Local sampling of the quantum phase-space distribution of a continuous-wave optical beam

It is shown how the quantum phase-space distribution of a continuous-wave (cw) optical beam can be obtained independently at each point in phase space by a combination of unbalanced homodyne and balanced-heterodyne techniques. The unbalanced homodyning allows for the local sampling of phase space, whereas the heterodyne part, although introducing a loss of detection efficiency, provides a highly efficient reduction of 1/f excess noise that is required for the sampling of cw fields. The method complements well-known techniques for light pulses by providing a robust method for cw optical beams.


Introduction
Balanced optical homodyning has been successfully used in the past to measure phasedependent quadrature statistics and to show reductions of the quadrature fluctuations below the standard quantum limit, which can then be attributed to incident squeezed light. Taking a closer look at the first observations of squeezed light [1][2][3][4], it becomes clear that beat signals were recorded. The continuous-wave (cw) squeezed light was superimposed by a frequency shifted local-oscillator (LO) field in order to be able to measure the beat signal by a spectrum analyser (SA). In this way 1/ f excess noise can be efficiently suppressed in the measured signal. Similar ideas have been exploited when squeezing was measured on frequency-modulated side bands or within the spectral range of the detected light [5,6]. As these methods are based on the measurement of a beat signal, they could equally well be denoted as heterodyne measurements.
Another way to suppress the influence of excess noise is by the use of light pulses instead of cw fields. Triggering the measurements with the pulses allows for an efficient suppression of noise at frequencies below the inverse pulse duration. This method has been used to demonstrate the first quantum-state reconstruction from quadrature statistics measured in balanced optical homodyne tomography [7], whereas for cw light heterodyne-type methods were used [6].
Alternatives to balanced-homodyne tomography [8], a method for the local sampling of the phase-space distribution of the light field, have been proposed in the past [9,10]. This method is based on displacing the signal light field in phase space by the use of an unbalanced beam splitter and an LO field. Subsequently, the photon statistics in the output channel of maximum transmittance of the signal field is measured. In this way, the quantum phase-space distribution can be sampled locally at each point in phase space. The method is optimal for obtaining the phase-space distribution according to a recent theoretical work [11]. This idea has been implemented for cw input states that generate very low count rates with an avalanche photo diode operating in the Geiger regime [12], and for pulsed light [13,14]. However, for general input states, other implementations must be searched for that combine a sufficient photon-number resolution together with a broad range of detectable photon numbers, such as, for example, time-multiplexing detectors as realized in [15] for pulsed light. A solution 3 to this problem has been proposed by replacing the photo detector (PD) by a phase-randomized balanced-homodyne detector [16]. The resulting setup is a cascaded homodyne scheme that includes an unbalanced and a balanced stage [17]. A modified version of it has been proposed later for obtaining the phase-space distribution of an intra-cavity field [18].
If balanced, unbalanced or cascaded homodyne setups are applied to cw light fields, the electronic signals generated by the amplified PDs are directly processed without spectral filtering or temporal triggering. As their full spectrum from dc up to the full bandwidth of the detector is recorded, practical problems emerge due to unavoidable technical noise sources, such as electronic noise, etc. These noise sources provide a noise background in the measured signal that typically increases with decreasing frequency, for which it is commonly denoted as 1/ f noise [19]. A tomographic reconstruction of the phase-space distribution of the cw light field is then strongly compromised by the presence of this typically dominant technical noise. On the other hand, for pulsed light or heralded photons the signal may be spectrally analysed at the repetition rate of trigger events, which typically lies well above the frequencies where 1/ f noise is dominant. In this way 1/ f noise can be largely suppressed.
In this paper, we show how the proposal of cascaded homodyning can be modified to be experimentally feasible for the local sampling of the quantum phase space of an incident cw light field. We show how this can be reached by combining unbalanced homodyne and balanced-heterodyne techniques. The unbalanced homodyning allows for the local sampling of phase space, which represents an advantage over non-local reconstruction methods such as inverse Radon transforms. And the heterodyne part, although introducing a loss of detection efficiency, provides a highly efficient reduction of 1/ f excess noise that is necessary in the case of cw detection.

