Tomography of a Mode-Tunable Coherent Single-Photon Subtractor

Tomography of a Mode-Tunable Coherent Single-Photon Subtractor PHYSICAL REVIEW X 7, 031012 (2017) Young-Sik Ra, Clément Jacquard, Adrien Dufour, Claude Fabre, and Nicolas Treps Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France, 4 place Jussieu, 75252 Paris, France (Received 7 February 2017; revised manuscript received 18 April 2017; published 19 July 2017) Single-photon subtraction plays important roles in optical quantum information processing as it provides a non-Gaussian characteristic in continuous-variable quantum information. While the conventional way of implementing single-photon subtraction based on a low-reflectance beam splitter works properly for a single-mode quantum state, it is unsuitable for a multimode quantum state because a single photon is subtracted from all multiple modes without maintaining their mode coherence. Here, we experimentally implement and characterize a mode-tunable coherent single-photon subtractor based on sum-frequency generation. It can subtract a single photon exclusively from one desired time-frequency mode of light or from a coherent superposition of multiple time-frequency modes. To experimentally characterize the time-frequency modes of the single-photon subtractor, we employ quantum process tomography based on coherent states. The mode-tunable coherent single-photon subtractor will be an essential element for realizing non-Gaussian quantum networks necessary to get a quantum advantage in information processing. DOI: 10.1103/PhysRevX.7.031012 Subject Areas: Photonics, Quantum Physics, Quantum Information I. INTRODUCTION preparation of coherent-state-superposition [16–19] and hybrid entanglement [20,21], noiseless linear amplification Optical quantum information processing can be [22], and entanglement concentration [23,24]. classified mainly into two approaches depending on the The conventional way of implementing the single- encoding of quantum information: One is based on con- photon subtraction is to detect a single photon tapped tinuous electric-field quadratures (thus referred to as off from an input light using a low-reflectance beam splitter continuous-variable quantum information), and the other (BS) [11,25]. Such a method works well for a single-mode is based on discrete photon numbers (discrete-variable state [16,17,19], but it is unsuitable for a multimode state quantum information). Each of the approaches has its because the detected photon comes from any mode in an own advantages compared with the other: e.g., in the incoherent way, which results in a complete mixture of continuous-variable approach, highly multimode entangled annihilation operators over the multiple modes [26].To states can be deterministically generated [1–5], and in the fully benefit from the highly multimode entangled states discrete-variable approach, quantum processes that cannot available in the continuous variable approach [1–5], one be classically simulated can be implemented [6–9]. accordingly requires a single-photon subtraction that is able Therefore, a new approach to combine both advantages to operate only in the desired modes by maintaining their has attracted much attention; it is called hybrid quantum mode coherence [18,23,27]. information processing [10]. One of the fundamental In this work, we implement and characterize a single- operations for the hybrid approach is single-photon sub- photon subtractor which can be tuned to subtract a single traction, mathematically described by the annihilation photon exclusively from one desired time-frequency mode operator a ˆ. It introduces a non-Gaussian characteristic of light or coherently from multiple time-frequency modes. (i.e., negativity of the Wigner function) in continuous- The single-photon subtractor is based on the detection of a variable quantum information [11], which plays essential single photon generated via a sum-frequency interaction roles in various quantum information processing, e.g., between an input beam and a strong gate beam in which universal [12,13] and genuine [14,15] quantum computing, the choice of the gate-beam modes determines the time- frequency modes of single-photon subtraction [28–31]. To characterize single-photon subtractions with various youngsikra@gmail.com choices of the gate beam modes, we measure the subtraction Published by the American Physical Society under the terms of matrix of each single-photon subtraction by employing the Creative Commons Attribution 4.0 International license. coherent-state quantum process tomography [32,33]:the Further distribution of this work must maintain attribution to subtraction matrix contains complete information about a the author(s) and the published article’s title, journal citation, and DOI. general single-photon subtraction (i.e., amplitude, phase, and 2160-3308=17=7(3)=031012(8) 031012-1 Published by the American Physical Society RA, JACQUARD, DUFOUR, FABRE, and TREPS PHYS. REV. X 7, 031012 (2017) coherence between different modes) and can be used to where χ ¼ p c c . It results in the output state ij n n ni nj quantify its performances. We furthermore discuss the S½ρ ˆ=trðS½ρ ˆÞ with success probability proportional to d−1 possible experimental imperfections in a single-photon sub- trðS½ρ ˆÞ ¼ χ ha ˆ a ˆ i. This formalism can also be ij i ij j tractor such as unwanted heralding events (e.g., dark counts obtained from single-photon subtraction based on a multi- or two-photon detection) and optical losses, and estimate mode beam splitter as reported in Ref. [26]. It is important their effect on preparing a non-Gaussian quantum state. to note that a single-photon subtraction S is uniquely determined by the subtraction matrix χ, which is analogous II. DESCRIPTION OF GENERAL to the density matrix representation for a quantum state. SINGLE-PHOTON SUBTRACTION The subtraction matrix is Hermitian and positive semi- definite with a trace of 1, and trðχ Þ quantifies the purity of We start by introducing a formalism describing general the operation, 1=trðχ Þ the effective number of orthogonal single-photon subtraction in multiple modes [26].Ina pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi single-mode case, single-photon subtraction is uniquely modes, and ðtr χμ χÞ the fidelity between two single- defined by the single-photon annihilation operator a ˆ, which photon subtractions described by χ and μ. lowers one excitation of a photon-number state jni∶ a ˆjni¼ As an example, a single-photon subtractor based pffiffiffi njn − 1i. This operation is intrinsically nondeterministic on the conventional method [11,25] makes a com- ðincohÞ (i.e., non-trace-preserving) [34], which succeeds only if a pletely incoherent single-photon subtraction S ½ρ ˆ¼ desired outcome is obtained by measuring an ancillary d−1 ð1=dÞa ˆ ρ ˆa ˆ , which gives rise to the identity subtrac- i¼0 i system [11,25]. In the multimode case, on the other hand, ðincohÞ tion matrix χ ¼ δ =d exhibiting purity of 1=d.On ij ij single-photon subtraction can be diverse because it can the other hand, a coherent single-photon subtraction consist of, for example, one annihilation operator from ðcohÞ d−1 ˆ ˆ ˆ S ½ρ ˆ¼ A ρ ˆA with A ¼ c a ˆ shows the sub- multiple modes or several annihilation operators from 0 0 i i 0 i¼0 ðcohÞ multiple modes, added as a superposition or as a mixture. traction matrix of χ ¼ c c exhibiting purity of 1. ij i j In general, single-photon subtraction can be described by Differently