Abstract
Many applications in photonics require alloptical manipulation of plasma waves^{1}, which can concentrate electromagnetic energy on subwavelength length scales. This is difficult in metallic plasmas because of their small optical nonlinearities. Some layered superconductors support Josephson plasma waves^{2,3}, involving oscillatory tunnelling of the superfluid between capacitively coupled planes. Josephson plasma waves are also highly nonlinear^{4}, and exhibit striking phenomena such as cooperative emission of coherent terahertz radiation^{5,6}, superconductor–metal oscillations^{7} and soliton formation^{8}. Here, we show that terahertz Josephson plasma waves can be parametrically amplified through the cubic tunnelling nonlinearity in a cuprate superconductor. Parametric amplification is sensitive to the relative phase between pump and seed waves, and may be optimized to achieve squeezing of the orderparameter phase fluctuations^{9} or terahertz singlephoton devices.
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Main
Cuprates are strongly anisotropic superconductors in which transport is made threedimensional by Josephson tunnelling between the Cu–O planes. Tunnelling reduces the superfluid density in the direction perpendicular to the planes, and hence the frequency of the plasmon to below the average pair breaking gap. Weakly damped oscillations of the superfluid sustain transverse Josephson plasma waves (JPWs) that propagate along the planes.
Consider a complex superconducting order parameter in the ith Cu–O plane ψ_{i}(x, y, t) = ψ_{i}(x, y, t) expiθ_{i}(x, y, t), which depends on two inplane spatial coordinates x and y and on time t. For a terahertzfrequency optical field polarized perpendicular to the planes, excitations above the superconducting gap are negligible and the modulus of the order parameter ψ ^{2} (number of Cooper pairs) is nearly constant in space and time. Hence, the electrodynamics is dominated by the orderparameter phase θ_{i}(x, y, t). Ignoring at first the spatial dependence of the phase, the local tunnelling strength can, from the Josephson equations^{10}, be expressed in terms of an equivalent inductance, L, which depends on the local interlayer phase difference θ_{i, i+1}(t) = θ_{i}(t) − θ_{i+1}(t) as L(θ_{i, i+1}(t)) ∼ L_{0}/cos(θ_{i, i+1}(t)) (i and i + 1 are the indices for two neighbouring layers). Here L_{0} = ℏ/2eI_{c} is the inductance at equilibrium, ℏ the reduced Planck’s constant, 2e the Cooper pair charge and I_{c} the critical current. Denoting the capacitance of the Cu–O planes with a constant C, we express the Josephson plasma resonance (JPR) frequency as ω_{JP}^{2} = 1/(L(θ_{i, i+1}(t))C) = ω_{JP0}^{2} cos[θ_{i, i+1}(t)], where ω_{JP0}^{2} = 1/L_{0}C is the equilibrium value. Correspondingly, the oscillator strength f ∼ ω_{JP}^{2} for the plasma oscillations^{11} is also a function of the interlayer phase, and scales as f = f_{0} cos[θ_{i, i+1}(t)]. The dependence of the oscillator strength f on the cosine of the superconducting phase corresponds to a thirdorder optical nonlinearity.
According to the second Josephson equation^{10}, the interlayer phase difference θ_{i, i+1}(t) advances in time with the time integral of the interlayer voltage drop, as ∂[θ_{i, i+1}(t)]/∂t = 2eV/ℏ. For an optical field made resonant with the Josephson plasma frequency E(t) = E_{0}sin(ω_{JP0}t), the interlayer phase oscillates as θ_{i, i+1}(t) = θ_{0} cos(ω_{JP0}t), where E_{0} is the field amplitude and θ_{0} = (2ed/ℏω_{JP0})E_{0} (d ∼ 1 nm is the interlayer distance). This implies that the oscillator strength f(t) = f_{0} cos(θ_{0} cos(ω_{JP0}t)) ≈ f_{0}(1 − (θ_{0}^{2} + θ_{0}^{2} cos(2ω_{JP0}t))/4) is modulated at a frequency 2ω_{JP0}, whenever the field E_{0} is large enough to make the phase excursion θ_{0} sizeable.
