Double-Fock superposition interferometry for differential diagnosis of decoherence

We consider double-Fock superpositions of the form

$$\begin{eqnarray}\begin{array}{rcl} \mid N:M\rangle & = & \frac{1}{\sqrt{2}}\left( \mid N\rangle _{1,\phi }^{{}}\mid M\rangle _{2,\tilde{\phi }}^{{}}+\mid M\rangle _{1,\phi }^{{}}\mid N{{\rangle }_{2,\tilde{\phi }}} \right) \\ {} & \equiv & \frac{1}{\sqrt{2}}\left( \mid N,M\rangle +\mid M,N\rangle \right), \\ \end{array}\end{eqnarray} \tag{ 3 }$$

where $0\leqslant M\lt N$, and $\mid K{{\rangle }_{l,\theta }}$ denotes the Fock state of K bosons in the interferometric arm l and in the internal state $\mid \theta \rangle .$ The latter pools all remaining relevant degrees of freedom by which particles can possibly be distinguished (besides the mode number): for a photon, $\mid \theta \rangle$ typically describes the polarization and the spatio-temporal mode function. Double-Fock superpositions comprise single-photon ($N=1,M=0$) and $N00N$-states ($N\gt 1,M=0$) as special cases.

After propagation through the interferometer, the component $\mid M,N\rangle$ in (3) acquires the relative phase $(N-M)\eta$ with respect to $\mid N,M\rangle$. This phase can be inferred by combining the two arms at a beam-splitter and measuring the probability $\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid N:M\rangle }$ to find $({{s}_{1}},{{s}_{2}}=N+M-{{s}_{1}})$ particles in the two output modes [10]. In practice, the paths might not be fully indistinguishable, which is reflected by a non-unity scalar product $\langle \phi \mid \tilde{\phi }\rangle$, i.e. by partial distinguishability, which complicates the computation of event rates. One approach to partially distinguishable bosons consists in replacing the ideal bosonic permanent by a sum of more general immanants [11, 12]. Alternatively, the initial state can be decomposed into orthogonal components of different degrees of distinguishability [13–16]. Neither method, however, offers a straightforward extension to mixed states—a prerequisite in our context—because the resulting expressions for the event probabilities feature a complicated dependence on the scalar product $\langle \phi \mid \tilde{\phi }\rangle$.

Here, we overcome this shortcoming by treating the coherent many-particle propagation via double-sided Feynman diagrams [17]. Our starting point is the expectation value of the projector ${{\hat{Q}}_{({{s}_{1}},{{s}_{2}})}}$, whose eigen-space is defined by the desired particle numbers in the output modes of the beam-splitter (figure 1)

$$\begin{eqnarray}&&\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid N:M\rangle }=|{{\hat{Q}}_{({{s}_{1}},{{s}_{2}})}}\hat{U}\mid N:M{{\rangle }|^{2}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&=\langle N:M\mid {{\hat{U}}^{\dagger }}{{\hat{Q}}_{({{s}_{1}},{{s}_{2}})}}{{\hat{Q}}_{({{s}_{1}},{{s}_{2}})}}\hat{U}\mid N:M\rangle \end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&=\langle N,M\mid {{\hat{U}}^{\dagger }}{{\hat{Q}}_{({{s}_{1}},{{s}_{2}})}}\hat{U}\mid N,M\rangle \end{eqnarray} \tag{ 6i }$$

$$\begin{eqnarray}&&+\Re \left[ {{{\rm e}}^{{\rm i}\eta (N-M)}}\langle N,M\mid {{{\hat{U}}}^{\dagger }}{{{\hat{Q}}}_{({{s}_{1}},{{s}_{2}})}}\hat{U}\mid M,N\rangle \right],\end{eqnarray} \tag{ 6ii }$$

where $\hat{U}$ describes the many-particle beam-splitter transformation in Fock-space, induced by

$$\begin{eqnarray}&&\hat{a}_{k,\theta }^{\dagger }\to \frac{1}{\sqrt{2}}\left( {\rm i}\hat{b}_{k,\theta }^{\dagger }+\hat{b}_{3-k,\theta }^{\dagger } \right),\end{eqnarray} \tag{ 7 }$$

where k refers to the beam splitter mode (see figure 1) and a phase-shift of $\pi /2$ is acquired upon reflection. In deriving equation (6), we used that, for a balanced beam-splitter, the states $\mid N,M\rangle$ and $\mid M,N\rangle$ lead to the same event probability. Equation (6) contains two contributions to the event probability: a main contribution (i) for which the bra-and ket-vectors are the same, and a swapped contribution (ii) with different bra-and ket-vectors. These two terms can be interpreted as double-sided Feynman-diagrams that combine propagation forwards and backwards in time, implicit in equation (6): the state $\mid N:M\rangle$ is propagated in time via $\hat{U}$, projected onto the measurement outcome described by the projector $\hat{Q}$, and propagated back via ${{\hat{U}}^{\dagger }}$ (see figure 2).

