Experimental single-photon pulse characterization by electro-optic shearing interferometry

Alex O. C. Davis1, Valerian Thiel1, Michał Karpinski1,2,∗, and Brian J. Smith1,3

1Clarendon Laboratory, University of Oxford, Parks Road, Oxford, OX1 3PU, UK2Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland and

3Department of Physics and Oregon Center for Optical, Molecular,and Quantum Science, University of Oregon, Eugene, Oregon 97403, USA

(Dated: February 21, 2018)

The ability to characterize the complete quantum state of light is essential for both fundamental and appliedscience. For single photons the quantum state is provided by the mode that it occupies. The spectral temporalmode structure of light has recently emerged as an essential means for quantum information science. Here weexperimentally demonstrate a self-referencing technique to completely determine the pulse-mode structure ofsingle photons by means of spectral shearing interferometry. We detail the calibration and resolution of themeasurement and discuss challenges and critical requirements for future advances of this method.

Quantum photonic technology research is increasingly fo-cusing on ultrashort optical pulsed modes due to their high in-formation content and their compatability with integrated op-tical platforms. The time-frequency (TF) degree of freedomconstitutes an infinite Hilbert space [1], allowing an informa-tion content per photon limited only by the encoder and de-tector resolution. Accessing the information contained in boththe spectral amplitude and phase domains of ultrafast pulsedmodes of quantum light raises the possibility of surpassingthe standard quantum limit in precision measurements suchas pulse time-of-flight [2] and atmospheric characteristics [3].Furthermore, the freedom to encode quantum information inmultiple spectral-temporal modes, and then to recover that in-formation, extends the usefulness of quantum secure informa-tion protocols such as quantum key distribution (QKD) [4].However, for these advantages to be exploited, complete andreliable characterization techniques of quantum light pulsesare required.

Quantum optical technologies involve three experimentalstages: state preparation, state evolution or active manipula-tion, and ultimately measurement. The ability to accuratelycharacterize the quantum state of light is crucial for develop-ing all three stages of optical quantum technologies. Firstly,the development of nonclassical light sources requires com-plete characterization of the optical field output to certify theoutput light matches the desired quantum state [5]. Secondly,verifying operations that manipulate the quantum state of lightrequires full characterization of a complete set of probe states[6]. This forms the basis of optical quantum process tomogra-phy. Finally, the development of new quantum optical detec-tors requires the ability to probe the detector response with acomplete set of probe states, a technique known as quantumdetector tomography. The set of probe states can be verifiedby first using a previously calibrated detector.

The pulse envelope of ultrashort optical pulses varies on atime scale far shorter than the response time of the best pho-todetectors, making direct sampling of the temporal intensityof ultrashort optical pulses infeasible. Moreover, full recon-struction of an optical pulse train necessitates knowledge ofits spectral or temporal phase even if the amplitude is knownin both the time and frequency domains [7]. Although signif-icant progress has been made in characterizing intense ultra-short light pulses using nonlinear optical interactions with a

shorter optical pulse, even enabling resolution of the carrierfrequency oscillations [8], techniques for single-photon levelcharacterization are in their infancy. Methods to estimate theoptical pulse shape of single-photon sources based on inter-ference with well-known reference pulses have been realized,but such approaches require stable, tunable, mode-matchedreference pulses [6, 9–11]. Such a tunable reference source isdifficult to obtain and requires a priori information about thepulse to be characterized to ensure proper overlap with the ref-erence. These approaches also require the reference pulse tobe scanned in multiple parameters, thus requiring long mea-surement times and many on-demand copies of the originalstate. We therefore seek a self-referencing method for char-acterizing the ultrafast pulse mode structure of single-photonsources without need for reconfiguration of the apparatus.

Spectral phase interferometry for direct electric field recon-struction (SPIDER) is an established technique for character-ising ultrashort pulses in the high field regime. The SPIDERprotocol involves interfering an ultrafast pulse with a copy ofitself to which a constant translation of the spectrum, or spec-tral shear, has been applied. Information about the spectralphase of the pulse can be recovered through measurementsof the spectral interference pattern between the two pulses[8]. In previous work with bright pulses, the spectral shearis typically obtained through second harmonic generation orother processes which are nonlinear in the incident opticalfield. Other methods of ultrafast pulse characterization suchas frequency-resolved optical gating (FROG) also rely on suchnonlinear processes [12]. However, in the quantum regimesuch nonlinear processes are not practical under general con-ditions due to the difficulty of achieving high probabilities ofnonlinear interactions at the single-photon level in a noise-free, easily reconfigurable setting [13].

A promising method of obtaining the spectral shift neededfor spectral-shearing interferometry is electro-optic tempo-ral phase modulation [14]. This involves passing the lightthrough an electro-optically responsive material such aslithium niobate, in which the refractive index is simultane-ously varied such that a temporally varying phase is appliedto the pulse. The effect of linear temporal phase modulationis the desired uniform frequency shift. This method has theadvantage that it is deterministic and largely independent ofthe incident pulse shape or amplitude, and is thus applicable

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FIG. 1. Schematic of EOSI. A pulse is coupled into a Mach-Zehnderinterferometer where one arm applies a spectral shear via electro-optic modulation, whilst the other applies a relative temporal delay.Spectrally-resolved detection at the output allows extraction of thespectral phase through Fourier analysis.

at the single-photon level. Recently, the feasibility of suchscheme has been shown at the single-photon level [15].

Previous work has demonstrated the feasibility of electro-optic spectral shearing intereferometry (EOSI) with classicallight [14]. We demonstrated the first single-photon pulse re-construction by EOSI [16]. Here, we provide a detailed de-scription of the methods and thoroughly discuss the scope andchallenges of this class of characterization scheme.

I. BACKGROUND

The quantum state of a single photon is given by the fieldmode that it occupies [17–19]. Here we focus on the pulsemodes of an optical beam, in which the source emits singlephotons into a well-defined transverse spatial and polarizationmode, such as that found in a single-mode optical fiber. Tounderstand how EOSI enables the reconstruction of the pulsemode structure of a single-photon source, we begin by il-lustrating how the spectral shearing interferometry algorithmextracts the spectral phase of an electromagnetic field mode[8, 20–22]. We will assume that successive pulses are iden-tical, with the same polarization and spectral mode. We willwork in terms of the spectral representation of the analyticfield

ε(ω) =√I(ω)eiφ(ω) = FT [ε(t)] , (1)

where FT refers to the Fourier transform, I(ω) is the spec-tral intensity, ε(t) and ε(ω) are complex-valued analytic func-tions, φ(ω) is the spectral phase and the electric field E(t) isrelated to ε(t) by

E(t) = Reε(t). (2)

Clearly, the electric field ε(ω) can be fully reconstructedfrom measurements of I(ω) and the spectral phase φ(ω).Measurement of I(ω) can be obtained using a single-photonspectrometer described in section V. However the spectralphase is more difficult to obtain and measurements must be

made indirectly.

