Photonic quantum information processing is a revolutionary tool used in a number of areas including quantum random number generation^[1], cryptography^[2,3], quantum simulators^[4], boson sampling^[5], and quantum computing^[6]. The rapid development of photonics-integrated quantum circuits^[7–9] enables the realization of quantum computing on a photonic chip, which is a scalable and programmable platform for quantum information processing. Research into quantum photonic integrated circuits has witnessed significant progress in the last years for large-scale quantum computing because of the precise programming of circuits^[7,10,11]. In these circuits, unitary transformations are commonly used to implement a variety of tasks, e.g.,  to operate quantum gates^[12]. However, there are some restrictions on the transformation matrix that only uses unitary transformation. For one thing, it is difficult to design quantum circuits for new and more advanced quantum algorithms. It is virtually impossible to solve the well-known non-deterministic polynomial (NP)-complete problem only using unitary operators, while it is possible for non-unitary transformations to be conducive to these problems such as the realization of Abrams–Lloyd’s gate^[13,14] and a more efficient Fredkin gate^[15]. For another, with regard to some important physical systems whose time-evolution is no longer unitary, non-unitary operators are crucial in modeling their interaction with the environment^[16]. In photonic quantum information processing, quantum two-photon interference (QTPI) is the central operation unit. However, the potent two-photon interaction in QTPI is intrinsically limited to the bosonic nature of photons and to the conventional unitary beam splitters^[17], so manipulating the useful photon–photon interaction is pivotal and highly desirable^[4,6].

Introducing non-unitary transformations can allow tunable quantum interference, which is at the heart of integrated quantum photonic processors and can unlock fascinating prospects for new transformations (such as the options of loss and gain^[18,19]) and advanced applications (such as the optical interference unit in optical neural networks^[20]). Moreover, non-unitary transformations are also useful in modeling the inevitable imperfections of real experimental optical components^[18]. A universal method for treating non-unitary transformations can be based on singular value decomposition^[21] (SVD), but limited attempts related to non-unitary transformations have still been made on photonics integrated quantum circuits.

In this work, we investigate and simulate the performance of a 4×4 integrated photonic circuit with loss behaviors from coupled waveguides modeled by a non-unitary transformation. We model this circuit using SVD and control the two-photon coincidence rates by tuning the loss between the coupled waveguides. We find that, with a proper design, one can manipulate the coincidence dynamics of biphotons using non-unitary transformations such as tuning the loss and adjusting the observation basis.

We start by outlining the structure of our 4×4 chip, as shown in Fig. 1.

If we first only focus on the up two mode subsystems, we achieve the desired non-unitary transformation T^. Using SVD, the transformation matrix T^ can be decomposed to T^=W^D^U^^[21], where D^ is a real-valued diagonal matrix denoting the part of the non-unitary transformation, and W^ and U^ are unitary matrices. Our circuit is a passive system with no gain, so without loss of generality, the form of D^ can be defined as D^=diag[1,η], with 0≤η≤1^[21]. When η=1, T^ is unitary, otherwise is non-unitary. Correspondingly, U^ and W^ can be denoted, respectively, as the following two matrices: U^=12[1ii1],W^=12[ii−11].(1)

The resulting T^ is deduced as T^=12[−η+1i+iηi+iηη−1]. Now, consider a 4×4 matrix S^. Taking the whole circuit into consideration, S^ can be decomposed to S^=S^WS^DS^U,(2)where S^U=[U^OOU^], S^W=[W^OOU^][1U^1], and S^D=[10000cosθisinθ00isinθcosθ00001], with cosθ≡η(0≤θ≤π2). We can see that in the circuit, the part of the asymmetric beam splitter (BS) between waveguide 2 and waveguide 3 plays the role of the loss channel, and the splitting ratio between waveguide 2 and waveguide 3 is proportional to the loss (1−η). Inserting U^ and W^ in Eq. (1) into Eq. (2), we have S^=[i+−cosθ+sinθ22−1+i(cosθ−sinθ)22−sinθ−cosθ22−isinθ−icosθ22−1+i(cosθ−sinθ)22−i+cosθ−sinθ22isinθ+icosθ22−sinθ−cosθ22−sinθ−cosθ22isinθ+icosθ22−1+cosθ−sinθ22i+i(cosθ−sinθ)22−isinθ−icosθ22−sinθ−cosθ22i+i(cosθ−sinθ)221+−cosθ+sinθ22].(3)

