Abstract

Electro-optic quantum coherent interfaces map the amplitude and phase of a quantum signal directly to the phase or intensity of a probe beam. At terahertz frequencies, a fundamental challenge is not only to sense such weak signals (due to a weak coupling with a probe in the near-infrared) but also to resolve them in the time domain. Cavity confinement of both light fields can increase the interaction and achieve strong coupling. Using this approach, current realizations are limited to low microwave frequencies. Alternatively, in bulk crystals, electro-optic sampling was shown to reach quantum-level sensitivity of terahertz waves. Yet, the coupling strength was extremely weak. Here, we propose an on-chip architecture that concomitantly provides subcycle temporal resolution and an extreme sensitivity to sense terahertz intracavity fields below 20 V/m. We use guided femtosecond pulses in the near-infrared and a confinement of the terahertz wave to a volume of ${V_{\rm THz}} \sim {10^{- 9}}{({\lambda _{\rm THz}}/2)^3}$ in combination with ultraperformant organic molecules (${r_{33}} = 170\,\,{\rm pm}/{\rm V}$) and accomplish a record-high single-photon electro-optic coupling rate of ${g_{\!{\rm eo}}} = 2\pi \times 0.043\,\,{\rm GHz}$, 10,000 times higher than in recent reports of sensing vacuum field fluctuations in bulk media. Via homodyne detection implemented directly on chip, the interaction results into an intensity modulation of the femtosecond pulses. The single-photon cooperativity is ${C_0} = 1.6 \times {10^{- 8}}$, and the multiphoton cooperativity is $C = 0.002$ at room temperature. We show ${\gt}{70}\;{\rm dB}$ dynamic range in intensity at 500 ms integration under irradiation with a weak coherent terahertz field. Similar devices could be employed in future measurements of quantum states in the terahertz at the standard quantum limit, or for entanglement of subsystems on subcycle temporal scales, such as terahertz and near-infrared quantum bits.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Terahertz (THz) waves have recently pushed the boundaries of temporal and spatial resolution in a multitude of areas. They are information carriers for the highest-bandwidth communication [1,2], provide picosecond-scale resolution for tunneling currents [3], or drive ultrafast motion of electrons in gases and solids [4]. Also, intense THz waves can now manipulate quantum degrees of freedom, such as spins [5], electrons [3], or molecular orbitals [6], on subpicosecond timescales. While these milestones reveal the importance of THz science for future ultrafast quantum applications, information encoded at these speeds can travel only a limited distance before being absorbed. Aid is provided by quantum coherent interfaces that transfer the information to a probe beam that can travel long distances with low loss, such as a carrier at telecom frequencies. In the microwaves, at lower energies than the THz, electro-optic quantum coherent interfaces are already crucial building blocks for quantum computing. Current realizations follow two main approaches: either a direct electro-optic coupling [712] or an indirect one, often via an intermediate mechanical oscillator [1315]. Their performance is typically described by two figures of merit: the electro-optic coupling rate ${g_{\!{\rm eo}}}$ (which describes the rate at which new photons are generated as a result of the coupling) and the single-photon cooperativity ${C_0}$ (the coupling rate related to the loss rates of the participating photons). Numerous theoretical proposals for direct microwave-to-optical state transfer [79] have been complemented by experimental demonstrations, e.g., by co-integrating a superconducting cavity with an optical resonator [16] (with ${g_{\!{\rm eo}}} = 2\pi \times 100\,\,{\rm Hz}$) or in an electro-optic comb [10,11] (with ${g_{\!{\rm eo}}} = 2\pi \times 7.4\,\,{\rm Hz}$ [10]). Typical cooperativities can reach unity. In indirect electro-optic coupling, the intermediate optomechanical stage [17] insures a linear response to a quantum field. The small zero-point motion of mechanical resonators typically yields optomechanical coupling rates of ${g_0} = 2\pi \times (1 - 1000)\,\,{\rm kHz}$ for nano-objects [18], with higher values of ${g_0}$ for ultracold atoms [19] due to their low mass. Typical optomechanical cooperativities reach even above unity. Yet most platforms suffer from low coupling rates. An increased coupling rate can be achieved by operating at THz frequencies, where high photon energies can be combined with an extreme vacuum field confinement.

 figure: Fig. 1.

Fig. 1. Detection principle of quantum THz fields with electro-optics. (a) A cavity mode at THz frequencies (depicted in green) is sampled in the time domain by a femtosecond pulse in a cavity that contains a nonlinear medium with second-order nonlinearity ${\chi ^{(2)}}$. (b) In practice, the THz cavity is constituted of a metallic antenna, which exhibits an extremely high vacuum electric field $E_{\rm THz}^{\rm vac}$. The latter can give rise to a high single-photon coupling rate with photons in an ultrafast probe pulse depicted in red. (c) In the frequency domain, electro-optic detection entails the cross-phase modulation (sidebands from sum- and difference-frequency generation) introduced by a THz electric field onto a broadband probe via the second-order mixing process. (d) In the time domain, a probe pulse propagating along the antenna gap experiences a phase delay proportional to the interaction time ${t_{\rm int}}$ and the coupling strength. (e) The phase is later transformed into an intensity modulation using homodyne mixing and measured at a slow photodiode. The subcycle resolution is achieved by use of femtosecond probe pulses.

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In the THz, electro-optic interfaces [20,21] that convert the complex amplitude of a THz quantum state to the near-infrared are so far limited to bulk crystals with second-order nonlinear susceptibility ${\chi ^{(2)}}$ [22]. At such high frequencies, the temporal evolution of quantum states cannot be monitored by electro-optic mixing and subsequent probe detection at a photodiode since they have their cutoffs well in the microwave range. Alternatively, a resolution below one cycle of oscillation of the THz wave can be achieved by using femtosecond short probing pulses. More importantly, electro-optic interfaces enable unique measurements, beyond the ability to transfer any state to a carrier. They can sense the ground state of electromagnetic radiation through a measurement of its field rather than the photon number. This lead to the first measurement of vacuum field fluctuations in free space and their electric field correlation across time and space [20,21]. Their detection inside a bulk crystal required long integration times in ultrastable setups due to a low interaction of the participating fields.

Detection of cavity confined THz fields has been elusive to date in the quantum regime. However, a cavity confinement could boost the detection sensitivity of THz fields down to that required by single photons at short integration times and room temperature. Such intracavity electric field measurements could provide the spectral composition of the vacuum field fluctuations also responsible for the Casimir effect [23]. An on-chip cavity can facilitate a geometry where a material resonance is additionally coupled to the light field [24,25] via the near field. To achieve this, a cavity needs to first provide an extremely high vacuum field confinement. An ultrashort probe pulse will efficiently probe the electric field inside the cavity through an excellent overlap with an electro-optic medium. The first realizations of cavity-based electro-optic detection schemes [26] provided good THz field confinement but had a poor overlap with the probe mode. Very recently, a hybrid plasmonic-organic platform [27] for electric field sensing [28] in silicon photonics was demonstrated for efficient broadband THz field detection [29].

In this work, we demonstrate a single-mode, plasmonic-organic, coherent interface that provides electro-optical conversion of THz-to-near-infrared fields with an extremely high electro-optic coupling rate. Through the developed quantum description of our system, we underline the potential behind such high coupling for ultrasensitive electric field measurements on subcycle timescales. We close by discussing their potential towards performing measurements at the standard quantum limit.

2. TIME DOMAIN ELECTRO-OPTICS

The considered scenario is shown in Fig. 1(a): the electric field of a THz cavity mode is sampled in the time domain by femtosecond probe pulses by the interaction of the two fields inside a medium with ${\chi ^{(2)}}$ nonlinearity that fills the cavity. The implementation is shown in Fig. 1(b), where the THz cavity is a metallic antenna, and the femtosecond probing pulses are routed into the antenna gap (where the nonlinear medium is located) via silicon waveguides.

