## Optically manipulating superconductors

In superconductors, electrons of opposite momenta pair to form a highly correlated state that manages to flow without encountering any resistance. Matsunaga *et al.* manipulated the wavefunction of these pairs in the superconductor NbN with an electromagnetic pulse that they transmitted through a thin layer of the material (see the Perspective by Pashkin and Leitenstorfer). The superconducting gap, which is the energy needed to break the pairs apart, oscillated at twice the frequency of the pulse's electric field. When they matched this frequency to half the gap, the authors excited a collective mode in the superconductor called the Higgs mode, a relative of the Higgs boson in particle physics.

## Abstract

Superconductors host collective modes that can be manipulated with light. We show that a strong terahertz light field can induce oscillations of the superconducting order parameter in NbN with twice the frequency of the terahertz field. The result can be captured as a collective precession of Anderson’s pseudospins in ac driving fields. A resonance between the field and the Higgs amplitude mode of the superconductor then results in large terahertz third-harmonic generation. The method we present here paves a way toward nonlinear quantum optics in superconductors with driving the pseudospins collectively and can be potentially extended to exotic superconductors for shedding light on the character of order parameters and their coupling to other degrees of freedom.

Macroscopic quantum phenomena, such as superconductivity and superfluidity, emerge in a variety of physical systems, such as metals, liquid helium, ultracold atomic quantum gases, and neutron stars. One manifestation of the macroscopic quantum nature is the appearance of characteristic collective excitations. Indeed, phenomena associated with collective modes, such as second sound and spin waves in condensates, have been revealed in superfluid helium (*1*, *2*) and in ultracold atomic gases (*3*, *4*).

Generally, collective modes in ordered phases arising from spontaneous symmetry breaking are classified into (i) gapless phase modes [Nambu-Goldstone (NG) mode] and (ii) gapped amplitude modes (Higgs mode) (*5*–*7*). In charged-particle systems such as superconductors with long-range Coulomb interactions, the gapless NG mode becomes massive; that is, its energy is elevated to the plasma frequency as a result of the coupling to the gauge boson (photon field), which is referred to as the Anderson-Higgs mechanism (*8*, *9*). The Higgs amplitude mode in superconductors has been also studied theoretically (*6*, *10*–*15*); because it is not accompanied by charge fluctuations, it does not couple directly to electromagnetic fields in the linear response regime. This is why the Higgs mode in conventional *s*-wave superconductors was observed only recently after a nonadiabatic excitation with a monocycle THz pulse (*16*); previous observations were in a special case where the superconductivity coexists with charge density wave that makes the Higgs mode Raman-active (*17*, *18*). Hence, many questions regarding the Higgs mode in superconductors remain unresolved: How does the mode couple to strong electromagnetic fields in a nonlinear regime? Is it possible to dynamically control the Higgs mode and therefore the superconducting order parameter?

Recent advances in the intense THz generation technique (*19*, *20*) open a new avenue for studying matter phases in nonequilibrium conditions. Amplitude- and phase-resolved spectroscopy using multi-THz pulses has been realized (*21*), enabling the study of coherent transients in many-body systems in low-energy ranges. The purpose of the present work is to explore coherent nonlinear interplay between collective mode in a superconductor and THz light field by investigating the real-time evolution of the order parameter under the driving field of a multicycle (as opposed to monocycle) THz pulse.

In order to study evolutions on a picosecond time scale, we performed THz pump–THz probe spectroscopy (*16*, *22*) (Fig. 1A). To generate an intense multicycle THz pulse as a coherent driving source, we first created an intense monocycle THz pulse by the tilted-pulse front method with a LiNbO_{3} crystal (*19*, *23*). The monocycle pulse then goes through a band-pass filter to produce a narrow-spectrum multicycle pulse. Three band-pass filters are used to generate the different center frequencies at 0.3, 0.6, or 0.8 THz, respectively, with their power spectra displayed in Fig. 1B. These photon energies are all below the superconducting gap of our NbN sample in the low-temperature limit, which is 1.3 THz (Fig. 1C); this implies that the pump pulse does not generate quasi-particles (QPs) in one-photon processes at low temperatures. The sample is an *s*-wave superconductor NbN thin film with 24-nm thickness grown on an MgO substrate (*24*) with superconducting critical temperature (*T*_{c}) = 15 K. The ultrafast dynamics of the superconducting order parameter driven by the multicycle pump pulse is then probed through the transmittance of a monocycle THz pulse that enters the sample collinearly with the pump pulse with a variable time delay. In general, we can detect the temporal waveform of the transmitted probe THz electric field, *E*_{probe}, by varying the time delay of another optical gate pulse and using the electrooptic (EO) sampling method. In this experiment, we fixed the timing of the optical gate pulse such that, in the absence of the pump, *E*_{probe} at this timing monotonically changes with temperature, reflecting the change of the order parameter. Temporal evolution of the order parameter induced by the THz pump is sensitively monitored through the change of *E*_{probe} relative to its value in the absence of the pump as a function of the pump-probe delay time, *t*_{pp} (*16*, *22*); we denote this change as δ*E*_{probe}. For details, see (*25*). In the present case, we investigated the order parameter dynamics in the presence of coherently oscillating multicycle pump fields. The temporal waveform of the pump THz electric field *E*_{pump} is displayed in Fig. 1D for the center frequency of ω = 0.6 THz and the maximum electric field of 3.5 kV/cm. Figure 1E shows δ*E*_{probe} with *t*_{pp} at *T *= 14 to 15.5 K, at which ω is greater than 2Δ(*T*). In this temperature range, the probe electric field gradually increased as a function of *t*_{pp} to reach an asymptotic value, which indicates a reduction of the order parameter resulting from QP excitations (*22*). By contrast, at temperatures below 13 K where 2Δ(*T*) exceeded ω (Fig. 1F), the long-term reduction of the order parameter became less prominent as temperature decreased, because the QP excitation is suppressed. We immediately noticed that an oscillatory signal emerges with a frequency of 1.2 THz (=2ω) during the pump pulse irradiation, which indicates that the order parameter oscillates with twice the frequency of the driving field.