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Inserting the transformation (7), we identify the permutations of the particles in the modes that yield the same summands. All the possibilities for distributing the particles among the modes need to be taken into account; using $_{\phi }{{\langle K\mid K\rangle }_{\tilde{\phi }}}={{\langle \phi \mid \tilde{\phi }\rangle }^{K}}$, we write the signal probability as a polynomial in the scalar product $\langle \phi \mid \tilde{\phi }\rangle$

$$\begin{eqnarray}\begin{array}{ccccccccccccccc} \mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid N:M\rangle } & = & \frac{M!N!}{{{s}_{1}}!{{s}_{2}}!{{2}^{M+N}}} \\ {} & {} & \times \mathop{\sum }\limits_{J=0}^{M}{{\mathcal{C}}_{J}}\left( {\overbrace{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2J}}}^{({\rm i})}}+{\overbrace{{{(-1)}^{{{s}_{1}}}}\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2(M-J)}}\Re \left[ {{({\rm i}\langle \phi \mid \tilde{\phi }\rangle {{{\rm e}}^{{\rm i}\eta }})}^{N-M}} \right]}^{({\rm ii})}} \right) \\ \end{array},\end{eqnarray} \tag{ 8 }$$

where the explicit form of the combinatorial factor ${{\mathcal{C}}_{J}}$ together with an illustration are given in the appendix.

Equation (8) reveals the dependence of the probability $\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid M:N\rangle }$ for the event (${{s}_{1}},{{s}_{2}}$) on powers of the indistinguishability parametrized by $\langle \phi \mid \tilde{\phi }\rangle$ up to order $N+M$. Absolute-square powers of the form $\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2J}}$ (term (i)) contribute without any dependence on η. A particle 'starting' and 'ending' in the same arm (horizontal single-colored arrows in figure 3) contributes a certain amplitude, a particle ending in a different arm (diagonal two-colored arrows) yields an amended amplitude that is attenuated by a factor $\langle \phi \mid \tilde{\phi }\rangle$ or $\langle \tilde{\phi }\mid \phi \rangle$. The many-particle paths for which all particles end in the arm they started from (marked by dotted edges) constitute the 'classical' contribution, which can be understood via interference-free classical combinatorics. The other terms contain non-vanishing powers of $\langle \tilde{\phi }\mid \phi \rangle$ and describe different exchange processes. Bosonic exchange processes are those with $J\ne 0$, i.e. J counts how many particles were actually exchanged between the arms within one component $\mid N,M\rangle$, such that, naturally, $J\leqslant M$. Exchanges between the $\mid N,M\rangle$ and $\mid M,N\rangle$-components lead to a phase-dependence on η (term (ii)).

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For single-photon and $N00N$-states, M = 0, and the sum (8) reduces to one term

$$\begin{eqnarray}&&\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid N:0\rangle }=\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{{\rm dist}}\left[ 1+{{(-1)}^{{{s}_{1}}}}\Re \left[ {{\left( {\rm i}\langle \phi \mid \tilde{\phi }\rangle {{{\rm e}}^{{\rm i}\eta }} \right)}^{N}} \right] \right],\end{eqnarray} \tag{ 9 }$$

where $\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{{\rm dist}}=\left( \frac{N}{{{s}_{1}}} \right)/{{2}^{N}}$ is the 'classical' combinatorially obtained probability to find $({{s}_{1}},{{s}_{2}})$ distinguishable particles in the output modes. Consistently with $J\leqslant M=0$, no bosonic exchange processes take place.

For twin-Fock states $\mid N,N\rangle$, the event probability can also be obtained using equation (8), but neglecting the second summand (ii), i.e. the phase-dependent contribution: twin-Fock states do not carry any phase-relation between the two modes, therefore, a phase acquired in one arm manifests itself only as a global, non-observable phase. As a consequence, twin-Fock states are immune to dephasing. The input state $\mid 1,1\rangle$ leads to Hong–Ou–Mandel interference [18]; for higher occupation, we obtain terms proportional to $\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2J}}$ with $J=0,1,...,N$ [15].

