ARTICLE DOI: 10.1038/s41467-018-04690-y OPEN Observation of an anti-PT-symmetric exceptional point and energy-difference conserving dynamics in electrical circuit resonators 1 2 1,3 1 Youngsun Choi , Choloong Hahn , Jae Woong Yoon & Seok Ho Song Parity-time (PT) symmetry and associated non-Hermitian properties in open physical sys- tems have been intensively studied in search of new interaction schemes and their appli- cations. Here, we experimentally demonstrate an electrical circuit producing key non- Hermitian properties and unusual wave dynamics grounded on anti-PT (APT) symmetry. Using a resistively coupled amplifying-LRC-resonator circuit, we realize a generic APT- symmetric system that enables comprehensive spectral and time-domain analyses on essential consequences of the APT symmetry. We observe an APT-symmetric exceptional point (EP), inverse PT-symmetry breaking transition, and counterintuitive energy-difference conserving dynamics in stark contrast to the standard Hermitian dynamics keeping the system’s total energy constant. Therefore, we experimentally conﬁrm unique properties of APT-symmetric systems, and further development in other areas of physics may provide new wave-manipulation techniques and innovative device-operation principles. 1 2 Department of Physics, Hanyang University, 222 Wangsimni-Ro, Seoul 04763, Korea. School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Avenue, Ottawa, ON K1N 6N5, Canada. Electronics and Telecommunications Research Institute, Daejeon 34129, Korea. These authors contributed equally: Youngsun Choi, Choloong Hahn, Jae Woong Yoon. Correspondence and requests for materials should be addressed to J.W.Y. (email: email@example.com) or to S.H.S. (email: firstname.lastname@example.org) NATURE COMMUNICATIONS (2018) 9:2182 DOI: 10.1038/s41467-018-04690-y www.nature.com/naturecommunications 1 | | | 1234567890():,; ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04690-y n spite of their non-conserving nature and extra complexities, illustrated in Fig. 1b. Importantly, no explicit physical symmetry non-Hermitian symmetries and associated dynamics have is found for this generic APT-symmetric system under the PT Itriggered unprecedented interest in open physical systems operation. Moreover, the environmental energy-exchange scheme 1,2 because of their novel properties and potential applications . For is completely different from the PT-symmetric counterpart. This example, development of the parity-time (PT) symmetry in optics argument suggests that essential dynamics in APT-symmetric have suggested new ways of controlling light propagation and systems might be remarkably different from the PT-symmetric (APT) (PT) conﬁnement involving spontaneous symmetry-breaking transi- counterpart even though the relation H = ±iH implies 3–8 tion and non-Hermitian singularities . As a plausible variant of mathematically indistinguishable eigensystem structures for H (PT) (APT) the PT symmetry, anti-PT (APT) symmetry has been treated in and H in principle. Therefore, a comprehensive study positive−negative index multilayers , optically dressed atom lat- on the stationary and dynamic properties of a generic APT- tices , and rapid coherent photo-atomic transport experi- symmetric system is presently of importance in search of inter- 11 (APT) ments . An APT-symmetric Hamiltonian H can be esting and useful interaction conﬁgurations from conceptually conveniently deﬁned in terms of a PT-symmetric Hamiltonian H diverse, open-system physics domains. (PT) (APT) (PT)11 such that H = ±iH . Consequently, PT-symmetric- Here, we experimentally implement a model circuit of a generic like eigensystem structures involving exceptional points (EP), APT-symmetric system and observe both stationary and dynamic spontaneous symmetry-breaking transition, and self-intersecting properties associated with an EP singularity, spontaneous energy-spectral topology appear and