We physically captured the 2ω oscillation of the order parameter in terms of the precession of Anderson’s pseudospins (*26*, *27*). In the Bardeen-Cooper-Schrieffer (BCS) ground state, two electrons with wave numbers ** k** and

**–**form a spin-singlet Cooper pair, with the BCS wave function given by (1)where and denote unoccupied and occupied, respectively,

*k***and**

*k***–**. In Anderson’s pseudospin formalism, the states and are represented by up and down pseudospins, respectively (

*k**25*), where the BCS ground state is thought of as a quantum superposition of up and down pseudospins with amplitudes

*v*

**and**

_{k}*u*

**, respectively, for each**

_{k}**(Fig. 2A). The normal state has the pseudospins all up for |**

*k***<**

*k*|*k*

_{F}and all down for |

**| >**

*k**k*

_{F}at

*T*= 0 (Fig. 2B), where

*k*

_{F}is the Fermi wave number. Within this representation, the BCS Hamiltonian simply reads , where

**σ**= (σ

_{k}

_{k}*, σ*

^{x}

_{k}*, σ*

^{y}

_{k}*) is the Anderson’s pseudospin (*

^{z}*26*) mapped onto the Bloch sphere (Fig. 2C), whereas

**= (–Δ’, –Δ”, ε**

*b*_{k}**) is a pseudomagnetic field acting on**

_{k}**σ**. Here, ε

_{k}**is the band dispersion measured from the Fermi energy, is the complex order parameter, and**

_{k}*U*(>0) is the pairing interaction. In equilibrium, each pseudospin is aligned along the pseudomagnetic field. The time evolution of the BCS state is then described by the Bloch equation for the pseudospins, (2)that is, the time evolution of the BCS state is represented as the motion of the pseudospins in the pseudomagnetic field. For a spatially homogeneous monochromatic electric field

**exp(**

*E**i*ω

*t*) irradiated onto a superconductor, the

*z*component of

**, in the nonlinear response regime, becomes (**

*b*_{k}*25*) (3)where

**(**

*A**t*) is the vector potential representing the electric field. We can see that the leading term in the energy variation is ~

**(**

*A**t*)

^{2}, which is intuitively because electrons with charge –

*e*hybridize with holes with charge +

*e*in the condensed pair, leading to a nonlinear coupling between light and the condensate and to the pseudospin precession with the frequency 2ω. The coherent collective precession of the pseudospins (Fig. 2D) manifests itself macroscopically as the order parameter oscillation, and the change of the order parameter in turn affects the pseudomagnetic field. We calculated the time evolution of the order parameter self-consistently by numerically solving the Bloch equation for the multicycle pulse [see (

*25*) for details]. The simulation indeed exhibits the order parameter oscillation with twice the frequency of the external electric field (Fig. 2E).

An analytic solution for the linearized Bloch equation can in fact be obtained (*28*), where the temporal variation of the order parameter amplitude [Δ(*t*) = Δ + δΔ(*t*)] turns out to behave around 2ω = 2Δ as
(4)with ϕ being a phase shift that depends on ω. The divergence of the amplitude at 2ω = 2Δ can be interpreted as a resonance between the induced pseudospin precession with frequency 2ω and the collective amplitude mode of the order parameter, namely the Higgs mode (*16*), with frequency 2Δ. We can then relate the nonlinear current density induced by the external ac field with the change in the order parameter δΔ(*t*) via (*25*) as
(5)which takes a form of the London equation. This enables us to regard the nonlinear current, which reflects the dynamics of the order parameter, as a part of the supercurrent.

