Quantum electrodynamics of a superconductor-insulator phase transition

R. Kuzmin,1 R. Mencia,1 N. Grabon,1 N. Mehta,1 Y.-H. Lin,1 and V. E. Manucharyan1, ∗

1Department of Physics, Joint Quantum Institute,and Center for Nanophysics and Advanced Materials,

University of Maryland, College Park, Maryland 20742, USA.(Dated: May 22, 2018)

A chain of Josephson junctions implements oneof the simplest many-body models undergoing asuperconductor-insulator quantum phase transi-tion between states with zero and infinite resis-tance [1–5]. This phenomenon is central to ourunderstanding of interacting bosons and fermionsin one dimension [6]. Apart from zero resistance,the superconducting state is always accompaniedby a sound-like mode due to collective oscillationsof the phase of the complex-valued order param-eter [7–10]. Surprisingly little is known aboutthe fate of this mode upon entering the insu-lating state, where the order parameter’s ampli-tude remains non-zero, but the phase ordering is“melted” by quantum fluctuations [11]. Here wereport momentum-resolved radio-frequency spec-troscopy of collective modes in nanofabricatedchains of Al/AlOx/Al tunnel junctions. Our keyfinding is that the GHz-frequency modes survivefar into the insulating regime, i.e. an insulatorcan superconduct AC currents. The insulatingstate manifests itself by a weak decoherence ofcollective modes with an unusual frequency de-pendence: longer wavelengths decohere faster,in fact suggesting the absence of DC transport.Owing to an unprecedentedly large kinetic induc-tance per unit length, the observed phase moderepresents microwave photons with a remarkablylow speed of light (below 8 × 105 m/s) and highwave impedance (above 23 kΩ). The latter ex-ceeds the transition value for the Bose glass insu-lator, expected in this system, by an order of mag-nitude [12–14], which challenges theory to revisitthe finite-energy condensate dynamics near thetransition. More generally, the high impedancetranslates into a fine structure constant exceedinga unity, opening access to previously impossibleregimes of quantum electrodynamics [15].

Our devices consist of two closely spaced parallel chainsof over 33,000 junctions fabricated on an insulating sil-icon chip (Fig. 1a,b). The chains are short-circuitedat one end and connected to a dipole antenna at theother end for coupling external signals. The chip is sus-pended in the center of a metallic waveguide box with

∗ manuchar@umd.edu

a single broadband microwave input/output port (Fig.1c). This wireless interface minimizes parasitic groundcapacitances seen by the junctions and allows collectionof all the energy radiated off-chip. In the supercon-ducting state, the device can be viewed as a telegraphtransmission line [16] defined by the capacitance c be-tween the chains and the Josephson inductance l per unitlength (Fig. 1d). The capacitance adds inertia to thephase degree of freedom and its value is approximatelygiven by the vacuum permittivity ε0 adjusted by the di-electric constant of silicon. The inductance l plays therole of the inverse phase-rigidity of the condensate andit can significantly exceed the vacuum permeability µ0.The “sound” waves associated with the collective phaseoscillations across the junctions are equivalent to thetransverse electromagnetic modes (one-dimensional pho-

tons) of the transmission line with a velocity v = 1/√lc

and a wave impedance Z =√l/c.

In close analogy with vacuum quantum electrody-namics, zero-point fluctuations of fields in our one-dimensional system are controlled by the effective finestructure constant α = Z/RQ, where RQ = h/(2e)2 ≈6.5 kΩ is the resistance quantum for Cooper pairs [17].The superconducting state is favored for α 1, whenthe line mimics the usual weak-coupling electrodynamicsof the free space, for which α = 1/137.0. This is nota coincidence: at a given frequency the Josephson rela-tion links the fluctuation of phases across the junctionswith the fluctuation of electric field between the chains,which in turn defines the strength of light-matter cou-pling. The model of a single chain coupled to a groundplane (Fig. 1e) predicts a transition to the gapped Mottinsulator phase at αMott

c ≈ 1/4 [2, 3]. The transitionbelongs to the celebrated Berezinski-Kosterlitz-Thouless(BKT) type and is driven by the proliferation of 2π-slips(instantons) in the phases across the junctions [18]. Amore realistic model should include an oxide capacitanceacross every junction and a random offset charge at ev-ery island. In this case the chain is modeled using a gen-eral framework of disordered Tomonaga-Luttinger liquidswith the fine structure constant α replacing the inverseLuttinger interaction parameter [13]. Here theory pre-dicts a transition to a compressible Bose glass insulatorstate at a slightly elevated value of the fine structure con-stant αBG

c = 1/3 (Z = 2.2 kΩ) [12].

