Time-bin to Polarization Conversion of Ultrafast Photonic Qubits
Connor Kupchak, Philip J. Bustard, Khabat Heshami, Jennifer Erskine, Michael Spanner, Duncan G. England, Benjamin J. Sussman
aa r X i v : . [ qu a n t - ph ] N ov Time-bin to Polarization Conversion of Ultrafast Photonic Qubits
Connor Kupchak,
1, 2
Philip J. Bustard, Khabat Heshami, Jennifer Erskine,
1, 2
Michael Spanner, Duncan G. England, and Benjamin J. Sussman
1, 2 Department of Physics, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada National Research Council of Canada, 100 Sussex Drive, Ottawa, Ontario, K1A 0R6, Canada
The encoding of quantum information in photonic time-bin qubits is apt for long distance quantumcommunication schemes. In practice, due to technical constraints such as detector response time,or the speed with which co-polarized time-bins can be switched, other encodings, e.g. polarization,are often preferred for operations like state detection. Here, we present the conversion of qubitsbetween polarization and time-bin encodings using a method that is based on an ultrafast opticalKerr shutter and attain efficiencies of 97% and an average fidelity of 0.827 ± The encoding of quantum information (QI) into pho-tons holds much promise in numerous future technolo-gies. The QI can be mapped onto various degrees of free-dom that are used as basis-states. One attractive optionis to encode onto qubits composed of two co-polarizedbut temporally distinct wave packets, or time-bins; thesebasis states are often labelled by their arrival time as early ( | e i ) and late ( | l i ) [1]. Time-bin encodings have re-cently been used in the successful transmission of qubitsover hundreds of kilometers [2], and in teleportation us-ing real-world fiber networks [3, 4].The disadvantage is that direct readout of informationencoded in time-bins can require the peak-to-peak sepa-ration ∆ τ el between | e i and | l i to be sufficiently greaterthan the response time of the detector, and can imposea minimum time for the bin separation. Typical detec-tor response times correspond to a bin separation of atleast nanoseconds [5]; this limits the available bandwidthfor encoding and can necessitate active interferometricstabilization when preparing and detecting qubits andqudits [6]. Recently, advanced methods have emergedthat utilize nonlinear techniques to creatively detect lightstates encoded in temporal modes [7, 8], however theseimplementations are constrained to operate at low effi-ciencies.Polarization encoding is a popular choice for various QIapplications [9–11] but can be problematic for long dis-tance implementations [12–14]. Ideally one would havethe flexibility to convert arbitrary photonic states be-tween encodings depending on the application, e.g. atime-bin encoding for transmission and a polarizationencoding for state detection and manipulation. Manyprevious schemes for time-bin to polarization qubit con-version are lossy and rely on post-selection using passiveoptics [15–18]. Another approach could rely on activeswitches involving Pockels cells or electro-optical mod-ulation to convert between encodings by rotating thepolarization state of a single bin [19]. For these ac-tive implementations, the rise time of the device lim- its the temporal separation of the bins and restricts thedata transfer rate. Typical switching devices that sellcommercially, have rise times on the order of nanosec-onds, and shorter times of <
100 ps are achievable innon-commercial waveguides [20], but usually exhibit in-sertion losses of 1-3 dB. Recently, all-optical solutionsbased on cross-phase modulation (XPM) for convertingtime-bins between encodings have been developed thatcan switch as fast as 50 ps [21]. It is therefore desir-able to progress these all-optical conversion methods tohigher bandwidths and operational speeds in the ultra-fast regime.Here, we realize an efficient scheme for the conversionof qubits between time-bin and polarization encodings,and demonstrate its potential using ultrafast laser pulsesattenuated to the single-photon level. Our approachis reversible, and capable of bandwidths greater than200 GHz. Devices of this functionality may find use inhigh-bandwidth quantum communication networks andenable the interfacing of time-bin qubits with ultrafastquantum memories [22] in local QI processing.Our scheme is based