The EOSI protocol for obtaining the spectral phase worksby splitting the pulse with a 50:50 beam splitter, applyinga spectral shift to one of the copies and a temporal shift tothe other, recombining the two at a second beam splitter, andanalysing the spectral interference pattern (see fig.1). If thespectral shear is given by Ω, the relative time delay by τ andthe intensity of the spectral interference pattern from the twooutputs by I+

Ω (ω) and I−Ω (ω), then

I±Ω (ω) =1

2|ε(ω)eiωτ ± ε(ω + Ω)|2 (3)

=1

2(I(ω) + I(ω + Ω))

±Reε(ω)ε∗(ω + Ω)eiωτ.

One can then take the Fourier transform of this intensitypattern:

I±Ω(T ) ≡ FT I±Ω (ω)

=1

2I(T )(1 + eiTΩ) (4)

± FT |ε(ω)ε(ω + Ω)| exp[i∆φ(ω,Ω)] ∗ δ(T − τ)

± FT |ε(ω)ε(ω + Ω)| exp[−i∆φ(ω,Ω)] ∗ δ(T + τ)

where ∗ indicates a convolution, ∆φ(ω,Ω) = φ(ω)−φ(ω+Ω)and I(T ) ≡ FT [I(ω)], and so has a width determined by thetransform-limited duration of pulses with the spectrum I(ω).

If τ is chosen to greatly exceed the pulse duration, then thethree terms of the above expression can be resolved from oneanother, and it is possible to computationally filter one of thetwo side bands, e.g.

|ε(ω)ε(ω + Ω)|expi(ωτ + ∆φ(ω,Ω)) (5)

Taking the argument of this yields

ωτ + ∆φ(ω,Ω) ≡ θ(ω,Ω, τ) (6)

from which the spectral phase can be faithfully reconstructedif τ is well-calibrated. Up to an irrelevant constant phase, eq.6can be re-written as

θ(ω,Ω, τ) = (ω − ω0)τ + φ(ω)− φ(ω + Ω) (7)

where ω0 is some angular frequency, taken henceforth to bethe center frequency of the interference term. We call thisterm the phase gradient, as it is similar, at the first order, to thederivative of the phase with respect to frequency.

The Fourier analysis method used to extract the envelopeand the phase from the interference pattern is similar to aHilbert transform. It is achieved by digitally filtering one side-band in the Fourier domain with a narrow filter and taking ei-ther the modulus or the argument of the inverse Fourier trans-form. This procedure is depicted using experimental data byFig. 4.

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A. Phase reconstruction

There are several mathematical approaches to extractingthe spectral phase φ(ω) from the measured phase gradientθ(ω). Although in principle any of these algorithms sufficesto reconstruct any spectral phase, a different reconstructionmethod might be appropriate depending on what the spectralphase profile is expected to look like. For many applications,only the low-order terms in ω of the spectral phase φ(ω) areof interest and so it is most appropriate to proceed by findinga polynomial expansion of φ(ω). It is helpful to reformulateeq.6 in terms of a Taylor series expansion of the second argu-ment about ω,

θ(ω,Ω, τ) = (ω − ω0)τ − Ωdφ

dω− Ω2

2

d2φ

dω2− · · · (8)

Now we express φ(ω) as a power series about ω0 and definingδω ≡ (ω − ω0),

φ(ω0 + δω) = φ0 + φ1δω + φ2δω2 + φ3δω

3 · · · (9)

where φn is the n-th term in the expansion for the spectralphase. Substituting into eq.8 we obtain

θ(ω,Ω, τ) = δω τ − Ω(φ1 + 2δωφ2 + 3δω2φ3 · · · )− Ω2(φ2 + 3φ3δω + · · · ) (10)− Ω3(φ3 + · · ·

At this point we remark that if Ω is small compared to thebandwidth of the pulse, then for most of the frequency rangewhere θ(ω,Ω, τ) is large compared to its error (i.e., the spec-tral overlap region of the pulses) eq.8 reduces to

θ(ω,Ω, τ) = τδω − Ωdφ

dω(11)

and hence φ(δω)can be directly extracted as

φ(δω) =1

Ω

∫[τδω − θ(ω,Ω, τ)]dω (12)

with the integration performed over the region whereθ(ω,Ω, τ) can be reliably measured. However, eq.10 showsthat in this limit

θ(ω,Ω, τ) = −Ωφ1 + δω(τ −2Ωφ2)−6Ωδω2φ3 + · · · (13)

and hence that any error in the proper calibration of τ mani-fests after the integration over ω as an error in the extractedvalue of φ2, the second-order spectral phase or “chirp”. It istherefore vital that τ is accurately characterized, with errormuch less than 2Ωφ2.

Within the regime of the shear Ω being much less than thebandwidth of the pulse, the errors in the phase reconstructionwill be smaller with increasing shear. If the shear were com-parable to the pulse bandwidth, the visibility of interferencefringes will be lost due to the reduced overlap of the shearedand unsheared pulses, which would negatively affect phase re-contruction. The spectral shear applied by the set-up can betuned up to a maximum of approximately 0.3 nm, for a pulse

bandwidth of several nanometers, and so the interferometeroperates comfortably within the regime where the overlap isclose to unity. However the shear is large enough relative tothe bandwidth that the successive terms in eq.10 contributesignificantly across much of the range of the integral in eq.12and may not be ignored. Therefore, to extract the phase terms,we note that eq.10 is linear in the spectral phase terms φi andhence may be expressed

δω τ − θ(δω,Ω) = A~φ (14)

where ~φ is the vector (φ1, φ2, φ3 · · · )T and A is a matrix de-fined with its columns as

A ≡ (Ω, (δω + Ω)2 − δω2, (δω + Ω)3 − δω3, · · · ) (15)

In our analysis we chose A to include terms up to (Ω/δω)4 ≈10−4 to keep calculation errors small. Eq.14 can be efficientlyand accurately solved numerically using a matrix division rou-tine to finally obtain the spectral phase.

For arbitrary spectral phase (provided the pulse’s temporalsupport is limited to the region [−π/Ω, π/Ω]), the phase canalso be recovered through a concatenation approach [22]. Thismethod is more suitable for spectral phase profiles that arenon-analytic in ω about the center frequency ω0 or containhigh-order polynomial contributions in ω that would not bereconstructed by the above approach. This approach beginsby setting the spectral phase at some value φ(ω0) to be someconstant arbitrary value, typically zero, and then iterativelyreconstructing the phase at every point ω + nΩ for integer n,such that

φ(ω0 + [n+ 1]Ω) = φ(ω0 + nΩ) + θ(ω0 + nΩ)−Ωτ (16)

According to the Shannon theorem, this concatenation ap-proach suffices to completely characterise the pulse if it has notemporal support outside the range [−π/Ω, π/Ω] [8]. If thiscondition is not met, however, a disadvantage of this methodis that it only provides the phase relationships between pointsseparated by an integer multiple of the shear, which is often acoarser sampling than that given by the resolution of the spec-trometers. Therefore, whilst this process gives the exact phaserelationships of discrete sets of points separated by integermultiples of the shear, the phase differences between pointsoffset by amounts other than an integer multiple of the shearis not directly determined. One approach to solving this is touse each series to determine the profile of a mode in the timedomain, then average these pulses. The resulting pulse pro-file contains information from all the contributing frequencymeasurements, and can be Fourier transformed back into thespectral domain to provide the complete set of phase relation-ships [22].

