Coincidence Landscapes for Three-Channel Linear Optical Networks

In this section we expound on the example of two monochromatic photons entering a four-port system, i.e., two input ports and two output ports. For $\hat{a}_{j,\text{in}}^{\dagger}(\omega)$ the creation operator for an input monochromatic photon in mode $j$, a monochromatic single-photon state in mode $j$ is

We use the parenthetical (rounded) bra-ket notation to denote the purely monochromatic states. The commutator relation for monochromatic creation and annihilation operators is

with $\mathds{1}$ the identity operator.

A beam splitter is equivalent to a four-port passive interferometer, with two input ports and two output ports. Mapping the two input modes to the two output modes is achieved by the photon-number-conserving linear transformations scattering single-input to single-output photons as

from which

is constructed with each entry $U_{ij}$ in the matrix $\bm{U}$ being treated as a frequency-independent quantity. In practical optical systems, this assumption is desirable and approximately valid for narrow-band optical fields.

Conservation of total photon number of photons requires that

hence $\bm{U}$ is unitary with determinant det$\bm{U}=\text{e}^{\text{i}\varphi}$. Therefore, $\bm{U}$ can be rewritten as

with the matrix

special and unitary, i.e., unitary with determinant $+1$ depending on the three parameters

The matrix $R$ in Eq. (10) is a three-parameter $2\times 2$ special unitary matrix. The unitarity of the matrix is evident by computing $RR^{\dagger}$ and $R^{\dagger}R$ and obtaining the $2\times 2$ identity matrix $\mathds{1}$ in each case. Therefore, $R\in$U($2$). Furthermore, det$R=1$ so $R$ is “special”, implying that $R\in$SU($2$). The fact that the group SU(2) represents the action of the beam splitter is because the passive lossless beam-splitter preserves the total photon number: the number of photons entering the beam-splitter input ports equals the number of photons exiting the beam-splitter output ports. The $2\times 2$ matrix representation arises for just one photon entering the beam-splitter. In general, the matrices are of size $(2j+1)\times(2j+1)$ for a total of $2j$ photons entering the beam-splitter, with $j$ either an integer or a half-odd integer Campos et al. (1989).

We now introduce the SU(2) $D$ function for the irreducible representation (irrep) $j$,

with $\hat{J}_{k}$ the $(2j+1)\times(2j+1)$ matrix representation of the three $\mathfrak{su}(2)$ algebra generators. We use a standard abuse of language and refer to the $\mathfrak{su}(2)$ algebra generators as “angular momentum operators”, although there is no connection with the physical angular momentum. In other words, the analogy with angular momentum reflects the abstract notation that the beam-splitter action on incoming photons can be regarded as an abstract photon-number- preserving “rotation” of the photonic state in the two (input or output) modes.

From this formalism of “angular momentum” operators $\{\hat{J}_{k}\}$, we see that $\Omega$ (11) is just the Euler-angle triplet for the SU(2) transformation

and the entries of the matrix $R(\Omega)$ are Wigner $D$ functions for the SU(2) representation of $j=1/2$:

General expressions for $D^{j}_{mm^{\prime}}(\Omega)$ are known, and tables of explicit functions for various $j$ and $m$ and $m^{\prime}$ exist Varshalovich et al. (1988).

If two photons of frequencies $\omega_{a}$ and $\omega_{b}$ enter ports $1$ and $2$, respectively, and exit in distinct ports – say ports $2$ and $1$ – this post selected output state is constructed by applying the product

to the vacuum. The diagonal matrix on the right-hand side of Eq. (9) is constant so, for this specific case,

The overall phase $\text{e}^{\text{i}\varphi}$ is not of operational importance and, hence, safely discarded.

We employ the obvious relationship

where the explicit dependence of each term on $\Omega$ is suppressed. Henceforth, we suppress explicit $\Omega$ dependence when the nature of the dependence on $\Omega$ is self-evident. The first term on the right-hand side of Eq. (II.1) is

for $R(\Omega)$ given in Eq. (13).