Double-Fock superpositions with $0\lt M\lt N$ combine the best of both worlds: phase-sensitivity and bosonic effects. Since equation (3) is a superposition of two two-mode Fock states, bosonic bunching governs the general statistics of the particles in the output modes [10, 16, 19]. Simultaneously, interference between the two components $\mid N,M\rangle$ and $\mid M,N\rangle$ permits to measure the phase η. The phase-sensitivity of the $\mid N:M\rangle$-state is enhanced by a factor $N-M$ with respect to the single-photon case, just like for $N00N$-states [20], which feature an enhancement of a factor N. The richness of interference effects is reflected by the four different contributions depicted in figure 3(d). In general, the state $\mid N:M\rangle$ leads to $M+N+1$ distinguishable events: $(N+M,0),(N+M-1,1),...(0,N+M)$, each of which exhibits a certain dependence on higher powers of the scalar product $\langle \phi \mid \tilde{\phi }\rangle$.

In the previous section, we took into account the possible deterioration of interference due to path distinguishability ($\mid \langle \phi \mid \tilde{\phi }\rangle \mid \ne 1$), but we assumed that the state of the particles is always the same when they reach the beam splitter. Due to non-unitary random processes, however, we need to assume that the particles in the upper (lower) arm are in the internal state $\mid {{\psi }_{j}}\rangle$ ($\mid {{\tilde{\psi }}_{k}}\rangle$) with probability p_j (${{\tilde{p}}_{k}}$), i.e. in a mixed state. One then experiences event probabilities corresponding to the classical average (weighted by the p_j and ${{\tilde{p}}_{k}}$) of the quantum-mechanical probability evaluated for $\mid {{\psi }_{j}}\rangle$ and $\mid {{\tilde{\psi }}_{k}}\rangle$

$$\begin{eqnarray}&&\mathcal{P}_{({{s}_{1}},{{s}_{2}}),{\rm mix}}^{\mid N:M\rangle }=\mathop{\sum }\limits_{j,k}{{p}_{j}}{{\tilde{p}}_{k}}\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid N:M{{\rangle }_{{{\psi }_{j}},{{\tilde{\psi }}_{k}}}}},\end{eqnarray} \tag{ 10 }$$

where we made the dependence on $\mid {{\psi }_{j}}\rangle ,\mid {{\tilde{\psi }}_{k}}\rangle$ explicit.

In other words, we can still use equation (8), but each power of a scalar product $\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}$ in (8) needs to be replaced by the ensemble-averaged scalar product power (ASPP), denoted by curly brackets $\{\}$

$$\begin{eqnarray}&&\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}} \right\}=\mathop{\sum }\limits_{j,l}{{p}_{j}}{{\tilde{p}}_{l}}\mid \langle {{\psi }_{j}}\mid {{\tilde{\psi }}_{l}}\rangle {{\mid }^{2m}}{{\langle {{\psi }_{j}}\mid {{\tilde{\psi }}_{l}}\rangle }^{k}},\end{eqnarray} \tag{ 11 }$$

which mathematically corresponds to a higher moment of the scalar product. ASPPs of higher order, $(m,k)\ne (0,1)$, play a crucial role in our subsequent analysis: deteriorating processes affect the ASPPs of different orders in a different way, such that these quantities can encode information about the decoherence process.

In general, a possibly complex ASPP cannot be written as a function of the single-particle density matrices ρ and $\tilde{\rho }$ that describe the particles in the upper and lower arm, because the coherences between the arms single out particular bases $\{\mid {{\psi }_{k}}\rangle \},\{\mid {{\tilde{\psi }}_{k}}\rangle \}$. Consider, for example, a qubit-like particle prepared in $\mid \phi \rangle =\mid 1\rangle$ and a random process that acts on the upper arm, which leaves the qubit in $\mid 0\rangle$ or $\mid 1\rangle$ with probability 1/2. The average scalar product with an unaffected qubit in the lower arm is then $(\langle 0\mid 1\rangle +\langle 1\mid 1\rangle )/2=1/2$. A process that leaves the qubit in the upper arm in $\mid +\rangle =(\mid 0\rangle +\mid 1\rangle )/\sqrt{2}$ or $\mid -\rangle =(\mid 0\rangle -\mid 1\rangle )/\sqrt{2}$, however, leads to an average scalar product of $(\langle +\mid 1\rangle +\langle -\mid 1\rangle )/2=0$, and, consequently, to a different interference pattern. That is, even though the respective single-particle density matrix is the fully mixed state

$$\begin{eqnarray*}&&(\mid 0\rangle \langle 0\mid +\mid 1\rangle \langle 1\mid )/2=(\mid +\rangle \langle +\mid +\mid -\rangle \langle -\mid )/2=\widehat{{\mathbb{1}}}/2\end{eqnarray*}$$

in both cases, the average scalar product differs.