on the optical Kerr effect: in-duced birefringence in a χ (3) nonlinear medium which isproportional to the irradiance of an applied pump field.We use this effect to map photons between polarizationstates. Typically, a χ (3) medium is placed between twoaxis-crossed polarizers so that the input probe light isblocked except in the presence of an applied pump field;such a setup is referred to as an optical Kerr shutter(OKS) [23–25]. The shutter efficiency η is given by [26] η = sin (2 θ ) sin (cid:18) ∆ φ (cid:19) , (1)where ∆ φ = 2 πn L eff Iλ probe (2)is the phase shift induced by the pump field of intensity I , n is the nonlinear component of the refractive index, λ /4 λ /2 λ /2
45º Polarizer
PolarizationQubit Preparation PulseSeparation Rotate to CommonPolarization
Time-Bin Qubit (a) λ /2 λ /4
0º PolarizerPulseOverlap Polarization Analysis
Input Time-BinQubit (b)
FIG. 1: (a) Schematic diagram for time-bin qubit preparation starting with a polarization state and (b) conversion of time-binqubits to the polarization degree of freedom using the OKS; (c) corresponding experimental setup. L eff is the effective length of the medium, λ probe is thewavelength of the probe field, and θ is the polarizationangle between the pump and probe field. The case of∆ φ = π corresponds to the probe field undergoing a full90 ◦ polarization rotation i.e. horizontal flipped to verti-cal.Our experimental scheme for the qubit conversion pro-cess can be divided into two main parts displayed inFigs. 1(a)-(b). First, the time-bin qubit preparationstage shown in Fig. 1(a) where qubits are initially en-coded into polarization states using a half-waveplate(HWP) and quarter-waveplate (QWP) combination. Thepolarization qubits then enter a birefringent medium totemporally separate the horizontal and vertical compo-nents; establishing the | e i and | l i time bins, respec-tively. In our setup, a 10 mm long α -BBO crystalsets the separation between the | e i and | l i time-binsto be ∆ τ el =4.3 ps. After this stage, the qubits passthrough a polarizer set to transmit diagonal linear po-larization (45 ◦ ) resulting in a 50% loss and an addi-tional HWP to prepare time-bins that are horizontallyco-polarized.The second part, shown in Fig. 1(b), is the procedurefor converting time-bins to a polarization encoding. Todo so, we temporally overlap the pump field with the | l i time-bin and focus both fields into the Kerr medium torotate the polarization of the | l i bin from a horizontal tovertical polarization due to the OKS operation. This isfollowed by transmission through a second, identical α -BBO crystal with its axis rotated by 90 ◦ with respect tothe first, such that the now orthogonally polarized timebins are overlapped into a single temporal bin. This com-pletes the mapping of the qubits to a polarization-basedencoding that is suited for measurement and manipula-tion by common polarization state analysis techniques.Note that the OKS could also be implemented to per-form the reverse operation i.e. from a polarization totime-bin encoding. In this case, a polarization qubit isfirst sent through a birefringent material to achieve tem-poral mode separation of the polarization states followedby an OKS operation on the | l i time-bin. A diagram of our experimental setup is given inFig. 1(c). The pump beam is derived from a 1 kHzrepetition rate, chirped pulsed amplifier laser emittingpulses with a 90 fs duration at a wavelength of 800 nm.The probe field is generated by splitting-off a portion ofthe original pump pulse for use in a white light sourcegenerated in sapphire [27]. Before collinear combinationon a non-polarizing beam splitter (NPBS), both pumpand probe beams are spectrally tailored using indepen-dent 4f-shapers constructed with adjustable razor bladesat the focal plane to serve as a mask [28]. The probebeam is set to a central wavelength of 710 nm and band-width of ∆ λ probe =5.7 nm with a full width at half max-imum (FWHM) duration of ∼