For the second term on the right-hand side of Eq. (II.1), we resort to the formula for the permanent of a matrix $2\times 2$ matrix $X:(x_{ij})$, which is

Thus,

From the expressions for the determinant and permanent of a $2\times 2$ matrix, we can rewrite the amplitude in Eq. (II.1) as

so the scattering

can be written, up to an overall and unimportant $\text{e}^{\text{i}\varphi}$, as

Equation (22) shows, on general grounds, that the amplitude for coincidence counts in distinct output ports can be written in terms of the permanent and the determinant of the matrix $R(\Omega)$; these are in turn expressed as combinations of products of elements of the matrix $R(\Omega)$.

Consider, instead of Eq. (14), scattering of the input state $\hat{a}_{1,\text{in}}^{\dagger}(\omega_{a})\hat{a}_{2,\text{in}}^{\dagger}(\omega_{b})\ket{0}$ to the other possible post selected output state

with the resulting scattering amplitude

This scattering amplitude is related to that resulting from Eq. (14) as follows. The output states are related by a permutation of the frequencies of photons in modes 1 and 2. To this end, we introduce the matrix

which is obtained by permuting rows of $R(\Omega)$ in Eq. (13) so that row 1 of $R(\Omega)$ is row 2 of $R^{21}(\Omega)$, and row 2 of $R(\Omega)$ is row 1 of $R^{21}(\Omega)$. In fact, we can rewrite Eq. (22) as

We see that this scattering amplitude has the same form (up to an overall phase) as the amplitude of Eq. (24) (they are “covariant”). The scattering amplitude of Eq. (26) is obtained (up to an overall phase) by substituting $R$ with $R^{21}$ in Eq. (24). Thus, permuting the output frequencies induces a permutation of the rows that transforms $R\to R^{21}$ but does not change the expression of the scattering amplitude when written in terms of the permanent and the determinant.

Another observation is linked to the permutation of frequencies. The effect of such a permutation can also be made explicit by introducing the states

which are clearly symmetric and antisymmetric with respect to the permutation group $S_{2}$ for the two frequencies $\omega_{a,b}$. States (27) are, respectively, the $\ell=1,m=0$ (triplet) state and the $\ell=0,m=0$ (singlet) state, which can be obtained from the usual theory of two-mode systems in terms of angular momentum. The interferometric input and output states can be expanded in terms of $|\Psi^{\pm}\rangle$, and the effect of permuting frequencies of the output state is determined from the permutation of frequencies on $|\Psi^{\pm}\rangle$.

The group $S_{2}$ contains two elements represented by the identity $\mathds{1}$ and the permutation $P_{12}$, which exchanges $\omega_{1}$ with $\omega_{2}$). Representations of $S_{2}$ are conveniently labeled using the method of Young diagrams Wybourne (1970); Tung (1985); Fulton and Harris (1991); Yong (2007): $\Yboxdim{6pt}\yng(2)$ for the symmetric representation, and $\Yboxdim{6pt}\yng(1,1)$ for the antisymmetric representation.

We emphasize the role of the permutation group by writing

i.e., we can explicitly identify the $\ell=1$ SU(2) state $|\Psi^{+}\rangle$ with one of the basis states for the $\Yboxdim{6pt}\yng(2)$ representation of $S_{2}$, and the $\ell=0$ SU(2) state $|\Psi^{-}\rangle$ with the basis state for the $\Yboxdim{6pt}\yng(1,1)$ representation. This identification of representations of the symmetric group and representations of SU(2) is an example of Schur-Weyl duality Weyl (1939); Lichtenberg (1978); Rowe and Wood (2010), which proves invaluable in our discussion of SU(3) irreps later.

The scattering amplitudes

and

are easily verified. In the case of the 50-50 beam-splitter, $\beta=\pi/2$ such that the scattering amplitude for indistinguishable single photons in Eq. (29) vanishes. Thus, we have determined the roles of the permanent and determinant of the matrix $R$, the $S_{2}$ permutation symmetry of the input state, and higher $D$ functions of the group SU(2) for the two-channel interferometer and the beam-splitter. Higher $D$ functions arise as linear combinations of products of basic $D^{1/2}$ functions entering in the $2\times 2$ matrix $R$ of Eq. (10), as per Eqs. (17) and (II.1).

