Integrated quantum polariton interferometry

ABSTRACT Exciton-polaritons are hybrid radiation-matter elementary excitations that, thanks to their strong nonlinearities, enable a plethora of physical phenomena ranging from room

temperature condensation to superfluidity. While polaritons are usually exploited in a high-density regime, evidence for quantum correlations at the level of few excitations has been

recently reported, thus suggesting the possibility of using these systems for quantum information purposes. Here we show that integrated circuits of propagating single polaritons can be

arranged to build deterministic quantum logic gates in which the two-particle interaction energy plays a crucial role. Besides showing their prospective potential for photonic quantum

computation, we also show that these systems can be exploited for metrology purposes, as for instance to precisely measure the magnitude of the polariton-polariton interaction at the

two-body level. Our results will motivate the development of practical quantum polaritonic devices in prospective quantum technologies. SIMILAR CONTENT BEING VIEWED BY OTHERS ALL-OPTICAL

TEMPORAL LOGIC GATES IN LOCALIZED EXCITON POLARITONS Article Open access 02 August 2024 FORMATION OF MATTER-WAVE POLARITONS IN AN OPTICAL LATTICE Article 31 March 2022 LIGHT–MATTER

INTERACTIONS WITH PHOTONIC QUASIPARTICLES Article 23 September 2020 INTRODUCTION Exciton-polaritons have been emerging as a promising platform to explore fundamental physics effects in an

analog solid state scenario, such as Bose-Einstein condensation1,2, superfluidity3, soliton propagation4, spontaneous formation of vortices5, Josephson oscillations and self-trapping6,

analog gravity7. In parallel, their nonlinear properties have been exploited to deliver more application-oriented devices, such as all-optical transistors8, resonant tunnel diodes9,

Mach-Zehnder interferometers10, routers11, couplers12, or ultra-low-threshold lasers13, just to mention a few. In addition, polariton integrated circuits have been proposed in view of

mimicking the key functionalities of electronic circuits14, or to implement artificial neural networks15. All these fascinating results and perspectives are a consequence of the microscopic

nature of polariton excitations, which are characterized by a coherent superposition of two very different fields, the electromagnetic (photon) field and an optically active polarization

(the exciton), i.e., a collective crystal excitation made of a bound electron-hole pair16. As such, the polariton field inherits the small effective mass (~ 10−5 relative to the free

electron mass) of bound photonic modes, e.g., in a planar microcavity or waveguide17,18. In addition, it also inherits the large nonlinearity (comparatively, estimated to be about 4 orders

of magnitude larger than bulk silicon) derived from the Coulomb interaction between excitons19,20. At high excitation density (but well below the saturation regime), the polariton mean field

is described in terms of a non-equilibrium Gross-Pitaevskii equation, which has been successfully applied to interpret most of the quantum fluid phenomenology observed, so far21. In this

regime, confined polaritons can be effectively treated as an out-of-equilibrium gas of weakly interacting bosons, whose steady state depends on the balance between driving and losses in the

system. Recently, exciton-polariton condensates have also been proposed for quantum information processing, with the proposal of encoding qubits into the quantum fluctuations on top of the

condensate22, or in the polariton spin degrees of freedom23. In the last few years experiments have shown the possibility to excite the quantized polariton field in confined geometries at

the level of single or few excitation quanta24,25, and evidence of quantum correlations at the level of few polariton excitations has been reported26,27. In such a regime, however, the mean

field treatment does not provide an accurate theoretical description, calling for a more appropriate open quantum system approach to describe strongly interacting polaritons21. Hence,

triggered by all these promising experimental results, as well as by the thrilling perspective to perform quantum computational tasks by means of polariton-based devices, we have been

motivated to provide an in-depth theoretical analysis of integrated polariton circuits in the quantum regime, with particular attention to the differences with respect to linear photonic

integrated circuits and to the unique possibilities enabled by nonlinearities introduced by polariton interactions. In our view, such _quantum polariton integrated circuits_ (QPIC) can be

assumed as complex networks resulting from the combination of _n_ coupled _nonlinear_ waveguides and interferometers, in which polaritons can be injected and propagate while interacting at

the level of few quanta. It is worth stressing, however, that the precise characterization of a device having an arbitrary large number _n_ of propagation channels with _m_ < _n_

single-photon inputs, even in the _linear_ regime and by means of advanced numerical tools, is already a challenging task to be solved. In fact, this problem has recently attracted a great

deal of attention, both in theory and experiments, aiming at exploiting quantum resources to demonstrate the solution of a problem that no classical computer can solve in a feasible amount

of time, the so called quantum supremacy28,29,30,31. It has been conjectured that the presence of nonlinearities may be beneficial to reach quantum supremacy in this context32. More in

general, nonlinearities can be a crucial resource in the quest for the development of photonic quantum gates. Indeed, while original proposals only employing linear optical elements lead to

a non-deterministic computing scenario33,34, novel ideas aimed at developing deterministic quantum gates based on effective photonic interactions have been put forward35,36,37. Realizing

deterministic quantum photonic gates would represent a significant milestone for quantum technologies in near-term devices38. In light of these developments, the intrinsically large

polariton nonlinearity may thus play a key role. In this respect, we remind that there has been a longstanding debate to quantitatively measure the polariton-polariton nonlinearity39, and

only recently the corresponding order of magnitude has been more precisely inferred from photon-photon correlation measurements27. An increase in the magnitude of this parameter, induced by

a strong electric field, has also been reported40,41. On the other hand, no quantitatively accurate measure of this parameter exists, to date. This is because most of these measurements are

performed in the regime of large polariton density, typically treated at the mean field level, and whose nonlinear behavior is fully characterized by the product of inter-particle

interaction energy and particle density. In practice, disentangling these two contributions is at the origin of large uncertainties in the determination of the inter-particle contribution

alone. Here we focus our theoretical description on a realistic experimental configuration in which single photons are coupled onto a semiconductor device in which polariton excitations

exist and can be guided on chip42. In such a situation, single photons are converted into single polariton wave packets that retain the quantum radiation field coherence properties24,25, and

can then be interfered through evanescently coupled waveguides alternated with regions of free propagation, as illustrated in Fig. 1. We show that it is possible to realize nonlinear

quantum devices by exploiting polariton-polariton interactions that are naturally present in a QPIC. In particular, we introduce a scheme to implement a deterministic two-qubits quantum