Unbalanced homodyning
A local sampling of the quasi-probability distribution in the phase space of an optical mode can be obtained by unbalanced homodyning [9,10]. The outline of the setup is shown in figure 1. Both signal (S) and LO fields have a fixed difference phase within the time of data acquisition, and the relative phase φ and amplitude A of the LO field are adjustable. Both fields are superimposed on a beam splitter with very low reflectivity, |r | 1, so that almost all of the signal light is transmitted into the detector. The effect of the LO light is to add a coherent amplitude r A e iφ to the field incident on the PD. The PD is ideally measuring the photo-electron statistics related to the incoming photon flux integrated over the detection response time τ det , which must be smaller than the coherence time of the light, τ det < τ coh , in order to properly map information on the photon statistics into the photo-electron statistics.
Given the quantum efficiency η PD of the PD, the overall detection efficiency is η = η PD |t| 2 , where t is the transmittance of the beam splitter, and the PD 'sees' a signal field displaced by α = (r/t)A e iφ . It can be shown that the photo-electron statistics P n (α; η) measured in this way contains all the information necessary for obtaining the value of the s-ordered quasi-probability distribution W at the phase-space point α, i.e.
[−ξ(s; η)] n P n (α; η), The main advantage of this setup over optical balanced-homodyne tomography is that the phase-space distribution is obtained at each point in phase space independently. Thus, a local sampling of the phase-space distribution is obtained. Unfortunately, the setup in figure 1 requires a highly efficient photon-number resolving detector, which is currently available only in the form of avalanche photo diodes in the Geiger regime. As such diodes may only resolve between zero or some photons, it must be guaranteed by other means that the incident photon flux integrated over the detector response time is not larger than 1. Practically, this means that only phase-space distributions can be measured that are sufficiently narrow. Under these conditions, an experimental implementation was shown in [12].

Cascaded homodyning
To overcome the lack of highly efficient PDs with output signals that resolve the actual photon number, it has been proposed in [17] to substitute the PD by a phase-randomized balancedhomodyne detector [16]. This has the advantages that highly efficient linearly responding photo diodes can be used and that the resolution of photon numbers is easily obtained for sufficiently large intensities of the additional second LO field. Such a cascaded homodyne scheme has the outline shown in figure 2. The displaced signal field is now superimposed by LO2 with the 50 : 50 beam splitter BS2. The two output intensities are measured with photo diodes PD1 and PD2, and their photo currents are then subtracted to obtain the statistics of quadratures. If the phase of LO2 is integrated over a 2π interval or if this integral is approximated by Cascaded homodyne setup. The single PD of the unbalancedhomodyne setup, cf figure 1, is replaced by a phase-integrating balancedhomodyne setup consisting of a phase controlled second local oscillator field LO2, beam splitter BS2 and photo detectors PD1 and PD2, whose difference photo currents are measured.
randomization, the photon statistics can be obtained from the phase-integrated quadrature statistics p(q; α, η) as [16] where f nn (x) are pattern functions that can be obtained from the regular and irregular eigenfunctions of the Schrödinger equation of the harmonic oscillator [20,21]. Inserting equation (3) into (1), together with equation (2), the s-ordered phase-space distribution at the point α determined by LO1 is obtained directly as where the unique sampling function is given as [17] with the pattern function being determined by the Dawson integral F(x) [22] as This setup, while resolving the issue of high efficiency combined with photon-number resolution, is still problematic for cw light. As the measured observable is the difference of two photo currents that are generated in the photo diodes by integrating the incoming photon flux over the response time τ det , no spectral filtering or selection may be performed. That is, all frequencies from dc up to the bandwidth of the PD are measured. As a consequence, the obtained histogram of quadratures is strongly affected by 1/ f excess noise that may be dominant at low frequencies. Thus, for cw light, the implementation of the cascaded homodyne setup is very demanding due to the need to beat the excess noise by ultra-low noise detectors and electronics [27].

