A new method for locking the signal-field phase

A new method for locking the signal-field phase difference in a type-II optical parametric oscillator above threshold Matthew Pysher,1 Yoshichika Miwa,2 Reihaneh Shahrokhshahi,1 Daruo Xie,1 and Olivier Pfister1,∗ 1 Department

of Physics, University of Virginia, 382 McCormick Road, Charlottesville, Virginia 22904-4714, USA 2 Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan *[email protected]

Abstract: We propose and demonstrate a new method for phaselocking the signal fields emitted above threshold by a nondegenerate, type-II optical parametric oscillator (OPO). This method is based on the observation that amplitude modulation of the pump beam produces a related modulation of the frequency difference of the OPO signals via the temperature-tuning of the index of refraction in the nonlinear crystal. We successfully use pump modulation as a correction for phase-difference locking of the OPO signals and observe a 1 kHz beat note stable over more than 10 s, both figures solely limited by the measurement time. This method eliminates the need for applying electronic phase-correction signals directly to the nonlinear crystal which caused crystal damage in a previous phaselocking technique. © 2010 Optical Society of America OCIS codes: (160.2100) Electro-optical materials; (190.4870) Photothermal effects; (190.4970) Parametric oscillators and amplifiers.

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Received 20 Oct 2010; revised 9 Dec 2010; accepted 13 Dec 2010; published 17 Dec 2010

20 December 2010 / Vol. 18, No. 26 / OPTICS EXPRESS 27858

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Introduction

Ultrastable quantum optics is a tool of choice for precision measurements at the ultimate quantum limit. A prime example is the future advanced version of laser-interferometer gravitationalwave observatories [1,2] which will incorporate single-mode squeezed light in conjunction with a servo-loop controlled Michelson interferometer and optical cavities [3–6]. Another example of interest is the use of two-mode squeezing produced by ultrastable phaselocked OPOs [7, 8]. Such OPOs have been used to create continuous-variable EPR states [9, 10], evidence macroscopic Hong-Ou-Mandel interference [11], and make Heisenberg-limited heterodyne polarimetry measurements below the shot noise [12]. In the polarimetry experiment, the orthogonally polarized twin beams emitted above threshold by a type-II OPO were electronically phaselocked to each other using the tuning of the OPO via the electro-optic (EO) effect of its Na:KTiOPO4 (Na:KTP) nonlinear crystal. Note that this classical noise reduction on the phase difference occurred in the DC-10 kHz servo bandwidth, while the quantum noise reduction (squeezing) on the amplitude difference took place within the OPO cavity linewidth which extended to several MHz. Direct detection of the OPO beams after a polarizer yielded a squeezed intensity#136909 - $15.00 USD

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difference, twin-beam signal that was highly sensitive to minute polarization rotations, with the latter giving rise to a 1 MHz beat signal over a squeezed background, 4.8 dB below the standard quantum limit [12]. Such an experiment could thus be applied to ultrasensitive detection of chiral molecules, as well as to fundamental studies [13]. One severe practical limitation of the heterodyne polarimetry experiment was that the electrooptic phase-difference correction on the Na:KTP crystal necessitated the use of a fairly high voltage (∼ 65 V) on the crystal, exceeding 20 V/mm and resulting in electric damage [14], as can be seen in Fig. 1. The necessity for such a high voltage arose from our observation of an unexpected decrease of the EO effect at low frequencies (below 1 kHz) of the phaselock error signal. We explored this decrease of the EO coefficients at low modulation frequencies, and in section 2 of this paper, we report measurements of the r23 and r33 electro-optic coefficients and the z-axis electrical conductivity in KTP, Na:KTP and RbTiOAsO4 (RTA). In section 3, we then go on to demonstrate a new, all-optical way to implement phase-difference locking of a type-II OPO, where the correction signal does not use the electro-optic effect on the OPO crystal, thereby eliminating the possibility of crystal damage. This phaselock method is based on the observation that a variation of the intensity or polarization of the OPO pump beam yields a change in the frequency difference of the signal beams via the photothermal effect [15]. This can be viewed as an analog of pump-intensity control of femtosecond optical frequency combs via the Kerr effect [16]. In this experiment, we use polarization modulation of the pump beam to modulate the amount of downconverted light created in the OPO. Residual absorption of this light then leads to a thermal change in the OPO’s nonlinear crystal, which varies the crystal’s refractive indices and leads to a change in the frequency difference between the twin beams output by the OPO.