gate, specifically the \(\sqrt{\,{{\mbox{SWAP}}}\,^{\prime} }\) gate introduced in ref. 43 and defined in Eq. (27). Such a gate can be used in combination with single qubit operations to

implement the CNOT gate (thus, _any_ quantum gate44). Therefore our results demonstrate that QPICs represent a promising platform for the development of integrated photonic devices for

universal quantum computing, going beyond currently accepted non-deterministic paradigms based on linear optical elements. The functionalities introduced by QPICs can then be exploited in

addition to (or in combination with) the usual linear optical interferometers employed in conventional photonic circuits (PCs)45,46,47. In addition, the QPICs are shown to hold promise also

in quantum metrology. In particular, we show how they potentially allow to measure with high precision the two-polariton nonlinear shift by exploiting Fock states interference. More

generally speaking, while the framework here developed may pave the way for the realization of quantum technologies specifically enabled by large polariton interactions, it should be noticed

that the same theoretical analysis fully applies to any potential quantum photonic platform possessing significant third-order nonlinearity. In this respect, applications of QPIC ranging

from quantum metrology, simulations, and sensing, to quantum information processing and computing, may ultimately be envisaged. RESULTS MODEL AND THEORY We start by briefly outlining the

theoretical model that will be exploited to describe polariton-polariton interactions at the two-particle level in QPICs. In what follows, we will only consider the Hamiltonian unitary

evolution, which already allows to grasp the effects of two-particle interactions on quantum correlations. In fact, long lived polaritonic excitations48, and propagation lengths in the order

of 400 μm25, justify our unitary approach if restricted to devices with limited length. However, a quantitative description of the effects of particle losses, unavoidably present in

realistic experimental situations, is given in Supplementary information (SI), see Supplementary Note 3, showing how such incoherent processes do not affect the output photon statistics

derived in the unitary case. MODELING THE TIME EVOLUTION The effective Hamiltonian model used to describe the propagation of polaritons in the two co-directional waveguides sketched in Fig. 

1 reads (_ℏ_ = 1) $$\hat{{{{{{\mathcal{H}}}}}}}=\mathop{\sum}\limits_{k}\left[{\hat{{{{{{\mathcal{H}}}}}}}}_{k}^{a}+{\hat{{{{{{\mathcal{H}}}}}}}}_{k}^{b}+J(x)\left({\hat{a}}_{k}^{{{\dagger}}

}{\hat{b}}_{k}+{\hat{b}}_{k}^{{{\dagger}} }{\hat{a}}_{k}\right)\right]\,,$$ (1) in which \({\hat{a}}_{k}^{({{\dagger}} )}\) and \({\hat{b}}_{k}^{({{\dagger}} )}\) denote the annihilation

(creation) operators of a polariton with wavevector _k_ in the waveguide _A_ and _B_ respectively, and \({\hat{{{{{{\mathcal{H}}}}}}}}_{k}^{\sigma }={\omega }_{k}{\hat{\sigma

}}_{k}^{{{\dagger}} }{\hat{\sigma }}_{k}+{U}_{k}\,{\hat{\sigma }}_{k}^{{{\dagger}} }{\hat{\sigma }}_{k}^{{{\dagger}} }{\hat{\sigma }}_{k}{\hat{\sigma }}_{k}\) (_σ_ = _a_,  _b_) represents

the Hamiltonian of each waveguide in the lower polariton branch. A detailed derivation of this model from the coupled exciton-photon fields is reported in Supplementary Note 1. In

particular, the interaction energy _U__k_ accounts for the two-body interaction arising from the exciton fraction in polaritons at given wave vector _k_. The last term in Eq. (1) describes

hopping between the two waveguides, which is due to the evanescent coupling between the photonic fractions of the co-propagating polaritons in _A_ and _B_ channels, respectively. This is

formally equivalent to a beam splitter Hamiltonian in quantum optics. We can generically assume that the tunnel coupling energy, _J_(_x_), is a space-dependent parameter only determined by

the physical distance between the two channels at position _x_ along the propagation direction. Furthermore, since the evanescent fields in each waveguide decay exponentially, hopping is

expected to be suppressed when increasing the separation between the two channels, while being effectively present only in the regions where the two waveguides are close enough. As a

consequence, it is reasonable to assume _J_(_x_) to be non-zero and equal to a constant value, _J_ > 0, only when the distance is _d_, and zero elsewhere. In our scenario, we envision

single-polariton quanta to be created in the integrated device by shining single photons on the leftmost side of the device, at _x_ = 0, and their properties (for instance,

polariton-polariton correlations) are subsequently measured by detecting photons emerging on the rightmost side of the apparatus, at _x_ = _x__D_. In particular, we consider the initial

state as a product state, that is $$\left|\psi (t=0)\right\rangle =\left|{\psi }_{A}(t=0)\right\rangle \otimes \left|{\psi }_{B}(t=0)\right\rangle ,$$ (2) with \(\left|{\psi

}_{A}(t=0)\right\rangle \) and \(\left|{\psi }_{B}(t=0)\right\rangle \) describing the initial state of the waveguide _A_ and _B_ respectively. While at _t_ = 0 the quantum system is in a

product state, at _t_ > 0 it will be in a superposition of different quantum states. In addition to the obvious dependence on geometrical aspects, such as the length of the interaction

regions ({_l__j_}) and the separation between the waveguides, the degree of correlation present in \(\left|\psi (t)\right\rangle \) does depend on several factors, ranging from the number of

initial photons injected into the system, to their temporal and spatial shape. In the present paper we will only consider the cases where the initial state corresponds to either (i) a

single polariton, or (ii) a product of two single-polariton Fock states. Furthermore, we assume in both cases to have particles with the same wavevector, _k_. More explicitly, in what

follows we consider single polariton states as $$\left|\psi (t=0)\right\rangle ={\hat{a}}_{k}^{{{\dagger}} }\left|{{\Omega }}\right\rangle \equiv \left|{1}_{k}^{A},\,{0}_{k}^{B}\right\rangle

