Experimental observation of real spectra in Parity-Time symmetric ZRC dimers with positive and negative frequencies
Stéphane Boris Tabeu, Fernande Fotsa-Ngaffo, Senghor Tagouegni, Kazuhiro Shouno, Aurélien Kenfack-Jiotsa
EExperimental observation of real spectra in Parity-Time symmetric ZRC dimerswith positive and negative frequencies
Stéphane Boris Tabeu , , , ∗ Fernande Fotsa-Ngaffo , SenghorTagouegni , , Kazuhiro Shouno , and Aurélien Kenfack-Jiotsa , (Dated: July 7, 2020)We present in this work the first experimental observation of oscillations in Parity-Time symmetricZRC dimers. The system obtained is of first order ordinary differential equation due to the use ofimaginary resistors. The coupled cells must share the same type of frequency: positive or negative.We observed the real and imaginary parts of the voltage across the components of a ZRC cell.Exceptional points are well identified. This work may be very useful in the generation of new typeof oscillators. It can also be used in the design of new optoelectronic devices for major applicationsin the transport of information and the mimics of two-level systems for quantum computing. Keywords: Non-Hermitian systems, Parity-time symmetry, Imaginary resistor, ZRC-Dimer, Negative fre-quency
I. INTRODUCTION
Non-Hermitian systems are today a major fields ofinvestigations in different specialities. In 1998, CarlBender demonstrated that a non-Hermitian systems canhave a real spectra without being Hermitian opening theroad to Parity-Time (PT) Symmetry [1–3]. In QuantumMechanics, this reality is related to the potential whichsatisfied V ( x ) = V ∗ ( − x ) with an even real part andodd imaginary part [1–6]. In Optics, the refractive in-dex of the medium must satisfied a relation of the samenature n ( x ) = n ∗ ( − x ) [7–13] and ε ( x ) = ε ∗ ( − x ) fordielectric permittivity in Metamaterials [14–17]. Thenotion of Parity-Time Symmetry is also extended tomany order fields such as Photonics [7–13], Mechanics[18–20], Optomechanics [20–23], Acoustics [24–26] andElectronics [27–31] just to name few. In 2010, Reuterrealized the first experimental demonstration of PT-symmetry in Optics by coupling two waveguides havingequal amount of gain and loss [7]. In 2011, Schindleret al used two active RLC circuits to introduce the no-tion in Electronics [27, 28]. The loss cell was repre-sented by a natural positive resistor and the gain cellby a negative resistor using Negative impedance con- ∗ Correspondence email address: [email protected] verter. In 2013, the team of Carl Bender experimen-tally demonstrate the real spectra in coupled mechanicoscillators [18]. The Rabi oscillations occurred nearthe the exceptional point in the response of the sys-tem. More after it was presented the counterpart ofParity-Time symmetry: The Anti-Parity Time symme-try. In this new Physics, the loss and the gain are exclu-sively in the dimers [30, 32–38]. In Quantum Mechan-ics, the potential follows the relation V ( x ) = − V ∗ ( − x ) , the refractive index n ( x ) = − n ∗ ( − x ) in Optics and ε ( x ) = − ε ∗ ( − x ) for the dielectric permittivity in Meta-materials. Now the real parts of the potential, therefractive index of the medium and the permittivityare odd and their imaginary parts even giving rise tothe coupled of Gain- Gain cells or Loss-Loss cells tomade Anti-Parity Time symmetry. In others fields,the rotating frame is used to create indirectly posi-tive and negative frequencies in the cells of the dimersuch as in lasers gyroscopes [36]. It also used the Cou-pled Mode Theory (CMT) and adiabatic eliminationin optical waveguides , slowly varying envelope approx-imation in Nanophotonics to achieve the same reality.Many others works are also presented in these directions[30–37]. Tabeu et al presented in [30] how to achieveParity-Time symmetry and Anti-Parity time symmetryin electronics using imaginary resistors [31, 39, 40] tohave directly the system of first order ordinary differ-ential equation without the