Now, we further explore two-photon interference in such circuits described by transformation S^. We choose a two-photon N00N state as the input state. A N00N state is a superposition state of N photons in the first mode and N photons in the second mode. A N00N state can be written as |ψN00N⟩=12(|N102⟩+eiNϕ|01N2⟩), where subscripts 1 and 2 indicate the input modes, and phase ϕ can be set with a phase shifter. The N00N state has an N-fold enhancement in phase sensitivity, which is a key point of quantum metrology^[22,23]. In our work, we choose N as 2, meaning a superposition of 2 photons in each waveguide. The N00N state can be generated by pumping two spontaneous parametric down conversion sources. In consideration of the imbalance of the photon-pair generation probability of the two spontaneous parametric down-conversion (SPDC) sources, our input state can be further written as |ψ⟩=c|2102⟩+e2iϕ1−c|0122⟩^[24], where parameter c indicates the balance property, and c=1/2 indicates that the two SPDC sources are balanced. With the transformation matrix S^, we have A†^=S^B†^, where A†^=[a1†,a2†,a3†,a4†]T represents the photon creation operators aj† at each input mode indexed by j, and similarly, B†^=[b1†,b2†,b3†,b4†]T is the creation operator bj† at each output mode. Through Eq. (3), we have $$\begin{gathered} a1†=(i+−cosθ+sinθ22)b1†+[−1+i(cosθ−sinθ)22]b2† \\ +(−sinθ−cosθ22)b3†+(−isinθ−icosθ22)b4†, \\ a2†=[−1+i(cosθ−sinθ)22]b1†+(−i+cosθ−sinθ22)b2† \\ +(isinθ+icosθ22)b3†+(−sinθ−cosθ22)b4†. \end{gathered}$$(4)

The density matrix is denoted by the expression ρ^=|ψ⟩⟨ψ|, which is $$\begin{gathered} ρ^=c(a1†)2|0⟩⟨0|(a1)2+(1−c)(a2†)2|0⟩⟨0|(a2)2 \\ +c1−ce2iϕ(a2†)2|0⟩⟨0|(a1)2 \\ +c1−ce−2iϕ(a1†)2|0⟩⟨0|(a2)2. \end{gathered}$$(5)

For the non-local correlations between output ports 1 and 2, we have P1,2=|⟨1112|ψ⟩|2=⟨0|b2b1ρb1†b2†|0⟩|. Next, inserting Eq. (5) into it, we obtain the non-local correlations between output ports 1 and 2, $$\begin{gathered} P1,2=c(2sinθcosθ−3)264+(1−c)(2sinθcosθ−3)264 \\ −c(1−c)(e2iϕ+e−2iϕ)(2sinθcosθ−3)264. \end{gathered}$$(6)

When c=1/2, we have P1,2=12[(2sinθcosθ−3)232−(e2iϕ+e−2iϕ)(2sinθcosθ−3)264].

Then, we plot the two-photon non-local correlation P1,2 in Fig. 2(a), predicted by Eq. (6), which is a function of η and ϕ.

From the simulation results, we can see that as η changes when T^ is transitioning between unitary and non-unitary, the correlation also changes. Through the appropriate adjustment of η, we can either increase or decrease the probability the way we want, indicating that the loss can be perceived as a variable to tune the correlations and impact the quantum interference.

Also, we can see a periodicity that the photons bunch with ϕ=nπ and anti-bunch with ϕ=(2n+1)π2, which does not change with η and can be adjusted by parameter ϕ. That periodicity indicates that the amazing features of the N00N state are conducive to quantum metrology^[22].

Further, we calculate the correlations between ports 1 and 3, P1,3=|⟨1113|ψ⟩|2=⟨0|b3b1ρb1†b3†|0⟩|, that is $$\begin{gathered} P1,3=c[(sinθ+cosθ)232+(sin2θ−cos2θ)264] \\ +(1−c)[(sinθ+cosθ)232+(sin2θ−cos2θ)264] \\ +c(1−c)(e2iϕ+e−2iϕ)⁢[(sinθ+cosθ)232−(sin2θ−cos2θ)264]. \end{gathered}$$(7)

When c=1/2, P1,3=(sinθ+cosθ)232+(sin2θ−cos2θ)264+(e2iϕ+e−2iϕ)2⁢[(sinθ+cosθ)232−(sin2θ−cos2θ)264].