from the incoherent case, the subtraction a mixture of annihilation operators A with weights p ,as n n matrix of a coherent single-photon subtraction contains ðcohÞ shown in Fig. 1, where A can be expressed as a linear nonzero off-diagonal elements χ ≠ 0 for i ≠ j, which ij combination of basis annihilation operators fa ˆ ; a ˆ ;…; 0 1 indicates coherence of single-photon subtraction between a ˆ g in a d-dimensional orthonormal mode basis: d−1 different modes. d−1 A ¼ c a ˆ . The bosonic commutation relation of n ni i i¼0 ˆ ˆ each annihilation operator ½A ; A ¼ 1 dictates that n n III. TOMOGRAPHY OF SINGLE-PHOTON jc j ¼ 1, but different annihilation operators are not i ni SUBTRACTION ˆ ˆ necessarily orthogonal, ½A ; A  ≠ 0. A single-photon sub- n m To experimentally characterize single-photon subtraction, traction S acting on an input state ρ ˆ can then be expressed we employ coherent-state quantum process tomography as a quantum map [32,33]. As an arbitrary quantum state can be expressed d−1 in terms of coherent states (the Glauber-Sudarshan X X † † ˆ ˆ S½ρ ˆ¼ p A ρ ˆA ¼ χ a ˆ ρ ˆa ˆ ; ð1Þ P function) [35,36], any quantum process can be completely n n n ij i j n i;j¼0 characterized by measuring the responses (the output state and the success probability) on various input coherent states. Single-photon subtractor In general, however, characterizing a multimode process Input Output p , A {} n n requires a large number of coherent states, which grows ˆ ˆ p A A n n n n exponentially with the number of modes [33]. For single- photon subtraction, on the other hand, the difficulty of Single photon multimode characterization can be circumvented because a coherent state is an eigenstate of any annihilation operator Detector [37]; i.e., it is not altered by single-photon subtraction. This fact implies that one can get enough information about FIG. 1. A single-photon subtractor removes exactly one photon single-photon subtraction by measuring only the success from an input state ρ ˆ, which is heralded by the detection of a probability without measuring the output state. Let us single photon in the ancillary path. Single-photon subtraction can consider the tensor product of coherent states in different be described, in general, by a mixture of annihilation operators ˆ ˆ modes, jβi¼jβ i jβ i …jβ i , where the ket subscript 0 0 1 1 d−1 d−1 A ; A ;…, with the corresponding weights p ;p ;…, where all 0 1 0 1 denotes a mode for the basis annihilation operators in weights sum to 1, and different annihilation operators are not 2 d−1 2 ˆ ˆ Eq. (1),and jβj ¼ jβ j is the average photon number. necessarily orthogonal, ½A ; A  ≠ 0. The normalization constant n m i¼0 ˆ ˆ When it is used as an input state of single-photon subtraction N is p hA A i, which is proportional to the heralding n n n probability. in Eq. (1), the output state becomes the same as the input 031012-2 TOMOGRAPHY OF A MODE-TUNABLE COHERENT SINGLE- … PHYS. REV. X 7, 031012 (2017) (a) HG modes state, and the success probability is proportional to Nonlinear Input crystal d−1 χ β β . Note that the success probability is inde- ij i SPD i;j¼0 j pendent of the global phase of jβi.Asthe successprobability depends on the subtraction matrix χ, its elements χ can be ij Gate Output Up-converted photon obtained by measuring the probabilities for various input coherent states: We use a coherent state only in the ith mode, gate gate gate (b) jβi , to interrogate diagonal elements χ , and coherent states HG0 ii qffiffi qffiffi HG0 HG1 +HG 1 1 up up up only in i and j modes, j βi j βi (0-phase difference) += i j 2 2 qffiffi qffiffiffiffiffiffi 1 1 and j βi j − βi (π=2-phase difference), to obtain the i j 2 2 input input input (c) real and imaginary values of off-diagonal elements χ ¼ χ , ij ji Femtosecond laser Input respectively. The total number of the measurements is d . Note that it is not necessary to investigate the subtraction NBF SMF SPD NDF PS BS matrix by varying the average photon number jβj because Gate BiBO Output the subtraction matrix χ is independent of the input state. PS Lens IV. IMPLEMENTATION OF A MODE-TUNABLE FIG. 2. A mode-tunable coherent single-photon subtractor. COHERENT SINGLE-PHOTON SUBTRACTOR (a) Conceptual sketch. Detection of the up-converted photon heralds a single-photon subtraction in the input beam, whose We have implemented a mode-tunable coherent single- time-frequency modes are determined by the spectral amplitude photon subtractor for Hermite-Gaussian (HG) time- of the strong gate beam. To characterize the single-photon frequency modes of an input beam based on nonlinear subtraction, weak coherent states are used as the input. Inset: interaction with a strong gate beam, as described in First four HG time-frequency modes, expressed in the wave- Fig. 2(a). Inside a second-order nonlinear crystal, photons length domain. (b) Joint spectral amplitudes of the input and the from the two beams give rise to an up-converted photon via up-converted beams with HG and HG gates. The joint spectral 0 1 sum-frequency generation (SFG). When the up-converted amplitudes can be decomposed into the product of the spectral photon is detected by a single-photon detector (SPD), amplitudes of the input and the up-converted beams, colored in subtraction of a single photon from the input beam is gray. As HG and HG gate beams give rise to the same spectral 0 1 heralded. In the nonlinear conversion process, the joint amplitude for the up-converted beam, the sum of the joint spectral spectral amplitude of the input and the up-converted beams amplitudes by HG and HG gates can also be decomposed into 0 1 the product of the spectral amplitudes of the input and the up- is engineered in such a way that the spectral amplitude converted beams. (c) Experimental setup: θ ¼ 2.5°, single-photon of the gate beam is directly mapped onto the spectral detector (SPD), nonpolarizing BS, pulse shaper (PS), neutral amplitude of the input beam without affecting the spectral density filter (NDF), narrow bandpass filter (NBF), and single- amplitude of the up-converted beam, as shown in Fig. 2(b). mode fiber (SMF). Such a spectral engineering is accomplished by the horizontal alignment of the joint spectral amplitude and by having a much narrower bandwidth for the up-converted ðHGÞ ðcohÞ d−1 i ˆ ˆ ˆ subtraction S ½ρ ˆ¼A ρ ˆA with A ¼ ð−1Þ c a ˆ , beam with respect to the one of the gate beam; the former 0 0 i 0 i¼0 i ðHGÞ condition is satisfied by group velocity matching between where a is the annihilation operator for the ith HG the input and the gate beams [28,30], and the latter mode, and c ð¼ γ =jγjÞ is the normalized coefficient. The i i condition is satisfied by narrow bandwidth phase matching additional coefficient ð−1Þ originates from the wavelength via a thick crystal and/or by narrow bandpass filtering of inversion with respect to the central wavelength by energy the up-converted beam. We can therefore tune the time- conservation of SFG [38], which makes the sign change frequency modes of the single-photon subtraction by only for antisymmetric HG modes. In practice, the single- controlling the gate beam: If the gate is in the ith HG photon subtraction can entail additional annihilation mode, a single photon is subtracted from the same ith HG operators A (e.g., due to a nonideal joint spectral nð≠0Þ mode. In addition, if the gate is in a superposition of amplitude): different HG modes, a single photon is subtracted coher- ently from those HG modes because the spectral