Appl. Phys. B 64, 203–212 (1997)

c Springer-Verlag 1997

Continuous-wave quantum nondemolition measurements with vacuum and nonclassical meter input R. Bruckmeier, K. Schneider, H. Hansen, M. Schalke, S. Schiller, J. Mlynek Fakult¨at f¨ur Physik, Universit¨at Konstanz, D-78434 Konstanz, Germany (Fax: +49 7531/883072, E-mail: [email protected], http: //quantum-optics.physik.uni-konstanz.de) Received: 25 July 1996

Abstract. QND measurements of the amplitude quadrature of continuous-wave light are performed with a monolithic dual-port degenerate optical parametric amplifier (OPA). Operated with a vacuum meter input, both output beams are squeezed and 33% correlated, demonstrating individually squeezed twin beams. The sum of the signal and meter transfer coefficients is 1.05, demonstrating operation as a quantum optical tap. The device exhibits quantum state preparation ability for both signal and meter output, reaching the conditional variances of −0.8 dB and −2.5 dB, respectively. An improved quantum measurement is realized by injecting 3.4 dB amplitude-squeezed light into the meter input port of the OPA. This achieves increased correlation and squeezing of the output beams, and both improved operation as a quantum optical tap and as a quantum state preparator. The all-solid-state system was operated for up to 5 hours with high stability. PACS: 42.50-p

Optical QND measurements are indirect measurements of one observable of a signal wave which leave this observable unperturbed. The indirect measurement is performed by implementing a coupling to a meter (probe) wave in a nonlinear medium, whereby the quantum properties of the signal observable are copied onto one observable of the meter wave, followed of a destructive read-out of the meter wave by photodetection. For thorough discussion of QND theory, see [1]. In practice QND measurements are performed using signal and probe waves that are single-frequency continuous laser waves or quasi-monochromatic laser pulses, in both cases containing a single spatial mode; the observables may be the phase or amplitude quadratures or the intensity. In quantummechanical terms, the absence of perturbation of the signal observable means that there is no change in the probability distribution of the values of the observable. Since 1986, a variety of experiments on QND-type measurements on optical waves have been realized. The first experiment employed the χ(3) nonlinearity of optical fibers for pulses [2]. Experiments using χ(2) nonlinearities in crystals followed [3], with both pulsed and continuous-wave operation. In addition, a sodium atomic beam was used success-

fully as a χ(3) medium [4]. A particulary interesting implementation of QND measurements has been in fibers, with solitons as signal and meter waves, a technique that may have future applications in the telecommunication field [5]. Optoelectrical coupling was applied to destruction and recreation schemes [6]. Electromechanical coupling was used in the first measurements on a nonlinear mechanical resonator [7]. Recently, the long-standing challenge for repeated QND measurements was successfully mastered [8]. The continuous wave approach relied on the system discussed here. Nondegenerate parametric amplifiers were the first to be put into practice for χ(2) -QND measurements. In contrast, our experiment follows a degenerate OPA scheme proposed more recently by Smith et al. [9]. Here signal and meter inputs and outputs are defined by two cavity ports of a degenerate optical parametric amplifier, rather than orthogonally polarized signal and meter waves in nondegenerate type-II parametric amplifiers. Notable about this scheme is that measurements of the phase or amplitude quadrature should be possible by a simple π change in the phase of the harmonic wave that pumps the parametric amplifier. This paper covers two major experimental aspects. First, we show that this OPA can be operated in the QND regime. The second is the first demonstration that the use of squeezed light allows an enhancement of the quality of a QND measurement, following a suggestion by Shapiro [10] from 1980. The motivation to consider such a scheme stems from the fact that due to the practical limitation of a finite pump power, the nonlinear interaction is small and some of the quantum fluctuations of the meter input wave are transferred to the output waves, reducing the back action evasion, the measurement precision and the output correlation of the QND measurement. A reduction of the meter input fluctuations (in the quadrature component on which the QND measurement is performed) can therefore improve the performance of a QND device. This is a sensible approach, since there is no other way to improve the measurement once the technique, the nonlinear medium (and in particular its nonlinearity and loss) and the pump source are given. The generation of the nonclassical light, if it is performed using a subthreshold OPO, does not in principle require any substantial additional laser power, since its pump wave can

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Fig. 1. Schematic of a monolithic dual port OPA. A coupling prism is used for in- and outcoupling of the meter wave into the cavity

be recovered to pump the QND device. To date, there is only one published experiment where squeezed light was injected to a nonlinear optical system, an optical amplifier based on a nondegenerate OPA, to improve its operation [11]. Aside the fundamental interest in demonstrating the two issues sketched above, a technical motivation for the present work was to explore the feasibility of a reliable system that can be operated automatically for long periods of time. This feature is crucial for bringing QND measurements to the arena of applications, e.g. precision experiments requiring very long integration times [12]. The operating wavelength of the QND system described here was chosen as 1064 nm, because reliable and powerful single-frequency diode-pumped Nd:YAG lasers are available. Finally, recent developments in lithium niobate crystal quality, mirror coating and resonator design [13] were available for implemention of the present QND scheme. This paper is structured as follows. In Sect. 1 we sketch the principle of QND measurement using a dual port OPA, followed by a description of the components of the experiment and their operation in Sect. 2. Results of QND measurements using a vacuum and a squeezed meter input are presented in Sect. 3.