When processing the EOSI interferogram, care must betaken since most spectrometers will sample the intensity pro-file in the wavelength, rather than frequency, domain. For nar-rowband pulses it is acceptable to assume that the bandwidthof the pulse is sufficiently smaller than the wavelength that alinear relationship is sufficient to translate the spectrum fromthe wavelength domain to the frequency domain. However

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FIG. 2. Experimental setup for single-photon generation, shaping and EOSI detection (see text for detailed description). Ti:Sa: Titanium-Sapphire femtosecond oscillator; BS: beamsplitter; PD: photodiode; DM: dichroic mirror; PBS: polarizing beam-splitter; APD: single-mode-fiber-pigtailed avalanche photodiode; DG: diffraction grating; SLM: spatial light modulator; PMF: polarization-maintaining fiber; EOM:electro-optic phase modulator; RF: radio-frequency signal; BSC: Babinet-Soleil compensator; λ/2: half-waveplate; SP: spectrometer; CFBG:chirped fiber Bragg grating; SPCM: single photon counting module; TRSPS: time-resolved single-photon spectrometer.

this approximation is not valid in our demonstration and thor-oughly corrupts the extracted spectral phase if implemented[23]. In our implementation we corrected for the problem byusing Fourier transform interpolation, which functions welleven up to fringe spacing at the Nyquist limit.

II. SOURCE OF SHAPED SINGLE PHOTONS

The full experimental setup is depicted by Fig. 2. Thelaser source is a commercial femtosecond oscillator (Spect-raPhysics Tsunami) delivering pulses of 100 fs full width athalf maximum (FWHM) at a repetition rate of 80 MHz, corre-sponding to a spectrum centered at 830 nm and a bandwidth of10 nm FWHM. An internal fast photodiode is used to generatefrom the repetition rate the clock signal that is used throughoutthe experiment. Second harmonic generation is achieved in a1mm-long BiBO crystal, generating a 3nm FWHM spectrumat 415nm. Heralded single photons are generated by collinear,type-II spontaneous parametric down conversion (SPDC) inan 8mm-long potassium dihydrogen phosphate (KDP) crystal[24]. The orthogonally-polarized signal and idler fields with,respectively, 3 nm and 12 nm bandwidths are separated at apolarizing beam splitter (PBS), with a photodetection event inthe idler mode heralding the creation of a photon in the sig-nal mode. The phase-matching of the source was chosen suchthat the complex joint amplitude of the two-photon state pro-duced by the source would be separable, such that detectionof a heralding photon in the idler arm projects the signal intoa spectrally-pure single-photon Fock state [25]. Double pairsof photons, which would cause problems at the output of the

interferometer, are shown to be negligible compared to singlephoton events (g(2) = 0.06± 0.04).

To test the EOSI with a range of spectral phase profiles,the signal photon is then directed to a fiber-coupled pulseshaper capable of performing arbitrary spectral phase opera-tions [26]. The shaper consists of a standard 4-f line built witha 2000 lines/mm diffraction grating (Spectrogon) and 200 mmfocal length cylindrical lens (Thorlabs) with a 2D phase mask(Hamamatsu SLM, 1272 x 1024 pixel mask) and is capable ofachieving arbitrary spectral phase shaping with a resolution of0.04 nm/pixel and near-uniform spectral intensity transmis-sion of ≈40%. Losses are equally distributed between thediffraction grating efficiency and the insertion losses in fibercoupling at the output of the device.

The shaper is calibrated by leaving the phase profile of theSLM uniform except for a single pixel with a π phase shiftscanned across the mask. This creates a spatial discontinu-ity in the phase over an aperture comparable in size to thewavelength, which causes diffraction at the wavelength cor-responding to the pixel position. The pixel position of theSLM can therefore be mapped to the wavelength by sendingin bright classical pulses through the SLM and then into a con-ventional spectrometer (Shamrock 303i monochromator, 1200mm−1 grating, Andor) and then monitoring the position of thedip in the spectral intensity caused by diffraction around thephase-shifted aperture.

Since the shaper itself introduces spectral phase, it neededto be calibrated. The exact phase introduced by the shaperin its unmodulated state was determined by standard spec-tral interferometry. A second, free-space beam path was builtaround the pulse shaper to form a Mach-Zehnder interfer-

5

ometer with the shaper in one arm, into which bright clas-sical pulses were sent. The relative spectral phase in the twoarms introduced by the pulse shaper was then extracted us-ing Fourier algorithms[7]. This was eventually used to createa default mask for the SLM, designed to compensate for thenonuniform spectral phase introduced by the device itself andto therefore allow complete control and manipulation of thespectral phase.

The SLM is polarization-sensitive, so it is important thatonly light linearly polarized in the correct orientation is sentinto the pulse shaper. However, due to slight misalignmentand imperfection in the grating, a small amount of the otherpolarization appears in the line. A polarizing beamsplitter wasadded to remove the unwanted polarization and to ensure thatno interference fringes due to polarization, which could af-fect the phase reconstruction, are observed at the output of thepulse shaper. Note that the device is also capable of adjustingthe spectral amplitude [27] which is used to reduce the band-width by the idler photon from 12 to 8 nm in order to fit withinthe acceptance window of the fiber Bragg grating used in oursingle-photon spectrometer (see Sec.V).

III. OBTAINING SPECTRAL SHEAR

To obtain the spectral shear we used an EOSpace LiNbO3-waveguide electro-optic phase modulator (EOM). The modu-lator is driven by a 10 GHz sinusoidal voltage emitted by aparametric dielectric resonant oscillator (PDRO) and ampli-fier (Aspen Electronics). The PDRO upconverts by a factorof 125 the 80 MHz signal from the laser cavity photodiode,ensuring that the 10 GHz output signal is phase-locked to theoptical pulse train. The modulator is designed such that anapplied radio-frequency (RF) signal propagates through thedevice with phase velocity equal to the group velocity of anoptical pulse centered at 830 nm wavelength. A pulse trav-elling through the device therefore acquires a time-dependentphase φ(t), depending on the temporally locked RF field am-plitude with which it co-propagates. This results in a tempo-ral variation in the refractive index of the waveguide viewedfrom the reference frame of the optical pulse. The voltagedifference necessary to achieve a relative phase shift of π be-tween orthogonal polarization propagating through the EOMis referred to as the half-wave voltage Vπ and depends on thelength of the modulator and the electro-optic coupling at agiven frequency.The time-varying refractive index of the waveguide can thenbe used for deterministic manipulation of the frequency spec-trum of the pulse, such as spectral shear [15] and bandwidthmanipulation [28]. If the centre of the pulse co-propagateswith the steepest part of the sinusoidal RF waveform, then theoptical field at the output of the modulator may be written