they result in PT-APT symmetry-breaking transition, and pseudo-Hermitian vector- conjugate phenomena such as refractionless propagation, ﬂat space properties. Using a resistively coupled amplifying LRC 9–11 total transmission bands, and continuous lasing spectra . resonators, we realize an APT-symmetric electrical system per- Within the context of non-conserving binary oscillator pro- mitting precise parametric controls. Spectral and time-domain blems, a PT-symmetric system has a characteristic Hamiltonian measurements reveal inverse PT-symmetry breaking transition of eigenvectors, associated bifurcation of complex eigenvalues at an ε þ iγκ ðPTÞ H ¼ : ð1Þ APT-symmetric EP, and unique energy-difference conserving κε iγ dynamics in stark contrast to the conventional systems described by Hermitian Hamiltonians. Therefore, we experimentally con- (PT) H describes two equally tuned oscillators at energy level ε ﬁrm essential consequences of the APT symmetry and associated and with their attenuation (or amplifying, equivalently) rates anomalous non-Hermitian properties. Importantly, we show that differing by 2γ. For non-dissipative inter-oscillator coupling the non-Hermitian quantum mechanics reveals the underlying where κ is purely real-valued, the PT symmetry is an essential physics of the observed properties in an intuitive manner consequence as the system is invariant under the simultaneous although speciﬁc features of such properties can be understood in parity inversion (P) and gain-loss exchange (T) operations. See the standard circuit theory. Fig. 1a for schematic illustration of this property. The APT- symmetric counterpart is described by a Hamiltonian of the form ε þ iγ iκ Results ðAPTÞ H ¼ ð2Þ Spectral properties of an APT-symmetric circuit. The proposed iκε þ iγ electrical circuit consists of two amplifying LRC resonators con- nected in parallel through a coupling resistor as shown in Fig. 2a. which implies two equally amplifying oscillators at an ampliﬁ- The system simulates the APT−symmetric environmental- cation rate γ and with their energy level differing by 2ε,as interaction scheme with negative resistor units (–R and –R ) 1 2 providing a gain mechanism and with a coupling resistor (R )as Real PT symmetry a loss mechanism. For R = R = R = R and C = C = C, C 1 2 1 2 coupling essential dynamics of the system is described by a Schrödinger- (APT) (APT) type equation d|v〉/dt = –iH |v〉, where H is given by Eq. Gain Loss Gain Loss PT reversal (2). Here, the state vector |v〉 is deﬁned such that [V V ] ≡ 0.5 1 2 * –1/2 –1/ [exp(–iω t)|v〉 + exp(iω t)|v〉 ] with ω = 0.5[(L C) + (L C) 0 0 0 2 1 ] being average uncoupled-resonance angular frequency. The –1/ Hamiltonian matrix elements are determined by ε = 0.5[(L C) 2 –1/2 –1 – (L C) ], γ = 0, κ = (2RC) . See Supplementary Note 1 for Environment mathematical treatment based on the Kirchhoff’s circuit laws. Therefore, a generic APT-symmetric Hamiltonian is readily rea- Imag. Anti-PT symmetry lized in this simple model circuit. coupling We assemble an APT-symmetric model circuit with para- i –i meters R = 400 Ω, C = 425 nF, L = 1.46 mH, L (variable) = 1 2 PT reversal Gain Gain Loss Loss 1.46~0.78 mH, and R = R = 10 kΩ. These circuit parameters A B yield constant κ = 0.468 kHz, variable ε in a range from 0 to 1.18 kHz, and variable ω in a range from 6.39 to 7.57 kHz. Using this conﬁguration, we investigate stationary APT-symmetric proper- ties by exciting the circuit with a harmonically oscillating source Environment signal injected at V and monitoring V (t) in the time (t) domain. 