Electro-optic interface for ultrasensitive intracavity electric field measurements at microwave and terahertz frequencies: supplementary material

electric field measurements at microwave and terahertz frequencies: supplementary material ILEANA-CRISTINA BENEA-CHELMUS1,2,3, YANNICK SALAMIN1,4,5, FRANCESCA FABIANA SETTEMBRINI3, YURIY FEDORYSHYN4, WOLFGANG HENI4, DELWIN L. ELDER6, LARRY R. DALTON6, JUERG LEUTHOLD4, AND JÉRÔME FAIST3,* 1Contributed equally to this work 2Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA 3ETH Zurich, Institute of Quantum Electronics, Zurich, Switzerland 4ETH Zurich, Institute of Electromagnetic Fields (IEF), Zurich, Switzerland 5Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA, USA 6University of Washington, Department of Chemistry, Seattle, WA, USA *Corresponding author: cristinabenea@g.harvard.edu,salamin@mit.edu, jerome.faist@phys.ethz.ch Published 12 May 2020

. The most important symbols used in this paper.

Symbol
Meaning ω probe frequency κ p total probe loss rate, a sum of the input coupling losses κ ex,p and intrinsic losses κ 0,p p dielectric permittivity at probe frequency p = 0 n 2 mat A p effective cross-section of the probe mode in the plasmonic waveguide  The power of the pump pulses is 76 mW, when not explicitly stated differently. The power of the probe pulses is indicated for each 51 of the measurements discussed in this article.

52
In the present case, the multimode probing beam has a temporal extent of ∼400 fs at the plasmonic detectors, which is mainly 53 determined by the dispersion introduced by the 1 m long input fiber we use. The silicon waveguide and the plasmonic wavguide have 54 merely no effect on the probe pulse length, as we show in Fig. S1c. The repetition rate of the laser is 90 MHz.      Table S2. Parameters of the different antenna designs. Geometric dimensions are given together with the average number of thermal photons stored in the THz cavityn th , as well as the total measured losses κ THz (for the different lengths l gap ). The 800 GHz antenna does not exhibit a clear resonance, as can be seen from Fig. 3c in the main manuscript. We note that the intrinsic losses κ 0,THz due to absorption in the dielectric and the gold are negligible compared to the radiative losses of the antenna.

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The electric field of a THz wave is measured in this work with sub-cycle temporal resolution via its interaction with a femtosecond 75 probing beam in a medium that enables a second-order nonlinear optical mixing. In the following, we derive the most basic equations 76 that describe this process, starting from a general formalism which will be then stepwise applied to our particular case. To ease the 77 readability, we summarise in table S1 the most important symbols used in this paper.

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In general, a nonlinear polarisation P (2) is generated by the mixing of two arbitrary input electric fields which we call E k . In the 79 following, we derive the Hamiltonian that describes this interaction [2]. We emphasize that the present platform is in certain aspects 80 different from the majority of experimental scenarios discussed in literature. In particular, the near-infrared beam is a pulse which 81 interacts with the THz wave on a sub-cycle temporal and spatial scale. We therefore describe it as a multimode beam, as opposed to a 82 single-mode of an optical cavity. The THz wave instead is treated as a cavity mode with a center frequency determined by the THz 83 antenna which has a total loss rate of κ THz . The THz wave frequency is in our experiment much smaller than the bandwidth of the 84 probe pulse and the mixing terms effectively introduce a cross-phase modulation of the probe beam. Overall, our experiments are best 85 described by the unresolved sideband regime.

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In our design, we exploit the χ omit the subscript z in the following to ease the notation. The nonlinear interaction energy arises from the nonlinear polarisation The interaction Hamiltonian can be derived from the expression of the energy by using the electric field operators in the second 91 quantisation. For the probe pulse we assume a multi-mode fieldÊ ). u p (x, y, z) and u THz (x, y, z) are the three-93 dimensional spatial field distributions that obey the normalisation dV|u THz (x, y, z)| 2 = V THz , dV|u p (x, y, z)| 2 = V p , p and THz 94 are the permittivities, V p and V THz are the effective mode volumes andâ(ω) (â(ω) † ) andâ(Ω) (â(Ω) † ) are the annihilation (creation) 95 operators at the two frequencies. We assume without loss of generality that u p (x, y, z) and u THz (x, y, z) are real-valued. The total field 96 is thereforeÊ(t) =Ê p (t) +Ê THz (t), and P (2) (t) = χ (2) 33 (Ê p (t) +Ê THz (t)) 2 . We find: We further consider only the terms which are linear withÊ THz , and neglect all other terms (mixing of the probe only, two photon 98 processes of the THz) and findĤ I = dV2χ THz (t). In addition, the energy conservation rule has to apply. We make use of 99 the fact that all frequency dependent quantities (V p , p , u p , ω) change slowly as a function of probe frequency within the bandwidth of 100 the pulse. The interaction Hamiltonian can be reduced to: We define g eo (ω), the electro-optic coupling rate between the participating waves as: with the integral that describes the effective overlap volume V overlap = dV|u p (x, y, z)| 2 u THz (x, y, z).