Cascaded homodyne-heterodyne setup
If LO2 was operated at a slightly different frequency than the light coming from BS1, the balanced-homodyne part would become a heterodyne one and PD1 and PD2 would detect the beat signal at the difference frequency. If this difference frequency is sufficiently high, the 1/ f excess noise could be efficiently suppressed, similar to the squeezing experiments [1][2][3][4]. To implement the integration or randomization of the phase of LO2, one may now use a second independent laser source for LO2. As there will be no phase relation between signal, LO1 and LO2, the phase will be effectively randomized on the time scale of the coherence time. In this way no phase control of LO2 is needed, as shown in the setup of this homodyne-heterodyne scheme, shown in figure 3. This setup combines now the advantages of the previous ones, resolving their respective caveats. Firstly, the method is a local one, meaning that the phase-space distribution is obtained in each point in phase space independently. Thus no inverse Radon transform of the measured data is required. The measured data are simply integrated with the unique sampling function (5). Secondly, the PDs should have high quantum efficiency but need not be photon-number and PD2 generate the photo currentsÎ 1 andÎ 2 from the incident photon flux 1 andˆ 2 , respectively. The currents are transformed into voltagesÛ 1 andÛ 2 by the trans-impedance amplifiers TIA1 and TIA2, respectively, both with transimpedance R and bandwidth δω det . The difference voltage is processed in the SA, which consists in a bandpass filter (BP) with resolution bandwidth δω rbw and mid-frequency ω 0 , followed by an envelope detector (ED).
resolving. The resolution of photon numbers is obtained by a sufficiently intense LO2. Thirdly, excess noise is efficiently suppressed due to the heterodyne stage, so that the measured data truly reflect the quantum statistics of the signal field. Finally, the use of an independent laser source for LO2 leads to a natural phase randomization so that a phase control of LO2 is not required. All this, however, comes with a well-known disadvantage that is due to the spectral filtering of the heterodyne beat signal: the quantum efficiency of the heterodyne stage as an effective PD is half the efficiency of the employed photo diodes [25]. That means that the local sampling can be performed only for s-ordered quasi-probability phase-space distributions with s −1, i.e. for the Husimi Q function or Gaussian convolutions of it. However, this phasespace distribution still contains the complete information on the quantum state of the signal light. Therefore, one may guess that by means of maximum-likelihood methods [23,24] a corresponding deconvolution to, e.g., the Wigner function may be performed.

Cascaded homodyne-heterodyne setup
In the following, a detailed analysis of the proposed cascaded homodyne-heterodyne setup is performed in order to obtain experimentally relevant parameters and to show its experimental feasibility.

