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States with sub-Poissonian photon-number statistics obtained by post-selection from twin beams are experimentally generated. States with Fano factors down to 0.62 and mean photon numbers around 12 are reached. Their quasi-distributions of integrated intensities attaining negative values are reconstructed. An intensified CCD camera with a quantum detection efficiency exceeding 20% is utilized both for post-selection and beam characterization. Experimental results are compared with theory that provides the optimum experimental conditions.
© 2013 Optical Society of America
1. Introduction
Nonclassical states of light have attracted a great deal of attention as soon as an experimental evidence of their existence has been given [1—3]. Their systematic study has resulted in an establishment of a new branch of optics called quantum optics. Sub-Poissonian photon-number distribution (PND) and anti-bunching of photons observed in fluorescence of single molecules [1] belong together with phase squeezing [4] to the most important manifestations of nonclassical light. Detailed theoretical studies of such fields have revealed that the crucial role in forming their non-classicality is played by states with well defined photon numbers but completely unknown phases. These Fock states represent a counterpart to ’classical’ or coherent states that allow interference due to their well defined phases.
For this reason the generation of the simplest Fock state with just one photon has become a challenge for the whole generation of experimental quantum opticians started by the experiments with photon pairs emitted from atomic cascades [5]. Due to a large progress in the field, one-photon Fock states can be obtained from several kinds of sources including molecules [6], super-conducting micro-cavities [7], NV centers [8, 9], and semiconductor hetero-structures [10] at present. One-photon Fock states of the highest quality are generated from the so-called heralded single-photon sources [11—13] that rely on post-selection from a photon pair arising in spontaneous parametric down-conversion (SPDC). These states of photon pairs have been found useful in showing many unexpected features of quantum physics (violation of Bell’s inequalities [14, 15], teleportation [16], dense coding, etc.). They have also been successfully exploited in precise metrology [17, 18] including absolute detector calibration [19—21] and quantum cryptography [22, 23] where they help to assure unconditional security. Also quantum computation [24] can be based upon the use of Fock states.
Properties of Fock states with larger numbers of photons are even more challenging [7]. However, their generation is even more difficult compared to the generation of one-photon Fock states which, despite a long effort devoted, is still not easy. States with sub-Poissonian PND have already been generated in resonance fluorescence [25] (Fano factor F ≈ 0.998), Franck–Hertz experiment [26] (F ≈ 0.99), high-efficiency light-emitting diodes [27] (F ≈ 0.96), in the process of second-subharmonic generation [28] as well as in experiments with atoms passing micro-cavities [7, 29]. Stronger sub-Poissonian fields containing many photons have been successfully generated by continuous intensity post-selection from the beams coming from optical parametric oscillators [30]. Also an alternative method based on a feed-forward action on the beam has been verified [31].
Post-selection by a photon-number resolving detector from a twin beam generated in SPDC and containing many photon pairs [32] represents one of the most prospective ways for sub-Poissoian-light generation at present [33, 34]. A photon-number resolving detector with a sufficiently high quantum detection efficiency (QDE) plays the crucial role in this scheme. Time-multiplexed systems with avalanche photodiodes [35—38], hybrid photomultipliers [39, 40], semiconductor detector arrays and matrices [41], super-conducting bolometers [42—45] and intensified CCD cameras [21, 46] seem to have sufficiently high QDEs and low noise for this task.
2. Conditional states generated from twin beams by post-selection
Here, we demonstrate the generation of sub-Poissonian light using twin beams and an iCCD camera that serves both as detector post-selecting the sub-Poissonian light and monitoring the photocount statistics of this light. In more detail, both the signal and idler fields comprising a twin beam are monitored using a photocathode of the iCCD camera. Registration of a given number of, e.g., signal photocounts (detected photons) in certain area of the photocathode post-selects the idler field that attains the sub-Poissonian PND under suitable conditions. PND in this field is measured again using a different area of the photocathode [47]. Photocount distribution obtained in this area then allows to recover the PND of the post-selected idler field using, e.g., the expectation-maximization reconstruction method.
