Higgs-mediated optical amplification in a non-equilibrium superconductor

Michele Buzzi,1 Gregor Jotzu,1 Andrea Cavalleri,1 J. Ignacio Cirac,2 Eugene A. Demler,3

Bertrand I. Halperin,3 Mikhail D. Lukin,3 Tao Shi,4 Yao Wang,3 and Daniel Podolsky∗5

1Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany2Max Planck Institute for Quantum Optics, 85748 Garching, Germany

3Department of Physics, Harvard University, Cambridge, MA 02138, USA4CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,

Chinese Academy of Sciences, Beijing 100190, China5Physics Department, Technion, 32000 Haifa, Israel

The quest for new functionalities in quantum materials has recently been extended to non-equilibrium states, which are interesting both because they exhibit new physical phenomena andbecause of their potential for high-speed device applications. Notable advances have been made inthe creation of metastable phases1–9 and in Floquet engineering under external periodic driving10–12.In the context of non-equilibrium superconductivity, examples have included the generation of tran-sient superconductivity above the thermodynamic transition temperature6–8,13,14, the excitation ofcoherent Higgs mode oscillations15–18 and the optical control of the interlayer phase in cuprates19,20.Here, we propose theoretically a novel non-equilibrium phenomenon, through which a prompt quenchfrom a metal to a transient superconducting state could induce large oscillations of the order param-eter amplitude. We argue that this oscillating mode could act as a source of parametric amplificationof the incident radiation. We report experimental results on optically driven K3C60 that are con-sistent with these predictions. The effect is found to disappear when the onset of the excitationbecomes slower than the Higgs mode period, consistent with the theory proposed here. These resultsopen new possibilities for the use of collective modes in many-body systems to induce non-linearoptical effects.

I. INTRODUCTION

The Higgs mode is a fundamental collective excitationof systems with spontaneous symmetry breaking. It is agapped excitation associated with oscillations of the am-plitude of the order parameter. Examples of the Higgsmode in condensed matter are plentiful: it has been ob-served in the superconducting phase of NbSe2

21–23 andof amorphous superconducting films24, in the dimerizedantiferromagnet TlCuCl3

25, in a variety of incommen-surate charge density wave (CDW) systems26–28, and incold bosonic gases near the superfluid to Mott insulatortransition29,30.

In a number of experiments reported recently, super-conductivity is created non-adiabatically after a rapidchange in microscopic interactions, as induced by theapplication of a terahertz or mid-infrared (MIR) pumppulse31–34. Although the dynamics of this process are notyet fully understood, we posit that the “Mexican hat”effective potential for the superconducting order param-eter is established promptly after optical excitation, seeFig. 1(a). The appearance of large Higgs oscillations isthen a natural attribute of such photo-induced supercon-ductivity if the quench is fast compared to the frequencyof the Higgs mode ωH , see Fig. 1(b) (the order parameterdynamics following a sudden quantum quench has beenstudied theoretically in a number of related contexts, see,e.g., Refs. 35–38).

∗Corresponding author: podolsky@physics.technion.ac.il

We argue that in this situation, the coherent collectivemode can act as a source of parametric amplification39

for oscillations of long-lived phase fluctuations resultingin an enhanced reflectivity for a time-delayed probe pulse.This phenomenon can be qualitatively understood as asuperconductor with an excited Higgs mode, which thenplaces a parametric modulation on low-frequency phasemodes. We predict that the reflected beam would thenfeature amplification of intensity at the original frequencyω1, as well as the generation of an idler signal at thecomplementary frequency ωH − ω1 (see Fig. 1(c)). Wedub this phenomenon “Higgs amplification”.

We also report experimental results on optically drivenK3C60, which support these predictions. Using a pump-probe technique, we illuminate our samples with a pumppulse at 41 THz, and a probe pulse spanning frequen-cies between 1 to 7 THz. Phase-resolved detection ofthe reflected signal allows us to reconstruct both the realand imaginary parts of the conductivity at the probe fre-quency. We find that when K3C60 is driven with pumppulses of suitable duration, the incident probe light islocally amplified near the surface for frequencies below10 meV/~. We analyze experimental results taking intoaccount penetration depth mismatch between the pumpand probe pulses. We observe an anomalous enhance-ment of the reflectance, which in an homogeneously ex-cited medium would result in 6% amplification. For thepenetration depth of the probe beam of 700 nm this cor-responds to the amplification coefficient α ∼ 103 cm−1.

Our underlying theoretical considerations are pre-sented in Sections II and III, below. Experimental resultsare presented in Section IV, and a comparison of theory

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Re ψ Im ψ

(c) incoming

ω1N photons

ω1ω2 ωH= −

outgoing

N + 1 photonssignal ω1idler ω2

1 photon

(a) (b)energy

1 Higgs mode ωH

t < 0t > 0

HiggsGoldstone

FIG. 1: (a) Schematic potential in a broken symmetry state.The collective excitations are Goldstone and Higgs modes.(b) A sudden change in system parameters at time t = 0leads to a rapid inversion in the potential, thus inducing Higgsoscillations. (c) Light amplification in a superconductor withan excited Higgs mode. Left panel: A probe pulse containingN photons of frequency ω1 is incident on a superconductorwith an excited Higgs mode (represented by a wave). Rightpanel: The output consists of N + 1 photons of frequency ω1,plus a single photon of frequency ω2 traveling “backward”,due to in-plane momentum conservation. When many Higgsexcitations undergo stimulated decay, a large number of bothω1 and ω2 photons are produced.

and experiment is given in Section V. Some implicationsof this work, and the outlook for further developments,are given in Section VI. Additional details of our analysisare given in three appendices.