270 fs while a narrowbandpump beam is created by filtering to a top-hat-shapedspectrum of ∆ λ pump =1.8 nm and a pulse duration of∆ τ pump ∼ h i -cut, 8 mmlong yttrium aluminum garnet (YAG) crystal of no in-herent birefringence, chosen for its relatively high n value [29]. The probe field is spatially filtered to achievea Gaussian spatial mode with a beam waist in the focalplane of 20 µ m compared to 60 µ m for the pump. A set ofwaveplates and a polarizing beam splitter (PBS) are situ-ated after the second α -BBO crystal for polarization stateprojection; this is succeeded by a series of spectral filtersto extinguish the pump field and permit measurementat the single-photon level via coupling to an avalanchephotodiode (APD) using a single-mode fiber (SMF). Single time-bin OKS operation.-
First, we character-ize the efficiency of our OKS operation using attenuatedpulses defined in a single temporal bin. Here, we set theenergy of the pump pulse to 840 nJ and fix the polariza-tion to 45 ◦ (diagonal) with respect to the horizontally po-larized time-bin. To investigate single-photon-level con-ditions, the mean photon number h n i of our probe pulseis set at 1.17 ± τ del with respect to the probe and settingthe analysis optics to transmit a vertical polarization, weidentify the peak shutter efficiency (Fig. 2(a)). (a) (b) Experimental Fit
Average Input"Switched" Probe, Pump OnExperimental Fit Probe Input, Pump OffNoise, Pump On dd − − Delay (fs) C oun t s / sec
20 40 60 80 100 120 140 160020406080100120
Polarization Angle (deg) C oun t s / sec d d FIG. 2: The OKS operating on a single time bin with(a) counts per second measured as the pump pulse delayis scanned. The counts corresponding to the transmitted,polarization-rotated, probe pulse (blue circles) with fit (bluedashed line) are compared to the original, horizontally polar-ized input pulses (black squares) and their mean (solid line)and the noise counts (red circles). (b) Dependence of theOKS switching on the relative polarization angle θ , betweenthe pump and probe pulses and corresponding fit. The errorbars are derived from Poisson statistics. Under these conditions, we observe near-perfect po-larization rotation of the probe pulse from horizontal tovertical with a peak shutter efficiency of η = 0 . ± . N OKS , the noisecounts due to the pump beam N noise , and the counts cor-responding to the original input pulse N input such that η = ( N OKS − N noise ) /N input . It is also important to attaina sufficiently high signal-to-noise-ratio (SNR) in order todistinguish the state of the qubit. A SNR of 9 . ± . N noise ; thisvalue is comparable to other quantum channels designedfor time-bin qubits [30]. From the fit in Fig. 2(a), we canalso evaluate the operating speed of the OKS and finda FWHM of ∆ τ OKS = 0 . ± .
01 ps. Combined with∆ τ el , this establishes a potential bandwidth of our de-vice of over 200 GHz when operating on THz-bandwidthphotons. Note that the sin (∆ φ/
2) response of the OKS(see Eq. (1)) yields a FWHM that is less than the pumppulse duration [31].In order to verify the OKS operation with respect tothe polarization angle θ , the pump polarization is rotatedover a range of 180 ◦ and the corresponding polarization-rotated probe pulses are collected on the APD. Here, thepump pulse energy remains at 840 nJ and the temporaldelay between the pulses is fixed to zero. As can be seenin Fig. 2(b), along with the expected sin (2 θ ) behaviorin accordance with Eq. (2), the noise counts follow thepolarization of the pump field. Lastly, we characterize the performance of the OKSas a function of the energy of the pump pulse (Fig. 3).For this analysis, we fix the pump delay and polarizationto the optimal values and measure the OKS efficiencyand noise. From Fig. 3 it is clear that a range of op-timal pump energies emerge between 800-900 nJ wherethe OKS efficiency approaches 100% and the noise pho-ton rate remains low enough to yield a SNR of ∼
10. Atenergies greater than this range we observe a sharp, non-linear increase in noise photons that can be attributedto spectral broadening of the pump pulse by self-phasemodulation (SPM) in the YAG crystal. N o i se C oun t s / sec Pump Pulse Energy ( µ J) E ff i c i e n cy EfficiencyNoise Counts
FIG. 3: OKS efficiency (left ordinate: green squares) andnoise counts per second (right ordinate: red circles) with re-spect to the energy of the pump pulse. Error bars on thenoise counts are ∼ ±
Time-bin to polarization qubit conversion.-
With theOKS operation characterized, we turn our attention tomapping time-bin qubits to a polarization encoding ac-cording to the scheme depicted in Fig. 1. The inputtime-bin qubits are prepared with a mean photon num-ber of h n i = 0 . ± .
