Characterization of conditional state-engineering quantum processes by coherent state quantum process tomography

Optics is an ideal platform for encoding, manipulating and transmitting quantum information—photons do not interact strongly with the environment enabling effectively long coherence times that can be harnessed for long-distance communications [1] and there is extremely low thermal excitation of the optical field at room temperature effectively eliminating background noise. However, the first apparent advantage also poses a serious challenge, namely that photon–photon interactions—which are crucial for conditional logic operations such as two-qubit gates—are extremely weak and require currently infeasible optical non-linear optical interaction strength [2, 3]. In 2001 Knill et al proposed the concept of linear optics quantum computation [4]. In this approach, a large probabilistic nonlinearity could be invoked with the aid of ancilla photons and projective measurements—thus enabling implementation of photonic logic gates.

Outside the realm of quantum computation, similar probabilistic interactions can be used to arbitrarily manipulate the modal properties [5] or photon-number statistics of light [6]. The basic building block of most schemes is the post-selected beam splitter [7, 8]. Such schemes employ a non-classical ancilla state to generate entanglement between the output modes of a beam splitter [9]. Performing a measurement on one output mode of the beam splitter projects the other mode into a state that depends on the specific measurement outcome. This concept has given rise to a host of quantum optical state engineering protocols including photon subtraction [10, 11], photon addition [12, 13], cat-state generation [14, 15], quantum scissors [16], photon catalysis [17, 18], noiseless amplification [19–21], entanglement distillation [22, 23] and Fock state filtration (FSF) [24, 25]. A thorough review of such photon-level manipulations on travelling modes of light can be found in [26].

Experimentally, photon-level manipulations of a single optical mode are impeded by non-ideal ancilla states, poor overlap between ancilla and target modes, and limited availability of low-loss photon-number-resolving detection. In this article a conditional quantum process representative of such measurement-induced non-linear operations is experimentally investigated. A model of the specific scheme, known as FSF [24, 25], is developed in the presence of experimental imperfections, which is then compared with experimental characterization of a realistic implementation via quantum process tomography. We begin in section 2, by developing a realistic model of the FSF process that incorporates three central experimental imperfections—non-number resolving herald detection, mixed state ancilla, and mode mismatch between ancilla and target modes of the FSF. The experimental implementation is then described in section 3, in which avalanche photodiode detectors (APD) and heralded single photons are used to implement the filter. Then in section 4, the approach to characterize the FSF with coherent-state quantum process tomography (csQPT) is detailed and the results are compared with those predicted by the model developed. In section 5 we summarize the results and draw conclusions that indicate directions for future research.

FSF was proposed by Sanaka et al in 2006 [24] and is schematically identical to the single-photon catalysis demonstrated by Lvovsky et al in 2002 [27]. A simplified depiction of the ideal FSF quantum operation is shown in figure 1(a). An arbitrary pure input quantum state $|\psi \rangle$ is combined with an ancilla single photon $|1\rangle$ at a beam splitter of power reflectivity R. One output mode of the beam splitter is registered by a photon-number-resolving detector. Only events corresponding to detection of one photon in this output conditioning mode are accepted; whereby the output state $|\psi ^{\prime} \rangle$ is generated in the other mode. Thus detection of a single-photon in the conditioning mode heralds the success of the filter operation. The filtered state $|\psi ^{\prime} \rangle$ contains a 'hole' [28] in the photon-number distribution for particular choices of the beam splitter reflectivity R [24].

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The term catalysis is sometimes used to describe such an arrangement because internal to the operation, an ancilla single photon is both utilized in the process and subsequently released upon successful implementation of the process heralded by detection of a single photon in the trigger mode of the beam splitter output [18, 27, 29]. However, owing to quantum interference between the possible paths leading to a single-photon in the conditioning mode, the input state $|\psi \rangle$ is transformed in a non-trivial manner to a different state $|\psi ^{\prime} \rangle$. FSF and the more general catalysis variants [18, 27] have been shown to be applicable to generate a range of non-classical states of light including photon-number states [30] and Schrödinger cat states [18, 31]. Furthermore, it has been proposed that information can be encoded and stored within 'holes' in the photon-number distribution [32]. Such optical states with one or more 'holes' in their photon-number distribution are necessarily non-classical [28], opening a new platform to study non-classical behaviour. Finally, FSF has been demonstrated as a technique for generating entanglement [25] and may be useful as a non-Gaussian operation [33], a central resource for continuous-variable quantum information processing [34, 35], when photon subtraction does not suffice.