II. FROM HIGGS OSCILLATIONS TOANOMALOUS REFLECTION

The coupling between the Higgs mode and light canbe understood by observing that in a general state withbroken continuous symmetry, the Higgs mode can de-cay into a pair of Goldstone modes. For neutral su-perfluids, this process determines the lifetime of Higgsexcitations40. In superconductors, the Goldstone modedescribes phase fluctuations and therefore a charge cur-rent, which interacts directly with photons. Deep insidethe material, these photons are gapped to the plasmafrequency by the Anderson-Higgs mechanism, thus pro-tecting the Higgs mode from decay. However, near thesurface, a Higgs excitation can decay into a pair of gap-less vacuum photons whose frequencies add up to ωH.

This corresponds to an effective term in the Hamiltonian

HHiggs/photons ∼∑

ω1+ω2=ωH

(h a†ω1a†ω2

+ h†aω1aω2

) (1)

where h is a quantum field that annihilates a Higgs exci-tation and aω annihilates a vacuum photon of frequencyω. Hence, the energy of the excited Higgs mode is con-verted into entangled photon pairs. In the absence ofthe probe pulse, all photon pairs are generated equally.However, the incoming probe enhances the generation ofpairs in which one of the photons matches the incidentphotons because of the bosonic stimulation factor.

One can understand the origin of the coupling inEq. (1) from the following consideration. In supercon-ductors, the diamagnetic coupling term between light and

matter, Hdia = e2~2

2mc2nsA2, plays a key role, giving rise to

the London equation for the current response to an elec-tromagnetic field and to the Meissner effect. Here, ns

is the superfluid density which, for a coherently excitedHiggs mode, is expected to oscillate at the Higgs fre-quency, ns = ns,0 + δn cos(ωHt). In a quantized descrip-tion of the electromagnetic field, the vector potential A isa linear combination of photon creation and annihilationoperators, schematically of the form A ∼

∑ω(aω + a†ω).

This implies that the diamagnetic term gives rise to theterm (1), where h is a quantum analog of the oscillat-ing part of the superfluid density δn. Interactions of thistype are known to give rise to stimulated emission andparametric down conversion of light41.

These considerations give rise to the following mecha-nism of Higgs amplification. Consider an incident probepulse composed of N photons at frequency ω1 < ωH asit is reflected from a superconductor. The term a†ω1

a†ω2

creates a pair of photons, leading to a state with N + 1photons at ω1 and one ω2 = ωH−ω1 photon, see Fig. 1(c).The amplitude of this process is enhanced by a Bose fac-tor of

√N + 1. As long as the Higgs mode remains ex-

cited, this process leads to amplification of ω1 photonsand generation of ω2 photons, until the Higgs mode isdepleted. The net effect is outgoing light with two fre-quencies, ω1 and ω2, related by ω1 + ω2 = ωH, and theoutgoing light at ω1 having a higher intensity than theincoming signal.

In typical superconductors, the Higgs frequency issmaller than the plasma frequency. Then, incident lightwith ω1 < ωH does not penetrate deeply into the ma-terial, and Higgs amplification is a surface effect. Inparticular, due to the evanescent nature of the wavesinside the material, there are no phase matching con-ditions in this process. This means that there isn’t adiscrete set of frequencies at which Higgs amplificationis resonantly enhanced, and hence the level of ampli-fication is expected to depend smoothly on the probefrequency ω1. The frequency scale for the Higgs mode iscomparable to the superconducting gap. Hence, for manysuperconductors, Higgs amplification is expected to oc-cur in the terahertz, a frequency range that is of greatcurrent interest for fundamental science and technology

3

applications42,43. The next section presents a calculationof Higgs amplification based on a semiclassical descrip-tion of photons using Maxwell’s equations. Our discus-sion is agnostic about the specific microscopic mechanismunderlying light-induced superconductivity and, assum-ing that Higgs oscillations have been coherently excited,directly studies their effect on optical properties.

III. OPTICAL PROPERTIES OF ASUPERCONDUCTOR WITH AN EXCITED

HIGGS MODE

The optical properties of a superconductor with an ex-cited Higgs mode are understood as follows. ConsiderMaxwell’s equations combined with the relation betweenthe electrical current in the material and electric field:

∇×B− ε

c2∂E

∂t= µ0j , (2)

∇×E +∂B

∂t= 0 . (3)

Here ε is the dielectric constant arising from boundcharges and j is the free current density. For z > 0, out-side the superconductor, ε = εout (in vacuum εout = 1,but we allow for interfaces with other media) and we as-sume the current j vanishes; for z < 0, ε = εs, and thecurrent satisfies the London equation,

j = Λ(t)vs . (4)

Here, vs = ~2e∇θ − A, where θ is the superconducting

phase and A is the vector potential. Note that ∂tvs = E.In an equilibrium superconductor, Λ(t) reduces to the

static value Λs = e2ns

m , where ns is the superfluid den-sity. More generally, in a time-modulated superconduc-tor, Eq. (4) describes accurately the total current in-duced by a vector potential A in the limit where A andΛ vary slowly compared to any microscopic frequenciessuch as the superconducting energy gap and the scatter-ing rate 1/τ for electrons in the normal state. We useit here under the assumption that it is at least a goodstarting approximation for the frequencies of interest tous.