For absolute-squared scalar products of the form $\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}$, the coherences between the arms are irrelevant (figure 3), which makes the corresponding ASPPs independent of the single-particle bases, e.g. for the ensemble-averaged absolute-square of the scalar product, $\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}} \right\}={\rm Tr}(\rho \tilde{\rho })$ . The ASPPs of different orders are widely independent of each other; averaged absolute values merely fulfil

$$\begin{eqnarray}&&{{\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{m}}\}}^{k/m}}\leqslant \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{k}}\}\leqslant \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{m}}\},\end{eqnarray} \tag{ 12 }$$

for every $k\geqslant m\geqslant 1$, where the lower bound is due to Jensen's inequality and the upper bound follows from $\mid \langle {{\psi }_{k}}\mid {{\tilde{\psi }}_{j}}\rangle \mid \leqslant 1$. For two mixed states with the same eigenvectors, $\mid {{\psi }_{k}}\rangle =\mid {{\tilde{\psi }}_{k}}\rangle$ ($[\rho ,\tilde{\rho }]=0$), the upper bound of (12) becomes exact; two pure, possibly distinguishable states $\rho =\mid \phi \rangle \langle \phi \mid ,\tilde{\rho }=\mid \tilde{\phi }\rangle \langle \tilde{\phi }\mid$ saturate the lower bound.

Distinguishability has various causes: on the one hand, we consider an observer with a meter initially prepared in the state $\mid 0{{\rangle }_{{\rm meter}}}$, coupled to the lower arm. If the particle takes the upper path, $\mid \phi \rangle \mid 0{{\rangle }_{{\rm meter}}}\to \mid \phi \rangle \mid 0{{\rangle }_{{\rm meter}}}$; if it takes the lower path, $\mid \phi \rangle \mid 0{{\rangle }_{{\rm meter}}}\to \mid \phi \rangle \mid \beta {{\rangle }_{{\rm meter}}}$. Formally, the leakage of which-path information can be accommodated in an amended internal state $\mid \tilde{\phi }\rangle$ of the particle in the lower arm that incorporates the meter [21], such that $\langle \phi \mid \tilde{\phi }\rangle \;:=\;\langle 0\mid \beta \rangle$. On the other hand, mis-alignment of the setup or any other influence on the interferometric arms that permits to distinguish a particle in the upper arm from a particle in the lower arm via its internal state ($\mid \phi \rangle$ and $\mid \tilde{\phi }\rangle$, respectively) leads to the same effect. We neglect here the systematic acquisition of a relative phase between the two arms, which induces a shift of the overall signal in η, and assume $\langle \phi \mid \tilde{\phi }\rangle ={{\gamma }_{{\rm dist}}}\geqslant 0$. The overall impact of path distinguishability then leads to

$$\begin{eqnarray}&&\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\}=\gamma _{{\rm dist}}^{2m+k}.\end{eqnarray} \tag{ 13 }$$

Mixing can be due to classical noise that disturbs the internal state of the particle in an incoherent manner. Here, we model mixing as follows: with probability ${{\gamma }_{{\rm mix}}}$, all particles in an arm remain unaffected; with probability $(1-{{\gamma }_{{\rm mix}}})$, all particles are left in an unknown state that is chosen randomly for each run. That is, our mixing process corresponds to the addition of white noise with strength $1-{{\gamma }_{{\rm mix}}}$, which leads to the following attenuation of the ASPPs:

$$\begin{eqnarray}\begin{array}{rcl} \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\} & \to & \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\}\gamma _{{\rm mix}}^{2}, \\ \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\} & \to & \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\}\gamma _{{\rm mix}}^{2}+\frac{1-\gamma _{{\rm mix}}^{2}}{d}, \\ \end{array}\end{eqnarray} \tag{ 14 }$$

where $k\geqslant 1$. The last term proportional to $1/d$ reflects that the average absolute-squared scalar product of two random states is finite in a finite-dimensional Hilbert-space, whereas the average complex scalar product vanishes due to isotropy; in the following, we assume $d\to \infty$, which allows us to neglect the corresponding terms proportional to $1/d$. In contrast to distinguishability, which affects different powers in a different manner (equation (13)), mixing amends each ASPP of any power by the same factor $\gamma _{{\rm mix}}^{2}$.