,$$ (3) or $$\left|\psi (t=0)\right\rangle ={\hat{b}}_{k}^{{{\dagger}} }\left|{{\Omega }}\right\rangle \equiv \left|{0}_{k}^{A},\,{1}_{k}^{B}\right\rangle ,$$ (4) and two-polariton states

as $$\left|\psi (t=0)\right\rangle ={\hat{a}}_{k}^{{{\dagger}} }{\hat{b}}_{k}^{{{\dagger}} }\left|{{\Omega }}\right\rangle \equiv \left|{1}_{k}^{A},\,{1}_{k}^{B}\right\rangle ,$$ (5) with

\(\left|{{\Omega }}\right\rangle \) denoting the polariton vacuum in each propagating channel. Before going any further in our theoretical discussion, it is worth stressing that it is

nowadays possible to engineer photon sources having the desired properties needed for the creation of the few polaritons states assumed as the inputs of the devices characterized in the

following. Indeed, it has been experimentally shown that single quantum emitters can be used as bright sources of single-photon Fock states, with very high photon-number purity, and

near-unity indistinguishability49,50. Therefore, we consider a realistic experimental setup exploiting single quantum emitters as input stages (similarly to what has already been shown25)

whose single photons are used to feed the QPICs described in the following sections, therefore initializing the interferometer with the polariton states defined in Eqs. (3), (4) and (5),

respectively. SINGLE-POLARITON DYNAMICS The single polariton states in Eqs. (3) and (4) are not affected by interaction terms proportional to _U__k_, and their evolution through the QPIC in

Fig. 1 is only sensitive to the regions where _J_(_x_) ≠ 0. Let us assume now that a polariton state with wavevector _k_ corresponds to a particle propagating at a constant speed given by

its group velocity, that is $${v}_{g}={\frac{\partial {\omega }_{k}}{\partial k}},$$ (6) with _ω__k_ being the single-polariton dispersion entering in the Hamiltonian model. This implies

that, in order to propagate across the _m_-th interaction region of length _l__m_, the quantum particle needs roughly a time given by $${t}_{m}=\frac{{l}_{m}}{{v}_{g}}.$$ (7) In addition, by

observing that such states are in one-to-one correspondence with the two polarization states of a spin-1/2 particle, that is $$\left|\uparrow \,\right\rangle

=\left|{1}_{k}^{A},\,{0}_{k}^{B}\right\rangle ,\quad \left|\downarrow \,\right\rangle =\left|{0}_{k}^{A},\,{1}_{k}^{B}\right\rangle ,$$ (8) their evolution through an interaction length _l_ 

= _v__g_ _t_ is prescribed by the following unitary operator $${U}_{IR}^{(1)}(t) ={e}^{-i\hat{{{{{{\mathcal{H}}}}}}}t}=\\ ={e}^{-i{\omega }_{k}t}\left(\cos (Jt){\mathbb{1}}-i\,\sin

(Jt){\sigma }_{x}\right),$$ (9) with \(\hat{{{{{{\mathcal{H}}}}}}}={\omega }_{k}{\mathbb{1}}+J{\sigma }_{x}\), with \({\mathbb{1}}\) and _σ__x_ being the identity and the Pauli-X matrices

respectively. The subscript IR stands for “Interaction region”. In particular, notice that in correspondence of the following values of the propagation time (_ℏ_ = 1)

$${T}_{dip}^{(n)}=\frac{\pi }{4J}(1+2\,n),$$ (10) with _n_ being any integer, the operator in Eq. (9) describes a 50:50 beamsplitter. In practice, this means that if one properly chooses the

interaction length _l_ and the group-velocity _v__g_ in such a way that the ratio _l_/_v__g_ equals \({T}_{dip}^{(n)}\) for some integer _n_, then any initial single-polariton state, such

as one of Eqs. (3) and (4), is mapped into an equally weighted superposition of the two. For instance, in correspondence of \({T}_{dip}^{(0)}\equiv {T}_{dip}\), one has that

$${U}_{IR}^{(1)}({T}_{dip})\left|{1}_{k}^{A},\,{0}_{k}^{B}\right\rangle =\frac{\left|{1}_{k}^{A},\,{0}_{k}^{B}\right\rangle -i\,\left|{0}_{k}^{A},\,{1}_{k}^{B}\right\rangle }{\sqrt{2}}$$

(11) and $${U}_{IR}^{(1)}({T}_{dip})\left|{0}_{k}^{A},\,{1}_{k}^{B}\right\rangle =\frac{\left|{0}_{k}^{A},\,{1}_{k}^{B}\right\rangle -i\,\left|{1}_{k}^{A},\,{0}_{k}^{B}\right\rangle

}{\sqrt{2}}.$$ (12) In the next section we discuss the propagation of the two-polariton state in Eq. (5) through a generic QPIC. We show that it is possible to map the polariton-polariton

dynamics into that of a two-level atom driven by external electromagnetic pulses. DYNAMICS OF TWO IDENTICAL POLARITONS Starting from a double excitation subspace (i.e., an initial

configuration with a total number of two polaritons in the system), the quantum state evolves in time as the following superposition $$\left|\psi (t)\right\rangle =\alpha

(t)\left|{1}_{k}^{A},\,{1}_{k}^{B}\right\rangle +\beta (t)\left|{2}_{k}^{A},\,{0}_{k}^{B}\right\rangle +\gamma (t)\left|{0}_{k}^{A},\,{2}_{k}^{B}\right\rangle \,,$$ (13) where we introduced

the states \(\left|{2}_{k}^{A},\,{0}_{k}^{B}\right\rangle ={({\hat{a}}_{k}^{{{\dagger}} })}^{2}\left|{{\Omega }}\right\rangle /\sqrt{2}\) and \(\left|{0}_{k}^{A},\,{2}_{k}^{B}\right\rangle

={({\hat{b}}_{k}^{{{\dagger}} })^{2}}\left|{{\Omega }}\right\rangle /\sqrt{2}\). Therefore, in the general case, the time evolution of any two-polariton input state can be effectively

described by means of a three level system, and the coefficients _α_(_t_), _β_(_t_), and _γ_(_t_) are only related by the normalization condition on \(\left|\psi \right\rangle \). However,

by considering that the Hamiltonian is symmetric under the exchange of \({\hat{a}}_{k}\) and \({\hat{b}}_{k}\), the input state in Eq. (5) evolves as: $$\left|\psi (t)\right\rangle =\alpha