In general, the value of Λ will depend on the value ofthe superconducting energy gap. (Although the specialcase of an ideal clean superconductor at T = 0 is anexception to this rule, we expect that the gap dependencewill be manifest in the systems of interest to us, dueto polaronic effects and coupling to impurities.44) Sincethe Higgs mode represents a modulation in the energygap, we expect it to induce a similar modulation in Λ.Consequently, we may write

Λ(t) = Λs + Λme−iωHt + Λ∗me

iωHt (5)

where Λm describes the amplitude of the modulation, andωH is the Higgs mode frequency. For a BCS supercon-ductor, ωH = 2∆, where ∆ is the quasiparticle gap22.

Different frequencies mix due to the time dependencein Λ: incident light with frequency ω1 will produce out-going light with frequencies ω1 and ω2 = ωH − ω1, seeFig. 1(c). Mixing of the Higgs modulation and the sig-nal beam is also expected to induce light at the frequencyω3 = ω1 + ωH . We note, however, that the ω3 frequencylies well above the quasiparticle threshold 2∆, and weexpect that this channel of mode mixing will be sup-pressed relative to the channel at ω2. This can be under-stood by observing that at such high frequencies the cur-rent should be carried primarily by quasiparticles, ratherthan supercurrents, and one might expect the quasipar-ticle current to be less sensitive than the supercurrent tomodulations of the superconducting amplitude. In theremainder of this paper, for the sake of simplicity, weassume that excitation of the ω3 mode is negligible, andwe omit it entirely from our considerations. Our resultsshould be qualitatively applicable, however, as long asexcitation of the ω3 mode is substantially weaker thanthat of the ω2 mode.

Prior to solving the reflection problem, it is useful tofirst consider the evanescent wave solutions inside thesuperconductor. To simplify our discussion, we assumehere that Λm is spatially uniform inside the superconduc-tor. Non-uniformities in Λm due to the short penetrationdepth of the exciting radiation will be taken into accountin the processing of the experimental data, as detailed inAppendix A. For the case of uniform Λm, the evanescentsolutions are characterized by the spacetime dependence(V stands for j, vs, E, or B),

V =(V1e

−iω1t + V∗2eiω2t)eκz + c.c. (6)

Note that the same spatial dependence appears for ω1

and −ω2 terms, since they mix with each other homo-geneously in space. This should be contrasted with thestatic case, where each frequency mode decays with itsown κ. Substituting Eq. (6) into Eq. (4) and collectingterms with frequencies ω1 and −ω2 yields,

j1 = Λsvs,1 + Λmv∗s,2 = ΛsE1

−iω1+ Λm

E∗2iω2

, (7)

j∗2 = Λsv∗s,2 + Λ∗mvs,1 = Λs

E∗2iω2

+ Λ∗mE1

−iω1. (8)

For linearly polarized light, Eν = Eν ex, Bν = Bν ey,and jν = jν ex, where ν ∈ 1, 2. Then, Eq. (3) yieldsBν = κ

iωνEν , and Eq. (2) becomes(κ2c2 + ω2

1ε1 Υω1/ω2

Υ∗ω2/ω1 κ2c2 + ω22ε2

)(E1

E∗2

)= 0 . (9)

Here, Υ ≡ Λm/ε0, ε1 ≡ ε(ω1), and ε2 ≡ ε(−ω2), whereε(ω) = εs−Λs/(ε0ω(ω+ i0+) is the dielectric function ofthe superconductor [Note that, by changing the form ofε(ω), one can generalize the discussion, e.g., to introducedissipation]. The allowed values of κ are obtained byrequiring the determinant of the above matrix to vanish,

κ2±c

2 = −ω21ε1 + ω2

2ε22

±

√(ω2

1ε1 − ω22ε2)

2

4+ |Υ|2, (10)

4

and the fields inside the superconductor satisfy,

Et1,±

Bt1,±

=iω1

κ±≡ η± (11)

Et∗2,±

Bt1,±

=−iω2

(κ2±c

2 + ω21ε1)

Υκ±≡ φ± (12)

Bt∗2,±

Bt1,±

=

(κ2±c

2 + ω21ε1)

Υ≡ γ± (13)

where the superscript t indicates that these act as trans-mitted fields in the reflection problem.