ε(t) = ε0(t)exp(−iπVmax

Vπsin(2πft)

), (17)

where Vmax is the maximum voltage of the applied RF field,f is the frequency of the RF signal and ε0(t) ≈ |ε0(t)|eiω0t is

the temporal profile of the incident pulse where ω0 is the cen-tral frequency of the wavepacket. If the pulse is short com-pared to the period of the modulation (in our case, 100 ps),then we can neglect higher-order terms in the expanded tem-poral phase, resulting in

ε(t) = |ε0(t)|exp(i

(ω0 −

2π2fVmaxVπ

)t

)(18)

The result is therefore an effective frequency shift of Ω =2π2fVmax/Vπ . Since the temporal support of the pulse mustbe within the region where the RF signal is linear, there is alimit on the maximum temporal duration of pulses to whicha uniform shift can be applied without modifying the spectralenvelope.

In practice, the phase-lock between the RF signal and theoptical pulse train will have some noise, resulting in eachpulse experiencing timing jitter with respect to the RF signal.However, the effect of this is simply to reduce the visibility ofthe fringes in the interferogram, and does not systematicallyaffect the calculated spectral phase [29].

The electro-optic response of the medium is much greaterin the crystal axis aligned with the applied electric field. Forlight polarized along the propagation direction, the coefficientof the electro-optic tensor has a magnitude of 33 pm/V, whiletheir values are ≈ 3 pm/V for other polarization. Hence,light propagating in the orthogonal polarization is less affectedsince the modulation of the index of refraction is directly pro-portional to the electro-optic coefficient. By propagating theunshifted pulse through the modulator in an orthogonal polar-ization, it is possible to achieve a common spatial path for thetwo pulses [23]. This common path approach has the advan-tage of making the set-up more resistant to vibrational andthermal fluctuations in path length and so improving inter-ferences visibility, as well as minimizing the relative spectralphase between the two arms of the interferometer. Overall inour set-up we were able to achieve a relative spectral shear ofapproximately 0.58 nm at 830 nm central wavelength.

Generally, when the RF source is turned on, it will lock toan arbitrary phase of the pulse train, meaning the pulse co-propagates at a random point of the RF waveform. Further-more, when the source is active, the phase lock is dependenton thermal fluctuations which could result in a slow drift of thelock point. It is therefore necessary to stabilize the tempera-ture environment. The amplitude and phase of the RF signalwave can be adjusted in real time through a variable time de-lay and attenuator [15]. For bright classical light, it suffices tosend the beam (of the modulated polarization) into the EOMand spectrally resolve the output, and then select the settingof the RF phase control such that the center wavelength of thepulse is extremized. However, the single photon signal pro-duces far fewer counts, making reliable alignment in real timevastly more challenging. To make the process easier, a brightcoherent beam line is coupled into the same beam path asthe downconverted light during the photon-source alignment.This bright classical beam line was constructed such that thedelay in the pulse train relative to the single-photon pulses wasan exact integer multiple of the RF waveform period (100 ps).By setting the delay to precisely 200 ps, it is known that the

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bright coherent pulses are aligned with the same point of theRF waveform as the single photon pulses, and therefore willreceive the same spectral shear. In this way it is possible toensure the single photons are spectrally shifted by inspectingthe spectrum of bright classical light.

IV. CALIBRATING THE INTERFEROMETER

Our implementation of the EOSI uses a polarization Mach-Zehnder configuration. This has several advantages over aspatial mode Mach-Zehnder or other interferometer designs:lower losses, spatial-mode matching, common-path stability,and a reduced need to account for internal spectral phase dif-ferences within the interferometer. One minor disadvantageis that the unmodulated pulse also passes through the mod-ulator, making it susceptible to residual electro-optic effects.This could affect the relative shear Ω or otherwise distort thereference pulse and thus corrupt the extracted spectral phase,although this proved negiligible in our demonstration. Sev-eral factors reduce the visibility of the interferogram: thermalfluctuations in the interferometer, the timing jitter of the RFsignal with respect to the optical pulse train, the mixedness ofthe single-photon state arising from optical fluctuations in thepump and pulse shaper, and mixedness arising from imperfectseparability of the two-photon state and the tracing over thestate of the herald photon. Besides optical effects, the limitedresolution caused by the point-spread function of the spec-trometers also creates the appearance of reduced visibility.

The value of τ must be chosen such that the spectral fringespacing is well below the sampling limit of the spectrome-ter (corresponding to τ ≈ 27 ps) to provide an interferogramwith good visibility, but large enough such that the interfer-ence term is clearly separable from the DC signal and thatspectral fringes appear with sufficient density to extract thephase to high order. In the experiments, a time delay of τ ≈ 5ps was chosen.

After passing through the pulse shaper, the signal photonis prepared by a half wave plate, acting as the input of thepolarization Mach-Zehnder interferometer (PMZI), at 45

polarization to the optical axis of the PM-fiber coupledelectro-optic modulator into which it is directed. The singlephoton is thus in a superposition of polarization states, oneof which experiences a constant spectral translation (theEOM applies only a marginal spectral shear on the ordinaryray). The pulses were then directed into a free-space pathcontaining birefringent calcite wedges (a Babinet-Soleilcompensator, or BSC), which introduce a variable relativedelay between the two polarizations. In practice the PMfiber, which is highly birefringent, and the modulator itselfintroduce more than enough delay to resolve the interferenceterms and the role of the calcite wedges is to easily set thespectral interference fringe spacing to below the samplinglimit of the spectrometer. The two pulses are then rotated inpolarization by a final half wave plate oriented at 22.5 totheir polarizations and interfered at a polarizing beam splitter(closing the PMZI), with the resulting spectral interferencepatterns resolved on time-resolved single photon spectrome-

- 148 - 98 - 49 0 49 98 147

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FIG. 3. a) Temporal cross-correlation pattern of unsheared classi-cal wavepackets with the calibrated time axis; dashed: envelope ofthe cross-correlations obtained by Hilbert transform. b) Compari-son of the envelope of the cross-correlation pattern for an unsheared(dashed) and sheared (plain) classical wavepacket, showing a delayintroduced by the group velocity change of approximately 200 fs.c) Superposition of carrier-wave fringes showing difference in fre-quency caused by the spectral shear Ω.

ters (see Sec.V).

