Conserved quantities in parity-time symmetric systems

Zhihao Bian1,2, Lei Xiao1,3, Kunkun Wang1,3, Xiang Zhan1,4, Franck Assogba

Onanga5, Frantisek Ruzicka5,6, Wei Yi7,8, Yogesh N. Joglekar5, Peng Xue11 Beijing Computational Science Research Center, Beijing 100084, China

2 School of Science, Jiangnan University, Wuxi 214122, China3 Department of Physics, Southeast University, Nanjing 211189, China

4 School of Science, Nanjing University of Science and Technology, Nanjing 210094, China5 Department of Physics, Indiana University Purdue University

Indianapolis (IUPUI), Indianapolis, Indiana 46202, USA6 Institute of Nuclear Physics, Czech Academy of Sciences, Rez 250 68, Czech Republic

7 Key Laboratory of Quantum Information, University of Scienceand Technology of China, CAS, Hefei 230026, China and

8 CAS Center For Excellence in Quantum Information and Quantum Physics

Conserved quantities such as energy or the electric charge of a closed system, or the Runge-Lenzvector in Kepler dynamics are determined by its global, local, or accidental symmetries. They wereinstrumental to advances such as the prediction of neutrinos in the (inverse) beta decay processand the development of self-consistent approximate methods for isolated or thermal many-bodysystems. In contrast, little is known about conservation laws and their consequences in open sys-tems. Recently, a special class of these systems, called parity-time (PT ) symmetric systems, hasbeen intensely explored for their remarkable properties that are absent in their closed counterparts.A complete characterization and observation of conserved quantities in these systems and theirconsequences is still lacking. Here we present a complete set of conserved observables for a broadclass of PT -symmetric Hamiltonians and experimentally demonstrate their properties using single-photon linear optical circuit. By simulating the dynamics of a four-site system across a fourth-orderexceptional point, we measure its four conserved quantities and demonstrate their consequences.Our results spell out non-local conservation laws in non-unitary dynamics and provide key elementsthat will underpin self-consistent analysis of non-Hermitian quantum many-body systems that areforthcoming.

Introduction:—In Lagrangian dynamics, conser-vation laws are tied to symmetries of a systemthrough Noether’s theorem [1, 2]. They give riseto global constraints that must be satisfied by ap-proximate methods that are used to model the dy-namics. Their influence is ubiquitous in the per-turbative, variational, and computational methodsfor interacting (many-body) systems: only approx-imations that satisfy conservation laws [3, 4] arephysically meaningful and computationally stable.In traditional quantum theory, an observable O iscalled conserved if it commutes with the HermitianHamiltonian H0 of the system. Due to the equiv-alence between conservation and commutation, aunitary symmetry transformation on the quantumstate-space is generated by each conserved observ-able O, thereby reducing complexity of the eigen-value problem as transformation block-diagonalizesthe Hamiltonian. A complete set of conservationlaws for a system is obtained by identifying all lin-early independent observables that commute withthe Hamiltonian. In particular, norm of a quantumstate is conserved under the unitary time evolutionby any Hermitian Hamiltonian because the operatorO = 1 trivially commutes with any Hamiltonian.

PT -symmetric systems [5–25] are open systems

with balanced gain and loss. They are described bya Hamiltonian HPT that is invariant under the com-bined operation of parity and time reversal, and un-dergoes a time evolution that does not conserve thestate norm [26]. The spectrum of HPT changes fromreal into complex-conjugate pairs when the gain-lossstrength is increased. This PT -symmetry break-ing transition occurs at an exceptional point (EP)where eigenvalues and the corresponding eigenmodescoalesce [27]. Non-unitary dynamics of this tran-sition have been observed in classical systems [5–14] and non-interacting quantum systems compris-ing single photons [15–22], ultracold atoms [23], sin-gle spins [24], and superconducting qubits [25].