Spectrally analysed heterodyne signal
To show the electronic processing of the heterodyne signal, as shown in figure 4, we assume for the moment a unit quantum efficiency of the photo diodes. Imperfect detection will then be treated in the following section.
The incident photon flux coming from the output channels of BS2, cf figure 3, 8 are transformed by PD1 and PD2 into the short-circuit photo-electron currentsÎ d (t) = eˆ d (t).
These currents are amplified with two identical trans-impedance amplifiers TIA1 and TIA2 with bandwidth δω det to obtain the output voltageŝ which are convolutions of the currents with the time response of the amplifiers f det (t) with d t f det (t) = 1. The dc trans-impedance R determines the conversion of the photo currents to measurable voltages. The voltages are then subtracted to obtain Û (t) =Û 1 (t) −Û 2 (t), which in terms of the incident photon flux reads The photo diodes, together with their trans-impedance amplifiers, determine the smallest time scale of measurable signal fluctuations, given by the inverse detection bandwidth τ det ∼ 2π/δω det . As the amplifiers' low-pass characteristics f det (t) can be well approximated by a timeintegrating circuit, the difference photon flux is basically integrated over the detector response time interval τ det , i.e.
where U det = eR/τ det is the voltage generated by a single photo-electron within the response time of the detector. Here where the incident photon numbers accumulated during τ det at the detectors arê The difference voltage (9) is then given on the SA, which consists of the band pass (BP) filter followed by the envelope detector (ED). The BP filter is employed to select the part of the signal that oscillates with the beat frequency = ω s − ω lo of the superimposed light fields, i.e. its mid-frequency ω 0 is set to the difference frequency of the light fields ω 0 ≈ . The spectral width of this filter, δω rbw , known as the resolution bandwidth (RBW), is chosen so as to let pass through the beating of all spectral components of the optical field. That is, δω rbw > δω l , where δω l is the spectral bandwidth of the light.
The filtered signal is still oscillating at the chosen mid-frequency ω 0 of the BP filter. The second stage of the SA is detecting the amplitude of this oscillation by the use of an ED, which measures the peak value within one period of the oscillation. In this way a non-oscillating signal is generated that is still random due to the randomness of the signal injected into the SA 2 .
Mathematically, the effect of the BP filter is a further temporal convolution where the time response of the BP filter is 9 The function g bp (t) satisfies d tg bp (t) = 1 and has a temporal width τ rbw = 2π/δω rbw with δω rbw and ω 0 being the above-mentioned RBW and mid-frequency of the SA. We may approximate the filter function as g BP (t) ≈ τ −1 rbw for t ∈ [−τ rbw , 0] and g BP (t) = 0 elsewhere, to obtain from equation (11) To obtain the envelope of this fast oscillation, the peak value within the period 2π/ω 0 is measured by the ED. The result becomes the SA signal where φ ed is the phase of the corresponding peak value of the fast oscillation.

Quantum statistics of the spectrally analysed signal
The photo diodes typically have non-unit quantum efficiency η, so that the measured photoelectron statistics is different from the statistics of incident photons. To take this into account, we employ the quantum theory of photo detection (see e.g. [26]) to obtain the statistics of the output voltage of the spectrum analyserÛ sa . The heterodyne signal is measured with PDs that integrate the incident photon flux over the time interval τ det . Therefore, in the spectrally analysed signal (14), we may replace the time integration over the interval τ rbw by a sum over discrete time intervals τ det , with t k = t − kτ det and K = δω det /δω rbw , where K 1 because the highest detectable frequency is determined by the detector, i.e. δω rbw , ω 0 δω det . Here U rbw = e R/τ rbw is the voltage generated by a single photo-electron within the temporal resolution of the spectral analyser τ rbw and ξ k = 2 cos(φ ed − ω 0 t k ).
Using the discretized representation (15), the probability density to measure at time t the spectrally analysed difference voltage U with overall quantum efficiency η, given a phase-space displacement α, can now be formulated as Here the joint probability for detecting during the time intervals {[t k − τ det , t k ]} the accumulated photon numbers {n 1,k , n 2,k } with quantum efficiency η reads as [26] P {n 1,k ,n 2,k } ({t k }; α, η) = : where the operators are defined aŝ and · · · α denotes the expectation value in the quantum state of the signal field, coherently displaced by the amplitude α.
Using the Fourier representation of the Dirac delta function, equation (17) can be rewritten as p(U, t; α, η) = 1 2πU rbw d x : with the operator-valued characteristic function for the difference counts m = n 1 − n 2 being defined asĈ where the difference-count statistics is defined by the operator Given LO2 in a coherent state with mean photon flux lo , the accumulated mean photon number within τ det is N lo,det = lo τ det . Supposing that this number is much larger than that of the signal field, N lo,det N s (t) , the operator (22) can be approximated as a function of the quasicontinuous variable m,P m (t; η) ≈P(m, t; η, N lo,det ) with the normalized Gaussian operator Using this approximation, in equation (21) the sum can be replaced by an integral. The scaled argument xξ k in equation (20) can then be incorporated by substitution of the integration variable m → m = m · ξ k , which together with the relationP(m/ξ, t; η, N ) = ξP(m, t; ξ η, ξ N ), cf equation (23), results aŝ Furthermore, using equations (23) and (24), equation (20) can be seen to be a convolution of Gaussians of the form (23), with m 0 = 0 and m K = U/U rbw . The result is again Gaussian and reads as where δU = U rbw ηN lo,det ξ 2 .
Here we made use of equation (15) and the width of the Gaussian (26) is modified by the time-dependent function This function can be obtained for ω 0 δω rbw as the constant ξ 2 (t) ≈ 2τ rbw /τ det . In turn this leads to the root mean square voltage spread where N lo,rbw = lo τ rbw is now the accumulated photon number of the LO field within the integration time τ rbw determined by the RBW of the SA. The additional factor √ 2 is well known to appear due to the use of a heterodyne instead of a homodyne setup [25]. It is due to the fact that the spectrum of the real-valued beat signal has symmetric components at frequencies ± , whereas the SA selects only the positive-frequency part and thus discards half of the signal. This translates into the additional factor 1/2 in the exponent of the Gaussian (26), which finally leads to a reduction of the overall quantum efficiency by 50%.