The goal of this work is to explore for the first time howthe superconductor-insulator transition manifests itself

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FIG. 1. (a) Optical photograph of the Josephson transmission line with a dipole antenna at the left end and a short-circuittermination at the right one. (b) Scanning electron micrograph of typical strong and weak junction chains. (c) Photograph ofa device chip mounted in a single-port copper waveguide. The antenna is oriented along the preferred waveguide electric fieldpolarization. (d) Linear circuit model for collective modes. Here the inductance l is of purely kinetic origin. (e) Minimal modelof a chain exhibiting a BKT transition to a Mott insulator phase (see text). (f) Minimal circuit model of the double-chaindevice shown in (a). Here the capacitance c is due to electrostatic coupling between the chains.

in the propagation of electromagnetic waves in progres-sively higher impedance transmission lines. The minimalcircuit model for our double-chain line (Fig. 1f) includesthe Josephson energy EJ , the charging energy EC =e2/2CJ due to the oxide capacitance CJ , and the inter-chain charging energy E0 = e2/2C0, where C0 is the perjunction capacitance between the two chains. Introduc-ing the size a of the junction, the propagation parametersare now defined as l × a = 2(~/2e)2/EJ and c× a = C0.The junction plasma frequency ωp =

√8EJEC/~ defines

the ultra-violet cut-off in our system. With these pa-rameters, the phase transition boundary formulated forsingle chains should be applicable to our devices [19].

The example results of momentum-resolved spec-troscopy are shown in Fig. 2. The experiment is per-formed using a standard two-tone dispersive reflectome-try, taking advantage of the weak Kerr non-linearity of aJosephson junction [20] (Methods). Data reveals an or-dered set of discrete resonances which we associate withthe standing wave modes of the transmission line (Fig.2a). By indexing the individual resonances and plot-ting the frequency as a function of index n = 1, 2, ...,we obtain the dispersion relation ωn(kn), where kn is thewavenumber defined as kn+1−kn = π/L, and L = 10 mmis the length of the line (Fig. 2b). The dispersion is in ex-cellent agreement with a simple two parameter expressionω(k) = vk/

√1 + (vk/ωp)2, describing ultra-slow pho-

tons with a velocity v = 1.88 × 106 m/s and a bandedge at the plasma frequency ωp/2π = 24.8 GHz. Then = 1 mode is clearly visible at 40 MHz. This frequencyis almost three orders of magnitude below the plasmaedge and several times lower than the thermal frequencyassociated with the 10 mK temperature of our setup. Inaddition, the n = 1 mode frequency is half the modespacing, which correctly reflects the additional π phase-shift due to the short-circuit boundary condition. Thisobservation confirms that wave propagation occurs alongthe entire length of the system and the spectrum is gap-

FIG. 2. (a) Reflection signal as a function of excitationfrequency. Discrete resonances point out the collective modefrequencies and can be indexed one by one starting from thefirst mode at about 40 MHz. The third resonance is a spu-rious mode and is discarded. (b) Reconstructed dispersionrelation (see text). Theoretical fit (solid line) represents pho-tons with a negligible dispersion below 10 GHz and a bandedge at the Josephson plasma frequency ωp. (c) Mode spac-ing as a function of mode index. The sharp periodic featuresoriginate from the stitching error of the electron-beam lithog-rapher, which are not visible in device images.

less. Fluctuations of the mode spacing as a function ofmode index are found to be within 5−10% up to a wave-length as low as L/100 (Fig. 2c), showing no signals ofAnderson localization [10].