Fig. 1. Electrochromic damage in a Na:KTP crystal resulting from excessive voltage. The electrodes are on the top and bottom sides of the crystal. Note that the “Y” is a pencil mark while the black lines intersecting it indicate damage inside the crystal.

2.

Measurement of electro-optic coefficients

We tested the electro-optic coefficients of the nonlinear crystals by applying an EO sinusoidal modulation to the crystal and monitoring the resulting FM spectrum on the signal-idler beat note. We used two methods to measure the dependence of the electro-optic coefficients on the frequency of the applied field. In the first method, the crystal was placed inside an optical resonator and the EO effect was easily observed by measuring the cavity resonance shift with an #136909 - $15.00 USD

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oscilloscope. The EO coefficients r23 and r33 of the crystal were then individually measured by polarizing the input light field along the y and z axes of the crystal, respectively. Although sensitive, due to amplification of the EO effect from multiple passes of the resonant field through the crystal, this measurement of the resonance shift was nevertheless fairly noisy. We therefore used an ellipsometry setup in order to confirm the initial results. This ellipsometry measurement was more precise than the first method, even though it only enabled access to the composite EO coefficient n3z r33 − n3y r23 . The two methods produced consistent results for Na:KTP. Also, in order to investigate the possible correlation of the EO coefficient frequency dependence with ion currents in the crystals, we measured the electrical conductivity of the crystals along their z-axis, by using an auto-balancing bridge. 2.1.

Cavity resonance modulation measurements in Na:KTP

Fig. 2. Experimental setup for the cavity resonance modulation EO measurement. FI: Faraday isolator. PZT: piezoelectric transducer. The voltage Vo sin ω t, where Vo = 365 V, was applied along the z-axis of the crystal, giving a maximum field Ez of 1.2 × 105 V/m. The cavity length was scanned at 40 Hz by the PZT.

The experimental setup for the cavity resonance modulation measurements is shown in Fig. 2. The optical cavity used to measure the electro optic coefficients was composed of two concave mirrors of 150 cm curvature radius and 99% reflectivity. The incident light at λ = 532 nm was emitted by a frequency-doubled Nd:YAG monolithic laser (Lightwave Electronics, model 142). We used an x-cut, 5 mm × 3mm × 3mm flux-grown Na:KTP crystal fabricated by Coherent Crystal Associates and AR coated at 1064 and 532 nm. The crystal was placed at the center of the cavity. An AC voltage was applied to the top and bottom x-y surfaces of the crystal, onto which were attached copper foil electrodes. Hence, an electric field Ez was established along the z crystal axis. The cavity’s optical length is Lk = lo + nk l,

(1)

where nk is the index of refraction with k = y, z, depending on the polarization of the beam, lo is the cavity length in air, and l = 5 mm is the length of the Na:KTP crystal (lo + l = 10 cm). The cavity free spectral range is c , (2) ∆k = 2Lk and the cavity resonance frequency νk = mk ∆k , where mk is an integer. The EO index change is given by [17] n3y r23 Ez 2 n3z r33 Ez δ nz = 2 Since the EO effect is small, one can write the cavity resonance change as

δ ny

=

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2mk l 2 ∆k δ nk . c

(3) (4)

(5)



Using Eqs. (3) and (4) in the previous equation, one obtains |r23 | =

δ νy c n3y l∆y νy Ez

(6)

|r33 | =

δ νz c . n3z l∆z νz Ez

(7)

Electrooptic Coefficient (pm/V)

The modulation amplitudes δ νk and the free spectral ranges ∆k are measured on a digital oscilloscope, taking great care not to choose commensurable frequencies for the field modulation and the PZT scan. The free spectral ranges for different light polarizations are taken equal to first-order approximation. Figure 3 displays the measurement results. One clearly sees that the EO coefficients of

35

|r33|

30 25 20

|r23|

15 10 5 0 1

2

4 6

10

2

4 6

100

2

4 6

1000

Electric Field Frequency (Hz) Fig. 3. Experimental determination of the magnitudes of the Na:KTP EO coefficients |r23 |, |r33 |, versus EO modulation frequency.

