Interferometer Readout Noise Below the Standard Quantum Limit of a Membrane

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PHYSICAL REVIEW A 85, 063806 (2012)

Interferometer readout noise below the standard quantum limit of a membrane

T. Westphal,1,2 D. Friedrich,1,2 H. Kaufer,1,2 K. Yamamoto,3 S. Goßler,1,2 H. Muller-Ebhardt,¨ 1,2 S. L. Danilishin,4 F. Ya. Khalili,5 K. Danzmann,1,2 and R. Schnabel1,2 1Institute for Gravitational Physics, Leibniz Universitat¨ Hannover, 30167 Hannover, Germany 2Max Planck Institute for Gravitational Physics (Albert Einstein Institute), 30167 Hannover, Germany 3Institute for Cosmic Ray Research, The University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa, Chiba 277-8582, Japan 4School of Physics, University of Western Australia, Crawley WA 6009, Perth, Australia 5Department of Physics, Moscow State University, Moscow RU-119992, Russia (Received 17 November 2011; published 6 June 2012) Here we report on the realization of a Michelson-Sagnac interferometer whose purpose is the precise characterization of the motion of, i.e., membranes showing signiﬁcant√ light transmission. Our interferometer has a readout noise spectral density (imprecision) of 3 × 10−16 m/ Hz at frequencies around the fundamental

resonance of a SiNx membrane at about 100 kHz, without using optical cavities. The readout noise demonstrated is more than 16 dB below the peak value of the membrane’s standard quantum limit (SQL). This margin is signiﬁcantly bigger than those of previous works with nanowires [Teufel et al., Nature Nano. 4, 820 (2009); Anetsberger et al., Nature Phys. 5, 909 (2009)]. We discuss the meaning of the SQL for force measurements and its relation to the readout performance and emphasize that neither our nor previous experiments achieved a total noise spectral density as low as the SQL due to thermal noise.

DOI: 10.1103/PhysRevA.85.063806 PACS number(s): 42.50.St, 42.50.Nn, 03.65.Ta, 42.50.Lc

Systems composed of mechanical probes and laser light Recently, a microwave and an optical readout imprecision are able to perform ultrasensitive measurements of external were demonstrated below the SQL’s peak value of nanowires forces by monitoring the probes’ displacements. A prominent [9,10]. With these achievements it was shown that the number example is a gravitational wave detector, which uses laser of photons per second inside the interferometers was in princi- light and quasi-free-falling test mass mirrors to search for ple high enough to reach the SQL at sideband frequencies close changes in the far field of accelerated gravitational sources to the mechanical resonance. Due to the thermal excitation of [1]. The standard quantum limit (SQL) is a consequence the nanowires and due to an above shot-noise imprecision, of Heisenberg’s Uncertainty Principle [2] and sets a limit however, the (harmonic oscillator) SQL was not reached. to the sensitivity of a continuous measurement (monitoring) The mechanical