103
Therefore, Using χ (2) 33 = − 1 2 n 4 mat r 33 (n mat is the material index at the probe wavelength, r 33 the electro-optic coefficient), and Γ c the overlap 105 between the two interacting fields, we find the single photon electro-optic coupling rate as follows: u p (x, y, z) = v p (x, y)w p (z). Similarly, the three-dimensional spatial field profile of the THz beam in the plasmonic can 108 be factorised as follows: u THz (x, y, z) = v THz (x, y)w THz (z), with w THz (z) = 1 everywhere in the gap. These approximations are 109 compatible with the field distributions in the gap shown in Fig. S2b and c. Both the THz field as well as the near-infrared probe are 110 well confined to the plasmonic gap. Consequently, the overlap Γ c can be approximated by the well-defined overlap of the transverse 111 mode profiles:

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The electro-optic coupling rate depends on the THz mode volume V THz and on the electro-optic coefficient r 33 . Simulations of the 114 electric field energy may be used to determine the effective mode volume that influences the vacuum field. In addition, it depends on 115 the overlap of the THz mode with the probe mode. 116 We can now use the interaction Hamiltonian above to determine how the mode amplitude of the probe pulse evolve due to the non-117 linear interaction. For this purpose we will compute the quantum fieldÊ (2) p generated by the interaction of the probe pulse with a quan-118 tum terahertz field. The probe field is assumed to be classical, henceâ(ω) = α(ω) and E p (t) = i dω . Without loss of generality, we assume α(ω) = √ n p to be real-valued. This generated quantum field interferes with the classical 120 probe field, for which we assume no depletion. Consequently, the interaction Hamiltonian can be simplified to: S4. Single-mode vs multi-mode mixing process a, In the case that the probe is in a well-defined single frequency mode, the non-linear interaction leads to a side-band formation that is well resolved. b, In the case that the probe is a broadband femtosecond pulse, the interaction creates unresolved sidebands which effectively introduces a phase delay of the femtosecond probing pulse.
We recognise here a linear combination between the beam-splitter Hamiltonian and a squeezing Hamiltonian, which lead to the 122 formation of side-bands through sum and difference frequency generation (SFG and DFG) as shown in Fig.S4. For sake of simplicity, 123 we derive the evolutions first in the single mode picture and will then add up all generated field to calculate the total probe field after 124 interaction. We start from the equations of motion in the Heisenberg picture: and find We impose that the interaction time t int is equal to the propagation time of the probe through the plasmonic gap t int = l gap n g c 0 . We find 127 that upon interaction, the side-bands are generated with the following amplitudes: Therefore, after interaction, the resulting electric field of the probe in one arm of the interferometer is: res,1 (t) = E p (t) +Ê (2) (2),+ p (t) +Ê (2),− p (t) describes the sum of all fields generated in all side-bands by SFG (Ê which obey the following relationships: From here, we combine the equations above and find the total generated field 132Ê (2) We recognise in the formula above the field operator of the THz mode. For simplicity, we defineÊ n THz (t) =Ê THz (t)/E vac THz = 133 i(a(Ω)e −iΩt − a † (Ω)e iΩt ) and define the following phase delay: ∆φ(t) = 1 2 n 2 mat r 33 ωΓ c t intÊTHz (t). For the case of ∆φ(t) << 1, we 134 can approximate −i∆φ(t) = e −i∆φ(t) − 1 and find that

135Ê
(2) Finally, the resulting field is therefore In the second arm, the poling of the organic nonlinear material has been done using the opposite but equal poling voltage, and 137 as such, the total field of the probe after the interaction with the second arm can be retrieved by replacing r 33 → −r 33 (and hence 138 g eo (ω) → −g eo (ω),Ê (2) p or ∆φ(t) → −∆φ(t) ). Therefore, here the resulting field isÊ res,2 with: We recognise from this final formula that the interaction effectively introduces a phase delay of the probe which is linearly 140 dependent on the incident THz quantum field.