For the ideal Fock state filter a pure input state $|\psi \rangle$ in the photon-number basis transforms as

$$\begin{eqnarray}&&|\psi \rangle =\mathop{\sum }\limits_{n=0}^{\infty }{{C}_{n}}|n\rangle \Rightarrow \mathcal{N}\mathop{\sum }\limits_{n=0}^{\infty }{{R}^{(n-1)/2}}\left[ R-n(1-R) \right]{{C}_{n}}|n\rangle =|\psi ^{\prime} \rangle ,\end{eqnarray} \tag{ 1 }$$

where $\mathcal{N}$ is a renormalization factor associated with the probabilistic nature of the filter operation. There are two important properties of the state transformation in equation (1). For certain values of the beam splitter reflectivity R the probability amplitude to have n photons in the output state $|\psi ^{\prime} \rangle$ is equal to zero. This occurs for values of R such that $R=n/(n+1)$. Thus FSF can selectively remove a particular Fock layer $|n\rangle$ from the input state by choosing the beam splitter reflectivity appropriately. Hence FSF can be thought of as a conditional n-photon absorber [24, 25], reflecting the measurement induced nonlinearity of this operation. Furthermore, the output state $|\psi ^{\prime} \rangle$ cannot contain any population in Fock layers which were not populated in the original state. Thus the FSF operation cannot re-populate Fock layers which have been filtered. This property enables preparation of Fock states from input coherent or thermal states by implementing a series of FSF operations with different beam splitter reflectivities—successively filtering out all but one targeted Fock layer [30].

Ideal operation of the filter requires a pure ancilla single-photon state perfectly matched to the optical mode to be filtered and conditioning with a perfect photon-number-resolving detector. However, both are currently challenging to requirements to meet in the laboratory. Motivated by this we develop a more realistic model of the FSF operation taking into account three key factors impacting the performance of FSF. Development of such a realistic model allows diagnosis of sources of imperfection in a realistic device exploiting such quantum operations. Figure 1(b) shows the realistic FSF operation including the three sources of imperfection: (1) projection onto one photon in the heralding mode is replaced by a more general positive operator-valued measure (POVM) element describing the conditioning event denoted ${{\hat{\Pi }}_{\checkmark }}$, assumed to be diagonal in the photon-number basis; (2) impurity of the ancilla single-photon state in terms of admixture of the vacuum state; (3) the multimode nature of the heralding detector. These are addressed in turn in the proceeding sections, culminating in a realistic model for FSF.

2.1. Realistic heralding detector

Ideally the filter only operates when precisely one photon is present in the heralding output mode of the beam splitter, figure 1(b), which corresponds to a successful heralding event being associated with projection onto a single-photon state with a POVM element of the form ${{\hat{\Pi }}_{\checkmark }}=|1\rangle \langle 1|$. To understand the effect of a general, diagonal POVM element of the form ${{\hat{\Pi }}_{\checkmark }}={{\sum }_{g}}{{\theta }_{g}}|g\rangle \langle g|$ on the filter operation, input–output matrix elements of the FSF beam splitter with the form $\langle n+1\;-g,g|\hat{U}|n,1\rangle$ are considered. Here $\hat{U}$ is the unitary transformation associated with the beam splitter [8], $|n,1\rangle$ is the input state to the beam splitter with an n-photon state $|n\rangle$ in the target mode and a single-photon state $|1\rangle$ in the ancilla mode, and $|n+1-g,g\rangle$ is the ouptut state of the beam splitter with g photons in the trigger mode and $n+1-g$ in the heralded mode. This matrix element gives the probability amplitude to have g photons in the heralding mode and therefore by the unitarity of the process, which preserves photon number, $n+1-g$ photons in the filter output mode, given n photons are incident in the filter input mode in addition to the ancilla single photon. The probability amplitude for this scenario, denoted $A_{n+1\,g,g}^{\prime }$ is given by

$$\begin{eqnarray}\begin{array}{rcl} A_{n+1-g,g}^{\prime } & = & \langle n+1\;-g,g|\hat{U}|n,1\rangle \\ {} & = & \sqrt{\frac{(n-g+1)!g!}{n!}}{{R}^{(n-g)/2}}{{({\rm i}\sqrt{1-R})}^{g-1}}\left[ \left( \frac{n}{g-1} \right)R-(1-R)\left( \frac{n}{g} \right) \right]. \\ \end{array}\end{eqnarray} \tag{ 2 }$$

When the conditioning detector is capable of resolving photon number, projecting the trigger mode onto a state $|g\rangle$ with $g\geqslant 1$ photons and neglecting dark counts, an arbitrary input superposition state will transform as

$$\begin{eqnarray}&&|\psi \rangle =\mathop{\sum }\limits_{n=0}^{\infty }{{C}_{n}}|n\rangle \Rightarrow \mathcal{N}\mathop{\sum }\limits_{n=g-1}^{\infty }{{C}_{n}}A_{n+g-1,g}^{\prime }|n+1-g\rangle =|\psi ^{\prime} \rangle .\end{eqnarray} \tag{ 3 }$$

