Pulsed homodyne Gaussian quantum tomography with low detection efficiency

Pulsed homodyne quantum tomography usually requires a high detection efficiency limiting its applicability in quantum optics. Here, it is shown that the presence of low detection efficiency ($<50\%$) does not prevent the tomographic reconstruction of quantum states of light, specifically, of Gaussian type. This result is obtained by applying the so-called"minimax"adaptive reconstruction of the Wigner function to pulsed homodyne detection. In particular, we prove, by both numerical and real experiments, that an effective discrimination of different Gaussian quantum states can be achieved. Our finding paves the way to a more extensive use of quantum tomographic methods, even in physical situations in which high detection efficiency is unattainable.

Here we report on the design, construction and characterization of a time domain Balanced Homodyne Detection apparatus operating in the regime of high electronic noise. The experimental setup has been commissioned by studying coherent ultrashort (∼ 80 fs) light pulses using Pattern-function Quantum Tomography. Furthermore, we proved that the electronic noise can be treated as a detector inefficiency showing the effectiveness of this approach by measuring light coherent states with different energies.
In Quantum Mechanics the measurement of the state of a physical system implies to estimate the expectation value of any observable of the system. For a generic operator the expectation value is obtained starting from the measurement of an ensemble of observables defined "quorum" 1 . In particular, for the electromagnetic field the quorum is a continuum set of the field quadratures 2,3 . Balanced Homodyne Detection (BHD) is an experimental method for measuring a discrete set of the field quadratures for a single mode cw field [4][5][6][7] . Such measurements can be analyzed by the Pattern-function Quantum Tomography (PFQT) technique in order to characterize the quantum state of the optical mode 8 . Instead, for pulsed radiation fields a BHD operating in the time domain must be used [9][10][11][12][13][14][15] . In this case the optical pulses are equally prepared quantum states and the set of measurements fed to PFQT comprises single pulse quadrature measurements. Although this is a standard method in quantum information, its effectiveness under high noise condition has to be proven. Here we report the characterization of a time domain BHD apparatus operating in a regime of high electronic noise (EN). We treat the low shot-to-electronic-noise ratio (≈ 0.8 dB) as a detector inefficiency 16 . In order to prove the applicability of this treatment to high EN homodyne detection, the PFQT technique has been exploited. The expectation value of the number operator of pulsed coherent states with different energies has been found consistent with the measured mean number of photons per pulse. Furthermore, the analyzed states are confirmed to be Gaussian and minimum uncertainty states.

Results
In BHD the signal, i. e. the field under investigation, is mixed with a strong coherent reference field, the local oscillator (LO), by a 50/50 beam splitter. The outputs are collected by two photodiodes and the difference photocurrent (homodyne photocurrent) is measured. It can be proven that, when the LO is significantly more intense than the signal, the homodyne photocurrent is proportional to the signal field quadrature 17 . Givenâ andâ † the field operators associated with the optical mode of the signal, the quadrature operator is defined as: where Φ is the relative phase between the signal and the LO. The continuum set of quadratures with Φ ∈ [0, 2π] constitutes a quorum characterizing the signal mode. Noteworthy the concept of light mode, usually associated with a monochromatic field, needs to be generalized for pulsed light. A thorough theoretical treatment for BHD in pulsed regime is presented herewith in the Methods section. The opto-mechanical scheme of our experimental apparatus is shown in Fig. 1. The laser source is a mode-locked Ti:Sapphire oscillator with 80 MHz repetition rate. A beam splitter divides the incoming beam in two parts which then interfere in a second beam splitter (NPBS in Fig. 1). The outputs are detected and subtracted by a differential photodetector (Thorlabs PDB430A). The latter is made up of two Si/PIN photodiodes with nominal quantum efficiency of 0.77. We verified that the response of the photodiodes is linear up to 0.6 mW LO power, while a non-linear response sets in at higher powers. The detector subtraction efficiency is quantified by the common mode rejection ratio (CMRR), defined as the ratio between the detector output power when both photodiodes are illuminated and the power when one of the two is screened. The measured CMRR for our apparatus is larger than 36 dB. The homodyne photocurrent is recorded by a digital oscilloscope (Tektronix TDS3000B ) with a bandwidth of 500 MHz and a sampling rate of 5 GSamples/s. The digitized output is numerically integrated over time intervals corresponding to the duration of the pulse. Each integral is associated with a single quadrature measurement. We checked the independence of each quadrature measurement by performing a correlation test between two subsequent pulses. The Correlation Coefficients (CCs) 14 between pulse n and n + m (m = 0, 1...9) are shown in Fig. 2 (a). In the inset a parametric plot of 2×10 4 subsequent pulses is shown, where the pulse n+1 is plotted against the pulse n. The weak correlation in this plot demonstrates that there is no significant impact on the measured integral of the pulse n + 1 from that of the pulse n. Each CC in Fig. 2 (a) is calculated using the data obtained from 200 difference pulses. The CC between pulse n and pulse n + 1 is 0.30 ± 0.08; this value is slightly higher with respect to other reported BHDs, i. e. smaller than 0.1 [12][13][14][15] . We can conclude that there is a small correlation between adjacent pulses, since the CC has approximately the same value for m = 2, ...9. This could be attributed to noise at a frequency lower than the repetition rate of the pulses.
In shot noise regime the homodyne detector (HD) noise variance is expected to change linearly in the LO power with a constant offset representing the electronic noise 18 . We calculated the HD noise variance of 8 × 10 3 pulses for different values of the LO power. The result is shown in Fig. 2 (b): the HD noise variance contains an electronic background and a linear contribution up to 0.6 mW LO power; for higher powers the non-linear effects due to the photodiodes start. We work at 0.6 mW LO oscillator power in order to have the maximum shot-to-electronic noise ratio (S = 1.2) achievable in the linear regime. The measurements reported in Fig.  2 are obtained with the signal beam blocked, i.e. with the signal in the vacuum state. When the signal beam is not blocked, it is attenuated with respect to the LO by means of a neutral density filter (F 1 in Fig. 1). The phase difference Φ between the two arms of the interferometer can be modulated in the [0, 2π] range using a piezoelectric translator in the LO arm (P ZT in Fig. 1). The homodyne photocurrent is acquired by the digital oscilloscope and each difference pulse is integrated. The obtained value N i , corresponding to a certain piezo position p i , is proportional to a quadrature measurement x Φ : The proportionality factor γ is obtained using the vacuum state as a reference. This is allowed since the quadrature variance for the vacuum state is σ 2 [x Φ ] |0 = 1/2 for any phase value. Defining γ as: the vacuum homodyne measurement has been used to extract the N i0 values. N 2 i0 is the variance of 8 × 10 4 experimental data of the vacuum. The homodyne data have been calibrated dividing N i by γ. Note that γ will give a misleading estimation of the actual mean photon number. Therefore, a correction taking into account the EN effect is necessary, as discussed in the following. The calibrated measurements for three optical coherent states with different energies are reported in Fig. 3. The optical density OD of the filter F 1 used for each measurement is given in the caption. The homodyne traces are composed of M = 8 × 10 4 experimental data: However, the effect of the EN, which adds a random quantity to each field quadrature measurement, must be accounted for. Appel at al. demonstrate 16 that this noise can be treated as an optical loss channel with equivalent transmission efficiency, where S is the ratio between shot and electronic noise at the chosen LO power. In particular, the vacuum state measurement is distorted by the EN and consequently the rescaling factor γ must be changed. Following the Appel's treatment 16 and considering equation (3), the following expression is obtained, where γ is the conversion factor in case of absence of electronic noise. Thus, in order to take into account the EN, the homodyne data have been normalized to γ .
These data have been used to characterize the quantum state of the signal field via PFQT 8 .