1 QND Measurements using a dual port OPA We consider a dual port OPA as shown schematically in Fig. 1. One port, a partially transmitting mirror with power transmission coefficient T1 , allows input and output of the signal. The second port, realized by a frustrated total internal reflection mirror with variable power transmission T2 , serves for input and output of a meter wave of the same frequency and polarization. Intra-cavity losses A are incorporated by coupling to a vacuum mode. The signal wave is resonantly injected into the resonator; inside the latter there is no distinction between signal and meter waves. A pump wave propagating colinear with the resonator mode interacts with the subharmonic field and mediates a parametric deamplification d per round trip. For QND measurements of the amplitude quadrature of the signal wave, the phase of the pump wave must be chosen for maximum deamplification of the resonant wave. Qualitatively, the capability of QND measurements may be understood as follows. To achieve QND, we require that the information (i.e. the quantum fluctuations or a classical modulation) carried by the signal-out and meter-out waves originated mostly from that of the signal-in beam. This occurs if A  T1  T2 ∼ d, the last relation indicating that

the the pump wave is close to, but below, threshold and acts to deamplify the intracavity wave. Then the meter-in wave is near impedance matched, and consequently the meter-out wave is dominated by the part of the signal-in wave that is transmitted through the cavity. Thus, a precise measurement is possible. In addition, the signal-out wave is dominated by the signal-in wave due to the high reflectivity of T1 which suppresses the contributions by the meter-in wave. This indicates that the signal information is well preserved and the measurement back-action is small. It is important to note that this scheme requires parametric, i.e. noiseless, rather than linear (e.g. absorbtive) deamplification, as the latter would introduce significant additional noise to disable QND operation. A simple quantitative description can be deduced from the theory given in Ref.[9] which is based on the linear input-output formalism [14]. For a realistic description it is crucial to take into account losses inside the resonator. The theory involves three basic quantities relevant to the parametric amplifier: the available pump power matched to the resonator mode P2 , the strength of the pump-subharmonic interaction Γ , and the intracavity power loss per round-trip A. Γ is related to the single pass second harmonic power The power generation by P2 = Γ P12 at low input power P1 . √ deamplification per round-trip is given by d = 2 Γ P2 . We can now define a figure of merit r−1 for the QND measurement, p (1) r ≡ A/d. If the port transmissions are chosen as T1 = rd and T2 = d + T1 , the overall QND performance can be considered optimal, though the individual QND properties (see below) are maximized at slightly different operation conditions. The leading terms of the input-output relations for the lowfrequency Fourier components of the signal and meter wave quadratures for the above transmissivities are:    Xsin  out   √ 3/2 Xs 1− − r −r in   Xm √r = . (2) out Xm r −r2 /2 r Xlin Here, Xs , Xm and Xl are the amplitude quadratures of signal, meter and vacuum (due to loss) modes, respectively. Clearly, for good QND operation r  1 is required. Specifically, in this limit and for vacuum meter input (Variance in ) = 1), the back-action evasion (signal channel transV (Xm fer coefficient Ts ) and the measurement precision (meter channel transfer coefficient Tm ) are given by Ts = 1 − r,

Tm = 1 − r .

(3)

The signal and meter power gains and the quantum noise powers (variances V ) of the output waves are gs = 1 − 2r,

gm = r,

Vs = 1 − r,

Vm = r . (4)

The correlation Co between the two outputs, and the conditional variances of the signal output given the knowledge of the meter-out state and vice-versa are, respectively, 1 − Co2 = 2r,

Vm|s = 2r2 ,

Vs|m = 2r .

(5)

Vm|s is smaller than Vs|m since the meter output is strongly squeezed while the signal output is not. Thus, in the limit

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r → 0 the device satisfies the commonly stated requirements for QND measurements. For finite r on the other hand, both outputs can be simultaneously squeezed and quantum correlated. We note that for the above choice of port transmissions the resulting oscillation threshold is P2th = P2 (1 + r)4 , confirming that the OPA is operated close to but below threshold. Based on the input-output relations (2) we can easily discuss the use of an amplitude-squeezed meter-in wave, in ) < 1. This will lead to a clear improvement in the V (Xm signal preservation (Ts ). A smaller improvement in the measurement precision (Tm ) is expected, since the loss fluctuaout is more important. The correlation tion contribution to Xm will be improved as well. It is useful to state under which assumptions expressions (2) are valid: signal and meter input powers negligible compared to the pump power, so that there is no pump depletion and no transfer of quantum noise from the pump to signal and meter; the measurement is performed on quadrature spectral components of frequencies much smaller than the cavity HWHM bandwidth; the pump phase is set for maximum parametric deamplification, so that amplitude and phase quadratures do not couple; the input quadratures are those corresponding to the external spatial modes matched to the cavity mode. Summarizing this section, we emphasize the √ very weak √ scaling of the figure of merit r−1 = 4 4Γ P2 / A with the system parameters. This is the main problem facing the experimenter. The figure of merit depends weakly on pump power and nonlinearity. The strongest dependence of r is on the crystal losses, which at present are the most severe limitation to the quality of QND measurements. Losses are a materials-quality issue; values around 0.05%/cm at wavelengths around 1 µm are currently the lower limit for MgO:LiNbO3 . The pump power level is a technical limit, given by the availability of powerful single-frequency lasers. This limit can be pushed back by resonating the pump wave in an external enhancement cavity. We have implemented such a scheme, but have not used it in the final experiments because of the increased complexity. Taking as an example a loss A = 0.1%, a nonlinearity Γ = 5/kW and a pump power P2 = 1 W, a figure of merit r−1 = 12 is obtained. While this value shows the potential of the present technique, we will see below that a number of limitations decrease r−1 substantially. We remark that the same figure of merit scaling has been found for a QND measurement scheme of the pump wave of a degenerate parametric amplifier [15]. In comparison to QND measurements, the generation of strongly squeezed light is a much simpler undertaking, as can be seen by expressing the variance V of the squeezed quadrature emitted by a subthreshold OPO in terms of r: V = r2 . The QND properties however scale only linearly with r. 2 Experimental set-up An overview of the experimental set-up is shown in Fig. 2. A laser serves both as the source of the signal wave and as the fundamental wave to be converted to the pump wave of the QND and squeezing parametric amplifiers. The signal wave is imparted an amplitude modulation at 14.0 MHz