However, for a measurement operator of the form ${{\hat{\Pi }}_{\checkmark }}={{\sum }_{g}}{{\theta }_{g}}|g\rangle \langle g|$, a pure input state is transformed to a mixed state. More generally, an impure input state $\hat{\rho }$ transforms as

$$\begin{eqnarray}\begin{array}{rcl} {\hat{\rho }} & = & \mathop{\sum }\limits_{m,n=0}^{\infty }{{C}_{mn}}|m\rangle \langle n|\Rightarrow \mathop{\sum }\limits_{g=1}^{\infty }{{\theta }_{g}}\mathop{\sum }\limits_{m,n=g-1}^{\infty } \\ {} & {} & {{C}_{mn}}A_{m+g-1,g}^{\prime }A^{\prime} _{{ng+1-g}^{*}}|m+1-g\rangle \langle n+1-g|=\hat{\rho }^{\prime} , \\ \end{array}\end{eqnarray} \tag{ 4 }$$

which is nothing but a mixture of transformations of the form given by equation (3), weighted by the coefficients ${{\theta }_{g}}$ describing the heralding POVM element in the Fock basis.

In general a quantum process can be uniquely described by a rank-4 tensor, which relates the matrix elements in the Fock basis of the input and output states through the relation

$$\begin{eqnarray}&&{{[{{\rho }_{{\rm out}}}]}_{jk}}=\mathop{\sum }\limits_{m,n}\mathcal{E}_{jk}^{mn}{{\rho }_{mn}}\end{eqnarray} \tag{ 5 }$$

where the input density operator $\hat{\rho }$ in the Fock basis, represented by the matrix ${{\rho }_{mn}}$, is mapped onto an output density operators ${{[{{\rho }_{{\rm out}}}]}_{jk}}$ through the action of the quantum process.

The process tensor describing the state transformation in equation (4) can be derived by considering the action of the process on input coherent states [36]. The process tensor for FSF, denoted $({{\mathcal{E}}_{1}})_{jk}^{mn}$, when employing a general non-number-resolving phase-insensitive heralding detector POVM is given by

$$\begin{eqnarray}\begin{array}{rcl} ({{\mathcal{E}}_{1}})_{jk}^{mn} & = & \sqrt{\frac{j!k!}{m!n!}}\mathop{\sum }\limits_{g=1}^{\infty }{{\theta }_{g}}g!{{R}^{(j+k-2)/2}}{{(1-R)}^{(g-1)}} \\ {} & {} & \times \left[ \left( \frac{j+g-1}{g-1} \right)R-(1-R)\left( \frac{j+g+1}{g} \right) \right] \\ {} & {} & \times \left[ \left( \frac{k+g-1}{g-1} \right)R-(1-R)\left( \frac{k+g-1}{g} \right) \right] \\ {} & {} & \times {{\delta }_{m,j+g-1}}{{\delta }_{n,k+g-1}}. \\ \end{array}\end{eqnarray} \tag{ 6 }$$

2.2. Ancilla state heralding efficiency

In the realistic FSF operation, depicted in figure 1(b), the ancilla state, ${{\hat{\rho }}_{{\rm a}}}$, is modelled as a mixture of the single-photon and vacuum states such that

$$\begin{eqnarray}&&{{\hat{\rho }}_{{\rm a}}}=(1-{{\eta }_{{\rm H}}})|0\rangle \langle 0|+{{\eta }_{{\rm H}}}|1\rangle \langle 1|,\end{eqnarray} \tag{ 7 }$$

where ${{\eta }_{{\rm H}}}$ is the ancilla single-photon state heralding efficiency—a commonly used parameter that describes the efficiency with which a single photon heralded from, for example, a spontaneous parametric down-conversion (SPDC) source, can be matched to a desired optical mode [37], which here is the mode of the input state. To model the effect of non-unit heralding efficiency we take account of the heralding events which occur when the ancilla state is in the vacuum state. The probability amplitudes are derived for input Fock states and are given by

$$\begin{eqnarray}&&A_{n-g,g}^{\prime\prime }=\langle n-g,g|\hat{U}|n,0\rangle =\sqrt{\left( \frac{n}{g} \right)}{{R}^{(n-g)/2}}{{({\rm i}\sqrt{1-R})}^{g}}.\end{eqnarray} \tag{ 8 }$$

This is the amplitude to find g photons at the trigger output mode of the beam splitter and $n-g$ photons in the other output when the target input mode has n photons and the ancilla input mode occupies the vacuum state. In exact analogy to equation (4), for a realistic heralding detector described by POVM element ${{\hat{\Pi }}_{\checkmark }}={{\sum }_{g}}{{\theta }_{g}}|g\rangle \langle g|$ and vacuum ancilla state, an arbitrary input state described by density matrix $\hat{\rho }$ will transform as

$$\begin{eqnarray}&&\hat{\rho }=\mathop{\sum }\limits_{m=0}^{\infty }\mathop{\sum }\limits_{n=0}^{\infty }{{C}_{mn}}|m\rangle \langle n|\Rightarrow \mathop{\sum }\limits_{g=1}^{\infty }{{\theta }_{g}}\mathop{\sum }\limits_{m=g}^{\infty }\mathop{\sum }\limits_{n=g}^{\infty }{{C}_{mn}}A_{m-g,g}^{\prime\prime }A_{n-g,g}^{\prime\prime *}|m-g\rangle \langle n-g|=\hat{\rho }^{\prime} ,\end{eqnarray} \tag{ 9 }$$