Phase-randomized quadrature distribution
As N (t) is a heterodyne signal composed of two signals with mid-frequencies ω S (signal field) and ω L (LO2 field), for perfect balanced detection it oscillates with the beat frequency = ω S − ω L as where for perfect mode matching the slowly varying operators read as withn lo =â † loâ lo . Here φ lo (t) is the phase of the LO2 field and the single-photon flux is defined as with E( r ) being the transverse mode of the beams. In this way, the LO2 photon flux becomes lo = 0 â † loâ lo . Given that the BP mid-frequency is tuned to the beat frequency ω 0 ≈ and that ω 0 δω rbw , the integration time ∼ τ rbw in equation (14) is sufficiently long to suppress the counterrotating terms. As a consequence, the two remaining resonant terms can be written aŝ where the signal quadrature is defined aŝ The variance of the fluctuations of the accumulated LO2 photon number within τ rbw is where Poissonian fluctuations of the LO have been used, As a consequence, these fluctuations can be equivalently written as N lo,rbw = N lo,rbw , so that we obtainÛ The LO phase φ lo (t) is a random variable that changes on the time scale given by the coherence time of the LO field τ coh ∼ 2π/δω l , where δω l is the spectral bandwidth of the laser light. That is, given that τ rbw τ coh (i.e. δω l δω rbw ), equation (26) represents the probability density for the quadrature with a well-defined LO phase. Performing now a time average of the measured voltage statistics over a time T τ coh reveals automatically the phase-randomized statistics p(U ; α, η) = p(U, t; α, η) with where equations (26) and (37) have been used. From this equation, the linear relationship between the measured voltage and quadrature can be seen to be so that the probability density p(q; α, η) for the phase-randomized quadrature can be obtained from p(U ; α, η) dU = p(q; α, η) dq as p(q; α, η) = p( √ ηδU q; α, η) √ ηδU , which becomes with δq = 1/ √ η, which again is √ 2 times larger than in the case of a homodyne measurement.