We will first analyze the interaction in the frequency domain. The THz cavity mode centered around a frequency $\Omega$, described by ${\hat E_{\rm THz}}(t) = i\sqrt {\frac{{\hbar \Omega}}{{2{\epsilon _{\rm THz}}{V_{\rm THz}}}}} ({u_{\rm THz}}(x,y,z)\hat a(\Omega){e^{- i\Omega t}} -{\rm h.c.})$, mixes with a broadband (multimode) probe of frequency $\omega$, described by ${\hat E_p}(t) = i\int {\rm d}\omega \sqrt {\frac{{\hbar \omega}}{{2{\epsilon _p}{V_p}}}} ({u_p}(x,y,z)\hat a(\omega){e^{- i\omega t}} - {\rm h.c.})$ inside the ${\chi ^{(2)}}$ medium, as depicted in Fig. 1(c). This leads to the creation of sidebands by sum- and difference-frequency generation around the coherent probe tones. ${\epsilon _p}$ and ${\epsilon _{\rm THz}}$ are the permittivities, ${V_p}$ and ${V_{\rm THz}}$ are the effective mode volumes, and $\hat a(\omega)$ (${\hat a^\dagger}(\omega)$) and $\hat a(\Omega)$ (${\hat a^\dagger}(\Omega)$) are the annihilation (creation) operators at the two frequencies. ${u_p}(x,y,z)$ and ${u_{\rm THz}}(x,y,z)$ are the three-dimensional spatial field distributions that obey the normalization $\int {\rm d}V|{u_{\rm THz}}(x,y,z{)|^2} = {V_{\rm THz}}$, $\int {\rm d}V|{u_p}(x,y,z{)|^2} = {V_p}$. Considering a coherent and broadband probe beam ($\hat a(\omega) = \alpha (\omega)$), we can describe the physics of our system with the Hamiltonian (derivation in Supplement 1 section S3)

$${\hat H_I} = i\int {\rm d}\omega \alpha (\omega)\hbar {g_{\!{\rm eo}}}(\omega)({\hat a^\dagger}(\omega + \Omega)\hat a(\Omega) - {\rm h.c.}),$$
$$\quad +\; i\int {\rm d}\omega \alpha (\omega)\hbar {g_{\!{\rm eo}}}(\omega)(\hat a(\omega - \Omega)\hat a(\Omega) - {\rm h.c.}).$$

We recognize a sum between a beam splitter and a squeezing Hamiltonian. ${g_{\!{\rm eo}}}(\omega)$ is the single-photon electro-optic coupling rate and describes the rate with which one single THz photon mixes with one single probe photon to produce a photon in the sideband. A high ${g_{\!{\rm eo}}}(\omega)$ characterizes a strong nonlinear interaction and is defined in our case as (see Supplement 1 section S3)

$${g_{\!{\rm eo}}}(\omega) = \frac{1}{2}n_{\rm mat}^2r\omega E_{\rm THz}^{\rm vac}{\Gamma _c},$$
and it depends linearly on the vacuum electric field of the THz mode at the cavity resonance $E_{\rm THz}^{\rm vac} = \sqrt {\frac{{\hbar \Omega}}{{2{\epsilon _{\rm THz}}{V_{\rm THz}}}}}$. ${n_{\rm mat}} = \sqrt {{\epsilon _p}}$ is the refractive index of the electro-optic material at frequency $\omega$; $r$ is the electro-optic coefficient, which will be kept general for the moment; and ${\Gamma _c}$ is the overlap factor between the THz wave and the probe beam inside the electro-optic medium [29]. A high electro-optic coupling requires a small cavity volume ${V_{\rm THz}}$, a perfect overlap, and a high intrinsic second-order nonlinearity. Moreover, the vacuum electric field grows with the frequency $\Omega$ and is thus inherently higher at THz frequencies than at microwaves.

In the time domain, the use of an ultrashort probe pulse is essential for the subcycle resolution, as it limits the interaction to the portion of the THz field contained in the space-time volume equal to the extent of the probe pulse (here 400 fs in the interaction region). The sidebands generated by sum- and difference-frequency generation overlap spectrally with the incident probe spectrum and entail a self- and cross-phase modulation of the probe. Consequently, the probe field accumulates a phase delay $\Delta \hat \phi (t) = \frac{1}{2}n_{\rm mat}^2r\omega {\Gamma _c}{t_{\rm int}}{\hat E_{\rm THz}}(t)$ [Fig. 1(d)] that is linearly dependent on the subcycle THz electric field. The resulting field after an interaction time ${t_{\rm int}}$ is

$$\begin{split}{\hat E_{\rm res}}(t) = i\int\! {\rm d}\omega \sqrt {\frac{{\hbar \omega}}{{2{\epsilon _p}{V_p}}}} {u_p}(x,y,z)\alpha (\omega)({e^{- i(\omega t + \Delta \hat \phi (t))}} - {\rm h.c.}).\end{split}$$

Details are presented in the Supplement 1 section S4. For a vacuum input, we can define $\Delta \phi _{\rm THz}^{\rm vac} = \langle 0|\Delta \hat \phi (t)|0\rangle = {g_{\!{\rm eo}}}{t_{\rm int}} = \frac{1}{2}n_{\rm mat}^2r\omega E_{\rm THz}^{\rm vac}{\Gamma _c}{t_{\rm int}}$. The mixing of the probe pulse with a local oscillator pulse transforms the phase delay $\Delta \hat \phi (t)$ into an intensity modulation $\Delta {\hat I_{\rm out}}(t)$ as shown in Fig. 1(e).

3. MONOLITHIC CAVITY DESIGN

A false-color electron-beam micrograph of the electro-optic THz-to-near-infrared quantum interface is shown in Figs. 2(a) and 2(b). The electro-optic THz cavity consists of a bowtie antenna connected to plasmonic phase shifters. The THz antenna has been made by gold metallization and enhances the vacuum field at a single cavity mode. The THz vacuum electric field is localized almost entirely in the gap between the electrodes and is polarized across the gap. The silicon waveguides of width 450 nm (shown in blue) route a mode-matched infrared probe field (shown in red) on chip. Details about the electromagnetic properties of our design are discussed in Supplement 1 section S2. The dimensions of the antenna gap are width $w = 0.15$ µm, height $h = 0.15$ µm, and the length ${l_{\rm gap}} = 10$, 20 or 30 µm. At the antenna, the probe is launched into the plasmonic gap via a taper [31] with excellent overlap and low losses (see Supplement 1 section S2). The metallic antenna electrodes form a plasmonic slot waveguide and also provide efficient confinement for the probe. Overall, this results in an overlap of the two interacting waves of ${\Gamma _c} = 0.35$. Organic electro-optic molecules [32] fill the gap, as nowadays they provide one of the highest possible ${\chi ^{(2)}}$ nonlinearities. We use the organic nonlinear molecules 3:1 HD-BB-OH/YLD124 [33] that enable the mixing of fields polarized across the gap via a second-order susceptibility $\chi _{33}^{(2)} = - \frac{1}{2}{r_{33}}n_{\rm mat}^4$ (measured ${r_{33}} = 170\,\,{\rm pm}/{\rm V}$, ${n_{\rm mat}} = 1.77$). The confined vacuum field imparts the phase delay $\Delta \phi _{\rm THz}^{\rm vac}$ to the near-infrared probe (shown in red, also polarized across the gap) that propagates through the gap area with a group index ${n_g} = 3.58$.

 figure: Fig. 2.

Fig. 2. Layout and operation principle of the on-chip detector. (a), (b) Electron micrograph images of the device, showing in false colors the monolithic MZI in silicon and the co-integration of the hybrid organic-plasmonic phase shifters in each arm of the interferometer. The zoom-in (b) highlights the plasmonic gap, which contains the nonlinear organic molecules. The THz antenna confines the incident wave to the plasmonic gap where the interaction occurs. (c) The theoretical amplitude of the vacuum field at the antenna resonance frequency is reported as a function of gap length. (d) The incident THz wave is detected using electro-optic detection. The probing pulses at 1550 nm are coupled via grating couplers [30] to on-chip waveguides and propagate through the interferometer. At the plasmonic phase shifters, they experience a phase delay that varies linearly with the incident THz field amplitude at the point of temporal overlap. The phase modulation is transformed into an intensity modulation after the interferometer and measured by a simple photodiode. (e) A passive imbalance of $\Delta \phi = \pi /2$ between the two arms of the interferometer yields a linear response to the phase introduced by the incident THz field. (f) The two interacting waves propagate in orthogonal directions ($x$ and $y$). The coherence length (here defined at phase difference $\pi /2$) is shorter for faster oscillating frequencies. Therefore, at given plasmonic length, the maximum frequency is determined by ${l_{\rm gap}}$. Operation at higher frequencies requires shorter plasmonic cavities, as was shown in previous work [29], and operating at a targeted frequency allows an optimal length.