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A. Homodyne mixing by the on-chip interferometer 142 We compute the intensity modulation introduced by the non-linear mixing in the interferometric scheme realized on-chip. The

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interferometer is operated at a π 2 nominal delay. With the two equations above, we find that the total transmitted intensity is given by We now assume for simplicity that α(ω) = α = √ n p , g eo (ω) = g eo is constant for all probe frequencies because the relative 145 bandwidth of the probe is limited and we are far away from any resonances. The phase delay now becomes ∆φ(t) = g eo t intÊ n THz (t). 146 We find that In addition, we define I out ∼ 2|E p | 2 as the output intensity under no incident terahertz field. We find that: ∆Î out (t) I out = n 2 mat r 33 k 0 Γ c l gap n g i The overall intensity modulation is thus linear to the THz vacuum field.

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In this section, we discuss the expectation value of an electric field measurement by electro-optic sampling on different types of 153 terahertz radiation. We emphasise that, contrary to continuous measurements, the scheme employed here samples a sub-cycle portion 154 of the terahertz wave with a rate equal to the repetition rate of the probe laser, 90 MHz. The time interval between two consecutive 155 measurements is thus 11 ns, much longer than the measured decoherence time of the THz antennae (τ THz = κ −1 THz ), which are on the 156 order of ∼1-10 ps. Therefore, two subsequent measurements are not correlated and also any back-action induced heating of the THz 157 cavity will have decayed until the next measurement.

158
First, we will consider the case of a statistical mixture of Fock states |n , such as is the case of thermal radiation. They are described 159 by a density matrixρ = ∑ n P(n)|n n|. The bare vacuum state is a sub-category of such a statistical mixture, where all probabilities 160 are zero besides P(0), which is associated with the vaccum state.
for any type of statistical mixture of Fock states. As such, also 0|Ŝ eo (t)|0 = 0.

162
Second, we will consider the case of a coherent state of terahertz radiation |α THz . In this case, the THz wave contains √ n THz = α THz 163 THz photons.
The above formula can be interpreted for a classical THz field E THz,g (t) = n THzh Ω 2 THz V THz sin Ωt in the gap of the plasmonic modulator.

165
In this condition, the THz wave introduces a time-dependent intensity modulation which is linearly dependent on the local THz 166 electric field in the gap E THz,g (t): where I out = I in 2 e −αl gap is related to the input probe intensity I in and is considered at ∆φ THz (t) = 0 and a passive imbalance of the 168 interferometer of π/2. ∆φ THz (t) is the phase delay introduced by the THz wave in one arm via the linear electro-optic effect in the 169 space-time volume in which it overlaps with the probe pulse. In the event of phase-matched detection, the phase delay has been 170 shown above to be equal to: 171 ∆φ THz (t) = 1 2 n 2 mat r 33 E THz,g (t)l gap k 0 Γ c n g (S36) If we consider phase matching, the phase delay has to be considered for each frequency component of the broadband THz pulse 172 individually as it additionally depends on a frequency-dependent term sinc( Ωn g l gap c 0 ):

173
∆φ THz (Ω) = 1 2 n 2 mat r 33 E THz,g (Ω)l gap k 0 Γ c n g sinc( The modulation efficiency is thus η = ∆I pp The modulated intensity is detected by a photodiode with a typical bandwidth of 2 kHz, and transformed into a linearly dependent 175 voltage δV = G∆P. G is the gain and ∆P the modulation in power. Clearly, all equations above have been redacted in units of intensity, 176 however, they take an equivalent form for power, or output voltage.

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C. Finding the experimental g eo by probing the interface with a weak THz coherent state 178 We calculate the experimental single-photon electro-optic coupling rate by probing the system with a weak coherent state of THz 179 radiation |α THz that contains an average number of photons n THz = |α THz | 2 . Under this assumption, the electric field amplitude of 180 the coherent state is √ n THz E vac THz . Therefore, the peak-peak phase modulation ∆φ pp THz and the peak-peak intensity modulation ∆I pp 181 depend on g eo as follows: 182 Finally, 183 g eo = ∆φ pp THz c 0 2 √ n THz l gap n g (S40) As can be seen from the formula above, the number of input THz photons is crucial to determine the experimental electro-optic 184 coupling rate. This has been now computed from the spectrum of the input terahertz pulse, assuming a cross-section at the focus of

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The single photon cooperativity C 0 and the cooperativity related to the probe photons C = n p C 0 depend on the loss rates of the 189 system. They in essence relate the rate at which the nonlinear mixing leads to the generation of a photon, to the rate at which the 190 participating photons are lost by the structure.

