Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction

applied sciences Article Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction Jesús Liñares *,† , Xesús Prieto-Blanco †, Gabriel M. Carral †and María C. Nistal † Quantum Materials and Photonics Research Group, Optics Area, Department of Applied Physics, Faculty of Physics /Faculty of Optics and Optometry, Campus Vida s/n, University of Santiago de Compostela, E-15782 Santiago de Compostela, Galicia, Spain; [email protected] (X.P.-B.); [email protected] (G.M.C.); [email protected] (M.C.N.) *Correspondence: suso.linar[email protected] † These authors contributed equally to this work. Received: 8 November 2020; Accepted: 7 December 2020; Published: 10 December 2020   Abstract: In this work, we present the physical simulation of the dynamical and topological properties of atom-field quantum interacting systems by means of integrated quantum photonic devices. In particular, we simulate mechanical systems used, for example, for quantum processing and requiring a very complex technology such as a spin-1 / 2 particle interacting with an external classical time-dependent magnetic field and a two-level atom under the action of an external classical time-dependent electric (optical) field (light-matter interaction). The photonic device consists of integrated optical waveguides supporting two collinear or codirectional modes, which are coupled by integrated optical gratings. We show that the single-photon quantum description of the dynamics of this photonic device is a quantum physical simulation of both aforementioned interacting systems. The two-mode photonic device with a single-photon quantum state represents the quantum system, and the optical grating corresponds to an external field. Likewise, we also present the generation of Aharonov–Anandan geometric phases within this photonic device, which also appear in the simulated systems. On the other hand, this photonic simulator can be regarded as a basic brick for constructing more complex photonic simulators. We present a few examples where optical gratings interacting with several collinear and/or codirectional modes are used in order to illustrate the new possibilities for quantum simulation. Keywords: integrated photonics; quantum optics; quantum simulation 1. Introduction One of the most promising tasks in quantum science and technology is the implementation of quantum simulations. Its physical foundation is based on the fact that the dynamics of a quantum system is governed by its Hamiltonian ˆ H (time evolution) or momentum operator ˆ M (spatial evolution), that is given a Hilbert space H and some input state |Ψ(0)i ∈ H , the full evolution of the system is given by the action of the evolution operator on such a state. The evolution operator can be either the time evolution one ˆ Ut=exp (−iˆ Ht/¯h) , which comes from the Schr ¨ o dinger equation −i¯h∂|Ψi/∂t=ˆ H|Ψi , that is |Ψ(t)i=ˆ Ut|Ψ( 0 )i , or its spatial counterpart ˆ Us=exp (iˆ Ms/¯h) , which comes from the momentum operator in the position representation, that is i¯h∂|Ψi/∂s=ˆ M|Ψi , where s is the spatial variable that defines the direction along which the system evolves (it can be, for instance, the z -direction) then |Ψ(s)i=ˆ Us|Ψ( 0 )i . One is perhaps more familiar with the temporal case |Ψ(t)i=ˆ Ut|Ψ(0)i , but note that the spatial case is totally analogous [ 1 , 2 ]. Now, as it occurs in nature that very different and unrelated quantum systems share analogous Hamiltonians or momentum operators, the dynamics of these systems will be analogous. This implies that if we have some system of interest, we will be able Appl. Sci. 2020,10, 8850; doi:10.3390/app10248850 www.mdpi.com/journal/applsci Appl. Sci. 2020,10, 8850 2 of 21 to mimic its behavior (evolution) by means of other very different system. This last system, which is required to be fully controllable by the experimenter, constitutes a quantum simulator [ 3 , 4 ] of the system of interest. On the other hand, the importance of making quantum simulations obeys various reasons. First of all, whether classical or quantum, simulation is often an indispensable tool in science. Physical systems can be very complicated to study. If we find a way to mimic their behavior in a manageable way, their dynamics can be analyzed with much less difficulty, much less expensively, and much faster [ 3 ]. For instance, we can imagine that the system of interest is one whose working conditions are fragile and very sensitive to small perturbations. It could also be that those conditions are very specific and/or hard (even impossible) to achieve in the laboratory. Simulating these complicated systems will, in principle, improve our knowledge about them with much ease. Secondly, quantum simulation offers a crucial advantage over classical simulation. Quantum systems have a greater information storing capacity than classical systems; thus, a quantum simulator, which is a quantum system itself, will require much fewer physical resources than a classical simulator to store the same amount of information. Furthermore, classical simulations find difficulties simulating quantum systems when strong entanglement is present [ 3 ]. Finally, it is possible that the potential applications of a quantum system may be better implemented using a quantum