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4. RESULTS

We investigated three antennae with resonances at 220, 500, or 800 GHz. The effective mode volume ${V_{\rm THz}} = \frac{{\int_V {\rm d}V{w_e}}}{{{w_{e,\max}}}}$ (${w_e}$ is the electric energy density) has been computed from CST simulations. For a 10 µm long plasmonic gap, we find that ${V_{\rm THz}} = 0.78\,\,\unicode{x00B5}{\rm m}^3$, $0.79\,\,\unicode{x00B5}{\rm m}^3$, and $0.79\,\,\unicode{x00B5}{\rm m}^3$ for 220, 500, and 800 GHz, respectively. The theoretical vacuum electric field is reported in Fig. 2(c). For a gap length of ${l_{\rm gap}} = 10\,\,\unicode{x00B5}{\rm m}$, the vacuum electric field at the three resonances is 1.8, 2.7, and 3.4 kV/m, respectively. As such, the expected single-photon electro-optic coupling rates are ${g_{\!{\rm eo}}} = 2\pi \times 0.032\,{\rm GHz}$, 0.049 GHz, and 0.062 GHz. At room temperature, the cavity is, however, populated by thermal photons with an average population of ${\bar n_{\rm th}} \sim 28,12,7$ at 220, 500, and 800 GHz, respectively.

We implement the homodyne detection by an on-chip Mach–Zehnder interferometer (MZI), depicted in Figs. 2(a) and 2(d). The MZI contains one metallic antenna in each arm. The two antennae work in a push-pull configuration, guaranteed by opposite poling of the active material. The phase delay is transformed into an intensity modulation at the output of the interferometer, according to the transfer function of the interferometer in Fig. 2(e) (Supplement 1 section S4). When the built-in length difference between the two arms introduces a nominal delay of $\pi /2$, the intensity response of the interferometer to the THz field is linear and can be described by the electro-optic operator ${\hat S_{\!{\rm eo}}}(t)$ [20], which is

$${\hat S_{\!{\rm eo}}}(t) = \frac{{\Delta {{\hat I}_{\rm out}}(t)}}{{{I_{\rm out}}}} = \sqrt B {\hat E_{\rm THz}}(t)$$
with $\sqrt B = n_{\rm mat}^2{r_{33}}{k_0}{\Gamma _c}{l_{\rm gap}}{n_g}$. The latter can be measured with a low-bandwidth photodiode (lower than the detected THz wave). A dependence study of the electro-optic signal on the nominal delay is shown in the Supplement 1 of Ref. [29]. ${I_{\rm out}} = \frac{{{I_{\rm in}}}}{2}{e^{- \alpha {l_{\rm gap}}}}$ is the output power of the probe after the interferometer at $\langle \Delta \phi (t)\rangle = 0$, and ${I_{\rm in}}$ is the input power of the probe. $\alpha$ is the loss in the plasmonic gap, and ${l_{\rm gap}}$ will be optimized for the optimal trade-off between losses and signal amplitude to yield maximum sensitivity. The probe and the THz wave propagate in orthogonal directions (in-plane and out-of-plane, respectively). The coherence length shown in Fig. 2(f) reveals that high frequencies require a short plasmonic length. The latter avoids a dephasing of the THz field during detection [29] but limits the interaction time ${t_{\rm int}} = \frac{{{l_{\rm gap}}{n_g}}}{{{c_0}}}$ between the two waves. Finally, the accumulated phase is simply $\Delta \phi _{\rm THz}^{\rm vac} = \frac{{{g_{\!{\rm eo}}}{l_{\rm gap}}{n_g}}}{{{c_0}}}$.

In the following, we perform linear spectroscopy on our detectors using electro-optic sampling. Our goal is first to validate the theoretical description above with experimental results. Second, we proceed to determine ${g_{\!{\rm eo}}}$ experimentally from the peak phase modulation $\Delta \phi _{\rm THz}^{\rm pp}$ achieved when the chip is irradiated with a weak coherent THz pulse of amplitude $|{\alpha _{\rm THz}}\rangle$. Electro-optic sampling as performed here yields ${\langle {\hat S_{\!{\rm eo}}}(t)\rangle _{\hat \rho}} = \sum\nolimits_n \langle n|\hat \rho {\hat S_{\!{\rm eo}}}(t)|n\rangle = 0$ when applied to a statistical mixture $\hat \rho$ as, e.g., thermal photons. Therefore, the thermal photons do not contribute to the measured electric field amplitude. A measurement of such free-running fields, also of, e.g., vacuum fields, would require a field correlation measurement of the type $\langle {\hat S_{\!{\rm eo}}}(t){\hat S_{\!{\rm eo}}}(t + \tau)\rangle$ as in Ref. [20,34].

A weak, broadband (up to 2.5 THz), and phase-locked THz transient is launched onto the chip as shown in Fig. 2(d) (details on the amplitude and spectral composition of the incident pulse are shown in Supplement 1 section S2). From the broadband pulse, only the frequencies at the cavity resonance couple efficiently into the cavity gap. Here, the THz field has a time-dependent amplitude ${E_{THz,g}}(\Omega ,t)$ and contains ${n_{\rm THz}} = \alpha _{\rm THz}^2$ THz photons. At every time point of the transient, the measured phase modulation $\Delta {\phi _{\rm THz}}(t)$ depends on the local field ${E_{THz,g}}(\Omega ,t)$ as $\Delta {\phi _{\rm THz}}(t)\; = \frac{1}{2}n_{\rm mat}^2{r_{33}}{E_{{\rm THz},g}}(\Omega ,t)\,{l_{\rm gap}}{k_0}\,{\Gamma _c}{n_g}\, {\rm sinc}\big(\frac{1}{{{c_0}}}\Omega {n_g}{l_{\rm gap}}\big)$. The introduced phase thus depends on the effective interaction length of ${l_{\rm eff}} = {l_{\rm gap}} {\rm sinc}(\frac{1}{{{c_0}}}\Omega {n_g}{l_{\rm gap}})$ due to phase matching.

 figure: Fig. 3.

Fig. 3. Performance of the detectors. (a) Simulated field enhancement as provided by three different geometries of the bowtie antennae depends on the center design frequency. (b) Time trace of the THz transients as detected by the three different antenna designs. Clear frequency selectivity is visible already in the temporal evolution. The peak-peak of the detected field amplitude decreases with center frequency due to a reduction of the field enhancement. (c) The spectra [Fourier transform of the time traces in (b)] are a convolution of the source emission and the detector response, and they follow well the field enhancement provided by the antenna. All plotted results correspond a plasmonic gap length ${l_{\rm gap}} = 10\,\,\unicode{x00B5}{\rm m}$ and an output probe power of ${P_{\rm out}} \sim - 40\,\,{\rm dBm}$. We report on a 35 dB dynamical range in field (70 dB in intensity) at only 500 ms integration time as opposed to the 20 s integration time reported earlier [29]. (d) The normalized peak-peak electro-optic intensity modulation is reported as a function of the plasmonic gap length ${l_{\rm gap}}$, measured for devices with different ${l_{\rm gap}}$. We observe a clear saturation or even decay of the signal for long interaction lengths for the high-frequency antennae. Instead, a $30\,\,\unicode{x00B5}{\rm m}$ long plasmonic gap yields the highest signal for the 220 GHz antenna.

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The bowtie antennae with resonance at 220, 500, and 800 GHz provide a field enhancement shown in Fig. 3(a). The enhanced field naturally leads to a proportional increase in the electro-optic coupling rate ${g_{\!{\rm eo}}}$ by a factor of 100–1000, when compared to a free-space configuration with the same spectral characteristics. The field enhancement at resonance decreases owing to the fact that, in the current geometry, the broadband characteristics of bowtie antennae are more pronounced at higher resonant frequencies (see inset). The temporal evolution of the weak THz transient is shown as measured by the antennae in Fig. 3(b). We measure a very clean signal of amplitude below 0.1 MV/m and low noise with detectors with a ${l_{\rm gap}} = 10\,\,\unicode{x00B5}{\rm m}$ and a probe power of ${P_{\rm out}} = 0.1\,\,\unicode{x00B5}{\rm W}$. The Fourier transform of the time-domain signals confirm a spectral response that matches well the design target frequency shown in Fig. 3(c). The dynamic range is 35 dB in amplitude (70 dB in power) at only 500 ms integration time per point, higher than that in recent reports [29]. Before we proceed to extract the experimental electro-optic coupling rate ${g_{\!{\rm eo}}}$, we investigate the influence of the gap length ${l_{\rm gap}}$ onto our results. We wish to evaluate in which configuration the interaction is phase-matched and the effective interaction length is equivalent to the geometrical gap length ${l_{\rm eff}} = {l_{\rm gap}}$. For this purpose, we investigate the measured modulation efficiency, which is $\eta = \frac{{\Delta {I_{\rm pp}}}}{{{I_{\rm out}}}} = 2\Delta \phi _{\rm THz}^{\rm pp}$, as a function of ${l_{\rm gap}}$ in Fig. 3(d). $\eta$ is defined as the peak-peak electro-optic signal of the time traces normalized to the average output intensity, the factor 2 comes from the two antennae in one MZI, and $\Delta \phi _{\rm THz}^{\rm pp}$ is the peak-peak phase change in one single antenna. Additionally, from our derivation in Supplement 1 section S4C, we know that $\eta = 2\sqrt B \sqrt {\frac{{{n_{\rm THz}}\hbar \Omega}}{{2{\epsilon _{\rm THz}}{V_{\rm THz}}}}} = 4{g_{\!{\rm eo}}}{t_{\rm int}}\sqrt {{n_{\rm THz}}}$. We find that for the 220 GHz antenna, the modulation efficiency increases linearly with the gap length, reaching a record high value of 0.7% at ${l_{\rm gap}} = 30\,\,\unicode{x00B5}{\rm m}$. The modulation efficiency does not increase beyond ${l_{\rm gap}} = 20\,\,\unicode{x00B5}{\rm m}$ for the 500 GHz and the 800 GHz antennae. It is therefore reasonable to extract the electro-optic coupling rate from the measurements with ${l_{\rm gap}} = 10\,\,\unicode{x00B5}{\rm m}$.