Figure 1 provides a pictorial representation of this physics. We plot a vector that represents both the phase difference θ_{i, i+1}(t) (vector angle) and the oscillator strength f(t) (vector length).
This picture shows how, for small driving fields, only θ_{i, i+1}(t) oscillates at the driving frequency ω_{JP0}, whereas for larger fields these oscillations are accompanied by a 2ω_{JP0} modulation of the oscillator strength f(t).
Note also that the phenomena discussed above can be cast in terms of a Mathieu equation (see Supplementary Information 2). Thus, a 2ω_{JP0} modulation of the oscillator strength can serve as a pump for the parametric amplification of a second, weak plasma wave at frequency ω_{JP0}. In this paper we demonstrate experimentally this effect in La_{1.905}Ba_{0.095}CuO_{4} (LBCO_{9.5}), a cuprate superconductor with the equilibrium JPR at ω_{JP0}≅0.5 THz.
Terahertz pulses, generated with a photoconductive antenna^{12}, were used as a weak probe of JPWs (a schematic drawing of the measurement geometry is reported in Supplementary Information 1). A typical terahertzfield trace^{13} reflected by the sample is shown in Fig. 2a. Two different measurements are displayed: one taken below (red line) and the other one above (black line) the superconducting transition temperature T_{c} = 32 K. In the superconducting state, longlived oscillations with ∼2 ps period were observed on the trailing edge of the pulse, indicative of the JPR at ω_{JP0}≅0.5 THz. Figure 2b (solid red line) displays the corresponding reflectivity edge in frequency domain. The solid lines in Fig. 2c, d are the complex dielectric permittivity ɛ(ω) and the loss function L(ω) = −Im(1/ɛ(ω)) = ɛ_{2}(ω)/(ɛ_{1}(ω) + ɛ_{2}(ω))^{2}. L(ω) peaks at ω_{JP0}, where the real part of the dielectric permittivity, ɛ_{1}(ω), crosses zero.
These optical properties could be reproduced well by solving the wave equation in the superconductor in one dimension^{8} (see Supplementary Information 3), which yields the space and timedependent orderparameter phase θ_{i, i+1}(x, t) (Fig. 2e) and the corresponding changes (negligible in linear response regime) of the oscillator strength f = f_{0} cos[θ_{i, i+1}(x, t)] (Fig. 2f). The reflectivity, complex permittivity, and loss function (dashed lines in Fig. 2b–d, respectively), calculated from these simulations by solving the electromagnetic field at the sample surface^{8}, are in good agreement with the experimental data.
Amplification of a weak JPW like the one above (probe field) was achieved by mixing it with a second, intense pump field, which resonantly drove the Josephson phase to large amplitudes. Quasisingle cycle terahertz pulses, generated in LiNbO_{3} with the tilted pulse front method^{14} (yielding field strengths up to ∼100 kV cm^{−1}), were used to excite these waves in the nonlinear regime. The spectral content of these pulses extended between 0.2 and 0.7 THz, centred at the JPR frequency (see Supplementary Information 4). Note that the pump field strength used in this experiment exceeds the expected threshold to access the nonlinear regime, defined by θ_{0} = (2eE_{0}d)/(ℏω_{JP0}) ∼1 and corresponding in this material to field amplitudes E_{0} = (ℏω_{JP0})/(2ed) ∼20 kV cm^{−1}.
In Fig. 3, we report the timedelaydependent, spectrally integrated pump–probe response of LBCO_{9.5}. Changes in the reflected probe field were measured at one specific probe sampling time (τ = τ_{0}) as a function of pump–probe time delay (t). For a system in which the optical properties are dominated by a single plasma resonance, the spectrally integrated response is proportional to the plasma oscillator strength f.