simulation [ 5 ], if it happens that the quantum simulation parameters are easier to handle, compared to those of the quantum system of interest, in the sense of greater tunability [ 4 ]. All of this strongly suggests the need for quantum simulations. Formally, one has a quantum device, called the quantum simulator, which reproduces the evolution of the system of interest, the quantum system. Following [ 4 ], let us call the initial input state of the quantum system |φ( 0 )i , which evolves to |φ(ζ)i , where ζ stands either for time or some spatial coordinate, under the evolution operator ˆ U=exp (iˆ Oζ/¯h) . Here, ˆ O can be a Hamiltonian ( −ˆ H ) or a momentum operator ( ˆ M ). The quantum simulator, on the other hand, starts in the state |ψ(0)i and evolves to |ψ(ζ)i under the action of the operator ˆ U0=exp (iˆ O0ζ/¯h) . Since we have a system and a simulator, there exists a correspondence relating these elements, that is a correspondence between |φ(0)i ↔ |ψ(0)i , |φ(ζ)i ↔ |ψ(ζ)i , ˆ U↔ˆ U0 , and thus, ˆ O↔ˆ O0 . This is revealed via measurements on both systems. The greater the accuracy of this correspondence, the more one can trust the simulator [ 3 ]. In this work, we present an integrated quantum photonic simulator for atom-field quantum interacting systems. It is based on optical gratings and can be regarded as a basic brick for constructing more complex photonic simulators. First of all, we would like to stress the high interest in quantum photonic simulators; for instance, integrated photonic structures allow simulating a number of condensed matter effects such as Anderson localization, Mott transition, etc. [ 6 ], and more recently, anyonic interaction has been simulated by using an array of channel optical waveguides with a helically bent axis [ 7 ]. In our case, we will simulate in an integrated photonic device the dynamical and topological properties of two very well-known quantum interacting systems. This is relevant since a way to check the fidelity of a simulation is testing the same physical model of the dynamics with different systems and then comparing the results [ 5 ]. The two systems to be simulated are the well-known interaction of a spin-1 / 2 particle under a time-dependent external classical magnetic field (see for instance [ 8 ]) and the interaction of a two-level atom with an external classical electric field (see for instance [ 9 ]) which are used, at present, for implementing quantum information devices for quantum processing, quantum computation, and so on [ 8 ]. It is also well known that these mechanical systems require a very complex technology. We will take a standard photonic device, that is a two-mode planar guide modulated by an integrated grating (see for instance [ 10 ]) in which a single-photon quantum state is propagated. We will show that it can emulate both the dynamical and geometrical properties of the two aforementioned systems. As mentioned, this photonic simulator based on integrated optical gratings can be regarded as a basic brick for more complex photonic simulators; thus, the results obtained can be extended to multi-level systems or to several concatenated two-mode systems simulating concatenated temporal operations. Addressing practical issues, this simulator will also benefit from some advantages of classical and quantum integrated photonics [6,11,12] , such as Appl. Sci. 2020,10, 8850 3 of 21 the fast and energetically efficient operation and miniaturization capabilities of integrated photonics, which favor scalability. Moreover, integrated photonic devices are becoming some of the most powerful and appealing technologies for classical and quantum information [ 13 , 14 ]. Finally, we must stress that although we present a photonic device for quantum simulation, it can be also used to implement quantum operations and/or effects with the photonic device itself such as logic gates, geometric phases, and so on. The plan of the paper is as follows: In Section 2, we briefly present the Hamiltonian of the quantum interacting systems, in a suitable form for their photonic simulation, along with the physical parameters and some useful solutions. In Section 3, starting from quantum states as single-photon states, an integrated photonic device for the quantum simulation of both a spin-magnetic field coupling and light-matter interaction is presented, along with the limits of this simulation. In Section 4, quantum simulation of geometric phases is studied. Finally, a summary is presented in Section 5. 2. Mechanical Interacting Systems In this section, we present the main results about the mechanical interacting systems whose dynamic and topological properties are going to be simulated with an integrated photonic device. The primary aim is to write the Hamiltonians of these systems in a suitable form to facilitate the study of their photonic simulations. 