oscillators in Refs. [9,10] were nanowires of a quantity that does not commute with itself at different of aluminum and SiN, respectively, with effective masses times [3]. With respect to the internal degrees of freedom of of a few picograms and resonance frequencies in the MHz optomechanical measurement devices, the SQL also defines regime. a benchmark noise spectral density at which the door is Here, we report on an optomechanical experiment with opened to new quantum experiments such as the generation of a readout noise below the SQL, similar to Refs. [9,10]. In optomechanical entanglement [4] and also purely mechanical contrast to previous works, our setup was composed of an entanglement [5]. interferometer without cavities and a commercially available So far, a total noise spectral density as low as the SQL of a mechanical probe—a SiNx membrane with an effective mass force measurement has not yet been achieved, since the thermal of about 100 ng and a resonance frequency of about 100 kHz. excitation of the mechanical probe is usually far above the The membrane was used as a partially transmitting mirror SQL. In general, reaching the SQL requires (i) the acceleration inside a free-space Michelson-Sagnac interferometer [11,12]. noise of the mechanical probe being dominated by quantum The experiment was performed at room temperature√ and back-action due to the (optical) readout, (ii) the readout noise achieved a total readout noise of 3 × 10−16 m/ Hz. The linear being dominated by (photon) shot noise, and (iii) the readout spectral density value was a factor of more than six (the power power being optimized such that for uncorrelated quantum spectral density was 16 dB) below the peak value of the SQL. back-action noise and quantum readout noise the two are of The quantum readout noise of an interferometer is given identical size and thus provide one half of the SQL’s noise by photon counting noise, also called shot noise (sn). A power each [6]. rather useful measure is Sx,sn, the linear spectral density In case of quantum correlations between the readout of the shot√ noise calibrated to an apparent displacement x, quadrature amplitude and its orthogonal quadrature, condition given in m/ Hz. Since the shot noise has a white spectrum, (iii) may vary, even going beyond the SQL is possible [7,8]. and our interferometer has a white signal transfer function, Generally, the SQL does not have a white spectrum but the normalized spectral density is frequency independent and depends on the observation frequency. It also depends on the reads [12] probe’s dynamics and on the observable chosen. In particular, in the case of a harmonic oscillator, the SQL of a position hcλ¯ Sx,sn = , (1) measurement shows a peak at the mechanical resonance, while 16πr2P the SQL for a force measurement has a minimum here as seen withh ¯ the reduced Planck constant, c the speed of light, λ in Eqs. (4) and (5). the laser wavelength, r2 the membrane’s power reflectivity,

