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Abstract
We reveal a novel topological property of the exceptional points in a two-level parity-time symmetric system and then propose a scheme to detect the topological exceptional points in the system, which is embedded in a larger Hilbert space constructed by a four-level cold atomic system. We show that a tunable parameter in the presented system for simulating the non-Hermitian Hamiltonian can be tuned to sweep the eigenstates through the exceptional points in parameter space. The non-trivial Berry phases of the eigenstates obtained in this loop from the exceptional points can be measured by the atomic interferometry. Since the proposed operations and detection are experimentally feasible, our scheme may pave a promising way to explore the novel properties of non-Hermitian systems.
© 2017 Optical Society of America
1. Introduction
Non-Hermitian Hamiltonian used to describe open or dissipative systems usually have complex eigenvalues. However, it is recently found that a series of non-Hermitian Hamiltonians have real eigenvalues if they are invariant under the parity (P) and time-reversal (T) union operation [1, 2]. Meanwhile, the eigenstates of these Hamiltonians are also commuted with the PT-symmetric operation [3]. Due to various intriguing properties, the PT-symmetric systems have raised broad attentions. Some PT-symmetric models have been experimentally realized in physical systems, such as active LRC circuits [4, 5], coupled waveguides [6–8], photonic lattices [9, 10], microwave billiards [11], transmission lines [12], whispering-gallery microcavities [13, 14] and single-mode lasing [15, 16]. Recently, the schemes for simulating the PT-symmetric potentials with cold atoms in optical systems have been theoretically proposed [17–20] and experimentally realized [21].
Exceptional point (EP) is a special point in parameter space of non-Hermitian systems where both eigenvalues and eigenstates coalesce into only one value and state [22, 23]. One of the most important properties of EPs is that the states of EPs are chiral [24], which has been experimentally observed [25, 26]. On the other hand, the chirality leads to a unique effect that the eigenstates exchange themselves but only one of them obtains a π Berry phase in a cyclic evolution [27, 28]. This chiral phenomenon of EPs has been experimentally demonstrated in microwave cavities for the first time [29, 30]. Very recently, a full dynamically encircling of EPs has been realized [31] and a non-reciprocal topological energy transfer due to dynamical encircling of such points has been measured [32]. In contrast, some theoretical and numerical results suggested that the eigenstates can change to each other but both of them obtain a π/2 Berry phase due to the linear dependence of eigenstates [33–35].
In this paper, we demonstrate that there are two different chiral EPs in parameter space and the non-trivial Berry phase of EPs emerge differently due to the chirality breaking when the eigenstates sweep through EPs in a cyclic evolution. We then propose a scheme to realize the PT-symmetric Hamiltonian in cold atomic systems where the parameters can be exactly controlled in time. Based on the method of the Naimark-Dilated operation [36] and the embedding quantum simulator [37], we show that a two-level PT-symmetric Hamiltonian can be constructed through a four-level Hermitian Hamiltonian in an embedding cold atomic system. Finally, we propose to detect the Berry phase through the observation of the phase difference between the atomic levels, which can be measured through the atomic interferometry. The proposed scheme provides a promising approach to realize the PT-symmetric Hamiltonian in cold atomic systems and to further explore the exotic properties of the EPs.
The paper is organized as follows. Section 2 describes the two-level PT-Hamiltonian and the topological properties of the intrinsic EPs. In Sec. 3, we propose an experimentally feasible scheme to emulate the non-Hermitian two-level Hamiltonian in a Hermitian four-level cold atomic system. In Sec. 4, we show that the topological EPs can be measured by the atomic interferometry in the cold atom system. Finally, a brief discussion and a short conclusion are given in Sec. 5.