2.1. Spin-1/2 Particle Interacting with a Magnetic Field The Hamiltonian of a spin particle interacting with a magnetic field is given by ˆ H=−µ·B , where µ is the magnetic moment of the particle, which is proportional to the spin operator, and B is the magnetic field. Let us consider a spin1 2 particle, then µ=1 2µσ , where µ=¯hγp , with γp the gyromagnetic ratio of the particle ( p= e for an electron, p= n for a neutron, etc.), and σ= (σx , σy , σz) are the Pauli matrices. Moreover, we choose a time-dependent magnetic field, which, by assuming a sinusoidal dependence in time with frequency ω , is written as B=B0(sin θcos ωt , −sin θsin ωt , cos θ) , representing a magnetic field of modulus B0 rotated an angle θ with respect to the z -axis and spinning around this same axis with a frequency ω . We can then write the well-known spin-magnetic Hamiltonian as (see for instance [8]): ˆ H(t) = −¯h 2γpB0cos θσz−¯h 2γpB0sin θ(cos ωtσx−sin ωtσy). (1) By taking into account the expressions of the Pauli matrices, then the above Hamiltonian gives rise to the following time-dependent Schr¨ odinger equation: i¯h∂ ∂t|Ψ(t)i=−¯h 2γpB0 cos θsin θexp (iωt) sin θexp (−iωt)−cos θ!|Ψ(t)i= ≡ˆ H(t)|Ψ(t)i. (2) This time-dependent Hamiltonian is enough for our simulation purposes. Note that for the neutron case, we have the well-known Nuclear Magnetic Resonance (NMR). The solution of the equation above can be obtained as a time-dependent linear combination of down and up spin states, | 0 i and | 1 i , of the particle, that is |Ψ(t)i=co(t)| 0 i+c1(t)| 1 i . Thus, by using the vector representation of |Ψ(t)i and the matrix Hamiltonian given by Equation (2), the Schr ¨ o dinger equation can be rewritten as follows: i¯hdcn(t) dt=Encos θcn(t) + i∑ n6=m Cnm(t)cm(t). (3) Appl. Sci. 2020,10, 8850 4 of 21 with n= 0, 1, Eo=−¯hγp 2Bo and E1=¯hγp 2Bo the eigenvalues of spin states when a static magnetic field along z -direction is applied, and C01(t) = C∗ 10(t) = −¯hγp 2Bosin θexp(iωt) the coupling coefficients. On the other hand, it is interesting, for the sake of expositional convenience, to present the main results about the dynamics of the system. First of all, in order to simplify things, we can shift to a reference frame that is rotating at a frequency ωby means of the following unitary transformation: |η(t)i=ˆ U(t,ω)|Ψ(t)i=exp (−iωσzt/2)|Ψ(t)i; (4) therefore, by inserting |Ψ(t)i=exp (iωσzt/2)|η(t)i into the Schr ¨ o dinger equation, we obtain, after some simple calculations, the following Hamiltonian acting on |η(t)i: ˆ H0|η(t)i=¯h 2 −γpB0cos θ+ω−γpB0sin θ −γpB0sin θ γpB0cos θ−ω!|η(t)i. (5) This Hamiltonian ˆ H0 can be rewritten as a product of the form −¯h 2γp(σ·Be f ) , with Be f an effective static magnetic field, that is, ˆ H0=−¯h 2γp(σ·Be f ) = −¯h 2γpB0∆cos θ0σz−¯h 2γpB0∆sin θ0σx=−¯h 2∆o cos θ0sin θ0 sin θ0−cos θ0!, (6) with: ∆o=γpB0∆=γpB0 o(7) and: ∆=γpB0s1−2ω γpB0 cos θ+ω2 γ2 pB2 0 ; (8) therefore, the effective magnetic field is given by Be f = (B0 0sin θ0 , 0, B0 0cos θ0) , where B0 0 and θ0 are related to B0 and θ by B0 0=B0∆ , with sin θ0=γpB0sin θ/∆o=sin θ/∆ and cos θ0=γpB0[cos θ− (ω/γpB0)]/∆o= [cos θ−(ω/γpB0)]/∆ . As seen from the expression for ∆ , it would seem that for this reparametrization to have physical meaning, restrictions on the parameters would appear, as ∆ contains a possible non-positive term under the square root. The worst case would be likely to happen if cos θ= 1, but one can find that the resulting quantity 1 −( 2 ¯hω/µB0)+(¯hω/µB0)2 is never negative, for any value of the parameters. Next, it is interesting to compute the eigenstates |ηi and eigenvalues Eη of Hamiltonian ˆ H0 given by Equation (6), that is |η(t)i=exp−i ¯hEηt|η(0)i. After a standard calculation, we have: |η+(0)i=cos θ0 2|0i+sin θ0 2|1i,|η−(0)i=sin θ0 2|0i−cos θ0 2|1i, (9) where | 0 i ≡ ( 1, 0 )T and | 1 i ≡ ( 0, 1 )T , with T denoting the transpose. The eigenvalues are E±=±¯h 2∆o , that is E±=±¯h 2γpB0 0=±µ 2B0 0 . Therefore, by taking into account Equation (4), the full evolved states |Ψ(t)ican be easily obtained. 2.2. Two-Level Atom Interacting with an Electric (Optical) Field On the other hand, let us consider a two-level atom coupled to a harmonic external classical electric field, which describes semiclassical light-matter interaction. As is usually done [ 9 ], we label the atomic levels as |gi (ground) and |ei (excited). They have energies ¯hωg and ¯hωe , respectively. Appl. Sci. 2020,10, 8850 5 of 21 Their energy difference is given by hω0=¯h(ωe−ωg) . The expression for this non-interacting part of the Hamiltonian can be formally written as follows: ˆ Ho=¯hωg|gihg|+¯hωe|eihe|=¯h¯ ωI+¯hωoσz/2, (10) where we used the matrix representation of |gihg| and |eihe| , with ¯ ω= (ωg+ωe)/ 2, and I the two-dimensional identity matrix. As for the interacting part, it is given by the dipole interaction (electric dipole approximation) between an external electric (optical) field and the atom. Indeed, by assuming, for the sake of simplicity, that the field is propagating along z , has a sinusoidal dependence in time with frequency ω , and is linearly polarized along the x -direction, then the electric (optical) field can be written as follows E=E0cos ω(z/c−tux . Moreover, we disregard the spatial dependence of the electric (optical) field because the wavelength λ=c/ 2 πω is considered much larger than the atomic dimensions. Therefore, we apply the dipole or long-wavelength approximation, that is by assuming without lost of generality ωz/c= 2 mπ , with m an integer, then E=E0cos ωtux , the electric-dipole interaction is given by ˆ HI=−d·E=−dxEx= e xEx , where d is the atomic dipole operator d=q·r , with q=− e, and r is the electron’s position vector (operator). This interaction term allows transitions between the two levels. The form of