1050-2947/2012/85(6)/063806(5)063806-1 ©2012 American Physical Society T. WESTPHAL et al. PHYSICAL REVIEW A 85, 063806 (2012) and P the light power inside the interferometer. Note that the membrane is serving as a common end mirror, thereby doubling the displacement signal compared to a sin- x gle movable end mirror in an ordinary Michelson interferometer. The quantum back-action noise of an interferometer is given by the quantum radiation-pressure noise (rpn) on the Input mechanical probe. The susceptibility of a mechanical oscillator Translucent (50/50) membrane reads Beam- x() −1 splitter Output χ() = = , (2) 2 − − 2 F () m( im/Q m) with x() the spectrum of the probe’s displacement, F () the spectrum of the external force, m the effective mass of the mechanical probe, Q the mechanical quality factor, and m = 2πfm the oscillator eigenfrequency. Far above m,the susceptibility is proportional to −2. For frequencies far below the resonance, χ is constant, whereas its absolute value on res- | |= 2 onance is given by χpeak Q/(mm). Due to the frequency dependence in Eq. (2), the signal normalized spectral density FIG. 1. (Color online) Schematic of the Michelson-Sagnac inter- of the back-action noise is not white. For the membrane in our ferometer. The SiN membrane is aligned such that transmitted and Michelson-Sagnac interferometer it is given by [12] x reflected light beams propagate along the same optical axis. The lower 16πhr¯ 2P row of plots shows the four possible light paths, which interfere at the Sx,rpn() =|χ()| . (3) interferometer output. The inset shows a photograph of the membrane cλ having an area of 1.5 mm2. The quantum noise spectral densities given in Eqs. (1) and (3) both set limitations on the precision of a force Si3N4 membranes, mechanical resonance frequencies are even measurement: while the shot noise originates from the phase higher and lie in the megahertz region [17]. High-tensile-stress quadrature uncertainty of the light inside the interferometer, SiN membranes are interesting mechanical oscillators since the radiation-pressure noise originates from its amplitude their large surface area makes them truly macroscopic objects. quadrature uncertainty [13]. Assuming that the two quadra- Their surface quality is quite high, i.e., high enough to be able tures are uncorrelated with each other, which is the case to place a membrane in the middle of an optical cavity with a for coherent states, their noise variances simply add up, finesse of 15 000 [16], and also high enough to use the mem- = + Sx,tot Sx,sn Sx,rpn. For every frequency /2π there exists brane in a high-precision interferometer as demonstrated here. = | | 2 an optimum laser power PSQL cλ/[16 χ() πr ] at which In order to determine the parameters of our membrane, the two spectral densities have the same values. In this case we measured its power reflectivity to 30.4% at 1064 nm. the uncorrelated sum is minimal defining the standard quantum The reflectivity is directly linked to the membrane’s index limit with a displacement normalized spectral density of of refraction of n = 2.2 + i 1.5 × 10−4 at 1064 nm [16] S () = 2¯h |χ()| . (4) and to its etalon effect, which we used to determine the x,SQL membrane’s thickness to 66 nm. This yields an effective Using Eq. (2) the SQL force spectral density can be written as mass for the fundamental oscillation mode of m = 0.25 ρV ≈ = 3 115 ng, with mass density ρ 3.1g/cm [18] and volume 2¯h V = (1.5mm)2 × 66 nm. To avoid residual gas damping of SF,SQL() = . (5) |χ()| the membrane oscillation, our interferometer was set up in a vacuum chamber at a pressure of 10−6 mbar. The membrane’s The spectral densities given above need to be reached (at least ≈ 6 at one observation frequency) to achieve a displacement or a quality factor Q was measured to 10 . Residual gas damping force measurement with sensitivity at the SQL. For instance, was verified to reduce the measured Q of the membrane by at the mechanical resonance frequency the total noise needs to less than 0.3%. be as low as the following SQL (peak) value: Figure 1 shows the Michelson-Sagnac topology of our free-space laser interferometer. It was invented to deal with the 2¯hQ rather high light transmission of the membrane. The reflected S ( ) = . (6) x,SQL m m2 as well as the transmitted light from the membrane is kept m inside the same combined Michelson-Sagnac interferometer The mechanical oscillator used in our work was a 1.5-mm- mode. First, the input laser beam is split up by a balanced sized SiNx membrane for x-ray spectroscopy [14]. These high- beam splitter into two counterpropagating paths. Both paths tensile-stress thin films deposited on a Si frame have previously are folded by highly reflective mirrors such that a shared focus been used as mechanical oscillators in optomechanical cavity lies at the membrane’s position. The light fields reflected setups [15,16]. Their low effective mass combined with off the membrane interfere at the balanced beam splitter, their rather high tensile stress results in high-fundamental- forming the “Michelson mode” of our interferometer. The light resonance frequencies around 100 kHz. For stoichiometric beams transmitted through the membrane propagate along