Na:KTP drop sharply when the frequency of the applied electric field is below 20 Hz. At higher frequencies, the measured values for the effective r23 and r33 coefficients in Fig. 3 are in agreement with the respective values of 15.7 pm/V and 36.3 pm/V, previously measured at 633 nm for KTP [18]. Although giving access to the individual EO coefficients, this method is fairly noisy and inconvenient (it could be improved by use of lock-in detection, for example). In order to be able to conveniently study and compare different crystals, we subsequently adopted a different approach. 2.2.

Ellipsometry measurements in Na:KTP, KTP, and RTA

The experimental setup for the ellipsometry measurements is shown in Fig. 4. The polarization of the incident light is now linear at a 45o angle from y and z. The relative phase shift between the two polarizations yields an elliptically polarized beam. Crossed polarizers thus allow us to measure the changes of the degree of ellipticity caused by the EO modulation in the crystal. For #136909 - $15.00 USD

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Fig. 4. Experimental setup for the ellipsometry measurement. FI: Faraday isolator. The voltage Vo sin ω t, where Vo = 380 V, was applied along the z-axis of the crystal.

an input field the output field is

˜ E(0) = E0 (yˆ + zˆ)e−iω t ,

(8)

! " ˜ = E0 yˆ + eiθ zˆ e−iω t+iΘ , E(l)

(9)

√ where θ = 2π (nz − ny )l/λ and Θ = 2π ny l/λ . In the diagonal basis dˆ± = (yˆ ± zˆ)/ 2, this gives #! " " $ ! ˜ = E0 1 + eiθ dˆ+ + 1 − eiθ dˆ− e−iω t+iΘ . (10) E(l)

The beam intensity of the polarization transmitted by the output polarizer (dˆ− ) therefore has the dependence |1 − eiθ (t) |2 = 2 − 2 cos θ (t), which allows us to determine the EO modulation δ θ ∝ δ nz − δ ny = (n3y r23 − n3z r33 )Ez /2 from Eqs. (3) and (4). We tuned the average θ to π /2. Figure 5 displays the measurement results for two different Na:KTP crystals, one KTP crystal, and one RTA crystal. The two Na:KTP crystals differed in that one (crystal A) was a new

90

RTA

| δny

- δ nz | / Ez (pm/V)

80

Na:KTP (A)

70 60 50

Na:KTP (A)

40 30

Na:KTP (B)

20

KTP

10 0 10

-1

10

0

1

10

10

2

10

Frequency (Hz)

3

10

4

Fig. 5. Experimental results of the ellipsometry measurements. The Na:KTP (A) trace with cross markers and large error bars corresponds to the determination of |δ ny − δ nz |/Ez from the results of Fig. 3 (cavity resonance modulation).

crystal, whereas the other (crystal B) had been previously used in a continuous-wave OPO. The #136909 - $15.00 USD

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crystal sizes were 5 mm × 3 mm × 3mm for the Na:KTP crystals, 7 mm × 4 mm × 4 mm for the KTP crystal, and 20 mm × 3 mm × 1 mm for the RTA crystal. All crystals were flux-grown by Coherent Crystal Associates. In all KTP crystals, the EO effect is seen to disappear at low frequency. In addition, results of Fig. 3 have been plotted on Fig. 5 for comparison: it is clear that the two methods give consistent results. Undoped KTP seems to display an even lower EO effect than Na:KTP, whereas RTA, remarkably, has a quasi constant EO effect at all frequencies. We also found that the measured value of |δ ny − δ nz |/Ez for RTA is very close to that determined from the RTA EO coefficients measured in [18]. 2.3.

Electrical conductivity measurements in Na:KTP, KTP, and RTA

It is well known that RTA has a much lower ionic conductivity than flux-grown KTP [18]. To confirm this, we decided to measure the electrical conductivity of our samples. Our initial hypothesis was that the KTP crystals would become more conductive at lower frequencies, thereby allowing for a decrease of the internal field in the crystal and therefore of the EO index change. Use was made of an auto-balancing bridge circuit based on a low-noise OPA134PA operational amplifier (Fig. 6) [19]. The alternate input resistor r was used for calibration purposes.

