Homodyne tomography with homodyne-like detection

Stefano Olivares,1, 2 Alessia Allevi,3, 4, ∗ Giovanni Caiazzo,3 Matteo G. A. Paris,1, 2 and Maria Bondani41Quantum Technology Lab, Department of Physics “Aldo Pontremoli”,

University of Milan, via Celoria 16, I-20133 Milano (MI), Italy2INFN, Sezione di Milano, via Celoria 16, I-20133 Milano (MI), Italy

3Department of Science and High Technology, University of Insubria, Via Valleggio 11, I-22100 Como (CO), Italy4Institute for Photonics and Nanotechnologies, CNR, Via Valleggio 11, I-22100 Como (CO), Italy

We show that data from homodyne-like detection based on photon-number-resolving (PNR) detectors may beeffectively exploited to reconstruct quantum states of light using the tomographic reconstruction techniques orig-inally developed for homodyne detection based on photodiodes. Our results open new perspectives to quantumstate reconstruction, and pave the way to the use of PNR-based homodyne-like detectors in quantum informationscience.

A quantum tomographic scheme is a technique to evalu-ate the expectation value of any observables and, in turn, toreconstruct the density matrix or the Wigner function of thesystem, by processing the outcomes of a quorum of observ-ables, measured on repeated preparations of the state underinvestigation [1–4]. In the continuous-variable domain, quan-tum states are usually reconstructed by means of optical ho-modyne tomography (OHT) [5]. From the technical point ofview, OHT is based on an interferometric scheme (see the leftpanel of Fig. 1) in which a state ρ (the signal) is mixed at a bal-anced beam splitter (BS) with a high-intensity coherent state|β〉 = ||β| eiφ〉 (the local oscillator, LO). The two outputs ofthe interferometer are detected by two pin photodiodes, whosedifference photocurrent is suitably amplified, rescaled by theLO amplitude |β|, and recorded as a function of the LO phaseφ. By properly processing the data and applying advanced re-construction algorithms, it is possible to achieve the completeknowledge of the state under investigation in terms of the den-sity matrix [6, 7] and to calculate the expectation values ofgiven observables [4]. Among them, we mention the inver-sion algorithms, which can be applied if the effective quan-tum efficiency of the detection apparatus is above 50 %, andthe maximum-likelihood reconstruction protocols, which canbe applied also in the presence of strong losses [8–11].

In this Letter we consider the homodyne-like (HL) detec-tion scheme shown in the left panel of Fig. 1. At variance withthe standard homodyne detection scheme, in our apparatus thepin photodiodes are replaced by photon-number-resolving de-tectors and the LO |β〉 is a low-intensity (few tens of photons)coherent state. The physical information is obtained from thedifference between the effective number of detected photons,rather than from the difference between two macroscopic pho-tocurrents. Recently, we have demonstrated that HL detectionmay be successfully exploited for state-discrimination of co-herent states [12, 13]. Moreover, one has additional degreesof freedom available with HL data, and this may be used tooutperform standard homodyne detection in quantum key dis-tribution with continuous variables [14].

Motivated by the results achieved so far, here we addressquantum tomography of CV systems by the HL detectionscheme [15]. We prove that the HL scheme may be effi-ciently employed to reconstruct the density matrix of single-

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signal

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FIG. 1: (Left) HL detection scheme: the signal ρ is mixed with acoherent state |β〉 (LO) at a balanced BS. After the interference, thedifference between the number of photons detected at the BS out-puts is evaluated. (Right) Sketch of the experimental setup for thereconstruction of coherent and PHAV states.

mode quantum states, and to evaluate the expectation valuesof the first moments of the quadrature operator. In particular,we show that robust and reliable results may be obtained witha relatively modest unbalancing between signal and LO. Wealso investigate the role played by quantum efficiency, whichis a crucial parameter in the reconstruction of nonclassicalstates.

In OHT, the elements of the density matrix ρ of a quantumstate can be reconstructed as follows [1–7]

ρnm =

∫ π

0

dφ

∫ +∞

−∞dx p(x, φ)Fnm(x, φ), (1)

where p(x, φ) = 〈xφ|%|xφ〉 are the homodyne probabilitydistributions and |xφ〉 are the eigenstates of the quadraturesxφ = (a e−iφ + a† e−iφ)/

√2, a being the annihilation op-

erator of the field, [a, a†] = 1. In Eq. (1), Fnm(x, φ) are aset of suitable sampling functions (see Ref. [7] for details). Inpractice, the probabilities p(x, φ) are retrieved from the differ-ence of the two macroscopic photocurrents exiting the homo-dyne detector [16–18], and the matrix elements are obtainedby sampling Fnm(x, φ) over data.