2. Topological exceptional points in a PT-symmetric system
If a Hamiltonian H is non-Hermitian for describing an open system, there are gain or loss effects in this system and the eigenvalues are generally complex values. However, if gain and loss of this system are balanced, this system remains stable and all eigenvalues are real numbers. This phenomenon is described by the PT-symmetric quantum theory. Supposing that in this case σi (i = x, y, z) are Pauli matrices, the parity operator P is σx and the time-reversal operator T is the complex conjugation operator, which is an anti-linear operator. A simple PT-symmetric Hamiltonian can be constructed as [2]
where S is a general scaling factor of the matrix. The angle α characterizes the non-Hermiticity of the Hamiltonian. When α = 0, the Hamiltonian H is a Hermitian operator, in contrast α ≠ 0, the Hamiltonian H becomes a non-Hermitian operator. In the case of α = ±π/2, the eigenvalues and the eigenstates coalesce into a single value and state, respectively.The eigenvalues of Eq. (1) are E±(α) = ±χ = ±S cos(α) and the corresponding eigenstates are given by
In addition, the non-Hermitian Hamiltonian HPT has a bi-orthogonal basis |E±(α)〉, |Λ±(α)〉 [24, 38]:
The completeness relation and the orthogonality relation are [24, 38]:
When α = ±π/2, the two eigenvalues become E± = 0 and the corresponding eigenstates coalescence at the same time. Although |E±(±π/2)〉 seem ill defined, they satisfy the completeness relation and the orthogonality relation in Eq. (4). In general, and are ignored and the eigenstates are rewritten as:
The signs before i in Eq. (5), which is the phase information of eigenstates, depend on the system and give the chirality of these specific degeneracy points dubbed as EPs [22, 24, 27, 28], such that the two EPs are different from each other. The Hamiltonian thus is restricted to have purely real eigenvalues, and the time evolution operator is unitary in the bi-orthogonal basis theory, which means :
Especially when α = ±π/2, we have:
Although the determinant of this is not ±1, this time evolution operator do not change the eigenstates in EPs, which means .
With the analytical solution of the PT-symmetric Hamiltonian, we can study the topological properties of the EPs. Considering the eigenvalues E±(α), we can find that the eigenstates |E+(α)〉 and |E−(α)〉 respectively represent the higher and lower levels when α ∈ [−π/2+2Nπ, π/2+ 2Nπ], but respectively represent the lower and higher levels when α ∈ [π/2+2Nπ, 3π/2+2Nπ], with N being a positive integer. This means that the definition of the domain in the system is [−π/2, π/2] and the eigenstates exchange themselves when sweeping α through the EPs every time. With the eigenvalues E±(α) = ±S cos(α), one can also find that the corresponding point of α in the other Riemann surface is the point of ±π − α. Due to the degeneracy of the EPs, the eigenstates obtain non-Abelian geometric phases after passing through the EPs. For this degenerate non-Hermitian system, the Berry phase in the cyclic evolution is [39]
where A is the non-Abelian Berry connection: with dα being the α derivative. For the PT-symmetric Hamiltonian HPT here, we can obtain the non-Abelian Berry phases for two different loops from α to ±π − α (which pass through two EPs of different chiralities) asConsequently, when α successively sweeps through two different EPs in the same evolutionary direction, the eigenstates become original with an additional π Berry phase. In this case, the eigenstates under the whole evolution can be written as
Unlike |E±〉→|E∓〉→ − |E±〉 or |E±〉→ − |E∓〉→ − |E±〉 in the general case, Eq. (11) shows that there is an obviously different behavior in the intermediate process when one passes through different EPs successively, which is our primarily novel conclusion in this work.
The above results show that in non-Hermitian systems, the eigenvalue surfaces exhibit a complex-square-root topology with a branch point named by EP, which can also be considered as a critical point where a transition from PT-symmetric phase to broken-PT-symmetric phase. A consequence of this topology is that encircling an EP once in parameter space results in the exchange of both eigenvalues and eigenstates. This means that one has to encircle an EP twice to recover the original eigenvalues and eigenstates. On the other hand, one of the eigenstates acquires a ±π Berry phase when encircling an EP once and the other one acquires the same phase in the second loop. However, because one not longer distinguishes the clockwise or anticlockwise direction of the state evolution, the results of passing through the EPs once will be different. In this system, the acquired Berry phase in each loop is determined by the chirality of the EP and the evolutionary direction of the eigenstates in parameter space.
3. Realization of the two-level PT-symmetric Hamiltonian in a four-level cold atomic system
The PT-symmetric Hamiltonian may be difficult to be realized in a practical non-Hermitian two-level system. Because of the conserved population of PT-symmetric system, is it possible to simulate PT-symmetric non-Hermitian system with the Hermitian system. Accordingly, one can use a four-level Hermitian system to simulate the two-level PT-symmetric Hamiltonian in Eq. (1) by using the Naimark-Dilation operator [36]. For an arbitrary objective state υ = (a, b)T and an undetermined ancilla state u = (c, d)T, we hope to establish the equation:
where and are the undetermined time evolution operators. In ref. [36], the required time evolution operators and ancilla state have been found out. In the basis (|1〉, |2〉, |3〉, |4〉)T, if the four-level Hermitian Hamiltonian takes the form the corresponding time evolution operator is given by and and the corresponding ancilla state takes the form u = ηυ, where . It means that Eq. (12) becomes as:So if we take the four-level states as (|E±(α)〉, η|E±(α)〉)T, we can simulate the evolution of the states in Eq. (2) under the two-level PT-symmetric Hamiltonian. Below we present an experimentally realizable scheme with cold atoms to achieve this goal.
