## Abstract

Optical homodyne tomography (OHT) is a tool that allows the reconstruction of Wigner functions for each detection frequency of a propagating optical beam. It can measure probability distribution functions (PDF’s) of the field amplitude for any given quadrature of interest. We demonstrate OHT for a range of classical optical states with constant and time varying modulations and show the advantage of OHT over conventional homodyne detection. The OHT simultaneously determines the signal to noise ratio in both amplitude and phase quadratures. We show that highly non-Gaussian Wigner functions can be obtained from incoherent superpositions of optical states.

© Optical Society of America

## 1. Introduction

Optical homodyne tomography (OHT) allows the reconstruction of the Wigner function *W*(*x*
_{1}, *x*
_{2}) [1] representation of an optical state. The theoretical foundation of quantum state reconstruction was outlined by Vogel and Risken [2] and has inspired a series of experiments, initially with optical pulses [3,4] and more recently with cw light [5, 6, 7].

In practical applications, such as sensing or communication, a laser beam is modulated in order to carry information. This can occur at different modulation frequencies or channels Ω* _{m}*. Each modulation frequency has its own Wigner function

*W*

_{Ωm}(

*x*

_{1},

*x*

_{2}), and can vary dramatically from channel to channel [8]. Hence, a realistic laser beam with signal modulations is a propagating multi-mode quantum state, which is quite different to the single mode intra-cavity description of an optical state. Such a realistic light beam can only be described by a spectrum of Wigner functions

*W*

_{Ω}, one for each frequency Ω.

In the early experiments [3] of pulsed laser OHT, only one single Wigner function was reconstructed. This Wigner function contains all statistical moments of the photon number in the pulses and individual Fourier components of a pulse could not be separated. In more recent experiments with cw lasers, Wigner functions *W*
_{Ωd} for squeezed and classical states were reported for specific detection frequencies Ω* _{d}*. In the case of squeezed light, Wigner functions with elliptical two-dimensional Gaussian quasi-probability distributions were demonstrated [5, 6, 7].

In this paper, we use the Wigner function representation to describe coherent states with different forms of classical sinusoidal phase modulation (PM). At the modulation frequency we observe that the Wigner functions are displaced away from the origin of the phase space and the results allow us to directly determine the signal to noise ratio (SNR) for both amplitude and phase signals. At all other frequencies, the Wigner functions are always centred at the origin. We demonstrate how the Wigner functions are controlled by the quadrature and depth of modulation.

One of the advantages of OHT over conventional homodyne measurements [9,10] is that OHT can show any non-Gaussian features in the Wigner functions of more complex optical states. Such features are contained in the higher order moments of the noise statistics with the result that a simple measurement of the noise variance (2nd order moment) alone is an insufficient description. The additional complexity means Wigner function reconstruction demands more sophisticated statistically analysis. Unfortunately, unlike experiments in atom optics [11], complex optical states are extremely difficult to generate [12]. They require highly nonlinear processes which have fluctuations comparable to the average steady state amplitude. To date only a few proposals for the generation of such non-classical states exist [13]. We test the ability of OHT to reconstruct highly non-Gaussian Wigner functions by using laser beams with time varying modulation to produce classical superpositions of optical states. We discuss the accuracy of the OHT technique and some of the practical limitations.

This work demonstrates the capability of OHT for the diagnosis of realistic laser beams and establishes techniques that will be important once highly non-Gaussian, eg. “SchrÖdinger’s cat”, states can be experimentally realised.

## 2. Standard homodyne detection

The conventional approach for the measurement of a laser beam is the balanced homodyne detection. The light generated by a cw laser with an optical frequency *ν* is processed by an interferometric arrangement (Black components in Fig. 1). The majority of the optical power (*P _{LO}*) is split off by mirror M1 as the optical local oscillator beam and sent via mirror M3 to the combining beamsplitter M4. A small part of the optical power (

*P*) reaches M4 via the mirror M2 as the test beam. The relative phase between the beams is controlled by moving M3 with a piezo position controller (PZT). The two optical outputs are converted into photocurrents

_{test}*i*

_{1}(

*t*) and

*i*

_{2}(

*t*) using matched, high efficiency detectors PD1 and PD2, respectively. The difference

*i*

_{-}(

*t*) =

*i*

_{1}(

*t*) -

*i*

_{2}(

*t*) between the currents is analysed. The fluctuations of the test beam can be described with the generalised quadratures

*x*̂(

*θ*) = (

*a*̂

*e*

^{-iθ}+

*a*̂

^{†}

*e*

^{iθ})/2. Here

*x*̂

_{1}=

*x*̂(0) is the amplitude quadrature,

*x*̂

_{2}=

*x*̂(

*π*/2) is the phase quadrature. Under the condition of

*P*≫

_{LO}*P*this device will provide a photocurrent

_{test}*i*

_{-}which contains information about the fluctuations of the test beam alone, and is immune to the fluctuations , or noise, of the local oscillator beam.