Fig. 2. Experimental setup. HD: homodyne detector; SA: spectrum analyzer; MOD: modulation system; RL: resonance lock; PL: phase lock; DL: distance lock; PZT: piezoelectric translator

by a modulation system before being injected into the ring OPA. The accurate measurement of this modulation on the signal-in, signal-out and meter-out waves is one central task of the set-up. The other is to measure the quantum noise of the output beams and their correlation. To this end three homodyne detectors are used, one (HD1) for the characterization of the input modulation and the other two (HD2, HD3) to detect the two outputs. All output signals are measured by spectrum analyzers and evaluated by a computer. The squeezing OPA generates squeezed light, which, in the second part of the experiment, is fed into the meter input port of the QND ring resonator OPA. To operate the system stably, the doubling cavity, the squeezing OPA cavity and the laser frequency are locked on resonance with the resonance frequency of the QND OPA cavity. The two pump waves are phase-locked for maximum deamplification and the squeezed meter-in wave is appropriately phase-locked with respect to the signal-in wave. The various components are described in more detail below.

2.1 Laser and second harmonic generator A diode laser-pumped Nd:YAG-laser with 500 mW output power (Lightwave Electronics, model 122) serves as the primary source of the system. Most of the laser power is frequency-doubled in a MgO:LiNbO3 semi-monolithic standing-wave cavity. The cavity is 29.9 mm long and contains a 7.5 mm long crystal with a plane face coated AR for 532 nm and 1064 nm and a convex back face with a radius of curvature of 10 mm, coated HR for 532 nm and 1064 nm. A 25 mm radius concave mirror with 4.0% transmissivity serves as the input coupler, and is mounted on a piezoelectric actuator to keep the cavity resonant for the fundamental wave. A dispersive error signal for the servo loop is obtained by the Pound-Drever technique as follows. The crystal is electro-optically modulated with a weak RF signal. The (weak) fundamental beam transmitted through the HR end face of the crystal is detected and the photocurrent is mixed to d.c. with a local oscillator of appropriate phase. The doubler generates 232 mW at 532 nm of which 113 mW are available to pump the QND resonator and 77 mW to pump the squeezer.

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Fig. 3. Reflectivity and transmissivity of the ring OPA cavity for an injected signal wave, as a function of T2 and pump power. Dots: experimental data; lines: theoretical fit

2.2 Preparation of the signal-in beam The signal input wave, with a power set at 1.05 mW, is obtained from the Nd:YAG laser beam, after passing two electro-optic modulators (EOM). EOM 1 is used as a polarization modulator followed by a polarizer and serves to stabilize the signal-in power and to apply the 14.0 MHz amplitude modulation. A dc voltage (0 − 800 V) applied to the EOM 1 crystal is regulated by a servo loop so as to keep the dc power reflected by the polarizer constant. To avoid large excursions of the dc voltage, the EOM is also actively temperature-stabilized. Since the amplitude modulator generates unwanted phase modulation at 14 MHz, EOM 2 is used as a compensating phase modulator driven at the same frequency and appropriate amplitude and phase. The cancellation of the phase modulation is checked at HD1. This homodyne detector, in contrast to the other two, can also measure the phase noise by employing a phase-locked optical local oscillator. Finally, EOM 2 is also used to phase modulate the signal wave at 16 MHz for Pound-Drever stabilization of the laser frequency to the ring OPA cavity.

2.3 The monolithic OPA The heart of the experiment is a miniature monolithic MgO:LiNbO3 ring resonator (type-I temperature phase matched), mediating the nonlinear interaction that gives rise to the nonclassical effects. The resonator (see Fig. 2) is 6.0 mm long, with the side faces enclosing an angle of 16◦ , resulting in a round-trip length of 12.7 mm. Both curved faces have radii of curvature of 5.0 mm. The crystal c-axis is oriented perpendicular to the ring plane, and the circulating infrared wave is p-polarized (ordinary). The resonator mode is confined by three total internal reflections and a dielectric mirror with T1 = 1.45% at 1064 nm and antireflecting at 532 nm. A ring cavity is used here, as opposed to a standing-wave cavity, so that input and output waves are spatially separated; this avoids the need of optical isolators for separation, which typically would introduce 5% loss into the two channels. Due to the compact OPA design, a large cavity linewidth of 54 MHz (hwhm) is obtained for a typical value of T2 = 5%. This allows QND measurements at

