Experimental demonstration of coherence flow in $\mathcal{PT}$- and anti-$\mathcal{PT}$-symmetric systems

Non-Hermitian parity-time ($\mathcal{PT}$) and anti-parity-time ($\mathcal{APT}$)-symmetric systems exhibit novel quantum properties and have attracted increasing interest. Although many counterintuitive phenomena in $\mathcal{PT}$- and $\mathcal{APT}$-symmetric systems were previously studied, coherence flow has been rarely investigated. Here, we experimentally demonstrate single-qubit coherence flow in $\mathcal{PT}$- and $\mathcal{APT}$-symmetric systems using an optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence. Particularly, we observe two complete coherence backflows in one period in the $\mathcal{PT}$-symmetric system, while only one backflow in the $\mathcal{APT}$-symmetric system. Moreover, in the symmetry broken regime, we observe the phenomenon of stable value of coherence flow. We derive the analytic proofs of these phenomena and show that most experimental data agree with theoretical results within one standard deviation. This work opens avenues for future study on the dynamics of coherence in $\mathcal{PT}$- and $\mathcal{APT}$-symmetric systems.

Although many counterintuitive phenomena in PT -or APT -symmetric systems were previously studied, the flow of coherence in PT -symmetric systems has not been fully and thoroughly investigated. Moreover, the coherence flow in APT -symmetric systems has not been studied either theoretically or experimentally. The study of coherence flow is interesting and meaningful because it can discover various phenomena different from Hermitian quantum mechanics and reveal the relationship between non-Hermitian systems and their environment.
In Hermitian quantum systems isolated from their environment, the coherence flow between the subsystems generally oscillates periodically over time, and the oscillation period depends on the coupling strength between the subsystems. Different from Hermitian quantum systems, most non-Hermitian physical systems typically involve gain and loss induced by the environment. In this case, the behavior of coherence flow in non-Hermitian physical systems is generally quite different from that of the coherence flow in Hermitian physical systems.
For example, the dissipative coupling between the system and the environment may disturb and even wash out quantum coherence. On the other hand, due to the gain effect, the coherence originally lost into the environment may return to the system, and thus oscillates periodically over time. By investigating the coherence flow between the system and its environment, one can obtain some important information, such as the coupling strength between systems and their environment, the type of environment (i.e., Markov or non-Markov environment), the presence of memory effects in the open quantum dynamics, etc..
In this study, we experimentally demonstrate the coherence flow of a single qubit in PT -and APT -symmetric systems using a simple optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence in both PT -and APT -symmetric systems. Double touch of coherence (DTC) (i.e., complete coherence backflow happening twice in one period) is revealed in the PTsymmetric system, while only one backflow exists in the APT -symmetric system. In addition, we observe the phenomenon of stable value (PSV) of coherence in the symmetry broken regime, which is independent of its initial state. Concretely, the coherence tends to a stable value 1/a in the PTsymmetric system, but it approaches 1 in the APT -symmetric system. We also provide the theoretical analytic proofs of these phenomena (see Supplementary Notes 1-4) and compare with previous relevant works. Our results imply that the coherence backflow and PSV are quite different for these two kinds of symmetric systems.

II. RESULTS
Principle and setup of the experiment. A non-trivial general PT -symmetric Hamiltonian for a single qubit takes the form [11,17] While a generic APT -symmetric Hamiltonian of a single qubit can be expressed as [50,52] Here, the parameter s > 0 is an energy scale, a = γ/s > 0 is a coefficient representing the degree of non-Hermiticity,σ x andσ z are the standard Pauli operators. In the PT -symmetric system, the eigenvalue ofĤ PT is which is an imaginary number for a > 1 (the PT symmetry broken regime), while a real number for 0 < a < 1 (the PT symmetry unbroken regime). However, in the APTsymmetric system, the eigenvalue ofĤ APT is which is an imaginary number for 0 < a < 1 (the APT symmetry broken regime), while a real number for a>1 (the APT symmetry unbroken regime). Note that the eigenvalues of both HamiltoniansĤ PT andĤ APT are zero for a = 1 (the exceptional point). For different s, the time evolution of quantum states under the HamiltonianĤ PT (Ĥ APT ) follows the same rules because s is an energy scale. Therefore, without loss of generality, we consider s = 1 for bothĤ PT andĤ APT [23,24]. In our experiment, the non-unitary operators U PT = exp(−iĤ PT t) and U APT = exp(−iĤ APT t) are realized by [23,53] where the loss-dependent operator is realized by a combination of two beam displacers (BDs) and two half-wave plates (HWPs) with setting angles ξ i and ξ j (see Supplementary Note 5) [23]. Above, R HWP and R QWP are the rotation operators of HWP and QWP (quarter wave plate), respectively. Here, the setting angles (θ 1 , ϕ 1 , ϕ 2 , ξ 1 , ξ 2 , ξ 3 ) depend on the initial state and are determined numerically by reversal design for each given time t, according to the timeevolution operators U PT and U APT . The dynamical evolution of the quantum states in the PTor APT -symmetric system is given by [18,23,54] where U (t) = U PT (t) or U APT (t), ρ(0) is the initial density matrix, and ρ(t) is the density matrix at any given time t.