coupling mode theory or a r X i v : . [ phy s i c s . c l a ss - ph ] J un the rotating frame. The configurations presented werewith Gain-Loss, Gain-Gain and Loss-Loss in the dimer.All these (Anti)-Parity-time systems have many appli-cations such as sensing and telemetry, non-reciprocaltransport of information, generation of qubits for quan-tum computing, wireless power transfer, switches, singlemode power Lasers [9–12, 17, 30, 31, 34, 38, 40–46] andothers .In this works, we proposed the experimental realiza-tion of ZRC cells and a PT-dimer based imaginary resis-tors. Since the Hamiltonian of the resulted system is thesame which the one encounter after some transforma-tions in Optics and Photonics, it can mimics easily suchthese systems and gather the bridge for new devices inOptolectronics and Photoelectronics with the avenue ofoptical and photonic quantum computing. The paper isorganized as follows: In Sect. II the RLC and ZRC arepresented with their frequencies. In Sect. III, the ZRCPT-dimer is studied. Its eigenvalues are calculated andsome simulations of its dynamic behavior are driven.In Sect. IV, the experimental realization of the dimer ispresented and the oscillations across the gain and losscells. The paper end with a conclusion. II. RLC AND ZRC CELLSA. Real volatge in RLC circuit (a) (b)
Figure 1: (a) The RLC cell is constituted of a realresistor R , an inductor L and a capacitor C . All thecomponents are mounted in parallel. The ZRC cell isconstituted of an imaginary resistor Z = jr , a realresistor R and a capacitor C all mounted in parallel.The RLC cell contains a real resistor, a capacitor andan inductor Fig.1(a) The Kirchhoff’s law applied at thenode gives us: I R + I L + I C = 0 (1)By applying the Ohm’s law at the borders of the com-ponents, we have a second order form of ordinary dif-ferential equation depicted as : d Vdt + γ dVdt + ω V = 0 (2) With γ = RC representing the damping rate and ω = (cid:113) LC the natural frequency of the cell. We looka solution in the form V ∝ e jαt and it derives from Eq.( 2) a second order equation in α : α − jγ − ω = 0 (3)The final voltage with an appropriate choice of initialconditions is : V ( t ) = 12 V e − ( γ ) t (cid:0) e jωt + e − jωt (cid:1) (4) α , = j (cid:16) γ (cid:17) ± (cid:114) ω − (cid:16) γ (cid:17) = 0 (5)with ω = (cid:113) ω − (cid:0) γ (cid:1) .The system is under-damped if ω > . Then, the sys-tem is pseudo-oscillatory with exactly the combinationof two voltages with positive and negative frequencies: V ( t ) = V ( ω ) + V ( − ω ) (6)The result is exactly a real value. V ( t ) = V e − ( γ ) t cos ( ωt ) (7) B. Complex voltage in ZRC circuit
The imaginary resistor is a resistor which resistance isa pure imaginary number. It is built indirectly by gyra-tors [39]. If it is associated to a resistor which resistanceis a real number, we can have a complex resistor whichimpedance is in the form R C = R + jr = | R C | e j Φ . R represents the real part and r the imaginary part (cid:0) | R C | = √ R + r ; tan (Φ) = rR (cid:1) . The ZRC cell isconstituted by a capacitor C , a real resistor R andat least an imaginary resistor Z = jr [30, 31]. Thevalue of components may be positive or negative with-out any restrictions. The cell is depicted in Fig. 1(b) .The Kirchhoff’s law applied at the node gives us: I Z + I R + I C = 0 (8)The Ohm’s law at the borders of resistors involves: V = RI R = ZI Z (9)From the combination of Eq. 8 and Eq. 9, we deducethe following: C dVdt + VR − j Vr = 0 (10)Figure 2: Experimental realization of ZRC circuits with positive and negative frequencies. Pseudo-oscillations ofthe real Re ( V ( ω )) = V e ( − γt ) cos ( ωt ) (in blue) and the imaginary Im ( V ( ω )) = V e ( − γt ) sin ( ωt ) (in cyan) partsof the voltage . The sign of the frequency is the one of the capacitors. the experiments run a f ≈ ± . kHz .The others values of components are : Z = j k Ω for the imaginary resistor ; R = ± k Ω for the real resistorsand C = ± nF for capacitors dVdt + (cid:18) RC − j rC (cid:19) V = 0 (11)The final voltage with an appropriate initial conditionsis: V ( t ) = V e( − RC t )e( j rC t ) (12)We obtained a complex voltage across the componentsof the circuit given as: V ( t ) = V e ( − γt ) e ( jωt ) = V e ( − γt ) (cos ( ωt ) + j sin ( ωt )) (13)with γ = RC and ω = rC which may be positive ornegative.The system with positive frequency does not describe the same reality with the one with negative frequencybecause cos ( ωt ) is even and sin ( ωt ) is odd. (cid:26) V ( ω ) = V e ( − γt ) (cos ( ωt ) + j sin ( ωt )) V ( − ω ) = V e ( − γt ) (cos ( ωt ) − j sin ( ωt )) (14) (cid:26) Re ( V ( ω )) = Re ( V ( − ω )) = V e ( − γt ) cos ( ωt )Im ( V ( ω )) = − Im ( V ( − ω )) = V e ( − γt ) sin ( ωt ) (15)Since this response of the system is complex, basedon the model of the imaginary resistor proposed inRef.[39] , one can probe the real part Re ( V ( ω )) = V e ( − γt ) cos ( ωt ) and the imaginary part Im ( V ( ω )) = V e ( − γt ) sin ( ωt ) of the system simultaneously. The signof the damping rate define the nature of the cell. De-pending on the signs of the components, When γ < , Figure 3: The circuit of the ZRC Dimer. Each cellcontains in parallel an imaginary resistor Z n = jr n , areal resistor R n , a capacitor C n . The ZRC-cells arecoupled by an imaginary resistor z = jr (cid:48) or a capacitor C (cid:48) or by all of them.we have a gain cell and when γ > , we have a losscell. The experimental verification of the behavior ofthe cell with positive and negative frequencies is givenin Fig. 2. The real and the imaginary parts of the volt-age are presented simultaneously for each case. The signof the frequency is the one of the capacitor in the cell.This gives the possibilities to achieve experimentallyPT-symmetric systems with cells of same frequency andAnti-Parity-Time (APT) symmetry with cells of oppo-site frequencies without the rotating frame condition asin existing systems in others fields of Physics. These re-sults open new avenues for experimental non-HermitianQuantum Mechanics and Quaternionic Quantum Elec-tronics. III. ZRC PARITY-TIME SYMMETRIC DIMERA. The ZRC model and equations of dynamics
The ZRC PT- symmetric dimer is made by two activeZRC-cells. The cells contain in parallel an imaginaryresistor, a capacitor and a real resistor. They are cou-pled by a capacitor or an imaginary resistor or by allof them. The values of components may be positive ornot without any restriction. In Fig. 3 is presented thesetup of the ZRC PT-symmetric dimer. By applyingthe Kirchoff’s laws at nodes 1 and 2, we have: (cid:26) I R + I C + I Z + I G + I C (cid:48) + I r (cid:48) = 0 I R + I C + I Z + I G − I C (cid:48) − I r (cid:48) = 0 (16)with I Cn = C n ddt V n ; I C (cid:48) = C (cid:48) ddt ( V − V ) ; I Rn = V n R n ; I Zn = − j V n r n ; I R (cid:48) = R (cid:48) ( V − V ) ; I r (cid:48) = − j r (cid:48) ( V − V ) (a)(b) Figure 4: The eigenvalues of the setup correspondingto three cases in which we have a positive and negativecapacitive couplings and at last a case with both thecapacitive and the imaginary couplings. (a) Real partsof the eigenvalues (b) Imaginary parts of theeigenvaluesfor all the components. These considerations lead onthe general form of first order ordinary differential equa-tions: dV dτ = [(1 + c ) ( j (1 + ν ) − γ ) − jc Γ ν ] V + [ − j (1 + c ) ν + c Γ ( j (1 + ν ) − γ )] V dV dτ = [ − j (1 + c ) Γ ν + c ( j (1 + ν ) − γ )] V + [(1 + c ) Γ ( j (1 + ν ) − γ ) − jc ν ] V (17)where τ = ω t ; ω n = r n C n ; Γ = ω ω ; c n = C (cid:48) C n ; ν n = r n r (cid:48) ; γ n = r n R n and ∆ = 1 + (cid:80) n =1 c n .Figure 5: Relative values of the modules of complexvoltages across the gain and loss cells I n = | V n ( τ ) | with different initial inputs. In the exact phase, thereare normal oscillations with an amplification of theinitial input due the presence of gain and loss in thesetup. In the broken phase, there is an exponentialgrowth and also a deep difference between