Finally, we report the single-photon electro-optic coupling rate ${g_{\!{\rm eo}}}$ in comparison with the theoretically calculated one in Fig. 4(a). The experimental values of $\eta$ depend on the single-photon coupling rate ${g_{\!{\rm eo}}}$ and on ${n_{\rm THz}}$ (see Supplement 1 section S4C). The input number of THz photons is ${n_{\rm THz}} \sim 350(\pm 100)$ at 220 GHz and ${n_{\rm THz}} \sim 230(\pm 100)$ at 500 GHz and has been calculated from the input THz pulse. We find that the experimental value is around ${g_{\!{\rm eo}}} \sim 2\pi \times 0.043\,\,{\rm GHz}$ and $2\pi \times 0.026\,\,{\rm GHz}$, respectively. This extremely high coupling rate is 10,000 times higher than in reported free-space experiments in quantum fields and entails that our detectors are highly sensitive. Indeed, they exhibit a very good signal-to-noise ratio when measuring phase-locked THz electric fields, which have an amplitude much below the ones of vacuum field fluctuations; see Fig. 4(b). We vary the electric field of the THz pulse such that it contains 350 photons down to $0.07 \times {10^{- 3}}$ photons per pulse. We use the 220 GHz antenna with 20 µm plasmonic length for detection. At a corresponding vacuum electric field amplitude of $E_{\rm THz}^{\rm vac} = \; {\sim} 2\,\,{\rm kV/m}$, the modulation efficiency is $\eta ={ 10^{- 4}}$. This quantity is independent of the power of the probe and characterizes the performance of the detector through ${g_{\!{\rm eo}}}$ solely. However, as any detection is limited by the shot noise of the probe, we show in Fig. 4(c) the signal-to-noise ratio (SNR) for a single measurement with a single pulse as a function of the in-gap THz field. The signal is found from measurements, and the shot noise is computed with the shot-noise formula derived in section S6 of the Supplement 1. For a vacuum input, we find a single-shot SNR of 0.004 for a probe power of ${- 44.2}\,\,{\rm dBm}$ as was used in the experiments. For a probe power of ${10^{- 3}}\,\,{\rm mW}$, which, as we will show readily, is the maximum probe power at which the current detectors work reliably, the expected SNR per pulse for vacuum fields is around 0.01—1 order of magnitude higher than reported in recent measurement of the electric field correlation on vacuum fields [20]. This result is groundbreaking, as it would reduce the integration in such experiments from days to minutes. For the weak THz transient that contains ${n_{\rm THz}} = 350$ photons, the SNR for one single pulse is 0.1. Finally, we are interested in the single-photon cooperativity ${C_0} = \frac{{4g_{\!{\rm eo}}^2}}{{{\kappa _p}{\kappa _{\rm THz}}}}$ that relates the electro-optic coupling rate to the loss rates of the probe and THz photons, ${\kappa _p}$ and ${\kappa _{\rm THz}}$, and the total cooperativity $C = {n_p}{C_0}$, related to the number of probe photons involved in the detection process. We find that ${C_0} = 1.6 \times {10^{- 8}}$, and details can be found in the Supplement 1 section S4D. We show that the devices can handle up to ${10^{- 3}} {\rm mW}$ probe power at the output in Fig. 4(d). Close to a probe power of ${10^{- 3}} {\rm mW}$ the performance decreases, and the electro-optic signal does not follow the linear trend described in the equation above. At this output power, the number of photons contained in the probe is estimated to be ${n_p} = 122873$. In summary, a total cooperativity as high as $C = 0.002$ is demonstrated. By optimizing the high-power resilience of the nonlinear material [35], the total cooperativity is expected to increase significantly above 1 (see Supplement 1 section S4E for guidelines on how to improve ${C_0}$ in current devices).

 figure: Fig. 4.

Fig. 4. Electric field sensitivity of the detectors. (a) The experimental single-photon electro-optic coupling rate ${g_{\!{\rm eo}}}$ is compared to the theoretically computed one. (b) The modulation efficiency is reported as a function of the THz electric field in the plasmonic gap for fields down to a few 10 V/m. The incident THz field is swept by changing the input power at the photoconductive emitter. The error bars represent the time-domain standard deviation for an integration time per point of 1 s for the first three points and 500 ms for the last three points. (c), The SNR for one single measurement with one single pulse is computed from the detected signal and the computed noise equivalent signal from shot noise. The measured data stems for a probe intensity of ${-}{44.2}\;{\rm dBm}$ and a plasmonic gap length of 20 µm. Also, we add the extrapolated SNR for a probe power of −30 dBm. (d) Detected electro-optic amplitude as a function of probe power shows an initial linear increase with probe power, which then deviates from a linear dependence for high probe powers.

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5. DISCUSSION AND OUTLOOK

On a final note, we point out that the reported coupling rate ${g_{\!{\rm eo}}}$ might provide a path in the area of measurements of quantum electric fields at the standard quantum limit (SQL). To explain, the measurement of the single-mode quantum electric field discussed here, ${\hat E_{\rm THz}}(t) = E_{\rm THz}^{\rm vac}({\hat X_{\rm THz}}\cos (\Omega t) + {\hat Y_{\rm THz}}\sin (\Omega t))$, via electro-optic interfaces entails the concomitant measurements of its two quadratures ${\hat X_{\rm THz}}$ and ${\hat Y_{\rm THz}}$. While such a singular measurement can be made infinitely strong, it will inherently introduce imprecision onto a second measurement of the same quantum state at a later time point. This is especially relevant for the characterization of free-running fields, e.g., thermal or vacuum fields, by electric field correlation measurements using two femtosecond probing pulses [20]. As a result, just like in optomechanical systems that deal with the measurement of zero-point motion of mechanical oscillators [36,37] (brief discussion in Supplement 1 S4F), the SQL imposes a limitation onto the precision of electric field correlation measurements. In section S5 of the Supplement 1, we derive in the first approximation that the SQL is expected in the current device to be around ${n_p}={ 10^8}$ photons, corresponding to 1 mW average output power. While the role of losses has to still be investigated in detail theoretically, and the high-power resilience and nonlinearity of the used organic nonlinear molecules have to be investigated at cryogenic temperatures experimentally, current correlation measurements were already demonstrated to be shot-noise limited at similar probe powers, even at long integration times and inside a cryogenic environment [20]. Cryogenic cooling of similar electro-optic polymers [38] was shown to affect the electro-optic coefficient only marginally. Moreover, a combination of last-generation electro-optic materials that demonstrate thermal stability through Diels–Alder cross-linking of molecular glasses [35] (${r_{33}} = 290\,\,{\rm pm}/{\rm V}$) or even higher nonlinear coefficients (${r_{33}} = 460\,\,{\rm pm}/{\rm V}$), a narrowband antenna design combined with a back-side reflector [39] could target a higher electro-optic coupling and thereby considerably lower the required power in future devices. In conclusion, we believe that similar devices are potential candidates for electric field measurements on quantum states at the SQL. Moreover, the record-high electro-optic coupling rate may already enable measurements of few thermal photons, and perhaps, in the future, enable experiments of cavity electrodynamics or near-field detection. The theoretical framework we have provided for cavity electro-optics with ultrashort pulses may facilitate the further development of such devices, e.g.,  for full-state tomography [40]. Beyond quantum applications, our detectors have great potential also in classical applications, owing to their increased sensitivity by 1 order of magnitude compared to state-of-the-art technology. The emergence of such devices and even more complex architectures thereof (e.g.,  the parallel engineering shown in Supplement 1 section S7) brings us closer to exploiting the richness of quantum mechanics in a broader range: for spectroscopy [4145], telecommunication [1,2,46,47], or manipulation of matter states.