modulation frequencies where the laser is nearly shot-noise limited (> 12 MHz at 1 mW). In addition, the ring design permits to implement one mirror, here chosen as the meter port mirror, using frustrated total internal reflection. By varying the distance between the resonator face and a Brewster-angle cut coupling prism mounted on a piezoelectric actuator, a variable transmission T2 is obtained, which allows optimization of the QND operation. The meter-in wave, which is either the vacuum state or the injected squeezed wave, enters the OPA through the coupling prism. The distance between the coupling prism and the ring resonator is critical. It can be changed to obtain a large range in T2 by computer control for the characterization of the OPA, see below, or individual QND measurements that do not require drift stability. In addition, the gap can be kept constant by means of a servo loop that measures the brightness at the center of Newton’s fringes that are generated by reflection of a He-Ne laser. The signal-in beam and the pump wave are injected colinearly into the OPA. The important mode overlap between signal wave and the elliptic and astigmatic TEM00 cavity mode is M = 89.3%. The pump wave is non-resonant and leaves the crystal after one round-trip. As discussed, good QND performance is predicted for large parametric deamplification, d  T1  A. To maximize d, the pump wave is accurately mode-matched to the cavity mode. Furthermore, to allow continuous operation, the relative phase between the pump and signal waves is actively controlled by a servo. It locks the pump phase using a piezo actuator for maximum parametric deamplification of the resonant wave by minimizing the power of the meter-out wave. The linear and nonlinear optical properties of the ring resonator are characterized by measuring the power reflectance and transmittance as a function of prism position and pump power, as shown in Fig. 3. T1 , A, and M are determined by with t pump off variation of T2 , curve (a). The intracavity losses amount to A = 0.58% for the p-polarized mode. In addition, the power calibration of HD2 relative to HD3 can be checked, and the power reflectivity of the beam splitter directly before HD1 is obtained. This takes into acount optical efficiencies for the propagation of the signal-in beam to the crystal, and the signal-out and meterout beams from the crystal to the detectors, which are 99.4%, 97.5% and 99.0%, respectively. With the pump beam turned on and phase locked for deamplification, the point of operation in Fig. 3 (i.e. the cavity reflectivity and transmissivity) moves to the left as the pump power and consequently d are increased. At a pump power of 113 mW and at best pump mode matching a nonlinear deamplification of d = 2.7% is obtained. Varying the coupling prism position (i.e. T2 ) at maximum pump power yields curve (b) in Fig. 3. We remark that for the range of T2 given in this paper, the amplitudes of the error signals vary by two orders of magnitude. Servo gains must be changed appropriately to operate safely at extreme values of T2 , as otherwise the servos will operate too slowly for large T2 or oscillate for low T2 .

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2.4 The squeezed light source A detailed account of the OPA used to generate the bright squeezed light can be found in Ref.[16]. The use of bright squeezed light rather than squeezed vacuum (note that in the framework of the simple input-output relations (2) there is no difference) permits to achieve phase locking of the relative phase between the signal and meter waves in a relatively simple manner. This lock is essential, since the squeezed quadrature of the meter input beam must be superimposed on the quadrature of the signal beam being measured. With a dither-and-lock technique, the squeezed light propagation phase shift is controlled so as to give destructive interference with the signal-in beam, as detected in HD3. Maximum constructive interference would also use the squeezed quadrature (i.e. the squeezing ellipse is turned by 180◦ ), however, due to imperfect modematching and opposite power interference, a slight degradation of QND performance is expected. The amplitude squeezed light used in our experiments exhibits 3.4 ± 0.3 dB quantum noise reduction around 14 MHz, a power of 30 µW, and remains stable for many hours. The mode match of the squeezed beam to the resonator mode is 74% as deduced from the interference fringes on the meter output beam. The imperfect modematch results from complex phase front imperfections and leads to a reduced amount of quantum noise suppression actually available to improve the QND characteristics. It also leads to additional quantum noise being detected on HD2 due to the finite squeezed beam power. The chosen meter input power is a compromise between adding little noise and obtaining large error signals for stable operation of the system. 2.5 Polarization coupling within the resonator In this section we describe an effect which can degrade significantly the operation of a nonlinear resonator. The effect is apparent when a phase matching curve of the SHG efficiency of the ring resonator is recorded as in Fig. 4. This is a necessary preparatory step in the QND measurement procedure since a determination of the optimum crystal temperature is necessary to obtain maximum parametric interaction. Many narrow dips are seen to occur in which the nonlinearity is severely degraded. For optimum operation the resonator temperature is confined to a 50 mK wide region. The origin of this effect is mixing of polarizations within the cavity. Due to slight misalignments of the side faces relative to the crystal c-axis and crystal inhomogeneities the eigenpolarizations of the cavity are not purely s and p, but mixtures. Thus, if a p-polarized TEM00 laser wave is injected into the cavity it may couple to a varying extent to frequency-offset s-polarized cavity modes, which do not give rise to frequency conversion. This causes a reduction in harmonic generation, as observed in Fig. 4. A successful model for the observed coupling assumes the p-polarized mode to be coupled in sequence to each of a set of N s-polarized modes TEMnm . Each coupling may be described by a unitary 2 × 2 matrix, resulting in a (N + 1) × (N +1) interaction matrix that relates the N +1 coupled modes to the N + 1 bare modes. This model is used to calculate the cavity reflectivity for an input p-wave. Figure 5 shows a

