Experimental simulation of anti-parity-time symmetric Lorentz dynamics

Qiang Li; Cheng-Jie Zhang; Ze-Di Cheng; Wen-Zheng Liu; Jun-Feng Wang; Fei-Fei Yan; Zhi-Hai Lin; Ya Xiao; Kai Sun; Yi-Tao Wang; Jian-Shun Tang; Jin-Shi Xu; Chuan-Feng Li; Guang-Can Guo

doi:10.1364/OPTICA.6.000067

1. INTRODUCTION

The Hermiticity of a Hamiltonian operator would guarantee real eigenenergies and promise a unitary evolution with conserved probability. This requirement was considered as one of the fundamental axioms of quantum mechanics [1]. However, there are efforts to extend the conventional quantum mechanics with relaxing the requirement for the Hermiticity of observables. One of the important cases is parity-time (PT) symmetry quantum mechanics [2–4]. In the symmetry-unbroken zone, Hamiltonians with PT symmetry have real eigenenergies, and the probability conservation conditions can also be satisfied by redefining the inner product [4–6]. An PT-symmetric Hamlitonian follows $[H, \hat{P} \hat{T}] = H \hat{P} \hat{T} - \hat{P} \hat{T} H = 0$ , with the parity-time operator ( $\hat{P} \hat{T}$ ). Although the impact of PT symmetry in the basic theory of quantum mechanics is still debated, it has been recently observed in real physical systems [7–11] and has obtained fruitful theoretical and experimental results [12–25]. Another important case is the counterpart anti-PT symmetric Hamiltonian, which satisfies ${H, \hat{P} \hat{T}} = H \hat{P} \hat{T} + \hat{P} \hat{T} H = 0$ . There are only a few results on the investigation of anti-PT symmetry [26–30]. Nevertheless, several intriguing properties have been shown in systems with anti-PT symmetry, such as spontaneous phase transition of the scattering matrix [26], refractionless or unit refraction [29], and constant refraction [30].

Recently, the Bogoliubov–de Gennes (BdG) equation is used to describe the dynamics of bosonic Bogoliubov quasi-particles [31]. The BdG Hamiltonian is pesudo-Hermitian [32–34] with anti-PT symmetry, and the dynamics are found to be a continuous complex Lorentz transformation in the complex Minkowski space, which is different from the usual unitary transformation in Hilbert space. Such Lorentz quantum dynamics are proposed to be observable in physical systems hosting bosonic Bogoliubov quasi-particles [31], which would provide new insights into the dynamical properties of quasi-particles [35–37].

In this work, we experimentally simulate the Lorentz quantum dynamics in an anti-PT symmetry region by using an optical open system. The information is encoded in single photons generated from a color center in a gallium nitride (GaN) film [38,39]. The state evolutions are found to be stabilized on the constant-energy surfaces of the BdG equation. The new defined inner product of states during the evolution remains constant under the Lorentz transformation. The evolution of states in three categories, corresponding to the space-like, light-like, and time-like states in the complex Minkowski space, is demonstrated. Our work could be helpful to investigate the dynamics of quasi-particles in different systems. Moreover, the demonstrated defect-based optical setup promises further practical quantum information processing with daily used materials.

2. THEORETICAL FRAMEWORK

In order to simulate the Lorentz transformation, we consider the following pseudo-Hermitian BdG Hamiltonian:

H_{BdG} = (\begin{matrix} α & i μ + ν \\ i μ - ν & - α \end{matrix}),

where

α

,

μ

, and

ν

are set to be real. The pseudo-Hermitian condition

H_{BdG}^{†} = η_{0} H_{BdG} η_{0}^{- 1}

holds, with

η_{0} = diag {1, - 1}

. When

ν = 0

, this system is just PT symmetric, with

η_{0}

matrix identified as the operator of

\hat{P}

. On the other hand, such a pseudo-Hermitian Hamiltonian also obeys the anti-PT symmetry property

\hat{P} \hat{T} H_{BdG} = - H_{BdG} \hat{P} \hat{T}

, with the Pauli matrix

σ_{x}

being the operator of

\hat{P}

. The time evolution operator of this pseudo-Hermitian BdG Hamiltonian can be deduced as