063806-2 INTERFEROMETER READOUT NOISE BELOW THE ... PHYSICAL REVIEW A 85, 063806 (2012) the same optical axis. They also overlap at the beam splitter 1.4 10 Measurement 8 and form the “Sagnac mode.” For a perfectly balanced beam 1.2 Theory Calibration splitter, the fields of a Sagnac mode interfere at the beam 6 /2) splitter in such a way that all the light goes back to the 1 4 λ laser source. Light leaking out of the other interferometer 0.8 2 DC port then corresponds to the signal port of the Michelson 0 0.6 -2 mode. Its photoelectric detection provides information about 0.4 -4 the membrane’s position. By placing the membrane such that -6 the two Michelson arms have identical lengths, the signal port 0.2 -8 Calibration (V can be made completely dark limited only by the interference 0 -10 contrast. For a perfect dark port, the interferometer signal port Normalized transmitted power 0 0.2 0.4 0.6 0.8 1 1.2 1.4 shows a perfect rejection of laser intensity noise. In the present Membrane displacement (units of λ/2) setup, however, we placed the membrane such that a small amount of light power leaked out of the Michelson signal port FIG. 2. (Color online) Comparison of the theoretical fringe in order to be able to detect the interferometer signal with pattern of a Michelson-Sagnac interferometer with measured output a single photodiode. Note that in contrast to previous setups data. The derivative (dashed red line) of the transmitted power serves with membranes [16,19] our present setup did not involve as calibration of the interferometer output in Fig. 3. optical cavities; however, the Michelson-Sagnac topology in principle allows for the implementation of a power-recycling The input-normalized light power at the photodiode is then as well as a signal-recycling resonator for further sensitivity given by improvements [12,20,21]. 2 Pout Aout As the light source we used a Mephisto Nd:YAG laser = = c1 + c2 cos (4φ) + c3 cos (2φ), (8) that provided up to 2 W light power at λ = 1064 nm. The Pin Ain laser source was equipped with a direct feedback to the pump with constants diode’s current driver to suppress its relaxation oscillation = 2 − 2 2 2 + 2 2 2 = 2 2 2 (a so-called noise eater). For a further reduction of laser c1 rbs tbs tm 2rbstbsrm,c2 2rbstbsrm, (9) intensity noise, 10 mW were picked off before the mode c = 4 r2 − t2 t r t r . cleaner and detected on a photodiode. The signal derived was 3 bs bs m bs bs m processed by a PID controller and fed back to an acousto-optic The derivative modulator placed into the optical path before the mode cleaner. ∂Pout 2c2πPin c3πPin The servo’s control bandwidth was about 1 MHz, yielding an =− sin (4φ) − sin (2φ) (10) ∂x λ λ additional noise suppression of 12 dB in the measurement band around 100 kHz. As a result, technical laser noise did not of this fringe equation provides the calibration of the interfer- influence the membrane displacement measurements reported ometer output in photoelectric voltage per unit length. here. About 53 mW of the stabilized light did eventually Note that, although the described interferometer can be circulate inside the interferometer. The photoelectric detection thought of as a composition of a Michelson and a Sagnac was performed with a single InGaAs photodiode of 500-μm mode, their interference provides a distinct fringe pattern. A diameter at the interferometer’s signal port. The membrane pure Sagnac interferometer can be dark at the output only = position was piezo controlled such that about 0.55 mW of if tbs rbs; however, a Michelson-Sagnac interferometer can = light was detected. This light served as a local oscillator for have a dark output port even if tbs rbs. In the latter case the the sideband fields produced by the membrane’s oscillation Michelson mode needs to be detuned from its own dark fringe appearing as an amplitude modulation of the local oscillator to provide an overall dark fringe [11]. The required differential beam. phase of the Michelson mode is given by In the following paragraph we derive the Michelson-Sagnac 2 2 r − t trmm interferometer’s input-output relation, which is then used to 2φdark = arccos bs bs . (11) 2r t r model the fringe pattern at the signal port, and√ calibrate the bs bs m interferometer’s noise spectral density in m/ Hz. Let αin Obviously, Eq. (11) is defined only if the unbalancing of the and αout be the complex amplitudes of the interferometer beam splitter is not too strong. In our experiment the beam = input and output fields, respectively, and let rbs (rm) and splitter was rather well balanced (rbs/tbs 0.486/0.514) and dark tbs (tm) be the amplitude reflectivity and transmissivity of φ was close to π/4. the interferometer beam splitter (of the membrane); then Figure 2 shows a measured fringe pattern of the Michelson- the input-normalized output field at the signal port is given Sagnac interferometer (bold, green line). For the calibration of by the x axis we fitted a second-order polynomial function to the response function of the piezo that drove the membrane α out = 2 − 2 + along the optical axis. The different heights of neighboring rbstm tbstm rbstbsrm exp (i2φ) αin maxima are due to the slight unbalancing of the interferometer + − tbsrbsrmexp ( i2φ) beam splitter. For comparison the theoretical fringe pattern = 2 − 2 + according to Eq. (8) is also given. The dashed curve in Fig. 2 rbs tbs tm rbstbsrm2 cos (2φ), (7) shows its derivative according to Eq. (10). It was used for the depending on the membrane’s detuning φ = 2πx/λ. absolute calibration of the interferometer’s sensitivity. For a