Next, we consider a normally-incident signal beam offrequency ω1, and reflected beams at frequencies ω1 andω2, with

Ei1,B

i1 ∝ e−iω1(t+z

√εout/c) , (14)

Erν ,B

rν ∝ e−iων(t−z√εout/c) . (15)

where ν ∈ 1, 2. The fields are linearly polarized asbefore, Eν = Eν ex, Bν = Bν ey, and Maxwell’s equa-tions constrain them to satisfy Ei

1 = −(c/√εout)B

i1 and

Erν = (c/

√εout)B

rν . Within the superconductor, the

transmitted fields are superpositions of two evanescentwaves of the form (6) with κ = κ±, whose amplitudesare fixed by boundary conditions imposed independentlyon each frequency:

Bi1 +Br

1 = Bt1,+ +Bt

1,− (16)

Ei1 + Er

1 = Et1,+ + Et

1,− (17)

Br∗2 = Bt∗

2,+ +Bt∗2,− (18)

Er∗2 = Et∗

2,+ + Et∗2,− (19)

Combining these with Eqs. (11), (12), and (13) yields

r ≡ Br1

Bi1

=(1 + ζ) +

√εout(η+ + ζη−)

(1 + ζ)−√εout(η+ + ζη−), (20)

r12 ≡Br∗

2

Bi1

=2(γ+ + ζγ−)

(1 + ζ)−√εout(η+ + ζη−), (21)

where

ζ = −γ+ −

√εoutφ+

γ− −√εoutφ−

. (22)

Eqs. (20) and (21) are the main theoretical results of thispaper: r is the reflection amplitude of the signal beam,whereas r12 is the amplitude of the emitted idler mode atthe down-converted frequency ω2 = ωH−ω1, normalizedby the amplitude of the incident signal beam.

Figure 2 shows R = |r|2 and R12 = |r12|2 as a function

of ω1, in the case ωH < ωps, where ωps =√

Λs/(ε0εs)is the superconducting plasma frequency of the mate-rial. Note that, in the absence of dissipation, there isamplification R > 1 over the entire range 0 < ω1 < ωH,and the maximum amplification occurs at ω1 = ωH/2.One can study the effect of dissipation by writing ε(ω) =

(a)

R

ω1/ωH

ωH = 0.8ωps

ωH = 0.8ωps, σr 6= 0

ωH = 0.5ωps

(b)

R12

ω1/ωH

ωH = 0.8ωps

ωH = 0.8ωps, σr 6= 0

ωH = 0.5ωps

FIG. 2: (a) Reflection coefficient R and (b) downconversionintensity R12 as a function of the signal frequency ω1. Weassume εs = 5, εout = 5.62 (corresponding to diamond),and Λm = 0.4Λs. Two different values of the Higgs fre-quency were taken, ωH = 0.5ωps and ωH = 0.8ωps, where

ωps =√

Λs/(ε0εs) is the superconducting plasma frequency.The green curves, corresponding to ωH = 0.8ωps, illustratethe effects of dissipation, taken into account by assumingε(ω) = εs − σ(ω)/(iε0ω) with σ(ω) = σr + iΛs/(ω + i0+),where σr is a real constant; the blue and orange curves in-clude no dissipation, σr = 0. Results are for the model inwhich Λm is uniform in space.

εs − σ(ω)/(iε0ω) in Eqs. (20) and (21) and adding areal part to the conductivity function σ(ω). This sup-presses the reflectivity and also reduces the frequency ω1

at which the maximum occurs, as shown in Fig. 2(a).Note that the net amplification in Fig. 2 is small, of the

order of a few percent. In order to understand this, focuson ω1 = ω2 = ωH/2, where the maximum is obtained,with value

Rmax = 1 +εout ω

2H (D+ −D−)

2

[εout ω2H +D+D−]

2 , (23)

where D± =√

4 (Λs ± |Λm|) /ε0 − εs ω2H . For ω2

H Λs

and Λm Λs, this can be expanded to obtain

Rmax = 1 +|Λm|2

Λ2s

εout ω2H

4 εsω2ps

. (24)

The factor |Λm|2/Λ2s expresses the fact that the amplifi-

cation is proportional to the intensity of the modulation.

5

The factor εoutω2H/(4εsω

2ps) is the square of the London

penetration depth divided by the wavelength of the inci-dent light; it expresses the fact that amplification is weakif the light cannot probe deeply into the superconduc-tor. This suggests that, in order to enhance the ampli-fication, one could consider instances in which the lightcan probe a larger region of the superconductor priorto being reflected. Two possible approaches to achievethis come to mind: First, by using incoming light witha shallow incidence angle to the sample, thus introduc-ing geometrical factors which enhance the effect; second,by studying systems in which the Higgs frequency ex-ceeds the plasma frequency. These possibilities will bediscussed elsewhere45. Note that, in a realistic supercon-ductor, the penetration of light is controlled by the totalplasma frequency ωp, which receives contributions fromall charge carriers, not just the superconducting ones. Itis this total plasma frequency that presumably controlsthe strength of the Higgs amplification in Eq. (24).