Complete characterization of conservation laws insuch systems is an outstanding question. What arethe conserved quantities in PT -symmetric open sys-tems? How do they constrain the (approximatemethods used to model the) dynamics, particularlyin the PT -broken region where amplifying eigen-modes occur? With an approach inspired by earlyworks on pseudo-Hermiticity [28–30], we conclu-sively address these questions. For a broad class ofPT -symmetric Hamiltonians encompassing all ex-perimentally relevant models, we analytically con-struct a complete set of linearly independent ob-

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servables whose expectation values do not changewith time. We demonstrate our construction witha four-site PT -symmetric Hamiltonian by encodingthe four sites in the path and polarization of a sin-gle photon and simulating its non-unitary dynam-ics. We track four constants of motion across thePT -transition, and demonstrate the consequencesof these non-local, conserved quantities throughthe dynamics of adjacent-site phase differences andconserved-observable eigenstates.

Conserved observables for HPT :—Consider a PT -symmetric system described by a d-dimensionalHamiltonian HPT with an energy scale J (~ = 1).An observable η is called an intertwining opera-tor [29, 30] for HPT , if it satisfies

ηHPT = H†PT η. (1)

It follows the non-unitary time evolution operatorG(t) = exp(−iHPT t) keeps the observable η un-changed, i.e. G†(t)ηG(t) = η. Expectation valueof η in an arbitrary quantum state is, therefore, aconserved quantity. The equivalence between con-servation and commutation breaks down for a non-Hermitian Hamiltonian. In principle, the completeset of conserved observables can be obtained bynumerically solving the set of d2 linear equations(1) [31, 32].

Instead, we analytically obtain the complete set ofintertwining operators for all PT -symmetric Hamil-tonians that are also transpose symmetric, i.e.HPT = HT

PT . This broad class includes all experi-mentally investigated PT -symmetric systems in theclassical [5–14] and quantum [15–25] domains, andmost of the tight-binding models [33]. The recursiveconstruction is as follows: the transpose symmetryimplies T HPT T = H†PT , where the time-reversaloperator T is given by complex conjugation. It fol-lows from the PT symmetry of the Hamiltonianthat the parity operator P is a conserved observ-able, i.e. η1 = P. We then construct a sequenceof linearly independent, dimensionless observablesηi = ηi−1HPT /J (i = 2, · · · , d). The intertwiningnature of ηi−1 implies that ηi is also an intertwin-ing operator or, equivalently, a conserved observable.This sequence terminates with ηd because the char-acteristic equation for HPT is a polynomial of orderd. Thus, ηd+1 is a linear combination of lower-orderconserved observables. By expressing η and HPTin the bi-orthogonal eigenbasis of HPT [30], thereare no additional intertwining operators when thespectrum of HPT is non-degenerate. Our procedureyields the set of d linearly independent conserved ob-servables for a d-dimensional PT -symmetric system.

Non-unitary dynamics of a four-site PT -symmetric system:—For an experimental demon-stration of conserved observables under non-unitarydynamics, we use the Hamiltonian

HPT (γ) =1

2

3iγ −

√3J 0 0

−√

3J iγ −2J 0

0 −2J −iγ −√

3J

0 0 −√

3J −3iγ

,

(2)

which is compactly written as HPT (γ) = −JSx +iγSz, where Sx and Sz are spin-3/2 representationsof the SU(2) group [34]. It describes a four-sitesystem with mirror-symmetric tunneling, a lineargain-to-loss profile, and the parity operator P =antidiag(1, 1, 1, 1). Four equally spaced eigenvalues

are given by {−3/2,−1/2,+1/2,+3/2}√J2 − γ2,

which give rise to a fourth-order EP at the PT -breaking threshold γ = J (Fig. 1a). The single