Dephasing is ubiquitous: to name two examples, unstable optical setups lead to phase fluctuations in photonic experiments, while atomic interferometers are affected by background gas collisions. We incorporate the loss of phase coherence between the arms of the interferometer by assuming that, with probability ${{\gamma }_{{\rm phase}}}$, all phases remain unaffected; with probability $1-{{\gamma }_{{\rm phase}}}$, all particles in the lower mode acquire a uniformly random phase $0\leqslant {{\eta }_{{\rm rand}}}\leqslant 2\pi$. The survival rate ${{\gamma }_{{\rm phase}}}$ is, thus, independent of the number of particles⁴ . Although each value of ${{\eta }_{{\rm rand}}}$ induces an interference pattern in η with high visibility, the origin of that pattern is shifted by $(N-M){{\eta }_{{\rm rand}}}.$ Since the shift is unknown and varies from run to run, the experimentally observed interference pattern is washed out.

We account for dephasing by amending the phase-dependent ASPPs

$$\begin{eqnarray}\begin{array}{rcl} \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\} & \to & \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\}{{\gamma }_{{\rm phase}}}, \\ \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\} & \to & \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\}, \\ \end{array}\end{eqnarray} \tag{ 15 }$$

where $k\geqslant 1$, i.e. phase-independent terms remain naturally unaffected by dephasing.

The three decoherence mechanisms commute, and the resulting ASPPs after all processes become the product of the survival rates

$$\begin{eqnarray}&&\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\}={{\gamma }_{{\rm phase}}}\gamma _{{\rm mix}}^{2}\gamma _{{\rm dist}}^{2m+k},\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\}=\gamma _{{\rm mix}}^{2}\gamma _{{\rm dist}}^{2m},\end{eqnarray} \tag{ 17 }$$

for $k\geqslant 1,m\geqslant 0$. Summarizing, on the one hand, observable signals $\mathcal{P}_{({{s}_{1}},{{s}_{2}})}^{\mid N:M\rangle }$ depend on various ASPPs (equation (8)). On the other hand, different ASPPs reflect the decoherence parameters in a different way (equations (16), (17)). This nourishes the hope that we can observe differences between decoherence mechanisms in many-particle interference signals.

The three decoherence mechanisms described in the previous section reduce the fringe visibility of every interferometric signal. For the single-particle state $\mid 1:0\rangle$, the visibility (1) is reduced to

$$\begin{eqnarray}&&\mathcal{V}_{(1,0)}^{\mid 1:0\rangle }=\left\{ \langle \phi \mid \tilde{\phi }\rangle \right\}=\gamma _{{\rm phase}}^{{}}\gamma _{{\rm mix}}^{2}\gamma _{{\rm dist}}^{{}},\end{eqnarray} \tag{ 18 }$$

i.e. one can only infer the product of definite powers of the three model parameters. Geometrically speaking, a given value of the visibility inferred from the observed signal (figure 4(a)) leaves room for a surface in the three-dimensional space $({{\gamma }_{{\rm phase}}},{{\gamma }_{{\rm mix}}},{{\gamma }_{{\rm dist}}})$ (figure 4(c)).

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For $N00N$-states $\mid N:0\rangle$, only the process in which all particles are exchanged between the modes is relevant (figure 3(b)), and we find

$$\begin{eqnarray}&&\mathcal{V}_{({{s}_{1}},{{s}_{2}})}^{\mid N:0\rangle }=\left\{ {{\langle \phi \mid \tilde{\phi }\rangle }^{N}} \right\}={{\gamma }_{{\rm phase}}}\gamma _{{\rm mix}}^{2}\gamma _{{\rm dist}}^{N}.\end{eqnarray} \tag{ 19 }$$

Even though the phase-sensitivity of $\mid N:0\rangle$ is enhanced with respect to $\mid 1:0\rangle$, there are no contributions from bosonic exchange processes ($J\gt 0$) in equation (8), and only the product of definite powers of the three model parameters can be inferred from the experimental data.