(t)\left|{1}_{k}^{A},\,{1}_{k}^{B}\right\rangle +\beta (t)\left(\left|{2}_{k}^{A},\,{0}_{k}^{B}\right\rangle +\left|{0}_{k}^{A},\,{2}_{k}^{B}\right\rangle \right)\,.$$ (14) Therefore, by

defining $$\left|g\right\rangle =\left|{1}_{k}^{A},\,{1}_{k}^{B}\right\rangle ,\quad \left|e\right\rangle =\frac{\left|{2}_{k}^{A},\,{0}_{k}^{B}\right\rangle

+\left|{0}_{k}^{A},\,{2}_{k}^{B}\right\rangle }{\sqrt{2}},$$ (15) the Hamiltonian model in Eq. (1) can be expressed on this basis, reading $$\hat{{{{{{\mathcal{H}}}}}}}=(2{\omega

}_{k}+{U}_{k}){\mathbb{1}}+2J(x){\sigma }_{x}-{U}_{k}{\sigma }_{z}\,,$$ (16) where \({\mathbb{1}}\), _σ__x_, and _σ__z_ are the identity and the Pauli-X and Z matrices, respectively. In

particular, in our representation \({\sigma }_{z}\left|\,g\,\right\rangle =\left|\,g\,\right\rangle \) and \({\sigma }_{z}\left|\,e\,\right\rangle =-\left|\,e\,\right\rangle \). This model

equivalently describes a two-level atom propagating in free space with transition frequency/detuning given by 2_U__k_, and effectively interacting with a transverse field,

\(\overrightarrow{F}\), producing a Rabi frequency 2_J_(_x_), and \(\left|g\,\right\rangle \) (\(\left|e\,\right\rangle \)) corresponds to the ground (excited) state of such fictitious atom.

In other words, the propagation of the state in Eq. (5) through the QPIC reported in Fig. 1, is equivalent to the physical situation pictured in Fig. 2. On the left (input side), there is

an atom initially prepared in the \(\left|\,g\,\right\rangle \) state. Such a particle is then sent at speed _v__g_ through a sequence of regions having different properties, i.e. with or

without the external transverse field \(\overrightarrow{F}\), and its properties are read on the rightmost side of the apparatus (output region). In particular, we assume that in between the

regions represented in Fig. 2 and characterized by a length _l__j_, there are regions where the atom can propagate freely, i.e. \(\overrightarrow{F}=\overrightarrow{0}\). Similarly to what

happens in the one particle sector, the faster the atom moves (i.e., the larger _v__g_), the smaller is the time spent by polaritons in each hopping regions, and thus the smaller is the

probability of observing a polariton jumping from a waveguide to the adjacent one. On the other hand, in the two-particle subspace of the Hamiltonian (1), the terms depending on _U__k_ play

a non negligible role. In particular, following Eq. (16), and by virtue of the properties of the Pauli matrices, the propagation within the polariton interferometer depicted in Fig. 1 is

completely determined by composing the proper number of times two unitary operators $${U}_{IR}^{(2)}(t)=\; {e}^{-i\phi t}\Bigg[\cos \left(\sqrt{{U}_{k}^{2}+4{J}^{2}}t\right){\mathbb{1}}+\\

+i\frac{\sin \left(\sqrt{{U}_{k}^{2}+4{J}^{2}}t\right)}{\sqrt{{U}_{k}^{2}+4{J}^{2}}}\left(-2J{\sigma }_{x}+{U}_{k}{\sigma }_{z}\right)\Bigg],$$ (17) and $${U}_{FP}^{(2)}(t)={e}^{-i\phi

t}\left[\cos ({U}_{k}t){\mathbb{1}}+i\sin ({U}_{k}t){\sigma }_{z}\right]\,,$$ (18) where _ϕ_ = 2_ω__k_ + _U__k_ and the subscript FP stands for “free propagation". The operator in Eq.

(17) describes the evolution of an initially symmetric state along an interaction length _l_ = _v__g__t_, while the one in Eq. (18) describes the evolution of the two-polariton system in the

free propagation region between two interaction regions. In particular, notice that \({U}_{FP}^{(2)}(t)\) introduces a phase difference between the states \(\left|g\right\rangle \) and

\(\left|e\right\rangle \), which is proportional to 2_U__k_. APPLICATIONS In this section we consider two paradigmatic cases of the generic QPIC, showing how these devices can be used for

either precisely measuring the interaction strength between two single polaritons or implementing nonlinear quantum gates. A generalized version of the solution corresponding to the case of

ideally lossless polariton propagation is reported in SI, see Supplementary Notes 2–3. SINGLE HOPPING REGION: NONLINEAR HOM EFFECT The most elementary building block of a QPIC, namely a

single region of evanescent coupling between waveguides _A_ and _B_, characterized by the length _l_, is depicted in Fig. 3. This configuration is typically exploited in conventional PCs to

realize a beamsplitter. Here, the main difference is the presence of the polariton number-dependent nonlinear interaction within the single waveguide channel, which is going to alter the

beam splitting condition, as well as influencing the many particle statistics in each channel. In order to characterize the quantum behavior of the device, we will consider the auto- and

cross-correlation functions at the output ports, defined as $${G}_{AA}(t,\,{U}_{k}/J)=\left\langle \psi (t)\right|{\hat{a}}_{k}^{{{\dagger}} }\,{\hat{a}}_{k}^{{{\dagger}}

}\,{\hat{a}}_{k}\,{\hat{a}}_{k}\left|\psi (t)\right\rangle \,,$$ (19) and $${G}_{AB}(t,\,{U}_{k}/J)=\left\langle \psi (t)\right|{\hat{a}}_{k}^{{{\dagger}} }\,{\hat{b}}_{k}^{{{\dagger}}

}\,{\hat{b}}_{k}\,{\hat{a}}_{k}\left|\psi (t)\right\rangle \,,$$ (20) where _t_ is the polariton arrival instant at the rightmost end of the device. Here, polaritons are converted into

photons. Therefore, the former amplitude represents the probability of having two photons emerging in pairs from the waveguide _A_, and _G__A__A_(_t_,  _U__k_/_J_) = _G__B__B_(_t_,  

_U__k_/_J_) by symmetry. The other amplitude, _G__A__B_(_t_,  _U__k_/_J_), accounts for events related to photons simultaneously emerging (and being detected) from different waveguides.