Fig. 4. Second harmonic output from frequency-doubling of the signal wave frequency-locked to the ring resonator as a function of its temperature. Pump is off. The strong, resonance-like structures are due to coupling between p-polarized and s-polarized cavity modes

Fig. 5. Power reflection of the signal input wave, obtained by electro-optic tuning of the ring resonance frequency. The p-polarized resonance couples to s-polarized resonances. Their smaller width is due mainly to the larger electro-optic tuning coefficient

comparison of the measured and modeled cavity reflection spectrum, where three s-polarized resonances appear in the vicinity of the p-resonance. The model contains as fitted parameters the input transmissivity T1s and loss As for the s-polarized modes (assumed to be equal for all s-modes) and 2N coefficients: a coupling angle and the (electro-optically tunable) bare s-resonance detuning relative to the bare presonance. The coupling angles can be interpreted as the product of the transverse mode overlap of the p-mode and the respective s-mode times the polarization rotation angle per round trip. For example, in Fig. 5 the coupling angles are 0.102◦ , 0.089◦ and 0.294◦ for the coupling of p to s1 , s2 , s3 . As there are many coupled s-resonances, the round trip polarization rotation amounts to a few degrees. The model also describes well the observed avoided crossings of the s-resonances with the p-resonance in the electro-optically tuned spectrum as a function of laser frequency[17].

2.6 The detection system The signal and noise properties of the optical beams are measured with self-homodyne detectors, HD1-HD3, in which the

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Fig. 6. Detection noise power spectra. The QND measurements are performed at frequencies above 14 MHz, where the excess laser noise is suppressed by more than 10 dB relative to the shot noise

incident beam is mixed with the vacuum mode and the resulting beams are detected by photodiodes. The reflectivity of each beam splitter is tuned to 50% by tilting it. The detectors use InGaAs photodiodes (ETX 500-T) with η = 97% quantum efficiency at 1064 nm and very low noise electronics (NE5211 transimpedance amplifiers). A quantum noise equivalent power of 200 µW at 14 MHz is achieved (37 µW at 4 MHz). For each homodyne detector a hybrid junction generates the sum and difference currents, which are proportional to the amplitude quadratures of the signal beam and the vacuum beam, respectively. All following measurements refer to amplitude rather than phase quadrature fluctuations. The intensity noise of the incident beam relative to the coherent state level is determined from the ratio of the sum current noise power, which is proprotional to the amplitude noise of the beam, and the difference current noise power, which provides the reference for the shot noise. We denote by ”squeezing” (S) the inverse of the relative quantum noise. Computer controlled high quality RF relays are used to route the currents of each detector in turn to two spectrum analyzers (HP 8591A, detection bandwidth 100 kHz, zero span time 0.5 s) where the noise powers are measured. A computer receives this information and automatically subtracts the electronic noise level, determined when the laser is off. Measurements of the various noise powers are given in Fig. 6. The electronic noise levels of the sum current channel (shown in the plot) and of the difference channel are nearly identical. The shot-noise level, as derived from the difference current noise power, varies linearly with the optical power over many orders of magnitude. To compensate asymmetries in the hybrid junctions and the relays and the spectrum analyzers, the readings of the spectrum analyzer that measures the difference currents are offset individually for each homodyne detector selected by the RF relays. The offsets were obtained most reliably by the following procedure. The relative intensity noise is recorded as a function of the laser power incident on the homodyne detector. A linear dependence with small slope is found. From the requirement that in the limit of vanishing optical power only its quantum noise contributes and technical noise is absent, the calibration factor is obtained. The small slope indicates the presence of technical noise, which can be de-

Fig. 7. Squeezing of the signal and meter output beams at 14.6 MHz as a function of outcoupling T2 . Circles: vacuum meter input; triangles: squeezed meter input; lines: theory

termined by subtracting the difference current noise from the sum current noise. Excess amplitude noise of the signal-in beam due to technical laser noise was mesured on HD1 to be 0.16 dB at 1 mW and 14.6 MHz, but increased strongly towards lower frequencies. This measurement procedure results in an almost perfect quadratic dependence over many orders of magnitude of the applied modulation signal on the optical power incident on the detector. We emphasize that this linearity of the shot noise and the quadratic dependence of modulation noise are a necessary condition for reliable SNR measurements. The reliability of the measurement system was also verified on the bright squeezed light as a function of its power in the generating OPA and independently by inserting an attenuator into the squeezed beam.

3 Results In this section we report the results of QND experiments performed with the ring resonator. We describe measurements using the vacuum state as a meter input beam and compare the results to the case where we inject squeezed light into the meter input port.