Finally, we can use Young diagrams to summarize neatly the contents of Eqs. (24) and (26) so as to display the covariance explicitly:

where ${\Yboxdim{4pt}\yng(2)}_{ij}:=\text{per}\,R^{ij}$ is the permanent of the matrix $R^{ij}$, and where ${\Yboxdim{4pt}\yng(1,1)}_{ij}:=\text{det}\,R^{ij}$ is the determinant of $R^{ij}$.

For realistic systems the input photons are not monochromatic, nor should they be. If photons are to be delayed relatively to each other, their temporal envelopes need to be of finite duration. This temporal-mode envelope is sometimes called the photon wavepacket. Detectors are not strictly monochromatic either, as the duration of the detection must be finite. In practice, detectors are often preceded by spectral filters so are close to monochromatic. Mathematically the source field and the detector response should be modeled not as monochromatic functions as in the previous section but, rather, in terms of the appropriate temporal mode.

Mathematically the state of one photon in each of $n$ modes is a complex-value-weighted multifrequency integral of monochromatic single-photon states in each mode (4) given by

for $\bm{\omega}:=(\omega_{1},\dots,\omega_{n})$ the $n$-dimensional frequency, d${}^{n}\bm{\omega}$ the $n$-dimensional measure over this domain, and $\tilde{\phi}(\omega)$ the spectral function. For the source spectral function, we choose a Gaussian Tan et al. (2013)

with $\omega_{0}$ the carrier frequency and $\sigma_{0}$ the half-width of the Gaussian distribution.

Previously we treated the detector as dual Tan et al. (2013) in the sense that photon counting corresponds to the Fock number-state projector $|n\rangle\langle n|$, for $|n\rangle$ the number state (32), but here we employ the flat-spectrum incoherent Fock-number state measurement operator Rohde and Ralph (2006)

that is applicable to detectors currently used in photon-coincidence experiments such as the BosonSampling type Metcalf et al. (2013); Broome et al. (2013); Spring et al. (2013); Tillmann et al. (2013); Crespi et al. (2013); Rohde et al. . A further adjustment accounts for the threshold nature of single-photon counting modules: due to saturation they either see nothing or measure inefficiently at least one photon without number-resolving capability Bartlett and Sanders (2002).

In the case of the two-photon Hong-Ou-Mandel dip experiment Hong et al. (1987), the coincidence rate is a linear combination of the determinant, (17), and permanent, (II.1), of the $2\times 2$ matrix. The coefficients of this combination are controlled through an adjustable time delay $\tau$ between the pulses arriving at respective times $\tau_{1}$ and $\tau_{2}$ such that $\Delta:=\tau_{1}-\tau_{2}$. For identical Gaussian pulses of unit width ($\sigma_{0}=1$) and the detector measurement, (34), the resultant coincidence rate is

For zero time delay $\Delta=0$, the pulses are indistinguishable. For $\Delta=0$, Eq. (II.2) reduces to

Thus, the antisymmetric part $D^{\Yboxdim{4pt}\yng(1,1)}_{00}(\Omega)$ of Eq. (II.2) has vanished, and only the symmetric part of the amplitude $D^{\Yboxdim{4pt}\yng(2)}_{00}(\Omega)$ survives: $D^{\Yboxdim{4pt}\yng(2)}_{00}(\Omega)=\cos\beta$. The balanced beam splitter has $\beta=\pi/2$ so $D^{\Yboxdim{4pt}\yng(2)}_{00}(\Omega)=0$ thereby leading to $\wp_{11}(\Delta=0)=0$.

This null amplitude thus results in the Hong-Ou-Mandel dip corresponding to a nil coincidence rate at $\Delta=0$ in the ideal limit; i.e., the nil coincidence rate shows that the two photons entering the beam splitter are operationally indistinguishable. Under these conditions the two photons are forbidden to yield a coincidence because the probability amplitudes for the two cases where both photons are transmitted and both photons are reflected (the two cases that would yield a coincidence) cancel each other, leaving only the noncoincidence events of both photons being in a superposition of going one way or the other.

In Sec. III.2 we generalize Eq. (9) to the case $3\times 3$ matrices. The SU(2) $R$ matrix of Eq. (9) become an SU(3) matrix. We report in Appendix A essential details on the Lie algebra $\mathfrak{s}u$(3) and their representations Baird and Biedenharn (1963); Slansky (1981). Representations of SU(3) are obtained by exponentiating the corresponding representations of $\mathfrak{su}$(3). In Sec. III.3 we briefly discuss the SU(3) $D$ functions using standard labeling and construction for SU(3) $D$ functions Rowe et al. (1999).