the dipole operator can be calculated by using twice the closure relationship ˆ I=|gihg|+|eihe|, that is ˆ I(−ex)ˆ I; therefore: dx=−hg|ex|gi|gihg|−he|ex|ei|eihe|−hg|ex|ei(|gihe|+|eihg|)≡ −c+I+c−σz−doσx, (11) with c±= (hg| e x|gi±he| e x|ei)/ 2, d0=hg| e x|ei , and where we have used the matrix representations (|gihe|+|eihg| ≡ σx , |gihg| ≡ (I+σz)/ 2, and |eihe|= (I−σz)/ 2. We must stress that sometimes, parity arguments [ 9 ] narrow the form of the dipole operator, leading up to the expression dx= −hg|ex|ei(|gihe|+|eihg|) = −doσx . Likewise, it is customary to introduce the Rabi frequency Ω=Eodo/¯h . In short, the total Hamiltonian is given by ˆ Ho+ˆ HI , and therefore, the corresponding Schr¨ odinger equation, by using this new variables, is given by: i¯h∂ ∂t|Ψ(t)i=¯h(¯ ω+C+)I−¯h(ω0 2−C−)σz+¯hΩσxcos ωt|ψ(t)i=ˆ H|Ψ(t)i, (12) with C±=c±cos ωt . The general solution of this Schr ¨ o dinger equation can again be obtained by using a time-dependent linear combination of fundamental and excited states, |gi≡| 0 i and |ei≡| 1 i , of the particle, that is |Ψ(t)i=co(t)| 0 i+c1(t)| 1 i . However, the high value of the frequency ω means that the electric field is rapidly oscillating, which suggests to make the following change |Ψ(t)i=f0(t)exp(iωt/ 2 )| 0 i+f1(t)exp(−iωt/ 2 )| 1 i ≡ exp (iωσzt/2)|η(t)i . Moreover, in many cases, by parity arguments, it is fulfilled that c±= 0. Therefore, by substituting this state into Equation (12), we obtain the following Hamiltonian for the state f0(t),f1(t)T≡ |η(t)i, ˆ H0|η(t)i ≈ ¯h¯ ωI−¯hδ 2σz+¯hΩ 2σx|η(t)i, (13) with δ= (ω0−ω) the detuning parameter and where we have neglected terms rapidly oscillating of the form exp(±iωt) ; that is, a temporal Rotating Wave Approximation (RWA) has been made, and a time independent Hamiltonian has thus been obtained. We must stress that under this approximation, the terms C± in Equation (12) can be also neglected independently of the wave-function parity. Finally, note that Hamiltonians given by Equations (6) and (13) have the same algebraic structure. 3. Quantum Photonic Simulations In this section, we present the quantum simulation, which can be implemented by an integrated photonic device, that is integrated optical gratings supporting two collinear guided modes, that is Appl. Sci. 2020,10, 8850 6 of 21 two mode guides assisted by a periodic perturbation. It can be considered the basic brick for constructing more complex simulators. 3.1. Classical Study of the Photonic Device Let us consider a standard integrated photonic device consisting of, for example, an integrated waveguide 1D (one-dimensional) with refractive index n(x) (slab guide) that supports two optical modes e0(x) and e1(x) . These modes are collinear and travel in the z -direction with propagation constants β0 and β1 , that is wave vectors in the z -direction defined as β= (ω/c)N , where ω is the frequency of the mode and N is the effective index [ 10 ]. The mentioned modes are coupled by an integrated grating present in a region of the slab guide, as shown in Figure 1a), where relevant parameters are indicated, that is substrate index ns , film index nf , cover index nc= 1, and film thickness d . The grating is represented, for example, by a periodic modulation (perturbation) of the electrical permittivity [10], ∆e(x,z) = ∆e(x)cos(γz+αo), (14) with ∆e(x) the modulation strength of the optical grating, αo an initial phase, if required, γ=2π/Λ the frequency of the perturbation, and Λ its period. This index profile can be obtained by different technologies of integrated optics, for instance ion-exchange in glass [15,16] could be used, or even by optical fiber technology [ 17 ]. Likewise, in crystals such as lithium niobate, these optical gratings can also be reconfigurable due to acousto-optic or electro-optic effects [18]. Figure 1. ( a ) Integrated optical grating with period Λ on a two-mode slab waveguide with modes e0(x) and e1(x) . The inset shows the simulated mechanical device: atom-field interaction. ( b ) Integrated optical grating with period Λ on a two-mode channel waveguide with modes, for example e00(x , y) and e10(x,y). As we assume that only two guided modes are excited in our integrated photonic structure, the perturbed electric field amplitude is given by e(x , z) = a0(z)e0(x) + a1(z)e1(x) . It is well known that the general set of equations that describe the coupling between n copropagating optical modes with propagation constants βn in a perturbed waveguide and that allow calculating the amplitude coefficients an(z)is given by (see for instance [10]): −idan(z) dz =˜ βn(z)an(z) + ∑ n6=m Cnm(z)am(z), (15) Appl. Sci. 2020,10, 8850 7 of 21 where ˜ βn(z) = βn+Cnn(z) are the corrected propagation constants due to the self-coupling coefficients Cnn and Cnm are the coupling coefficients between the n mode and each of the other m modes. All these coefficients are calculated as follows [10]: Cnm(z) = ω 2Z∆e(x,z)en(x)e∗ m(x)dx, (16) with ω the temporal frequency of the modes and en(x) and em(x) the normalized optical n and m modes of a planar guide. The study with 2D guides (integrated channel guides or optical fibers) can be also made, but no new relevant result would be obtained. Indeed, a channel guide can be defined starting from a planar guide whose width is reduced up to a size a of the same order as its depth, that is a≈d , as shown in Figure 1b). In such a case, the optical modes are characterized by two subscripts, one for each spatial direction, that is enp(x,y)and emq(x,y); therefore, the general coupling coefficients are given by: Cnpmq(z) = ω 2Z∆e(x,y,z)enp(x,y)e∗ mq(x,y)dxdy, (17) where we assumed that the grating modulation can also have a y -dependence. Finally, the mode coupling equations can be obtained by applying the formal changes n→np , m→mq in Equation (15). Accordingly, the results for planar guides can be easily transferred to channel guides. 