191
The single photon cooperativity C 0 is: with κ p , κ THz the loss rates of the probe and THz photons which we discuss below. 193 We can define a total cooperativity of the coherent probe with a single THz photon as 194 C = n p C 0 (S42) In our case, the total number of photons in the probe is equal to n p = P out hω f rep , with P out the probe power at the output of the 195 plasmonic detector.

196
The THz loss rate κ THz are reported for our two antenna designs at 220 GHz and 500 GHz as shown in Fig. S3c and d and in 197   table S2.

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E p E res κ 0,p κ ex,p b a κ ex,p : photons that are recorded κ 0,p : photons lost e.g. due to absorption The total probe loss rate κ p = κ 0,p + κ ex,p is constituted of the sum between intrinsic and extrinsic probe losses, see gap. This rate can be computed from the propagation loss shown in Fig. S6b. For a plasmonic length of 10 µm and a plasmonic width 202 of 150 nm, the absorption is 3 dB, hence I out I in = 0.5. Therefore, κ 0,p = − 1 t int ln( I out I in ) = 2π × 0.95 THz.

203
The extrinsic probe loss rate is defined as the rate at which the probe photons escape from the interaction region. In the present 204 case, this is inversely proportional to the propagation time through plasmonic gap κ ex,p = 1 t int . For a 10 µm long plasmonic gap, the 205 loss rate is κ ex,p = 2π × 1.59 THz.

206
With this value, we have that the measured single photon cooperativity for the 220 GHz antenna design is C 0 = 1.6 × 10 −8 for 207 220 GHz, l gap = 10 µm. To determine the cooperativity related to the probe, we use the input probe power, which is equivalent to 208 doing a post-selection on the measurement. At P out = 10 −3 mW, the total number of probe photons is n p = 122873 we find the 209 cooperativity C = 0.002.

210
E. Analysis of electro-optic coupling rate and single photon cooperativity 211 We are providing in the following a numerical analysis of the device parameters' influence on the electro-optic coupling rate g eo and 212 single photon cooperativity C 0 . This should serve as a guideline for new designs.

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Of particular interest are the plasmoinc gap width and length, as these two parameters have the strongest influence on the 215 performance of the detector. In particular, they both have a direct impact on the terahertz mode volume and the losses. For simplicity, 216 we approximate the terahertz mode volume by V THz = w gap l gap h gap F, where F is a correction factor determined by numerical 217 simulations. The overlap factor which is a function of the gap width is approximated by Γ c = 15w −1 gap + 0.25 as found from numerical 218 simulations. Using eq.(8) we can plot the electro-optic coupling rate for a terahertz frequency Ω = 2π × 220GHz.  The equivalence can be noticed already in the analogy of the interaction Hamiltonian in the two cases. We have shown above that the 231 interaction Hamiltonian that describes the nonlinear electro-optic interaction in the single mode regime is given by a sum of a beam 232 splitter Hamiltonian and a squeezing Hamiltonian with g eo (ω) = 1 2 n 2 mat r 33 ω Likewise, in optomechanics, we find the same type of Hamiltonian, bearing in mind that, unlike in our example, the optical beam 235 used to measure the displacement is a cavity mode of a resonator with a high quality factor. In the case of zero detuning of the probe 236 wave from the cavity resonance, the interaction Hamiltonian that describes the system can be found e.g. in equation (33) where (corresponding toÊ THz (0) andÊ THz (τ)), and, as such, back-action of the first measurement onto the second one at time τ later is 253 essential. At the standard quantum limit, the total uncertainty of the measurement is minimized through a fine balance between 254 measurement-based imprecision and imprecision noise due to back-action.

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The noise in the first measurement will be determined by the imprecision noise S In the case of the 220 GHz antenna with l gap = 30 µm, we find that we reach the standard quantum limit over a measurement time 268 τ = 0.2π/Ω if the probe contains ∼ 10 8 for photons, which corresponds to an average power of 1 mW. For comparison, in the vacuum 269 field measurements of ref.
[3], with the given experimental values of the g eo = 2π × 1.6 kHz, t int = 32 ps we find that the standard 270 quantum limit is reached at n p = 10 13 , which is experimentally unfeasible. At the standard quantum limit, the total noise is two times 271 higher than the imprecision noise. Much below the standard quantum limit, as the case here, the noise is equal to the imprecision 272 (shot) noise. The electro-optic signal depends linearly on the intensity (and therefore total power) of the probe pulse that is detected at the output of

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As a final note, we wish to underline on the incredible richness that such a device architecture opens up by discussing a multi-frequency