For the double-Fock state $\mid 1,1\rangle$, the depth of the resulting phase-independent Hong–Ou–Mandel dip [18] is proportional to

$$\begin{eqnarray}&&\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\}={\rm Tr}(\rho \tilde{\rho })=\gamma _{{\rm mix}}^{2}\gamma _{{\rm dist}}^{2},\end{eqnarray} \tag{ 20 }$$

which does not allow any differentiation between mixing and distinguishability, which makes the actual purity of a single photon only accessible through advanced analyses [22]. In general, two identical mixed states $\rho =\tilde{\rho }\ne \mid \phi \rangle \langle \phi \mid$ cannot be distinguished from two pure distinguishable states $\mid \phi \rangle ,\mid \tilde{\phi }\rangle$ that lead to the same ASPP.

In principle, the visibilities for the three initial states $\mid 1:0\rangle ,\mid 2:0\rangle$ and $\mid 1,1\rangle$ depend on the three decoherence model parameters in a different way (compare equations (18)–(20)) and, by combining the data from three experiments with different initial states, the parameters ${{\gamma }_{{\rm dist}}},{{\gamma }_{{\rm mix}}},{{\gamma }_{{\rm phase}}}$ can be extracted. However, when the initial state is changed, it is difficult to assess which deteriorating effects are due to the possibly imperfect state preparation and which are caused by the actual decoherence in the interferometer. In the following, we show how to circumvent this problem by extracting the decoherence parameters in one single experiment.

Twin-Fock states [23] with $M=N\gt 1$ clearly bring out the qualitative difference between mixing and distinguishability, and are routinely generated in the experiment [14, 15, 24–26]. The simplest example is provided by $M=N=2$. For twin-Fock states, the phase-dependent summand (ii) in (8) is absent, and we find

$$\begin{eqnarray}&&\mathcal{P}_{(0,4)}^{\mid 2,2\rangle }=\frac{1}{16}\left( 1+4\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}} \right\}+\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{4}} \right\} \right),\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&\mathcal{P}_{(1,3)}^{\mid 2,2\rangle }=\frac{1}{4}\left( 1-\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{4}} \right\} \right),\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&\mathcal{P}_{(2,2)}^{\mid 2,2\rangle }=\frac{1}{8}\left( 3-4\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}} \right\}+3\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{4}} \right\} \right),\end{eqnarray} \tag{ 23 }$$

which match the results obtained via the orthonormalization of single-particle wave-functions [14, 15]. The dependence of event probabilities on different powers of $\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}$ stems from different bosonic exchange processes (figure 3(e)). The absence of a second-order term in $\mathcal{P}_{(1,3)}^{\mid 2,2\rangle }$ is responsible for the narrowing of the width of the (1, 3)-signal [15] with respect to the single-photon coherence length, the alternating signs in $\mathcal{P}_{(2,2)}^{\mid 2,2\rangle }$ induce the non-monotonicity of the (2, 2)-signal [14].

Under the decoherence model above, we use (17), in addition to (20), find

$$\begin{eqnarray}&&\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{4}} \right\}=\gamma _{{\rm dist}}^{4}\gamma _{{\rm mix}}^{2},\end{eqnarray} \tag{ 24 }$$

which allows us to read off ${{\gamma }_{{\rm dist}}}$ and ${{\gamma }_{{\rm mix}}}$ from the combined signal $(\mathcal{P}_{(1,3)}^{\mid 2,2\rangle },\mathcal{P}_{(2,2)}^{\mid 2,2\rangle })$ (note that $\mathcal{P}_{(0,4)}^{\mid 2,2\rangle }$ is fixed by the latter two), as illustrated in figure 5.

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As a result, the quantum-to-classical transitions induced by mixing and by distinguishability differ strongly: pure mixing (in general: ${{p}_{j}}={{\tilde{p}}_{j}}$ and $\mid {{\psi }_{j}}\rangle =\mid {{\tilde{\psi }}_{j}}\rangle$; here, in our model: ${{\gamma }_{{\rm dist}}}=1$) implies that ASPPs of higher powers $\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\}$ all take the same value, saturating the upper bound of equation (12). Mixing therefore always induces a linear interpolation between quantum and classical probabilities, as evident from equation (8) (straight blue dashed line denoted by ${{\gamma }_{{\rm dist}}}=1$ in figure 5). In contrast, pure distinguishability (i.e. the particles are described by pure states $\mid \phi \rangle$ and $\mid \tilde{\phi }\rangle$, the lower bound of equation (12) is saturated; here, ${{\gamma }_{{\rm mix}}}=1$), leads, in general, to more intricate, non-monotonic transitions [14] (curved red solid line for ${{\gamma }_{{\rm mix}}}=1$ in figure 5). This qualitative difference between these two decoherence mechanisms reinforces the role of non-monotonicity as a witness of a distinguishability-induced quantum-to-classical transition [6, 27].