In our scheme, we replace p(x, φ) with the HL probabilitydistributions pHL(∆φ), where ∆φ = ∆/(

√2|β|) and ∆ is the

(phase-dependent) difference between the number of photonsdetected at the two outputs of the homodyne detector [13]. We

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notice that, while the outcome x of standard homodyne detec-tion can assume any real value, the quantity ∆φ is intrinsicallydiscrete. However, this discretization does not represent a lim-itation for the proper reconstruction of the states under exam-ination, as shown hereafter. The information coming from theHL scheme is contained in the joint statistics q(n,m) of thenumber of photons detected at the outputs of the BS. To findthe theoretical expression of q(n,m), it is useful to considerthe Glauber-Sudarshan P representation of the input state ρ

ρ =

∫Cd2z P (z) |z〉〈z| , (2)

where P (z) is the P -function associated with ρ. After theinterference of the input state with the LO (the coherent state|β〉) at the balanced BS, the two-mode input state Rin = ρ ⊗|β〉〈β| becomes

Rout =

∫Cd2z P (z)

∣∣∣β + z√2

⟩⟨β + z√2

∣∣∣⊗ ∣∣∣β − z√2

⟩⟨β − z√2

∣∣∣(3)

and the corresponding joint photon-number statistics can bewritten as [19, 20]:

q(n,m) =∫Cd2z P (z) e−µc(z,β)−µd(z,β)

[µc(z, β)]n

[µd(z, β)]m

n!m!(4)

where µc(z, β) = 12 |β + z|2 and µd(z, β) = 1

2 |β − z|2. Wenote that, thanks to this formalism, we can easily add the effectof quantum efficiency ηk of the detector on the output modesk = c, d, by replacing µk(z, β) with ηkµk(z, β). Startingfrom the joint probability (4) it is possible to calculate thetheoretical probability distribution of the quantity ∆ = n−m,namely

pHL(∆) =

{ ∑∞k=0 q(∆ + k, k) if ∆ ≥ 0 ,∑∞k=1 q(k, k −∆) if ∆ < 0 ,

(5)

and the corresponding “rescaled” version pHL(∆φ), where∆φ = ∆/(

√2|β|). In order to investigate the performance

of the HL detector, we consider some paradigmatic opticalstates: the coherent state, which is phase-sensitive, its phaseaveraged counterpart (PHAV), which is of course phase in-sensitive, and the single-photon Fock state. While for thecoherent and the PHAV states we provide an experimentaldemonstration, for the Fock state we performed Monte Carlosimulated experiments and we studied the effect of a non-unit quantum efficiency η < 1 with respect to the ideal case,namely η = 1.

For a coherent state ρ = |α〉〈α|, the theoretical jointphoton-number statistics, and the HL distribution, can be ob-tained by setting P (z) = δ(2)(z − α), where δ(2)(ζ) is thecomplex Dirac’s delta function. The experimental setup issketched in the right panel of Fig. 1. The second harmonicpulses (∼ 5 ps pulse duration) at 523 nm of a mode-lockedNd:YLF laser regeneratively amplified at 500 Hz were divided

at a Mach-Zehnder interferometer into two parts in order toyield the signal and the LO. The length of one arm of the in-terferometer was changed in steps by means of a piezoelectricmovement (Pz) in order to vary the relative phase φ betweenthe two arms in the interval [0, π]. Two variable neutral den-sity filter wheels (ND) were used to change the balancing be-tween the two beams, which were then recombined in a bal-anced BS. At the BS outputs two multi-mode 600-µm-corefibers (MF) delivered the light to a pair of photon-number-resolving detectors. In detail, we employed two hybrid pho-todetectors (HPDs, mod. R10467U-40, Hamamatsu Photon-ics), whose outputs were amplified, synchronously integratedand digitized. HPDs are commercial detectors endowed withpartial photon-number-resolving capability and a good linear-ity up to 100 photons. As already demonstrated elsewhere (seefor instance [13, 21, 22]), HPD response can be characterizedin a self-consistent way with the same light under examina-tion. Here we just remark that from the experimental data it ispossible to calculate the gain of the detection apparatus in or-der to recover the distribution of detected photons at each BSoutput. In addition, this kind of detector allows us to obtaininformation on the relative phase between the two arms of theinterferometer by simply monitoring the mean number of de-tected photons measured at each BS output as a function of thepiezoelectric movement [13, 23]. In the present case, 3× 106

data corresponding to subsequent laser pulses were recordedand used to reconstruct the density matrix of the coherent stateand of the PHAV state, which is a diagonal state obtained byrandomizing the phase of a coherent state |α〉 = ||α| eiθ〉