As we can see in Fig. 2(b), this time, a dip appears instead of the peak in P1,2, i.e., the position of the dip and peak has inversed due to the choice of observation basis, indicating that the coincidence statistics of the biphotons can be changed by an appropriate choice of the observation basis. In addition, as we can see in the figures, on each different basis, we can both tune the quantum interference by altering the loss η.

We can further take the balance property into consideration. Without loss of generality, here we make θ=0 for P1,2 and θ=0.8 for P1,3. Then, we have P1,2=9−9(e2iϕ+e−2iϕ)c(1−c)64 and P1,3=[(j+k)232+(j2−k2)264]+c(1−c)(e2iϕ+e−2iϕ)[(j+k)232−(j2−k2)264], where j=sin0.8, and k=cos0.8. As we can see in Figs. 2(c) and 2(d), the more unbalanced between the two SPDC sources, the weaker the visibility of the quantum interference is, which is a notable point in the real experiment.

Next, we also change the input state into the entangled photon pairs, signal, and idler photons, which can be generated from SPDC. The density matrix is denoted by the following expression^[25]: ρ^=12(a1,s†a2,i†|0⟩⟨0|a2,ia1,s+a2,s†a1,i†|0⟩⟨0|a1,ia2,s)+e−τ2T22(a1,s†a2,i†|0⟩⟨0|a1,ia2,s+a2,s†a1,i†|0⟩⟨0|a2,ia1,s), where τ describes a time delay generated by a delay line between the two photons. We assume that the second-order coherence between the two photons can be modeled by a Gaussian function of the time delay τ^[25,26], where T is the coherence time, and the signal and idler photons are labeled using subscripts s and i. Similarly, the non-local correlations between output ports 1 and 2 is P1,2=|⟨11,s12,i|ψ⟩|2+|⟨12,s11,i|ψ⟩|2, $$\begin{gathered} P1,2=[2−(cosθ−sinθ)2]232+(sinθ−cosθ)28 \\ +e−τ2T2[2−(cosθ−sinθ)2]216. \end{gathered}$$(8)

We can see in Figs. 3(a) and 3(b), that we can tune the quantum interference by adjusting η, increasing the value of correlation, and moreover, improve the visibility of quantum interference optimal in loss=1 of P1,2 and optimal in loss=0.3 of P1,3, which is necessary for performing high-fidelity quantum gate^[27]. Also, we can see that there is a peak when τ=0, which is more of a fermionic behavior. Now, let us turn to the correlations between ports 1 and 3, P1,3=|⟨11,s13,i|ψ⟩|2+|⟨13,s11,i|ψ⟩|2, $$\begin{gathered} P1,3=(sinθ+cosθ)216+(sin2θ−cos2θ)232 \\ −e−τ2T2(sinθ+cosθ)216+e−τ2T2(sin2θ−cos2θ)232. \end{gathered}$$(9)

Similarly, from the simulation results, we can also see the analogous changes between different observation bases and different losses.

In conclusion, we find that quantum interference can be impacted not only by quantum nature but also by the appropriate design of non-unitary transformation in a 4×4 chip. Using the SVD approach, we can have a clear routine for developing non-unitary transformations and go further to control these fascinating correlation characteristics and improve the visibility of quantum interference for high-fidelity quantum gates. These findings have great potential in designing more complex integrated photonic circuits with more novel functions and more potential quantum computing performance. For practical experimental setups, multimode interference (MMI) is a potential element for the configuration of the lithium niobate on insulator (LNOI) chip, and the phase shift can be realized through the electro-optical effect of the lithium niobate (LN). By designing different MMI structures, it’s possible to precisely tune the loss. However, there still exist some practical limitations to controlling quantum correlations through loss such as the effect of different polarization modes. While lithium niobate thin film is emerging as a promising platform in the pursuit of efficient quantum information processing on a chip scale^[28–30], with our previous experimental results, which have been investigated a lot using lithium niobate thin film^[31–34], it is of great possibility to provide theoretical support and extend our theoretical results to experimental practice in the future.