Because the coherent interaction between the superconductor and the THz electromagnetic radiation results in the nonlinear (2ω, 4ω, 6ω…) oscillations of the order parameter (Eq. 3), the nonlinear current in Eq. 5 should accommodate higher odd-order harmonics in the transmitted pump THz pulse. To confirm this, we performed a nonlinear transmission experiments for the pump THz pulse (without the probe pulse) (Fig. 3A). Figure 3B shows the waveforms of the transmitted pump THz pulse above (15.5 K) and below (10 K) *T*_{c} = 15 K; the waveform of the transmitted pulse below *T*_{c} is considerably distorted. The power spectra of the transmitted pump THz pulse are shown in Fig. 3C on a logarithmic scale and in Fig. 3D on an expanded linear scale at various temperatures. Below *T* = 13 K, a prominent peak appears around 1.8 THz, which indeed coincides with 3ω. The intensity at 3ω as a function of the pump electric field strength, depicted in Fig. 3E on a log-log scale, obeys |*E*_{pump}|^{6} dependence, endorsing that the signal arises from the third harmonic generation (THG). The THG intensity at 10 K normalized by that of the incident pump pulse reaches 8 × 10^{−5}, which is high for a film with only 24-nm thickness and 3.5-kV/cm peak electric field (*29*). We could increase the interaction length up to about 0.2 μm, the penetration depth of the sample at 0.6 THz (*30*), which would result in even higher conversion efficiency. A shift of THG peak energy with temperature is discerned in Fig. 3D, which is attributed to the softening of the Higgs mode [2Δ(*T*)] toward *T*_{c}. The THG signal disappears before the softening completes because the resonant enhancement is rapidly suppressed when Δ(*T*) moves out of the narrow bandwidth of the incident pump field.

The 2ω oscillation of the order parameter and the THG were also observed for ω = 0.3 and 0.8 THz pumping. Figure 4B summarizes the temperature dependence of the THG intensities for ω = 0.3, 0.6, and 0.8 THz. For ω = 0.3 and 0.6 THz, the THG signal peaks at 13.5 and 10 K, respectively, whereas the THG signal for ω = 0.8 THz monotonically increases with decreasing temperature. Comparing the temperature dependence of the order parameter 2Δ(*T*) (Fig. 4A) with twice the pump frequency 2ω (=0.6, 1.2, and 1.6 THz), one can deduce that the peak in the THG does fall on 2ω = 2Δ(*T*). The THG intensity in Eq. 5 depends on the change of the order parameter amplitude, which is resonantly enhanced when 2ω approaches the inherent Higgs amplitude mode 2Δ(*T*). Indeed, the temperature dependence of the THG intensity calculated with Eq. 5 and shown in Fig. 4C agrees qualitatively with experiment in Fig. 4B. We conclude that the resonance of the Anderson’s pseudospin precession in the superconductor is achieved by irradiation of THz pump, which results in large THG. The theoretical results in Fig. 4C exhibit sharp resonance peaks, which result from the lifetime of the Higgs mode assumed to be infinite (i.e., power-law decay) within the BCS approximation (*10*, *12*). In contrast, the observed resonance widths in Fig. 4B are finite, which may be caused by decaying channels for the Higgs mode and the finite spectral width of the pump pulse (Fig. 1B). There are in fact various possible decay processes—including scattering with single-particle excitations, impurities, phonons, or low-frequency NG mode that emerges near *T*_{c} (*31*)—for which systematic studies are desirable (*32*).

We last note that superconductors are known to exhibit highly nonlinear responses near the critical field or temperature, giving rise to nonlinear *I-V* characteristics and higher-order harmonics in transport measurements with a frequency range from a few hertz to microwave (*33*–*35*). By contrast, the large nonlinear optical effect revealed here originates from resonance of ac fields to the collective amplitude mode of the order parameter, which leads to the strong THG emission in THz frequency range.

The time-resolved observation of the THz higher-order harmonics will provide a unique avenue for probing ultrafast dynamics of the order parameter in out-of-equilibrium superconductors. It is highly intriguing to explore the quantum trajectories of the pseudospins on Bloch sphere in the nonperturbative light-matter interaction regime with much higher THz fields, which would result in a dynamics of superconducting order parameter not attained in conventional regimes. The present scheme using the nonlinear coupling between pseudospins and light can be also extended to unconventional superconductors, such as the cuprate or iron-pnictide, which would provide new insight about the high-*T*_{c} superconductivity and the interplay between the superconducting phase and other coexisting or competing orders.

**Correction (8 September 2014):** Equation 2 has been updated.

## Supplementary Materials

www.sciencemag.org/content/345/6201/1145/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S3

References

## References and Notes

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Supplementary materials are available on
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Recently, a time-resolved Raman experiment in a high-
*T*_{c}cuprate has been reported and accounted for by the dynamics of individual pseudospins corresponding to the charge fluctuations induced by an impulsive stimulated Raman process (*38*). - ↵
- ↵ The conversion efficiency was estimated without compensating the frequency dependence of the EO-sampling detection coefficient for the 1-mm-thick ZnTe crystal. The correction would make the conversion efficiency even higher.
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**Acknowledgments:**This work was supported by a Grant-in-Aid for Scientific Research (grant nos. 25800175, 22244036, 20110005, 25104709, 25800192, and 26247057) and Advanced Photon Science Alliance by the Photon Frontier Network Program from Ministry of Education, Culture, Sports, Science, and Technology.