In the next section, we will present experimental ev-idence for Higgs amplification in the superconductorK3C60. This is a natural candidate system for Higgsamplification. In this material, the low density of elec-trons (three per C60 molecule) and the weak hoppingbetween C60 molecules conspire to yield an anomalouslysmall plasma frequency ωp = 72meV/~. Optical exci-tation at mid-infrared wavelengths has been shown totransform the high-temperature (T Tc) metallic phaseof K3C60 into a transient non-equilibrium state with thesame optical properties as the low temperature supercon-ductor (T < Tc). The transient state, which is thought tobe a photoinduced non-equilibrium superconductor, dis-plays a saturated reflectivity (R = 1), a gap in the realpart of the optical conductivity σ1 and a divergent low-frequency imaginary conductivity σ2

7. In order to esti-mate the zero-temperature gap, 2∆, at the light-inducedstate, we use the onset temperature for light-inducedsuperconductivity, Tlight−induced ≈ 100 K, which yields2∆ ≈ 30 meV7. This gives an estimate for ωH, whichis expected to be reduced from this value at non-zerotemperature (in the fit to our theory, we find ωH is of theorder of 24 meV/~, see Sec. V). Hence, K3C60 combines arelatively large value of ωH/ωp, as necessary to enhancethe reflectance in Eq. (24), together with the possibil-ity for rapid quenches into the superconducting state bypump pulses, as needed to induce large Higgs oscillations.In particular, by modifying the type of quench protocolused, one may control the amplitude of Higgs oscillations.

The Higgs mode should be rather insensitive to thelack of long-range phase coherence, which is expected notto exceed a few coherence lengths in the photoinducedsuperconducting state. Hence, we expect the mechanismof Higgs amplification to occur in this case, despite thelack of global phase coherence.

IV. EXPERIMENTAL RESULTS

We follow the same protocol described in Ref. 7,8, tophoto-induce the transient optical properties of a non-equilibrium superconductor in K3C60 but using shorterand more intense pulses. To test the hypotheses discussedin the theory section above, we additionally analyzed con-ditions in which the quench was made slower with respectto the Higgs mode frequency of the photo-induced state.The pump pulse FWHM duration τ was tuned to dif-ferent values between 100 fs and 1.8 ps. This range ofpulse durations is interesting because it crosses a char-acteristic time scale τ∗ = h/(24 meV) ∼ 172 fs, whichcorresponds to the period of the amplitude (Higgs) modein the photo-induced superconductor. Note that in ourexperimental geometry the idler mode is not detected. Infuture studies, this mode could provide a measurementof the Higgs frequency and would allow for distinguish-ing Higgs amplification from other types of parametricamplification46,47.

In the excitation regime explored in this experiment,when the pump pulse is significantly longer than τ∗, thetransient state displays a reflection coefficient at the sur-face that is saturated at R = 1 for all frequencies be-low ∼10meV. By reconstructing the complex optical con-ductivity with the same procedure used previously8, weextracted a gapped real optical conductivity σ1 at allfrequencies ω < 10 meV and a divergent low-frequencyimaginary conductivity σ2, indicative of a light-inducedsuperconducting state. Our experiment reveals that asthe pulse duration is made progressively shorter than τ∗,the transient state acquires a reflection coefficient that islarger than R = 1 immediately after the pump, indica-tive of optical amplification through the non-adiabaticcreation of a superconducting state. The real part ofthe optical conductivity σ1 is negative at all frequen-cies ω < 10 meV while its imaginary part σ2 remainsdivergent. Conceptually, the observed dependence onthe pulse duration can be understood from the follow-ing consideration. Although the underlying mechanismof photo-induced superconductivity is still the subject ofdebate, we assume as a first approximation that the “ef-fective” final state Hamiltonian experienced by the low-energy electrons depends only on the total pulse energy,and not its duration (later, we will show evidence thatthe shorter pulses actually do drive the superconductiv-ity more strongly and induce a slightly larger superfluiddensity, although this is a weak effect). We furthermoreassume that the superconducting state lasts longer thanthe probe sequence. Hence qualitatively we assume thatby controlling the pump pulse duration we preserve thefinal effective Hamiltonian for electrons but change therate at which the microscopic parameters are modified.In the case of a superconductor we expect that the Higgs-amplitude mode gets strongly excited when the interac-tion strength is modified on a time scale shorter thanτ∗ = h/2∆.

K3C60 polycrystalline powders were excited at normal

6

FIG. 3: Sketch of the experimental geometry. K3C60 is ex-cited with vertically polarized mid-infrared pulses. The THzprobe pulses are polarized in the horizontal plane.

incidence with 170 meV linearly-polarized mid-infraredpulses. Their duration was tuned from 100 fs to 1.8 ps bychirping them through linear propagation in transparentand highly dispersive CaF2 rods. For all pulse durations,the pulse energy and the number of incident photons weremaintained constant. The transient low-frequency opti-cal properties of photo-excited K3C60 were retrieved asa function of pump-probe delay using transient terahertztime-domain spectroscopy using THz pulses with a band-width that ranged from 1 to 7 THz. These probe pulseswere made to strike the sample at near normal incidence,with a 7 incidence angle (see Figure 3), and they werep-polarized, that is, with the electric field perpendicu-lar to that of the MIR pump pulses. The measurementof the electric field reflected from the sample yielded aphase-resolved measurement of the reflection coefficientand through it the complex optical properties. The pen-etration depth of the mid-infrared pump (200nm) wasshorter than that of the THz probe (600-900nm). To ac-count for this, the data was analyzed as discussed in Ap-pendix A in order to obtain the reflectivity correspondingto an effective semi-infinite and homogeneously pumpedmedium.