ρPHAV =

∫ 2π

0

dθ

2π|α〉〈α| = e−|α|

2∞∑n=0

|α|2nn!|n〉〈n|. (6)

In order to process the data corresponding to the coherentstate, once obtained the number of detected photons at eachBS output, we calculated the shot-by-shot photon-number dif-ference, ∆. The corresponding phase value φ was determinedby acquiring a suitable data sample (5 × 104 pulses) for eachone of the 60 piezo positions. As the piezo moves in regularsteps, the measured mean number of detected photons followsa sinusoidal trend due to the interference at the beam splitter,from which the effective value of φ for each piezo positioncan be evaluated. The 5× 104 data corresponding to the sameφ were uniformly distributed around that value with a step of1/(5× 104).

The typical behavior of ∆φ as a function of φ is shown inFig. 2(a) (3× 105 data). Note that, due to irregularities in thepiezoelectric movement, the data are not uniformly distributedin φ. The reconstruction procedure was applied to 10 sets of3×105 data drawn from the whole sample: the modulus of theaverage density matrix is shown in Fig. 2(b). The correspond-ing mean number of photons is 〈n〉 =

∑n nρn,n = 1.06.

The fidelity [24] between the reconstructed coherent state anda coherent state with the same amplitude is F = 99.4 %.

As a matter of fact, the fidelity may not fully assess thequality of a reconstruction scheme [25] and therefore, in or-

3

FIG. 2: Experimental reconstruction of a coherent state with mea-sured amplitude α = 1.03. The LO has an amplitude |β| = 3.82. (a)∆φ as a function of LO phase, φ. The experimental data (green dots)are shown together with the mean value (solid line) and the standarddeviation (dashed lines) of the quadrature, as obtained through thepattern function tomography. (b) Reconstructed density matrix. Thefidelity with the expected state is F = 99.4 %.

der to test the quality of HL reconstruction, we also considerthe HL evaluation of the expectation values of some observ-ables. To this aim, we remind that given a generic observable,we may obtain its expectation value 〈A〉 = Tr[ρA] from thedistribution of homodyne data as [4]

〈A〉 =1

π

∫ π

0

dφ

∫Rdx p(x, φ)R[A](x, φ) (7)

where R[A](x, φ) is a kernel, or pattern function, associatedwith the operator A. For instance, in the following we willuse the fist two moments of the quadrature operator, whichmay be obtained from the kernels [4]:

R[xθ](x, φ) = 2x cos(φ− θ) , (8)

R[x2θ](x, φ) =

(4x2 − 1

η

)[4 cos2(φ− θ)− 1

] 1

4+

1

4.

(9)

By using the HL probabilities pHL(∆φ) instead of p(x, φ)and setting θ = 0 from Eq. (7) we got: 〈x0〉 = 1.451 ±0.003 and var[x0] = 〈x20〉 − 〈x0〉2 = 0.688 ± 0.015 forthe case under investigation. These values must be com-pared to those expected in the limit of a HL scheme, namely〈x0〉theo =

√2α =

√2 1.03 = 1.456 and var[x0]theo =

12 + 1

2 |α|2/|β|2 = 0.536. The last result explicitly showsthe contribution to the variance due to the low-intensity LO,which becomes negligible as |β| � |α|. Notice that sincewe are dealing with classical states, the quantum efficiencyjust rescales the energy of the detected states, and we cansafely set η = 1: the results we obtain are thus referredto the detected states. We see that while the mean valueof the quadrature is well reconstructed by the experimentaldata, the variance is larger than expected. Such discrepancyis likely due to phase fluctuations occurred during the mea-surements session. In order to check whether this interpreta-tion holds, we consider the PHAV state, which is phase in-sensitive and thus should not be influenced by the presenceof possible phase fluctuations. As in the case of the coher-ent state, we saved 10 sets of 3 × 105 data by calculating the

FIG. 3: Experimental reconstruction of a PHAV state with amplitude|α| = 1.08. The LO has an amplitude |β| = 3.82. (a) ∆φ as a func-tion of φ. The experimental data (green dots) are shown togetherwith the mean value (solid line) and the standard deviation (dashedlines) of the quadrature as obtained through the pattern function to-mography. (b) Reconstructed density matrix. The fidelity with theexpected state is F = 99.9 %.