Combining the measured values of ωp, v, and theknown dimensions of the chains, we reliably extract thewave impedance along with other chain parameters inmultiple devices (Suppl. Mat.). For the device from Fig.2, we get Z = 11.7 kΩ. This is equivalent to a strikingly

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large value of the fine structure constant α = 1.8, exceed-ing the theoretical transition value by more than a factorof 5. Yet, this large value of α is fully consistent with theindividual junction parameters expected from the fabri-cation process. Spectroscopy thus concludes that downto a frequency of ωp/600 the propagation of collectivemodes is completely unaffected by the insulating phase.

To reveal the insulating phase we explore the decoher-ence of the collective modes in devices with progressivelyhigher wave impedance (Fig. 3). This requires an ac-curate measurement of the real and imaginary parts ofthe reflection coefficient at the single-photon excitations.For technical reasons, this was possible in our setup inapproximately 4− 12 GHz band. The higher impedancewas achieved by reducing the chain width while keepingall other dimensions the same (Fig. 1b). We define themode quality factor Q as the ratio of mode frequencyto its linewidth after subtracting the small contributiondue to the radiation of photons into the measurementport (Methods). For devices with Z . 12 kΩ the qual-ity factor grows with reducing the normalized frequencyω/ωp independently of other device parameters (Fig. 3b- red and orange markers; see also Suppl. Mat.). To-wards the lower end of the band, it reaches a relativelyhigh value of Q = 105. The suppression of the Q-factortowards the plasma frequency is likely linked to the pres-ence of a large (of order 104) number of modes at ω ≈ ωp.The key observation of our experiment is that this depen-dence of Q-factor on frequency undergoes a remarkablereversal in weaker junction chains (Fig. 3b - blue and vi-olet markers). For some intermediate chain width (device“c”) the Q-factor becomes flat in frequency and it devel-ops a clear tendency to drop towards lower frequenciesin smaller width chains (device “d” and “e”). In otherwords, in sufficiently weak junction chains, the longerthe wavelength of an excitation, the faster it decoheres.Such a behavior is highly unusual for materials-relatedloss, but it is consistent with the insulator phase: ex-trapolating to zero frequency, the observed decoherencewould indeed inhibit DC transport.

For devices with EJ/EC < 7 (devices “e” and“f”)the mode spacing fluctuations are dramatically enhancedalong with device-to-device variations (Suppl. Mat.),making it difficult to accurately recover the chain pa-rameters. Nevertheless, the Q-factor in such devicescan still exceed 103. Interestingly, reducing the waveimpedance Z by about 20% without modifying the junc-tion dimensions – achieved by shrinking the spacing be-tween the chains from 10 µm to 2 µm – showed no effecton the Q-factor in these devices. This observation sug-gests that the observed decoherence is much less sensitiveto impedance Z than to the EJ/EC ratio.

As a control experiment, we demonstrate the reversibletransition between the “superconducting” and “insulat-ing” frequency dependence of the Q-factor in a singledevice (Fig. 4). A fresh Josephson transmission line wasfabricated with relatively small value of EJ/EC such thatit still shows Q-factor growing towards the low frequen-

FIG. 3. (a) Direct one-tone reflection magnitude as a func-tion of frequency for devices with reducing (top to bottom)junction area. (b) Extracted internal resonance quality factorQ plotted as a function of the mode frequency normalized tothe plasma frequency for each device. The extracted deviceparameters are in the inset of the plot.

FIG. 4. The Q-factor (a) and the mode spacings (b) as afunction of frequency for the three subsequent incarnations ofa single device: fresh after fabrication (blue), aged (green),and annealed (red).

cies. The device is then aged for about 1000 hours at am-bient conditions, which reduces EJ by about 25%. Thisis confirmed by the mode spacing data clearly showingabout a 10% reduction in both v and ωp without a dra-matic enhancement of frequency disorder. This turnedout to be enough to observe the reversed, “insulating”

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frequency dependence of Q. In a final step, we annealedthe device on a hot plate, also at ambient conditions,which recovered the fresh value of EJ together with theoriginal “superconducting” frequency dependence of Q.Note that at ω/2π ≈ 5 GHz the quality factor under-gone a remarkable swing by an order of magnitude. Weconfirm that same aging test performed on devices withEJ/EC & 70 had no effect on the decoherence.