Figures 4(a) and 4(b) compare the transient opticalproperties of K3C60 upon photoexcitation with intensemid-infrared pulses of 1.8 ps and 100fs duration, respec-tively. The red curves report the optical properties ofthe equilibrium metallic state, whilst the light-blue dotsrepresent those of the transient state, measured at thepeak of the response. For longer excitation pulses [1.8 ps,Fig. 4(a)] the transient optical properties resemble thoseof the equilibrium superconductor with a reflectivity sat-urated at R = 1, a gapped real conductivity (σ1 ≈ 0)and a divergent imaginary conductivity for all frequen-cies below ∼10meV. For τ = 100fs [Fig. 4(b)] we findthat below ∼10 meV, the reflectivity becomes larger thanR = 1, reaching an average value in the gapped regionof ∼ 1.04, with a maximum of ∼ 1.06. The extracted

real part of the optical conductivity is correspondinglynegative, indicative of negative dissipation. These twoobservations suggest amplification of the incoming lowfrequency THz probe light. Importantly, the imaginarypart of the optical conductivity maintained a 1/ω be-havior below ∼ 10meV, indicating the superconductingnature of the transient state.

The evolution of the optical properties for pump pulseseither shorter or longer than τ∗ can be captured by theaverage value of the reflectivity in the 5-8meV-frequencyrange and the superfluid density extracted by a 1/ω fitto the imaginary part of the optical conductivity at lowfrequencies. Figure 5(a) and 5(b) show the evolution ofthese two quantities as a function of the duration of theexcitation pulse (Fig. 5(b)). The average reflectivity inthe gapped region decreases from ∼ 1.04 to ∼ 1, as thepump pulse duration varies from 100 fs to 1.8 ps. Theblue shaded area in the top panel highlights the regimewhere light amplification is observed. At the same time,the superfluid density of the photoinduced superconduc-tor does not appear to strongly depend on the pulse du-ration of the excitation pulse.

V. COMPARISON OF THEORY ANDEXPERIMENT

The experimental data does not give a direct measure-ment of the Higgs frequency – in the light-induced super-conducting state, the gap measured in σ1(ω) coincideswith the lower edge of the mid-infrared absorption peakof K3C60, suggesting that the superconducting gap 2∆is hidden under the spectral weight of this broad peak.However, there are two independent observations thatgive consistent estimates for ωH. First, as argued ear-lier, the maximum of light amplification occurs slightlybelow ωH/2. The reflectivity measured with the shortestexcitation pulse (τ = 100 fs) shown in Fig. 4(a) wouldthen suggest that ωH is somewhat bigger than 20meV/~.Second, as mentioned earlier, the onset temperature ofthe light-induced superconductivity corresponds to a gap2∆(T = 0) of 30meV/~ for the zero-temperature tran-sient superconductor. This gives an upper bound on ωH,which tracks 2∆(T ).

The measured optical conductivity σ(ω) has Higgsmodulations built in. In order to make a comparisonbetween theory and experiment, we need to model theoptical conductivity of the static superconducting state,σ. We parametrize σ as a sum of Drude and Lorentzianpeaks:

σ(ω) =Λs

γD − iω+

3∑n=1

Bn ω

i(Ω2n − ω2) + γn ω

(25)

The Lorentzians represent the broad mid-infrared ab-sorption peak in K3C60

48. For simplicity, we assume thispeak to be unaffected by the onset of superconductivity,and fix the parameters Bn, Ωn, and γn by fitting to the

7

FIG. 4: (a) Reflectivity and complex optical conductivity (sample-diamond interface) of K3C60 measured at equilibrium(red solid curves) and at the peak of the pump-probe response (light-blue dots) with pump pulse duration of 1800fs. Fornon-equilibrium systems we present inferred local quantities at the sample-diamond interface as discussed in Section V andAppendix A. (b) Same quantities, measured with pump pulse duration of 100fs. The shaded area highlights the frequencyrange where amplification is observed. All data were taken at T=100K and at a fluence of 4.5mJ/cm2. The blue solid curvesare the the optical conductivity and reflectivity calculated from the theoretical model taking into account Higgs modulations.The corresponding fit parameters are shown in Table I.

conductivity of the equilibrium normal state, as discussedin App. C. For the light-induced superconducting phase,we maintain the same values for the Lorentzian peaks,while for the Drude peak we replace γD → 0+ and allowΛs, which plays the role of the static superfluid density,to vary.

Given the static conductivity σ, we use the resultsof Section III to model the Higgs amplification phe-nomenon. We apply Eq. (20) to compute the complexreflection coefficient r at the signal frequency, in whichwe take ε(ω) = εs − σ(ω)/(iε0ω). By inverting the Fres-nel equation, r is then expressed as an effective opticalconductivity σ(ω), see Sec. B, allowing us to make a com-parison of the full complex response of the system.