Here, we use the l 1 norm of coherence [55,56] to quantify the coherence of ρ(t), i.e., where ρ(t) i,j denotes the matrix element obtained from ρ(t) by deleting all diagonal elements. In the single-qubit case, Eq. (9) is simplified as Here, ρ(t) 1,2 and ρ(t) 2,1 are the two off-diagonal elements of the single-qubit density matrix.
As shown in Fig. 1, our experimental setup consists of four parts (photon source, state preparation, implementation of the Fig. 1: Experimental setup. Blue area to the left: Pairs of 808 nm single photons are generated by passing a 404 nm laser light through a type-I spontaneous parametric down conversion, and using a nonlinear-barium-borate (BBO) crystal. Orange area: After photons pass through the 3 nm interference filter (IF), one photon serves as a trigger and the other signal photon is prepared in an arbitrary linear polarization state. Grey area: Two sets of beam displacers (BDs), together with half-wave plates (HWPs) and quarter-wave plates (QWPs), are used to construct the operators U PT and U APT . In the measurement part, the density matrix is constructed via quantum state tomography. PBS: polarization beam splitter.
operator U PT or U APT , and measurement). In the photonsource part, we generate heralded single photons via type-I spontaneous parametric down-conversion, with one photon serving as a trigger and the other as a signal photon (blue area). Because of the disturbance of the single-mode fiber to polarization, the signal photon needs to pass through the sandwich structure (QWP-HWP-QWP) to eliminate this influence, and then goes through various optical elements. In the orange area, we finish preparing the single-qubit arbitrary quantum state α|H + βe iϕ |V (|α| 2 + |β| 2 = 1, α, β ∈ R) after the HWP and QWP. Before the signal photon enters the gray region, we separately prepare three initial quantum states |H , (|H + |V ) / √ 2, and |H + √ 3|V /2, by appropriately choosing the rotation angles of the HWP and the QWP in the state preparation part.
The gray part has the function of simulating the U PT or U APT . The loss operator L can be implemented with two sets of BD and two HWPs between BDs. For the HWP along the up (bottom) path, the angle is ξ 1 (ξ 2 ). In order to simulate U PT , we choose the plate combinations in the solid green wireframe. While the plates in the dotted green wireframe are used to simulate U APT .
In the measurement part (green area), the density matrix at any given time t can be constructed via quantum state tomography after the signal photon passes through the gray region.
Essentially, we measure the probabilities of the photon in the bases 2} through a combination of QWP, HWP, and PBS (polarization beam splitter), and then perform a maximum-likelihood estimation of the density matrix (tomography). The outputs are recorded in coincidence with trigger photons. The measurement of the photon source yields a maximum of 30,000 photon counts over 3 s after the 3 nm interference filter (IF).