Funding

European Research Council (670478, 340975); National Science Foundation (DMR-1303080); Air Force Office of Scientific Research (FA9550-15-1-0319).

Acknowledgment

We acknowledge the insightful comments of Giacomo Scalari.

Disclosures

The authors declare no competing interests.

 

See Supplement 1 for supporting content.

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References

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  1. J. Ma, R. Shrestha, J. Adelberg, C.-Y. Yeh, Z. Hossain, E. Knightly, J. M. Jornet, and D. M. Mittleman, “Security and eavesdropping in terahertz wireless links,” Nature 563, 89–93 (2018).
    [Crossref]
  2. K. Sengupta, T. Nagatsuma, and D. M. Mittleman, “Terahertz integrated electronic and hybrid electronic–photonic systems,” Nat. Electron. 1, 622–635 (2018).
    [Crossref]
  3. T. Rybka, M. Ludwig, M. F. Schmalz, V. Knittel, D. Brida, and A. Leitenstorfer, “Sub-cycle optical phase control of nanotunnelling in the single-electron regime,” Nat. Photonics 10, 667–670 (2016).
    [Crossref]
  4. H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions,” Nature 561, 507–511 (2018).
    [Crossref]
  5. S. Schlauderer, C. Lange, S. Baierl, T. Ebnet, C. P. Schmid, D. C. Valovcin, A. K. Zvezdin, A. V. Kimel, R. V. Mikhaylovskiy, and R. Huber, “Temporal and spectral fingerprints of ultrafast all-coherent spin switching,” Nature 569, 383–387 (2019).
    [Crossref]
  6. T. L. Cocker, D. Peller, P. Yu, J. Repp, and R. Huber, “Tracking the ultrafast motion of a single molecule by femtosecond orbital imaging,” Nature 539, 263–267 (2016).
    [Crossref]
  7. C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
    [Crossref]
  8. M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, “Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators,” Phys. Rev. A 96, 043808 (2017).
    [Crossref]
  9. A. Rueda, W. Hease, S. Barzanjeh, and J. M. Fink, “Electro-optic entanglement source for microwave to telecom quantum state transfer,” npj Quantum Inf. 5, 108 (2019).
    [Crossref]
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    [Crossref]
  11. M. Zhang, B. Buscaino, C. Wang, A. Shams-Ansari, C. Reimer, R. Zhu, J. M. Kahn, and M. Lončar, “Broadband electro-optic frequency comb generation in a lithium niobate microring resonator,” Nature 568, 373–377 (2019).
    [Crossref]
  12. F. Lindel, R. Bennett, and S. Y. Buhmann, “Probing the sculpted quantum vacuum: quantum optics of nonlinear crystals,” arXiv:1905.10200v2 (2019).
  13. S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
    [Crossref]
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    [Crossref]
  15. L. Shao, M. Yu, S. Maity, N. Sinclair, L. Zheng, C. Chia, A. Shams-Ansari, C. Wang, M. Zhang, K. Lai, and M. Lončar, “Microwave-to-optical conversion using lithium niobate thin-film acoustic resonators,” Optica 6, 1498–1505 (2019).
    [Crossref]
  16. L. Fan, C.-L. Zou, R. Cheng, X. Guo, X. Han, Z. Gong, S. Wang, and H. X. Tang, “Superconducting cavity electro-optics: a platform for coherent photon conversion between superconducting and photonic circuits,” Sci. Adv. 4, eaar4994 (2018).
    [Crossref]
  17. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
    [Crossref]
  18. J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
    [Crossref]
  19. F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
    [Crossref]
  20. I.-C. Benea-Chelmus, F. Settembrini, G. Scalari, and J. Faist, “Electric field correlation measurements on the electromagnetic vacuum state,” Nature 568, 202–206 (2019).
    [Crossref]
  21. C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer, “Direct sampling of electric-field vacuum fluctuations,” Science 350, 420–423 (2015).
    [Crossref]
  22. A. S. Moskalenko, C. Riek, D. V. Seletskiy, G. Burkard, and A. Leitenstorfer, “Paraxial theory of direct electro-optic sampling of the quantum vacuum,” Phys. Rev. Lett. 115, 263601 (2015).
    [Crossref]
  23. C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, “Observation of the dynamical Casimir effect in a superconducting circuit,” Nature 479, 376–379 (2011).
    [Crossref]
  24. G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, “Magneto-transport controlled by Landau polariton states,” Nat. Phys. 15, 186–190 (2019).
    [Crossref]
  25. S. De Liberato, “Electro-optical sampling of quantum vacuum fluctuations in dispersive dielectrics,” Phys. Rev. A 100, 031801 (2019).
    [Crossref]
  26. I.-C. Benea-Chelmus, T. Zhu, F. Settembrini, C. Bonzon, E. Mavrona, D. Elder, W. Heni, J. Leuthold, L. Dalton, and J. Faist, “Three-dimensional phase modulator at telecom wavelength acting as a terahertz detector with an electro-optic bandwidth of 1.25 terahertz,” ACS Photon. 5, 1398–1403 (2018).
    [Crossref]
  27. W. Heni, Y. Kutuvantavida, C. Haffner, H. Zwickel, C. Kieninger, S. Wolf, M. Lauermann, Y. Fedoryshyn, A. F. Tillack, L. E. Johnson, D. L. Elder, B. H. Robinson, W. Freude, C. Koos, J. Leuthold, and L. R. Dalton, “Silicon–organic and plasmonic–organic hybrid photonics,” ACS Photon. 4, 1576–1590 (2017).
    [Crossref]
  28. Y. Salamin, W. Heni, C. Haffner, Y. Fedoryshyn, C. Hoessbacher, R. Bonjour, M. Zahner, D. Hillerkuss, P. Leuchtmann, D. L. Elder, L. R. Dalton, C. Hafner, and J. Leuthold, “Direct conversion of free space millimeter waves to optical domain by plasmonic modulator antenna,” Nano Lett. 15, 8342–8346 (2015).
    [Crossref]
  29. Y. Salamin, I.-C. Benea-Chelmus, Y. Fedoryshyn, W. Heni, D. L. Elder, L. R. Dalton, J. Faist, and J. Leuthold, “Compact and ultra-efficient broadband plasmonic terahertz field detector,” Nat. Commun. 10, 5550 (2019).
    [Crossref]
  30. T. Watanabe, M. Ayata, U. Koch, Y. Fedoryshyn, and J. Leuthold, “Perpendicular grating coupler based on a blazed antiback-reflection structure,” J. Lightwave Technol. 35, 4663–4669 (2017).
    [Crossref]
  31. J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
    [Crossref]
  32. B. H. Robinson, L. E. Johnson, D. L. Elder, A. A. Kocherzhenko, C. M. Isborn, C. Haffner, W. Heni, C. Hoessbacher, Y. Fedoryshyn, Y. Salamin, B. Baeuerle, A. Josten, M. Ayata, U. Koch, J. Leuthold, and L. R. Dalton, “Optimization of plasmonic-organic hybrid electro-optics,” J. Lightwave Technol. 36, 5036–5047 (2018).
    [Crossref]
  33. D. L. Elder, C. Haffner, W. Heni, Y. Fedoryshyn, K. E. Garrett, L. E. Johnson, R. A. Campbell, J. D. Avila, B. H. Robinson, J. Leuthold, and L. R. Dalton, “Effect of rigid bridge-protection units, quadrupolar interactions, and blending in organic electro-optic chromophores,” Chem. Mater. 29, 6457–6471 (2017).
    [Crossref]
  34. I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93, 043812 (2016).
    [Crossref]
  35. H. Xu, F. Liu, D. L. Elder, L. E. Johnson, Y. de Coene, K. Clays, B. H. Robinson, and L. R. Dalton, “Ultrahigh electro-optic coefficients, high index of refraction, and long-term stability from Diels–Alder cross-linkable binary molecular glasses,” Chem. Mater. 32, 1408–1421 (2020).
    [Crossref]
  36. V. B. Braginsky, “Classical and quantum restrictions on the detection of weak disturbances of a macroscopic oscillator,” J. Exp. Theor. Phys. 26, 831 (1968).
  37. D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, “Continuous force and displacement measurement below the standard quantum limit,” Nat. Phys. 15, 745–749 (2019).
    [Crossref]
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2020 (3)

