1. Introduction

It is well known that quantum-correlated twin beams are an elegant candidate for verifying the foundations of quantum physics. Since the twin beams were first generated experimentally [1], they have been extensively studied [2–7] and have also been applied to sub-shot-noise [6] and quantum nondemolition measurements [7]. Recently, measurements of the probability distribution P{x _θ} of the quadrature-field amplitude x _θ = X cos θ + Y sin θ of an optical field have been used for the experimental measurement of optical quantum states. The measurement is based on the fact that knowledge of the field-quadrature probability distribution for all θ allows one to reconstruct the Wigner function W(X,Y) for completely describing the quantum system by use of the inverse Radon transform [8–10]. However, in an experiment involving the measurement of bright twin beams we encountered the issue of mode matching; i.e., there are no suitable local oscillators to match the frequency-nondegenerate twin beams. If one is interested only in the photon-number distribution, this issue can be solved by use of the direct-detection technique. This is because the bright twin beams can overcome both Johnson and dark-current noise; fast photodiodes having high quantum efficiency can be employed. Using this technique, we investigated the photon-number statistics of continuous-wave twin beams [11], following the measurement sub-shot-noise correlation of the total photon number using twin pulses of light [12]. A complete description of the quantum state of multimode fields requires simultaneous measurements of the joint statistics of all the modes. Joint statistics yield information about the correlations between the modes that cannot be obtained from single-mode measurements. An experimental application of optical homodyne tomography to the measurement of the joint photon-number distribution of the twin-beam state emerging from a nondegenerate optical parametric amplifier (NOPA) has recently been demonstrated [10]. Furthermore, conditional distribution is another important property of multimode fields. Conditional distributions of pairs of noncommuting observables between the modes represent the spirit of the original Einstein-Podolsky-Rosen (EPR) gedanken experiment in its truest form. However, conditional probability distributions, which are narrower than those of uncorrelated separable states, have not, to our knowledge, been experimentally established. In this paper we investigate the intensity fluctuation joint probability and conditional probability distributions of the twin beams, which were generated with an optical parametric oscillator (NOPO) operating above threshold.

According to the extension of the EPR argument [13–15], we consider two spatially separated subsystems at A and B. One can predict the result of a measurement of an observable x at A, on the basis of the result performed at B. The predicted results for the measurement at A, based on the measurement at B, are described by a set of conditional distributions P(x|$x_i^B$ ) (throughout this paper, the subscript i is used to indicate the possible result, discrete or otherwise, of the measurement x^B ) giving the probability of a result for the measurement at A, conditional on a result $x_i^B$ for the measurement at B. The probability that subsystem A is in the variable designated $x_i^B$ is P($x_i^B$ ), the probability of the result at B, since through locality the action of measuring x at B could not have induced the result at A. On the basis of the measurement $X_i^B$ at B, we attribute to the inferred element of x the inference variance ${\mathrm{\Delta }}_{\text{inf}}^2$ x = Σ _iP($x_i^B$ )${\mathrm{\Delta }}_i^2$x where µ _i and ${\mathrm{\Delta }}_i^2$x are the mean and the variance, respectively, of the conditional probability distribution p(x|$x_i^B$ ). For a given experiment one could in principle measure the individual mean (µ_i) and variance (${\mathrm{\Delta }}_i^2$x) of the conditional distribution for the various quantum states.

Quantum inference would be demonstrated in a convincing manner if we could measure each of the conditional distributions and establish that each of the distributions is very narrow, in fact constrained so that

Furthermore, δ, which depends on the variance ${\mathrm{\Delta }}_i^2$x, is less than the standard quantum limit. In this case the measurement x^B at B will always imply that the result of x at A is within the range µ _i ± δ so that the result of the measurement at A is predetermined to be within a bounded range of width 2δ. After considering the below-standard quantum limit δ, the width of the conditional distribution of the quantum inferable state is narrower than that of the separable state; that is, quantum inference with a precision is better than the standard quantum limit. To our knowledge, the narrow distributions over the outcome domain have not been experimentally established. In this paper we will give the narrower conditional distribution of twin beams, which were generated by a NOPO.

Let us note the intensity fluctuations of the twin beams. The experiments of Gao et al. [6] achieved a quantum-noise reduction of 88% (-9.2 dB) in the intensity difference between twin beams. Our experiment reports the conditional distribution between signal and idler of the twin beams. In direct detection by using a photodetector, assuming that most electrons in the current are photoelectrons, the mean current can be expressed in terms of the mean number 〈n〉 of photoelectrons being counted as 〈i〉 = 〈n〉e/T, where e and T are the charge of an electron and the time interval, respectively. The variance of the current will be 〈(Δi)²〉 = 〈i ²〉 - 〈i〉² = (e/T)²[〈n ²〉 - 〈n〉²] = (2eΔƒ)²〈(Δn)²〉, where Δƒ = 1/(2T) is the electrical bandwidth of the detection system. Thus the variance of the photocurrent fluctuation at a particular frequency is simply proportional to the variance of the photoelectron counts, which depends on the state of light.