The variance *V*
_{Ωd}(*θ*) describes the properties of the test beam at one detection frequency Ω_{d} and is normalised to the standard quantum limit so that for a coherent state, *V*
_{Ω}(*θ*) = 1. The difference photocurrent *i*
_{-}(*t*) is analysed using an RF spectrum analyser. The electric noise power *P*
_{iΩd} is proportional to the optical power of the local oscillator *P _{LO}* and the variance

*V*

_{Ωd}(

*θ*)

Due to the strong attenuation in the neutral density filter ND (transmission ~ 1%), the test beam does not contain the actual noise spectrum of the laser but rather fluctuations introduced by the vacuum. If the output of the laser was squeezed, (*V*
_{Ωd} < 1) or slightly noisy (*V*
_{Ωd}> 1) the test beam would still have a noise spectrum close to the quantum noise limit of *V*
_{Ωd} = 1. This device is not particularly sensitive to laser noise. However, any modulation or squeezing generated inside the interferometer is clearly detectable. For example, a phase modulation introduced by driving the electro-optical modulator (EOM) within the interferometer at frequency Ω_{m} would increase *V*
_{Ωm}(*θ* = *π*/2).

## 3. Modulation

The output of the modulator is given by *E*
_{out} = *E*
_{0} cos[2*πνt* + *δ*sin(Ω_{m}*t*)] and the phase modulation generates sideband pairs at *ν _{L}* ±

*n*Ω

_{m}. For

*δ*≪ 1, only the first order sidebands (

*n*= 1) are important. The linearized annihilation operator for the quantum mode after the modulator is given by

$$\phantom{\rule{1.2em}{0ex}}\approx {E}_{0}-2\mathit{i\beta}{E}_{0}\mathrm{sin}\left({\Omega}_{m}t\right)+\delta \hat{a}\left(t\right),$$

where *β* = *J*
_{1}(*δ*), the first order Bessel function, and the operators contain the quantum fluctuations. The output current from the homodyne detector for a projection angle *θ* is

The factor of proportionality depends on the efficiencies of the detectors, losses and the electronic gain. *δX*̂(*θ*,*t*) describes the quantum fluctuation in this particular quadrature. In conventional experiments, the analysis of a particular Fourier component of the photocurrent is done by an RF spectrum analyser with the phase *θ* of the local oscillator kept constant. The result is a spectrum as given in Eq. (1). As a consequence the phase variance *V*
_{Ωm}(*θ* = *π*/2) increases proportional to the modulation depth while all other parts of the spectrum *V*
_{Ω}(*θ* = *π*/2) with Ω ≠ Ω_{m} remain unchanged.

## 4. Optical homodyne tomography

In order to obtain the Wigner function of a light beam, only small modifications to the balanced homodyne apparatus are required [6]. Phase synchronous detection is introduced by replacing the spectrum analyser with a mixer demodulator (Blue components in Fig. 1). The mixer is gated by an electronic local oscillator signal derived from the same generator that drives the EOM. This electronic signal is shifted by a phase *ψ* giving a mixed down difference current *i*
_{Ωm}(*θ*,*ψ*;*t*). Starting with Eq. (2) and using Fourier transforms we can derive the output current from the mixer as :

where *δX _{c}*(

*θ*,Ω

_{m};

*t*) can be understood as the total quantum fluctuations centered around ±Ω

_{m}.

*δX*(

_{ci}*θ*,Ω

_{m};

*t*) and

*δX*(

_{cr}*θ*,

*Ω*;

_{m}*t*) are the imaginary and real parts of

*δX*(θ, Ω

_{c}_{m};

*t*) respectively. For

*ψ*= 0, we obtain

The first term contains all the modulation and the second term all the quantum fluctuations. Note that for synchronous detection, the phase of the modulation has to be known to the observers. In practice, this is not always possible.

If the modulation phase *ψ* is unavailable to the detection system, as would be the case for the monitoring of remotely generated signals, then either a phase recovery technique or asynchronous demodulation is required. Asynchronous demodulation can be simulated by equation (4) where the demodulation phase *ψ* is a linear function of time. This corresponds to an uncorrelated demodulation generator operating at Ω_{1} in Fig. 1.

A typical synchronously demodulated photocurrent plot is shown in Fig. 2(a), where the phase angle *θ* is repetitively scanned. By selecting data that correspond to the same value inside the vertical intervals *δθ* , we obtain measurements of, *w*
_{Ωd}(*θ*,*t*), for any given quadrature interval (*θ*, *θ* + *δθ*). Next a histogram of this current is formed by binning the data in intervals *δθ* for a coherent state. This results in the PDF *w*
_{Ωd}(*x*, *θ*) of the quadrature amplitude for various projection angle *θ*. Fig. 2(b) shows a series of such PDF’s for a full scan of *θ*.

The width of a PDF corresponds to the variance *V*(*θ*) of the given quadrature. For any realistic, and thus linearisable state with photon number *N* ≫ 1, the shape of the PDF is a Gaussian. For a coherent state the width of the Gaussian is equal to the photon number, thus identical to a Poissonian distribution. Squeezed states have sub-Poissonian PDF’s at one particular angle *θ _{s}*, the squeezing quadrature.