We employ appropriate basis states, endowed with “nice” properties under permutation of output modes, to obtain the $D$ functions Rowe et al. (1999). Some of these “nice” properties are given explicitly in Eqs. (A) and (114) in Appendix A. The required states are either symmetric or antisymmetric under permutation of modes 2 and 3. The action of elements of the permutation group of three objects ($S_{3}$) on these basis states and thus on the $D$ functions has been discussed earlier Rowe et al. (1999).

The connection between scattering amplitudes and $D$ functions is given in Eq. (III.3) and Table 1. As in Sec. II, we eventually label the SU(3) irreps with Young diagrams. For an interferometer containing 3 photons, the Young diagram have 3 boxes. Young diagrams with 3 boxes also label representations of the permutation group $S_{3}$.

We briefly discussed in Subsec. II.1 the effect on scattering amplitudes of permuting two of the output photons. An important part of our work is to generalize this discussion to the three-photon case. We start this in Subsec. III.4 where we introduce the permutation group $S_{3}$ of three objects.

The permutation group $S_{3}$ has a richer structure than does the permutation group of two objects. In addition to defining the permanent and the determinant of a $3\times 3$ matrix, we define additionally another type of matrix function known as an immanant Littlewood and Richardson (1934); Littlewood (1950); Valiant (1979); Bürgisser (2000). The permanent and the determinant are in fact special cases of immanants. The immanants of the $3\times 3$ matrix $R$ are constructed as linear combinations containing in general six triple products of entries of $R$. Here we provide few explicit expression as the expressions are excessively complicated to include in full. Whereas the permanent and determinant can be expressed in terms of a single SU(3) $D$-function, thereby generalizing Eqs. (29) and (30) of the previous section, the last immanant of the SU(3) matrix is a linear combination of SU(3) $D$-functions as introduced earlier Rowe et al. (1999).

In Subsec. III.5 we generalize our previous discussion of the effect of permutation of frequencies on rates in the two-mode problem to the effect of permuting the frequencies for the three-mode case. We also discuss the connection between $D$-functions and immanants of matrices $R^{ijk}$ where rows have been permuted. The permanent and determinant transform back into themselves (up to maybe a sign in the case of the determinant) under such a permutation of rows. In general immanants do not satisfy such a simple relation: their transformation rules are more complicated. We provide in Eq. (71) of Subsec. III.5 the explicit expression of the ${\Yboxdim{6pt}\Yvcentermath 1\yng(2,1)}_{ijk}$ immanants in terms of SU(3) $D$-functions Tan et al. (2013).

Finally, in Sec. III.6, we provide details on the relation between rates and immanants. Just as the scattering rate for two monochromatic photons can be expressed in terms of the permanent and the determinant of the appropriate $2\times 2$ scattering matrix, the scattering rate for three monochromatic photons can be written in terms of the immanants of the appropriate $3\times 3$ scattering matrix.

We have shown explicitly in Sec. II.1 how the scattering rates can be written in a covariant form by using the permanent and determinant of the $2\times 2$ matrix $R(\Omega)$. Previously we found in Tan et al. (2013) that the same observation holds for the case of three photons in a three-channel interferometer. This result is summarized in Eq. (75), which leads to the result that, for monochromatic photons, the rates have simple expressions in terms of immanants of the matrices $R^{ijk}$.

A general interferometer with three input and three output ports transforms the creation operators for input photons to the output creation operators, where the action on the basis vectors of each photon is

where $\bm{U}$ must be a $3\times 3$ matrix and is treated here as being frequency independent. For photon-number-conserving scattering, $\bm{U}$ is now a $3\times 3$ unitary matrix with determinant $\text{e}^{i\xi}$. Thus, the unitary matrix can be expressed as

with $R(\Omega)$ now a special unitary $3\times 3$ matrix, i.e., an SU(3) matrix. In contrast to Sec. II, $\Omega$ now labels the parameter element of SU(3).