shot-by-shot photon-number difference between the two BSoutputs. Since the PHAV state is phase-insensitive, we ran-domly assigned a phase value to each experimental value ofd. A typical trace of d vs. φ is shown in Fig. 3(a), whereasin panel (b) the modulus of the reconstructed density matrixis presented. As expected, the off-diagonal elements are ab-sent. In order to compare the reconstructed matrix to thetheoretical prediction in Eq. (6) with |α| = 1.08, we calcu-lated the fidelity and obtained F = 99.9 %. For what con-cerns the first and second moment of the quadrature, we ob-tained 〈xθ〉 = 0.004 ± 0.003 and var[xθ] = 1.725 ± 0.013,∀θ. In this case, the expected values are 〈xθ〉theo = 0 andvar[xφ]theo = 1

2 + |α|2 + 12 |α|2/|β|2 = 1.706, respectively.

The very good agreement between theory and experiment forthis phase-insensitive state confirms our considerations aboutthe variance of the reconstructed coherent state.

We now turn the attention to the Fock state |1〉, for whichwe show the results obtained by using Monte Carlo simulatedexperiments setting β =

√20. The state |1〉 has a highly sin-

gular P function given by:

P (z) = δ(2)(z) +∂2

∂z ∂z∗δ(2)(z) , (10)

and the joint probability in Eq. (4) reads:

qη(n,m) =

e−η|β|2

n!m!

(η|β|2

2

)n+m [1 +

(n−m)2 − η|β|2|β|2

], (11)

where we assumed that both the detectors have the same quan-tum efficiency η. The corresponding pHL(∆) is quite clumsyand is not explicitly reported here.

In Fig. 4(a), we show a typical HL trace, i.e. ∆φ as afunction of φ, obtained from a Monte Carlo simulation (with5 × 104 data) assuming ideal detection, i.e. η = 1. It is clearthat, as one may expect, ∆φ does not depend on the relativephase φ since the Fock state is phase insensitive. The mod-ulus of the reconstructed density matrix is plotted in panelFig. 4(b). In this case the density matrix is diagonal and

4

FIG. 4: Reconstruction of the Fock state |1〉 (Monte Carlo simulatedexperiment), with LO amplitude β =

√20 and quantum efficiency

η = 1. (a) ∆φ as a function of φ; (b) Reconstructed density matrix.The fidelity with the expected state is F = 99.9 %.

thus the fidelity of the state coincides with that of the photon-number distribution. In particular, for the density matrix inFig. 4(b) we have F = 99.9 %.

Up to this point, we have shown the reliability of the HLdetection scheme and of the reconstruction strategy underideal detection conditions. However, since photon-number-resolving detectors are real detectors, it is worth investigatingwhether the HL scheme also works in the presence of an over-all quantum efficiency η < 1. When standard HD is adopted,it is well known that a non-unit quantum efficiency rescalesthe mean value of the reconstructed density matrix in the caseof classical states of light, such as coherent and PHAV states,whereas it deeply modifies the properties of the reconstruc-tion in the case of nonclassical states, such as Fock states[17]. In order to check if analogous results can be achievedby means of the HL scheme, we present the results obtainedby simulated experiments for the Fock state |1〉. We assumethe same value of LO adopted above, i.e. β =

√20, but now

we set η = 0.4, a realistic value for many kinds of photon-number-resolving detectors [21, 26]. In Fig. 5, we plot themodulus of the reconstructed density matrix, panel (a), andthe photon-number statistics, panel (b), for the Fock state |1〉.It is clear from the reconstructed density matrix that the effectof a non-unit quantum efficiency is to add a vacuum compo-nent to the state: in this case the expected density matrix isρ = η|1〉〈1| + (1 − η)|0〉〈0| [17] and its fidelity with respectto the reconstructed one is F = 99.0 %.

FIG. 5: Reconstruction of the Fock state |1〉 (Monte Carlo simulatedexperiment), with LO amplitude β =

√20 and quantum efficiency

η = 0.4. (a) Reconstructed density matrix; (b) Photon-number statis-tics of the reconstructed state. (the error bars have been obtained av-eraging over 10 simulated experiments). The fidelity with respect tothe expected distribution is F = 99.0 %.

In conclusion, our numerical and experimental resultsdemonstrate that the HL detection scheme may be used, in-stead of standard (pin-based) homodyne detectors, to recon-struct the density matrix of quantum states of light, as well asthe first two moments of the quadrature operator. Our resultsopen new perspectives to quantum state reconstruction, andpave the way to the use of PNR-based homodyne-like detec-tors in quantum information science.

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