U_{BdG} = e^{- i H_{BdG} t} = (\begin{matrix} x & y^{*} \\ y & x^{*} \end{matrix}),

where

x = Cosh (β t) - i α Sinh (β t) / β

,

y = (μ + i ν) Sinh (β t) / β

, and

β = \sqrt{μ^{2} + ν^{2} - α^{2}}

. We can check that

{| x |}^{2} - {| y |}^{2} = 1

, and this time evolution operator is pseudo-unitary, i.e.,

U_{BdG}^{†} η_{0} U_{BdG} = η_{0}

. Equation (2) is the same as the Lorentz transformation for (1,1)-spinor described in Ref. [31]. Without loss of generality, we let

x = 1 / \sqrt{1 - υ^{2}}

and

y = - υ / \sqrt{1 - υ^{2}}

, so that the

U_{BdG}

matrix becomes a more intuitionistic form,

U_{BdG} = \frac{1}{\sqrt{1 - υ^{2}}} (\begin{matrix} 1 & - υ \\ - υ & 1 \end{matrix}),

where

- 1 < υ < 1

and

υ \in R

. Equation (3) is a Lorentz transformation with the speed of light

c = 1

. For an arbitrary initial state

| ψ_{i} ⟩ = {[m_{i}, n_{i}]}^{T}

, the final state is represented as

| ψ_{f} ⟩ = {[m_{f}, n_{f}]}^{T}

after the

U_{BdG}

evolution. The new inner product defined by

η_{0}

is invariant under the time evolution generated by the pseudo-Hermitian Hamiltonian

H

, i.e.,

m_{i}^{2} - n_{i}^{2} = ⟨ ψ_{i} | η_{0} | ψ_{i} ⟩ = ⟨ ψ_{f} | η_{0} | ψ_{f} ⟩ = m_{f}^{2} - n_{f}^{2}

. This inner product is the same as the one defined by the Minkowski metric involving one-dimensional time and one-dimensional space.

The non-unitary $U_{BdG}$ can be simulated in an open system, which is realized by introducing an ancilla qubit and performing the projective operation [25]. The evolution of the total system, including the carrier qubit $(| 0_{s}, 1_{s} ⟩)$ and the ancilla qubit $(| 0_{a}, 1_{a} ⟩)$ , can be expressed as

U_{tot} = | φ_{0_{s}} ⟩ ⟨ 0_{s} | \otimes | 0_{a} ⟩ ⟨ 0_{a} | + | φ_{1_{s}} ⟩ ⟨ 1_{s} | \otimes | 1_{a} ⟩ ⟨ 0_{a} | + | φ_{0_{s}}^{⊥} ⟩ ⟨ 1_{s} | \otimes | 0_{a} ⟩ ⟨ 1_{a} | + | φ_{1_{s}}^{⊥} ⟩ ⟨ 0_{s} | \otimes | 1_{a} ⟩ ⟨ 1_{a} |,

where

| φ_{0_{s}} ⟩ = (1 / \sqrt{1 + υ^{2}}) (| 0_{s} ⟩ - υ | 1_{s} ⟩)

,

| φ_{1_{s}} ⟩ = (1 / \sqrt{1 + υ^{2}}) (- υ | 0_{s} ⟩ + | 1_{s} ⟩)

, while

| φ_{0_{s}}^{⊥} ⟩

and

| φ_{1_{s}}^{⊥} ⟩

are their orthogonal states, respectively. The projective operator can be written as

P = l \otimes \frac{| 0_{a} ⟩ + | 1_{a} ⟩}{\sqrt{2}} \frac{⟨ 0_{a} | + ⟨ 1_{a} |}{\sqrt{2}},

where

1

represents the identity operator. The evolution after the projection operation (or post-selection) can be calculated as

U_{ps} = P U_{tot} = \frac{1}{\sqrt{2}} (| φ_{0_{s}} ⟩ ⟨ 0_{s} | + | φ_{1_{s}} ⟩ ⟨ 1_{s} |) \otimes \frac{| 0_{a} ⟩ + | 1_{a} ⟩}{\sqrt{2}} ⟨ 0_{a} |,

considering that the initial state of the ancilla qubit in our experiment is always

| 0_{a} ⟩

. After projection, the remaining state becomes

| φ_{0_{s}} ⟩ ⟨ 0_{s} | + | φ_{1_{s}} ⟩ ⟨ 1_{s} | \propto U_{BdG} | ψ_{i} ⟩

.