To reconstruct the spectral phase, it is necessary to have pre-cise knowledge of certain parameters of the interferometer. Inparticular the birefringent time delay τ , which must be knownto much higher accuracy than the chirp-induced effective de-lay Ωφ2 for accurate reconstruction, and the spectral shear Ω.The value of Ω can be chosen and measured by sending classi-cal light of the modulated and unmodulated polarizations into

7

the interferometer and resolving the spectrum. However, thevalue of τ is more challenging to determine and much moreimportant to the accuracy of the reconstruction. The error ofthe value of τ that would lead to errors in the reconstructedphase comparable to the typical values one would wish tomeasure is on the order of several hundred femtoseconds, outof an absolute value of approximately 5 ps. It is therefore notpossible to perform direct measurements of τ to satisfactoryaccuracy with a conventional photodiode. Attempts to cal-culate the value of τ theoretically are challenging due to theseveral processes within the modulator that contribute to thedelay. The main contribution to τ is from the birefringence ofthe LiNbO3 in the modulator, which is strongly temperaturedependent. Furthermore, LiNbO3 is a highly dispersive ma-terial, and so the difference in group velocity of the shearedand unsheared pulses and, again, the temperature dependencethereof, would need to be taken into account. One would alsoneed to simulate how the group velocity of the sheared pulsevaried as its spectrum was translated during its propagationthrough the modulator. We therefore used an interferometricmeans of directly measuring τ .

Calibration of the calcite wedges is done by injecting brightclassical pulses derived from the laser into the interferometerwith the EOM switched off and measuring with a silicon pho-todiode the intensity at the output whilst scanning the positionof the calcite wedges. This method only requires knowledgeof the carrier frequency of the incident light to map the wedgeposition to delay. Firstly it is necessary to create a mappingof wedge position onto relative time delay introduced by thebirefringence of the calcite. The interference signal at oneoutput of the interferometer is given by:

Iout =1

2

∫dω I0(ω) cos (φrel(ω)) (19)

where I0 is the spectral intensity of the incident field and φrelis the relative spectral phase within the PMZI. With the EOMoff, the relative phase φrel(ω) only contains a linear term thatcorresponds to a global delay τ between the polarizations, andso it is possible to approximate the signal as:

Iout(τ) = Icc(τ) cos (ω0τ) (20)

where Icc is the envelope of the cross-correlation and ω0 isthe carrier frequency of the pulse. It contains all the de-lay from propagation through the EOM and the polarization-maintaining fiber, and is also directly proportional to thelength of calcite through which light travels. The latter allowsto control the delay between the two polarizations. Hence,the first moment of the envelope gives the point of zero delay,while the fringe spacing gives the temporal delay as a functionof wedge position, as shown by Fig. 3a). To maximize over-lap between the pulses in the two polarizations, this time-axiscalibration would ideally be done with the EOM switched off.However, due to the aforementioned thermal effects on therefractive indices of the modulator, the axes can only be cali-brated across the range of interest with the EOM being driven.Therefore, the phase of the RF relative to the optical pulsetrain was chosen such that the pulse sits at the extremum of

the refractive index modulation. This leaves the center wave-length of the pulse unchanged, allowing extraction of the timedelay by the method described above. Note that any change infringe spacing due to mismatch of higher order spectral phase,such as dispersion, will only appear in the wings of the inter-ferogram, and can be easily accounted for with Fourier anal-ysis. This method then allows to unambiguously calibrate thetemporal delay caused by the calcite wedge.

With the axis calibration in hand it is then possible to ex-tract the time delay τ of the interferometer. Since the pulse re-construction experiment involves spectrally shearing one armwithin the interferometer, this results in a change in the timedelay τ between the two polarizations due to group delay dis-persion. In this case, the cross-correlation signal is given by

Icc(τ) = Icc(τ − τ0)cos [(ω0 + Ω/2)(τ − τ0)] (21)

where τ0 is the contribution to the delay caused by group de-lay dispersion of the sheared pulse as derived in eq.13. Theinterferogram measured when the EOM is shearing is thendelayed by a constant value τ0, and the difference in fringespacing also allows measurement of the spectral shear. Asshown in figure 3b), the zero-delay point is shifted by about200 fs when maximum shear is applied on the EOM. The signof this delay depends on the sign of the shear (positive forlower wavelength), which is consistent with the index pro-gression (n decreases when λ increases). By defining the zerodelay point as the new reference, the stage position is thenfully calibrated and the time delay τ between the two armsof the interferometer is fully characterized. The advantage ofthis calibration method is that it does not require any ultrafastdetection: the sampling rate of the acquisition simply has to besufficient to resolve temporal fringes created by the movementof the stage, which is typically have a period on the order ofa few kHz. It does however require phase stability, but this isobtained directly from the common-path polarization interfer-ometer. Finally, as depicted by fig.3c), when the two interfer-ograms (shear vs no shear) are superimposed, the difference inoptical frequency due to the difference in wavelength can beclearly seen. Performing the scan multiple times and averag-ing the phase extracted from the Fourier analysis conclusivelyallows the extraction of the shear Ω. Though directly mea-suring the shear using a spectrometer is more robust, in theabsence of a high-resolution conventional spectrometer thecross-correlation method allows a direct measurement at theoutput of the interferometer, taking into account every opticalelement.

V. SPECTRALLY-RESOLVED DETECTION

Since spectral-shearing interferometry requires spectrally-resolved measurement, it is necessary to simultaneouslygather enough single photon counts in multiple spectral chan-nels before contrast is lost in the interferometer. Conven-tional spectrometers often work by coupling spectral modesinto some other basis, such as spatial modes, by means ofa prism or diffraction grating, and monitoring each of those

8

2.26 2.27 2.280

10

20

30

Optical frequency (fs- 1)

Counts

Interferogram

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

Delay (ps)

Normalizedamplitude

Fourier domain

2.26 2.27 2.28

-4

0

4

-2

0

2

Optical frequency (fs- 1)

Phasegradient(rad)

Reconstructedphase(rad)

a) b) c)

FIG. 4. Phase extraction process for an unshaped heralded single photon. a) output of the interferometer, showing a 20 seconds acquisition.b) Fourier transform of the interferogram, showing a peak centered around the delay τ of 5ps. Overlay: filter used to isolate the sideband.c) argument of the inverse Fourier transform of the filtered sideband (black) and reconstructed phase (blue). The delay τ has been removed,showing a linear slope due to the quadratic phase accumulated through fiber propagation.

modes with a separate detector. However, spectrometers capa-ble of concurrently monitoring multiple spectral modes withsingle-photon level intensity resolution have been difficultto realize due to the size and expense of arrays of multi-ple single-photon counting detectors. One solution for shortpulsed modes with a well-characterized repetition rate is dis-persive Fourier-transform spectroscopy, whereby the pulseundergoes frequency-to-time mapping in a dispersive mediumsuch that the temporal envelope of the output pulse corre-sponds to the spectral envelope of the original pulse [30–32].Temporally-resolved detection with just one single photoncounting module (SPCM) then provides spectrally-resolveddetection. The spectrometers implemented here use extremelyhigh dispersion fiber Bragg gratings to map the frequency ofa temporally-short optical pulse onto the temporal envelopeof the output pulse. We have recently shown that by usingstate-of-the-art chirped fiber Bragg grating (CFBG) as the dis-persive element and low-timing jitter single-photon detectors,high resolution spectrometers can be built [33].