We find that it is possible to describe the experimen-tal data well by taking ωH = 24 meV/~. Then, for eachvalue of τ , there are only two parameters we allow to

change, the superfluid density Λs and the amplitude ofHiggs modulations, Λm/Λs. We emphasize that all otherparameters in the model are determined in an unbiasedmanner by comparison with the equilibrium optical prop-erties of the normal state. The blue solid curves in fig-ure 4(a,b) shows the complex optical conductivity andthe reflectivity of the superconductor with parametersas chosen in Table I. These results show good agreementwith measurements, despite the limited number of fittingparameters used and the simplicity of the model, whichdoes not take into account the full microscopic detailsof the system. In our fit, we find that the Higgs mod-ulation amplitude increases with decreasing τ , in agree-ment with the expectation that shorter pulses give riseto more rapid quenches into the superconducting state,and therefore to larger Higgs oscillations. In addition, wefind that Λs is slightly larger than its equilibrium value

8

(a)

(b)

FIG. 5: (a) Inferred local transient reflectivity (sample-diamond interface), averaged between 5-8meV, as a functionthe pump pulse duration. The blue-shaded region indicatesthe range of pulse durations where amplification is observed.(b) ωσ2(ω → 0) = Λs, which is proportional to the transientsuperfluid density, extracted by a low frequency 1/ω fit toσ2(ω). All data were taken at T=100K and a constant pumpfluence of 4.5mJ/cm2.

in the normal state Λs,eq. This effect is larger for theshorter pulses. Even though the total pump pulse en-ergy is maintained fixed, the shorter pulses have higherpeak intensities, and can drive the superconductor morestrongly since the pump drives the system nonlinearly.Note that, in order to preserve sum rules, this requiresspectral weight to transfer from higher energies and maybe an indication that the plasmonic peak must be mod-ified in a complete microscopic description of the phe-nomenon.

τ (fs) Λs/Λs,eq Λm/Λs

1800 1.05 0.21000 1.11 0.37100 1.17 0.47

TABLE I: Model parameters for different pump pulse widths.

The goal of our theoretical analysis was to provide thesimplest physical picture of Higgs amplification. There-fore, we limited our discussion to the simplest case,in which the Higgs modulations were assumed to bemonochromatic and spatially uniform. In practice, onemay expect broadening of the Higgs excitation in fre-

quency, due to its finite lifetime, and in momentum, dueto the non-equilibrium character of photoinduced super-conductivity. This more comprehensive picture couldbe obtained by performing frequency and angle resolvedmeasurements. However, these experiments would re-quire more advanced instrumentation, including THz andIR free electron laser radiation. Importantly, this wouldallow for the separate measurement of both signal andidler amplification. This would provide direct access tothe underlying non-linear dynamics of the photoinducedsuperconducting order parameter.

VI. OUTLOOK

We envision several potentially interesting applicationsof the Higgs amplification phenomenon. Of particular in-terest for quantum information is the possibility of gen-eration of entangled photon pairs at THz frequencies, asexpected from Eq. (1). Properties of the entangled pho-tons may be controlled by tuning the intensity, duration,and angle of incidence of the pump beam.

The notions introduced above can be generalized to thenon-linear dynamics of other kinds of non-equilibriumcondensates, including charge and spin density waves,and excitonic condensates.

Systems with several competing orders should exhibitmultiple finite energy collective modes, leading to an ad-ditional richness of the order parameter dynamics. Theinteraction between light and strongly excited collectivemodes opens a new frontier in the study of light-matterinteraction in many-body quantum states.

Acknowledgments

We thank R. Averitt, I. Carusotto, J. Faist, M. Ha-tridge, A. Imamoglu, A. Georges, A. Millis, D. Pekker,and P. Zoller for illuminating discussions. We ac-knowledge financial support from the European ResearchCouncil under the European Union’s Seventh Frame-work Programme (FP7/2007-2013)/ERC Grant Agree-ment No. 319286 (QMAC), the Deutsche Forschungs-gemeinschaft via the excellence cluster ‘The HamburgCentre for Ultrafast ImagingStructure, Dynamics andControl of Matter at the Atomic Scale’ and the prior-ity program SFB925, the National Science Foundation(NSF), the Israel Science Foundation (grant 1803/18),the Harvard-MIT Center for Ultracold Atoms, DARPADRINQS program (award D18AC00014), and the Van-nevar Bush Faculty Fellowship. YW is supported by thePostdoctoral Fellowship of the Harvard-MPQ Center forQuantum Optics and the AFOSR-MURI Photonic Quan-tum Matter (award FA95501610323). DP is grateful forthe hospitality of ITAMP at the Harvard-SmithsonianCenter for Astrophysics, and of the Aspen Center forPhysics, which is supported by NSF grant PHY-1607611.

9

Appendix A: Determining the optical conductivityof an equivalent homogenous medium

In the time-resolved experiments one measures thepump-induced difference in the complex reflected electric

field ∆Er(ω). The “raw” complex reflection coefficient inthe photo-excited state rpumped(ω) can then be extractedby inverting the following equation:

∆Er(ω)

E0r (ω)

=rpumped(ω)− r0(ω)

r0(ω)(A1)

where r0(ω) is the unperturbed complex reflection coef-ficient of K3C60 known from broadband FTIR measure-ment and E0

r (ω) is the reflected electric field in the unper-turbed state. If the pump light penetrates in the sampleseveral times more than the probe light, one can assumethat the probe pulse samples a volume in the materialthat has been homogenously transformed by the pump.In this case it is possible to directly extract the complexvalued optical response functions by inverting the Fresnelequations.