3.1 Squeezing measurements The first nonclassical effect considered are the squeezings Ss and Sm of the signal-out and the meter-out beams, respectively. The results of the squeezing measurements at 14.6 MHz when varying T2 are shown in Fig. 7. It can be seen clearly that the fluctuations of the meter output beam are further reduced by using a squeezed meter wave. The functional dependence can be understood qualitatively for the case of a vacuum meter-in: The squeezing of the signal output beam is degraded for higher T2 as both the outcoupling efficiency decreases and the OPO threshold increases. For the meter-out squeezing there is an optimum, since for low T2 the outcoupling efficiency for the meterout beam limits the squeezing, while for large T2 the OPO threshold grows compared to the pump power. The quantum

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noise reduction in the meter output is larger than that of the signal because T2 > T1 . Regarding the theoretical values for the squeezing and the other properties discussed below, we note that the theory given in Ref.[9] must be extended to provide a satisfactory description of the experiment. Effects that are taken into account are the imperfect modematch of the signal-in and the meter-in beams to the resonator, the large outcoupling T2 beyond the good cavity approximation, and a consistent way to include measurement frequencies comparable to the cavity linewidth. Details of the calculations, which predict some dramatic effects that also have been observed experimentally, will be presented in a forthcoming paper [18]. Good agreement of theory and experiment as shown in the following figures is achieved after assuming increased losses of A = 1.1% when the QND OPA is pumped. This may be an indication of the presence of green induced infrared absorbtion. An approach that is consistent with all power measurements leads to adjusted values of d = 2.4%, Γ = 1.3/kW and r−1 = 1.5.

3.2 Quantum correlation and conditional variance For a QND device the meter output reading must be correlated with the signal output reading: “the needles move together” [19]. The optical correlation Co of the amplitude fluctuations of the output beams leads to an electronic correlation of the detector sum currents Cc , which is reduced compared to Co due to electronic noise of the photodetectors. Denoting the noise power contribution from the light by Vo and that from the detector noise in the i+ -channel of HD2 by Vel (corrected for the baseline noise of the spectrum analyzer), it can be shown that the optical and electronic correlations are related by Cc2 /Co2 = Vo /(Vo + Vel ). The current correlation coefficient Cc of the two output beams is measured at 14.6 MHz via the noise N (g, φ) present on the difference current between the attenuated (g) and phase shifted (φ) sum current of HD2 (i+2 ) and the sum current of HD3 (i+3 ), N (g, φ) = V (i+3 − g eiφ i+2 ). The reduction of the noise compared to that corresponding to two uncorrelated waves, n, is obtained by normalizing to the noise of HD3, n(g, φ) = N (g, φ)/N (0, φ). n varies quadratically with g, and its minimum allows the determination of the current correlation: Ming,φ [n(g, φ)] = 1 − Cc2 . The experimental procedure leading to Cc consists in optimally phase shifting i+2 by selecting an appropriate cable length, passing it through a variable attenuator and mixing it with i+3 in a directional coupler. For each attenuation, the noise powers of both coupler outputs, N (+g, φopt ) and N (−g, φopt ), are compared to the results for infinite attenuation (g = 0), returning n(±g, φopt ). The measured dependence on the attenuation g is shown in Fig. 8 for the case of a vacuum meter input wave. A fast strategy to measure the correlation with small statistical error is to evaluate n(±g, φopt ) for g ∼ 0.3, which is larger than the optimum g, and to infer Cc and Co from the quadratic dependence. The results for the optical correlation are given in Fig. 9 for both vacuum and squeezed meter input beams, also showing clearly the improved correlation when using a squeezed meter input beam.

Fig. 8. Correlation between signal and meter ouput detector currents as a function of the gain (attenuation) applied to the sum current of HD2. Points: data; line: quadratic fit. The noise reduction factor 1 − Cc2 is obtained at the minimum of the parabola

Fig. 9. Optical correlation between signal and meter ouput beams. Circles: vacuum meter input. triangles: squeezed meter input; lines: theory

From the strong correlation and the near shot-noise limited input beam one can conclude that the two output beams are quantum correlated, or entangled. Furthermore, as both beams are squeezed, these measurements represent the first demonstration of individually squeezed twin beams [20]. The QND scheme used here has the particular property of permitting quantum state preparation for both output beams: measuring one beam will lead to a state reduction of the entangled state and therefore to a state preparation of the other beam. This will reduce the variance of the second beam by the factor 1 − Cc2 , leading to the conditional variances Vs|m = (1 − Cc2 )/Ss and Vm|s = (1 − Cc2 )/Sm for the signalout and the meter-out beams, respectively. The conditional variances resulting from the measured squeezings and correlations are shown in Fig. 10. Both conditional variances are below unity and therefore both fulfill the quantum state preparation condition. An improvement of up to 0.3 dB at the

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Fig. 10. Conditional variances Vm|s and Vs|m at 14 MHz as a function of outcoupling transmission for a vacuum (diamonds) and a squeezed meter input (triangles)

point of optimimum output coupling occurs upon injection of the squeezed meter. In the context of QND measurements a conditional variance below unity ensures that after the amplitude measurement of one output beam the future evolution of the other can be predicted with precision beyond the standard quantum (coherent state noise) limit. Given such a well-prepared state, a signal that modulates this state can be measured with improved accuracy.