where dark counts in the heralding detector are neglected. The process tensor describing the state transformation of equation (9) can be obtained in the same manner as for equation (6). Denoted $({{\mathcal{E}}_{0}})_{jk}^{mn}$, the tensor elements associated with the vacuum ancilla and non-number resolving phase-insensitive conditioning detector is given by

$$\begin{eqnarray}\begin{array}{rcl} ({{\mathcal{E}}_{0}})_{jk}^{mn} & = & \frac{1}{\sqrt{m!n!}}\mathop{\sum }\limits_{g=1}^{\infty }{{\theta }_{g}}\sqrt{(j+g!)(k+g)!}{{\sqrt{R}}^{j+k}}{{\sqrt{1-R}}^{2g}}\sqrt{\left( \frac{j+g}{g} \right)\left( \frac{k+g}{g} \right)} \\ {} & {} & \times {{\delta }_{m,j+g}}{{\delta }_{n,k+g}}. \\ \end{array}\end{eqnarray} \tag{ 10 }$$

By the linearity of quantum mechanics, a mixed ancilla state ${{\hat{\rho }}_{{\rm a}}}$ will lead directly to a mixture of the two process tensors derived in this section and the preceding section, i.e. a mixture of ${{\mathcal{E}}_{0}}$ and ${{\mathcal{E}}_{1}}$, weighted according to the heralding efficiency ${{\eta }_{{\rm H}}}$. Thus the process map for realistic FSF is given by

$$\begin{eqnarray}&&{{\mathcal{E}}_{{\rm FSF}}}={{\eta }_{{\rm H}}}{{\mathcal{E}}_{1}}+(1-{{\eta }_{{\rm H}}}){{\mathcal{E}}_{0}}.\end{eqnarray} \tag{ 11 }$$

2.3. Mode mismatch with FSF heralding detector

Photon-counting detectors such as APDs and time-multiplexed detectors [38] are general spatially, temporally and polarization multimode. A well-defined spatial and polarization mode is typically selected by using a single-mode fiber (SMF) and polarizing element in front of the detector to filter a single spatial and polarization mode. However, since the response time of a typical photon-counting detector is often far greater than the optical pulse duration, such detectors are inherently temporally multimode [39]. Tualle-Brouri et al studied the impact of using a multimode heralding detector in a scheme to prepare Schrödinger cat states by photon subtraction from squeezed states of light [40]. It was shown that for such a scheme, the problem could be reduced to two effective modes, thus aiding analysis of the multimode effects. In the realistic FSF scheme depicted in figure 1(b) a similar two-mode decomposition may be performed. It is assumed that the FSF input state $\hat{\rho }$ (e.g. a coherent state) occupies a single well-defined spatial-temporal-polarization mode. The impact of the multimode heralding detector arises when there is mismatch between the ancilla state ${{\hat{\rho }}_{{\rm a}}}$ mode and the FSF input state mode. In this situation, part of the ancilla mode which is not overlapped with the FSF input mode can nevertheless trigger events in the heralding detector.

Assuming the FSF input state $\hat{\rho }$ occupies a single mode the problem can be decomposed into two effective modes: (1) the mode defined by the input state and (2) the mode orthogonal to this in which the ancilla has some population [40]. Ideally, the ancilla state mode would perfectly match the FSF input, in which case the problem reduces to a single mode. In practice, a fraction of the ancilla mode may not be overlapped with the FSF input but may still lead to detection events in the heralding detector. In figure 1(b) this is represented pictorially by the green-shaded part of the ancilla-state mode. A mode mismatch parameter M is defined such that

$$\begin{eqnarray}&&M=\frac{{{\eta }_{{\rm H}}}}{\eta _{{\rm H}}^{\prime }},\end{eqnarray} \tag{ 12 }$$

where ${{\eta }_{{\rm H}}}$ is the heralding efficiency of the ancilla single-photon state into the same mode as the FSF input and $\eta _{{\rm H}}^{\prime }$ is the heralding efficiency registered by the multimode FSF heralding detector, such that $\eta _{{\rm H}}^{\prime }\geqslant {{\eta }_{{\rm H}}}$. For M = 1 the problem is single-mode and thus the process tensor of equation (11) applies. Events where the FSF heralding detector 'clicks' due to detection of the ancilla single photon in the mode which is not overlapped with the input state correspond to there having been no interaction between the input state $\hat{\rho }$ and the ancilla. In these events the beam splitter will simply attenuate the input state. This can be modelled as a probabilistic attenuation process, where the input state intensity is attenuated by a factor of $1-R$. The process tensor describing attenuation is given by [36]

$$\begin{eqnarray}&&({{\mathcal{E}}_{{\rm att}}})_{jk}^{mn}=\sqrt{\frac{m!n!}{j!k!}}\frac{{{\eta }^{(j+k)/2}}{{(1-\eta )}^{m-j}}}{(m-j)!}{{\delta }_{m-j,n-k}},\end{eqnarray} \tag{ 13 }$$

where η is the fraction of the incoming intensity which is passed to the output mode of the beam splitter, i.e. $\eta =R$, figure 1(b).

