Experimental simulation of the Parity-Time-symmetric dynamics using photonics qubits

The concept of parity-time (PT) symmetry originates from the framework of quantum mechanics, where if the Hamiltonian operator satisfies the commutation relation with the parity and time operators, it shows all real eigen-energy spectrum. Recently, PT symmetry was introduced into optics, electronic circuits, acoustics, and so many other classical fields to further study the dynamics of the Hamiltonian and the energy of the system. Focusing on the dynamical evolution of the quantum state under the action of PT symmetric Hamiltonian, here we experimentally demonstrated the general dynamical evolution of a two-level system under the PT symmetric Hamiltonian using single-photon system. By enlarging the system using ancillary qubits and encoding the subsystem under the non-Hermitian Hamiltonian with post-selection, the evolution of the state can be observed with a high fidelity when the successfully parity-time symmetrically evolved subspace is solely considered. Owing to the effectively operation of the dilation method, our work provides a route for further exploiting the exotic properties of PT symmetric Hamiltonian for quantum simulation and quantum information processing.


I. INTRODUCTION
The Hermiticity of the operators is considered as one of the fundamental axioms of quantum mechanics, of which guarantees the real energy spectra of the quantum system and the unitary evolution with conserved probability [1,2]. However, a quantum version of non-Hermitian Hamiltonian satisfying parity-time (PT ) symmetry [3,4] (where P and T denote the parity and time reversal operators, respectively) can still exhibit the real energy spectrum and the probability conservation conditions by redefining the inner product [5,6].
For the case considered here, given that a Hamiltonian H with P (T ) symmetry which obeys the relation PH P=H (T H T =H ) and that with PT symmetry obeys [H, PT ]=0. Later, the concept of PT symmetry was first experimentally observed in electrical circuits system [7,8], and then extended to other systems which usually consists of balanced gain and loss, such as the optical waveguides [9][10][11][12][13], mechanical systems [14], optical microcavities [15,16], and optical systems with atomic media [17][18][19]. These systems exhibit the properties of conservative systems. Recently, the phenomena for growing interest in exploring novel effects on PT -symmetric classical optical systems is not only refers to the simulation of PT -symmetric theory itself but also opening the doors for novel photonic applications, striking examples include the exceptional points [20][21][22], unidirectional light transport [13,15] and the single-mode lasers [23,24].
However, experimental study on PT -symmetric physics in quantum regime remains huge challenge. Some non-trivial and undebatable effects show that the most possible approach for realizing the PT -symmetric Hamiltonian is to utilize an open quantum system. Yet, it is difficult to achieve a controllable PT -symmetric Hamiltonian by controlling the environment [25]. Some progress associated with PT -symmetric Hamiltonian has been made with this approach in the system of photons [26,27], ultracold atoms [28], nitrogen-vacancy(NV) centers [29], nuclear spins [30,31], and superconducting qubits [32]. However, unambiguously observing the evolution of a quantum state (qubit) under the the PT -symmetric Hamiltonian has not been realized which is critical not only for further applications of PT -symmetric theory but also providing great insight for the fundamentals of quantum physics [3].
In this study, we designed a linear optical circuit to simulate the PT -symmetric and experimentally simulated the evolution of the quantum state under the generalize PTsymmetric Hamiltonian. The dynamic evolution of the qubit could be observed under the PT -symmetric Hamiltonian over the time effectively by using the conventional quantum gates and post-selection. Our results show that the evolution of high fidelity of the qubit (quantum state) governed by PT -symmetric system not only reveals the exotic properties of the PT theory but also provides a route for further exploiting PT -theory framework to investigate the fundamental problems.

II. METHODS
We know, in the realm of quantum computation, the operators must be unitary that means the Hamiltonian is Hermitian. However, we now construct a two-level PT -symmetric Hamiltonian Eq.(S1) as a subsystem in a Hilbert space with higher dimensions to simulate the non-unitary operator U PT (detail see Supplementary Material 1 Eq.(S2)) [33]. The system, used to simulate the PT time evolution in the case of considering real variables r = µ = s , consists of a work qubit w and an ancillary qubit a, which implies a two-qubit system. The initial state of the system is prepared in |ϕ ini. = |0 a |0 w . First, only the ancillary qubit a of the initial state undergoes the unitary operation and the work qubit w remains unchanged which can be described as U 1 : where cos ϑ a = ω 2 cos 2 (ωt/2 ) + (µ + s) 2 sin 2 (ωt/2 ) sin ϑ a = (µ − s) 2 + 4r 2 sin 2 θ sin(ωt/2 ) Then, the ancillary qubit a acts as a control qubit and performs the U 2 operation on the work qubit w only when the a is |0 a : And that performs the U 3 operation on the work qubit w only when the a is |1 a : where At final, a Hadamard operation U 4 = H is applied to the ancillary qubit a. After the above series of unitary operations, the initial state |ϕ ini. = |0 a |0 w will evolve into: where c is a non-zero coefficient and only affects the possibility to obtain the ancillary qubit a being in state |0 a but does not influence the law of PT -symmetric evolution that work qubit w obeys. Until now, the two-qubit system is observed. If the ancillary qubit a is measured in state |0 a , the work qubit w does evolve into state e −i t H PT |0 w , which indicates that the evolution associated with the work qubit w is charged by the PT -symmetric Hamiltonian in Eq. (S2).