2 1 Fig. 1 Schematic diagrams of parity-time and anti-PT symmetric binary A characteristic resonance spectrum is obtained by taking a systems. a Parity-time (PT)-symmetric coupled oscillators implied by the Fourier-transformed intensity W (f) = |F[V (t)]| in the frequency n n (PT) PT-symmetric Hamiltonian H in Eq. (1). b Anti-PT (APT)-symmetric (f) domain. Measured W (f) spectrum as a function of the energy- (APT) (PT) binary system derived from the relation H = ±iH . In the two detuning parameter ε is shown in Fig. 2b. Here, we clearly notice diagrams, κ denotes inter-resonator coupling constant and vertical arrows a branch-point splitting of the resonance peak at a threshold indicate directions of energy exchange between the coupled-resonator point of ε = κ. Remarkably similar spectral effects are found for system and environment nonlinear coupled-oscillator systems associated with the 2 NATURE COMMUNICATIONS (2018) 9:2182 DOI: 10.1038/s41467-018-04690-y www.nature.com/naturecommunications | | | Frequency (kHZ) NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04690-y ARTICLE exactly in parallel with the eigenvalue property near a PT- 2,16 Resonator 1 Resonator 2 symmetric EP . V V 1 2 The threshold condition ε = κ corresponds to the APT- L L 1 2 symmetric EP where eigenvalues and eigenvectors simultaneously coalesce. In Fig. 3c, we show measured amplitude ratio v /v of C C 1 2 1 2 the state vector |v〉 at the resonance-center frequency as a function of ε. The amplitude ratio is uniquely deﬁned for a state R R R C R R R B A 1 2 A B and the measured values clearly coalesce in the Gauss plane at ε –R –R 1 2 + = κ, conﬁrming that this threshold condition represents an APT- symmetric EP. On top of the v /v plot in Fig. 3c, we indicate (v , 1 2 1 v ) for the corresponding eigenvectors |λ 〉. For ε ≤ κ, both |λ 〉 2 ± + Negative resistor units and |λ 〉 are invariant under simultaneous P (exchange of the two arrows each other) and T (complex conjugation of the arrows) operations, i.e., the eigenvectors are in the exact PT-symmetry phase. In contrast, the PT symmetry in |λ 〉 is broken for ε > κ. Therefore, the stationary response of the APT-symmetric circuit undergoes a spontaneous PT-symmetry breaking at the EP even though the system does not have any explicit physical symmetry as pointed out earlier. In the eigensystem analysis so far, we have experimentally showed that a binary APT-symmetric EP involves a spontaneous PT-symmetry breaking in the eigenvector conﬁguration and, in addition, the real/imaginary eigenvalue splitting property is reversed with respect to the PT-symmetric counterpart, i.e., real-eigenvalue splitting for the broken PT-symmetry phase (ε > κ) and imaginary-eigenvalue splitting for the exact PT-symmetry phase (ε < κ). In this respect, APT-symmetric binary systems and associated phenomena can be treated in a manner similar to the PT symmetry as far as their stationary responses are treated. Fig. 2 APT-symmetric model circuit and its spectral property. a Circuit diagram of APT-symmetric LRC resonators. The circuit consists of two Dynamic properties of an APT-symmetric circuit. We further resistively coupled amplifying LRC resonators with negative resistor units. A study dynamic properties of a generic APT-symmetric system in variable inductor is used for L to precisely control the energy-detuning our resistively coupled LRC resonators. In particular, we inves- parameter ε. b ε-dependent resonance-excitation spectrum W (f) tigate temporal responses for the broken PT-symmetry phase where the eigenvalue splitting is purely real-valued and the sys- 12–15 injection-locking and pulling phenomena . However, the tem’s time-evolution does not involve a measurement-instability resonance-peak coalesce observed in this case is obtained in a problem due to rapid exponential growth of the probe-voltage purely linear-gain regime and thereby it does not involve any signals. The experimental procedures include following steps in nonlinear relaxation processes causing the conventional injection- the temporal order: Isolation of resonator 2 with the remaining locking phenomena. Instead, this property originates from an parts including the coupling resistor R and resonator 1; con- APT-symmetric EP and spontaneous symmetry-breaking