amplitude † † ðSFGÞ ˆ ˆ ˆ ˆ S ½ρ ˆ¼ p A ρ ˆA þ p A ρ ˆA ð2Þ of the up-converted beam is independent of the spectral 0 0 n n n n¼1 amplitude of the gate beam [see Fig. 2(b)]. Let us assume that the gate beam is a strong coherent state jγi¼jγ i jγ i …jγ i , where the average with p ¼ 1. The weight of A , i.e., p , is defined as 0 1 d−1 0 1 d−1 n 0 0 n¼0 2 d−1 2 photon number jγj ð¼ jγ j Þ ≫ 1. Detection of an mode selectivity of single-photon subtraction [39], which i¼0 up-converted photon heralds the coherent single-photon becomes unity for the ideal case. 031012-3 RA, JACQUARD, DUFOUR, FABRE, and TREPS PHYS. REV. X 7, 031012 (2017) Figure 2(c) describes the experimental setup developed to V. EXPERIMENTAL RESULTS implement and characterize the mode-tunable coherent We start by implementing the single-photon subtraction single-photon subtractor. A femtosecond laser (central wave- for the HG mode by sending a gate beam in the HG mode 0 0 length, 795 nm; full width at half maximum (FWHM), (central wavelength: 795 nm; FWHM: 4 nm). To represent 11 nm; repetition rate, 76 MHz) is split into input and gate its subtraction matrix, we choose a wavelength-band mode beams at a BS. The spectral amplitudes of the two beams are basis, which consists of 25 different wavelength bands from individually controlled by PS having a spectral resolution 786 nm to 804 nm (see Ref. [40] for their spectrums). We of 0.2 nm. A NDF attenuates the input beam to prepare a characterize the implemented subtraction by using input coherent state, having the average photon number of one per coherent states to the number of 625 in the wavelength- pulse, and the gate beam has 1 mW power (corresponding to band modes, and the average photon number of the input around 5 × 10 photons per pulse). The two beams (beam coherent states is increased up to 90 for fast data acquis- diameter: 1.6 mm) are focused by a single plano-convex lens ition. To construct a physical subtraction matrix (positive (focal length: 190 mm) onto a bismuth borate (BiBO) bulk and semidefinite), we have employed the maximum like- crystal (thickness: 2.5 mm), which generates frequency up- lihood technique [41] for all the following tomography converted light (central wavelength: 397.5 nm; FWHM: results. Figure 3(a) shows the obtained subtraction matrix 0.6 nm) via SFG. The phase difference between the beams by using a HG gate beam. Note that not only diagonal need not be fixed as it does not affect the operation of terms but also off-diagonal terms exist around a 795-nm the single-photon subtractor. To achieve a high mode wavelength, manifesting coherent single-photon subtrac- selectivity, the group velocities of the beams inside the tion from different wavelength-band modes; the imaginary crystal are matched by using the same central wavelength part of the matrix shows negligibly small values because and the same polarization [30], and a narrow bandwidth of the phase is almost zero over all the wavelength-band the up-converted light is generated by a phase matching modes. The dominant eigenvalue of the subtraction matrix, via the thick crystal, which is further filtered by a narrow obtained via diagonalization, corresponds to the mode bandpass filter (FWHM: 0.4 nm). The up-converted light selectivity p . Since this value is close to 1, we can is then collected into a single mode fiber, and it is detected associate the corresponding eigenvector with the dominant by an on-off type SPD (Hamamatsu C13001-01; quantum single-photon annihilation operator A . This is shown in the efficiency: 40%; dark count rate: 10 Hz). To measure the last row of Fig. 3(a), which agrees well with the spectral success probability of single-photon subtraction for the amplitude of the gate beam and shows a high mode quantum process tomography, we record the count rates selectivity and purity [42]. of the SPD with various input coherent states as described We next tune the single-photon subtractor by adjusting in Sec. III. the spectral amplitude of the gate beam. Figure 3(b) shows (a) HG (b) HG (c) HG (d) HG + HG (e) HG + i HG 0 1 2 0 1 0 1 786 795 804 786 795 804 786 795 804 786 795 804 786 795 804 786 786 786 786 786 Re[ ] 795 795 795 795 795 0.2 0.1 804 804 804 804 804 786 786 786 786 786 0.1 0.2 Im[ ] 795 795 795 795 795 804 804 804 804 804 Prob. Phase Prob. Phase Prob. Phase Prob. Phase Prob. Phase p0=0.92 p0=0.92 p0=0.90 p 0=0.92 p 0=0.85 0.2 0.2 0.2 0.2 0.2 s=0.86 s=0.84 s=0.81 s=0.84 s=0.73 0.1 0 0.1 0 0.1 0 0.1 0 0.1 0 A0 0 - 0 - 0 - 0 - 0 - 786 795 804 786 795 804 786 795 804 786 795 804 786 795 804 Wavelength (nm) Wavelength (nm) Wavelength (nm) Wavelength (nm) Wavelength (nm) FIG. 3. Tomography of single-photon subtraction based on 25 wavelength-band modes. The first, second, and third rows are real and imaginary parts of the subtraction matrix χ, and the mode of the dominant annihilation operator A , respectively. In the third row, bars and points represent the probability and phase of each wavelength band, respectively, and the line is a visual guide. Note that p is mode selectivity, and s is purity. 031012-4 TOMOGRAPHY OF A MODE-TUNABLE COHERENT SINGLE- … PHYS. REV. X 7, 031012 (2017) the subtraction matrix obtained by using a HG gate beam. whose measured spectrums are provided in Ref. [40].For The two negative areas in the real part are due to the sign the characterization, we use up to 49 input coherent states difference of the HG mode with respect to the central by maintaining the average photon number of one per wavelength [see the inset of Fig. 2(a)]; this also confirms pulse. For a HG gate beam, the subtraction matrix, shown the coherence between the longer and the shorter wave- in Fig. 4(a), has its dominant element in the HG mode. It ðHGÞ length parts. The sign change also appears as the π-phase also exhibits high fidelity with the ideal operation a ˆ as jump in the dominant annihilation operator A , shown in 0 well as a high mode selectivity and purity. As the gate mode the last row of Fig. 3(b). Similarly, we implement and is shifted to higher order, the dominant element in the characterize single-photon subtraction for the HG mode, 2 subtraction matrix is shifted accordingly [see Fig. 4(b) and shown in Fig. 3(c). Figures 3(d) and 3(e) are obtained by Ref. [40]]. When the gate beam is in a superposition sending a gate beam in a superposition of HG and HG 0 1 of HG and HG , a coherent single-photon subtraction 0 1 modes, (d) with 0-phase difference and (e) with π=2-phase takes place, as shown in Fig. 4(c) for the same phase and ðHGÞ Fig. 4(d) for π=2-phase difference between the two HG difference. As HG implements −a ˆ , as discussed in modes. The off-diagonal elements between HG and HG Sec. IV, the sum of HG and HG gate modes with 0 1 0 1 pffiffiffi ðHGÞ ðHGÞ ðHGÞ ðHGÞ modes clearly show the coherence between a ˆ and a ˆ 0-phase difference implements ð1= 2Þða ˆ − a ˆ Þ, 0 1 0 1 and the tunability of their relative phase. The fidelities which makes the subtraction matrix distributed at lower pffiffiffi ðHGÞ ðHGÞ wavelengths than the central wavelength. The π=2-phase with the ideal operations ð1= 2Þða ˆ − a ˆ Þ and 0 1 pffiffiffi ðHGÞ ðHGÞ difference between the gate modes results in imaginary ð1= 2Þða ˆ − ia ˆ Þ, respectively, are also high. The 0 1 values in the subtraction matrix because of the phase single-photon subtractor can also be tuned to act on difference between wavelength-band modes. We provide pffiffiffi P ðHGÞ 4 i multiple HG modes coherently, ð1= 5Þ ð−1Þ a ˆ i¼0 i additional subtraction matrices using different gate beams pffiffiffi ðHGÞ 6 i in Ref. [40]. in Fig. 4(e) and ð1= 7Þ ð−1Þ a ˆ in Fig. 4(f), i¼0 i As our single-photon subtractor is designed for para- respectively. To investigate the independence of the sub- metric multimode sources [3,43], we now characterize it traction matrix on the input state, we characterize the with a mode basis approximating the eigenmodes of this single-photon subtraction for Fig. 4(f) using input coherent process: HG modes fHG ; HG ;…; HG g. The input and states with much higher average photon number amounting 0 1 6 the gate beams are based on the same HG-mode basis to 90. The obtained matrix, shown in Fig. 4(g), is almost (central wavelength: 795 nm; FWHM of HG : 4 nm), identical to the subtraction matrix measured by an average (a) (b) (c) (d) HG HG HG + HG HG +i HG 0 1 0 1 0 1 2 2 6 6 Re[ ] Re[ ] 1 1 5 5 1 1 4 4 Re[ ] Re[ ] 0 0 3 3 2 2 -1 0 -1 0 1 1 1 1 1 0 2 2 0 0 2 2 Im[ ] Im[ ] 0 0 1 1 1 1 1 1 2 2 -1 -1 0 0 3 3 0 0 4 4 5 5 F=0.98 F=0.94 -1 0 -1 0 6 6 1 1 p0=0.98, s=0.96 p0=0.95, s=0.90 2 2 F=0.96 F=0.93 p =0.96, s=0.93 p =0.96, s=0.92 0 0 (e) (f) (g) HG + … + HG HG + … + HG HG + … + HG n ˆ = 90 0 4 0 6 0 6 6 6 6 5 5 5 4 4 4 Re[ ] Re[ ] Re[ ] 3 3 3 2 2 2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2 2 2 -1 -1 -1 3 3 3 4 4 4 5 5 5 F=0.94 F=0.96 F=0.97 6 6 6 p0=0.96, s=0.92 p0=0.98, s=0.96 p0=0.98, s=0.96 FIG. 4. Tomography of single-photon subtraction based on seven HG modes. An index in the horizontal plane denotes the order of a HG mode from 0 to 6. In diagrams (a, b, e, f, g), we present only the real part of the subtraction matrix χ as the imaginary part is negligibly small. For the same reason, we present only a truncated part of a matrix in diagrams (c) and (d). The average photon number of the probe beam is 1 in diagrams (a–f) and 90 in diagram (g). Note that F is fidelity with the ideal subtraction matrix, p is mode selectivity, and s is purity. See Ref. [40] for the full data. 031012-5 RA, JACQUARD, DUFOUR, FABRE, and TREPS PHYS. REV. X 7, 031012 (2017) Initial Single-photon Final photon number of 1 in Fig. 4(f), exhibiting fidelity of 0.99 loss subtraction loss between them. We provide additional subtraction matrices (in) (fi) Input Output using different gate beams in Ref. [40]. VI. DISCUSSION FIG. 5. The input state sequentially experiences initial loss ðinÞ ðfiÞ L , realistic single-photon subtraction R, and final loss L . We discuss here the possible imperfections of the single- photon subtractor by taking into account undesired herald- ing events. In practice, a single click by an on-off SPD does ðinÞ ðfiÞ modes, L ¼ B ⊗ B ⊗ … and L ¼ B ⊗ ðinÞ ðinÞ ðfiÞ T T T not always herald single-photon subtraction because it may ðinÞ ðfiÞ B ðfiÞ ⊗ …, where T (T ) is the transmittance of a originate from an accidental event by detector dark counts fictitious beam splitter for initial (final) loss. If the input or detection of two photons [25]. A realistic single-photon state is a multimode state ρ ˆ ¼ σ ˆ ⊗ σ ˆ ⊗ …, which consists subtraction R is then described as of identically squeezed vacuum σ ˆ in each mode [3], the final quantum state reduced to the dominant subtraction R½ρ ˆ¼ w ρ ˆ þ w S½ρ ˆþ w S½S½ρ ˆ; ð3Þ 0 1 2 mode reads which results in the output state R½ρ ˆ=trðR½ρ ˆÞ with the ðfiÞ ðinÞ tr ðL ½R½L ½ρ ˆÞ ðfiÞ 12… success probability proportional to trðR½ρ ˆÞ. The first term ρ ˆ ≡ ðfiÞ ðinÞ tr ðL ½R½L ½ρ ˆÞ represents the identity operation due to an accidental click, 012… the middle term is the desired single-photon subtraction S † ˆ ˆ A B ½σ ˆA ðovrÞ T 0 ðfalseÞ ðcorrÞ ¼ r B ½σ ˆþ r ; ð4Þ in Eq. (1), and the last term is the double application of the ðovrÞ ˆ ˆ trðA B ðovrÞ½σ ˆA Þ T 0 single-photon subtraction due to two-photon detection. Therefore, their respective weights w , w , and w ð¼ 1 − 0 1 2 ðovrÞ ðfiÞ ðinÞ where T ð¼ T T Þ is the overall transmittance of the w − w Þ are an important factor to assess the quality of the 0 1 entire setup, and the two-photon detection weight w of R single-photon subtractor. These weights can be measured is set to zero, as it is negligible. The first term, B ½σ ˆ,is a ðovrÞ using input coherent states. If a coherent state jβi in the squeezed vacuum mixed with the vacuum noise heralded dominant subtraction mode is used, the success probability by a false click in R. It originates from the accidental of the operation is proportional to trðR½ρ ˆÞ ¼ w þ click (quantified by w ¼ 1 − w ) and single-photon click 2 2 4 0 1 w p jβj þ w p jβj ; thus, measuring the success proba- 1 0 2 from other modes (quantified by mode selectivity p ) and bility with respect to jβj can reveal the weights w , w , and 0 1 ðfalseÞ ðinÞ has a ratio of r ¼ f½ð1 − w Þþ w ð1 − p ÞT hn ˆi = 1 1 0 σ ˆ w . Note that the mode selectivity p can be obtained 2 0 ðinÞ ½ð1 − w Þþ w T hn ˆi Þg, where hn ˆi is the average 1 1 σ ˆ σ ˆ through the tomography method presented in Secs. III photon number of σ ˆ. The second term is the single-photon and V. The implemented single-photon subtractor exhibits subtracted state from B ½σ ˆ heralded by a correct click ðovrÞ a dominating contribution of single-photon subtraction in R, which originates from single-photon subtraction (w ¼ 0.99), a very small contribution of the identity exclusively from the dominant subtraction mode; it has a operation (w ¼ 0.01), and negligible two-photon subtrac- ðcorrÞ ðinÞ ðinÞ −3 ratio of r ¼fðw p T hn ˆi Þ=½ð1−w Þþw T hn ˆi g. 1 0 1 1 σ ˆ σ ˆ tion (w < 10 ) (see Ref. [40] for the experimental data). Based on the characteristics of the implemented single- The significant suppression of two-photon subtraction is photon subtractor (w ¼ 0.99, p ¼ 0.9) and the typical −3 1 0 due to a low conversion ratio (10 ) of the input beam to the experimental conditions (initial and final losses of 10%, up-converted beam for a 1-mW gate beam, which still respectively, which incorporate 2% optical loss of the provides a moderate heralding rate of around 2 kHz, with implemented single-photon subtractor), one can estimate an input state with an average photon number of 1. that a non-Gaussian state exhibiting a negativity of the Based on this realistic model of single-photon subtrac- Wigner function amounting to −ð0.3=2πÞ can be obtained tion, we can estimate its performance (e.g., negativity of from an input state of 4-dB multimode squeezed the Wigner function) in a general experimental condition vacua [44]. including losses. Figure 5 depicts the sequence of oper- ðinÞ ations: Initial loss L accounts for imperfection of VII. CONCLUSIONS quantum state preparation (e.g., excess noise of squeezed vacuum) and the propagation loss before single-photon We have experimentally implemented a mode-tunable ðfiÞ subtraction, and final loss L accounts for the propagation coherent single-photon subtractor and characterized it by loss after single-photon subtraction and the inefficiency of employing coherent-state quantum process tomography. quantum state measurement (e.g., homodyne detection). We could readily tune the time-frequency modes of single- Such optical losses can be modeled as a coupling with photon subtraction by adjusting the spectral modes of the vacuum by a fictitious beam splitter B (transmittance: T). gate beam, which does not require a physical reconstruction For simplicity, let us consider homogeneous loss for all the of a mode-coupling device [6–9]. 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Tomography of a Mode-Tunable Coherent Single-Photon Subtractor