As shown in Fig. 3a, b, this integrated response exhibits a reduction of the signal and oscillations at a frequency ∼2ω_{JP0}. Note that the oscillation frequency did not depend on the pump electric field strength E_{0}, whereas the frequency reduced when the base temperature of the experiment was increased, consistent with the reduction of the equilibrium ω_{JP0} (see Supplementary Information 5 and 6). The effect completely disappeared at T > T_{c}.
Hence, the theoretically predicted 2ω_{JP0} modulation of the total oscillator strength f (see above) is reproduced well by the data in Fig. 3. This response could also be simulated using the space and timedependent sineGordon equation (see Fig. 3c, d), yielding good agreement between experiment and theory (see blue lines in Fig. 3a, b).
Note that here we only analyse pump–probe delays t ≳ 0 ps, because the response at the earliest times suffers from perturbed free induction decay^{15}. This effect consists in the deformation of a coherent signal, which occurs when the pump strikes the sample during the oscillatory relaxation of the probe (for t ≲ 0 ps in our case).
Selected timedomain probe traces measured before and after excitation are displayed in Fig. 4. Crucially, at specific time delays the probe field is amplified (Fig. 4a), whereas at other delays it is suppressed (Fig. 4b) with respect to that measured at equilibrium.
In Fig. 5 we report the timedelaydependent and frequencydependent loss function L(t, ω) = −Im(1/ɛ(t, ω)) = ɛ_{2}(t, ω)/(ɛ_{1}(t, ω) + ɛ_{2}(t, ω))^{2}, a quantity that peaks at the zero crossing of ɛ_{1}(t, ω) and is always positive for a dissipative medium (that is, a medium with ɛ_{2}(t, ω) > 0). The experimental data of Fig. 5a show that, after excitation, L(ω) acquires negative values around ω_{JP0} (red regions). This is indicative of a negative ɛ_{2}(t, ω), and hence amplification. The effect is strong near zero pump–probe time delay, then disappears after ∼1 ps, and is observed again periodically with a repetition frequency of ∼2ω_{JP0}. The same effect appears also in the simulations (Fig. 5b), yielding periodic amplification at a repetition frequency of 2ω_{JP0}.
In Supplementary Information 7 we report additional quantitative estimates of the degree of amplification. We include a negative absorption coefficient and a reflectivity larger than 1 at ω ≃ ω_{JP0}. The extracted values are α = (−0.090 ± 0.003) μm^{−1} and R = (1.042 ± 0.008), respectively.
The data and simulations reported here demonstrate that terahertz JPWs can be parametrically amplified, exhibiting the expected sensitivity to the relative phase of strong and weak fields mixed in this process and the oscillatory dependence at twice the frequency of the drive.
Parametric amplification of terahertz light based on nonlinear optical techniques has already been shown in the past^{16}. However, the physics demonstrated here extend beyond potential applications in photonics, directly leading to coherent parametric control of the superfluid in layered superconductors, and providing a means to manipulate the properties of the material or to probe them in new ways^{17}.
Moreover, the ability to amplify a plasma wave could lead to terahertz singlephonon manipulation devices that may operate above 1 K temperatures. These would exploit concepts that to date have been developed only at microwave frequencies and in the millikelvin regime^{18,19,20,21}. Finally, the parametric phenomena discussed here can also potentially be used to squeeze^{9,22,23} the superfluid phase, and may lead to control of fluctuating superconductivity^{24}, perhaps even over a range of temperatures above T_{c} (refs 25,26).
Methods
Laser pulses with 100 fs duration and ∼5 mJ energy from a commercial Ti:Sa amplifier were split into three parts (92%, 7%, 1%). The most intense beam was used to generate strongfield terahertz pulses with energies up to ∼3 μJ via optical rectification in LiNbO_{3} with the tilted pulse front technique. These were collimated and then focused at normal incidence onto the sample (with polarization perpendicular to the Cu–O planes, that is, along the c axis) using a Teflon lens and a parabolic mirror, with focal lengths of 150 mm and 75 mm, respectively. The pump spot diameter at the sample position was ∼2.5 mm. The pump field strength was calibrated with electrooptic sampling in a 0.2mmthick GaP crystal, yielding a maximum value of ∼100 kV cm^{−1} (see Supplementary Information 4).