3.3 Transfer coefficients Besides the quantum state preparation ability, which is a property of the output beams alone, an optical measurement system is also characterized by how well the information encoded on the signal-in beam is transferred to the read-out beam and to what extent the original information is preserved on the output signal beam. This is described quantitatively by the signal and meter transfer coefficients Ts and Tm introduced in Sect. 1. They are defined as the ratio of the signal and the meter output signal-to-noise ratios (SNR) relative to the signal input SNR (SNRin s ). The SNRs of the output beams are measured directly with HD2 and HD3. The signal power and noise power that yield the SNR are measured zero span at the signal modulation frequency and at 14.6 MHz, respectively. Figure 11 illustrates the kind of signals detected in a QND measurement. The SNR of the signal-in beam is monitored at HD1 by picking off a fraction with a R = 22.60% beamsplitter from the main beam. The SNR of the signal-in beam is therefore inferred from the SNR measured at HD1 by SNRin s = SNRHD1 ηopt η (1 − R)/R, where ηopt = 98.5% is the optical efficiency from the beam splitter to the detector. For any phase-insensitive device, such as a beam splitter, the sum of both transfer coefficients is at most unity if the input beams are shot noise limited: Ts + Tm ≤ 1. However, a quantum optical tap succeeds in breaking this barrier by operating phase-sensitively, ideally transfering the full SNR from the input to both outputs: Ts + Tm = 2, albeit only for one quadrature. Figure 12 shows the results of our mea-

Fig. 11. Spectra of the amplitude noises detected by HD1-3 (thick lines) and shot-noise level (thin lines). Vertical scaling is equal in all plots. The 14 MHz intensity modulation applied to the signal input wave is transferred to both outputs. Note that the signal-to-noise ratio of the meter output modulation (HD3) has a contribution from the amplitude squeezing present and leads to an improved transfer coefficient Tm

surements as T2 is varied. It can clearly be seen that the signal transfer is improved when the vacuum meter input is replaced by a squeezed beam. For vacuum meter input, there is a wide range for T2 for which the device operates as a quantum optical tap, the optimum being at 6.9%, where the transfer coefficients are Ts = 0.60 and Tm = 0.45. Injecting a squeezed meter input, the sum of the transfer coefficients reaches the optimum at T2 = 7.6%, with Ts = 0.69 and Tm = 0.43. Much care was taken to operate the experiment at RF frequencies where the input beam was almost perfectly shot-noise limited, as additional technical noise causes an overestimation of the transfer coefficients. The linearized theory of this experiment predicts that the amplitude and phase fluctuations should be decoupled. In that case, each transfer coefficient is identical to the absolute square of the correlation coefficient of the fluctuations of the

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ideal device would in addition fulfill Co → 1, Tm → 1 and Vs|m → 0. However, both previous criteria have become widely accepted because there is a sharp boundary between classical and nonclassical operation that allows to classify experiments by their performance. As the signal-out beam is squeezed in our case, the input state is clearly modified and therefore this is not an ideal QND measurement. However, like in many other experiments, ideal QND measurements are predicted in the limit of vanishing passive losses. It is interesting to note that in this case the nonclassical property of a squeezed signal-out beam arises because of, and not in spite of, losses. Fig. 12. Ts + Tm versus the meter outcoupling for vacuum (circles) and for squeezed meter input (triangles). For both cases there is a wide operation regime where the sum is larger than unity and the device qualifies as a quantum optical tap

Fig. 13. Summary of the results. The conditional variance Vm|s is plotted versus Ts + Tm as the outcoupling transmission T2 is varied. The system acts both as a quantum state preparator and as a quantum optical tap when in the grey region. Significantly improved performance is demonstrated when squeezed light is injected into the meter input port of the resonator (triangles) as opposed to using a vacuum meter input beam (circles)

signal-in and the respective output beam [21], which gives additional insight to the nature of the transfer coefficients. 3.4 QND operation In the previous sections this device was demonstrated to exhibit two independent nonclassical properties, i.e. operation as a state preparator and as a quantum optical tap. These two conditions are widely accepted to qualify a single device for QND operation. Both criteria are used to generate Fig. 13: The conditional variance is plotted against the total transfer coefficient and the lower right sector is termed the QND region. Here the data has been slightly averaged for clarity. Achieving operation in the QND region is not alone sufficient to infer true QND operation. A more complete set of necessary parameters is Ts = 1, gs = 1, Ss = 1. An

4 Summary and conclusion In conclusion, we have implemented a new scheme for continuous QND detection of amplitude fluctuations by signalprobe coupling in a dual-port degenerate parametric amplifier. A figure of merit, r−1 , has been identified which characterizes the quality of the QND measurement possbile with this approach. To maximize r−1 , a monolithic MgO:LiNbO3 ring resonator has been developed which exhibits relatively low loss (0.58%) and a nonlinearity close to the maximum value expected for the nonlinear material used. A 113 mW pump wave for the OPA was generated by highly efficient (70%) frequency doubling of a miniature diode-pumped monolithic ring laser in an external enhancement cavity. These efforts resulted in a figure of merit r−1 = 1.52, indicating the difficulty of achieving strong QND properties. In particular, lower values of intracavity loss seem realistic. On the technical side, it is worth emphasizing that both frequency-degenerate OPAs used in the experiment were operated with phase-locked pump waves, which represents, to the best of our knowledge, a novelty. The ability to vary the meter output coupling in real time during the course of the experiment permitted not only to optimize the performance, but also a fairly complete understanding of the system. The detailed performance of the device is in satisfactory (and for such a complex system, gratifying) agreement with the theoretical predictions of a model that takes into account a large number of parameters describing additional quantum noise effects. With regard to the quantum properties, the output beams of the ring OPA were shown to be individually squeezed and quantum correlated, which constitutes the first observation of individually squeezed twin beams. As a consequence, both output beams satisfy the state preparation condition and permit a measurement precision beyond the standard quantum limit, achieving a conditional variance of -2.5 dB. Furthermore, the dual-port OPA is a quantum optical tap, transfering input information to its outputs better than any phase-insensitive device. Thanks to the frequency-stable laser and stable monolithic ring OPA cavity, the system was fully stabilized and continuously run for up to 5 hours. Reliable and reproducible measurements of the quantum effects could thus be performed. Due to this reliability, we could demonstrate that the system performance can be improved by replacing the vacuum meter-in beam by a phase-locked amplitude squeezed beam, which reduces the amplitude noise injected by the