2. Experimental setup

 

Fig. 1. Experimental setup. D1 and D2, detectors; PBS, polarizing beam splitter; LPF, low pass filters; G, amplifier; λ/2, Half-wave plate; rf (ψ) and rf(θ), rf local oscillators.

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Our experimental setup is shown in Fig. 1. The twin beams generator, here a nondegenerate optical parametric oscillator (NOPO) pumped above its threshold by a diode-pumped CW frequency-doubled YAG laser, emits quantum-correlated twin beams. A more detailed description of the NOPO can be found in the references [16]. The NOPO is pumped by the second harmonic wave of 40 mW at 532 nm, which is about seven times the oscillation threshold of 6 mW. The 1064-nm orthogonally polarized twin beams emerging from the NOPO are detected separately by two high efficient detectors (ETX300 InGaAs). The detection efficiencies have been measured to be around 92%. Since the photon fluxes are very high, the detectors do not resolve individual photons but generate large average currents with small fluctuations due to fluctuations in the waves. An electronic circuit generates the photocurrents i ₁(t) and i ₂(t), which are proportional to particular intensities (or photon number) of the signal and idler, respectively. Electronic noise is negligible (more than 10 dB below the shot noise). After passing through a 21.4-MHz low-pass filter, the photocurrent is amplified by a low-noise 46-dB-gain amplifier. Rather than measuring the currents i ₁(t) and i ₂(t) directly, for technical reasons we measured their spectral components in a small bandwidth around the radiofrequency of Ω = 4 MHz by mixing photocurrents with rf local oscillators of frequency Ω. The near-dc region down-converted frequency outputs of the mixers, i ₁(Ω) and i ₂(Ω), are further amplified and low-pass filtered. Their fluctuations are detected within the bandwidth of 100 kHz through the low-pass filters. The resulting photocurrents are a measure of the photon numbers of the input beams ${\mathit{â}}_1^{\psi }$ = [â₁ (ω+Ω)e ^-iψ + â₁ (ω-Ω)e ^iψ] and ${\mathit{â}}_2^{\mathrm{\theta }}$ = [â₂ (ω+Ω)e ^-iθ + â₂ (ω-Ω)e ^iθ], respectively. Here, â₁ (â₂) is the corresponding beams detected by detector D₁ and D₂, ω is the optical frequency, and ψ (θ) is the phase of the rf local oscillators driving the mixer. The phase ψ is carefully chosen, and the relative phase between ψ and θ is fixed to make the measurement fixed on the same component. The i ₁(Ω) and i ₂(Ω) are subsequently sampled by an A/D board (AT-MIO-16E-2). A coherent light is input into the polarizing beam splitter (PBS) on the other input port for calibration of the criterion of the separable state. A half-wave plate is inserted in the twin beams before the PBS. As shown by Heidmann et al. [1], when the polarization of the twin beams is rotated by an angle of 0° to the PBS axis, the measured photocurrents are corresponding to the signal and idler.

3. Experimental results

 

Fig. 2. Record of noise trace of channel 1 for separable coherent state and signal of twin beams in a 100kHz bandwidth at 4MHz.

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Typical traces of the resulting photocurrent fluctuations (In the following text, X is used to label the photocurrent fluctuation) of channel 1 for coherent light and the signal of twin beams are shown in Fig. 2. Although the noise current of signal light is larger than that of coherent light, the noise of the intensity difference between the signal and idler is less than that of coherent light. The variance of the photocurrent fluctuation difference between channel 0 and channel 1 determines the amount of correlation between the signal and idler. It describes the correlation of twin beams and can be observed by measuring difference in photocurrent fluctuation with a spectrum analyzer [1–7] or fluctuations variances with a data acquisition system [11]. The variances ${\mathrm{\Delta }}_{\mathit{\text{coh}}}^2$(X^A -X^B ) = 58.5 for coherent light and ${\mathrm{\Delta }}_{\mathit{\text{twin}}}^2$(X^A -X^B ) = 19.0 for twin beams are obtained [11], or in other words, quantum correlation in amount of -4.9 dB (19.0/58.5=0.32) was detected. These values agree with the results of simultaneous measurement with a spectrum analyzer. Based on these data, the photon-number statistics were obtained and reported in Ref. [11]. After the data are stored, the joint probability and conditional distributions can also be reconstructed.