3.2. Quantum Study of the Photonic Device A canonical quantization procedure [ 1 , 2 ] proves that the a(z) coefficients become the photon absorption (or emission) operators ˆ a(z) (correspondence principle). Therefore, the coupled mode equations are in fact the Heisenberg equations, which, in general, give the evolution of the operators in time. In this case, they give the spatial evolution of the operators. Moreover, the relevant operator here responsible for the spatial propagation of quantum states of the device is the momentum operator, which is the generator of spatial translations [ 2 , 19 ], and not the Hamiltonian, which is the generator of temporal translations. In short, we can study the integrated photonic device in a fully quantum mechanical way, by solving the equations for absorption operators (spatial Heisenberg equations). That is, by performing the change a(z)→¯hˆ a(z)in Equation (15), we obtain [2,20]: −i¯hdˆ an(z) dz =¯h˜ βnˆ an(z) + ¯h∑ n6=m Cnm(z)ˆ am(z). (18) We must stress that modal coupling preserves energy; therefore, Equation (18) corresponds to a unitary transformation, and accordingly, Cnm =Cmn . On the other hand, we only consider single-photon states, which is enough for our simulation purposes. We must stress that the linear momentum of a single photon without modal coupling, that is ∆e(x , y) = 0, is given by p(0,1)=¯hβ(0,1) depending on whether the photon is excited in mode β0 or mode β1 [ 19 , 20 ]. Obviously, more general quantum states could be used such as multiphoton states, entangled states of two photons, and so on. These states would give rise to more complex quantum simulations, which fall outside of the scope of this work. In general, the single-photon state is a quantum superposition because the photon can either be excited in the mode β0 , that is | 1 0i , or in the mode β1 , that is | 1 1i . Hence, the general quantum state is given by: |L(z)i=a0(z)|10i+a1(z)|11i, (19) where a0(z) and a1(z) are the quantum complex amplitudes and fulfil the normalization condition |a0(z)|2+|a1(z)|2= 1. States | 1 0i and | 1 1i must be understood as single-photon states at a distance (plane) z . It can be checked that the solutions of Equation (18) for the spatial propagation of emission operators are the same as for single-photon states |L(z)i [ 21 ]. The main reason is that Appl. Sci. 2020,10, 8850 8 of 21 single-photon states are proportional to emission operators, that is |10i=ˆ a† 0| 0 i , |11i=ˆ a† 1| 0 i . Indeed, let us consider free propagation, that is non-coupling case Cnm = 0, then the solutions of Equation (18) are ˆ a0,1(z) = exp(iβ0,1z)ˆ a0,1( 0 ) . Now, let us consider, for the sake of simplicity, a single-photon state at z= 0, for instance |10i , then the optical propagation can be obtained as follows: |10i=ˆ a† 0(0)|0i (one-photon emission), but by taking into account the z -propagation, we have |L(z)i=exp(iβ0z)ˆ a† 0(z)| 0 i=exp(iβ0z)|10i ≡ a0(z)|10i ; therefore, the coefficients a0,1(z) of a single-photon state have the same optical propagation solution as the operators ˆ a0,1(z). This can be proven for a general unitary transformation after a certain algebra. We will take advantage of this property for simulation purposes. Accordingly, for single-photon states, Equation (18) can formally be rewritten as follows: i¯hd|L(z)i dz =−¯h ˜ β0(z)C12(z) C21(z)˜ β1(z)!|L(z)i=−ˆ M(z)|L(z)i, (20) where |L(z)i is given, in vector representation, by (ao(z) , a1(z))T and ˆ M is the matrix representation of the so-called momentum operator. This is equivalent to the matrix equation given by Equation (2) for the Hamiltonian. We must stress that Equations (2), (3), (13), (18) and (20) are the main results for implementing the quantum simulations in this work. 3.3. Photonic Simulation of Spin-Magnetic Field Interaction Let us consider an integrated optical grating characterized by the function given by Equation (14). It will be useful to work with the slowly varying operators ˆ An , defined as ˆ An=ˆ anexp(−iβnz) . On the other hand, the coupling coefficients for a grating with initial phase αo= 0 can be written as C00 =c00 cos γz , C11 =c11 cos γz , and C01 =C10 =Cocos γz , where cnm = (ω/ 2 )R∆e(x)en(x)· e∗ m(x)dx , and so on. Therefore, from Equation (18), we obtain, for the two-mode case, the following spatial Heisenberg equations: idˆ A0(z) dz =−ˆ A0c00 cos γz−ˆ A1C0 2e−i∆β01z[eiγz+e−iγz], (21) idˆ A1(z) dz =−ˆ A1c11 cos γz−ˆ A0C0 2ei∆β01z[eiγz+e−iγz], (22) where ∆β01 =β0−β1 . The above equation reveals that we have oscillating terms coming from the cosine of the self-coupling terms with arguments ±γz and also terms with arguments (γ−∆β01)z and (γ+∆β01)z . We make the assumption that γ is of the same order as ∆β01 , so that (γ−∆β01) is small. The other