thebehavior of gain and loss cellsThe system of first order ordinary differential equationis PT- symmetric when the several conditions are sat-isfied: (cid:26) Γ = 1 ; c = c = cν = ν = ν ; γ = − γ = γ (18)We can associate to our system a non-Hermitian ef-fective Hamiltonian such that : j ddτ | Ψ ( τ ) (cid:105) = H eff | Ψ ( τ ) (cid:105) (19)where: j is the imaginary unit (cid:0) j = − (cid:1) ; | Ψ ( τ ) (cid:105) =( V ( τ ) , V ( τ )) T ; H eff = k I + k x σ x + k y σ y + k z σ z . I is the × matrix unit and σ x , σ y and σ z are Pauli’smatrices. σ x = (cid:18) (cid:19) ; σ y = (cid:18) − jj (cid:19) ; σ z = (cid:18) − (cid:19) (20)The complex coefficients k , k x , k y , and k z are functionof the independent real parameters of the system de-duced as : (cid:26) k = − (1 + c + ν ) / ∆ ; k x = − ( c − ν ) / ∆ k y = − cγ/ ∆ ; k z = − j (1 + c ) γ/ ∆ (21) Since the system is PT-symmetric, it satisfied the fol-lowing relation: [ PT , H eff ] = 0 (22)where P = σ x is the Parity operator and T = K is thetime reversal operator with K being the complex con-jugation operation. That is k , k x and k y must be realwhereas k z must be purely imaginary. The eigenvaluesof the Hamiltonian resulted from det ( H eff − ωI ) = 0 are given as follow: ω ± = k ± (cid:113) k x + k y + k z (23) B. The eigenvalues of the ZRC PT-dimer andoscillations
Function of the real parameters of the system, theeigenvalues of the effective Hamiltonian are : ω ± = − (1 + c + ν ) ± (cid:112) (1 + 2 c ) ( γ P T − γ ))1 + 2 c (24)with γ P T = ± | c − ν |√ c .The system is PT-unbreakable if c < − and PT-breakable c > − . When c = ν , one have a non-Hermitian diabolic point in the unbreakable Parity-Time symmetry and a thresholdless point in the break-able Parity-Time symmetry.The exceptional points are where both eigenvaluesand eigenvecors coalesce. There are well identifiedin Fig. 4. Three cases are presented. In the firstcase only the positive capacitive coupling is activated ( c = 0 . ν = 0) . When γ < γ P T , all the eigenfre-quencies are real (Re ( ω ± ) (cid:54) = 0 ; Im ( ω ± ) = 0) and weare in the exact phase of Parity-Time Symmetry. When γ = γ P T , the eigenvalues coalesce ( ω + = ω − ) : thisis the exceptional point. Its presence in a systemcalls for enhancement of sensing than the one with di-abolic point, for chiral properties which are revealedwhen the point is encircled with time varying compo-nents in the setup, opening the road to explain theasymmetric transport of information in Optics andin Photonics. It is at this point that happens uni-directional invisibility in the scattering properties ofwaveguides with Parity-Time symmetry. At last, when γ > γ P T , the real parts of eigenvalues are identical (Re ( ω + ) = Re ( ω − )) and the imaginary parts emergeopposite in sign (Im ( ω + ) = − Im ( ω − )) : we are in thebroken phase Parity-Time symmetry. In this phasethe system is nonreciprocal and there is a deep dif-ference between gain cell and loss cell when the ini-tial input is set to one or another cell. The Co-herent Perfect Absorber-Laser (CPA-L) occurs in the (a) (b)(c) (d)(e) (f) Figure 6: Experimental observation of real part of voltages in the ZRC PT-dimer at γ = 0 . γ P T . The first casewith positive frequency and capacitive coupling C = 10 nF ; C (cid:48) = 2 nF ; Z = j k Ω; c = 0 . ν = 0 : (a) The realvoltage dynamic across the loss and the gain cells ( f ≈ . kHz ; R ≈ . k Ω) (b) The Lissajous’s curve. Thesecond case with negative frequency and capacitive coupling C = − nF ; C (cid:48) = 2 nF ; Z = j k Ω; c = − . ν = 0 : (c) The real voltage dynamic across the loss and thegain cells ( f ≈ − . kHz ; R ≈ . k Ω) (d) The Lissajous’s curve. The third case with positive frequency ,capacitive and imaginary couplings C = 10 nF ; C (cid:48) = 2 nF ; Z = j k Ω; c = 0 . ν = 0 . : (e) The real voltagedynamic across the loss and the gain cells ( f ≈ . kHz ; R ≈ . k Ω) (f) The Lissajous’s curve.Figure 7: View of the experiment environmentthis phase when the