energy gap√J2 − γ2 in the spectrum of (2) leads

to dynamics with period T (γ) = 2π/√J2 − γ2 for

γ < J .We encode four sites of the system in the path and

polarization degrees of freedom of a single photon,and label them as |1〉 = |UH〉, |2〉 = |UV 〉, |3〉 =|DH〉, |4〉 = |DV 〉. Here {|H〉, |V 〉} are horizontaland vertical polarizations, and {|U〉, |D〉} denote up-per and lower paths, which undergo gain and loss,respectively. Mapping HPT into a Hamiltonian withsite-selective loss HL(γ) = HPT (γ) − 3iγ1/2, weimplement the operator GL(t) = exp(−iHLt) viaa lossy linear optical circuit, which is related toG(t) through G(t) = GL(t) exp(3γt/2) [15]. Sucha transformation adds an overall gain to experi-mental measurements, which enables the experimen-tal system with passive PT symmetry to simulateideal PT -symmetric models [15–22]. By project-ing the time-evolved state |ψ(t)〉 = G(t)|ψ(0)〉 ontothe site-modes |k〉 (k = 1, · · · , 4), time-dependentstate norms are obtained (Fig. 1b). To probe con-served quantities ηi(t) ≡ 〈ψ(t)|ηi|ψ(t)〉, quantum-state tomography is carried out on time-evolvedstates (Fig. 1c). We reconstruct the time evolutionover multiple time scales and a wide range of γ forfour initial states given by |ψ1〉 = |1〉, |ψ2〉 = (|1〉 +|2〉 + |3〉 + |4〉)/2, |ψ3〉 = (|1〉 +

√2|4〉)/

√3, |ψ4〉 =

(|1〉 + |4〉)/√

2. Conserved quantities ηi(t) can alsobe directly probed via projective measurements [35].

Figures 1d-1f exemplify the non-unitary dynam-ics generated by HPT (γ) through a time-dependentstate norm N(t) = 〈ψ(t)|ψ(t)〉. Constant in the Her-mitian limit (γ = 0), N(t) oscillates with a periodT (γ) in the PT -symmetric region (Fig. 1d). State

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PBS

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State tomography

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D

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FIG. 1. Non-unitary dynamics of a four-site PT -symmetric system. a Schematic of a four-site system with nearest-neighbor tunneling, gain for sites |1〉 and |2〉, and corresponding loss for sites |4〉 and |3〉. Spectrum of HPT . bSchematic of the optical circuit used to measure the state norm and the inner product between the initial andfinal states. BS: beam splitter; HWP: half-wave plate; QWP: quarter-wave plate; APD: avalanche photodiode;PBS: polarizing beam splitter; CC: compensated crystal; GL(t): lossy time evolution circuit [35]. c Schematic ofquantum-state tomography used to reconstruct the time-evolved state |ψ(t)〉 [35]. d In the Hermitian limit withγ = 0, measured state norm N(t) is conserved and remains unity (open symbols). In the PT -symmetric region withγ = 0.2J , N(t) oscillates with a period T (γ) (filled symbols). e At the PT transition point γ = J , state norm growsalgebraically, N(t) ∝ t6. f In the PT -symmetry broken region with γ = 1.2J , state norm grows exponentially withtime. Experimental errors are due to photon-counting statistics; when not shown, error bars are smaller than thesymbol size.

norms do not oscillate around unity, due to the non-othogonality of eigenvectors of the non-HermitianHamiltonian HPT (γ) [35]. At the fourth-order EP,the state norm N(t) grows algebraically with timeas t6 (Fig. 1e). Such a scaling is dictated by the or-der of the exceptional point: at the EP, H4

PT = 0and the power-series expansion of G(t) terminatesat the third order, giving rise to the t6 dependencefor the norm. In the PT -symmetry broken region,the measured norm grows exponentially with time(Fig. 1f).Measurement of four non-local conserved

quantities:—For our four-site system, the re-cursive procedure gives four conserved observables.Expressing the time-evolved state in the site-basis|ψ(t)〉 =

∑4k=1 ak(t)|k〉, the expectation value of η1

is given by [28–30]

η1(t) =

4∑k=1

a∗5−k(t)ak(t), (3)

with similar, non-local expressions for the remain-ing three conserved observables η2, η3 and η4 [35].

Therefore, the global conservation of ηi(t) does nottranslate into local densities that obey continuityequations. This is in stark contrast with the Her-mitian or thermal cases where a globally conservedquantity, such as the momentum, gives rise to alocal continuity equation involving the momentum-density and the corresponding stress tensor [1].