transi- necting the V terminal to the ground to set V = 0; time- 1 1 tion as we will conﬁrm in the following analyses in a quantitative harmonic excitation of resonator 2 at the resonance-center fre- manner. quency by gain-assisted self-oscillation; disconnection of the V Loci and bandwidths of the resonance peaks in the measured terminal off the ground; connection of resonator 2 with the spectral proﬁles correspond to the real and imaginary parts of the remaining parts; and acquiring V (t) and V (t) to determine the 1 2 eigenvalue dynamic state |v(t)〉 = [v (t) v (t)] evolved from the initial state | 1 2 v(0)〉 = [0 1] . We conduct this experiment for ε = 1.48κ and the pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 λ ¼ ± ε κ ð3Þ result is summarized in Fig. 4a. Measured electric energy E = 2 2 0.5C V in capacitor C , magnetic energy M = 0.5L I in n n n n n n inductor L , and total energy T = E + M for resonator n are n n n n (APT) of H . In the measured spectral proﬁles, the resonance-peak plotted as functions of time. An unprecedented property revealed location and full-width at half-maximum (FWHM) in the in the measured time-domain response is beating patterns that –1 frequency domain follow f = (2π) [ω + Re(λ )] and peak 0 ± conserve the energy difference ΔT = T –T . This is in stark 2 1 –1 Δf = π [δ + Im(λ )], respectively. Here, δ denotes a residual peak ± contrast to the standard Hermitian dynamics keeping the sys- background absorption rate due to parasitic internal resistance in tem’s net energy T + T constant and also to the PT-symmetric 1 2 * * the constituent elements and its empirical value is around 25 Hz dynamics conserving a cross-conjugate product v v + v v . 1 2 1 2 in our experiment. Therefore, Re(λ ) and Im(λ ) in experiment ± ± Within the context of the classical circuit theory, this unusual can be inferred from the peak location and bandwidth, property is explained as originating from a speciﬁc conﬁguration respectively, as shown in Fig. 3a, b. Excellent agreement of the of the circuit and associated energy-variation rate properties. experimental values with the theory conﬁrms that a generic APT- Conﬁguring the circuit for the APT-symmetric environmental symmetric system is indeed realized in the model circuit with energy-exchange scheme in Fig. 1b, we required a set of circuit high degree of precision. In addition, a PT-APT conjugate constant conditions such that R = R = R = R and C = C = C. C 1 2 1 2 property in the eigenvalue spectrum is evident therein. λ shows a These conditions result in the resonator’s energy-variation rate purely imaginary splitting for ε < κ, merging (λ = λ )at ε = κ, + – dT V V and a purely real splitting for ε > κ. Resulting from the n 1 2 ð4Þ (APT) (PT) fundamental relation of H = ±iH , this characteristic is dt R NATURE COMMUNICATIONS (2018) 9:2182 DOI: 10.1038/s41467-018-04690-y www.nature.com/naturecommunications 3 | | | 1.5 0.5 1 Re (v /v ) 1 2 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04690-y 8.5 EP | 〉 Re + 8.0 | 〉 7.5 Experiment Theory v 7.0 6.5 | 〉 4 0.4 0.2 | 〉 0.0 | 〉 –0.2 2 | 〉 1.5 –0.4 0.5 0.0 0.5 1.0 1.5 2.0 2.5 –1 Fig. 3 Eigensystem structure of an APT-symmetric circuit. a, b Real and imaginary eigenvalues inferred from the resonance-excitation spectra in Fig. 2bin comparison with theory. The theoretical curves are obtained by a binary Hamiltonian model derived from Kirchhoff’s circuit laws. The shaded area for ε/κ > 1 indicates the parametric region of the broken PT-symmetry (PTS) phase. c State amplitude ratio v /v as a function of the energy-detuning parameter ε, 1 2 where v denotes complex amplitude of the state vector |v〉 such that |v〉 = [v v ] . Circles indicate experimentally obtained values and dashed curves are n 1 2 obtained from the binary Hamiltonian model. The upper inset panels show corresponding eigenvector |λ 〉 coordinates on the Gauss plane for ε = 0, 0.5κ, κ, and 1.5κ which is