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PHYSICAL REVIEW X 7, 031012 (2017) Young-Sik Ra, Clément Jacquard, Adrien Dufour, Claude Fabre, and Nicolas Treps Laboratoire Kastler Brossel, UPMC-Sorbonne Universités, CNRS, ENS-PSL Research University, Collège de France, 4 place Jussieu, 75252 Paris, France (Received 7 February 2017; revised manuscript received 18 April 2017; published 19 July 2017) Single-photon subtraction plays important roles in optical quantum information processing as it provides a non-Gaussian characteristic in continuous-variable quantum information. While the conventional way of implementing single-photon subtraction based on a low-reflectance beam splitter works properly for a single-mode quantum state, it is unsuitable for a multimode quantum state because a single photon is subtracted from all multiple modes without maintaining their mode coherence. Here, we experimentally implement and characterize a mode-tunable coherent single-photon subtractor based on sum-frequency generation. It can subtract a single photon exclusively from one desired time-frequency mode of light or from a coherent superposition of multiple time-frequency modes. To experimentally characterize the time-frequency modes of the single-photon subtractor, we employ quantum process tomography based on coherent states. The mode-tunable coherent single-photon subtractor will be an essential element for realizing non-Gaussian quantum networks necessary to get a quantum advantage in information processing. DOI: 10.1103/PhysRevX.7.031012 Subject Areas: Photonics, Quantum Physics, Quantum Information I. INTRODUCTION preparation of coherent-state-superposition [16–19] and hybrid entanglement [20,21], noiseless linear amplification Optical quantum information processing can be [22], and entanglement concentration [23,24]. classified mainly into two approaches depending on the The conventional way of implementing the single- encoding of quantum information: One is based on con- photon subtraction is to detect a single photon tapped tinuous electric-field quadratures (thus referred to as off from an input light using a low-reflectance beam splitter continuous-variable quantum information), and the other (BS) [11,25]. Such a method works well for a single-mode is based on discrete photon numbers (discrete-variable state [16,17,19], but it is unsuitable for a multimode state quantum information). Each of the approaches has its because the detected photon comes from any mode in an own advantages compared with the other: e.g., in the incoherent way, which results in a complete mixture of continuous-variable approach, highly multimode entangled annihilation operators over the multiple modes [26].To states can be deterministically generated [1–5], and in the fully benefit from the highly multimode entangled states discrete-variable approach, quantum processes that cannot available in the continuous variable approach [1–5], one be classically simulated can be implemented [6–9]. accordingly requires a single-photon subtraction that is able Therefore, a new approach to combine both advantages to operate only in the desired modes by maintaining their has attracted much attention; it is called hybrid quantum mode coherence [18,23,27]. information processing [10]. One of the fundamental In this work, we implement and characterize a single- operations for the hybrid approach is single-photon sub- photon subtractor which can be tuned to subtract a single traction, mathematically described by the annihilation photon exclusively from one desired time-frequency mode operator a ˆ. It introduces a non-Gaussian characteristic of light or coherently from multiple time-frequency modes. (i.e., negativity of the Wigner function) in continuous- The single-photon subtractor is based on the detection of a variable quantum information [11], which plays essential single photon generated via a sum-frequency interaction roles in various quantum information processing, e.g., between an input beam and a strong gate beam in which universal [12,13] and genuine [14,15] quantum computing, the choice of the gate-beam modes determines the time- frequency modes of single-photon subtraction [28–31]. To characterize single-photon subtractions with various youngsikra@gmail.com choices of the gate beam modes, we measure the subtraction Published by the American Physical Society under the terms of matrix of each single-photon subtraction by employing the Creative Commons Attribution 4.0 International license. coherent-state quantum process tomography [32,33]:the Further distribution of this work must maintain attribution to subtraction matrix contains complete information about a the author(s) and the published article’s title, journal citation, and DOI. general single-photon subtraction (i.e., amplitude, phase, and 2160-3308=17=7(3)=031012(8) 031012-1 Published by the American Physical Society RA, JACQUARD, DUFOUR, FABRE, and TREPS PHYS. REV. X 7, 031012 (2017) coherence between different modes) and can be used to where χ ¼ p c c . It results in the output state ij n n ni nj quantify its performances. We furthermore discuss the S½ρ ˆ=trðS½ρ ˆÞ with success probability proportional to d−1 possible experimental imperfections in a single-photon sub- trðS½ρ ˆÞ ¼ χ ha ˆ a ˆ i. This formalism can also be ij i ij j tractor such as unwanted heralding events (e.g., dark counts obtained from single-photon subtraction based on a multi- or two-photon detection) and optical losses, and estimate mode beam splitter as reported in Ref. [26]. It is important their effect on preparing a non-Gaussian quantum state. to note that a single-photon subtraction S is uniquely determined by the subtraction matrix χ, which is analogous II. DESCRIPTION OF GENERAL to the density matrix representation for a quantum state. SINGLE-PHOTON SUBTRACTION The subtraction matrix is Hermitian and positive semi- definite with a trace of 1, and trðχ Þ quantifies the purity of We start by introducing a formalism describing general the operation, 1=trðχ Þ the effective number of orthogonal single-photon subtraction in multiple modes [26].Ina pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi single-mode case, single-photon subtraction is uniquely modes, and ðtr χμ χÞ the fidelity between two single- defined by the single-photon annihilation operator a ˆ, which photon subtractions described by χ and μ. lowers one excitation of a photon-number state jni∶ a ˆjni¼ As an example, a single-photon subtractor based pffiffiffi njn − 1i. This operation is intrinsically nondeterministic on the conventional method [11,25] makes a com- ðincohÞ (i.e., non-trace-preserving) [34], which succeeds only if a pletely incoherent single-photon subtraction S ½ρ ˆ¼ desired outcome is obtained by measuring an ancillary d−1 ð1=dÞa ˆ ρ ˆa ˆ , which gives rise to the identity subtrac- i¼0 i system [11,25]. In the multimode case, on the other hand, ðincohÞ tion matrix χ ¼ δ =d exhibiting purity of 1=d.On ij ij single-photon subtraction can be diverse because it can the other hand, a coherent single-photon subtraction consist of, for example, one annihilation operator from ðcohÞ d−1 ˆ ˆ ˆ S ½ρ ˆ¼ A ρ ˆA with A ¼ c a ˆ shows the sub- multiple modes or several annihilation operators from 0 0 i i 0 i¼0 ðcohÞ multiple modes, added as a superposition or as a mixture. traction matrix of χ ¼ c c exhibiting purity of 1. ij i j In general, single-photon subtraction can be described