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meter-in beam. For 3.4 dB squeezing all quantum properties of the system improved: We obtained increased squeezing and correlation of the output beams. The maximum of Ts +Tm rose from 1.05 to 1.12 and the minimum conditional variance of the meter-out beam was reduced from −2.5 dB to −2.8 dB. Acknowledgement. It is a pleasure to thank J.-Ph. Poizat, R. Paschotta, G. Breitenbach, G. Rempe, A. G. White, A. Levenson, S. F. Pereira and H. J. Kimble for stimulating discussions and helpful suggestions. Financial Support has been provided by the Deutsche Forschungsgemeinschaft, ESPRIT projects BRA 6934 QUINTEC, LTR 20029 ACQUIRE. R. B. and H. H. were supported by fellowship of the Studienstiftung des deutschen Volkes.

References 1. V. B. Braginsky, F. Khalili, ”Quantum Measurements”, Cambridge University Press, 1992. V. B. Braginsky, Y. I. Vorontsov, K. S. Thorne, Schience 209, 547 (1980). C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980). M. J. Holland, M. J. Collett, D. F. Walls, M. D. Levenson, Phys. Rev. A 42, 2995 (1990) 2. M. D. Levenson, R. M. Shelby, M. Reid, D. F. Walls, Phys. Rev. Lett. 57, 2473 (1986) 3. A. La Porta, R. E. Slusher, B. Yurke, Phys. Rev. Lett. 62, 28 (1989). S.F. Pereira, Z.Y. Ou, H.J. Kimble, Phys. Rev. Lett. 72, 214 (1994) 4. P. Grangier, J.-F. Roch, G. Roger, Phys. Rev. Lett. 66, 1418 (1991). J.-Ph. Poizat, P. Grangier, Phys. Rev. Lett. 70, 271 (1993) 5. S. R. Friberg, S. Machida, Y. Yamamoto, Phys. Rev. Lett. 69, 3165 (1992); S. Sp¨alter, P. van Loock, A. Sizmann, G. Leuchs, this issue, 213–218

6. E. Goobar, A. Karlsson, G. Bj¨ork, Phys. Rev. Lett. 71, 2002 (1993). J.-F. Roch, J.-Ph. Poizat, P. Grangier, Phys. Rev. Lett. 71, 2006 (1993). E. Goobar, A. Karlsson, G. Bj¨ork, Opt. Lett. 18, 2138 (1993) 7. D. S. Greywall, B. Yurke, P. A. Busch, A. N. Pargellis, R. L. Willett, Phys. Rev. Lett. 72, 2992 (1994) 8. K. Bencheikh, J. A. Levenson, P. Grangier, O. Lopez, Phys. Rev. Lett. 75, 3422 (1995). R. Bruckmeier, H. Hansen, S. Schiller, J. Mlynek, in preparation 9. P. Smith, M. Collett, D. F. Walls, Opt. Comm. 102, 105 (1993); Opt. Comm. 106, 288 (1994); F. Harrison, D.F. Walls, P. Smith, M. Collett, M.J. Dunstan, this issue, 225–233 10. J. H. Shapiro, Opt. Lett. 5, 351 (1980) 11. Z. Y. Ou, S. F. Pereira, H. J. Kimble, Phys. Rev. Lett. 70, 3239 (1993) 12. D. Bakalov et al, Nucl. Phys. (Proc. Suppl.) B35, 180 (1994) 13. S. Schiller and R.L. Byer, J. Opt. Soc. Am. B 10, 1696 (1993) 14. M. J. Collett, C. W. Gardiner, Phys. Rev. A 30, 1386 (1984) 15. S. Schiller, S. Kohler, R. Paschotta, and J. Mlynek, Appl. Phys. B 60, S77 (1995) 16. K. Schneider, R. Bruckmeier, H. Hansen, S. Schiller, J. Mlynek, accepted for publication in Opt. Lett 17. R. Bruckmeier, Dissertation Universit¨at Konstanz (1996). Ring resonators with increased specified manufacturing precision and 3.90 mm radius mirrors were fabricated, with the goal of reducing the coupling coefficients and achieving a bunching of the transverse cavity mode spectrum into nine groups, respectively. However, no relevant improvement of the polarization effect was obtained. In a different experiment the effect of polarization coupling was also observed in a monolithic standing-wave resonator 18. R. Bruckmeier, H. Hansen, S. Schiller, J. Mlynek, publications in preparation 19. B. Yurke, J. Opt. Soc. Am. B 2, 732 (1985) 20. M. A. M. Marte, Phys. Rev. Lett. 74, 4815 (95) 21. J.-Ph. Poizat, J.-F. Roch, P. Grangier, Ann. Phys. Fr. 19, 265 (1994)