Rabi oscillations at the exceptional point in anti-parity-time symmetric diffusive systems

Gabriel Gonzalez(Dated: January 1, 2021)

The motivation for this theoretical paper comes from recent experiments of a heat transfer system of two ther-mally coupled rings rotating in opposite directions with equal angular velocities that present anti-parity-time(APT) symmetry. The theoretical model predicted a rest-to-motion temperature distribution phase transitionduring the symmetry breaking for a particular rotation speed. In this work we show that the system exhibits aparity-time (PT ) phase transition at the exceptional point in which eigenvalues and eigenvectors of the corre-sponding non-Hermitian Hamiltonian coalesce. We analytically solve the heat diffusive system at the excep-tional point and show that one can pass through the phase transition that separates the unbroken and brokenphases by changing the radii of the rings. In the case of unbroken PT symmetry the temperature profiles ex-hibit damped Rabi oscillations at the exceptional point. Our results unveils the behavior of the system at theexceptional point in heat diffusive systems.

Keywords:

A closed or conservative system evolves according to a Her-mitian Hamiltonian in contrast with open or non conservativesystems which are described by non-Hermitian Hamiltonians.There are a special class of non-Hermitian systems in whichthe energy exchange between the system and the environmentis balanced. The entire balanced system exhibits a symmetrycalled PT symmetry where the symbol P stands for parityand interchanges the gain and loss components of the totalsystem and T represents the operation of time reversal andhas the effect of turning a system with loss into a system withgain and viceversa.[1–3]Non-Hermitian PT symmetric systems can exhibit a rich andunexpected behavior and have broad applications in classi-cal and quantum physics.[4–7] PT symmetric systems havebeen intensively studied in optics in which many intrigu-ing phenomena haven been experimentally confirmed andhas led to the development of new ways of controlling lightpropagation.[8–11]Recently, anti-PT (APT) symmetric systems have attracted alot of attention because they exhibit noteworthy effects differ-ent from the PT counterpart. An APT symmetric Hamilto-nian can be defined in terms of a PT symmetric Hamiltonianby H(APT ) = ±iH(PT ), but physically it is really difficultto implement it in the laboratory since it requires the couplingbetween the two subsystems to be a purely imaginary value, incontrast with the PT systems which requires a real coupling.Anti-PT symmetry has been demonstrated by using dissipa-tively coupled atomic beams,[12] cold atoms,[13] electricalcircuits,[14] and optical devices.[15–17] These breakthroughshave initiated the field of exploring unique APT effects. Morerecently, Li et al reported the experimental realization of anAPT symmetric diffusive system in Ref.[18]. The system in-vestigated in Ref.[18] is depicted in Fig. 1 and consists of twoidentical solid rings with inner and outer radius given by Rand R + δR, respectively. The thickness is b. The upper ringis rotating with angular velocity ω1, while the lower ring is ro-tating with angular velocity ω2 = −ω1. There is an interfaceof thickness d and thermal conductivity ki between the tworings. The temperature distribution along the inner edges ofthe upper and lower rings is given by the following diffusion

(a) (b)

FIG. 1: The figure (a) shows two identical rotating rings with equalbut opposite angular velocities joined together by a stationary inter-mediate layer and (b) the imaginary and real parts of the eigenfre-quencies as a function of the tangential velocity where the dottedline represents the exceptional point vEP = hc/κ.

coupled partial differential equations

∂T1∂t

= D∂2T1∂x2

− v ∂T1∂x

+ hc(T2 − T1)

∂T2∂t

= D∂2T2∂x2

+ v∂T2∂x

+ hc(T1 − T2) (1)

where x is the coordinate along each edge, D = k/ρc is thediffusivity, v is the tangential velocity in the inner edge of therings,hc = h/ρcb is the rate of heat exchange coupling, ρ isthe density, c is the heat capacity and h = ki/d is a coefficientthat represent the heat exchange between the two rings. Usingplane wave solutions, i.e. Ti = Aie

i(κx−ωt), the system givenin Eq. (1) can be cast into an APT symmetric Hamiltoniangiven by

H(APT ) =

(−i(κ2D + hc) + κv ihc

ihc −i(κ2D + hc)− κv

)(2)

where κ is the wave number and ω are the eigenvalues of the

arX

iv:2

012.