A clear and unambiguous differentiation of distinguishability, mixing and dephasing is possible using the double-Fock superposition $\mid 2:1\rangle$. Such a state can be generated experimentally by annihilating a single photon in a twin-Fock state $\mid 2,2\rangle$, where the photon is extracted from either mode with the same probability [28], a technique that was experimentally demonstrated for the $\mid 3:1\rangle$ state [29]. Using (8), the two pertinent probabilities become

$$\begin{eqnarray}\begin{array}{rcl} \mathcal{P}_{(0,3)}^{\mid 2:1\rangle } & = & \frac{1}{8}\left( 1\mp \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\langle \phi \mid \tilde{\phi }\rangle \}{\rm sin} (\eta ) \right) \\ {} & {} & +\frac{1}{4}\left( \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\}\mp \{\langle \phi \mid \tilde{\phi }\rangle \}{\rm sin} (\eta ) \right), \\ \end{array}\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}\begin{array}{rcl} \mathcal{P}_{(1,2)}^{\mid 2:1\rangle } & = & \frac{3}{8}\left( 1\pm \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\langle \phi \mid \tilde{\phi }\rangle \}{\rm sin} (\eta ) \right) \\ {} & {} & -\frac{1}{4}\left( \{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\}\pm \{\langle \phi \mid \tilde{\phi }\rangle \}{\rm sin} (\eta ) \right), \\ \end{array}\end{eqnarray} \tag{ 26 }$$

where the upper signs refer to the event $({{s}_{1}},{{s}_{2}})$ and the lower ones to $({{s}_{2}},{{s}_{1}})$. The dependence on three different ASPPs can be understood from figure 3(d): the phase-independent classical contribution is modified by a phase-independent bosonic exchange contribution, weighted by $\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\}$, and two differently weighted phase-dependent terms, corresponding to exchange of one or three particles.

The three independent observables that characterize interference are the visibility of $(3,0)$-signals $\mathcal{V}_{(3,0)}^{\mid 2:1\rangle }$, the visibility of $(2,1)$-signals $\mathcal{V}_{(2,1)}^{\mid 2:1\rangle }$ and the total probability to find all particles in one mode, $\mathcal{P}_{30+03}^{\mid 2:1\rangle }=\mathcal{P}_{(3,0)}^{\mid 2:1\rangle }+\mathcal{P}_{(0,3)}^{\mid 2:1\rangle }$ (the total probability to find the particles in the (2, 1) or (1, 2)-channel is the complement $1-\mathcal{P}_{30+03}^{\mid 2:1\rangle }$). An additional binary degree of freedom is the phase relation between the $(1,2)$ and the $(3,0)$-signal, which is formally accounted for as follows: the visibility $\mathcal{V}_{(2,1)}^{\mid 2:1\rangle }$ is set to its negative value when the $(1,2)$ and $(0,3)$ signals are out of phase (we excluded complex scalar products $\langle \phi \mid \tilde{\phi }\rangle$, such that we never encounter phase-shifts other than 0 and π). We combine equations (18) and (20) with equation (16)

$$\begin{eqnarray}&&\left\{ \mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2}}\langle \phi \mid \tilde{\phi }\rangle \right\}=\gamma _{{\rm dist}}^{3}{{\gamma }_{{\rm phase}}}\gamma _{{\rm mix}}^{2},\end{eqnarray} \tag{ 27 }$$

to find the observables as a function of the decoherence model parameters. Inserting equations (18, 20, 27) into the probabilities (25, 26), we can express the decoherence parameters $({{\gamma }_{{\rm dist}}},{{\gamma }_{{\rm phase}}},{{\gamma }_{{\rm mix}}})$ as a function of the observables $\left( \mathcal{V}_{(2,1)}^{\mid 2:1\rangle },\mathcal{V}_{(3,0)}^{\mid 2:1\rangle },\mathcal{P}_{30+03}^{\mid 2:1\rangle } \right)$.