The high sensitivity of the quality factor on both EJ

and EJ/EC signals that the observed decoherence isprobably caused by the scattering of photons on thephase-slip fluctuations [18]. Phase-slips are indeed thekey ingredient behind the quantum BKT transition [13].The main result of our experiment is that, whether elas-tic or inelastic, the scattering of photons on phase-slips isfrequency-dependent and can be dramatically suppressedat GHz-frequencies. Consequently, we observed low-losswave propagation in transmission lines with a measuredwave impedance over 23 kΩ (α > 3.5), an order of mag-nitude above the theoretical transition value.

Can the lack of insulating behavior in our chains bemerely a finite size effect despite the large number ofjunctions N > 33, 000? A recent transport measure-ment on chains an order of magnitude shorter than oursreported strongly insulating behavior of resistance forEJ/EC < 16 [21]. Another recent experiment, alsoon chains an order of magnitude shorter than ours, re-ported voltage gaps as large as 0.5 meV (equivalent toa frequency of 120 GHz) for EJ/EC = 5 [14]. Further-more, the scaling of breakdown voltages agreed with theBose glass insulator predictions, i.e. αc = 1/3. In ourexperiment, the characteristic frequency ωS associatedwith the phase-slip process can be estimated in the limitα αc [18]. Mott insulator scenario assumes construc-tive interference of phase slips at every junction, suchthat ωS/ωp ≈ N exp(−

√8EJ/EC). Taking EJ/EC = 10

(Fig. 4), we get ωS/2π ≈ 100 GHz, which seems inconsis-tent with microwave propagation. In a Bose glass insu-lator, by contrast, phase-slips add with random phases,caused by offset charges, and hence N →

√N . This

yields a more reasonable estimate ωS/2π = 500 MHz.However, even this frequency is two orders of magnitudelarger than the measured linewidth of collective modes.We conclude that the non-zero frequency of the pho-ton plays a key role in decoupling it from phase-slips.Although superconductivity of finite-wavelength excita-tions by an insulator does not contradict the renormal-ization group view of a phase transition [11], theory ofthe observed decoherence is missing.

Collective mode spectroscopy measurement presentedhere can be used to explore superconductor-insulatortransition in a broad variety superconducting nanowiresand thin films, where many fundamental questions re-main unresolved from transport data [22–27]. The avail-ability of a practically infinite one-dimensional mediawith the fine structure constant α > 1 and a speed oflight reduced by over two orders of magnitude is a uniqueresource for quantum technology. It can be used to ex-

plore extreme regimes of light-matter coupling using su-perconducting circuits [28, 29], quantum dots [30], spinqubits [31], and possibly trapped Rydberg atoms or polarmolecules [32]. An immediate application is the analogsimulation of the strongly-correlated dynamics of quan-tum impurity models [33–35].

We acknowledge discussions with L. Glazman, M.Goldstein, M. Houzet, I. Protopopov, and J. Sau.The work was supported by US NSF (DMR-1455261),US-Israel BSF, and ARO-MURI (W911NF15-1-0397).

METHODS

Here we briefly summarize the important technicaldetails of our experiment. The measured devices andtheir extracted parameters are summarized in the tablebelow. Supplementary materials are available uponrequest.