Fig. 6. Experimental setup for z-axis conductivity measurements.

For such a circuit, one has Zc = −RT

Vin . Vout

(11)

The conductivity σ of the crystal is a function of the angular frequency ω of the input voltage, L σ = Cω , S

(12)

where C is the crystal’s capacity, S is its x-y face area, and L is its thickness. One would therefore expect to see a slope change in the measurement data, if the capacity were changing. Figure 7 displays the measurement results for frequencies above 500 Hz. Our simple measurement technique did not allow us to obtain decent enough signal-to-noise ratio for lower frequencies. For all crystals, the slope of the conductivity is found to be constant. Unsurprisingly, the RTA sample exhibited the lowest conductivity. No capacity change could be seen, even for the KTP

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-6

-1

Conductivity σ (10 Ω m-1)

120

KTP Na:KTP (A) Na:KTP (B) RTA

100 80 60 40 20 0 0

1000 2000 3000 4000 5000

Frequency ν (Hz) Fig. 7. Experimental results of z-axis conductivity measurements. Solid lines are linear regression fits.

crystal for which the EO effect drops at a few kHz, well above 500 Hz. However, the existence of nonzero ordinate intercepts in Fig. 7 indicates the non-Ohmic nature of KTP and Na:KTP crystals at low frequencies: one could therefore still be in the presence of site-hopping K+ ion currents, which would effectively decrease the internal electric field and thereby the EO effect. We have observed a decrease of the EO efficiency below 100 Hz modulation frequency for KTP and Na:KTP crystals. RTA crystals do not show this effect. Measurements of the electrical conductivity did not show any related variation, in particular for the KTP crystal in which the EO effect decreased well in the range of the conductivity measurements. We hypothesize that an explanation for this decrease in the electro-optic effect might still be related to internal field decrease due to ion migration in the crystal, but of a non-ohmic nature. These results are of importance for the realization of ultrastable OPO’s for quantum optics applications, including continuous-variable entanglers for quantum information processing [20] and ultraprecise measurements [12]. We find that RTA (or periodically poled RTA) crystals are better suited than KTP for these systems, with similar nonlinear coefficients and no EO abnormalities. However, PPRTA crystals are currently difficult to obtain since they contain arsenic [21], thereby making systems capable of using KTP favorable. 3.

Phase-locking experiment

The need to apply excessive voltage to the crystal in order to obtain an effective phaselock led us to search for a new method, which did not directly use the EO effect of the nonlinear crystal. One possibility, which has been used by other groups to phaselock the output beams of above threshold OPO’s, involves using a quarter waveplate placed inside the optical cavity to induce degenerate self phaselocking [7,10,22]. While effective, this method introduces an extra loss mechanism inside the OPO which can lead to a reduction in squeezing. More importantly, it is also limited to creating degenerate output beams, thereby making it unsuitable for the creation of the nondegenerate output beams necessary for heterodyne polarimetry experiments.

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We settled on using a new, all-optical method which utilizes the temperature tuning of the nonlinear crystal caused by changes in the lasing power of the OPO.

Fig. 8. Experimental setup for the phaselocking experiment.