by Differently from the incoherent case, the subtraction a mixture of annihilation operators A with weights p ,as n n matrix of a coherent single-photon subtraction contains ðcohÞ shown in Fig. 1, where A can be expressed as a linear nonzero off-diagonal elements χ ≠ 0 for i ≠ j, which ij combination of basis annihilation operators fa ˆ ; a ˆ ;…; 0 1 indicates coherence of single-photon subtraction between a ˆ g in a d-dimensional orthonormal mode basis: d−1 different modes. d−1 A ¼ c a ˆ . The bosonic commutation relation of n ni i i¼0 ˆ ˆ each annihilation operator ½A ; A ¼ 1 dictates that n n III. TOMOGRAPHY OF SINGLE-PHOTON jc j ¼ 1, but different annihilation operators are not i ni SUBTRACTION ˆ ˆ necessarily orthogonal, ½A ; A  ≠ 0. A single-photon sub- n m To experimentally characterize single-photon subtraction, traction S acting on an input state ρ ˆ can then be expressed we employ coherent-state quantum process tomography as a quantum map [32,33]. As an arbitrary quantum state can be expressed d−1 in terms of coherent states (the Glauber-Sudarshan X X † † ˆ ˆ S½ρ ˆ¼ p A ρ ˆA ¼ χ a ˆ ρ ˆa ˆ ; ð1Þ P function) [35,36], any quantum process can be completely n n n ij i j n i;j¼0 characterized by measuring the responses (the output state and the success probability) on various input coherent states. Single-photon subtractor In general, however, characterizing a multimode process Input Output p , A {} n n requires a large number of coherent states, which grows ˆ ˆ p A A n n n n exponentially with the number of modes [33]. For single- photon subtraction, on the other hand, the difficulty of Single photon multimode characterization can be circumvented because a coherent state is an eigenstate of any annihilation operator Detector [37]; i.e., it is not altered by single-photon subtraction. This fact implies that one can get enough information about FIG. 1. A single-photon subtractor removes exactly one photon single-photon subtraction by measuring only the success from an input state ρ ˆ, which is heralded by the detection of a probability without measuring the output state. Let us single photon in the ancillary path. Single-photon subtraction can consider the tensor product of coherent states in different be described, in general, by a mixture of annihilation operators ˆ ˆ modes, jβi¼jβ i jβ i …jβ i , where the ket subscript 0 0 1 1 d−1 d−1 A ; A ;…, with the corresponding weights p ;p ;…, where all 0 1 0 1 denotes a mode for the basis annihilation operators in weights sum to 1, and different annihilation operators are not 2 d−1 2 ˆ ˆ Eq. (1),and jβj ¼ jβ j is the average photon number. necessarily orthogonal, ½A ; A  ≠ 0. The normalization constant n m i¼0 ˆ ˆ When it is used as an input state of single-photon subtraction N is p hA A i, which is proportional to the heralding n n n probability. in Eq. (1), the output state becomes the same as the input 031012-2 TOMOGRAPHY OF A MODE-TUNABLE COHERENT SINGLE- … PHYS. REV. X 7, 031012 (2017) (a) HG modes state, and the success probability is proportional to Nonlinear Input crystal d−1 χ β β . Note that the success probability is inde- ij i SPD i;j¼0 j pendent of the global phase of jβi.Asthe successprobability depends on the subtraction matrix χ, its elements χ can be ij Gate Output Up-converted photon obtained by measuring the probabilities for various input coherent states: We use a coherent state only in the ith mode, gate gate gate (b) jβi , to interrogate diagonal elements χ , and coherent states HG0 ii qffiffi qffiffi HG0 HG1 +HG 1 1 up up up only in i and j modes, j βi j βi (0-phase difference) += i j 2 2 qffiffi qffiffiffiffiffiffi 1 1 and j βi j − βi (π=2-phase difference), to obtain the i j 2 2 input input input (c) real and imaginary values of off-diagonal elements χ ¼ χ , ij ji Femtosecond laser Input respectively. The total number of the measurements is d . Note that it is not necessary to investigate the subtraction NBF SMF SPD NDF PS BS matrix by varying the average photon number jβj because Gate BiBO Output the subtraction matrix χ is independent of the input state. PS Lens IV. IMPLEMENTATION OF A MODE-TUNABLE FIG. 2. A mode-tunable coherent single-photon subtractor. COHERENT SINGLE-PHOTON SUBTRACTOR (a) Conceptual sketch. Detection of the up-converted photon heralds a single-photon subtraction in the input beam, whose We have implemented a mode-tunable coherent single- time-frequency modes are determined by the spectral amplitude photon subtractor for Hermite-Gaussian (HG) time- of the strong gate beam. To characterize the single-photon frequency modes of an input beam based on nonlinear subtraction, weak coherent states are used as the input. Inset: interaction with a strong gate beam, as described in First four HG time-frequency modes, expressed in the wave- Fig. 2(a). Inside a second-order nonlinear crystal, photons length domain. (b) Joint spectral amplitudes of the input and the from the two beams give rise to an up-converted photon via up-converted beams with HG and HG gates. The joint spectral 0 1 sum-frequency generation (SFG). When the up-converted amplitudes can be decomposed into the product of the spectral photon is detected by a single-photon detector (SPD), amplitudes of the input and the up-converted beams, colored in subtraction of a single photon from the input beam is gray. As HG and HG gate beams give rise to the same spectral 0 1 heralded. In the nonlinear conversion process, the joint amplitude for the up-converted beam, the sum of the joint spectral spectral amplitude of the input and the up-converted beams amplitudes by HG and HG gates can also be decomposed into 0 1 the product of the spectral amplitudes of the input and the up- is engineered in such a way that the spectral amplitude converted beams. (c) Experimental setup: θ ¼ 2.5°, single-photon of the gate beam is directly mapped onto the spectral detector (SPD), nonpolarizing BS, pulse shaper (PS), neutral amplitude of the input beam without affecting the spectral density filter (NDF), narrow bandpass filter (NBF), and single- amplitude of the up-converted beam, as shown in Fig. 2(b). mode fiber (SMF). Such a spectral engineering is accomplished by the horizontal alignment of the joint spectral amplitude and by having a much narrower bandwidth for the up-converted ðHGÞ ðcohÞ d−1 i ˆ ˆ ˆ subtraction S ½ρ ˆ¼A ρ ˆA with A ¼ ð−1Þ c a ˆ , beam with respect to the one of the gate beam; the former 0 0 i 0 i¼0 i ðHGÞ condition is satisfied by group velocity matching between where a is the annihilation operator for the ith HG the input and the gate beams [28,30], and the latter mode, and c ð¼ γ =jγjÞ is the normalized coefficient. The i i condition is satisfied by narrow bandwidth phase matching additional coefficient ð−1Þ originates from the wavelength via a thick crystal and/or by narrow bandpass filtering of inversion with respect to the central wavelength by energy the up-converted beam. We can therefore tune the time- conservation of SFG [38], which makes the sign change frequency modes of the single-photon subtraction by only for antisymmetric HG modes. In practice, the single- controlling the gate beam: If the gate is in the ith HG photon subtraction can entail additional annihilation mode, a single photon is subtracted from the same ith HG operators A (e.g., due to a nonideal joint spectral nð≠0Þ mode. In addition, if the gate is in a superposition of amplitude): different HG modes, a single photon is subtracted coher- ently from those HG modes