The optical component of the TRSPS, consisting of achirped fiber Bragg grating (Teraxion) spliced onto port 2 ofan optical circulator (see inset of fig.2), has a transmission ef-ficiency of approximately 10%. The CFBG introduces a fre-quency dependent delay of approximately 950 ps/nm on thephoton, which is then detected by a fast SPCM (PDM, MicroPhoton Devices) with approximately ∆t = 30 ps timing res-olution. The SPCM signal is processed by a time-to-digitalconverter (TDC, Pico- Quant, HydraHarp 400) triggered by afast photodiode signal that samples the laser pulse train actingas the experiment clock. The TDC has timing resolution downto 12 ps, below the detector timing resolution. The triggerphotodiode (Alphalas, UPD-15- IR2-FC) has approximately15 ps rise time and timing variation less than 1 ps. Since thespectral resolution is ultimately limited by the timing resolu-tion of the SPCM, the device is able to detect single photonswith approximately 55 pm spectral resolution at 830 nm. TheBragg grating is written to reflect light across a wavelengthrange of 825–835 nm and therefore for broadband pulses alsoacts as a bandpass filter.

VI. RESULTS

A. Data acquisition

The procedure for a typical experimental single-photonstate characterization is depicted by Fig.4. An acquisitionconsists of accumulating a spectrogram at the output of theTRSPS over approximately 20 seconds (see Fig. 4a), whichwas found to be a sufficiently long acquisition time for pre-cise phase extraction given the final coincidence count rates.The visibility of the interference fringes is typically greaterthan 70% and is limited by phase instabilities within the inter-ferometer and the resolution of the spectrometer. Performingthe Fourier analysis detailed in Sec.I then allows extraction ofthe phase gradient θ shown in Fig. 4c from the sideband inthe Fourier domain (Fig. 4b). The spectral phase informationis finally obtained by following the algorithm from Sec.I A.Note that the spectral range over which the phase is recon-structed must be restricted to the region of the spectrum inwhich fringes are observed, since the extracted phase outsideof this region is not recoverable (see the edge of Fig. 4c)

To build statistics, this procedure is repeated for multipleacquisitions, each consisting of 20 seconds of data, and theextracted phases are averaged. However, the individual in-terferograms themselves may not always be summed becauselong-term instabilities, such as slow temperature drifts, resultin a slow shift of the global phase of the interference fringesand so reduce the contrast of the resultant interferomgram. In-terferograms that do not reach a threshold contrast of 50% arediscarded and do not contribute to the final reconstruction.

B. Reconstruction of phase that is polynomial in frequency

The experiment was conducted with various spectral phaseprofiles imprinted on the pulse shaper in order to test the fi-delity of the reconstruction. In the absence of deliberately ap-plied spectral phase, the device recovers a spectral phase witha quadratic coefficient of 87000 ± 1000 fs2 and a cubic of(5± 1) · 105 fs3. This value is found to be in good agreementwith the theoretically predicted spectral phase the single pho-

9

tons in our experiment receive from their generation and prop-agation, taking into account the approximately 2.5 meters ofpolarization-maintaining fiber, the KDP crystal, and the rela-tive phase within the interferometer (calcite wedges and EOMwaveguide). This is referred to as the “uncompensated phase”.

2.26 2.27 2.28

- π2

0

π2

Phasegradient

(rad)

2.26 2.27 2.280

π

2 π

3 π

Phase

(rad)

Uncompensated

2.26 2.27 2.28

0

π20

π10

Phasegradient

(rad)

2.26 2.27 2.28- π4

0

π4

Phase

(rad)

Compensated

2.26 2.27 2.28

0

π2

π

Phasegradient

(rad)

2.26 2.27 2.280

2 π

4 π

6 π

Phase

(rad)

Excess Cubic

2.26 2.27 2.28

- π4

0

π4

π2

Phasegradient

(rad)

2.26 2.27 2.28

-2 π

0

2 π

Phase

(rad)

Excess Cubic Compensated

2.26 2.27 2.28-π

- π2

0

π2

π

Phasegradient

(rad)

2.26 2.27 2.280

2 π

4 π

Phase

(rad)

Excess Quartic

2.26 2.27 2.28- π2

- π4

0

π4

π2

Optical frequency (fs-1)

Phasegradient

(rad)

2.26 2.27 2.280

π

Optical frequency (fs-1)

Phase

(rad)

Excess Quartic Compensated

FIG. 5. Left column: average of the extracted phase gradients cor-rected for delay; right column: reconstructed phases from the leftcolumn using the concatenation algorithm for a frequency shearΩ < 0. The error bars represent the standard deviation. The error iszero at the center frequency since it is constrained by the algorithm.The left column is approximately the derivative of the right one.

This quadratic spectral phase is then removed by applyinga compensating phase mask on the pulse shaper. The cu-

bic component is not compensated for. Similarly, additionalphases are tested, such as excess cubic or quartic phase, withand without the compensation for the second-order phase. Thephases that are extracted from the interference pattern are allshown on Fig. 5 for the different configurations of the pulseshaper, as well as the reconstructed phases. In the first case,without any applied phase on the shaper, we can see that theextracted phase gradient, θ(ω,Ω), is mostly linear and nega-tive and the reconstructed phase itself is quadratic and posi-tive. The difference in sign comes from the sign of the spec-tral shear, which was negative (i.e. positive in wavelength,δλ = 0.58 nm), and extracted phase can be approximatedto the first order as the gradient of the real phase, hence aquadratic phase has a linear gradient. This trend can be veri-fied in all the different phases that are imprinted on the pulseshaper. One can also verify that in the compensated case,the remaining phase is quite small in comparison to the largequadratic phase. A cubic component can be extracted, thevalue of which is found to be equal to that recovered in theuncompensated case, implying the expected additivity of theapplied phase.

Polynomial phase profiles up to quartic spectral phase wereimprinted on the single photon by the pulse shaper to testthe limit of the reconstruction, in every case with or withoutcompensation of the quadratic part. In each case, it can beseen that the reconstruction is consistent with the phase ap-plied on the shaper. The uncompensated cases always showa quadratic part around the same value (∼ 90000 fs2), whichfalls to zero in the compensated case, while the higher ordercoefficients remain. The retrieved coefficient for cubic phaseis (2.5 ± 0.3) · 107 fs3 and (2.0 ± 1.0) · 109 fs4 for quarticphase.

To further validate the calibration of the setup, we can com-pare the phase reconstruction to the known phase that was im-printed on the pulse shaper. This is achieved by computing therelative phase between the shaped and unshaped pulse. Theresults depicted by Fig. 6 show a good agreement betweenthe reconstruction and the original phase. The discrepancycan be attributed to the inability of the pulse shaper to writelarge polynomial coefficients, since the phase is too steep onthe edge of the spectrum, thus introducing errors. Indeed, itcan be seen that the reconstruction for quartic phase is moreunstable than the other cases, since such a phase shows verysteep phase gradient close to the wings of the spectrum. How-ever, it can still be reconstructed with high accuracy even inthe uncompensated case where a large quadratic componentis mixed. It is worth noting that both algorithms described inSec.I A can be used to reconstruct polynomial phases, givingsimilar results. This shows that the setup and the algorithm isindeed able to reconstruct higher order phase structures at thesingle photon level.