3. EXPERIMENTAL SETUP AND RESULTS

Figure 1 shows our experimental setup. The information carrier is encoded in the polarization of single photons. We utilize a home-built confocal imaging system to prepare a single photon source by exciting an intrinsic point defect in a gallium nitride (GaN) film epitaxially grown on sapphire with a 532 nm continuous-wave laser. The single photon source is shown to be stable with a narrow width, and the minimum autocorrelation value is about $g^{2} (0) = 0.242$ , which confirms the single-photon emission. The detailed setup and analysis can be found in Supplement 1.

Fig. 1. Experimental setup. (a) State preparation. The polarization of a single photon generated from an intrinsic defect in a gallium nitride (GaN) film is first initialized to be horizontal by a half-wave plate (H1) and a polarization beam splitter (PBS1). The polarization state is further prepared as $| ψ_{in} ⟩ = (m | H ⟩ + n | V ⟩)$ with a half-wave plate (H2) and a quarter-wave plate (Q1), where $| H ⟩$ and $| V ⟩$ represent the horizontal and vertical polarizations, and $m$ and $n$ are two real amplitudes, respectively. (b) The change in photon number. The state is separated into two paths $p$ and $q$ with a beam displacer (BD1). The polarization in path $q$ is rotated to be horizontal with H3, which is the same as that in path $p$ . The transmitted photon number is controlled by H4 and PBS2. (c) The non-orthogonal state transformation. In path $p$ , $| H ⟩$ is transformed to $| φ_{H} ⟩$ with H5 and Q2, whereas in path $q$ , $| V ⟩$ is transformed to $| φ_{V} ⟩$ with H6 and Q3. BD2, H7, H8, and BD3 are used to combine different components denoted as $V p$ , $V q$ , $H p$ , and $H q$ . The post-selection state is chosen by H9, H10, and PBS3. The final two paths are combined by H11 and BD4, where the polarization state is rotated by H12, and the output state becomes $| ψ_{out} ⟩$ .

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To purify the initial state, the polarization of the photon is first rotated by half-wave plate 1 (H1) to be horizontal, namely, $| H ⟩$ , which is then selected by polarization beam splitter 1 (PBS1). Arbitrary pure states $| ψ_{i n} ⟩ = m | H ⟩ + n | V ⟩$ ( $m$ and $n$ are real coefficients, and $| V ⟩$ represents the vertical polarization) can be prepared by using half-wave plate 2 (H2) and quarter-wave plate 1 (Q1), as shown in Fig. 1(a).

In our scheme of simulating the anti-PT symmetric Lorentz dynamics, two subsequent stages are involved. In the part where there is a change in photon number corresponding to the open-system evolution [Fig. 1(b)], the photon is separated into two paths by beam displacer 1 (BD1), in which the horizontal component is horizontally separated by about 3 mm (path $p$ ), while the vertical component remains unchanged (path $q$ ). The polarization in path $q$ is first rotated to be horizontal with H3, and the transmitted rate is controlled by H4 and PBS2, which represents the proportion of photons flowing into the system. In order to carry out the non-orthogonal state transformation [Fig. 1(c)], the states in paths $p$ and $q$ are transformed to be $| φ_{H} ⟩ = (| H ⟩ - υ | V ⟩) / \sqrt{1 + υ^{2}}$ and $| φ_{V} ⟩ = (- υ | H ⟩ + | V ⟩) / \sqrt{1 + υ^{2}}$ by H5 and Q2, H6 and Q3, respectively. After BD2, four components are obtained, in which the components denoted as $V p$ and $V q$ , $H p$ and $H q$ are recombined by the BD3, respectively. The post-selection on the path state $1 / \sqrt{2} (| p ⟩ + | q ⟩)$ is realized by setting H9 and H10 to 22.5°, which is selected by PBS3. The two paths are combined again into one, and the output state is represented as $| ψ_{out} ⟩ = \frac{1}{\sqrt{2}} q_{0} (m | φ_{H} ⟩ + n | φ_{V} ⟩) = \frac{1}{\sqrt{2}} q_{0} [(m - υ n) | H ⟩ + (- υ m + n) | V ⟩] / \sqrt{1 + υ^{2}}$ ( $q_{0}$ is a coefficient when considering the loss of photons to the environment), which is exactly the same as the evolved result of $U_{BdG}$ . The state information is reconstructed by a quantum state tomography setup, which consists of a quarter-wave plate, a half-wave plate, and a polarization beam splitter (not shown in Fig. 1).