Following Eq. (17), after some algebra, analytic expressions can be obtained, which read: $${G}_{AA}(t,\,{U}_{k}/J)=\frac{2{J}^{2}}{{U}_{k}^{2}+4{J}^{2}}{\sin

}^{2}\left(\sqrt{{U}_{k}^{2}+4{J}^{2}}t\right)$$ (21) and $${G}_{AB}(t,\,{U}_{k}/J)=1-2{G}_{AA}(t,\,{U}_{k}/J)\,.$$ (22) The dependence of _G__A__A_ and _G__A__B_ on _U__k_/_J_ is shown in

Fig. 4, where we plot the dependence of these correlation functions on the time spent by the two polariton wave packets in the region of length _l_. This time is compared to the parameter

_T__d__i__p_, which corresponds to the time for which the device would behave as a regular 50:50 beam splitter in the absence of polariton interactions, i.e. for _U__k_/_J_ = 0. In fact,

this can be interpreted as the time for which the interferometer results in the well known Hong-Ou-Mandel (HOM) effect in conventional PCs45,51,52,53,54. Such an established phenomenon is

related to the bosonic nature of two indistinguishable particles going through a 50:50 beam splitter from two different input ports, and it is characterized by a dip in the probability of

simultaneously detecting the two particles emerging from different output ports, ideally going to zero, due to the destructive interference of their probability amplitudes55. In our

notation, this corresponds to the condition for which _G__A__B_(_T__d__i__p_,  0) = 0. At the same time, this condition also implies that the probability of observing two bosons emerging in

pairs from the same waveguide is maximized (boson bunching), i.e. _G__A__A_(_T__d__i__p_,  0) = _G__B__B_(_T__d__i__p_,  0) = 0.5. This is shown in the plots of Fig. 4 for _U__k_/_J_ = 0 and

_t_/_T__d__i__p_ = 1, with an obvious periodicity as a function of _t_/_T__d__i__p_. However, while the HOM effect has been mostly explored in linear interferometers, so far, our framework

allows to straightforwardly address the effect of polariton nonlinearities. In particular, the HOM dip tends to be suppressed on increasing _U__k_/_J_, i.e. strong polariton correlations

inhibit the quantum interference at the beam splitting condition. More in detail, as suggested by the results in Eqs. (21) and (22), for _U__k_/_J_ ≫ 1 (i.e., what we define the strongly

correlated polariton regime) the probability of observing photons emerging in pairs from the same waveguide becomes negligible, while photons are mostly expected to be simultaneously

detected from the two different output channels, i.e. $${G}_{AA}(t,\,{U}_{k}/J)\to 0,\quad {G}_{AB}(t,\,{U}_{k}/J)\to 1$$ (23) The detailed convergence to this regime as a function of the

relevant scale _U__k_/_J_ is explicitly shown in Fig. 5. Here we plot the behavior of the two amplitudes defined in Eqs. (21) and (22) for _t_/_T__d__i__p_ = 1, and the maximum and minimum

value they respectively reach while varying the ratio _t_/_T__d__i__p_, i.e. corresponding to an interaction time \(\tilde{t}\equiv \pi /(2\sqrt{{U}_{k}^{2}+4{J}^{2}})\). Essentially, this

behavior can be interpreted as follows: on increasing the on-site interaction within each propagating channel, the bunching probability is suppressed. In quantum photonics, this is

reminiscent of the polariton blockade effect56,57, recently evidenced in confined geometries26,27. Evidently, if such a regime could be accessed in a propagating geometry, the device would

implement a kind of nonlinear beam splitter in which bosonic coalescence is destroyed in favor of “fermionization” induced by strong interactions58,59. More practically, even being limited

to the realm of weak nonlinearities that do not allow to access such strong correlation regime, this configuration may still allow to directly assess interaction-induced deviations from the

HOM condition. However, as it will be detailed in the “Discussion" section, polariton interaction energies have been reported in the few _μ_eV range, which are too small to be detected

on the scale of the plot in Fig. 5 (_J_ is typically in the meV order of magnitude, thus _U__k_/_J_ < 1). The scheme in Fig. 3, on the other hand, is just the simplest building block for

QPICs. Nevertheless, as discussed in the following section, by exploiting a slightly more complex scheme it is possible to design polariton circuits showing enhanced sensitivity to the

critical parameter _U__k_/_J_. The latter could be exploited, in turn, to implement quantum gates for polaritons encoding information even in the presence of small nonlinearities, as pointed

out in the following. TWO HOPPING REGIONS: POLARITON RAMSEY INTERFEROMETER Motivated by the nonlinear HOM results shown before, and by the atom-polariton correspondence previously discussed

in the “Model and Theory” subsection, in relation to the “Dynamics of two identical polaritons”, here we explore the dynamics and the resulting quantum correlations at the output of the

device depicted in Fig. 6. In this case, the hopping is split into two separate regions having the same length, _l_, and separated by a free propagation region within each waveguide. As

previously done, we define the time spent in each interaction region, _t_, and the time spent in the region _x_2 < _x_ < _x_3, _T_, as the relevant parameters. Given these definitions,

the action of the device in Fig. 6 on the two polariton state defined in Eq. (5) is described by the unitary operator $$U(t,\,T)\equiv

{U}_{IR}^{(2)}(t){U}_{FP}^{(2)}(T){U}^{(2)}_{IR}(t)\,.$$ (24) In particular, a direct analytic computation shows that the probability of having two photons being detected from the same

output channel after the second hopping region reads $${G}_{AA}(t,\,T;\,{U}_{k}/J)= \frac{2{J}^{2}}{({U}_{k}^{2}+4{J}^{2})}{\sin }^{2}(2\theta t){\cos }^{2}({U}_{k}T)\\