Thus, the complete process tensor describing the realistic FSF operation as depicted in figure 1(b) is given by

$$\begin{eqnarray}&&{{\mathcal{E}}_{{\rm FSF}}}=M\left[ {{\eta }_{{\rm H}}}{{\mathcal{E}}_{1}}+(1-{{\eta }_{{\rm H}}}){{\mathcal{E}}_{0}} \right]+(1-M){{\eta }_{{\rm det} }}R{{\mathcal{E}}_{{\rm att}}},\end{eqnarray} \tag{ 14 }$$

where R is the FSF beam splitter reflectivity and ${{\eta }_{{\rm det} }}$ is the quantum efficiency of the heralding detector—both of which must be included explicitly as multiplicative factors for the attenuation component to correctly account for the success probability of this event.

FSF was first demonstrated experimentally by Sanaka et al in 2006 [24] and later by Resch et al in 2007 [25]. Both experimental demonstrations were similar in that the non-linear absorption property of the filter was inferred from two- and four-fold coincidence measurements performed at the output of the FSF for incident one- and two-photon input states to the filter. The beam splitter reflectivity R was varied and the Hong–Ou–Mandel visibility [41] recorded for the two different input Fock states. Operation of the filter consistent with theory was inferred by noting that for an incident one-photon Fock state the best visibility occurred for $R=1/2$ and for an incident two-photon Fock state for $R=2/3$ [24, 25]. Although these experiments go some way towards demonstrating the filter operation, such coincidence measurements are inherently post-selected on preservation of photon-number, and are thus loss insensitive. This masks the effect of non-unit heralding efficiency and to some extent enables a degree of number resolution in the APD heralding detector. Furthermore, probing the filter with Fock states does not enable preservation of coherence between Fock layers to be verified, which requires input states with a superposition of photon number states, e.g. a coherent state.

The technique of csQPT [36, 42, 43] does not require the use of difficult to prepare non-classical number-state superpositions to probe a quantum process, but rather readily available coherent states emitted from a stable laser system. Ideal filter operation requires a perfect number-resolving heralding detector and a pure ancilla photon state, perfectly mode-matched to the mode of the quantum state to be filtered. We probe an implementation of the process with realistic components to assess the effect of such imperfections. Thus to verify the model of FSF developed above, equation (14), csQPT is performed on an experimental implementation of the FSF operation, enabling reconstruction of an estimate of the process tensor ${{\mathcal{E}}_{{\rm FSF}}}$ from experimental data. This constitutes only the second experimental demonstration of csQPT on a conditional quantum process, the first being that of Kumar et al who recently demonstrated the experimental reconstruction of the bosonic creation and annihilation operators [44].

The complete optical schematic of the setup used to perform csQPT of the FSF operation is shown in figure 2 and is split into four distinct stages: (a) preparation of the ancilla single photon ${{\hat{\rho }}_{{\rm a}}}$, (b) preparation of coherent states $\hat{\rho }=|\alpha \rangle \langle \alpha |$ used to probe the FSF operation, (c) the FSF beam splitter and heralding detection and (d) homodyne detection of the FSF output state $\hat{\rho }^{\prime}$ to perform csQPT. Each stage is detailed in the following sections.

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3.1. Ancilla state preparation

The ancilla single-photon state ${{\hat{\rho }}_{{\rm a}}}$ is derived in a heralded fashion from a SPDC source [37, 45]. An 80 MHz train of 100 fs pulses at 830 nm central wavelength from a Ti:sapphire oscillator (Spectra-Physics Tsunami) is frequency doubled with a conversion efficiency of approximately 30% in a 700 μm long β-barium borate crystal cut for type-I collinear phase matching resulting in 85 fs pulses at 415 nm. The residual fundamental is filtered out using dichroic mirrors and Schott glass filters, resulting in a maximum second-harmonic power of 500 mW. The second-harmonic beam is subsequently spatially filtered with a pinhole (PH) and focused into an 8 mm thick potassium dihydrogen phosphate (KDP) crystal cut for type-II collinear degenerate down conversion [37, 45]. The source produces a nearly two-mode squeezed vacuum state, which is described in the photon-number basis as

$$\begin{eqnarray}&&|\psi \rangle \approx \sqrt{1-{{\lambda }^{2}}}\left[ |0,0\rangle +\lambda |1,1\rangle +{{\lambda }^{2}}|2,2\rangle +O({{\lambda }^{3}}) \right],\end{eqnarray} \tag{ 15 }$$