During the experiment, the reconstructed output states $σ$ are generally mixed. We then obtain the corresponding nearest pure state with the form of $ρ = (m_{0} | H ⟩ + n_{0} | V ⟩) (m_{0} ⟨ H | + n_{0} ⟨ V |)$ by minimizing the trace distance between them, which is defined as $T (ρ, σ) := Tr [\sqrt{{(ρ - σ)}^{†} (ρ - σ)}] / \sqrt{2}$ . The average value of fidelity is $0.996 \pm 0.003$ , which confirms the precision of our experiment. Due to the fact that there is a step in the change of photon numbers to simulate the corresponding $U_{BdG}$ evolution, there is a ratio coefficient $\sqrt{p_{1} / p_{0}}$ to obtain the non-normalized state $U_{BdG} | ψ_{in} ⟩$ from the normalized state reconstructed through the state tomography process, in which $p_{0}$ and $p_{1}$ represent the initial and final photon numbers, respectively. In order to extend the evolution steps, a second initial state that is close to the first final state is prepared and is sent to the same $U_{BdG}$ operation. We can then obtain the second final state. In this way, all the subsequent evolutionary results in each step can be acquired in principle when the first initial state is given.

Figure 2 shows the experimental results of the state evolution under the $U_{BdG}$ operation, in which the value of $υ$ is chosen as $\pm 0.4$ . The initial states $| H ⟩$ , $| V ⟩$ , and $(| H ⟩ \pm | V ⟩) / \sqrt{2}$ with the new inner products of 1, $- 1$ , and 0, respectively, are prepared and are denoted by red circles in Fig. 2. The states located on hyperbolic curves gradually approach the asymptote $m - n = 0$ when $υ_{-} = - 0.4$ , whereas they gradually approach the asymptote $m + n = 0$ when $υ_{+} = 0.4$ . The inner products of corresponding evolved states are found to be the same as those of the initial states, which confirms the Lorentz transformation. The states with different new inner products corresponding to the space-like, time-like, and light-like states in the complex Minkowski space are located in the light blue region, the light red region, and their boundaries, respectively. It is found that the evolved states move faster from the zero points ( $m = 0$ or $n = 0$ ) with larger absolute $m$ and $n$ . Due to the high-precision control of the optical simulator, experimental results agree well with the theoretical predictions represented as solid lines. The dashed lines are symmetric theoretical results with a global $π$ phase difference.

Fig. 2. State evolution when $υ = \pm 0.4$ . The points in red circles represent the first initial states during the evolution. The arrows represent the evolution directions with the corresponding $υ$ . The blue rhomboids and dark cyan pentagons represent evolved states with the initial state setting to be $| H ⟩$ . The purple rhomboids and pink pentagons represent evolved states with the initial state setting to be $| V ⟩$ . The olive hexagons and wine rhomboids represent evolved states with the initial state setting as $(| H ⟩ + | V ⟩) / \sqrt{2}$ . The red hexagons and orange rhomboids represent evolved states with the initial state setting as $(| H ⟩ - | V ⟩) / \sqrt{2}$ . The solid lines represent corresponding theoretical predictions. The dashed lines are symmetric theoretical results with a global $π$ phase difference. The light blue and light red regions correspond to the space-like and time-like regions, respectively. The boundary between these two regions corresponds to the light-like case. Error bars are estimated from the Poissonian counting statistics, which are small and are within the size of the symbols.