terms are rapidly oscillating and, thus, will average to zero on a sufficiently large z -scale. We must stress that what we do here is essentially a spatial RWA, which is well-known in light-matter interaction in the time domain. Therefore, the above equations become, to a good approximation, idˆ A0(z) dz=−ˆ A1 Co 2exp [−i(∆β01 −γ)z], (23) idˆ A1(z) dz=−ˆ A0 Co 2exp [i(∆β01 −γ)z]. (24) Next, we make the following relabeling ∆β01 ≡∆β and define the new absorption operators ˆ A0=ˆ b0exp(−i∆βz/2) and ˆ A1=ˆ b1exp(i∆βz/ 2 ) . We rewrite the Heisenberg equations in terms of the ˆ b(z)operators, idˆ b0 dz=−∆β 2ˆ b0−Co 2ˆ b1exp (iγz), (25) idˆ b1 dz=∆β 2ˆ b1−Co 2ˆ b0exp (−iγz). (26) Appl. Sci. 2020,10, 8850 9 of 21 It is interesting to note that ˆ a0,1 and ˆ b0,1 are, after the spatial RWA, the same operators, except a global phase, that is ˆ a0,1 =b0,1 exp(i¯ βz) , ¯ β= (β0+β1)/ 2. Finally, as mentioned above, we can write the above equation in a matrix form acting on the single-photon state |Lb(z)i=b0(z)| 1 0i+b1(z)| 1 1i , where |L(z)i=|Lb(z)iexp(i¯ βz), that is, i¯h∂ ∂z|Lb(z)i=−¯h 2 ∆βCoexp (iγz) Coexp (−iγz)−∆β!|Lb(z)i= =−¯h 2B cos αsin αexp (iγz) sin αexp (−iγz)−cos α!|Lb(z)i=−ˆ M(z)|Lb(z)i, (27) with cos α=∆β/ B, sin α=Co/ B, and B =p∆β2+C2 o . This equation is just the spatial equivalent of Equation (2). A Hamiltonian operator is replaced by a momentum operator. Obviously, the dynamic properties are identical under the formal changes: ω↔γ , γpBo↔ B, θ↔α . Note that a rotating equivalent vector (γpB)eq ≡B= B (sin αcos γz , −sin αsin γz , cos α) is obtained. In short, we have achieved a photonic simulator of aspin-magnetic field coupling. However, we must stress some limitations for this simulator. The momentum operator ˆ M defined by (27) and simulating the coupling spin-magnetic field only is valid under the spatial RWA, that is the values of ∆β and γ have to be high and not too different. Therefore, we will be able to simulate the interaction with magnetic fields with a large z-component and oscillating with a frequency ωof the same order as the term γpBocos θ. Finally, it is interesting to obtain a constant momentum operator by applying a unitary transformation (rotating reference system) to the quantum state |Lb(z)i, that is, |l(z)i=ˆ U(z,γ)|Lb(z)i=exp (−iγσzz/2)|Lb(z)i. (28) Therefore, by using |Lb(z)i=exp (iγσzz/2)|l(z)i in Equation (27), we obtain, after a certain, but straightforward calculation, the following momentum operator for the state |l(z)i: −ˆ M0|l(z)i=¯h 2 −∆β+γ−Co −Co∆β−γ!|l(z)i=−¯h 2Do cos α0sin α0 sin α0−cos α0!|l(z)i, (29) where cos α0= (∆β−γ)/ D o , sin α0=Co/ D o , and D o=p(∆β−γ)2+C2 o . As shown later, these results are important for simulating both dynamical and topological properties. By comparison between Equations (5) and (29), we obtain the following simulation parameters: (ω−γpBocos θ)↔(∆β−γ),γpBosin θ↔Co. (30) Next, it is interesting to compute the eigenstates |l(z)i and eigenvalues βl of the momentum operator ˆ M0 given by Equation (6), that is |l(z)i=exp i ¯hplz|l(0)i=exp iβlz|l(0)i . After a standard calculation, we have: |l+(0)i=cos α0 2|10i+sin α0 2|11i,|l−(0)i=sin α0 2|10i−cos α0 2|11i, (31) with eigenvalues β±=± D o/ 2, that is linear momentums p±=±¯h 2Do=±po . Therefore, by taking into account Equation (28), the full evolved states |Lb(z)i can be easily obtained. Note that we are simulating the quantum state Ψ(t) given by Equation (4), and thus, for example, quantum processing based on NMR could be simulated by this photonic device. For the sake of expositional convenience, we will return to this question in the next subsection. Appl. Sci. 2020,10, 8850 16 of 21 In short, a single-photon state acquires an AA geometric phase under propagation in an integrated photonic grating. The same expression is found for a spin-1/2 particle in a magnetic field; therefore, topological simulations can be made. Thus, by applying the simulation parameters given by Equation (30), geometric phases can be obtained. 4.2. Elimination of the Dynamical Phase in Spin-Magnetic Field Photonic Simulation It is well known in the mechanical case that the geometric phase is hidden in the spin-magnetic field interaction because it is combined with the dynamical phase within φ± ; therefore, the dynamical phase has to be eliminated in order to take advantage of the properties of a geometric phase. We present a photonic solution, which is similar to the one used in the mechanical case, that is if the quantum state, after evolution under a first Hamiltonian ˆ H1=ˆ H for time T , finds a second Hamiltonian ˆ H2=−ˆ H , then the dynamical phase are mutually canceled; however, the eigenstates do not change, therefore neither does the geometrical phase. In the photonic case, we have to find a new momentum operator, that is a new integrated optical grating, such as ˆ M2=−ˆ M . We assume that such a new optical grating has the same frequency γ ; therefore, we have the same rotating system, that is the same transformation (28). Accordingly, the condition for eliminating the dynamical phase is obtained from the operator ˆ M0, that is, ˆ M0 2=¯h 2 ∆β0−γC0 o C0 o−∆β0+γ!