TABLE I. Josephson transmission line parameters

Device v, 106

m/secωp/2π,GHz

Λ Z,kOhm

EJ/EC

a 2.76 27.0 38.4 7.4 440b 1.88 24.8 28.4 12.6 80c 1.12 20.8 20.2 19.0 15.8d 1.16 22.3 19.4 21.3 11.3d′ 0.99 19.3 19.3 23.6 8.7d′′ 1.12 21.6 19.4 21.8 10.6e 0.98* 20.8* 17.7* 23* 7*f 0.83* 20.8* 15* 19* 7* *g 22.6 26.7 4.7 0.7 712h 8.20 21.5 8.4 2.3 211i 2.11 20.9 37.9 7.0 484

Device fabricationThe chains were fabricated using the standard Dolanbridge technique involving a MMA/PMMA bi-layerresist patterned by electron beam lithography withsubsequent double-angle deposition of aluminium withan intermediate oxidation step. The substrate is ahigh-resistivity silicon wafer. Due to the large number ofjunctions in the chain, patterning was done by stitchingmultiple fields of view with a size of 100 µm. Thestitching error is invisible in device images. Curiously,it can be clearly seen in Fig. 2c as sharp periodic shiftsin the mode spacing data. We use this information toconfirm the conversion between the standing wave indexand the wavelength.Wireless RF-spectroscopy setupThe chip hosting the chains is mounted at the centerof a copper waveguide (Fig. 1c). In order to launchmicrowaves we have designed a coaxial-to-waveguidetransition launcher with a good matching in the range7−12 GHz. The antenna attached to the chain is smallerthan the free space wavelength at these frequencies.Therefore, the combination of the chip antenna, thewaveguide box, and the launcher can be viewed as asemi-transparent“mirror” with a frequency-dependent

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finesse. Note that the two chains of the transmissionline are spaced by only a few micrometers, whereas thedistance between a chain and a wall of the copper box isat least 5 mm, comparable to the full length of the chain.Such a setup minimizes typical parasitic capacitancesdue to the measurement circuitry seen by the junctions.One-tone spectroscopyWe used a Rohde & Schwarz ZNB network analyzer tomeasure the frequency-dependent reflection amplitudeand phase in a single-port reflection experiment. To fitthe data in the vicinity of each resonance, we use thecommonly known expression:

S11(ω) =2i(ω − ω0)/ω0 −Q−1

ext +Q−1int

2i(ω − ω0)/ω0 +Q−1ext +Q−1

int

, (1)

where ω0 is the resonance frequency, Qint ≡ Q is theinternal quality factor, plotted in Figs. 3-4, and Qext isthe external quality factor, which in general is a complexnumber. It’s real part can be viewed as a measure ofopacity of the mirror at the antenna end of the chain. Wefound that at frequencies below 7 GHz, the opacity growsupon reducing the frequency which is consistent with thepropagation cut-off of our copper waveguide. The opacityalso has a tendency to increase as frequency grows above10 GHz which may be related to a partial Anderson local-ization of microwaves in the chain and their decouplingfrom the antenna end. As a result, one-tone spectroscopybecomes inefficient far outside the 7− 12 GHz pass-bandof our coaxial-to-waveguide launching system. For thisreason, the frequency range in Fig. 3 is limited. Finally,

we note that our reflection data fits exceptionally wellto the above expression for S11(ω), as described in theSupplementary material, which allows very accurate ex-traction of both ω0 and Q.Two-tone spectroscopyWe use a weak cross-Kerr interaction between the pho-tons in different modes in order to perform broadbandspectroscopy shown in Fig. 2. First, the readout modeis selected in the pass-band. Reflection amplitude andphase at a properly chosen frequency near the resonanceare measured as a function of the frequency of the sec-ond tone, which is scanned to look for other modes. Thecross-Kerr effect results in the shifting of the frequencyof the readout mode due to the photon occupation ofevery other mode in the system. Since expressions forfrequency shift per photon are readily calculable, we usethis information to approximately calibrate the measure-ment power down to a single-photon level. In chains withEJ/EC ≈ 10, the spectrum below a frequency of about1 GHz becomes difficult to interpret. Nevertheless, thereare resonances down to a frequency of about 100 MHzExtracting Z from dispersion relationWe tried two methods for extracting Z. First is basedon our knowledge of junction areas and the fact that theoxide capacitance has a rather device-independent valueof 45 fF/µm2. The second method is based on the knownformulas for the capacitance of two infinitely long copla-nar strips. Both methods yield consistent results within20%. We used the more conservative result (smaller val-ues of Z and higher values of EJ/EC) in the main textof the manuscript.

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