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We further compare the cases with different $υ$ . The experimental results are shown in Fig. 3. The initial state is set to be $(| H ⟩ + | V ⟩) / \sqrt{2} (m = 1 / \sqrt{2}, n = 1 / \sqrt{2})$ with $υ$ chosen as $- 0.3$ , $- 0.4$ , and $- 0.5$ in Figs. 3(a)–3(c), respectively. The other initial state is set to be $| H ⟩ (m = 1, n = 0)$ with $υ$ being $- 0.3$ , $- 0.4$ , and $- 0.5$ in Figs. 3(d)–3(f), respectively. It clearly shows that, as long as the initial state is the same, all the subsequent evolved states are located on the same curve, which agrees well with the prediction of the Lorentz transformation. With the increase of the absolute value of $υ$ , the movement distance between two adjacent points on the curve increases. Error bars are estimated from the counting statistics, which are assumed to follow a Poisson distribution. They are small and are within the size of the symbols.

Fig. 3. State evolution with different $υ$ . The initial states located in the red circles are set to be $(| H ⟩ + | V ⟩) / \sqrt{2}$ in the first row with (a) $υ = - 0.3$ , (b) $υ = - 0.4$ , and (c) $υ = - 0.5$ , respectively, while the initial states located in the red circles are set to be $| H ⟩$ with (d) $υ = - 0.3$ , (e) $υ = - 0.4$ , and (f) $υ = - 0.5$ , respectively. Error bars are due to the Poissonian counting statistics, which are within the size of the symbols.

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Figure 4 shows the new defined inner product of the evolved states when the initial state is set to be $| H ⟩$ and the value of $υ$ is chosen as $- 0.3$ . It is clearly shown that the new inner product of states evolving under the $U_{BdG}$ gate remains unchanged within experimental errors.

Fig. 4. New defined inner product of states as a function of $n$ . The value is defined as ( $m^{2} - n^{2}$ ). The initial state is set to be $| H ⟩$ with $υ$ chosen as $- 0.3$ . The dashed line represents the theoretical prediction of 1. Error bars are estimated from counting statistics.

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4. DISCUSSION

In this work, we experimentally simulate the Lorentz quantum dynamics of bosonic Bogoliubov quasi-particles with anti-PT symmetry, in which the non-unitary operation of $U_{BdG}$ is demonstrated in an optical quantum simulator with single photons generated from a point defect in a gallium nitride film. The $U_{BdG}$ operator is effectively simulated in a subsystem of a full Hermitian system with post-selection. The anti-PT system proposed here provides a distinct way to connect anti-PT systems with Lorentz dynamics. The state evolutions are found to be stabilized on the constant-energy surfaces of the BdG equation. Three different kinds of states, i.e., space-like, light-like, and time-like, are demonstrated, which correspond to states in the Minkowski space. Moreover, the new inner product of states in the evolution is found to be constant within the experimental errors. The results of our work would be helpful to investigate the dynamics of quasi-particles in diverse physical systems and offer new insights into the non-Hermitian theory and quantum mechanics in open systems. The defect-based optical quantum simulation shows the instructive application of daily used material for quantum information processing.

Funding

National Key Research and Development Program of China (2016YFA0302700); National Natural Science Foundation of China (NSFC) (11504253, 11734015, 11774335, 11821404, 61327901, 61490711, 61725504); The Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-SLH003); Anhui Initiative in Quantum Information Technologies (AHY020100, AHY060300); The Fundamental Research Funds for the Central Universities (WK2470000020, WK2470000026).

Acknowledgment

This work was partially carried out at the USTC Center for Micro and Nanoscale Research and Fabrication.

See Supplement 1 for supporting content.

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Experimental simulation of anti-parity-time symmetric Lorentz dynamics

Abstract

1. INTRODUCTION

2. THEORETICAL FRAMEWORK

3. EXPERIMENTAL SETUP AND RESULTS

4. DISCUSSION

Funding

Acknowledgment

REFERENCES

Supplementary Material (1)

Cited By

Figures (4)

Equations (6)

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