III. THE EXPERIMENTAL SIMULATION OF DYNAMICAL EVOLUTION UN-DER PARITY-TIME-SYMMETRIC HAMILTONIAN
The sketch the experimental configuration is shown in Fig.1 which includes three modules: the preparation module, the evolution and the detection part. In the preparation module, we use a 30-mm-long, 2-mm-wide,1-mm-thick flux-grown PPKTP crystal (Raicol Crystals) with a grating period of 10.0 µm for frequency-degenerate type-Π quasi-phase-matched collinear parametric down-conversion, which is pumped by 45-mW vertically polarized beam at 405 nm (Omicron) to generate heralded single photons in the framework of Sagnac interferometer. One of the entangled photons is set as target qubit for coincidence detection while the other photon, with polarization degree of freedom(H (horizontal) and V (vertical)), is identified as working qubit w for evolution. Then, a beam displacer(BD) is used to generate the spatial mode degree of freedom(marked with path p and path q) which is encoded as ancillary qubit a. The initial state of the two-qubit system is prepared in the form |ϕ ini. = |0 a |0 w , wherein the H (V ) and path p(path q) are encoded as |0 w (|1 w ) and|0 a (|1 a ), respectively. It is worth mentioning that |0 w (|1 w ) and |0 a (|1 a ) are synergistically related(work together) due to the constructive contribution of BD.
In the evolution module, four subsequent stages are involved. In the first stage, the unitary operation U 1 acts on the ancillary qubit a(Eq.(1)). That is, the photon in path   to the inevitable experimental imperfections, we present the dynamical evolution of the PTsymmetric Hamiltonian along with four time points: t 0 = 0s, t 1 = 0.7876s, t 2 = 0.9894s,  The comparison between the theoretical results and the experimental results for the evolved subspace is shown in Fig.2, which corresponds to the Hamiltonian U PT with different values of t . From Fig. 2(a) to Fig. 2(d), the dynamic evolution process of the initial work qubit w |ρ w ini. are shown which undergoes the PT -symmetric Hamiltonian U PT over time t 0 , t 1 , t 2 , t 3 , where |ρ w ini. can be obtained by tracing out the ancillary qubit a from |ϕ ini. . The real and imaginary parts of the final reconstructed density matrix via a quantum state tomography are found to be the same as those of the theoretically predicted density matrix, which confirms the perfect combination of our theoretical framework and experimental results. Subsequently, the formula F (ρ the. , ρ exp. ) = T r(ρ the. ρ exp. )/( T r(ρ 2 the. ) T r(ρ 2 exp. )) [34] are employed to calculate the fidelities between the theoretical expectation values and the experimental results with respected to different t as shown in Fig.3 (a), from which we can calculate that the average value of fidelity is 0.9992 ± 0.0002.
Furthermore, the dynamical evolution of the state under U PT are also studied via monitoring P |0 w in the time range from t 0 to t 3 which is shown in Fig.3 (b). Considering the form of U PT in Eq.(S1), the dynamics of P |0 w could be always observed in the range of unbroken PT symmetry. It can be obtained that the measured results (blue line) dependence of the state evolution under U PT with t are well coincide with the theoretical predictions (red line), and the system eventually approaches to a steady state.

IV. SUMMARY
Considering the practical condition in our experiment, the successfully evolved subspace is only a simulation of the PT -symmetric system as a part of the full Hermitian system on account of: 1) Observing the evolution of the broken -PT -symmetry zone is poor at our optical system; 2) Most quantum computation and quantum simulation problems can be implemented in the unbroken -symmetry zone, such as no-signalling principle [26]. In fact, we do not have to construct a complete PT Hermitian, because our purpose is to observe and characterize the evolution of single-photon qubit through the framework of PT , which is expected to be used to implement one of the basic systems of dedicated quantum information processing.
In summary, we experimentally investigate the quantum simulation of the dynamical behaviors under the PT -symmetric Hamiltonian using a single-photon system. The U PT operator is effectively simulated in a subsystem of a full Hermitian system using post-selection.
The results show that the state during the evolution can be observed with a high fidelity when the PT -symmetrically evolved subspace is solely considered. Owing to the effectively operation of the dilation method, our work provides a route for further exploiting the ex-