M. Forsch, R. Stockill, A. Wallucks, I. Marinković, C. Gärtner, R. A. Norte, F. van Otten, A. Fiore, K. Srinivasan, and S. Gröblacher, “Microwave-to-optics conversion using a mechanical oscillator in its quantum ground state,” Nat. Phys. 16, 69–74 (2020).
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H. Xu, F. Liu, D. L. Elder, L. E. Johnson, Y. de Coene, K. Clays, B. H. Robinson, and L. R. Dalton, “Ultrahigh electro-optic coefficients, high index of refraction, and long-term stability from Diels–Alder cross-linkable binary molecular glasses,” Chem. Mater. 32, 1408–1421 (2020).
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P. Sulzer, K. Oguchi, J. Huster, M. Kizmann, T. L. M. Guedes, A. Liehl, C. Beckh, A. S. Moskalenko, G. Burkard, D. V. Seletskiy, and A. Leitenstorfer, “Determination of the electric field and its Hilbert transform in femtosecond electro-optic sampling,” Phys. Rev. A 101, 033821 (2020).
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2019 (11)

D. Mason, J. Chen, M. Rossi, Y. Tsaturyan, and A. Schliesser, “Continuous force and displacement measurement below the standard quantum limit,” Nat. Phys. 15, 745–749 (2019).
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L. Shao, M. Yu, S. Maity, N. Sinclair, L. Zheng, C. Chia, A. Shams-Ansari, C. Wang, M. Zhang, K. Lai, and M. Lončar, “Microwave-to-optical conversion using lithium niobate thin-film acoustic resonators,” Optica 6, 1498–1505 (2019).
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I.-C. Benea-Chelmus, F. Settembrini, G. Scalari, and J. Faist, “Electric field correlation measurements on the electromagnetic vacuum state,” Nature 568, 202–206 (2019).
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S. Schlauderer, C. Lange, S. Baierl, T. Ebnet, C. P. Schmid, D. C. Valovcin, A. K. Zvezdin, A. V. Kimel, R. V. Mikhaylovskiy, and R. Huber, “Temporal and spectral fingerprints of ultrafast all-coherent spin switching,” Nature 569, 383–387 (2019).
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A. Rueda, W. Hease, S. Barzanjeh, and J. M. Fink, “Electro-optic entanglement source for microwave to telecom quantum state transfer,” npj Quantum Inf. 5, 108 (2019).
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2018 (8)

M. Piccardo, P. Chevalier, T. S. Mansuripur, D. Kazakov, Y. Wang, N. A. Rubin, L. Meadowcroft, A. Belyanin, and F. Capasso, “The harmonic state of quantum cascade lasers: origin, control, and prospective applications [Invited],” Opt. Express 26, 9464 (2018).
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Y. Salamin, B. Baeuerle, W. Heni, F. C. Abrecht, A. Josten, Y. Fedoryshyn, C. Haffner, R. Bonjour, T. Watanabe, M. Burla, D. L. Elder, L. R. Dalton, and J. Leuthold, “Microwave plasmonic mixer in a transparent fibre–wireless link,” Nat. Photonics 12, 749–753 (2018).
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H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions,” Nature 561, 507–511 (2018).
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I.-C. Benea-Chelmus, T. Zhu, F. Settembrini, C. Bonzon, E. Mavrona, D. Elder, W. Heni, J. Leuthold, L. Dalton, and J. Faist, “Three-dimensional phase modulator at telecom wavelength acting as a terahertz detector with an electro-optic bandwidth of 1.25 terahertz,” ACS Photon. 5, 1398–1403 (2018).
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B. H. Robinson, L. E. Johnson, D. L. Elder, A. A. Kocherzhenko, C. M. Isborn, C. Haffner, W. Heni, C. Hoessbacher, Y. Fedoryshyn, Y. Salamin, B. Baeuerle, A. Josten, M. Ayata, U. Koch, J. Leuthold, and L. R. Dalton, “Optimization of plasmonic-organic hybrid electro-optics,” J. Lightwave Technol. 36, 5036–5047 (2018).
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2017 (7)

D. L. Elder, C. Haffner, W. Heni, Y. Fedoryshyn, K. E. Garrett, L. E. Johnson, R. A. Campbell, J. D. Avila, B. H. Robinson, J. Leuthold, and L. R. Dalton, “Effect of rigid bridge-protection units, quadrupolar interactions, and blending in organic electro-optic chromophores,” Chem. Mater. 29, 6457–6471 (2017).
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W. S. L. Lee, K. Kaltenecker, S. Nirantar, W. Withayachumnankul, M. Walther, M. Bhaskaran, B. M. Fischer, S. Sriram, and C. Fumeaux, “Terahertz near-field imaging of dielectric resonators,” Opt. Express 25, 3756 (2017).
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W. Heni, Y. Kutuvantavida, C. Haffner, H. Zwickel, C. Kieninger, S. Wolf, M. Lauermann, Y. Fedoryshyn, A. F. Tillack, L. E. Johnson, D. L. Elder, B. H. Robinson, W. Freude, C. Koos, J. Leuthold, and L. R. Dalton, “Silicon–organic and plasmonic–organic hybrid photonics,” ACS Photon. 4, 1576–1590 (2017).
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T. Watanabe, M. Ayata, U. Koch, Y. Fedoryshyn, and J. Leuthold, “Perpendicular grating coupler based on a blazed antiback-reflection structure,” J. Lightwave Technol. 35, 4663–4669 (2017).
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I.-C. Benea-Chelmus, M. Rösch, G. Scalari, M. Beck, and J. Faist, “Intensity autocorrelation measurements of frequency combs in the terahertz range,” Phys. Rev. A 96, 033821 (2017).
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Q. Sun, Y. He, K. Liu, S. Fan, E. P. J. Parrott, and E. Pickwell-MacPherson, “Recent advances in terahertz technology for biomedical applications,” Quant. Imaging Med. Surg. 7, 345–355 (2017).
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2016 (5)

D. H. Park, V. Yun, J. Luo, A. K. Jen, and W. N. Herman, “EO polymer at cryogenic temperatures,” Electron. Lett. 52, 1703–1705 (2016).
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I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93, 043812 (2016).
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T. Rybka, M. Ludwig, M. F. Schmalz, V. Knittel, D. Brida, and A. Leitenstorfer, “Sub-cycle optical phase control of nanotunnelling in the single-electron regime,” Nat. Photonics 10, 667–670 (2016).
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T. L. Cocker, D. Peller, P. Yu, J. Repp, and R. Huber, “Tracking the ultrafast motion of a single molecule by femtosecond orbital imaging,” Nature 539, 263–267 (2016).
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C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
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2015 (3)

C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer, “Direct sampling of electric-field vacuum fluctuations,” Science 350, 420–423 (2015).
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A. S. Moskalenko, C. Riek, D. V. Seletskiy, G. Burkard, and A. Leitenstorfer, “Paraxial theory of direct electro-optic sampling of the quantum vacuum,” Phys. Rev. Lett. 115, 263601 (2015).
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Y. Salamin, W. Heni, C. Haffner, Y. Fedoryshyn, C. Hoessbacher, R. Bonjour, M. Zahner, D. Hillerkuss, P. Leuchtmann, D. L. Elder, L. R. Dalton, C. Hafner, and J. Leuthold, “Direct conversion of free space millimeter waves to optical domain by plasmonic modulator antenna,” Nano Lett. 15, 8342–8346 (2015).
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2014 (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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2011 (3)

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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C. M. Wilson, G. Johansson, A. Pourkabirian, M. Simoen, J. R. Johansson, T. Duty, F. Nori, and P. Delsing, “Observation of the dynamical Casimir effect in a superconducting circuit,” Nature 479, 376–379 (2011).
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S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
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2009 (1)

J. Tian, S. Yu, W. Yan, and M. Qiu, “Broadband high-efficiency surface-plasmon-polariton coupler with silicon-metal interface,” Appl. Phys. Lett. 95, 013504 (2009).
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2008 (2)

G. I. Groma, J. Hebling, I. Z. Kozma, G. Varo, J. Hauer, J. Kuhl, and E. Riedle, “Terahertz radiation from bacteriorhodopsin reveals correlated primary electron and proton transfer processes,” Proc. Natl. Acad. Sci. USA 105, 6888–6893 (2008).
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F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
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1998 (1)

S. Hunsche, M. Koch, I. Brener, and M. Nuss, “THz near-field imaging,” Opt. Commun. 150, 22–26 (1998).
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1968 (1)

V. B. Braginsky, “Classical and quantum restrictions on the detection of weak disturbances of a macroscopic oscillator,” J. Exp. Theor. Phys. 26, 831 (1968).