A histogram f(cs, ci) giving the number of experimental realizations with cs signal and ci idler photocounts is available after a sufficiently high number of measurement repetitions. The normalized histogram fi(ci; cs) ≡ f(cs, ci)/∑cif(cs, ci) then describes the measurement on the idler field conditioned by the detection of cs signal photocounts. The idler-field conditional photon-number distribution (CPND) pc,i(ni; cs) arising after detecting cs signal photocounts can easily be reconstructed. The method of expectation maximization allows to find this distribution as a steady state available by the following iteration procedure [48]:
In Eq. (1), the matrix elements T (ci, ni) give probabilities of having ci photocounts when detecting a field with ni photons. These probabilities valid for an iCCD camera with Ni active pixels, QDE ηi and dark-count rate per pixel Di have been derived in [48]:The reconstructed CPNDs pc,i(ni; cs) can be compared with the ’theoretical’ ones obtained from the reconstructed joint signal-idler PND psi(ns, ni) along the formula:
The matrix elements Ts(cs, ns) characterize detection in the signal field and are given by the formula analogous to that in Eq. (2). The joint PND psi(ns, ni) of a twin beam can be expressed as a two-fold convolution of three Mandel-Rice PNDs [49] characterizing paired, signal noise and idler noise parts of the field [21, 50]: p(n;M,b) = Γ(n+M)/[n!Γ(M)]bn/(1+b)n+M. Symbol Γ stands for the Γ-function. Each part a has its own number Ma of independent modes and mean number of photons (or photon pairs) ba per mode, a = p, s, i. Appropriate values of these quantities as well as QDEs ηs and ηi can be derived from the histogram f(cs, ci) using a fitting procedure (for details, see [21, 51]).3. Sub-Poissonian-light generation
Sub-Poissonian PNDs have been experimentally detected using an iCCD camera and twin beams (see Fig. 1) centered at the wavelength 560 nm. Twin beams originated in a non-collinear type-I interaction in a 5-mm long BaB2O4 crystal pumped by the third harmonics of a femtosecond cavity dumped Ti:sapphire laser with pulse duration 150 fs at 840 nm (for details, see [47]). Ns = Ni = 6272 pixels of the photocathode with mean dark-count rates Ds = Di = 0.04/Ns have been used in detecting each field. The histogram f(cs, ci) has been built after 2 × 105 measurement repetitions. Its analysis has revealed the following values of parameters: ηs = 0.235, ηi = 0.243, bp = 0.056, Mp = 180, bs = 9.8, Ms = 0.012, bi = 29, and Mi = 0.0009. Covariance between the signal and idler photon-numbers ns and ni was 91%. There were 〈cs〉 = 2.39 and 〈ci〉 = 2.45 photocounts on average in the signal and idler detection areas, respectively. The field in front of the camera was on average composed of 〈np〉 = 9.87 photon pairs, 〈ns〉 = 0.12 noise signal photons and 〈ni〉 = 0.03 noise idler photons.
Fig. 1 Scheme of the experiment. An output of a femtosecond cavity dumped Ti:sapphire laser is converted to its third harmonics (THG, 280 nm) and used to pump a BaB2O4 nonlinear crystal. Nearly degenerate signal and idler (steered by high-reflectivity mirror HR) beams are selected using 14 nm wide bandpass filter IF and detected using an iCCD. Long-pass (above 490 nm) filter diminishes the noise. The intensity of the pumping beam is actively stabilized (rms below 0.3%) using motorized half-wave plate HWP and polarizer P based on the remnant UV intensity read by detector D.
The analysis of experimental data has revealed that sub-Poissonian idler-field CPNDs pc,i can be obtained for signal-field photocount numbers cs in the range from 2 to 7 [see Fig. 2(a)] [50]. We note that sub-Poissonian statistics is quantified by its Fano factor F [F = 〈(Δn)2〉/〈n〉] attaining values lower than one [52]. The graph in Fig. 2(a) shows that sub-Poissonian PND is qualitatively preserved during detection and so also sub-Poissonian photocount statistics is observed. This behavior occurs as the number of noisy photons is much lower than the number of photon pairs. As a consequence, an imperfect QDE ηi only causes an increase in the values of Fano factor Fi maintaining sub-Poissonian character of the distribution. Both theoretical and experimental results show that the lowest values of Fano factor Fi are reached after post-selecting by the detected photocount numbers cs considerably greater than the mean value 〈cs〉. Greater values of the used photocount numbers cs lead to greater mean values of conditional idler photon numbers 〈nc,i〉, as documented in Fig. 2(b). However, sub-Poissonian idler-field generation conditioned by great values of cs is not practical as the probability of detecting such photocount numbers is low [see Fig. 2(c)]. According to Fig. 2(c) the photocount numbers cs up to 6 are acceptable in this respect. Values of the Fano factor Fi around 0.8 have been reached for cs ∈ (3, 4, 5, 6) for fields containing around 12 photons on average. The lowest experimental value of Fi equals to 0.62 ± 0.18 and was observed for cs = 6. We note that the experimental values of Fi were obtained with a relatively large uncertainty as the post-selection probability is low. However, the uncertainty can be substantially reduced just by increasing the number of measurement repetitions. The sub-Poissonian CPND pi(ni; cs = 4) with Fano factor Fi = 0.74 ± 0.08 conditioned by the detection of cs = 4 signal photocounts is shown in Fig. 3(a). This field is nonclassical by definition as its Glauber-Sudarshan quasi-distribution (QD) Pc,i of integrated intensity W attains negative values. Even the QD related to the symmetric operator ordering keeps negative values [see Fig. 3(b)]. We note that QDs Wc,i in Fig. 3(b) were obtained from the CPNDs pc,i using the decomposition into Laguerre polynomials [49].