We consider a cloud of 87Rb cold atoms in a magenton optical trap with the temperature of hundreds of µK [40]. with five internal states in the ground-state manifold denoted by |j〉 (j = 1, 2, 3, 4, 5) as shown in Fig. 1. In experiments, they can be the hyperfine states |F = 1, mF = −1, 0, 1〉 and |F = 2, mF = −1, 0〉, which are separated by the hyperfine splitting ωHF and the Zeeman splitting ωZ induced by a uniform static magnetic field. In addition, we can use resonant microwaves Ω1,2 to couple |1〉 ←→ |3〉 and |2〉 ←→ |4〉 and use radio-frequency fields Ω3,4 to couple |1〉 ←→ |2〉 and |3〉 ←→ |4〉 respectively. Here Ω1,2,3,4 and ω1,2,3,4 denote the Rabi frequencies and the frequencies of the microwave radios or the radio-frequency fields, respectively. All the Rabi frequencies can be independently tuned through the strengths of the microwave radios or the radio-frequency fields. Supposing that ωei are the energies of |i〉 and the energy of |1〉 is the zero of energy, the total Hamiltonian can be written as H = H0 + Hint with
Fig. 1 Schematic representation of the light-atom interaction configuration of the four-level Hamiltonian. The spin state can be mapped into the ground states of 87Rb, of which |1>=|F=1,mF=0>,|2>=|F=1,mF=−1>,|3>=|F=2,mF=0>,|4>=|F=2,mF=1>. The sublevels can be linked by microwave fields or radio frequency fields.
In the bare-state basis (|1〉, |2〉, |3〉, |4〉)T, one has
and the Hamiltonian in the rotating frame becomesTo ensure the Hamiltonian being time-independent, we choose ω1 +ω2 = ω3 +ω4. On the other hand, considering the resonance condition ∆1 = ω1 + ωe1 − ωe3 = 0, ∆2 = ω2 + ωe3 − ωe4 = 0, ∆3 = ω3 + ωe2 − ωe4 = 0, ∆4 = ω4 + ωe1 − ωe2 = 0, with ω1 = ω2 = ωHF, ω3 = ω4 = ωZ. In particular, we choose the corresponding Rabi frequencies Ω1 = −Ω3 = iS sin(α) cos(α) and Ω2 = Ω4 = S cos2(α). Under these conditions, the Hamiltonian can be represented as HF in Eq. (13):
Up to this step, we have proposed a method to realize the required four-level Hamiltonian for simulating the two-level PT-symmetric Hamiltonian in a cold atomic system. It is noteworthy that in this system, we can precisely and easily control the non-Hermitian parameter α by independently adjusting the four Rabi frequencies in experiments [40].
4. Detecting the Berry phase in the PT-symmetric system
In this section, we show how to detect the Berry phases of the mimicked EPs in the four-level cold atomic system. First, we need an additional atomic level for this measurement, which is denoted by |0〉 in the cold atomic system as shown in Fig. 1. We assume the atoms are initially pumped to |0〉 and the transitions |0〉 → |i〉 (i = 1, 2, 3, 4) can be realized successively through the stimulated-Raman-adiabatic-passage (STIRAP) [41, 42]. It is noted that the microwave radios and the radio-frequency fields must be phase-locking between each STIRAP for keeping the coherence between the states. On the other hand, only the phase difference between |0〉 and |i〉 is needed to be concerned, so it is nonsignificant what the population differences between |0〉 and other levels are. Under this condition, we make |0〉 → |0〉 + η|ψ±(α0)〉T from the very beginning, where η is an arbitrary real number, for the preparation of the PT-symmetric initial state. The phase difference between |1〉 and |2〉 is the only distinction between |E±(α)〉 for a given α, and the phase differences between |0〉 and |1〉 (|2〉) can be used to detect the Berry phases of |E±(α)〉. Thus we can detect the Berry phases by measuring the phase differences between the corresponding atomic levels.
The phase difference between two atomic levels can be measured through the atomic interferometry [43, 44]. For an arbitrary state denoted by |a〉 + e−iφ|b〉, a π/2-pulse operation takes the form
with ϕ being a controllable phase of the π/2-pulse. After applying a π/2-pulse to the state |a〉 + e−iφ|b〉, one can find the relationship between the atomic populations and the phase differences asConsidering Eq. (2) and Eq. (20), the atomic populations of the different levels are given by
For a given α, the function of φ in Eq. (21) can be used to distinguish |E+(α)〉 and (E−(α)〉, so we can verify whether the eigenstates exchange themselves through measuring the function of ϕ in Eq. (21) when sweeping an EP.