because the spectral amplitude † † ðSFGÞ ˆ ˆ ˆ ˆ S ½ρ ˆ¼ p A ρ ˆA þ p A ρ ˆA ð2Þ of the up-converted beam is independent of the spectral 0 0 n n n n¼1 amplitude of the gate beam [see Fig. 2(b)]. Let us assume that the gate beam is a strong coherent state jγi¼jγ i jγ i …jγ i , where the average with p ¼ 1. The weight of A , i.e., p , is defined as 0 1 d−1 0 1 d−1 n 0 0 n¼0 2 d−1 2 photon number jγj ð¼ jγ j Þ ≫ 1. Detection of an mode selectivity of single-photon subtraction [39], which i¼0 up-converted photon heralds the coherent single-photon becomes unity for the ideal case. 031012-3 RA, JACQUARD, DUFOUR, FABRE, and TREPS PHYS. REV. X 7, 031012 (2017) Figure 2(c) describes the experimental setup developed to V. EXPERIMENTAL RESULTS implement and characterize the mode-tunable coherent We start by implementing the single-photon subtraction single-photon subtractor. A femtosecond laser (central wave- for the HG mode by sending a gate beam in the HG mode 0 0 length, 795 nm; full width at half maximum (FWHM), (central wavelength: 795 nm; FWHM: 4 nm). To represent 11 nm; repetition rate, 76 MHz) is split into input and gate its subtraction matrix, we choose a wavelength-band mode beams at a BS. The spectral amplitudes of the two beams are basis, which consists of 25 different wavelength bands from individually controlled by PS having a spectral resolution 786 nm to 804 nm (see Ref. [40] for their spectrums). We of 0.2 nm. A NDF attenuates the input beam to prepare a characterize the implemented subtraction by using input coherent state, having the average photon number of one per coherent states to the number of 625 in the wavelength- pulse, and the gate beam has 1 mW power (corresponding to band modes, and the average photon number of the input around 5 × 10 photons per pulse). The two beams (beam coherent states is increased up to 90 for fast data acquis- diameter: 1.6 mm) are focused by a single plano-convex lens ition. To construct a physical subtraction matrix (positive (focal length: 190 mm) onto a bismuth borate (BiBO) bulk and semidefinite), we have employed the maximum like- crystal (thickness: 2.5 mm), which generates frequency up- lihood technique [41] for all the following tomography converted light (central wavelength: 397.5 nm; FWHM: results. Figure 3(a) shows the obtained subtraction matrix 0.6 nm) via SFG. The phase difference between the beams by using a HG gate beam. Note that not only diagonal need not be fixed as it does not affect the operation of terms but also off-diagonal terms exist around a 795-nm the single-photon subtractor. To achieve a high mode wavelength, manifesting coherent single-photon subtrac- selectivity, the group velocities of the beams inside the tion from different wavelength-band modes; the imaginary crystal are matched by using the same central wavelength part of the matrix shows negligibly small values because and the same polarization [30], and a narrow bandwidth of the phase is almost zero over all the wavelength-band the up-converted light is generated by a phase matching modes. The dominant eigenvalue of the subtraction matrix, via the thick crystal, which is further filtered by a narrow obtained via diagonalization, corresponds to the mode bandpass filter (FWHM: 0.4 nm). The up-converted light selectivity p . Since this value is close to 1, we can is then collected into a single mode fiber, and it is detected associate the corresponding eigenvector with the dominant by an on-off type SPD (Hamamatsu C13001-01; quantum single-photon annihilation operator A . This is shown in the efficiency: 40%; dark count rate: 10 Hz). To measure the last row of Fig. 3(a), which agrees well with the spectral success probability of single-photon subtraction for the amplitude of the gate beam and shows a high mode quantum process tomography, we record the count rates selectivity and purity [42]. of the SPD with various input coherent states as described We next tune the single-photon subtractor by adjusting in Sec. III. the spectral amplitude of the gate beam. Figure 3(b) shows (a) HG (b) HG (c) HG (d) HG + HG (e) HG + i HG 0 1 2 0 1 0 1 786 795 804 786 795 804 786 795 804 786 795 804 786 795 804 786 786 786 786 786 Re[ ] 795 795 795 795 795 0.2 0.1 804 804 804 804 804 786 786 786 786 786 0.1 0.2 Im[ ] 795 795 795 795 795 804 804 804 804 804 Prob. Phase Prob. Phase Prob. Phase Prob. Phase Prob. Phase p0=0.92 p0=0.92 p0=0.90 p 0=0.92 p 0=0.85 0.2 0.2 0.2 0.2 0.2 s=0.86 s=0.84 s=0.81 s=0.84 s=0.73 0.1 0 0.1 0 0.1 0 0.1 0 0.1 0 A0 0 - 0 - 0 - 0 - 0 - 786 795 804 786 795 804 786 795 804 786 795 804 786 795 804 Wavelength (nm) Wavelength (nm) Wavelength (nm) Wavelength (nm) Wavelength (nm) FIG. 3. Tomography of single-photon subtraction based on 25 wavelength-band modes. The first, second, and third rows are real and imaginary parts of the subtraction matrix χ, and the mode of the dominant annihilation operator A , respectively. In the third row, bars and points represent the probability and phase of each wavelength band, respectively, and the line is a visual guide. Note that p is mode selectivity, and s is purity. 031012-4 TOMOGRAPHY OF A MODE-TUNABLE COHERENT SINGLE- … PHYS. REV. X 7, 031012 (2017) the subtraction matrix obtained by using a HG gate beam. whose measured spectrums are provided in Ref. [40].For The two negative areas in the real part are due to the sign the characterization, we use up to 49 input coherent states difference of the HG mode with respect to the central by maintaining the average photon number of one per wavelength [see the inset of Fig. 2(a)]; this also confirms pulse. For a HG gate beam, the subtraction matrix, shown the coherence between the longer and the shorter wave- in Fig. 4(a), has its dominant element in the HG mode. It ðHGÞ length parts. The sign change also appears as the π-phase also exhibits high fidelity with the ideal operation a ˆ as jump in the dominant annihilation operator A , shown in 0 well as a high mode selectivity and purity. As the gate mode the last row of Fig. 3(b). Similarly, we implement and is shifted to higher order, the dominant element in the characterize single-photon subtraction for the HG mode, 2 subtraction matrix is shifted accordingly [see Fig. 4(b) and shown in Fig. 3(c). Figures 3(d) and 3(e) are obtained by Ref. [40]]. When the gate beam is in a superposition sending a gate beam in a superposition of HG and HG 0 1 of HG and HG , a coherent single-photon subtraction 0 1 modes, (d) with 0-phase difference and (e) with π=2-phase takes place, as shown in Fig. 4(c) for the same phase and ðHGÞ Fig. 4(d) for π=2-phase difference between the two HG difference. As HG implements −a ˆ , as discussed in modes. The off-diagonal elements between HG and HG Sec. IV, the sum of HG and HG gate modes with 0 1 0 1 pffiffiffi ðHGÞ ðHGÞ ðHGÞ ðHGÞ modes clearly show the coherence between a ˆ and a ˆ 0-phase difference implements ð1= 2Þða ˆ − a ˆ Þ, 0 1 0 1 and the tunability of their relative phase. The fidelities which makes the subtraction matrix distributed at lower pffiffiffi ðHGÞ ðHGÞ wavelengths than the central wavelength. The π=2-phase with the ideal operations ð1= 2Þða ˆ − a ˆ Þ and 0 1 pffiffiffi ðHGÞ ðHGÞ difference between the gate modes results in imaginary ð1= 2Þða ˆ − ia ˆ Þ, respectively, are also high. The 0 1 values in the subtraction matrix because of the phase single-photon subtractor can also be tuned to act on difference between wavelength-band modes. We provide pffiffiffi P ðHGÞ 4 i multiple HG modes