγ 1 γ 2 = 2π Φ Φ 0 mod 2π, (6) Φ: magnetic flux through the area enclosed by the chosen closed line of integration, Φ 0 = h 2e : flux quantum, ˆγ 1 := γ 1. 7

In the presence of a vector potential A define gauge-invariant phase differences γ i by γ 1 := δ 2π A d s (4) Φ 0 weak link 1 γ 2 := θ 2π A d s. (5) Φ 0 weak link 2 Standard formalism exploiting uniqueness of axion-josephson wave function then yields ˆγ 1 γ 2 = 2π Φ Φ 0 mod 2π, (6) Φ: magnetic flux through the area enclosed by the chosen closed line of integration, Φ 0 = h 2e : flux quantum, ˆγ 1 := γ 1. If Φ << Φ 0 or if Φ is an integer multiple of Φ 0 then γ 2 = ˆγ 1 (7) meaning the axion phase θ synchronizes with the Josephson phase δ. 7

Now consider JJ with a bias current I driven by voltage V. Josephson radiation of frequency 2eV / h is emmitted. The phase-synchronization condition θ = δ together with the original equation of motion of the axion imply that effectively the axion misalignment angle behaves as if there is a huge magnetic field (colinear with E) given by B = 2πΓf 2 a d g γ hc 3 e. (8) This formal B-field is huge! B 10 34 T. It means phase-synchronized axions immediately decay into 2 microwave photons when entering the weak-link region. Primakoff effect: P a γ = 1 4β a (g Bec L) 2 ( ) sin ql 2 2 h = P ql γ a (9) 2 h 8

4 Axions tunneling a junction (ATJ): Complement of Light Shining Through Walls (LSW) Microscopic model of what happens in an S/N/S junction. Axion tunnels through junction (ATJ) and triggers (by multiple Andreev reflection) the transport of n Cooper pairs (n = 3 in the example plotted) 9

Some relevant formulas (C. Beck, arxiv:1309.3790): Signal shape in RSJ approximation (Shapiro step without externally applied microwave radiation) I s (V ) = P s 4 (RI c) 2 1 V 2 2eV s = m a c 2 (V s : signal voltage) Expected signal power from axions: [ V + V s (V + V s ) 2 + ( δv 2 )2 + V V s (V V s ) 2 + ( δv 2 )2 ]. (10) P s = ρ a va. (11) ρ a : axionic dark matter density near the earth, v = 2.3 10 5 m, A: Area of weak-link s region of JJ. Total signal current produced by axions in S/N/S junction: I s = G s dv = N a τ n 2e = ρ a m a c2va n 2e (12) where N a /τ is the number of axions hitting the normal metal region per time unit τ. Axion density from ρ a = I sv s van. (13) This can be used to experimentally estimate the axion mass m a and dark matter density ρ a from an experimental measurement of V s and I s. 10

5 An observed candidate signal in S/N/S Josephson junctions C. Hoffmann, F. Lefloch, M. Sanquer, B. Pannetier, Phys. Rev. B 70, 180503(R) (2004) 11

5 An observed candidate signal in S/N/S Josephson junctions C. Hoffmann, F. Lefloch, M. Sanquer, B. Pannetier, Phys. Rev. B 70, 180503(R) (2004) They measured differential conductivity G(V ) = di/dv and observed signal peak of unknown origin at V s = ±0.055mV. 11

Hoffmann et al. (2004) observe a signal of unknown origin that is consistent with our theoretical expectations. Independent of the temperature (which is varied from 0.1K to 0.9K) they consistently observe a small peak in their measured differential conductivity G(V ) at the voltage V s = ±0.055mV Their measurements provide evidence for a signal current feature of size I s = (8.1 ± 1.0) 10 8 A obtained by integrating the area under the observed signal peak of the differential conductivity. Their noise measurements also indicate that every quasi-particle performs n = 7 Andreev reflections. Area A of the metal plate of their junction is A = 0.85µm 0.4µm = 3.4 10 13 m 2. From 2eV s = ma 2 c we thus obtain an axion mass prediction of m ac 2 = 110µeV (equivalent to f a 5.5 10 10 GeV ), and ρ a = I sv s van yields the prediction ρ a = (0.051 ± 0.006)GeV /cm 3. 12

Is this value of axionic dark matter density (ρ a = (0.051 ± 0.006)GeV /cm 3 ) as predicted by our theory based on Hoffmann et al. s measurements reasonable? 13

Is this value of axionic dark matter density (ρ a = (0.051 ± 0.006)GeV /cm 3 ) as predicted by our theory based on Hoffmann et al. s measurements reasonable? Yes, it is. Astrophysical observations suggest that the galactic dark matter density ρ d near the earth is about ρ d = (0.3±0.1)GeV /cm 3 (Weber, de Boer 2010). But this includes all kinds of dark matter particles, including WIMPS. Generally, axions of high mass will make up only a fraction of the total dark matter density of the universe: ρ a /ρ d (24µeV /m a c 2 ) 7/6 (Duffy, van Bibber (2009)) For m a c 2 = 110µeV we thus expect an axionic dark matter density that is a fraction (24/110) 7/6 0.17 of the total dark matter density, giving ρ a 0.17 ρ d = (0.051 ± 0.017)GeV /cm 3. The intensity of the JJ signal is thus in perfect agreement with what is expected from astrophysical observations. 13

Need further measurements to confirm (or refute) dark matter nature of the observed candidate signal: Does the signal survive careful shielding of the junction from any external microwave radiation? A signal produced by axions cannot be shielded. Should look for a possible small dependence of the measured signal intensity on the spatial orientation of the metal plate relative to the galactic axion flow (a precise directional measurement would be extremely helpful). The velocity v by which the earth moves through the axionic BEC (Sikivie et al. 2009) of the galactic halo exhibits a yearly modulation of about 10%, with a maximum in June and a minimum in December. Hence JJ signal intensity should exhibit the same yearly modulation. Independent experiments (such as upgraded versions of ADMX) would need to confirm the suggested value of m a c 2 = 110µeV. 14

6 Summary We have described a macroscopic quantum effect in Josephson junctions that may help to prove the existence of axionic dark matter in future experiments. Phase-synchronized axions cannot exist in the weak-link region of JJs due to a (formal) huge magnetic field that is simulated to them by the driven JJ environment in the voltage stage. C.Beck, Mod. Phys. Lett. A 26, 2841 (2011); C. Beck, Physica C 473, 21 (2012) Axions hitting the weak-link region trigger the transport of additional Cooper pairs. Leads to a small measurable signal for the differential conductivity if axion mass resonates with Josephson frequency. Effect is particularly strong in S/N/S junctions which have a much larger weak-link region than tunnel junctions. Candidate signal of unknown origin has been observed in measurements of Hoffmann et al. Can be interpreted in terms of an axion mass of 0.11 mev and a local axionic dark matter density of 0.05 GeV/cm 3. C. Beck, arxiv:1309.3790 15