The experimental setup for the phaselocking experiment is displayed in Fig. 8. The OPO consisted of an X-cut, linearly and noncritically quasiphasematched, periodically poled KTP crystal placed in a bow-tie ring resonator. The output coupler of the OPO transmitted approximately 2 percent of light at 1064 nm, while the other three mirrors were coated for highest possible reflectivity. The crystal, which was obtained from Raicol Crystals, was temperature stabilized to a few tenths of a millidegree Celsius by use of a modified commercial temperature-controller circuit (Wavelength Electronics HTC-1500). A frequency-doubled Nd:YAG laser (Innolight Diabolo) provided a narrowband (∼1 kHz/100 ms spectral linewidth, ∼2 MHz/min frequency drift) CW 532 nm pump beam, frequency-modulated at 12 MHz and polarized along the Y KTP axis, which downconverted to crosspolarized (Y ,Z) signal beams at about 1064 nm. The OPO cavity was locked on single-mode oscillation by a variant of the Pound-Drever-Hall technique [23] that used the pump depletion signal demodulated at 12 MHz to create an error signal [11]. (Due to the nonlinear coupling to the IR, this green pump signal displays the narrow IR resonance shape of the OPO cavity, even though the cavity has minimal finesse at 532 nm.) This signal was then processed by a home-made loop filter, amplified, and used to control the OPO cavity length via a piezo-electric transducer. This cavity servo loop was always closed when the phaselock servo operated. The OPO cavity was locked on the nearly frequency-degenerate mode, and the frequency difference was temperature-tuned to about 15 MHz. The phaselock servo relied on an electro-optic modulator (EOM) (Conoptics 350-52) to create polarization modulation of the pump beam as well as implement the phaselock servo correction. The overlapping OPO signal beams were mixed by a polarizer oriented at 45◦ from the (Y ,Z) axes and sent on a fast photodetector that recorded the beat note at the frequency difference of the signal beams. To obtain the phase-difference error signal, weak OPO signal beams were extracted from a “leak” through one of the high-reflectivity mirrors of the cavity. The beams were separated by polarization and one of them was frequency-shifted by an acousto-optic modulator (AOM) at 70 MHz. After the beams were recombined and interfered on a fast photodetector, the shifted beat note was then “superheterodyned” with a stable radiofrequency reference at a doubly-

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Fig. 9. IR modulation at various average output powers. Modulation was obtained by applying a 10 Vpp sine wave at 1 kHz to the EOM. Each trace corresponds to a different pump polarization (and therefore a different effective pump power), with the topmost trace corresponding to pure Y polarization (maximum power), the bottommost trace to any effective power below the lasing threshold (0 pump power was used), and the remaining traces to intermediate pump polarizations. Effective pump powers were estimated by multiplying the total power by the square of the cosine of the polarization angle of the pump beam relative to the Y axis of the crystal.

balanced mixer. This involved mixing the signal from the optical beat note with an RF signal at a frequency given by the difference between the 70 MHz shift arising from the AOM and the desired frequency difference of the OPO twin beams. For example, in order to lock the frequency difference of the twin beams at 15 MHz, we tuned the beat note between the AOMshifted beam and the unshifted beam to 55 MHz (or 85 MHz). This signal was then mixed with a 55 MHz (or 85 MHz) RF signal to create a DC error signal. The mixer’s output was then processed by a Vescent Photonics D2-125 laser servo system and used as the correction signal input to the EOM, which controlled the pump polarization, and therefore the IR power in the OPO, so that the temperature tuning of the OPO from the residual IR absorption yielded the set beat note value (in this example 15 MHz). Approximately 510 mW of 532 nm light was sent through the EOM at a polarization angle of 45◦ relative to the axis of the EO crystal. One component of this light was then phase modulated by an amount dependent upon the correction signal applied to the EOM. The quarter waveplate immediately following the EOM was set to 45◦ to create linear polarization. The half waveplate in front of the OPO was then adjusted to rotate the polarization of the pump beam to an angle that gave the desired amount of average lasing power. These waveplates converted phase modulation of the EOM to polarization modulation, with larger correction signals on the EOM corresponding to larger pump polarization rotations at the OPO. Here, polarization modulation effectively acted as amplitude modulation for the parametric downconversion process, while also allowing us to neglect pump absorption changes (assuming pump absorption to be independent of polarization) and attribute all observed heating effects to absorption of infrared (IR) light. The pump beam polarization after the EOM and waveplates can be derived by Jones calculus as follows:

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%

% & % −i2θ & 1 1 −i cos(2φ ) sin(2φ ) e √ −i 1 sin(2φ ) − cos(2φ ) 0 2

% & % & & cos(θoffset + ∆θ ) 0 1 1 √ ∝ , (13) sin(θoffset + ∆θ ) 1 2 1

where the first three Jones matrices respectively correspond to the φ ◦ -angled half waveplate, 45◦ -angled quarter waveplate, and EOM imparting a phase delay of 2θ , where θ = θ0 + ∆θ . θo f f set = (2φ + θ0 − 45◦ ), while ∆θ represents the shift in output polarization arising from the signal applied to the EOM. The effective pump power is then sin2 (θo f f set + ∆θ ). As can be seen in Fig. 9, a maximal amount of modulation on the output lasing power was observed when the pump polarization was set so the OPO was operating just above threshold. Two factors contributed to this. First, modulation of the pump beam is maximized when the pump polarization is at an angle of 45◦ relative to the axis of the nonlinear crystal. Since the lasing threshold of the degenerate mode was 330 mW, which was more than half of our available power, moving closer to threshold always increased the modulation. Second, the change in lasing power for a given change in pump power is also maximized at threshold because the OPO conversion efficiency is not linear with the pump power and has its steepest variation just above threshold [24]. Simply maximizing the amount of modulation obtained for a given voltage applied to the EOM does not, however, yield the optimum settings for the phaselock, as the stability of the OPO lock increases as the OPO is moved farther above threshold. With the phaselock loop turned off, we used a maximum hold measurement to observe a beat note frequency drift of approximately 35 kHz over 10s, as shown in Fig. 10. In addition, the beat

Fig. 10. Single-sweep and maximum-hold traces of the beat note signal, with the phaselock turned off. The maximum hold trace had a 10 s measurement time. Both the resolution bandwidth and video bandwidth were set to 1 kHz.

note had a fast jitter that can be seen on a single sweep of the spectrum analyzer (blue trace). Note the very high signal to noise ratio (40 to 50 dB) over very short times (where jitter doesn’t contribute) due to the excellent mode-matching of the OPO beams emitted by the same cavity. With the phase-difference lock loop closed, all drift and jitter disappear from the frequency difference signal, as evidenced in Fig. 11 where the phaselocked beat note shows no observable drift or broadening. Its linewidth is measured to be less than 1.5 kHz FWHM, limited by the 1 kHz resolution bandwidth of the measurement. Measurements over longer times were limited #136909 - $15.00 USD

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Fig. 11. Single-sweep and maximum-hold traces of the beat note signal, phaselocked at 15 MHz. The maximum hold trace had a 10 s measurement time. Both the resolution bandwidth and video bandwidth were set to 1 kHz.

by the stability of the OPO cavity, which was not as good as in previous experiments which used extremely stable super-invar resonator structures [11]. The next step of this work will be to implement this new phaselock technique on these intrinsically stable OPO cavities. The beat note was measured to shift approximately 65 MHz per degree Celsius change in the crystal temperature. In the power region in which we were operating our lock (∼3-5 W of intracavity IR power), we observed a frequency shift of approximately 26 kHz per mW change in output power. Although it appears to be quite fast, we were not able to measure the exact speed of this thermal effect, since direct power spectrum measurements of sidebands of the beat note cannot distinguish between the amplitude modulation arising directly from changes in optical power and the frequency modulation arising from photothermal changes to the crystal. While it is generally of interest to minimize the amount of IR absorption in nonlinear crystals, this phaselocking method does require some absorption. However, in crystals with minimal IR absorption, it would still be possible to use this method by performing amplitude modulation on the pump. The heating could then arise solely from green photon absorption which often occurs at a much higher rate than that of IR. 4.

Conclusion

We have demonstrated a novel technique to lock the phase difference of the signal beams of a type-II OPO above threshold. This technique is tunable and does not require placing an electric field across the nonlinear crystal of the OPO, which previously lead to severe crystal damage. Although a thermal method, our pump-polarization-mediated index change in the OPO crystal appears to be very fast, probably because the crystal volume concerned with heat transfer is only that of the focused pump beam. This result paves the way for applying our previous sub-shotnoise-sensitivity results [12] to technological and fundamental studies, as well as for realizing ultrastable, frequency-difference tunable, entanglement sources. Acknowledgements MP, RS, and OP were supported by U.S. National Science Foundation grants Nos. PHY0855632 and PHY-0960047. DX and OP were supported by NSF grants No. PHY-0240532

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and No. EIA-0323623. YM was supported by G-COE commissioned by the MEXT of Japan.

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