where the parameter λ is related to the squeezing parameter and depends on the pulse energy of pump and $|m,n\rangle$ describes the state with m (n) photons in the trigger (ancilla) mode. Since the KDP crystal is cut for type-II phase matching, the trigger and ancilla modes are orthogonally polarized and easily separated using a polarizing beam splitter (PBS). The trigger mode is coupled into a SMF to select a well-defined spatial mode from the SPDC source and detected with an APD (Perkin–Elmer SPCM-AQ4C). Although the APD is a binary detector, since λ is extremely small in the experiment ($\lambda \approx 0.07$), when the APD registers a click the ancilla mode collapses to a state very closely approximating a single-photon Fock state with negligible higher-order photon number terms.

In practice, it is generally difficult to exactly mode match the heralded ancilla single-photon state both spatially and temporally with, for example, the local oscillator (LO) used to perform balanced homodyne detection [37]. Even if the SPDC source produces only a single two-mode squeezed state in the spatial modes examined (i.e. the joint spectral amplitude for the trigger and ancilla fields is factorable [45]) there is no guarantee the ancilla mode can be well matched to the desired mode. This leads to an ancilla state overlapped with the target mode consisting of an admixture of the heralded single-photon state with vacuum, as given in equation (7), where ${{\eta }_{{\rm H}}}$ is the heralding efficiency into the target mode. During the experiment ${{\eta }_{{\rm H}}}$ is monitored by setting the half-wave plate (HWP) in the FSF beam splitter to direct the ancilla state directly to the balanced homodyne detector. This enables quantum state tomography (QST) of the ancilla state from which ${{\eta }_{{\rm H}}}$ can be extracted, figure 3. The heralding efficiency during the experiment was found to be ${{\eta }_{{\rm H}}}=0.45\pm 0.04$, after correcting for the efficiency of the BHD, which has a value of ${{\eta }_{{\rm BHD}}}=0.86$ [46].

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To determine the parameter M characterizing the effect of using a multimode heralding detector to condition the FSF operation, the heralding efficiency ${{\eta }_{{\rm H}}}^{\prime}$ in the FSF heralding mode is determined using APD 2, figure 2. The HWP in the FSF beam splitter is set to direct the ancilla state to conditioning detector APD 2, where the heralding efficiency ${{\eta }_{{\rm H}}}^{\prime}$ is directly monitored using a coincidence counting program implemented in a field-programmable gate array (FPGA). After correcting for the APD efficiency ${{\eta }_{{\rm APD}}}=0.45$ (determined from the APD datasheet after correction for losses associated with using a non-anti-reflection-coated input fiber) the heralding efficiency registered by APD 2 is determined to be ${{\eta }_{{\rm H}}}^{\prime} =0.62$. This gives a value of the mode mismatch parameter $M=0.45/0.62=0.73$.

3.2. Probe state preparation

3.3. FSF beam splitter and balanced homodyne detection

Since FSF is a conditional process, heralded by the simultaneous generation of an ancilla single-photon state ${{\hat{\rho }}_{{\rm a}}}$ and the detection of a photon in the FSF heralding mode, it is necessary to record the success probability of the operation, $P(\alpha )$, which is in general a function of the input probe state, to perform csQPT [43, 48]. The data captured by the BHD do not contain information about the success probability. The success probability is monitored using a coincidence counting program implemented in a FPGA, figure 2(d). The output transitor–transitor logic (TTL) pulses from APD 1 and APD 2 are split between the DSO for triggering data acquisition from the BHD and the FPGA to record the success probability. The measured success probability as a function of the probe state intensity $|\alpha {{|}^{2}}$ is shown in figure 4 in addition to the predicted success probability derived from the single-mode and multimode models of FSF presented in section 2, where the parameter values measured in section 3 are used. The experimentally measured success probability $P(|\alpha {{|}^{2}})$ is consistent with the multimode model of FSF—while the single-mode model predicts a consistently lower probability of success. The emergent discrepancy between the multimode model success probability and the experimental success probability for $|\alpha {{|}^{2}}\gt 1.25$ is an indication that the probe state mode and LO mode are not perfectly overlapped—as is assumed in order to decompose the problem into two effective modes. The mode overlap between the LO and probe state mode was measured independently to be 97%. However, in principle these modes can be made to match perfectly since they are derived from the same laser.