Figure 2 shows measured expectation values ηi(t)for the symmetric initial state |ψ2〉. While generi-cally dependent on γ, the expectation values ηi(t)remain constant regardless of whether the systemis Hermitian (γ = 0), in the PT -symmetric region(γ = 0.2J), at the fourth-order EP (γ = J), ordeep in the PT -symmetry broken region (γ = 1.2J).While η1(t) is positive for all γ due to the symmet-ric nature of |ψ2〉, it is not positive-definite over theentire quantum state space.

Results in Fig. 2 clearly demonstrate that despitetheir non-unitary evolution, open systems governedby PT -symmetric Hamiltonians support non-local,conserved quantities. Their constancy at the EP andin the PT -symmetry-broken region, where the state

4

0 . 0 6 0 . 8 1 . 2 4 8 1 20- 4- 3- 201 2| � ⟩

γ= 0 γ= 0 . 2 Jγ= J γ= 1 . 2 J

η 4(t)

t i m e t

1 . 0 9- 0 . 2 0

- 3 . 0 2- 3 . 1 4

0 . 0 6 0 . 8 1 . 2 4 8 1 20

0 . 61 . 21 . 82 . 43 . 0

0

2| � ⟩γ= 0 γ= 0 . 2 Jγ= J γ= 1 . 2 J

η 3(t)

t i m e t

0 . 8 72 . 0 7

2 . 1 2

0 . 3 2

0 . 0 6 0 . 8 1 . 2 4 8 1 20- 2 . 0- 1 . 5- 1 . 0- 0 . 5 2| � ⟩

γ= 0 γ= 0 . 2 Jγ= J γ= 1 . 2 J

η 2(t)

t i m e t

- 1 . 3 7

0 . 0 6 0 . 8 1 . 2 4 8 1 20

0 . 51 . 01 . 5

γ= 0 γ= 0 . 2 Jγ= J γ= 1 . 2 J2| � ⟩

η 1(t)

t i m e t

1

dc

ba

FIG. 2. Non-local, conserved observables across the PT -transition. a-d For a symmetric initial state |ψ(0)〉 = |ψ2〉,measured expectation values ηi(t) of the four, dimensionless observables depend on γ, but remain time invariant.The symmetry of the initial state is responsible for the γ-independent behavior of the positive η1(t) and negativeη2(t). Error bars are smaller than the symbol size and not shown. Conserved quantities ηi(t) for t < 1.2 and t > 4 inthe unitary and PT -symmetric cases, and that for t < 1.2 in the PT -broken case (and at the exceptional point) areobtained by direct measurement [35]. By contrast, ηi(t) for 1.2 < t < 4 are measured through state tomography. Theexperimental results of conserved quantities obtained from both methods are consistent with each other, and matchwell with theoretical predictions.

norm increases with time algebraically or exponen-tially, leads to surprising consequences that we nowdiscuss.

Consequences of conservation laws:—First, con-stancy of conserved quantities at the EP or in thePT -symmetry broken region gives rise to a phase-locking phenomenon. Writing the time-evolved statein terms of site amplitudes rk(t) and phases φk(t)

as |ψ(t)〉 =∑4k=1 rke

iφk |k〉, we track the phase-difference between adjacent sites, θk(t) ≡ φk(t) −φk−1(t) (k = 2, 3, 4) for the four initial states.Figures 3a-3d show that irrespective of the initialstate, the phase differences θk(t) reach the steady-state value of π/2 at the EP (a,b) and in thePT -symmetry-broken region (c,d). Steady-statevalues are determined by the site distribution ofgain and loss, but are independent of the initialstate [36]. The phase locking is the result of an ex-ponential separation between the time dependenceof 〈ψ(t)|η|ψ(t)〉 and state norm 〈ψ(t)|ψ(t)〉. Thus,it is also a generic feature of quantum (and clas-sical) systems with site-selective loss, where thePT -symmetry-breaking transition is signaled by theemergence of slowly decaying eigenmodes.

Second, since equivalence between conservationand commutation breaks down for a PT -symmetricsystem, conserved observables ηi and the Hamilto-

nian HPT cannot be simultaneously diagonalized(except at γ = 0). Thus, non-orthogonal eigenmodes|Ej(γ)〉 (j = 1, · · · , 4) of the Hamiltonian, and or-thogonal eigenstates |vj〉 of a conserved observablehave qualitatively different dynamics.