Experimental results. Figure 2(a, b, c) demonstrate the timeevolution dynamics of the coherence of three initial quantum states |H , (|H + |V )/ √ 2, and (|H + √ 3|V )/2 in the PT -symmetric system. Coherence varies over time t for: (i) a = 0.31 (blue curve), a = 0.47 (red curve) (0 < a < 1); and (ii) a = 1.5 (blue curve), a = 2.8 (green curve) (a > 1). For 0 < a < 1 (the PT symmetry unbroken regime), coherence oscillates (see blue and red curves) suggesting a coherence complete recovery and backflow. There are two complete backflows of coherence in one period, i.e., double touch of coherence (DTC), which is observed in our experiment and agrees with our theoretical results (see Supplementary Note 3). However, for a > 1 (the PT symmetry broken regime), a PSV of coherence occurs (see dark and green curves). Extracted from the experimental data, the recurrence time fits the theoretical value given by and the stable value for the PSV agrees well with the theoretical value 1/a (see Supplementary Note 1). For the same three initial quantum states in the H APT case, the dynamical characteristics of coherence are shown in Fig. 3, where Figs. 3(a, b, c) are respectively for the initial states |H , (|H + |V )/ √ 2 and (|H + √ 3|V )/2. In contrast to the H PT case, coherence oscillations occur for a > 1 (the APT symmetry unbroken regime), while PSV occurs for 0 < a < 1 (the APT symmetry broken regime), as verified in Figs. 3(a, b, c). Different from the PSV in the PT -symmetric system, the stable value for the PSV in the APT -symmetric system is 1 (see blue and red curves). Figure 3(b) shows that the saturated coherence does not change over time t for any value of a. As demonstrated in Figs. 3(a, c), there exits only a single backflow in one period (see dark and green curves), i.e. the DTC phenomenon does not occur in the APT -symmetric system (the theoretical proof is in Supplementary Note 4).
The oscillating period observed in the experiment is consistent with the theoretical value given by and the stable value for the PSV observed in the experiment is in a good agreement with the theoretical value 1 (see Supplementary Note 2). To better understand Fig. 3(b), we plot   shows the trajectory evolution of the initial quantum state (|H + |V ) / √ 2 on the Bloch sphere in the APT -symmetric system. Figure 4 shows that the evolved quantum state travels over time along the outer edge of the XY plane, which is independent of a. Thus, it can intuitively reflect why the coherence of the quantum state [shown in Fig. 3(b)] remains unchanged during the time evolution.

III. DISCUSSION
Our setup provides a simple platform to investigate both PT -and APT -symmetric systems. First, the gain and loss, associated with dissipative coupling between the system and environment, can be readily simulated with optical elements. By selecting the appropriate combination of optical elements with adjustable angles, both PT -and APT -symmetric systems can be realized with this setup. Second, our setup can be used to demonstrate the dynamics of PT -and APTsymmetric systems for each given evolution time t, by per-forming the corresponding nonunitary gate operations on the initial states. The dynamics of PT -and APT -symmetric systems for each given evolution time t is stable and the coherence time of photons is long enough, thus one can accurately extract the critical information from the nonunitary dynamics.
Let us briefly recall the difference between Rabi oscillations and coherence flow oscillations. In our work, we only consider a single qubit, with the usual two logical states |0 and |1 . Rabi oscillations refer to the dynamical evolution of the population probability of the logical state |0 or |1 of the qubit. For example, this occurs when the qubit is placed inside a cavity and the cavity-qubit coupling is sufficiently strong, so there is an exchange of energy between the qubit and the photons bouncing back and forth many times inside the cavity. Also, Rabi oscillations occur when a classical driving field is applied to a qubit, where there is an exchange of energy between the qubit and the drive, and the Rabi frequency is proportional to the applied driving field amplitude. On the other hand, for a single qubit, the coherence of quantum states is defined as the sum of the two off-diagonal elements of the single-qubit density matrix, according to Eq. (9). Coherence flow oscillations refer to the oscillations of the coherence of quantum states. Different from Rabi oscillations, coherence flow oscillations do not require the qubit to exchange energy with photons located in a cavity or exchange energy with a classical pulse. Clearly, Rabi oscillations and coherence flow oscillations are completely different notions. Now let us make a brief comparison with previous works [18,23,37,52], which are most relevant to this work: (i) A theoretical and experimental research on the dynamics of coherence under PT -symmetric system has been recently presented by Wang et al. [37] in a single-ion system. There, the coherence evolution was discussed by the time average of the coherence and the diagonal element (e.g., ρ 00 ) of the quantum state density matrix [37]. In our work, we provide a simple platform to demonstrate PT -and APT -symmetric systems in experiments. We discuss the l 1 norm of the coherence (i.e., the summation of off-diagonal elements of the density matrix), which is quite different from the time average of the coherence. Furthermore, we theoretically predict some phenomena (the DTC and PSV phenomena) which were not reported by Wang et al. [37], and give an experimental demonstration in a linear optical system.