scattering properties are investi-gated in PT-waveguides. We have the same descrip-tion with the second case with negative capacitive cou-pling ( c = − . ν = 0) but more than − / to be inthe breakable PT-symmetry and the third case withboth positive capacitive coupling and imaginary cou-pling ( c = 0 . ν = 0 . . For the third case, analyticalsolutions and numerical simulations with the algorithmof Runge-Kutta order 4 are presented in Fig. 5. Thebehaviors of oscillations when the input is set from theloss is equal to the one of the initial input from gainafter a beat of time. This confirms the character ofnon-reciprocity which is a signature of non-Hermitiansystems. IV. EXPERIMENTAL REALIZATION OFOSCILLATIONS IN ZRC PT-DIMER
In this section is presented the experimental realiza-tion of the ZRC Parity-Time symmetric dimer withpositive and negative frequencies. The imaginary re-sistor used is in the model of Ref.[39]. One can eas-ily probe the real and imaginary parts of the volt-ages across the components. We have use the Neg-ative Immitiance Converter to achieve negative val-ues of components with resistors, capacitors and op-erational amplifiers. The realizations are those of thethree configurations listed in the previous section at γ = 0 . γ P T . Only the real parts of voltages are pre-sented in Fig. 6. In the first case, the positive natural frequency of independent cells is f = 7 . kHz . Thevalues of different components are: Z = Z = Z = j k Ω ; R = 59 . k Ω R = − . k Ω; C = C = C = 10 nF ; C (cid:48) = 2 nF ; jR (cid:48) = 0 . The frequencyof oscillations obtained is f ≈ . kHz (Fig. 6(a-b)).In the second case, the negative natural frequencyof independent cells is f = − . kHz . The val-ues of different components are: Z = Z = Z = j k Ω ; R = 38 . k Ω R = − . k Ω; C = C = C = − nF ; C (cid:48) = 2 nF ; jR (cid:48) = 0 . The frequencyof oscillations obtained is f ≈ − . kHz (Fig. 6(c-d)). In the third case, the positive natural frequencyof independent cells is f = 7 . kHz . The val-ues of different components are: Z = Z = Z = j k Ω ; R = 39 . k Ω R = − . k Ω; C = C = C = 10 nF ; C (cid:48) = 2 nF ; jR (cid:48) = j k Ω . The frequencyof oscillations obtained is f ≈ . kHz (Fig. 6(e-f)).Operational amplifiers (LF356), metal-film resistorsand polystyrene capacitors are used in the experimen-tal circuits. All the elements are tuned to be within . with respect to the desired values. For injectingthe initial conditions, we have used DG200ABA analogswitches and and an external standard voltage source.All the waveforms are acquired by Tektronix TDS3014Bdigital oscilloscope. Agilent 33120A function generatoris used to generate the trigger signal for the experimen-tal circuits (See Fig. 7) V. CONCLUSION
We have reported in this work the experimental real-ization of the ZRC PT-symmetric Dimer. A compara-tive study is made between RLC and ZRC cells. Con-trary to the RLC cell which dynamic is of second orderordinary differential equation with a real output, theZRC is of first order ordinary differential equation witha complex output. We succeed the measurements of thereal and imaginary parts of the voltage in positive andnegative frequencies. Oscillations of the PT-dimers arealso probed in the exact phase of the Parity-Time sym-metry without any restriction or approximation. Thiswork paves the way for the design of new electronic andoptolectronic devices. It opens the investigation on theexploitation of the new band of negative frequencies inTelecommunications and the generation of new oscilla-tors for simulations of qubits in quantum computing. [1] C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT-Symmetry, Physi-cal Review Letters, vol. 80, no. 24, pp. 5243-5246, Jun.1998. [2] C. M. Bender, S. Boettcher, and P. N. Meisinger, PT-symmetric quantum mechanics, Journal of Mathemati-cal Physics, vol. 40, no. 5, pp. 2201-2229, May 1999.[3] C. M. Bender, M. V.Berry, and A. Mandilara, General-ized PT symmetry and realspectra, Journal of Physics