In fact $\Omega$ is an 8-tuple of angles, as the matrix $R(\Omega)$ can be written as the product  Klimov and de Guise (2010)

with

the octuple of SU(3) Euler-like angles. The set $\{T_{ij}\}$ comprises SU(2) subgroup matrices

or

depending on the values of $(ij)$. Also

Factorizing Eq. (III.2) into SU(2) submatrices corresponds physically to a sequence of SU(2) phase-shifter/beamsplitter/phase-shifter transformations on modes (23), (12), and (23), with SU(2) parameters defined by the Euler angles Yurke et al. (1986).

Representations of SU(3) are labeled by two non-negative integers $(p,q)$ Baird and Biedenharn (1963); Slansky (1981). This two-integer labeling is a natural extension from the SU(2) labeling of representations by a single non-negative integer $2j$, which is the total photon number in the two-mode case and is thus analogous to twice the angular momentum. The $(p,q)$ labels are defined explicitly in Eq. (57), but the essence of this definition is that, for three photons entering three ports, outputs depend on interference between various outcomes (generalizing the idea that the Hong-Ou-Mandel dip is due to destructive interference between both photons being transmitted and both being reflected as discussed at the end of Sec. II). These inferences are accounted for by considering how to partition the cases where three photons are divided according to a partition $[\lambda_{1},\lambda_{2},\lambda_{3}]$ into three output ports such that

Only the difference between total photon numbers in the three partitions are needed so just the pair $(p,q)$ defined in Eq. (57) is needed, not a triple; hence $(p,q)$ serves as a good labeling for states corresponding to partitioning photons into channels.

The $3\times 3$ matrices of the form given in Eq. (III.2) carry the SU(3) irrep $(1,0)$. The generalization to SU(3) of Eq. (13) is thus

with dependence of $R$ and $D$ on $\Omega$ implicit.

An expression for the matrix entries in Eq. (45) is easily obtained by explicit multiplication of the matrices of Eqs. (41)-(43) per the sequence of Eq. (III.2); e.g.,

In $D^{(1,0)}_{\bm{\nu},\bm{n}}$, the triple

is the occupancy of the output state in channels (1,2,3), and

is the occupancy of the input channel.

Consequently, for specified $\Omega$, $D^{(1,0)}_{(010),(100)}$ is the amplitude for scattering with one photon entering port 1 and exiting port 2. The input state

can scatter to $3^{3}=27$ possible output states:

If the output state is one of the six possible states containing photons in distinct ports (here post selected for outputs in ports $i$, $j$, and $k$),

then the amplitude for scattering from the initial to this final state is, up to a constant overall phase $\text{e}^{i\xi}$, given by

where $\text{e}^{i\xi/3}D^{(1,0)}_{i,(100)}$ denotes the scattering amplitude

To avoid repetitions of products like

we introduce the shorthand $R_{ij}$ as the entry $(i,j)$ in the unitary matrix $R$ of Eq. (45) and introduce a shorthand notation:

Thus, for instance,

Products of the type (51) can be expanded in terms of SU(3) $D^{(p,q)}_{\bm{\nu},\bm{n}}$-functions for higher representations [compare Eq. (29)]. Which values $(p,q)$ to use in the expansion of Eq. (51) can be determined as follows.

Because each monochromatic photon state

is a basis state for the (three-dimensional) representation $(1,0)$, with the SU(3) scattering matrix given in Eq. (45), the product of three photon states is an element in the Hilbert space that carries the tensor product $(1,0)\otimes(1,0)\otimes(1,0)$ of SU(3). This Hilbert space decomposes into the sum of SU(3) irreps given by Baird and Biedenharn (1963); Speiser (1963); Lichtenberg (1978); O’Reilly (1982)

where, in addition to the labeling of SU(3) irreps by non-negative integers $(p,q)$, we also provide the labeling and decomposition in terms of Young diagrams. The connection between the partition $[\lambda_{1},\lambda_{2},\lambda_{3}]$ such that the sum (44) holds and

the Young diagram containing $\lambda_{i}$ boxes on row $i$ and the labels $(p,q)$ and is simple:

From this decomposition we infer that, in general, only functions with $(p,q)=(3,0)$,$(1,1)$ or $(0,0)$ can occur, so that