The conditions assumed in the previous paragraph arenot correct for the experiments presented here, becausethe penetration depth of the mid-infrared pump (220 nm)is at least three times shorter than that of the THz probe(600-900 nm). Nevertheless, it is instructive to see whatresults for the optical conductivity would be obtained ifthe assumption is made.

In the left panel of Fig. 6, we show raw data for thereflectivity, in equilibrium and after photoexcitation bypulses of three different lengths. In the second and thirdpanels, we show the values of σ1(ω) and σ2(ω) that onewould obtain from the observed reflectivities using theFresnel equations, assuming optical conductivities inde-pendent of distance from the surface.

In contrast, the curves in Fig. 4 were obtained aftertaking account that the penetration depth of the proberadiation is longer than the that of the pump pulse, usingthe following approach. As the pump penetrates in thematerial, its intensity is reduced and it will induce pro-gressively weaker changes in the refractive index of thesample. This is modeled by “slicing” the probed thick-ness of the material into thin layers where we assumethat the pump-induced changes in the refractive indexare proportional to the pump intensity in the layer, i.e.n(ω, z) = n0(ω) + ∆n(ω)e−αz where n0(ω) is the un-perturbed complex refractive index, α is the attenuationcoefficient at the pump frequency, and z is the spatialcoordinate along the sample thickness.

For each probe frequency ωi, the complex reflectioncoefficient r(∆n) of such multilayer stack is calculatedwith a characteristic matrix approach49 keeping ∆n asa free parameter. As equation (A1) relates directly the

measured quantity ∆Er(ω)/E0r (ω) to the changes in re-

flectivity we can extract ∆n by minimizing numerically:∣∣∣∣∣∆Er(ωi)E0r (ωi)

− r(ωi,∆n)− r0(ωi)

r0(ωi)

∣∣∣∣∣ (A2)

Note that ∆n(ω) represents the pump-induced change inthe refractive index at the surface, where the pump hasnot yet been attenuated by the absorption in the mate-rial. By taking n(ω) = n0(ω) + ∆n(ω) one can recon-struct the optical response functions of the material as ifit had been homogeneously transformed by the pump.

The blue data points in Fig. 4 show the data pro-cessed in this manner to obtain the optical propertiesapplicable to the region closest to the surface, where theexcitation pulse is strongest. Specifically, panels in thesecond and third columns show the effective values ofσ1(ω) and σ2(ω) deduced for this region, while the firstcolumn shows the reflectivity that would be expected ifthese values of the optical conductivity were to hold in-dependent of depth. The blue solid curves show valuesobtained from the theoretical model with Higgs modu-lation. Comparing Figs. 4 and 6, we see that while thecurves differ in detail, the enhanced optical response forthe shortest pulse data, seen in both figures, tends to sup-port our theoretical model. In particular, while we do notactually observe amplification in the raw reflected signal,our analysis suggests that amplification would have beenobserved if the pump pulse were able to penetrate moredeeply into the sample.

Appendix B: Expressing the reflectivity in terms ofa complex conductivity function

In the theory, we computed the complex reflection am-plitude r, using Eq. (20). To convert this to a conductiv-ity function, we used the Fresnel relation for the reflectionamplitude at the interface between two media

r(ω) =

√εout − i

√−ε(ω)

√εout + i

√−ε(ω)

. (B1)

For us, the first medium was diamond, which to agood approximation has a frequency-independent dielec-tric function

√εout = 2.37. This relation was inverted,

ε(ω) = εout

(1− r(ω)

1 + r(ω)

)2

, (B2)

in order to obtain the complex conductivity using σ(ω) =−iε0ω[ε(ω)− εs].

Appendix C: Optical conductivity in the normalstate

We model the optical conductivity σ(ω) of the equilib-rium normal state as the sum of a Drude peak, represent-ing the conduction band, and a sum of three Lorentzians,

10

FIG. 6: The left panel shows raw data for the reflectivity (sample-diamond interface), in equilibrium and after photoexcitationby pulses of three different lengths. The second and third panels show the values of σ1(ω) and σ2(ω) that one would obtain fromthe observed reflectivity via the Fresnel equations, assuming optical conductivities independent of distance from the surface.Note that this neglects that the intensity of the pump pulse decays as a function of distance from the surface of the sampleand the probe is sampling an inhomogenously transformed volume.

representing a broad mid-infrared absorption peak48.

σeq(ω) =Λs,eq

γD − iω+

3∑n=1

Bn ω

i(Ω2n − ω2) + γn ω

(C1)

A fit to the optical conductivity of the equilibriumnormal state in the measured range 4meV/~ < ω <

100meV/~ (see Figure 7) gives the following values:Λs,eq = 3, 470 Ω−1cm−1meV/~, γD = 3.56 meV/~,B1 = 18, 300 Ω−1cm−1meV/~, Ω1 = 70.4 meV/~,γ1 = 86.6 meV/~, B2 = 4, 600 Ω−1cm−1meV/~,Ω2 = 26.1 meV/~, γ2 = 34.0 meV/~, B3 =2, 400 Ω−1cm−1meV/~, Ω3 = 102.6 meV/~, γ3 =35.0 meV/~.

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