After confirming the two states |E±(α)〉, we can measure the phase difference between |0〉 and |1〉 to determine the Berry phases γ± for |E±(α)〉, which is related to the value of α and the evolution loop. For |0〉 and |1〉, we can also find the relationship between atomic populations and total phase differences:
where the total phases between the corresponding atomic levels in the case of different eigenstates ϕ+ = γd − α/2 − θ0(α)π/2 and φ− = γd − (π − α)/2 − θ0(α)π/2 with γd = ∓S cos(α)t + ϕ ωZt being the dynamic phase due to the evolution , π/2-pulse and the Zeeman energy difference, respectively, and θ0(α) = θ[−cos(α)] being a Heaviside unit step function whose value is 0 for −cos(α) < 0 and 1 for −cos(α) < 0. It is clear that the topology of the EPs is the source of the Heaviside unit step function. Due to the symmetry of the trigonometric functions, one needs two values of γd to confirm |E+ (α)〉 and |E−(α) in Eq. (22).For implicity, here we choose the dynamic phase as 0 and π/2. The phases φ± and the atomic populations N0,1 in the different eigenstates are shown in Fig. (2). For a given eigenstate, one can measure the phases φ± to obtain the Berry phases of the eigenstates from Eq. (22). In particular, for a given eigenstate and the parameter α, we first measure N1,2 in Eq. (21) to determine which eigenstate is through the function of φ. Then one can detect the total phase ϕ± by measuring N0,1 in Eq. (22), as shown in Fig. 2. After that, one can control the systemic parameter α to sweep an EP in parameter space that the eigenstate is predicted to obtain a non-Abelian phase in Eq. (10). Again one can successively measure N1,2 and N0,1 to determine whether the eigenstate has been changed and the variation of total phase. Here we can find that: i) The eigenstates obtain a phase of +π/2 or −π/2 alternately when α sweeps through the different EPs successively. ii) The eigenstates exchange themselves when α sweeps through an EP and the phase differences from |E±(α)〉 to |E∓(π − α)〉 are always ±π/2. iii) The eigenstates obtain a ±π Berry phase when α sweeps ±2π. In short, the measurement of the phase differences between |i〉 (i = 0, 1, 2) provides a simple way to experimentally verify Eq. (11) and demonstrate the Riemann sheet structure and the intrinsic properties of the EPs.
Fig. 2 The phase difference and atomic population versus angle. The red, green, yellow and blue line denote the |E+(α)〉, |E−(π − α)〉, E−(α)〉 and E+(π − α)〉,respectively. (a,c) The dynamic γd = 0. (b,d) The dynamic phase γd = π/2.
5. Discussion and summary
In the above case, there is no dynamical phase contribution between the four states |i〉 (i = 1, 2, 3, 4), but the evolution of |i〉 must be still adiabatic in order to avoid non-adiabatic transitions. To be specific, one should keep the adiabatic cyclic evolutions with the four Rabi frequencies in parameter space, such that the change rate of α are significantly smaller than the typical Rabi frequencies. This means that the evolution period of the system is much larger than the inverses of the energy gaps between the atomic levels. On the other hands, with the non-adiabatic transition between |E±(α)〉 being avoided, the change rate of α is also smaller than 2χ, which is the difference of eigenvalue. Thus, the adiabatic condition takes the form
To fulfill this condition, one should assure the system remains in the eigenstate of the Hamiltonian all the time.
In summary, we have revealed the different properties between encircling and passing through EPs in a two-level PT-symmetric system and further proposed an experimental scheme to realize the system through an embedding four-level cold atomic system. Moreover, we have demonstrated that the change of the eigenstates and the relevant topological phase when passing through EPs can be confirmed by probing the phases of the atomic levels from the standard phase measurement. Our work proposes a method to realize PT-symmetric Hamiltonian in cold atomic systems and therefore provides a useful tool to explore the properties of PT-symmetry and EPs.
Funding
National Key Research and Development Program of China (NKRDP) (2016YFA0301803); National Natural Science Foundation of China (NNSFC) (11647109, 11604103); Natural Science Foundation of Guangdong Province (2015A030310277, 2016A030310462, 2016A030313436); Project of Enhancing School With Innovation of Guangdong Ocean University (GDOU2017052602); Startup Foundation of GDOU (A16419); SRFYTSCNU (15KJ15); Startup Foundation of SCNU.
Acknowledgments
We thank Profs. Hui Yan, Shi-Liang Zhu and Dr. Feng Mei for helpful discussions.
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