1457

1v1

[qu

ant-

ph]

29

Dec

202

0

2

APT Hamiltonian which are given by

ω± = −i[(κ2D + hc)±

√h2c − κ2v2

]. (3)

The exceptional point where the two eigenvectors coalesceis when v2EP = h2c/κ

2, i.e. ω+ = ω−. The sudden col-lapse of the eigenvectors and eigenvalues at the exceptionalpoint leads to an abrupt reduction in dimensionality. Manyof the interesting properties of non Hermitian systems arefound at or close to the exceptional point which have ledto many novel and exotic phenomena. Exceptional pointsare currently the subject of many interesting and counter-intuitive phenomena associated with them such as topologi-cal mode switching,[19, 20] reflection and transmission,[21–23] instrinsic single-mode lasing[24, 25] and coherent perfectabsorption.[26]

In this work we study the APT symmetric diffusive systemgiven by Eq. (1) when v = vEP and show that the systembehaves as a pair of coupled linear oscillators one with gainand the other one with loss. The noteworthy feature of theexceptional point vEP is that it exhibits damped Rabi oscil-lations in the unbroken PT phase transition that depends onthe radii of the rotating rings. We obtain the analytical tem-perature distribution of each ring at the exceptional point andobtain the conditions that have to be fulfill in order for thesystem to be in equilibrium. We start our investigation bymaking the following variables change in Eq. (1): τ = hct,z =

√hc(λ− 1)/Dx where λ > 1 is an auxiliary constant

to be determined and ∆T = Ti − T0 where T0 is a referencetemperature. Rewriting Eq. (1) in terms of the new variableswe have

hc∂∆T1∂τ

= hc(λ− 1)∂2∆T1∂z2

− vEP

√hc(λ− 1)

D

∂∆T1∂z

+ hc(∆T2 −∆T1)

hc∂∆T2∂τ

= hc(λ− 1)∂2∆T2∂x2

+ vEP

√hc(λ− 1)

D

∂∆T2∂z

+ hc(∆T1 −∆T2) (4)

Looking for solutions of the form ∆Ti = e−λτfi(z) in Eq.(4) we end up with the following system of coupled ordinarydifferential equations

d2f1dz2

− vEP√Dhc(λ− 1)

df1dz

+ f1 +1

λ− 1f2 = 0

d2f2dz2

+vEP√

Dhc(λ− 1)

df2dz

+ f2 +1

λ− 1f1 = 0. (5)

Inspection of Eqs.(5) reveals that they are invariant undercombined parity, i.e. f1 ↔ f2, and time reversal t → −ttransformation. To solve the system of equations analyticallywe first differentiate one of the equations and then use theother equation to eliminate f2 in order to get the followingfourth order differential equation(

d4

dz4+

(2− εv2EP

Dhc

)d2

dz2+ (1− ε2)

)f1(z) = 0. (6)

where ε = 1/(λ − 1). By assuming a solution of the formf1(z) ∝ cosh(χz) for Eq. (6) we get the following conditionover χ:

χ4 + (2− a2)χ2 + (1− ε2) = 0, (7)

where a2 = εv2EP /Dhc. The solution of Eq. (7) is given by

χ2 =1

2

(a2 − 2±

√a4 − 4a2 + 4ε2

). (8)

In order to have an oscillatory behavior we must demand thatχ2 < 0, which implies that

(i) a4 − 4a2 + 4ε2 > 0

(ii) a2 − 2 +√a4 − 4a2 + 4ε2 < 0.