essentially identical for the two resonators because the Resonator 1 E M T 1 1 1 –1 voltage-product energy rate R V V on the right-hand side of Resonator 2 E M T 1 2 2 2 2 Eq. (4) is independent of the resonator index n. See Supplemen- tary Note 2 for details. The energy-difference conserving dynamics implies that there is no net energy exchange between the two coupled resonators. In conventional coupled-resonator systems following the Hermitian dynamics, beating is a natural consequence of an inter-resonator energy exchange. In this 0.4 respect, the speciﬁc beating patterns appearing in Fig. 4a for the T proﬁles are not explained by the standard interpretation. 0.2 The counterintuitive beating pattern in the APT-symmetric circuit is induced by a periodic energy exchange between the 0.0 whole resonator system and the environment. According to Eq. (4), the resonator’s energy ampliﬁcation or dissipation is –0.2 determined by the sign of the voltage-product V ·V . Therefore, 1 2 the speciﬁc beating patterns in Fig. 4a are understood by the –0.4 periodic change of the V V -product sign for the two resonators 1 2 oscillating at slightly different frequencies. Note that the electric current I through the coupling resistor R is high for V V <0 C C 1 2 and the Ohmic dissipation exceeds the system’s net gain, resulting 60 in attenuation of the resonant excitation. In contrast, I for V V C 1 2 > 0 is low and the response of the system is led by the gain that renders the excitation stronger. As key evidences, we provide –1 measured voltage-product energy rate R V V in Fig. 4b and its 1 2 –1 integrated proﬁle J(t ) = Σ [R V (t )V (t )]Δt in Fig. 4c, where a b 1 b 2 b 0.0 0.5 1.0 1.5 2.0 2.5 3.0 the dummy sampling-time index b is running over 0 to a. The Time (ms) beating pattern in J(t) in Fig. 4c shows a quantitative agreement Fig. 4 Energy-difference conserving dynamics in time domain. a with the T (t) patterns in Fig. 4a, conﬁrming the validity of the Measured time evolution of the total energy T , electric energy E , and n n energy-variation rate relation given by Eq. (4). magnetic energy M for an initial state |v(0)〉 = [0 1] . We set ε = In a more fundamental viewpoint, the energy-difference 1.48κ in this measurement. b Measured voltage-product energy rate conserving dynamics is associated with a bi-orthogonal vector- V V /R. c Integrated voltage-product energy rate Σ(V V /R)Δt 1 2 1 2 space property. Following the pseudo-Hermitian representation showing an exact correlation with the beating patterns in T . Here, the of quantum mechanics for our case with an APT-symmetric sampling time interval Δt = 4 μs 4 NATURE COMMUNICATIONS (2018) 9:2182 DOI: 10.1038/s41467-018-04690-y www.nature.com/naturecommunications | | | Broken PTS phase Exact PTS phase Im (v /v ) 1 2 Σ (V V /R)Δt (pJ) V V /R (μJ/s) 1 2 1 2 Energy (pJ) –1 –1 (2π) lm() (kHz) (2π) [Re()+ ] (kHz) Im NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04690-y ARTICLE (APT) 21 22 Hamiltonian H , the inner-product that permits a prob- transmission and opto-mechanical oscillator experiments. abilistic description of state vectors should take a The subject is presently attracting a special attention because of form (ψ,ϕ) = 〈ψ|η|ϕ〉, where η is a Hermitian metric operator its potential for robust time-asymmetric or nonreciprocal devices. –1 (APT)† (APT) satisfying η H η = H . In our case with a binary H In the potential APT-symmetric circuit conﬁgurations, required (APT) , the metric operator is given by time-varying topological operations with an APT-symmetric EP can be readily created by introducing electrical tunability in the resistor and capacitor elements with appropriate transistors and η ¼ ð5Þ 0 1 varactor diodes. Furthermore, realization of optical systems involving mathematically identical non-Hermitian Hamiltonians and it yields a conserved quantity should be feasible in coupled resonators or integrated-waveguide systems where