06 and the pump pulse energy set to825 nJ. A mean photon number of ∼ ±
1% when also including the fiber coupling efficiency,APD response and transmission losses.To quantify the performance of our qubit conversionscheme we perform quantum process tomography [33] onthe converted time-bin qubits. Process tomography is ac-complished by using our preparation waveplates to gener-ate 6 input polarization states ( | H i , | V i , | D i = √ ( | H i + | V i ), | A i = √ ( | H i − | V i ), | R i = √ ( | H i + i | V i ), | L i = √ ( | H i− i | V i ) where H indicates horizontal polar-ization and V vertical polarization; these form three mu-tually unbiased bases in the qubit Hilbert space. These Im (a) (b) (c) I X Y ZI X Y Z00.20.40.60.81 00.20.40.60.81 I X Y ZI X Y Z00.20.40.60.81 00.20.40.60.81
750 800 850 9000.650.70.750.80.85
Pump Pulse Energy (nJ) A v e r age F i de li t y FIG. 4: (a) Real and (b) imaginary components of the experimentally reconstructed process tensor (filled bars) in the basisof Pauli matrices as compared to the ideal process tensor values (wire-grid). On-diagonal elements represent components foreach of the Pauli operators { , X, Y, Z } . (c) Behavior of the average fidelity (black circles) at pump energies near the optimumcompared to the classical thresholds; this is 2/3 for the case of single photons (dotted line) and 0.70 (dashed line) for the meanphoton number and efficiency used in our study. six states are first converted to their corresponding time-bin counterparts (i.e. | H i goes to | e i and | V i goes to | l i ) as shown in Fig. 1(a). Upon conversion back to thepolarization degree of freedom by the OKS scheme inFig. 1(b), the output state is projected onto all six po-larization states using the analyzer waveplates. This 36-element data set forms an over-complete basis and allowsus to experimentally reconstruct the process tensor χ exp in the Pauli operator basis σ i=1..4 ≡ { , X, Y, Z } , similarto previous work [34]. Proper conversion corresponds toan identity operator that is defined by unity at the ( , )element and zeros otherwise.Fig. 4(a-b) shows the real and imaginary componentsof the experimentally-reconstructed process matrix forthe qubit conversion, where elements of χ exp determine acompletely positive map E ( ρ in ) = P χ exp ij σ i ρ in σ j = ρ out that characterizes the quantum channel. The fidelityof the reconstructed process matrix χ exp compared tothe ideal case χ is calculated by F proc ( χ , χ exp ) = (cid:0) Tr p √ χ χ exp √ χ (cid:1) to produce a process fidelity of0.740 ± F avg =(2 F proc +1) / F avg =0.827 ± | l i bin and leads to an unintended ratio of horizontal and vertical components. As a result, thecombined temporal mode contains an improper polar-ization when projected onto a measurement basis. Fur-thermore, the remaining, non-polarization rotated pho-tons in the | l i bin can also be erroneously recorded onthe APD and reduce the visibility between orthogonalstates. At the higher pump energies, the increase in self-phase modulation-related [27] noise reduces the abilityto correctly discriminate the polarization state, therebydecreasing the SNR and fidelity. In future implementa-tions, the fidelity of the process could be increased by us-ing shorter probe wavelengths due to the reduction in thepump energy needed to achieve ∆ φ = π . Here, the SNRwould likewise increase with spectral separation betweenpump and probe fields due to fewer SPM noise photonscreated at the probe wavelength. Overall improvementsto our scheme are possible by using anti-reflective coat-ings on the faces of the YAG sample and by decreasingthe pump power required for conversion by moving to afiber system.In summary, we present a platform for ultrafast polar-ization rotation that enables conversion of qubits betweentime-bin and polarization encodings. The technique isreversible, highly efficient, and leaves the spectrum ofthe photon unchanged and thus adds a valuable toolto the suite of ultrafast protocols designed to measuretime-bin qubits [7, 8]. Our switch operates at picosecondtimescales to allow time-bin encodings that are ordersof magnitude faster than typical detector response times( >
100 ps) and permits high-bandwidth quantum com-munication without requiring complex stabilized interfer-ometers [37]. In addition to communications, we expectour scheme to find applications in photonic quantum in-formation processing, such as linear quantum computingin a single spatial mode [38] and to offer a path towardsarchitectures with hybridized encodings and higher di-mensional quantum states that can benefit from efficientand ultrafast operations. Beyond quantum optics, ourOKS properties could be applied to areas where efficient,ultrafast switching of weak signals at low noise would beof benefit, for example time-resolved spectroscopy [31, 39]or nonlinear microscopy [40, 41]. Our study of the noiseprocesses in the OKS at the single photon level provide abenchmark for these applications. Implementation of ourapproach in a waveguide will enable low-power operationand integration into more compact setups for a range ofphotonic applications.
Funding Information
This work is supported by the Natural Sciences andEngineering Research Council of Canada.
Acknowledgments
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