Condition (ii) gives ε < 1 and condition (i) gives

a < acrit = 2(

1−√

1− ε2). (9)

If ε < 1 and a < acrit we get the following oscillatory solu-tion for f1

f1(z) = A1 cos (χ1z) +B1 cos (χ2z) , (10)

where χ1,2 =√|χ2

±| and A1 and B1 are constants to be de-termined. In order to obtain the value of ε we must considerthe periodicty of f1(z), i.e. f(0) = f(2πR

√hc/Dε), which

gives us the following conditions

χ1,2R

√hcDε

= n, (11)

where n = ±1,±2, . . .. Solving Eq. (11) we get the followingvalue for ε

ε =hcR

2

Dn2. (12)

Using the fact that ε = 1/(λ − 1) we get the following valuefor λ

λ = 1 +Dn2

hcR2(13)

3

Equation (13) is in agreement with Eq. (3) when v2EP =h2c/κ

2 and k = n/R. Interestingly, Eq. (13) is valid onlywhen conditions (i) and (ii) are fulfilled.Once we know the value of ε we can substitute in a2 =εv2EP /Dhc in order to get a = ε, substituting this value intoEq. (9) we have the conditions that have to be satisfied in or-der to have unbroken-PT symmetry at the exceptional pointwhich are 0 < ε < 1 and 0 < ε < 2(1 −

√1− ε2), which

gives us the following solution

4

5< ε < 1 (14)

Equation (14) is the main result of this study which states thattwo phase transitions take place at the exceptional point anddepends only on the radii of the rotating rings.Substituting a = ε in Eq. (8) we get χ2

+ = ε2 − 1 andχ2− = −1, therefore f1(z) = A1 cos(

√1− ε2z) + B1 cos(z)

which means we have to choose A1 = 0 in order to fulfill theperiodicity condition. Substituting f1 into Eq. (5) we obtainthe following ordinary differential equation for f2:

d2f2dz2

+ εdf2dz

+ f2 = −εB1 cos(z). (15)

The general solution for Eq. (15) is given by

f2(z) = A2e−z/2ε cos

(√1− (ε/2)2z + φ

)−B1 sin(z),

(16)where A2, B1, φ and n are constants to be determined by theinitial conditions.If we impose the following initial conditions over the temper-ature profiles in the rings

T1(x, 0) = T2(x, 0) = T0 +A cos(x/R) (17)

we need to choose n = 1 and B1 = A in order to get

f1(x) = A cos(x/R) (18)

and

f2(x) = Ae−Dx/2R3hc sec(φ) cos

(αx

R+ φ

)−A sin(x/R)

(19)where α =

√1− (hcR2/2D)2 and

φ = arctan[cot(2πα)− csc(2πα)eπD/hcR

2]. (20)

The solution given in Eq. (18) means that ∆T1(x, 0) = f1(x),therefore the temperature distribution in the first ring will notchange in position but will only decay on time, in contrastwith the solution given in Eq. (19) which is different from theinitial condition, therefore the temperature profile will changein position and decay on time. In Fig. (2) we show the Rabioscillations as a function of position for different values of ε.If we impose the following new conditions over the tempera-

ture profiles in the rings

T1(x, 0) = T0+A cos(x/R) and T2(x, 0) = T0+A sin(x/R)(21)

0 50 100 150 200 250 300-10

-5

0

5

10

x/R

f 1,f2

0 50 100 150 200 250 300

-10

-5

0

5

10

x/R

f 1,f2

(a) (b)

FIG. 2: Numerical solution of the coupled equations given in Eq.(5). The graph shows the Rabi oscillations for (a) ε = 0.86 and (b)ε = 0.96, respectively. The Rabi oscillations are harder to see nearthe upper end of the unbroken PT symmetry.

we need to choose n = 1, B1 = −A, A2 = 0, which meansthat ∆Ti(x, 0) = fi(x), therefore both temperature distribu-tions will remain invariant and will only decay on time.