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The raw quadrature samples for the FSF output states recorded by the BHD are binned with 30 phase bins in the interval $\theta \in [0,\pi ]$ and 601 quadrature bins in the interval $X\in [-5,5]$. The reconstruction is performed up to a maximum photon number n = 6, and incorporates correction for the BHD efficiency ${{\eta }_{{\rm BHD}}}=0.86$. Thus the reconstructed process tensor is able to predict the evolution of an arbitrary input state $\hat{\rho }=\sum _{m,n=0}^{6}{{\rho }_{mn}}|m\rangle \langle n|$ on the truncated input Hilbert space $\mathcal{H}$. The maximum-likelihood reconstruction algorithm [43], implemented in MATLAB, took approximately 6.5 h to perform 150 iterations on a multi-core desktop computer. The iterations are stopped when the change in the likelihood approaches the machine precision corresponding to negligible change of the process tensor elements. Furthermore, dilution of $\mu =0.5$ was used [49] to curb oscillations in the likelihood at the start of the reconstruction [43]. Figure 5(a) shows diagonal elements $\mathcal{E}_{kk}^{nn}$ of the model process tensor (equation (14)), where the model parameters are those determined in section 3, and figure 5(b) of the reconstructed process tensor.

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On inspection the model and reconstructed diagonal elements are similar, both in terms of the structure within each input Fock layer and the sum of each input Fock layer, which represents the success probability for a particular input Fock state. A more quantitative analysis is afforded by calculating the fidelity between the reconstructed and model process tensors according to

$$\begin{eqnarray}&&F={{\left[ {\rm Tr}{{\left( {{({{{\hat{E}}}_{{\rm recon}}})}^{1/2}}{{{\hat{E}}}_{{\rm model}}}{{({{{\hat{E}}}_{{\rm recon}}})}^{1/2}} \right)}^{1/2}} \right]}^{2}},\end{eqnarray} \tag{ 16 }$$

where the processes are represented as operators on the combined input–output Hilbert space $\mathcal{H}\;\otimes \;\mathcal{K}$ according to the Jamiołkowski–Choi isomorphism [48]. It is necessary to normalize each operator $\hat{E}$ [50] such that the fidelity F satisfies $0\leqslant F\leqslant 1$. This preserves the relative success probability for each input Fock layer and effectively amounts to changing the overall 'gain' of the process. The fidelity thus defined is a function of the whole process tensor, i.e. not just the diagonal elements displayed in figure 5.

The fidelity is a multi-dimensional function of the model parameters and would ideally be maximal for the parameter values experimentally determined in section 3. To this end, two cuts of the fidelity as a function of the model parameters were calculated. Figure 6(a) shows the fidelity as a function of the ancilla state heralding efficiency ${{\eta }_{{\rm H}}}$ and APD efficiency ${{\eta }_{{\rm APD}}}$ used in the model of equation (14), with the beam splitter reflectivity R = 0.5 and the mode mismatch parameter M = 0.73, as determined in section 3. The fidelity peaks for the expected parameter values of ${{\eta }_{{\rm H}}}=0.45\pm 0.04$ and ${{\eta }_{{\rm APD}}}=0.45$. Figure 6(b) shows the fidelity as a function of the ancilla-state heralding efficiency ${{\eta }_{{\rm H}}}$ and the beam splitter reflectivity R for fixed M = 0.73 and ${{\eta }_{{\rm APD}}}=0.45$. In the experiment, the beam splitter reflectivity was set to R = 0.5, i.e. the filter was set to null out the n = 1 component [24]. The fidelity peaks in the region of R = 0.5, adding further confirmation that the reconstructed process tensor is consistent with that predicted by the full model, equation (14).

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A further test to compare the reconstructed and modelled process tensors is based upon calculating the fidelity between output quantum states predicted by the model and by the reconstructed tensor. To this end, 10⁵ input density matrices, randomly drawn from the Wishart distribution, were generated for increasing values of maximum photon number ${{n}_{{\rm max} }}$. The random input density matrices were evolved according to the reconstructed and model tensors to give two normalized output states for each random input state, according to

$$\begin{eqnarray}&&\hat{\rho }_{{\rm recon}}^{\prime }=\frac{{{\mathcal{E}}_{{\rm recon}}}(\hat{\rho })}{{\rm Tr}\left[ {{\mathcal{E}}_{{\rm recon}}}(\hat{\rho }) \right]},\quad {\rm and}\quad \hat{\rho }_{{\rm model}}^{\prime }=\frac{{{\mathcal{E}}_{{\rm model}}}(\hat{\rho })}{{\rm Tr}\left[ {{\mathcal{E}}_{{\rm model}}}(\hat{\rho }) \right]}.\end{eqnarray} \tag{ 17 }$$

The standard definition for fidelity between two quantum states is employed [3], similar to equation (16), with the process 'super-operators,' $\hat{E}$, replaced with the state density operators, $\hat{\rho }$. From the set of input states both the mean fidelity and standard deviation could be estimated as a function of ${{n}_{{\rm max} }}$, blue data in figure 7(a). First, note that the fidelity between the output states is effectively unity when the model process tensor corresponds to the realistic FSF model developed in section 2, with the parameter values taken as those experimentally determined in section 3. As a demonstration that the reconstructed tensor is distinct from the process of 50% attenuation, the same fidelity was calculated where the model tensor was taken to be attenuation by 50%, red data in figure 7(a). The results indicate that the full model describing realistic FSF developed in section 2 provides a far better description of the state evolution than a simple 50% attenuation process. The reason why this result is important can be understood by examining the schematic of FSF in figure 1(b). If the heralding measurement and ancilla state would have no effect on the filter input state then the effect of the operation would simply be to attenuate the input state by a factor $1-R$. Thus by demonstrating that the reconstructed tensor corresponding to the experimental filter is distinct from such an attenuation process the presence of a non-trivial state transformation due to the FSF operation is identified.