-\frac{2{J}^{2}{U}_{k}}{{({U}_{k}^{2}+4{J}^{2})}^{3/2}}{\sin }^{2}(\theta t)\sin (2\theta t)\sin (2{U}_{k}T)\\ +\frac{8{J}^{2}{U}_{k}^{2}}{{({U}_{k}^{2}+4{J}^{2})}^{2}}{\sin }^{4}(\theta

t){\sin }^{2}({U}_{k}T),$$ (25) with \(\theta =\sqrt{{U}_{k}^{2}+4{J}^{2}}\). Similarly to the previous case, we have _G__B__B_ = _G__A__A_ and

$${G}_{AB}(t,\,T;\,{U}_{k}/J)=1-2{G}_{AA}(t,\,T;\,{U}_{k}/J)\,.$$ (26) Here, it is worth highlighting the explicit dependence of the correlation functions on _t_, for which the degree of

correlation of the output photons can be modulated by either changing the interaction length, _l_, or the separation between the two hopping regions, i.e., the time _T_. Globally, the

behavior of _G__A__A_(_t_,  _T_;  _U__k_/_J_) and _G__A__B_(_t_,  _T_;  _U__k_/_J_) for different values of _U__k_/_J_, and as a function of both _t_/_T__d__i__p_ and _T_/_T__d__i__p_, is

reported in Fig. 7. Correlation functions display a periodic dependence on _T_/_T__d__i__p_. In particular, for all the values of _U__k_/_J_ considered, the probability of having photons

emerging simultaneously from the same waveguide, quantified by _G__A__A_(_t_,  _T_;  _U__k_/_J_), is maximized close to the condition _t_/_T__t__i__p_ = 0.5. The first maximum is located in

all the cases at _T_/_T__d__i__p_ = 0, i.e. the condition for which the analog setup reduces to the one characterized in the previous section (2_t_ = _T__d__i__p_ and _T_ = 0). The following

peaks of _G__A__A_(_t_,  _T_;  _U__k_/_J_) as a function of _T_/_T__d__i__p_ explicitly depend on _U__k_/_J_. Due to their complementarity, _G__A__B_(_t_,  _T_;  _U__k_/_J_) is minimized

whenever _G__A__A_(_t_,  _T_;  _U__k_/_J_) is maximized, as it is evident from Fig. 7(d–f). In addition, these results suggest that the geometry assumed for the device in Fig. 6 is

considerably more sensitive to small values of the ratio _U__k_/_J_, if compared to the elementary interferometer with a single hopping region. In order to better clarify this point, let us

consider a selection of results plotted in Fig. 8, where we show both _G__A__A_(_t_,  _T_;  _U__k_/_J_) and _G__A__B_(_t_,  _T_;  _U__k_/_J_) as a function of _T_/_T__d__i__p_, at fixed

_t_/_T__d__i__p_ = 0.5 (i.e., following the vertical dashed lines in Fig. 7). In analogy to the Ramsey interferometer, here significant variations in the degree of correlation between the

photons emerging at the output of the device are obtained by changing the ratio _T_/_T__d__i__p_. The latter essentially coincides with varying the distance _L_ = _x_3 − _x_2 between the two

interaction regions for a given choice of the hopping distance (_d_) and the group velocity of the propagating mode (see, e.g., Supplementary Note 1). As a possible application of this QPIC

device, we propose it can be used to extract information about the value _U__k_ (knowing the value of _J_), by considering the output correlations obtained with devices having an

interaction length equivalent to _t_/_T__d__i__p_ ≈ 0.5 and different values of _T_/_T__d__i__p_. Notice that _t_ = 0.5_T__d__i__p_ corresponds to an interfering region that is only half of

the length for the beamsplitting condition in the linear regime (_U__k_ = 0), which occurs for _t_ = _T__d__i__p_. The reason for the increased sensitivity to the small nonlinear shift can

then be traced back to the same reason giving the high sensitivity to small energy shifts in the conventional Ramsey interferometer, given the formal analogy reported above60. QPICS FOR

QUANTUM COMPUTING: THE \(\SQRT{SWAP^{\PRIME} }\) GATE Here we analyze the potential application of the QPICs characterized before in quantum information processing. The starting point of our

discussion is the behavior displayed in the limit _U__k_ ≫ _J_ by the QPIC discussed in “Single hopping region: nonlinear HOM effect” of this subsection. In such a regime, since the

presence of two polaritons in the same waveguide is suppressed due to the strong repulsive nonlinearity, the QPIC would implement a \(\sqrt{\,{{\mbox{SWAP}}}\,^{\prime} }\) quantum gate43.

Indeed, when a single rail qubit encoding is assumed (each waveguiding channel is in logic \(\left|0\right\rangle \) state if no polariton is present, and logic \(\left|1\right\rangle \)

with a single propagating polariton state), it is straightforward to verify that for _t_ = _T__d__i__p_, when using \(\{\left|{0}_{k}^{A},\,{0}_{k}^{B}\right\rangle

,\,\left|{1}_{k}^{A},\,{0}_{k}^{B}\right\rangle ,\,\left|{0}_{k}^{A},\,{1}_{k}^{B}\right\rangle ,\,\left|{1}_{k}^{A},\,{1}_{k}^{B}\right\rangle \}\) as the two-qubit basis set, the QPIC in

Fig. 3 is described by the following matrix operation $$\sqrt{SWAP^{\prime} }=\left(\begin{array}{llll}1&0&0&0\\ 0&\frac{1}{\sqrt{2}}&\frac{-i}{\sqrt{2}}&0\\

0&\frac{-i}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\ 0&0&0&1\end{array}\right)\,,$$ (27) which allows to implement universal quantum computing when combined with arbitrary

single qubit rotations44. However, in order to observe such a behavior one would require a value of _U__k_ that is definitely larger than any nonlinearity expected for polariton systems in

conventional material platforms. Hence, the relevant question is: Is there a way to design a device that (i) acts on single polariton states as the nonlinear beam-splitter described in

“Single hopping region: nonlinear HOM effect” of this subsection, and also (ii) suppresses polariton bunching in favor of a _G__A__B_ = 1, even in the presence of small _U__k_/_J_? In fact,

the simplest device implementing both functionalities is actually the analog Ramsey interferometer characterized before. For what concerns the first operation, it is sufficient to consider

the unitary operator defined in Eq. (24) to realize that such a QPIC behaves like a (non-linear) beam-splitter when fed with single-polariton states. Then, regarding the (more challenging)

two-qubit operation, it is sufficient to analyse the results of Fig. 8: depending on the value of _U__k_, there always exists some value for the ratio _T_/_T__d__i__p_ for which _G__A__A_ = 

0 and _G__A__B_ = 1. In particular, it can be recognized that such values are roughly determined as _t_ = _T__d__i__p_/2 and _T_/_T__d__i__p_ ≈ 2_J_/_U__k_ when _U__k_/_J_ ≪ 1. This is

obtained by performing a series expansion of Eq. (25) in terms of _U__k_/_J_, which after little algebra yields $${G}_{AA}(t,\,T;\,{U}_{k}/J)\approx \frac{1}{2}{\cos }^{2}\left(\frac{\pi