## 5. Wigner function reconstruction

Wigner functions can be reconstructed from the PDF’s using the inverse Radon transform [2, 14].

The resulting Wigner function *W*
_{Ωd}(*x*
_{1}, *x*
_{2}) is shown in Fig. 2(c). For a coherent state the function is symmetric - with concentric contour lines. For a squeezed state the function is asymmetric, with elliptical contour lines. For all coherent or squeezed states, without modulation (*β* = 0), the Wigner function is centered at the origin. This can be seen from equation Eq. (5) where only the second term contributes in this case. The orientation of the ellipse gives the squeezing quadrature. Since the Wigner function is normalized to the standard quantum limit, its position and size is independent of the optical power. Thus, a squeezed vacuum state has the same Wigner function as a bright squeezed state, provided that *P _{LO}* ≫

*P*. Note that the Wigner function should not be confused with the widely used picture of a “ball on a stick” which tries to describe several properties of an optical state simultaneously. The “stick” indicates the average, DC optical power while the “ball” represents high frequency, AC fluctuations. The Wigner function which can be measured at a particular frequency depends only on the noise and signals at that frequency. Hence, Wigner functions can only be displaced from the origin by introducing a modulation.

_{test}The distance of the centre of the Wigner function from the origin is a direct measurement of the modulation depth. Amplitude modulation causes displacement along the *x*
_{1} axis whilst PM along the *x*
_{2} axis. The Wigner function gives us immediately information about the signal quadrature, strength and noise. It provides the conventional signal to noise ratio, where both signal and noise are measured in the same quadrature, as well as the relative size of the noise in the orthogonal quadrature. The later is of interest in applications where some degree of crosstalk between the quadratures is unavoidable. Amongst others, this includes any application of resonantly locked cavities where a small imperfection of the locking can introduce cavity detunings and thus a mixing of the quadratures.

## 6. Experimental results

#### 6.1 Varying the depth of phase modulation

The process of reconstructing Wigner functions and the effect of modulation depth is clearly demonstrated in Fig. 3. Here we use phase modulation and synchronous detection for four different modulation depths. For *β* = 0 (no modulation, i.e. a coherent state) the Wigner function is circular and centred at the origin. As the frequency modulation depth is increased, the Wigner function is displaced along the “*x*
_{2}” (phase variance) axis. For large modulation, it is apparent that the phase modulation process has introduced significant amplitude modulation, resulting in the Wigner function being displaced vertically from the *x*
_{1} axis (amplitude variance). This is due to the imperfection of the phase modulator.

It is important to note that synchronous demodulation requires the optimization of the demodulation phase *ψ*. This ensures that the modulation component is detected with maximum efficiency and results in the optimum SNR being recorded on the Wigner function.

#### 6.2 Switched phase modulation

In order to demonstrate the ability of our OHT system to record the details of a highly non-Gaussian distribution, we added a low frequency modulation to gate the PM on and off. By selecting this gating frequency at 200 Hz, within the detection bandwidth ( 100 kHz) we can then record the resulting distribution. The resulting Wigner function is plotted in Fig. 4. As can be seen, the gating process (square wave signal generator in Fig. 1) produces a Wigner function with 2 peaks: one peak corresponds to zero modulation and is located at the origin, whilst the other corresponds to phase modulation.

#### 6.3 Asynchronous detection (variable phase ψ).

Finally we demonstrate the results of asynchronous detection. This is achieved experimentally by using a separate Ω_{1} as the demodulation signal, which is different from the modulation frequency Ω_{m} used to drive the EOM. It is necessary to ensure that Ω_{m} and Ω_{1} differ in frequency by an amount small comparing with the detection bandwidth. Under these conditions the detected Wigner function then represents the weighted average of all possible demodulation phase values *ψ*. The resulting Wigner function, for phase modulation, is shown in Fig. 5. The distribution is now centred on the origin and spread symmetrically along the phase quadrature axis. The peaks at the extreme of the modulation correspond to the turning points where the dwell time of the modulation, as a function of phase, is greatest.

## 7. Discussion and summary

In conclusion, we have demonstrated that the reconstruction of Wigner functions provides clear information about the modulation and information carried by a laser beam at a given detection frequency. For the coherent and the squeezed states with modulations, the Wigner functions contain the same information as could be obtained with homodyne detection but they are easier to interpret. In particular, the displacement of a Wigner function *W*
_{Ωd} provides information on the quadrature and strength of the modulation and the width of *W*
_{Ωd} describes the noise. The signal to noise ratio at any given quadrature can be read directly.

In general, most optical states which can be generated can be described by a linearised theory and thus has a two dimensional Gaussian *W*
_{Ωd}(*x*
_{1},*x*
_{2}). In order to demonstrate the ability of measuring highly non-Gaussian Wigner functions, an asynchronous demodulation scheme is used.

## Acknowledgements

We wish to acknowledge helpful discussions with U. Leonhardt, A. G. White, G. Breit-enbach, T. C. Ralph and B. C. Buchler. This research is supported by the Australian Research Council.

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