=−ˆ M0=−¯h 2 ∆β−γCo Co−∆β+γ!; (48) therefore, C0 o=−Co and (∆β−γ) = −(∆β0−γ) . The first condition can be achieved by introducing an initial phase in the second grating, that is ∆e(x , z) = ∆e(x)cos(γz+π) , and the second one is achieved if ∆β0= 2 γ−∆β . These results indicate that we need an additional grating with a new difference between propagation constants (linear momentum of the photon) ∆β0= (β0 0−β0 1) . In Figure 5the system for eliminating the dynamical phase is shown . Therefore, the total phases are φ±=ν(∓pΛ/¯h−π), and the dynamical phases are: φ± d=p± ¯hνΛ∓νπ cos α0(49) Therefore, the total dynamical phase is Φd= 0, and the total geometrical phase after the single-photon state propagates through the two integrated gratings is twice the value acquired in the first grating, that is, Φ± g=∓2νπ(1−cos α0) = ∓Φg. (50) Alternatively, propagation constants can be unchanged, and the grating frequency can be modified, that is γ0= 2 ∆β−γ ; however, in this case, the transformation (28) must be applied with the factor γ0 . In short, we eliminated the dynamical phase; therefore, these results could be used for implementing logic gates or transformations based on topological phases, which are much more insensitive to fabrication errors, unlike dynamical phases, which as mentioned are of order ω . With these results, robust P-gates (Phase gates) can be designed; thus, the following transformation is implemented between the eigenstates: P= exp(−i2νπ cos α0)0 0 exp(i2νπ cos α0)!=exp(−iΦgσz)(51) Note that for 4 νcos α0=± 1, a Z-gate is obtained, and for 4 νcos α0=± 1 / 2, an S-gate is implemented, and so on. Appl. Sci. 2020,10, 8850 17 of 21 Figure 5. Elimination of the dynamical phase by using two consecutive integrated optical gratings. The second grating has an initial phase π . Likewise, a prism is used to make a projective measure of states |10iand |11ifor obtaining the probabilities P0and P1and, therefore, the geometric phase Φg. 4.3. Geometric Phases with Other Quantum States On the other hand, the eigenstates given by Equation (31) can be written as single-photon states excited in rotated optical modes, that is e+(x , y) = cos α0 2eo(x , y) + sin α0 2e1(x , y) and e−(x , y) = −sin α0 2eo(x,y) + cos α0 2e1(x,y); therefore: |l+(0)i=cos α0 2|10i+sin α0 2|11i=|1+i,|l−(0)i=−sin α0 2|10i+cos α0 2|11i=|1−i(52) The elimination of the dynamical phase means that these eigenstates have undergone the transformation e±iΦg|1±i; therefore, the following formal relationships can be written: eiΦg|1+i=eiΦgˆ a† +|0±i,e−iΦg|1−i=e−iΦgˆ a† −|0±i(53) with Φg=− 2 νπ( 1 −cos α0) . Accordingly, the following transformations, induced by geometric phases, for the absorption operators are obtained: ˆ a±(z=nΛ) = e±iΦgˆ a±(0)(54) This result can be used for obtaining geometric phases of other quantum light states. As an example, we present two states, that is the number photon state or Fock state |n+i and the coherent state |α+i , where subindex + indicates that the state is excited in the optical mode e+(x , y) . The Fock state under propagation becomes: |n+(0)i=1 √n+!ˆ a+(0)†n+|0i −→ 1 √n+!ein+Φg(ˆ a† +n+|0i=ein+Φg|n+i(55) Therefore, the quantum state has acquired a geometric phase n+Φg . Likewise, the coherent state can be rewritten by using the complex displacement operator, that is, |α+(0)i=e(α+ˆ a† +(0)−c.h.)|0i −→ e(α+eiΦgˆ a† +−c.h.)|0i=|eiΦgα+i(56) Therefore, the geometric phase is Φg , that is the same as the one acquired by a single photon. The same procedure can be applied to any other quantum light state excited in the integrated photonic grating. Appl. Sci. 2020,10, 8850 18 of 21 4.4. Optical Measurement of Geometric Phases Finally, we present how to measure the geometric phase starting from the measurements of the single-photon detection probability, which can be extracted by a prism-waveguide coupler [ 10 ], as shown in Figure 5. We focus on the spin-magnetic field interaction simulation case. By assuming that the input state is a single photon excited in the mode e0(x , y) and taking into account the relationships given by Equation (52), the following final state is obtained after the two gratings: |L(z)i= (cos2α0 2+sin2α0 2eiΦg)|10i+cos α0 2sin α0 2(1−eiΦg)|11i(57) The probability of the detection of a photon in Mode 0, that is P0 , or in Mode 1, P1 , is a function of the geometric phase, that is, P1=sin2α 2(1−cos Φg),P0=1−P1(58) If Φg= 2 π , then P1= 0 and P0= 1, and if Φg=π , then P1=sin2α and P0=cos2α . Therefore, from the measurement of P1 and P0 , the phase Φg is obtained. In Figure 5it is shown the projective measure of states | 1 0i and | 1 1i by a prism-waveguide coupler, which projects these state in different spatial directions. Finally, note that by using a coherent state, these probabilities are proportional to the intensity of the light, what can be called a semiclassical optical characterization, or in the most technical way, the geometric phase is also acquired by the classical fields, but would rigorously correspond to the so-called Hannay phase [33]. 