Experimental requirements
For a proper functioning of the heterodyne stage, specific frequency ranges of the laser fields and adjustments of the SA are required. Suppose that the light sources are external-cavity diode lasers with line widths of the order of ω l /2π ∼ 1 MHz and coherence times τ coh ∼ 1 µs. In order that the photo diodes generate photo currents that contain information on the photon statistics, the detection time should be much smaller τ det τ coh , which may be satisfied by δω det /2π = 100 MHz trans-impedance amplifiers, corresponding to τ det ∼ 10 ns. Furthermore, we have seen that the BP-filter bandwidth of the SA should be δω l δω rbw < δω det and that δω rbw < ω 0 < δω det , so that the time scales are ordered as These conditions can be satisfied by choosing the bandwidth δω rbw /2π ∼ 10 MHz and the filter mid-frequency as ω 0 /2π ∼ 20 MHz, the latter being matched by the laser detuning = ω 0 . The overall quantum efficiency of the cascaded homodyne-heterodyne setup is For a given measurement error, this condition can be typically satisfied either by increasing the amplifier trans-impedance R or by increasing the intensity of LO2, i.e. N lo,rbw . Strictly speaking, the measurement error generates a convolution of the voltage distribution (38). However, as we require condition (46) for a proper sampling, it is guaranteed that U e δU , so that the modified spread of the operator function in equation (38) can be estimated as being unaffected.
Typical values for condition (46) are obtained for an LO2 power 10 mW at 800 nm, transimpedance R ∼ 100 k and δω rbw /2π ∼ 10 MHz as δU rbw ∼ 1.6 nV and N lo,rbw ∼ 4 × 10 12 so that the measurement error should be U e 1 mV, which is feasible in experiment.
A further source of error is in the adjustment of the point in phase space α = r exp(i φ) where the quasi-probability is sampled. Typically, the intensity and phase of LO1 are controlled independently. Suppose, therefore, that we may fix amplitude and phase with uncertainties r e and φ e . Whereas the radial error may be supposed to be relative to the actual radius, r e /r = ε r , we assume that the phase error φ e is constant. The chosen phase-space point is therefore precise within an area S e = r 2 ε r φ e . As a reference we may use the Gaussian phasespace distribution of a coherent state |α 0 , which reads as The size of the uncertainty area of equation (47) is δS coh = (1 − s)/2, so that for resolving the details of this Gaussian, the precision of the adjustment of the phase-space point by LO1 should be S e δS coh .
The maximum value s max , cf equation (43), corresponds to the smallest uncertainty area of the coherent state, δS coh (1 − s max )/2, which together with equation (42) leads to r 2 ε r φ e 1.
This condition becomes more and more stringent for sampling at increasing distances from the origin of phase space. For the phase-space distribution (47) we may define the maximum radius r max ∼ √ 1 − s of the area that is required to be sampled 3 . Assuming that one would try to come close to the maximum value s max , this results in the simple relation ε r φ e η (coherent state).
This requirement on the precision of the adjustment of amplitude and phase of LO1 is easily satisfied in experiment.

Summary and conclusions
In summary, we have shown that a setup consisting of an unbalanced optical homodyne stage followed by a balanced optical heterodyne stage can be employed to locally sample the phase space of the incident signal field. The local sampling means that for each choice of amplitude and phase of the first LO a specific point in the phase space of the signal field is addressed. Integrating the measured quadrature histograms with a unique sampling function leads then to the value of the quasi-probability distribution of the quantum state of the signal field at this particular point in phase space. The heterodyne stage has been shown to provide for an efficient suppression of excess noise by allowing for a spectral analysis of the measured beat signal. Furthermore, as the second LO is an independent laser source the necessary phase integration is naturally obtained due to the lack of correlation of signal and LO phase. Whereas the heterodyne technique, on the one hand, reduces the overall quantum efficiency of the detection by 50%, on the other hand, it provides for a very efficient suppression of excess noise in the measured data. Furthermore, as the quasi-probability distribution is obtained locally by scanning over the area of interest in phase space, the remaining noise in the data is not amplified as it possibly would be in an inverse Radon transform. Thus, we may expect rather noise-free phase-space distributions, which may then be used in maximum-likelihood methods to estimate the corresponding quasi-probability distribution with a larger value of the operatorordering parameter.