The 7% beam was used to generate the terahertz probe pulses with a photoconductive antenna. These had a dynamic bandwidth of 0.1–3 THz, corresponding to a time resolution of ∼250 fs. The caxis optical properties of the superconductor (both at equilibrium and throughout the pumpinduced dynamics) were probed in reflection geometry, with a probe incidence angle of 45° and a spot diameter at the sample position of ∼2 mm. The reflected probe pulses were electrooptically sampled in a 1mmthick ZnTe crystal, using the remaining 1% of the 800 nm beam. This measurement procedure returned the quantity E_{probe}(t, τ), with t being the pump–probe delay and τ being the electrooptic sampling time coordinate.
The sample used in our experiment was a single crystal of La_{1.905}Ba_{0.095}CuO_{4} cut and polished along an ac oriented surface ∼3 × 3 mm in size. Its equilibrium optical response in the superconducting state was determined by measuring the complexvalued E_{probe}(ω) (pump off) both at T < T_{c} and T > T_{c}, and by referencing it to the normalstate reflectivity measured in another crystal coming from the same batch of samples^{27}.
The spectrally integrated pump–probe traces of Fig. 3 were measured by scanning the pump–probe delay t at a fixed sampling time τ = τ_{0}. This was chosen to be on the trailing edge of the pulse, where the JPR oscillations are present. Note that the observed dynamics, and in particular the 2ω_{JP0} oscillations, did not depend significantly on the specific τ_{0} value at which the scan was performed.
The frequency and timedelaydependent loss function of Fig. 5 (as well as all complex optical properties of the perturbed material) was determined by applying Fresnel equations^{11} to the pumpinduced changes in the reflected electric field. These were normalized by independently recording E_{probe}(t, τ) in the presence and absence of the terahertz pump field. Note that there was no need to take into account any pump–probe penetration depth mismatch in the calculation.
In the simulations, the Josephson phase evolution θ_{i, i+1}(x, t) was determined through the onedimensional sineGordon equation^{6}:
γ being a damping constant, c the speed of light, ɛ_{r} the dielectric permittivity, and ω_{JP0} the equilibrium JPR frequency. This equation was solved numerically, with the terahertz pump and probe fields overlapping at the vacuum/superconductor interface. For more details on this topic, we refer the reader to Supplementary Information 3.
Data availability.
The data supporting the plots within this paper and other findings of this study are available from the corresponding author on request.
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Acknowledgements
The research leading to these results received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/20072013)/ERC Grant Agreement no. 319286 (QMAC). We acknowledge support from the Deutsche Forschungsgemeinschaft via the excellence cluster ‘The Hamburg Centre for Ultrafast Imaging—Structure, Dynamics and Control of Matter at the Atomic Scale’ and the Priority Program SFB925. Work performed at Brookhaven was supported by US Department of Energy, Division of Materials Science under contract no. DEAC0298CH10886.
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A.C. conceived the project together with S.R. S.R. built the terahertz pump–probe experimental setup, performed the measurement and analysed the experimental data with the support of D.N. The simulations were performed by S.R. and E.C., with input from Y.L., S.R.C. and D.J. The results were discussed and interpreted by S.R., Y.L. and A.C. The sample was grown and characterized at Brookhaven by G.D.G. The manuscript was written by A.C., S.R. and D.N., with input from all authors.
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Rajasekaran, S., Casandruc, E., Laplace, Y. et al. Parametric amplification of a superconducting plasma wave. Nature Phys 12, 1012–1016 (2016). https://doi.org/10.1038/nphys3819
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DOI: https://doi.org/10.1038/nphys3819
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