(ii) In previous works on information flow [18,23], the trace distance was introduced to characterize the information flow; while in our work the l 1 norm of the coherence, described by Eq. (9), is introduced to characterize the coherence flow. The concept of coherence flow is different from information flow. Second, the trace distance D (ρ 1 (t), ρ 2 (t)) generally measures the distinguishability of two quantum states, while the l 1 norm of coherence quantifies the coherence of a quantum state. In this sense, the physical meaning of the coherence flow is different from that of the information flow. Third, the physical phenomena revealed by the coherence flow and the information flow are not exactly the same. For example, we found that in the PT -symmetry unbroken regime, there are two complete coherence backflows in one period; while, the previous works [18,23] showed that in the PT -symmetry unbroken regime, there exists only one information flow within one period. Last, quantum coherence is an intriguing property of quantum states, which is a key resource in quantum computing, quantum communication, and quantum metrology; and the PT /APT systems have attracted considerable interest. Thus, we believe that it is significant to study the evolution of coherence of quantum states in PT and APT systems.
(iii) It is obvious that our work differs from the work by Wen et al. [52]. Their work studied the information flow in an APT -symmetry system, which is different from the coherence flow; while, the present work focuses on the coherence flow.

IV. CONCLUSIONS
In summary, we have experimentally demonstrated the coherence flow in both PT -and APT -symmetric systems by using a single-photon qubit. In this paper, the DTC phenomenon in one period in the PT -symmetric unbroken regime has been demonstrated, which however does not occur in the APT -symmetric system. Moreover, the PSV has been observed in the PT /APT -symmetric broken regime, which is independent of the initial state. As an extension of this work, we have numerically simulated the dynamics of coherence for two-qubit PT /APT systems (for details, see Supplementary Note 6). The simulations show that for both twoqubit PT /APT systems, there exist different periodic oscillations of coherence (including one coherence backflow, two coherence backflows, and multiple coherence backflows in one period) in the unbroken regime; while there exists PSV in the broken regime, which is independent of the initial state. Our work merits future study on the multi-qubit coherence flow in PT -and APT -symmetric systems, which is left as an open question.

V. METHODS
Device parameters. The photon-source system of the singlequbit, the pump laser power is 130 mW. In the state preparation part, three initial quantum states |H , (|H + |V ) / Analysis of experimental imperfections. Due to the accuracy of the rotation angle, and the imperfection of the interference visibility between BDs, several points of experiment data do not fit well with our theoritical data. To solve this problem, we improve the extinction ratio of interference between BDs for a high interference visibility. Instead of manual adjustment, we use motorized precision rotation mount to ensure the higher accuracy of the plate rotation angle. Meanwhile, the experimental errors are estimated from the statistical variation of photon counts, which satisfy the Poisson distribution.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Code availability
The code used for simulations is available from the corresponding authors upon reasonable request.
Here θ 4 = ω 4 st. When t → ∞, cosh θ 4 ∼ sinh θ 4 → ∞. Thus, it follows from Eq. (S22) that: Accordingly, it follows from Eq. (S21) that: Equation (S24) shows that the phenomenon of stable value (PSV) of coherence occurs after a long time evolution; that is, the coherence tends to 1, which is independent of the initial states.
First, we consider the case when dC l 1 (|φ(t) ) dm3 = 0. According to Eq. (S18), we have Because the period of tan 2θ 3 is π 2 and the period of C l1 (|φ(t) ) is T θ3 =π (i.e., T APT = π s √ a 2 −1 ), one period of C l1 (|φ(t) ) includes two periods of tan 2θ 3 . Thus, there exist two different values of θ 3 (or t) satisfying Eq. (S44) or Eq. (S38) within one period of C l1 (|φ(t) ). From the above discussion, one can conclude that dC l 1 (|φ(t) ) dt has two zero points in one period (i.e., T APT = π s √ a 2 −1 ) of coherent evolution. Therefore, the coherent oscillation of quantum states in the APT -symmetric-unbroken regime has only one backflow within one period (eg., see Supplementary Figure 2). Supplementary Note 6: Coherence flow for two-qubit PT -and anti-PT -symmetric systems We have numerically simulated the dynamics of coherence for two-qubit PT /APT systems. As shown in Supplementary  Figures 4(a, c), there exist different periodic oscillations of coherence (including one coherence backflow, two coherence backflows, and multiple coherence backflows in one period) for PT /APT -symmetric systems in the unbroken regime. In addition, as illustrated in Supplementary Figures 4(b, d), there exists PSV for both PT -and APT -symmetric systems in the broken regime, which are independent of the initial states.