The volume of physically allowed combinations of $(\mathcal{V}_{(2,1)}^{\mid 2:1\rangle },\mathcal{V}_{(3,0)}^{\mid 2:1\rangle },\mathcal{P}_{30+03}^{\mid 2:1\rangle })$ is shown in figure 6. Dephasing, distinguishability and mixing lead to very different trajectories, which define the edges of the volume. When one decoherence mechanism fully destroys coherence (${{\gamma }_{{\rm dist}}}=0$ or ${{\gamma }_{{\rm phase}}}=0$ or ${{\gamma }_{{\rm mix}}}=0$), it is not possible to differentiate the other two: full mixing or full distinguishability (${{\gamma }_{{\rm dist}}}=0$ or ${{\gamma }_{{\rm mix}}}=0$, respectively) lead to classical behavior, independently of the value of the other decoherence parameters. When phase coherence is fully lost (${{\gamma }_{{\rm phase}}}=0$), distinguishability cannot be differentiated from mixing, as also evident from equations (20): when the expectation values of the scalar product (18) and the third power (27) both vanish, only the product of ${{\gamma }_{{\rm dist}}}$ and ${{\gamma }_{{\rm mix}}}$ can be inferred. If the relationship between the decoherence parameters and the observables were linear, we would observe a cube-like volume in figure 6; the pathological cases explain why we instead deal with a four-sided wedge. Outside the realm of full decoherence, each choice of $({{\gamma }_{{\rm dist}}}\gt 0,{{\gamma }_{{\rm phase}}}\gt 0,{{\gamma }_{{\rm mix}}}\gt 0)$ leads to exactly one point $(\mathcal{V}_{(2,1)}^{\mid 2:1\rangle },\mathcal{V}_{(3,0)}^{\mid 2:1\rangle },\mathcal{P}_{30+03}^{\mid 2:1\rangle })$, i.e. decoherence rates can be inferred unambiguously from the observables, and differential diagnosis is possible (figures 4(b), (c)). In particular, each value of the single-particle visibility $\mathcal{V}_{(1,0)}^{\mid 1:0\rangle }$ resulting from a single-particle interference experiment is compatible with a surface in the three-dimensional space (yellow surfaces in figure 6), the exact position on that surface then clearly reveals all three decoherence parameters. Remarkably, the visibility of the $(2,1)$-signal $\mathcal{V}_{(2,1)}^{\mid 2:1\rangle }$ can fully vanish for non-vanishing values of the decoherence parameters, due to the competition of the different phase-dependent terms of opposite sign in equation (26) [28]. In general, full dephasing does not lead to the classical behavior of distinguishable particles: even though both visibilities vanish for ${{\gamma }_{{\rm phase}}}\to 0$, bosonic statistics survive, favouring the $(3,0)$ and (0, 3) signal over the $(2,1)$ and (1, 2) signal [19].

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For the state $\mid 2:1\rangle$, the three measured observables match the three physical parameters of the decoherence model presented in section 3, such that the latter can be extracted with confidence outside pathological cases. This bijective relationship, however, is not a trivial artifact of scaling to larger particle numbers: in $N00N$-state interferometry, one also measures several independent signals, but due to the unique dependence on $\{{{\langle \phi \mid \tilde{\phi }\rangle }^{N}}\}$ (equation (9)), different decoherence processes cannot be distinguished.

Decoherence processes that act on many particles may impact on the many-body density matrix in a complex fashion, beyond the three-parameter model of section 3: the mixing process may affect the upper and lower arm differently and act in a more intricate way than by the addition of white noise and dephasing can occur in a nonlinear fashion that impinges on different particle numbers in a different way. Moreover, non-ideal beam splitters, particle loss and imperfect detectors with finite detection efficiency and dark counts will additionally degrade the measured signals. As a general framework, a decoherence model predicts the ASPPs $\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}{{\langle \phi \mid \tilde{\phi }\rangle }^{k}}\}$ as a function of its model parameters.

By increasing the number of particles N and M, we can control a larger set of observables, which allows us to keep up with the complexity of more sophisticated decoherence models and eventually infer the model parameters: the double-Fock superposition $\mid N:M\rangle$ yields signals that permit to infer $2M+1$ different ASPPs (equation (8)). We checked for $N=M+1$ that all $2M+1$ independent ASPPs can be inferred unambiguously from experimental observables up to M = 11. For twin-Fock states $\mid N,N\rangle$, all powers $\{\mid \langle \phi \mid \tilde{\phi }\rangle {{\mid }^{2m}}\}$ for $m=1...N$ can be inferred, which we checked up to N = 10. It remains open, however, whether the relationship between experimental observables and ASPPs is always invertible.