}{4}\frac{{U}_{k}}{J}\frac{T}{{T}_{dip}}\right),$$ (28) and $${G}_{AB}(t,\,T;\,{U}_{k}/J)\approx {\sin }^{2}\left(\frac{\pi }{4}\frac{{U}_{k}}{J}\frac{T}{{T}_{dip}}\right)\,.$$ (29) Let us

stress the advantages of the proposed scheme: no post-selection is required to implement the quantum gate, and no errors due to state occupancy beyond the computational basis are expected to

occur after proper initialization of the system. Thus, the present result holds great promise to realize prospective devices for photonic quantum computing with deterministic gates. In the

next section we are going to discuss the effects of inevitable losses in a realistic implementation of this device with propagating polaritons, showing that it is actually feasible in

state-of-the-art technology. DISCUSSION Here we assess the relevance of the previous theoretical analysis in view of potential applications and realistic experiments in quantum polaritonics.

As already mentioned in the “Introduction” section, the polariton-polariton interaction energy has proven to be a quantity that is hard to be probed directly in experiment, although a few

reliable estimates exist in the literature26,27. Moreover, such nonlinearities have been shown to be strongly enhanced in suitably engineered nanostructures and exploiting the application of

an external electric field40, although the precise enhancement has later been reduced41. Following recent estimates, we assume a polariton nonlinear interaction energy as large as _U_ = 15 

μeV in standard III–V semiconductor technology27, when the polariton field is confined in a spot on the order of 1 μm2. In a propagating geometry, even if a polariton wave packet is somehow

delocalized in space, this value might be assumed of the same order of magnitude if the wave form spreads by about a wavelength along the propagation direction. In addition, this value might

be further enhanced by a factor of 5–10 due to electric field tuning, depending on the different systems reported in the literature40,41, or even exploiting different mechanisms based,

e.g., on coupling to the biexciton state61. In this section we provide some estimates on the values of _U__k_ that can be detected by a QPIC in realistic scenarios. First, we consider an

experimental setup with a single interaction region. A similar device has been experimentally realized12. The QPIC model based on these system parameters is reported in SI, see Supplementary

Note 1. An alternative realization would require working with totally internally reflected propagating polaritons, such as the ones obtained in a planar waveguide62, or through a partially

etched planar waveguide41. We notice that the main considerations reported here would not change. In ref. 12 the tunneling rate between two coupled waveguides has been studied as a function

of the spatial separation _d_. A realistic estimate for the tunneling energy, _J_, is measured to be on the order of _J_ = 0.5 meV. Hence, the relevant ratio _U__k_/_J_ would range from 10−2

to 10−1 in state-of-art devices. These values appear too small to provide a significant deviation from the device operated in a linear regime, as shown in Figs. 4 and 5. On the other hand,

such values could be easily detected from correlation measurements in the analog Ramsey setup of Fig. 6, according to the results reported in Figs. 7 and 8, by increasing _T_. Practically

speaking, due to the relation reported in Eq. (26), in order to extrapolate information about the polariton-polariton interaction energy, _U__k_, it is sufficient to consider the behavior of

the cross-correlation function, i.e., double-click events obtained at the output of devices having free propagation region of different lengths _L_. By collecting data obtained on

increasing _L_ (to which _T_ is linearly related through the polariton group velocity, _v__g_), the double-click statistics would reproduce an oscillating behavior compatible with those

reported in Fig. 8(b). Finally, by means of Eqs. (25) and (26), a best fit estimate for the nonlinearity _U__k_ could be straightforwardly extracted. A further point of discussion is related

to the finite polariton lifetime, for which the probability of observing photons emerging from the output channel of the QPIC will necessarily decay on increasing _L_. The problem of losses

has been addressed in detail in Supplementary Note 3, where the finite lifetime of polaritons is treated by means of a Lindblad master equation. On the one hand, as expected, numerical

results reveal that the inclusion of a decay rate Γ in the system evolution, such that _τ_ = _ℏ_/Γ, leads to an exponential decay of either populations or correlations with the time spent in

the QPIC (i.e., the total length of the device). On the other hand, once correlation functions are properly normalized, i.e., $${G}_{\alpha \beta }\to {g}_{\alpha \beta }=\frac{{G}_{\alpha

\beta }}{\langle {\hat{n}}_{k}^{\alpha }\rangle \langle {\hat{n}}_{k}^{\beta }\rangle }\quad \alpha ,\beta =A,\,B$$ (30) it is found that the behavior in the presence of losses exactly

reproduces the one expected for the lossless case. In particular, by considering realistic values for the model parameters12, such as _J_ = 0.5 meV (corresponding to _T__d__i__p_ = 

_ℏ__π_/(4_J_) ≈ 1 ps), polariton decay rate Γ = 0.01 meV (which corresponds to _τ_ ≃ 66 ps), and a group velocity in the range from 4 to 5 _μ_m/ps, (see also Supplementary Note 1), the range

_T_/_T__d__i__p_ ∈ [0,  65] could be explored by means of devices having a total length _L__t__o__t_ = _v__g_(_T__d__i__p_ + _T_) of _at most_ 330 _μ__m_. In particular, for such a device

length a population decay by _at most_ a 1/_e_ factor may be expected in each waveguide. Depending on the value of the ratio _U__k_/_J_, such a range of parameters would be large enough to

observe (more than) a complete oscillation of the cross-correlations (i.e., for _U__k_/_J_ = 0.1, corresponding to 50 _μ_eV), or to explore the range _g__A__B_ ∈ [0,  0.25] in correspondence

of a nonlinearity _U__k_ = 5 _μ_eV. Finally, concerning the specific application of the Ramsey-like interferometric configuration to realize a SWAP-type quantum gate, we explicitly report

some realistic device parameters in Fig. 9. In particular, we address the dependence of the free propagation region length for the specific operation in Eq. (27), defined as

\({L}_{\sqrt{SWAP^{\prime} }}\), as a function of _U__k_, for some values in the supposedly realistic range 5−50 _μ_eV. This length is the _L__t__o__t_ corresponding to the condition for

which _G__A__A_ = 0.0 (or _G__A__B_ = 1.0, see Fig. 8). Similarly to the previous discussion, we consider two values for the group velocity (as explicitly given in the Figure). A device with

a total length _L__t__o__t_ ~ 330 _μ_m would be sufficient to implement a full \(\sqrt{SWAP^{\prime} }\) gate, in the presence of a nonlinearity _U__k_ ≃ 15 _μ_eV, which can be

realistically expected on the basis of experimental results reported in the literature. This propagation length is fully within the polariton lifetime of _τ_ ~ 66 ps assumed before.