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Y. Salamin, B. Baeuerle, W. Heni, F. C. Abrecht, A. Josten, Y. Fedoryshyn, C. Haffner, R. Bonjour, T. Watanabe, M. Burla, D. L. Elder, L. R. Dalton, and J. Leuthold, “Microwave plasmonic mixer in a transparent fibre–wireless link,” Nat. Photonics 12, 749–753 (2018).
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Adelberg, J.

J. Ma, R. Shrestha, J. Adelberg, C.-Y. Yeh, Z. Hossain, E. Knightly, J. M. Jornet, and D. M. Mittleman, “Security and eavesdropping in terahertz wireless links,” Nature 563, 89–93 (2018).
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Alegre, T. P. M.

J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, “Magneto-transport controlled by Landau polariton states,” Nat. Phys. 15, 186–190 (2019).
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Aspelmeyer, M.

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
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J. Chan, T. P. M. Alegre, A. H. Safavi-Naeini, J. T. Hill, A. Krause, S. Gröblacher, M. Aspelmeyer, and O. Painter, “Laser cooling of a nanomechanical oscillator into its quantum ground state,” Nature 478, 89–92 (2011).
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Avila, J. D.

D. L. Elder, C. Haffner, W. Heni, Y. Fedoryshyn, K. E. Garrett, L. E. Johnson, R. A. Campbell, J. D. Avila, B. H. Robinson, J. Leuthold, and L. R. Dalton, “Effect of rigid bridge-protection units, quadrupolar interactions, and blending in organic electro-optic chromophores,” Chem. Mater. 29, 6457–6471 (2017).
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Awari, N.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions,” Nature 561, 507–511 (2018).
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Ayata, M.

Baeuerle, B.

B. H. Robinson, L. E. Johnson, D. L. Elder, A. A. Kocherzhenko, C. M. Isborn, C. Haffner, W. Heni, C. Hoessbacher, Y. Fedoryshyn, Y. Salamin, B. Baeuerle, A. Josten, M. Ayata, U. Koch, J. Leuthold, and L. R. Dalton, “Optimization of plasmonic-organic hybrid electro-optics,” J. Lightwave Technol. 36, 5036–5047 (2018).
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Y. Salamin, B. Baeuerle, W. Heni, F. C. Abrecht, A. Josten, Y. Fedoryshyn, C. Haffner, R. Bonjour, T. Watanabe, M. Burla, D. L. Elder, L. R. Dalton, and J. Leuthold, “Microwave plasmonic mixer in a transparent fibre–wireless link,” Nat. Photonics 12, 749–753 (2018).
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S. Schlauderer, C. Lange, S. Baierl, T. Ebnet, C. P. Schmid, D. C. Valovcin, A. K. Zvezdin, A. V. Kimel, R. V. Mikhaylovskiy, and R. Huber, “Temporal and spectral fingerprints of ultrafast all-coherent spin switching,” Nature 569, 383–387 (2019).
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G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, “Magneto-transport controlled by Landau polariton states,” Nat. Phys. 15, 186–190 (2019).
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A. Rueda, W. Hease, S. Barzanjeh, and J. M. Fink, “Electro-optic entanglement source for microwave to telecom quantum state transfer,” npj Quantum Inf. 5, 108 (2019).
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S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
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Beck, M.

G. L. Paravicini-Bagliani, F. Appugliese, E. Richter, F. Valmorra, J. Keller, M. Beck, N. Bartolo, C. Rössler, T. Ihn, K. Ensslin, C. Ciuti, G. Scalari, and J. Faist, “Magneto-transport controlled by Landau polariton states,” Nat. Phys. 15, 186–190 (2019).
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I.-C. Benea-Chelmus, M. Rösch, G. Scalari, M. Beck, and J. Faist, “Intensity autocorrelation measurements of frequency combs in the terahertz range,” Phys. Rev. A 96, 033821 (2017).
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I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93, 043812 (2016).
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P. Sulzer, K. Oguchi, J. Huster, M. Kizmann, T. L. M. Guedes, A. Liehl, C. Beckh, A. S. Moskalenko, G. Burkard, D. V. Seletskiy, and A. Leitenstorfer, “Determination of the electric field and its Hilbert transform in femtosecond electro-optic sampling,” Phys. Rev. A 101, 033821 (2020).
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Belyanin, A.

Benea-Chelmus, I.-C.

Y. Salamin, I.-C. Benea-Chelmus, Y. Fedoryshyn, W. Heni, D. L. Elder, L. R. Dalton, J. Faist, and J. Leuthold, “Compact and ultra-efficient broadband plasmonic terahertz field detector,” Nat. Commun. 10, 5550 (2019).
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I.-C. Benea-Chelmus, F. Settembrini, G. Scalari, and J. Faist, “Electric field correlation measurements on the electromagnetic vacuum state,” Nature 568, 202–206 (2019).
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I.-C. Benea-Chelmus, T. Zhu, F. Settembrini, C. Bonzon, E. Mavrona, D. Elder, W. Heni, J. Leuthold, L. Dalton, and J. Faist, “Three-dimensional phase modulator at telecom wavelength acting as a terahertz detector with an electro-optic bandwidth of 1.25 terahertz,” ACS Photon. 5, 1398–1403 (2018).
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I.-C. Benea-Chelmus, M. Rösch, G. Scalari, M. Beck, and J. Faist, “Intensity autocorrelation measurements of frequency combs in the terahertz range,” Phys. Rev. A 96, 033821 (2017).
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I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93, 043812 (2016).
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C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
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Bhaskaran, M.

Bonjour, R.

Y. Salamin, B. Baeuerle, W. Heni, F. C. Abrecht, A. Josten, Y. Fedoryshyn, C. Haffner, R. Bonjour, T. Watanabe, M. Burla, D. L. Elder, L. R. Dalton, and J. Leuthold, “Microwave plasmonic mixer in a transparent fibre–wireless link,” Nat. Photonics 12, 749–753 (2018).
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Y. Salamin, W. Heni, C. Haffner, Y. Fedoryshyn, C. Hoessbacher, R. Bonjour, M. Zahner, D. Hillerkuss, P. Leuchtmann, D. L. Elder, L. R. Dalton, C. Hafner, and J. Leuthold, “Direct conversion of free space millimeter waves to optical domain by plasmonic modulator antenna,” Nano Lett. 15, 8342–8346 (2015).
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Bonn, M.

H. A. Hafez, S. Kovalev, J.-C. Deinert, Z. Mics, B. Green, N. Awari, M. Chen, S. Germanskiy, U. Lehnert, J. Teichert, Z. Wang, K.-J. Tielrooij, Z. Liu, Z. Chen, A. Narita, K. Müllen, M. Bonn, M. Gensch, and D. Turchinovich, “Extremely efficient terahertz high-harmonic generation in graphene by hot Dirac fermions,” Nature 561, 507–511 (2018).
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Bonzon, C.

I.-C. Benea-Chelmus, T. Zhu, F. Settembrini, C. Bonzon, E. Mavrona, D. Elder, W. Heni, J. Leuthold, L. Dalton, and J. Faist, “Three-dimensional phase modulator at telecom wavelength acting as a terahertz detector with an electro-optic bandwidth of 1.25 terahertz,” ACS Photon. 5, 1398–1403 (2018).
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I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93, 043812 (2016).
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Braginsky, V. B.

V. B. Braginsky, “Classical and quantum restrictions on the detection of weak disturbances of a macroscopic oscillator,” J. Exp. Theor. Phys. 26, 831 (1968).

Brener, I.

S. Hunsche, M. Koch, I. Brener, and M. Nuss, “THz near-field imaging,” Opt. Commun. 150, 22–26 (1998).
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Brennecke, F.