4.5. Geometric Phases in Light-Matter Photonic Simulation Finally, we check that light-matter simulation can be also used to obtain geometric phases. For the sake of simplicity, we show geometric phases for wedge circuits, although more general cases can be studied. Let us consider an initial state |L( 0 )i=| 1 0i , that is the point ( 0, 0, 1 ) on the photonic Bloch sphere. Next, let us consider an asynchronous optical grating with δsCo ; therefore, according to Equation (37) (phase gate), for δszo= 3 π/ 2, we obtain |L(zo)i= ( 1 /√2)(| 1 0i+i| 1 1i) , that is the state reaches the Bloch sphere point ( 0, 1, 0 ) . Next, we consider that the grating has a greater coupling coefficient Co, then the single-photon state is given by the expression: |L(z)i=1 √2[(cos δrz 2+bsin δrz 2+ia sin δrz 2)|10i+ (asin δrz 2+icos δrz 2−ib sin δrz 2)|11i](59) where a=δs/δr and b=Co/δr . If we choose a distance z=z1 such as δrz1/ 2 =π/ 2, then, after a certain calculation, the following state is obtained: |L(z1)i=eiφo √2(|10i+i|11i)(60) where φo=atn(a/b) = atn(δs/Co) , that is a=sin φo and b=cos φo . The state has reached the point ( 0, − 1, 0 ) of the photonic Bloch sphere. Now, we show that φo is a geometric phase. Indeed, for the sake of symmetry, the state before reaching the above state has crossed the meridian y= 0 when δrz/2 =π/4, that is, the state: |L0i=(1+b+ia) 2|10i+(a+i(1−b)) 2|11i ≡ m0eie0|10i+m1eie1|11i. (61) It is easy to prove that both states have the same phase, that is e0=e1 , and the modulus is given by m0=sin(φo/ 2 ) and m1=cos(φo/ 2 ) ; therefore, the state crosses the point P= (sin φo , 0, cos φo) of the photonic Bloch sphere as indicated in Figure 4for ϕ=φo . Now, we must recall that partial Appl. Sci. 2020,10, 8850 19 of 21 cycles also generate geometric phases, which can be calculated by closing the end points of the open cycle by a geodesic line [ 28 , 30 ]. In our case, the corresponding geodesic lies along the meridian at the plane x= 0, from point ( 0, − 1, 0 ) to initial point ( 0, 0, 1 ) . In short, the state has followed a wedge circuit Cw , as shown in Figure 4. Now, we calculate the subtended solid angle by this wedge circuit. It is easy to check that a wedge circuit with angle φo subtends a solid angle Ω(Cw) = 2 φo ; therefore, the geometric phase is Φg= ( 1 / 2 )Ω(Cw) = φo=atn(δs/Co) , which is just the global phase obtained in Equation (60). It is worth underlining that in this case, the geometric phase is not hidden, and therefore, the dynamical phase does not have to be eliminated. Moreover, this geometric phase can be also obtained in a atom-optical field system out of resonance by using Θ -pulses, with the first optical field with low amplitude Eo (Z gate) and the next with a higher amplitude Eo . By using the simulation parameters given by Equation (34), the geometric phase would be Φg=atn(δ/Ω). 5. Conclusions We propose a quantum photonic device based on integrated optical gratings in a two-mode slab guide to simulate the interaction between external fields and atoms. By using single-photon states, we study the simulations of a spin-(1/2)-magnetic field system, as for example nuclear magnetic resonance, and a two-level atom-optical field system corresponding to light-matter simulation. Both dynamical and geometric properties are simulated, in particular the geometric phases obtained by the mentioned systems. We prove that dynamical properties can be simulated for a wide range of cases with practical interest, although in the spin-(1/2)-magnetic field system it is restricted to relatively high values of frequency and magnetic field amplitude Bo . Overall, atom-optical field interaction does not present these restrictions. This study of integrated optical gratings opens up possibilities to more general simulations if several modes are used. Thus, spin ( s )-magnetic field interaction simulations could be implemented by using a number N= 2 s+ 1 of codirectional optical modes assisted by optical gratings; multilevel atom ( n )-optical field interaction can be simulated by using N=n collinear optical modes coupled by optical gratings, which can in turn simulate, for example, two-qubit single photon logic gates, which has a high interest in quantum information systems; likewise, optical gratings allow interaction between d collinear modes, and thus, simulators based on N codirectional modes can reduce the number of paths used up to N/d , which improves the optical integration of the photonic simulator. On the other hand, AA geometric phases have been also obtained for both systems. The spin-(1/2)-magnetic field system requires dynamic phase cancellation, which is simulated by using two optical gratings; however, in the atom-optical field system, such cancellation is not required. Obviously, we must emphasize that although the proposed integrated photonic device is intended for quantum simulation, it can also be used to implement quantum operations and/or effects with the photonic device itself, such as logic gates, geometric phases, and so on, by using single-photon states or more general quantum states, as shown. Author Contributions: All authors contributed equally to this work. All authors read and agreed to the published version of the manuscript. Funding: Xunta de Galicia, Consellería de Educación, Universidades e FP, Grant GRC Number ED431C2018/11; Ministerio de Economía, Industria y Competitividad, Gobierno de España, Grant Number AYA2016-78773-C2-2-P. Acknowledgments: One of the authors (G.M.C.) wishes to acknowledge the financial support by Xunta de Galicia, Consellería de Educación, Universidades e FP, by a predoctoral grant co-financed with the European Social Fund. Conflicts of Interest: The authors declare no conflict of interest. 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