Discussions
The PFQT technique can be summarized as follows.
For a generic observableÔ of the optical mode it is possible to find a pattern function RÔ(x; Φ) for which the statistical average over the experimental data, gives an estimation of the expectation value Ô with an error This technique is used to estimate the expectation value of the number operator (â †â ), which can be evaluated in an independent way. The pattern function of the number operator is: where the factor η is the quantum efficiency of the homodyne detection apparatus. In order to take into account the EN loss effect, we put η = η eq given in equation (4). Finally we estimate the mean photon number as: The error on this estimation is calculated using equation (7). The results for the three different homodyne traces are reported in  Fig. 3 taking into account the equivalent quantum efficiency η eq as in equation (4).
calculated using the PFQT are consistent, in the order of magnitude, with the mean number of photons per pulse obtained from the nominal OD of the filter. In this way we verify that the treatment of low EN 16 is valid also for high EN detection systems. Now we focus on the optical state with the lowest energy [ Fig. 3 (c)]. The analysis for the other states is analogous.
First we verify the Gaussianity of the experimental data through the evaluation of the kurtosis excess κ, which is zero for Gaussian distributions 19 . Considering 10 3 experimental data in the first phase range of the homodyne trace in Fig. 3 (c) and we obtain a kurtosis excess κ = −0.008. Since all the phase ranges lead to a κ ≈ 0, this proves the Gaussianity of the homodyne probability distribution function. Then we use the PFQT to evaluate the expectation values of the the two quadratures associated with the position and momentum operators (x (Φ=0) =q; x (Φ=π/2) =p) and of their variances (σ 2 [q]; σ 2 [p]). The results, shown in   Fig. 3 (c).
Finally we evaluate the covariance matrix C, that fully characterizes Gaussian states and directly depends on the field quadratures 20 . The elements of the covariance matrix are defined as: whereR = (q,p) is the first-moments vector of the optical state and {Â,B} =ÂB+BÂ is the anti-commutator between two operators 20 . From the PFQT we obtain the following estimations: C = 0.53 ± 0.5 −0.02 ± 0.03 −0.02 ± 0.03 0.53 ± 0.05 .
This is consistent with what is expected for a coherent state. The other two homodyne measurements, reported in Fig. 3 (a) and (b), lead to similar results.
In conclusion, we reported the characterization of a time domain BHD apparatus operating in the high electronic noise regime. The apparatus has been characterized via the PFQT of optical coherent states with different mean number of photons per pulse. We treated the electronic noise trough the approach used by Appel at al. 16 and we showed that such a treatment is valid for high EN homodyne detectors. This result is of relevance in the perspective of using BHD and PFQT in experiments performed in high noise environments such as time domain pump-probe spectroscopy. In particular, ultrashort light pulses could be used to prepare coherent states into matter and BHD to have a full characterization of the light pulses which have interacted with transient matter states.
to the pulse: in other words, we have a Poissonian distribution not with respect to the number of photons in a monochromatic wave, but to the number of photons in the superposition |1ᾱ . The normalized operatorŝ