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
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Phys. Rev. A (7)

S. Barzanjeh, D. Vitali, P. Tombesi, and G. J. Milburn, “Entangling optical and microwave cavity modes by means of a nanomechanical resonator,” Phys. Rev. A 84, 042342 (2011).
[Crossref]

C. Javerzac-Galy, K. Plekhanov, N. R. Bernier, L. D. Toth, A. K. Feofanov, and T. J. Kippenberg, “On-chip microwave-to-optical quantum coherent converter based on a superconducting resonator coupled to an electro-optic microresonator,” Phys. Rev. A 94, 053815 (2016).
[Crossref]

M. Soltani, M. Zhang, C. Ryan, G. J. Ribeill, C. Wang, and M. Loncar, “Efficient quantum microwave-to-optical conversion using electro-optic nanophotonic coupled resonators,” Phys. Rev. A 96, 043808 (2017).
[Crossref]

S. De Liberato, “Electro-optical sampling of quantum vacuum fluctuations in dispersive dielectrics,” Phys. Rev. A 100, 031801 (2019).
[Crossref]

P. Sulzer, K. Oguchi, J. Huster, M. Kizmann, T. L. M. Guedes, A. Liehl, C. Beckh, A. S. Moskalenko, G. Burkard, D. V. Seletskiy, and A. Leitenstorfer, “Determination of the electric field and its Hilbert transform in femtosecond electro-optic sampling,” Phys. Rev. A 101, 033821 (2020).
[Crossref]

I.-C. Benea-Chelmus, C. Bonzon, C. Maissen, G. Scalari, M. Beck, and J. Faist, “Subcycle measurement of intensity correlations in the terahertz frequency range,” Phys. Rev. A 93, 043812 (2016).
[Crossref]

I.-C. Benea-Chelmus, M. Rösch, G. Scalari, M. Beck, and J. Faist, “Intensity autocorrelation measurements of frequency combs in the terahertz range,” Phys. Rev. A 96, 033821 (2017).
[Crossref]

Phys. Rev. Lett. (1)

A. S. Moskalenko, C. Riek, D. V. Seletskiy, G. Burkard, and A. Leitenstorfer, “Paraxial theory of direct electro-optic sampling of the quantum vacuum,” Phys. Rev. Lett. 115, 263601 (2015).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

G. I. Groma, J. Hebling, I. Z. Kozma, G. Varo, J. Hauer, J. Kuhl, and E. Riedle, “Terahertz radiation from bacteriorhodopsin reveals correlated primary electron and proton transfer processes,” Proc. Natl. Acad. Sci. USA 105, 6888–6893 (2008).
[Crossref]

Quant. Imaging Med. Surg. (1)

Q. Sun, Y. He, K. Liu, S. Fan, E. P. J. Parrott, and E. Pickwell-MacPherson, “Recent advances in terahertz technology for biomedical applications,” Quant. Imaging Med. Surg. 7, 345–355 (2017).
[Crossref]

Rev. Mod. Phys. (1)

M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452 (2014).
[Crossref]

Sci. Adv. (1)

L. Fan, C.-L. Zou, R. Cheng, X. Guo, X. Han, Z. Gong, S. Wang, and H. X. Tang, “Superconducting cavity electro-optics: a platform for coherent photon conversion between superconducting and photonic circuits,” Sci. Adv. 4, eaar4994 (2018).
[Crossref]

Science (2)

C. Riek, D. V. Seletskiy, A. S. Moskalenko, J. F. Schmidt, P. Krauspe, S. Eckart, S. Eggert, G. Burkard, and A. Leitenstorfer, “Direct sampling of electric-field vacuum fluctuations,” Science 350, 420–423 (2015).
[Crossref]

F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, “Cavity optomechanics with a Bose-Einstein condensate,” Science 322, 235–238 (2008).
[Crossref]

Other (1)

F. Lindel, R. Bennett, and S. Y. Buhmann, “Probing the sculpted quantum vacuum: quantum optics of nonlinear crystals,” arXiv:1905.10200v2 (2019).

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Detection principle of quantum THz fields with electro-optics. (a) A cavity mode at THz frequencies (depicted in green) is sampled in the time domain by a femtosecond pulse in a cavity that contains a nonlinear medium with second-order nonlinearity ${\chi ^{(2)}}$. (b) In practice, the THz cavity is constituted of a metallic antenna, which exhibits an extremely high vacuum electric field $E_{\rm THz}^{\rm vac}$. The latter can give rise to a high single-photon coupling rate with photons in an ultrafast probe pulse depicted in red. (c) In the frequency domain, electro-optic detection entails the cross-phase modulation (sidebands from sum- and difference-frequency generation) introduced by a THz electric field onto a broadband probe via the second-order mixing process. (d) In the time domain, a probe pulse propagating along the antenna gap experiences a phase delay proportional to the interaction time ${t_{\rm int}}$ and the coupling strength. (e) The phase is later transformed into an intensity modulation using homodyne mixing and measured at a slow photodiode. The subcycle resolution is achieved by use of femtosecond probe pulses.
Fig. 2.
Fig. 2. Layout and operation principle of the on-chip detector. (a), (b) Electron micrograph images of the device, showing in false colors the monolithic MZI in silicon and the co-integration of the hybrid organic-plasmonic phase shifters in each arm of the interferometer. The zoom-in (b) highlights the plasmonic gap, which contains the nonlinear organic molecules. The THz antenna confines the incident wave to the plasmonic gap where the interaction occurs. (c) The theoretical amplitude of the vacuum field at the antenna resonance frequency is reported as a function of gap length. (d) The incident THz wave is detected using electro-optic detection. The probing pulses at 1550 nm are coupled via grating couplers [30] to on-chip waveguides and propagate through the interferometer. At the plasmonic phase shifters, they experience a phase delay that varies linearly with the incident THz field amplitude at the point of temporal overlap. The phase modulation is transformed into an intensity modulation after the interferometer and measured by a simple photodiode. (e) A passive imbalance of $\Delta \phi = \pi /2$ between the two arms of the interferometer yields a linear response to the phase introduced by the incident THz field. (f) The two interacting waves propagate in orthogonal directions ($x$ and $y$). The coherence length (here defined at phase difference $\pi /2$) is shorter for faster oscillating frequencies. Therefore, at given plasmonic length, the maximum frequency is determined by ${l_{\rm gap}}$. Operation at higher frequencies requires shorter plasmonic cavities, as was shown in previous work [29], and operating at a targeted frequency allows an optimal length.
Fig. 3.
Fig. 3. Performance of the detectors. (a) Simulated field enhancement as provided by three different geometries of the bowtie antennae depends on the center design frequency. (b) Time trace of the THz transients as detected by the three different antenna designs. Clear frequency selectivity is visible already in the temporal evolution. The peak-peak of the detected field amplitude decreases with center frequency due to a reduction of the field enhancement. (c) The spectra [Fourier transform of the time traces in (b)] are a convolution of the source emission and the detector response, and they follow well the field enhancement provided by the antenna. All plotted results correspond a plasmonic gap length ${l_{\rm gap}} = 10\,\,\unicode{x00B5}{\rm m}$ and an output probe power of ${P_{\rm out}} \sim - 40\,\,{\rm dBm}$. We report on a 35 dB dynamical range in field (70 dB in intensity) at only 500 ms integration time as opposed to the 20 s integration time reported earlier [29]. (d) The normalized peak-peak electro-optic intensity modulation is reported as a function of the plasmonic gap length ${l_{\rm gap}}$, measured for devices with different ${l_{\rm gap}}$. We observe a clear saturation or even decay of the signal for long interaction lengths for the high-frequency antennae. Instead, a $30\,\,\unicode{x00B5}{\rm m}$ long plasmonic gap yields the highest signal for the 220 GHz antenna.
Fig. 4.
Fig. 4. Electric field sensitivity of the detectors. (a) The experimental single-photon electro-optic coupling rate ${g_{\!{\rm eo}}}$ is compared to the theoretically computed one. (b) The modulation efficiency is reported as a function of the THz electric field in the plasmonic gap for fields down to a few 10 V/m. The incident THz field is swept by changing the input power at the photoconductive emitter. The error bars represent the time-domain standard deviation for an integration time per point of 1 s for the first three points and 500 ms for the last three points. (c), The SNR for one single measurement with one single pulse is computed from the detected signal and the computed noise equivalent signal from shot noise. The measured data stems for a probe intensity of ${-}{44.2}\;{\rm dBm}$ and a plasmonic gap length of 20 µm. Also, we add the extrapolated SNR for a probe power of −30 dBm. (d) Detected electro-optic amplitude as a function of probe power shows an initial linear increase with probe power, which then deviates from a linear dependence for high probe powers.

Equations (5)

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H ^ I = i d ω α ( ω ) g e o ( ω ) ( a ^ ( ω + Ω ) a ^ ( Ω ) h . c . ) ,
+ i d ω α ( ω ) g e o ( ω ) ( a ^ ( ω Ω ) a ^ ( Ω ) h . c . ) .
g e o ( ω ) = 1 2 n m a t 2 r ω E T H z v a c Γ c ,
E ^ r e s ( t ) = i d ω ω 2 ϵ p V p u p ( x , y , z ) α ( ω ) ( e i ( ω t + Δ ϕ ^ ( t ) ) h . c . ) .
S ^ e o ( t ) = Δ I ^ o u t ( t ) I o u t = B E ^ T H z ( t )

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