Furthermore, longer lived polaritons might be obtained by using polariton waveguides41, thus allowing to realize an experimental proof-principle demonstration of this quantum gate even in

the presence of smaller nonlinearities. In particular, putting together the effects of losses on the measured coincidences, the following count rates can be expected in realistic

experimental scenarios. Measured coincidence rates in the order of _C__R_ ~ 1 MHz can be considered for state-of-art single photon sources based on quantum dots, including both

demultiplexing and detection efficiencies30. Thus, by assuming in- and out-coupling efficiency to and from the QPIC in the order of _η__i__n_,_o__u__t_ ~ 0.3 per channel (which is within

reach by using state-of-art grating or edge couplers to the polariton chip), and considering a population decay by a 1/_e_ factor in each of the two arms, an estimated number of coincidence

counts rate can be given as \({({\eta }_{in}{\eta }_{out}/e)}^{2}{C}_{R}\simeq 1\) kHz. This is a very promising result considering that coincidence rates in the order of 1 Hz or lower can

nowadays be measured in quantum integrated photonics experiments30. We conclude this section by discussing the state preparation considered in this work. In our theoretical framework, we

assumed the initial two-particles states to correspond to (i) identical and simultaneous polaritons, which (ii) propagate within the QPICs with the same dynamical properties, namely the same

wavevector and group velocity. Deviations from these assumptions may result in non-ideal visibility of the correlation signals measured at the detectors. As already observed in “Model and

Theory” subsection of the “Results” section, single quantum emitters can be considered to a large extent as near-optimal single photon sources of indistinguishable quanta49,50. Simultaneous

or delayed arrival of these single photon states at the two input ports can be controlled by external means to a large extent. As a consequence, we do believe that the creation of the

radiation states needed as input stages for testing the dynamics in QPICs is possible by means of the state-of-the-art technology. Regarding the second issue, since the coupling to the

external sources is usually achieved by means of grating couplers, we do observe that only pre-determined wavevectors will enter into the propagation channels. In particular, since the

realization of waveguides is subject to lithography and etching processes, it can be controlled to sub-nm precision. This implies that QPICs can be fabricated with almost ideal symmetric

single-mode devices. CONCLUSION We have introduced a quantum technology platform that extends current performances of photonic integrated circuits, by exploiting effective interactions

between polaritons. We have shown that they allow to build nonlinear quantum devices and deterministic quantum logic gates. As a first step, we have analyzed how polariton interactions can

induce a significant deviation from the typical Hong-Ou-Mandel behavior in an integrated beam splitter. Starting from this result, we explicitly designed an integrated polariton

interferometer that allows to straightforwardly implement a deterministic \(\sqrt{\,{{\mbox{SWAP}}}\,^{\prime} }\) gate. By considering realistic experimental conditions and parameters, the

proposed QPIC represents a viable route for prospective photonic-based quantum information processing. Noticeably, the interferometric nature of QPICs makes these devices very well suited

also for metrology and sensing purposes. As a targeted example, we have shown how an analog integrated Ramsey interferometer could be used for measuring polariton interaction energies at

two-particle level. We thus believe these results will motivate the realization of complex polariton interferometers, which will set the basis of future experiments exploiting the relatively

strong nonlinearities of these systems for metrology applications, sensing, quantum simulations, and quantum information processing. METHODS Numerical simulations for this work have been

performed by home made Python codes solving for either the time evolution of the Schrodinger equation or the density matrix master equation. DATA AVAILABILITY All the data employed for this

work will be made available from the authors upon reasonable request. CODE AVAILABILITY All the codes employed for this work will be made available from the authors upon reasonable request.

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waveguides. _Appl. Phys. Lett._ 102, 012109 (2013). Article  ADS  Google Scholar  Download references ACKNOWLEDGEMENTS We acknowledge financial support from the Italian Ministry of Research

(MIUR) through the PRIN 2017 project “Interacting photons in polariton circuits” (INPhoPOL). Useful discussions with V. Ardizzone, D. Ballarini, A. Gianfrate, E. Maggiolini, D.

Suárez-Forero are gratefully acknowledged. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Dipartimento di Fisica, Università di Pavia, via Bassi 6, I-27100, Pavia, Italy Davide Nigro & 

Dario Gerace * Dipartimento di Fisica, Università di Napoli Federico II, Complesso Universitario di Monte Sant’Angelo, Via Cintia, 80126, Napoli, Italy Vincenzo D’Ambrosio * CNR NANOTEC,

Institute of Nanotechnology, Campus Ecotekne, Via Monteroni, 73100, Lecce, Italy Daniele Sanvitto Authors * Davide Nigro View author publications You can also search for this author inPubMed

 Google Scholar * Vincenzo D’Ambrosio View author publications You can also search for this author inPubMed Google Scholar * Daniele Sanvitto View author publications You can also search for

this author inPubMed Google Scholar * Dario Gerace View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS D.N. proposed the original idea of the

analog Ramsey interferometer for polaritons, and performed all the numerical simulations. D.G. supervised the work; D.N., V.D’A., D.S., and D.G. discussed the results, their interpretation,

and the physical realization of the proposed devices. D.N. and D.G. drafted the first version of the manuscript, with key inputs from all the authors to produce the final version.

CORRESPONDING AUTHORS Correspondence to Davide Nigro or Dario Gerace. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. PEER REVIEW PEER REVIEW INFORMATION

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ARTICLE Nigro, D., D’Ambrosio, V., Sanvitto, D. _et al._ Integrated quantum polariton interferometry. _Commun Phys_ 5, 34 (2022). https://doi.org/10.1038/s42005-022-00810-9 Download citation

* Received: 13 July 2021 * Accepted: 21 December 2021 * Published: 03 February 2022 * DOI: https://doi.org/10.1038/s42005-022-00810-9 SHARE THIS ARTICLE Anyone you share the following link

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