Figures 3e-3g show the measured expectation val-ues of the four conserved observables ηi for differentγ, with the system initialized in different eigenmodes|Ej(γ)〉. The experimental result (colored boxes) isobtained by averaging the measured values ηi(t) overall time-points [35]; theoretical predications are rep-resented by open boxes. At the fourth-order EP, allexpectation values ηi for the only eigenstate of HPTvanish (Fig. 3f). Deep in the PT -symmetry-brokenregion (γ = 1.2J), ηi are zero for each of the four,non-orthogonal eigenmodes |Ej(γ)〉. In a Hermitiansystem, energy eigenmodes can have multiple, non-zero constants of motion. In contrast, non-local con-served quantities for a PT -symmetric system vanishidentically for all eigenmodes that participate in thePT -symmetry breaking transition.

Lastly, in a closed system, a coherent superposi-tion of eigenstates of a conserved quantity cannot begenerated from a single, initial eigenstate. In a PT -symmetric system, however, the eigenstates of a con-served observable exhibit nontrivial dynamics. Asan example, we consider the dynamics of orthonor-

5

a b dc

fe g h

FIG. 3. Consequences of non-local, conserved observables. a-b At the EP, the adjacent-site phase differences θk(t)reach a steady-steady value π/2 irrespective of the initial state. c-d The same phase locking occurs in the PT -symmetry broken region. The diverging state norm and the constancy of ηi(t) are responsible for this phenomenon.e Conserved quantities ηi measured in the PT -symmetric eigenstates |Ej〉 for γ = 0.2J . f At the fourth-order EP,γ = J , all conserved quantities ηi are zero for the sole eigenstate of HPT . g In the PT -symmetry broken region,γ = 1.2J , all ηi vanish for each eigenstate |Ej(γ)〉. h Measured time-evolution of the angle Φj(t) between |vj〉 and|ψ(t)〉.

mal eigenstates |vj〉 of a conserved observable η1.Starting with the initial state |ψ(0)〉 = |v1〉, the ex-perimentally measured time-evolution of the angleΦj(t) between |vj〉 and |ψ(t)〉 is shown in Fig. 3h.Fixed, for all times, at zero or π/2 in the Hermitianlimit (γ = 0), angles Φj(t) vary periodically with pe-riod T (γ) in the PT -symmetric region (γ = 0.2J),but reach steady-state value at the EP (γ = J) or inthe PT -symmetry-broken region (γ = 1.2J). Theseresults also hold for quantum (and classical) systemswith mode-selective loss.

Outlook:—Recent advances have led to the real-ization of PT symmetry in minimal quantum sys-tems such as a single spin through Hamiltonian di-lation [24] and a single superconducting transmonthrough post-selection [25]. These systems are inte-gral to near-term [37] or well-established [38] large-scale quantum simulators that will address funda-mental, hard questions in strongly correlated, many-body systems. Modeling the dynamics of suchsystems with PT symmetry is a challenging openquestion. Approximate methods such as tensornetworks [39], the density matrix renormalizationgroup [40], or the quantum Monte Carlo [41], arebased on conservation laws for unitary (or thermal)time evolution, and do not apply to strongly corre-lated, many-body PT -symmetric systems.

By theoretically revealing and experimentally con-firming non-local conservation laws in non-unitary

dynamics, our work provides key elements for self-consistent analysis of quantum, many-body, opensystems with PT symmetry, which are forthcomingin the near future. Understanding conserved observ-ables in these systems offers useful insight into quan-tum dynamics therein, which stimulates the develop-ment of new, self-consistent approximation methodsfor open systems in general.

This work has been supported by the Natural Sci-ence Foundation of China (Grant Nos. 11674056and U1930402), the Natural Science Foundationof Jiangsu Province (Grant No. BK20190577),the National Key R&D Program (Grant Nos.2016YFA0301700,2017YFA0304100), the NationalScience Foundation grant DMR-1054020 and thestartup funding of Beijing Computational ScienceResearch Center. YJ thanks Andrew Harter andJoshua Feinberg for discussions.

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