063806-3 T. WESTPHAL et al. PHYSICAL REVIEW A 85, 063806 (2012)

frequencies the measured noise spectral density corresponds 10-11 Calibrated shot noise Measured dark noise √to the readout noise (imprecision) being more than a factor -12 Total readout noise 40 below the peak of the SQL. The broad peaks around 10 Hz) the membrane resonance were due to beat signals with the √ Modelled thermal noise

-13 Sum (total noise) mechanical resonances of the membrane mounting. 10 Measurement To conclude, the Michelson-Sagnac interferometer of the SQL( f ) -14 present work shows a readout noise power spectral density (in 10 SQL( 2 fm) m /Hz) being a factor of more than 40 (16 dB) below the peak

-15 value of a SiNx membrane’s SQL. This factor is considerably 10 higher than that achieved in previous works with nanowires,

-16 in which factors of about 1.25 (1 dB) [9]and2(3dB)[10,22] 10 were achieved, respectively. The absolute calibration of the Equivalent displacement (m/

-17 interferometer’s sensitivity was based on the measurements of 10 the absolute light power and its noise at the interferometer’s 98 100 102 104 106 108 110 signal port. It was found in very good agreement with Frequency f=Ω/2π (kHz) the modeled thermal excitation of our room-temperature membrane. We emphasize that, just like all previous works, FIG. 3. (Color) Measured (green) and modeled (yellow) noise our optomechanical arrangement did not reach its SQL of a spectrum of the Michelson-Sagnac laser interferometer containing force measurement, leaving this an all-time goal. In order to aSiNx membrane at room temperature. The total readout√ noise reach the SQL at the membrane’s resonance frequency we × −16 (imprecision) has a level of slightly above 3 10 m/ Hz and don’t need to further reduce the absolute value of the readout is given by the sum of electronic dark noise and shot noise.√ It lies noise, but rather the relative contribution of detector dark noise × −15 well below the peak value of the SQL of 2 10 m/ Hz at the has to be lowered. But first of all, the membrane needs to be f membrane’s resonance frequency m. For our modeling we assumed cooled to temperatures belowh ¯ /2k ≈ 2.5 μK, with k all noise sources to be uncorrelated. m B B the Boltzmann constant. This rather low temperature is related to the rather low resonance frequency of the membrane. We fixed input power, the fringe pattern and the dc output power note that the regime of quantum radiation-pressure coupling directly provide the (dc) membrane position, i.e., the operating should be possible at significantly higher temperatures around point of the interferometer, and the corresponding fringe 1K[12]. In principle our interferometer can be enhanced by derivative. The measured power spectrum of the high-pass adding a mirror to the output port. In this configuration, the filtered photocurrent was first divided by the independently interferometer might enable the demonstration of dissipative measured ac gain of the photodiode electronics and by the cooling [23]. The same configuration could also be used resolution bandwidth of the measurement in order to obtain the to establish a signal-recycling cavity [12,20]. In this case 2/ power spectral density of the photoelectric voltage in V Hz. the quantum readout noise is reduced without increasing the The result was then converted into a linear spectral density by laser power at the membrane. However, the interferometer’s taking the square root. Finally the calibration from Fig. 2 was broadband signal bandwidth would be reduced down to the used to convert to a linear displacement noise spectral density. signal-recycling cavity bandwidth. A lowered readout noise in Figure 3 shows the measured and modeled linear noise our setup would certainly be valuable in order to enable the spectral densities of the membrane’s displacement inside observation of the membrane’s thermal noise spectrum below the Michelson-Sagnac interferometer. The spectral densities its resonance frequency. This might allow the identification of include technical and quantum noise of the interferometer’s the underlaying loss channel in this kind of high-Q oscillator, readout (shot noise) and the room-temperature thermal noise giving insight in solid-state physics. of the membrane.√ The shot noise has a spectral density of 2 × 10−16 m/ Hz and was derived from a noise measurement of the 0.55-mW output field using the interferometer ACKNOWLEDGMENTS calibration described above. It was found in accordance with Eq. (1) for an intrainterferometer light power of 53 mW. This work has been supported by the International Max The thermal noise was calculated following Ref. [12]. Its Planck Research School for Gravitational Wave Astronomy absolute values agreed with the independent calibration of and by the Centre for Quantum Engineering and Space-Time the interferometer’s sensitivity described above. The spectral Research, QUEST. The work of F. Ya. Khalili was supported density of the membrane’s SQL was calculated according to by NSF and Caltech Grant No. PHY-0967049. We thank A. Eq. (4). The peak value of the SQL is highlighted√ as a dashed Xuereb, H. Miao, and A. Rudiger¨ for helpful comments on our horizontal line at a level of 2 × 10−15 m/ Hz. At off-resonant manuscript.

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