Fig. 2 (a) Fano factor Fi and (b) mean photon number 〈nc,i〉 [photocount 〈ci〉] of conditional idler photon-number [photocount] distribution as they depend on the number cs of signal photocounts. (c) Marginal signal-field photocount distributions fs(cs) = ∑ci f(cs, ci) and . Asterisks give experimental values, triangles mark values obtained in the maximum-likelihood reconstruction and solid curves arise from the theory. In plots (b) and (c) experimental errors are smaller than symbol sizes.
Fig. 3 (a) Sub-Poissonian idler-field CPNDs pc,i(ni) revealed by the maximum-likelihood reconstruction (▴) and the theoretical CPND (plain curve) for cs = 4. (b) Corresponding QDs Pc,i of integrated intensity Wi for the symmetric operator ordering (▴ → dash-dot curve). Poissonian PND (○) and its distribution of integrated intensity (dashed curve) are shown for comparison.
There exist four important parameters that determine available values of Fano factor Fi: mean photon-pair number (〈np〉), QDE of the camera η, mean number of noise signal photons (〈ns〉) used for post-selection, and mean number of noise idler photons in the post-selected field (〈ni〉). The dependence of Fano factor Fi on mean photon-pair number 〈np〉 is weak, as the curves in Fig. 4(a) showing the least available values (optimized with respect to cs) as well as the values obtained with the maximum post-selection probability document. They indicate that moderate values of 〈np〉 are optimum, which is the case of the performed experiment. Whereas fields with low values of 〈np〉 suffer from the noise (〈ns〉) when detecting the signal photons, greater values of 〈np〉 are handicapped by a lower signal QDE ηs. The QDE ηs crucially limits the available values of Fano factor Fi. The greater the values of ηs the smaller the values of Fi, as shown in Fig. 4(b). We note that the greater the values of ηs the smaller the post-selection probabilities ps(cs). Nonzero noise in the post-selecting signal field (〈ns〉) degrades the post-selection process and, naturally, greater values of Fano factor Fi occur [Fig. 4(c)]. If this noise is negligible the values of Fi close to zero can be reached even for non-unit values of QDE ηs. However, this occurs when post-selecting by a greater number cs of signal photocount. As such events are rarely observed this regime is practically not useful. Finally, the noise in idler field (〈ni〉) only conceals the sub-Poissonian character of the conditional idler field. The greater the values of 〈ni〉 the greater the values of Fano factor Fi. This analysis shows that the used experimental conditions, namely the chosen parameters of the twin beam, have allowed to reach the nearly optimum values of Fano factor Fi allowed by the used iCCD camera. The improvement of camera’s QDE opens the door for reaching lower values of Fano factor Fi.
Fig. 4 The least available value of idler-field Fano factor (plain curve) and the most probable value of Fano factor (dashed curve) for the conditional idler field ps(cs) as they depend on (a) mean photon-pair number 〈np〉 [Mp = 180], (b) QDE η ≡ ηs = ηi, and (c) mean number of noise signal photons 〈ns〉 [Ms = 0.012]; values of parameters are taken from the experiment.
A lower post-selection probability can be increased by considering the generation of several conditional idler fields differing in cs simultaneously. Also post-selection does not necessarily have to be based upon the measurement of a fixed number cs of signal photocounts – more sophisticated post-selection patterns are possible. This allows, for example, the generation of highly nonclassical CPNDs composed of several peaks provided that sufficiently low values of Fano factors are experimentally reached.
The confinement of the obtained sub-Poissonian pulsed field into a temporal window typically several tens or hundreds fs long represents a great advantage over other approaches. Among others it allows precise synchronization with other pulses or processes in quantum systems. Suppression of intensity fluctuations in sub-Poissonian fields has been found useful in optical communications where it enhances channel capacities [53]. The reduced intensity fluctuations are important in general in precise metrology where they allow to overcome classical limits in precision [54].
4. Conclusions
Sub-Poissonian light with Fano factors down to 0.62 containing on average around 12 photons has been generated using post-selection from a twin beam. Suitable experimental conditions for this method relying on photon-number-resolved detection of an iCCD camera have been theoretically found. Perspectives of this method have been experimentally confirmed.
Acknowledgments
Support by projects P205/12/0382 of GA ČR and CZ.1.05/2.1.00/03.0058 of the Czech Ministry of Education is acknowledged. The authors thank M. Hamar for his help with the experiment. J.P.Jr. thanks J. Perina for discussions.
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