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For this experiment the FSF beam splitter reflectivity is R = 0.5, indicating that the filter should preferentially filter out the n = 1 photon-number component of the input state $\hat{\rho }$ [24]. The unnormalized photon-number statistics at the filter output given that n photons were incident are given by

$$\begin{eqnarray}&&\tilde{P}(k|n)=\mathcal{E}_{kk}^{nn},\end{eqnarray} \tag{ 18 }$$

where as usual $\mathcal{E}_{kk}^{nn}$ are the diagonal elements of the process tensor and $\tilde{P}$ denotes the fact that the distribution is not normalized. Summing over k gives the success probability of the FSF operation given that n photons were incident. For the following analysis, the photon-number statistics at the filter output are of interest only when the operation is successful, i.e. a heralding event is observed. Thus the distribution $\tilde{P}(k|n)$ can be normalized according to

$$\begin{eqnarray}&&P(k|n)=\frac{\tilde{P}(k|n)}{\mathop{\sum }\limits_{k}\tilde{P}(k|n)},\end{eqnarray} \tag{ 19 }$$

where $P(k|n)$ is the normalized photon-number distribution at the filter output given that n photons were incident to the filter and a heralding event was observed.

From the reconstructed tensor, $P(1|1)=0.101$, and $P(0|1)=0.899$—in other words, if a single photon is sent into the filter and a heralding event is observed, approximately nine times out of ten the filter output state is the vacuum state—thus the filter exhibits approximately 90% loss for a single-photon state. If the filter were a passive optical element exhibiting linear loss then one would expect

$$\begin{eqnarray}&&P(n|n)=P{{(1|1)}^{n}},\end{eqnarray} \tag{ 20 }$$

since for linear loss if a single photon has a probability P of transmission then a two-photon state has a probability P² of being preserved and so forth for the higher-order Fock states. This linear loss behaviour is plotted in figure 7(b) (red circles), based on the value $P(1|1)=0.101$ extracted from the reconstructed tensor, and applying equation (20) to predict the behaviour of the filter if it were considered as linear loss. This is compared against the actual values $P(n|n),\;n\gt 1$, extracted from the reconstructed process tensor—plotted as the blue asterisks in figure 7(b).

Remarkably, the reconstructed process tensor predicts an approximately flat probability of 'survival' $P(n|n)$ for the higher-order input Fock states, whereas the values based on a linear loss model decrease factorially. Thus the filter effectively exhibits significantly less loss for Fock states of order $n\gt 1$ than for the n = 1 Fock state—consistent with the beam splitter reflectivity R = 0.5. This demonstrates that the filter is preferentially removing the n = 1 photon-number component from the input state [24, 25]—i.e. the filter exhibits a non-linear response to the input state. $P(n|n)$ was also calculated from the model process tensor of equation (14) using the parameter values obtained in section 3—plotted as the blue circles in figure 7(b). The predicted values for $P(n|n)$ are in excellent agreement with those extracted from the reconstructed tensor and again are completely distinct from the behaviour of linear loss.

In this article the photon-level quantum operation known as FSF [24] was investigated when implemented with realistic components. A model of the filter operation was developed incorporating experimental imperfections and put to the test by performing quantum process tomography of an experimental implementation of the Fock state filter. Although the practical filter operation was found to be non-ideal, we have identified key challenges to be met for improved performance. The full process tensor reconstruction goes beyond previous demonstrations based on Hong–Ou–Mandel interference [24, 25], and clearly shows the coherence-preserving property and non-linear behaviour of this operation, indicating the potential for this class of operations for quantum state engineering. Three key experimental challenges were identified—with the main issue being the unambiguous determination of photon-number in the FSF heralding mode. With the emergence of high-efficiency photon-number-resolving detectors [51, 52] and improved sources of photonic Fock states [53] it should be possible to achieve near-ideal FSF and hence also photon catalysis [18]—enabling the tailored generation of arbitrary quantum optical states in the continuous-variable domain. The results presented here show that csQPT [36, 43] should serve as a reliable diagnosis tool for such photon-level conditional quantum operations.

This work was supported by the University of Oxford John Fell Fund and EPSRC grant no. EP/E036066/1. BJS was partially supported by the Oxford Martin School programme on Bio-Inspired Quantum Technologies and EPSRC grant EP/K034480/1. MK was supported by a Marie Curie Intra-European Fellowship no. 301032 within the European Community 7th Framework Programme.