coherently, ð1= 5Þ ð−1Þ a ˆ i¼0 i additional subtraction matrices using different gate beams pffiffiffi ðHGÞ 6 i in Ref. [40]. in Fig. 4(e) and ð1= 7Þ ð−1Þ a ˆ in Fig. 4(f), i¼0 i As our single-photon subtractor is designed for para- respectively. To investigate the independence of the sub- metric multimode sources [3,43], we now characterize it traction matrix on the input state, we characterize the with a mode basis approximating the eigenmodes of this single-photon subtraction for Fig. 4(f) using input coherent process: HG modes fHG ; HG ;…; HG g. The input and states with much higher average photon number amounting 0 1 6 the gate beams are based on the same HG-mode basis to 90. The obtained matrix, shown in Fig. 4(g), is almost (central wavelength: 795 nm; FWHM of HG : 4 nm), identical to the subtraction matrix measured by an average (a) (b) (c) (d) HG HG HG + HG HG +i HG 0 1 0 1 0 1 2 2 6 6 Re[ ] Re[ ] 1 1 5 5 1 1 4 4 Re[ ] Re[ ] 0 0 3 3 2 2 -1 0 -1 0 1 1 1 1 1 0 2 2 0 0 2 2 Im[ ] Im[ ] 0 0 1 1 1 1 1 1 2 2 -1 -1 0 0 3 3 0 0 4 4 5 5 F=0.98 F=0.94 -1 0 -1 0 6 6 1 1 p0=0.98, s=0.96 p0=0.95, s=0.90 2 2 F=0.96 F=0.93 p =0.96, s=0.93 p =0.96, s=0.92 0 0 (e) (f) (g) HG + … + HG HG + … + HG HG + … + HG n ˆ = 90 0 4 0 6 0 6 6 6 6 5 5 5 4 4 4 Re[ ] Re[ ] Re[ ] 3 3 3 2 2 2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2 2 2 -1 -1 -1 3 3 3 4 4 4 5 5 5 F=0.94 F=0.96 F=0.97 6 6 6 p0=0.96, s=0.92 p0=0.98, s=0.96 p0=0.98, s=0.96 FIG. 4. Tomography of single-photon subtraction based on seven HG modes. An index in the horizontal plane denotes the order of a HG mode from 0 to 6. In diagrams (a, b, e, f, g), we present only the real part of the subtraction matrix χ as the imaginary part is negligibly small. For the same reason, we present only a truncated part of a matrix in diagrams (c) and (d). The average photon number of the probe beam is 1 in diagrams (a–f) and 90 in diagram (g). Note that F is fidelity with the ideal subtraction matrix, p is mode selectivity, and s is purity. See Ref. [40] for the full data. 031012-5 RA, JACQUARD, DUFOUR, FABRE, and TREPS PHYS. REV. X 7, 031012 (2017) Initial Single-photon Final photon number of 1 in Fig. 4(f), exhibiting fidelity of 0.99 loss subtraction loss between them. We provide additional subtraction matrices (in) (fi) Input Output using different gate beams in Ref. [40]. VI. DISCUSSION FIG. 5. The input state sequentially experiences initial loss ðinÞ ðfiÞ L , realistic single-photon subtraction R, and final loss L . We discuss here the possible imperfections of the single- photon subtractor by taking into account undesired herald- ing events. In practice, a single click by an on-off SPD does ðinÞ ðfiÞ modes, L ¼ B ⊗ B ⊗ … and L ¼ B ⊗ ðinÞ ðinÞ ðfiÞ T T T not always herald single-photon subtraction because it may ðinÞ ðfiÞ B ðfiÞ ⊗ …, where T (T ) is the transmittance of a originate from an accidental event by detector dark counts fictitious beam splitter for initial (final) loss. If the input or detection of two photons [25]. A realistic single-photon state is a multimode state ρ ˆ ¼ σ ˆ ⊗ σ ˆ ⊗ …, which consists subtraction R is then described as of identically squeezed vacuum σ ˆ in each mode [3], the final quantum state reduced to the dominant subtraction R½ρ ˆ¼ w ρ ˆ þ w S½ρ ˆþ w S½S½ρ ˆ; ð3Þ 0 1 2 mode reads which results in the output state R½ρ ˆ=trðR½ρ ˆÞ with the ðfiÞ ðinÞ tr ðL ½R½L ½ρ ˆÞ ðfiÞ 12… success probability proportional to trðR½ρ ˆÞ. The first term ρ ˆ ≡ ðfiÞ ðinÞ tr ðL ½R½L ½ρ ˆÞ represents the identity operation due to an accidental click, 012… the middle term is the desired single-photon subtraction S † ˆ ˆ A B ½σ ˆA ðovrÞ T 0 ðfalseÞ ðcorrÞ ¼ r B ½σ ˆþ r ; ð4Þ in Eq. (1), and the last term is the double application of the ðovrÞ ˆ ˆ trðA B ðovrÞ½σ ˆA Þ T 0 single-photon subtraction due to two-photon detection. Therefore, their respective weights w , w , and w ð¼ 1 − 0 1 2 ðovrÞ ðfiÞ ðinÞ where T ð¼ T T Þ is the overall transmittance of the w − w Þ are an important factor to assess the quality of the 0 1 entire setup, and the two-photon detection weight w of R single-photon subtractor. These weights can be measured is set to zero, as it is negligible. The first term, B ½σ ˆ,is a ðovrÞ using input coherent states. If a coherent state jβi in the squeezed vacuum mixed with the vacuum noise heralded dominant subtraction mode is used, the success probability by a false click in R. It originates from the accidental of the operation is proportional to trðR½ρ ˆÞ ¼ w þ click (quantified by w ¼ 1 − w ) and single-photon click 2 2 4 0 1 w p jβj þ w p jβj ; thus, measuring the success proba- 1 0 2 from other modes (quantified by mode selectivity p ) and bility with respect to jβj can reveal the weights w , w , and 0 1 ðfalseÞ ðinÞ has a ratio of r ¼ f½ð1 − w Þþ w ð1 − p ÞT hn ˆi = 1 1 0 σ ˆ w . Note that the mode selectivity p can be obtained 2 0 ðinÞ ½ð1 − w Þþ w T hn ˆi Þg, where hn ˆi is the average 1 1 σ ˆ σ ˆ through the tomography method presented in Secs. III photon number of σ ˆ. The second term is the single-photon and V. The implemented single-photon subtractor exhibits subtracted state from B ½σ ˆ heralded by a correct click ðovrÞ a dominating contribution of single-photon subtraction in R, which originates from single-photon subtraction (w ¼ 0.99), a very small contribution of the identity exclusively from the dominant subtraction mode; it has a operation (w ¼ 0.01), and negligible two-photon subtrac- ðcorrÞ ðinÞ ðinÞ −3 ratio of r ¼fðw p T hn ˆi Þ=½ð1−w Þþw T hn ˆi g. 1 0 1 1 σ ˆ σ ˆ tion (w < 10 ) (see Ref. [40] for the experimental data). Based on the characteristics of the implemented single- The significant suppression of two-photon subtraction is photon subtractor (w ¼ 0.99, p ¼ 0.9) and the typical −3 1 0 due to a low conversion ratio (10 ) of the input beam to the experimental conditions (initial and final losses of 10%, up-converted beam for a 1-mW gate beam, which still respectively, which incorporate 2% optical loss of the provides a moderate heralding rate of around 2 kHz, with implemented single-photon subtractor), one can estimate an input state with an average photon number of 1. that a non-Gaussian state exhibiting a negativity of the Based on this realistic model of single-photon subtrac- Wigner function amounting to −ð0.3=2πÞ can be obtained tion, we can estimate its performance (e.g., negativity of from an input state of 4-dB multimode squeezed the Wigner function) in a general experimental condition vacua [44]. including losses. Figure 5 depicts the sequence of oper- ðinÞ ations: Initial loss L accounts for imperfection of VII. CONCLUSIONS quantum state preparation (e.g., excess noise of squeezed vacuum) and the propagation loss before single-photon We have experimentally implemented a mode-tunable ðfiÞ subtraction, and final loss L accounts for the propagation coherent single-photon subtractor and characterized it by loss after single-photon subtraction and the inefficiency of employing coherent-state quantum process tomography. quantum state measurement (e.g., homodyne detection). We could readily tune the time-frequency modes of single- Such optical losses can be modeled as a coupling with photon subtraction by adjusting the spectral modes of the vacuum by a fictitious beam splitter B (transmittance: T). gate beam, which does not require a physical reconstruction For simplicity, let us consider homogeneous loss for all the of a mode-coupling device [6–9]. 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