required complex-index proﬁles can be effectively 2 2 hj v ηji v ¼jv j jv j ¼ hi T hi T ; ð6Þ 1 2 1 2 generated using impurity doping, functional thin ﬁlms, and 3–8 photonic nanostructures . where 〈···〉 indicates a time average of its argument over an oscillation cycle at ω . Here, we use a relation 〈T 〉 = 0.5 C|v | Data availability. The data that support the ﬁndings of this study 0 n n implied from the deﬁnition of v (see Supplementary Note 1). are available from the corresponding authors on request. Therefore, the energy-difference conserving time-evolution is understood as a general property that applies to any APT- Received: 21 November 2017 Accepted: 15 May 2018 symmetric system regardless of system’s details. Discussion In conclusion, we have experimentally demonstrated an electrical circuit that simulates a generic APT-symmetric system. Stationary References and dynamic properties were investigated using a resistively 1. Bender, C. M. & Boettcher, S. Real spectra in non-Hermitian Hamiltonians coupled amplifying-LRC-resonator circuit where precise para- having PT symmetry. Phys. Rev. Lett. 80, 5243–5246 (1998). 2. Klaiman, S., Günther, U. & Moiseyev, N. Visualization of branch points in PT- metric control and time-resolved measurement are highly feasible. symmetric waveguides. Phys. Rev. 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Further study of signiﬁcant of adiabatic state ﬂips generated by exceptional points. J. Phys. A: Math. Theor. interest within this context is to realize time-varying APT-sym- 44, 435302 (2011). 23,24 metric systems enabling topological operations around an EP . 24. Gilary, I., Mailybaev, A. A. & Moiseyev, N. Time-asymmetric quantum-state- This unique non-Hermitian interaction scheme was recently exchange mechanism. Phys. Rev. A 88, 010102(R) (2013). established for PT-symmetric EPs in the microwave- NATURE COMMUNICATIONS (2018) 9:2182 DOI: 10.1038/s41467-018-04690-y www.nature.com/naturecommunications 5 | | | ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-018-04690-y Acknowledgements Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in This research was supported in part by the Global Frontier Program through the published maps and institutional afﬁliations. National Research Foundation of Korea funded by the Ministry of Science, ICT & Future Planning (NRF-2014M3A6B3063708), the Basic Science Research Program (NRF- 2018R1A2B3002539), and the Presidential Post-Doc Fellowship Program (NRF- 2017R1A6A3A04011896). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give Author contributions appropriate credit to the original author(s) and the source, provide a link to the Creative Y.C., C.H., J.W.Y., and S.H.S. conceived the original concept and initiated the work. Y.C. Commons license, and indicate if changes were made. The images or other third party and J.W.Y. developed the theory and model. C.H. and Y.C. performed experiment. Y.C., material in this article are included in the article’s Creative Commons license, unless C.H., and J.W.Y. analyzed the theoretical and experimental results. All authors discussed indicated otherwise in a credit line to the material. If material is not included in the the results. J.W.Y. and Y.C. wrote the manuscript. Y.C., C.H., and J.W.Y. contributed article’s Creative Commons license and your intended use is not permitted by statutory equally to this work. regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ Additional information licenses/by/4.0/. Supplementary Information accompanies this paper at https://doi.org/10.1038/s41467- 018-04690-y. © The Author(s) 2018 Competing interests: The authors declare no competing interests. Reprints and permission information is available online at http://npg.nature.com/ reprintsandpermissions/ 6 NATURE COMMUNICATIONS (2018) 9:2182 DOI: 10.1038/s41467-018-04690-y www.nature.com/naturecommunications | | |
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