 

Fig. 3. Measured joint probability distribution for twin beams (a) and separable state (b) with equal mean photon numbers, viewed in 3D and as contour plots.

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The joint probability distributions P(X^A,X^B ) and their contours for the quantum-correlated twin beams (a) and the coherent light (b), of the same mean photon number as that of the quantum-correlated twin beam, are shown in Fig. 3. Note that no corrections for detection and escape efficiencies were made. To relate the values found for the average occupation number to the actual measured powers, we note that photon number is the average photon flux per unit bandwidth. A state with the same photon number 〈n〉 implies the same dc photocurrent 〈i〉, which is monitored by an oscilloscope, of the detectors. The joint probability distributions are sampled by subdividing the data trace of 200 000 points into 120×120 equal bins within which the $X_i^A$ and $X_i^B$ are approximately constant. The histograms of photocurrents are performed. In each bin the number of points is counted, and the probability distributions are constructed. Correlation between the signal and idler is clearly seen in Fig. 3 (a) and is better than that of separable states.

To investigate conditional distributions, the first step is constructing the conditional distribution of separable state to determine the standard quantum limit. The conditional probability of result X at A given a simultaneous measurement of X^B at B with result $X_i^B$ is defined as

where P($X_i^B$ ) is the probability of result $X_i^B$ at B. For P(${X\mathit{,}X}_i^B$ ), we divided the trace of coherent light into 120 sections corresponding to 120 different $X_i^B$ , in each of which $X_i^B$ is approximately constant. For each $X_i^B$ interval the X^A was sorted into 120 bins. Thus a set of probability distributions P(${X\mathit{,}X}_i^B$ ) was obtained. As expected, all distributions were found to be Gaussians with the same standard deviation. Probability P($X_i^B$ ) could be obtained by forming histograms of 120 bins for the trace of measurement X^B . The conditional distributions of the separable state for $X_i^B$ =-10, 0 and 10 are shown in Fig. 4 (a). All the conditional distributions of subsystem A have Gaussian distributions with the same mean and the same variance, which equals the mean photon number of light ${\mathrm{\Delta }}_{\mathit{\text{coh}}}^2$(X^A ) = 〈n〉, indicating the inherently character of coherent state. A theoretical fit is also shown for comparison. The measured mean and variance of conditional distributions for separable state were µ_i = 0 and ${\mathrm{\Delta }}_i^2$X_coh = 30.1±0.1, respectively. We can normalize this variance for the inference variance δ=1. These conditional distributions imply that the probability of obtaining a result X at A cannot be obtained upon measurement of X^B at B, in other words, the measurement at B does not allow an inference of the result of at A.

Using the same process as the above-mentioned, the conditional distributions of twin beams for $X_i^B$ =-10, 0 and 10 were performed and shown in Fig. 4 (b). Obviously, the conditional distributions of subsystem A are, no longer a set of distributions with zero mean. The mean becomes strongly dependent on the measurement results $X_i^B$ at subsystem B; the mean µ _i equals the measurement at subsystem B, $X_i^B$ . The measured variance of conditional distribution is ${\mathrm{\Delta }}_i^2$X_twin = 19.0±0.1, which is variance of difference between the signal and idler, ${\mathrm{\Delta }}_{\mathit{\text{twin}}}^2$(X^A -X_B ). A separable state conditional distribution (solid line) with the same observed mean photon number is also shown for comparison. It indicates that the conditional distributions of twin beams are narrower than that for separable states. This indicates the inherent quantum character of the twin beams. These properties of the conditional distributions of twin beams imply that the probability of obtaining a result X at A is dependent upon measurement of $X_i^B$ at B, in other words, the measurement at B allows an inference of the result at A to be performed with a precision better than given by the uncertainty bound 1. By normalizing to the separable state, we can get the inference variance, δ=0.62±0.02. This result agrees with what was calculated from the measured quantum correlations between the signal and idler.

 

Fig. 4. Conditional probability distribution of separable state (a) and twin-beam state (b)

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4. Conclusion

In summary, we have presented what to our knowledge is the first measurement of the conditional distributions of twin beams. The measured variances in photocurrent fluctuation exhibit up to -4.9±0.2 dB of quantum correlation between the signal and the idler, but the conditional distributions imply an inference of 0.62±0.02. After former successful experiments on EPR [17–19], this experiment will stimulate additional efforts to verify the foundations of quantum theory, to measure bright quantum states, and to carry out quantum information experiments. A recent experiment on the investigation of criteria for continuous variable entanglement using bright EPR beams has been carried out [20].