C. Arbitrary phase reconstruction

A final set of measurements were achieved to test the recon-struction capabilities of the setup when the phase that is im-printed on the single photons may not be expanded on a poly-

10

2.26 2.27 2.28- 4π

- 2π

0

Optical frequency (fs- 1)

Phasegradient(rad)

Compensated

2.26 2.27 2.28

- 2π

0

2π

Optical frequency (fs- 1)

Phasegradient(rad)

Cubic

2.26 2.27 2.28

- 6π

- 4π

- 2π

0

Optical frequency (fs- 1)

Phasegradient(rad)

Cubic Compensated

2.26 2.27 2.280

π

Optical frequency (fs- 1)

Phasegradient(rad)

Quartic

2.26 2.27 2.28

- 2π

- π

0

Optical frequency (fs- 1)

Phasegradient(rad)

Quartic Compensated

FIG. 6. Comparison between the phase written on the pulse shaper(dashed) and the reconstructed phase (plain). The shaded area repre-sents the standard deviation of the reconstruction, which is naturallylarger the further from ω0, where the phase is constrained. The insetnumbers represent the fidelity of the reconstruction.

nomial basis. As detailed in [16], two sets of arbitrary phaseswere chosen, which resulted in similar spectral and temporalintensities, but actually formed two orthogonal modes. Theyare defined by a linear spectral phase proportional to |ω − ω0|,where the positive (negative) linear phase is labelled V-phase(Λ-phase). The effect of such a phase on the wavepacket is todelay or advance the long-wavelength part of the spectrum inan opposite direction to the short-wavelength part.

The value imprinted on the pulse shaper was±750 fs, and areconstruction was made. Creating phase profiles of this sort,which present a non-continuous spatial derivative of the phaseon the mask of the SLM, can introduce diffraction, resultingin a small hole in the spectrum at the non-analytic point. Sim-ilarly to the previous section, these phases were reconstructedwith and without compensation of the dispersion. The ex-tracted and reconstructed phases are shown in Fig. 7. Theextracted phase gradients ∆φ(ω,Ω) closely resemble a stepfunction where the steepness is determined by the value of theshear, and the value of the plateau reflect the amount of lin-ear phase. Furthermore, we can clearly see an additional lin-ear phase gradient in the uncompensated cases, showing againthe presence of dispersion. The reconstructions show that thesetup is able to easily distinguish the two phases. Moreover,a fit in the reconstruction gives an absolute linear phase of+750 ± 25 fs for the V-phase and −775 ± 50 fs for the Λ-phase, and the same quadratic phase found in the previoussection for the uncompensated and compensated phases. TheΛ-phase is more difficult to reconstruct compared to the pre-

2.26 2.27 2.28

- π2

0

π2

Phasegradient

(rad)

2.26 2.27 2.280

2 π

4 π

Phase

(rad)

V-phase

2.26 2.27 2.28

- π4

0

π4

Phasegradient

(rad)

2.26 2.27 2.280

π

2 π

Phase

(rad)

V-phase compensated

2.26 2.27 2.28

- π4

0

π4

Phasegradient

(rad)

2.26 2.27 2.28

- π2

- π4

0

π2

Phase

(rad)

Λ-phase

2.26 2.27 2.28

- π4

0

π4

Optical frequency (fs-1)

Phasegradient

(rad)

2.26 2.27 2.28

-2 π

-π

0

Optical frequency (fs-1)

Phase

(rad)

Λ-phase compensated

FIG. 7. Left column: average of the extracted phase gradients cor-rected for delay; right column: reconstructed phases from the leftcolumn using the concatenation algorithm for a frequency shearΩ < 0. The error bars represent the standard deviation. The error iszero at the center frequency since it is constrained by the algorithm.The left column approximately shows the derivative of the right one.

vious phases due to the constraints put on the pulse shaper.Indeed, this phase has a steep negative slope, and compensat-ing for dispersion adds an extra negative quadratic phase onthe mask, resulting in additional wrappings.

D. Time-frequency distributions

With the phase now reconstructed, it is possible to cre-ate time-frequency distributions to fully describe the spectral-temporal wave function of the single photon. Indeed, as it isthe case with classical pulses, a complete description may beobtained by showing how the spectral parts are mapped in thetime domain. Multiple ways to describe this distribution exist,and we choose the chronocyclic Wigner function:

W (ω, t) =

∫dθ ε∗(ω − θ/2) e−iθt ε(ω + θ/2) (22)

The chronocyclic Wigner function has the advantages thatit provides a complete characterization of the single-photonspectral-temporal density matrix, is normalized and real-valued everywhere, and the overlap of two states can be easily

11

calculated from the overlap of the respective Wigner functions[34]. It is often referred to as a “quasiprobability distribution”since its marginals correspond to the probability distributionsof measurement outcomes in those bases, although the func-tion itself can locally take negative values. To compute suchdistribution, one needs the complex spectral amplitude |ε(ω)|in order to reconstruct the complete spectral wave functionwith the phase that was measured. In principle this can bedirectly measured during the acquisition by summing the twooutputs of the interferometer, which removes the interferencefringes. The result is quite close to the true spectrum for smallvalues of the shear. It is also possible to independently mea-sure the amplitude directly without sending pulses into the in-terferometer. The latter method was used here. The centerfrequency of the spectrum (i.e. the first moment of the dis-tribution) is used as the experimental value of ω0 throughoutthe algorithm, notably to constrain the center frequency of thephase concatenation and the fits. The time-frequency distribu-tions for the polynomial and the arbitrary phases written withthe pulse shaper are shown on Fig. 8.

Summing the distribution along one dimension yields ei-ther the temporal or the spectral marginals. As expected forpolynomial phase, the temporal wave function of the uncom-pensated single photon is much broader than that of the com-pensated one, cubic phase creates replicas, and quartic phasecreates a pedestral. In the case of the V-phase and Λ-phase,both marginals are similar, yet the two modes are orthogonal.Computing the overlap between these modes yield a value of0.06 ± 0.01, while the overlap of their marginal temporal in-tensities is 0.95± 0.01 and the spectral overlap is unity.

VII. APPLICABILITY AND CHALLENGES

A. Mixed state reconstruction

The methods developed thus far are aimed at reconstruc-tion of the pulse-mode state of pure single-photon sources, i.e.single-photon sources that emit photons into only one pulsemode. For sources that emit photons into a mixture of pulsedmodes, we must generalize the state reconstruction processto reconstruct the full density operator ρ. The problem ofcharacterizing a mixed state single-photon source is similar tocharacterization of a partially-coherent pulse train [35]. Theresult of a measurement with the EOSI is defined as the quan-tum counterpart of Eq. 4 as I±Ω (ω) = Tr

[ρΠ±(ω)

], where

Π±(ω) is the positive-operator valued measure (POVM) thatdescribes the measurement of a single photon in a mixed stateusing the experimental scheme. It is defined by

Π±(ω) =1

2(|ω〉 〈ω|+ |ω + Ω〉 〈ω + Ω|)

± |ω〉 〈ω + Ω| eiωτ ± |ω + Ω〉 〈ω| e−iωτ (23)

For a pure state, it is straightforward to show that this POVMis the one associated to Eq. 4. For a mixed state, by defin-ing ω′ = ω + Ω and scanning the shear, the 2-dimensional

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Uncompensated

0

100

200

300

400

500

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Compensated

0

100

200

300

400

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Cubic

-100

0

100

200

300

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Cubic Compensated

-100

0

100

200

300

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Quartic

-100

0

100

200

300

400

500

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Quartic Compensated

-100

0

100

200

300

400

500

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

V-Phase

-200

0

200

400

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

V-Phase Compensated

-200

0

200

400

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Λ-Phase

-200

0

200

400

-2480. -1240. 0. 1240. 2480.

835.

832.

829.

827.

825.

Time (fs)

Wavelength(nm)

Λ-Phase Compensated

-200

0

200

400

FIG. 8. Chronocyclic Wigner function for the phases that were im-printed with the pulse shaper.

intensity pattern is then given by

I±Ω (ω, ω′) =1

2(ρ(ω, ω) + ρ(ω′, ω′))

± Reρ(ω, ω′)eiωτ

(24)

Hence, by scanning the spectral shift Ω, it is possible to mea-sure the spectral density matrix ρ(ω, ω+ Ω) using 2D-Fourieranalysis similar to the one presented in Sec. I. Moreover,it may be easier to use the signal associated to the differ-

12

ence of the photocurrents at the outputs of the interferometerIdiffΩ (ω, ω′) = I+

Ω (ω, ω′)−I−Ω (ω, ω′) = 2 Reρ(ω, ω′)eiωτ

,

which directly samples the off-diagonal entries of the spec-tral density matrix. Note that adding a π/2 phase shift in theinterferometer yields 2 Im

ρ(ω, ω′)eiωτ

, which allows full

reconstruction of the density matrix. Even though the full re-construction would prove more difficult due to the increase indimension and need to scan the spectral shift, it is still feasiblewith this method.

It is worth noting that this only samples elements of thespectral density matrix that are within the shear of the maindiagonal. Full characterisaiton of an unknown state would re-quire scanning the frequency shift on a range that is on the or-der of the spectral bandwidth. With the capabilities outlinedin this paper, a mixed state of narrower bandwidth could inprinciple be reconstructed.

B. Challenges

While the general scheme depicted by Fig. 1 is not particu-larly complex, the experimental implementation places somelimitations on the input that EOSI can realistically charac-terise. Altogether, these constraints are in general much morelax for EOSI than those required by externally-referencingmethods. Here, we detail these constraints, and discuss howto choose the proper settings for the experimental setup.

The device is based on a polarization Mach-Zehnder con-figuration, which requires an input with well-defined polariza-tion. Polarization filtering may be employed to achieve this,but this incurs losses and precludes the device being used toexplore polarization-time/frequency correlations.

A main building block of the EOSI is a high-resolution single-photon spectrometer which we construct us-ing frequency-to-time conversion. While modern detectorsprovide sufficient temporal resolution, the methods for obtain-ing large chirped pulses are still limited. It is not feasible tostretch pulses sufficiently in free space to enable frequency-to-time conversion, so a fiber setup is needed. In the nearinfrared, optical fibers show typical dispersion on the order of100 ps/nm/km and propagation losses of more than 3 dB/km,rendering large chirps with fiber propagation largely imprac-tical. CFBGs prove to be the most efficient way of achievingthe dispersion needed by the setup, but typically act in reflec-tion over a limited spectral region, which is mainly definedby the length of the fiber and the specified dispersion. Largerspectral windows and larger chirp parameters could in prin-ciple be obtained by concatenating CFBG together. The as-sociation of the CFBG parameters and the timing jitter of thedetectors then limit both the spectral range and the spectralresolution[33], thus requiring a priori knowledge of the cen-ter wavelength and bandwidth of the unknown single photonstate.

The other main limitation comes from the EOM, withwhich the spectral shift is obtained. Various methods can beused to apply a high-power linear RF signal phase-locked tothe pulse train. While we use frequency multiplication anda phase-lock loop, we propose it is possible to amplify the

bandpass-filtered signal from a large bandwidth photodiodeand use it as the RF-wave which is automatically locked tothe pulse train. As described by Eq. 18, the amplitude ofthe frequency shear is directly proportional to both the fre-quency and the amplitude of the RF wave. The amplitude islimited by the amplifiers and by the EOM itself, while thefrequency is bounded by the temporal width of the linear re-gion of the RF-wave. If it is too large, the temporal pulseshape will be changed and thus introduce errors into the re-construction. Therefore the device input must be a train ofsingle photons with well-defined arrival times and it cannotbe applied to continuous-wave single-photon sources. Such aproblem could be tackled by replacing the sinusoidal RF waveby a serrodyne wave [36], which would not limit the temporalaperture of the device. The need for a large frequency shear isobvious from Eq. 8, since it provides a larger amplitude of thephase gradient. It also needs to be larger than the resolutionof the spectrometer. This parameter is however also boundedby the unknown state, as any phase structure smaller than theshear cannot be reconstructed. One could, in principle, con-catenate several modulators (or multi-pass a single modulator)to increase the shear. However, this results in further insertionloss, timing jitter and dispersion that will adversely affect thecalibration.

VIII. CONCLUSIONS

In summary, we have demonstrated a self-referencing tech-nique to reconstruct the spectral-temporal component of apulsed single photon wave function. This method is appli-cable across a wide range of wavelengths and pulse charac-teristics, and is deterministic in that only technical losses pre-vent every photon entering the device contributing to the finalreconstruction. The method can be readily extended to thecharacterization of mixed state single-photon sources, as wellas multi-photon states. In its present design our device haswide applications in single-photon source and detector char-acterization and quantum metrology. We anticipate that fu-ture experiments will utilize this technology for more diversepurposes such as spectral-temporal entanglement characteri-zation and state purity determination.

ACKNOWLEDGMENTS

We are grateful to C. Dorrer, C. Radzewicz, M. G. Raymer,and I. A. Walmsley for fruitful discussions. This projecthas received funding from the European Union’s Horizon2020 research and innovation programme under Grant Agree-ment No. 665148, the United Kingdom Defense Science andTechnology Laboratory (DSTL) under contract No. DSTLX-100092545, and the National Science Foundation under GrantNo. 1620822. This work was partially funded by the HOM-ING programme of the Foundation for Polish Science (projectno. Homing/2016-1/4) co-financed by the European Unionunder the European Regional Development Fund.

13

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