Using the same experimental values given in Ref. ??, i.e.

(a) (b)

FIG. 3: The graph shows the temperature profiles for both rings forthe following values D = 100mm2/s, ρ = 1000Kg/m3, c =1000J/Kg◦K, ki = 1W/m◦K, a = 100mm, b = 5mm, d =1mm and for the radii: (a) R = 21mm and vEP = 4.2mm/s and(b) R = 22mm and vEP = 4.4mm/s.

D = 100mm2/s, ρ = 1000Kg/m3, c = 1000J/Kg◦K,ki = 1W/m◦K, a = 100mm, b = 5mm and d = 1mm, wefind that Rabi oscillations take place for the fundamental waveif the inner ring radius is between 20mm < R < 22mm andthe rings are rotating with equal but opposite velocities givenby vEP = hcR. In Fig. (3) we show the temperature fieldsfor the unbroken-PT regime where damped Rabi oscillationsoccur in which the maximum and minimum temperatures are90◦ out of phase.Let us now consider the case when the rings are rotating withdifferent velocities close to the exceptional point, specificallywe will like to solve the following system

∂T1∂t

= D∂2T1∂x2

− (vEP + δv)∂T1∂x

+ hc(T2 − T1)

∂T2∂t

= D∂2T2∂x2

+ (vEP − δv)∂T2∂x

+ hc(T1 − T2) (22)

4

where δv << vEP . At first it seems that the system givenin Eq. (22) is not APT symmetric, however if we make thefollowing transformation ξ = x − δvt we obtain a system ofequations identical to the one given in Eq. (1) replacing x→ ξand v → vEP . The solution for Eq. (22) is given by

∆T1 = e−λhct cos((x− δvt)/R) (23)

∆T2 = −e−λhct sin((x− δvt)/R) (24)

which means that the temperature profiles are moving. Thisresult shows that we can have a rest-to-motion temperatureprofile without having equal opposite rotating velocities.In summary, we have predicted the existence of Rabi os-cillations at the exceptional point in the diffusive systemproposed by Li et al.. We showed that at the exceptionalpoint the system exhibits two PT phase transitions whichtakes place at critical values for the radii of the rotating rings.Specifically, if the rings are rotating in opposite directionswith equal tangential velocity given by vEP = hc/|κ| andthe ring radius lies between

√4D/5hc < R <

√D/hc for

the fundamental wave, i.e. κ = ±1/R, the temperature fieldsexhibit damped Rabi oscillations. We have shown also that itis not essential to have identical opposite rotating velocitiesfor the rings in order to have a rest-to-motion temperaturetransition, we can do this also by increasing/decreasing theupper/lower ring velocity away from the exceptional point inorder to obtain traveling wave solutions with positive/negativevelocity. Our work reveals the rich structure of exceptionalpoints in anti-parity-time symmetric diffusive systems.

I would like to acknowledge support by the programCatedras Conacyt through project 1757 and from project A1-S-43579 of SEP-CONACYT Ciencia Basica and LaboratorioNacional de Ciencia y Tecnologıa de Terahertz.

[1] C. M. Bender and S. Boettcher, Real Spectra in Non-HermitianHamiltonians Having PT Symmetry, Phys. Rev. Lett., 80, 5243-5246 (1998).

[2] C.M. Bender, S. Bottcher, S. and P. N. Meisinger, PT-symmetricquantum mechanics, J. Math. Phys. 40, 2201–2229 (1999).

[3] C. M. Bender, Making sense of non-Hermitian Hamiltonians,Rep. Prog. Phys., 70, 947 (2007).

[4] C. M. Bender, D. C. Brody and H. F. Jones, Complex Extensionof Quantum Mechanics, Phys. Rev. Lett. 89, 270401 (2002).

[5] A. Mostafazadeh, Pseudo-Hermiticity versus PT symmetry:The necessary condition for the reality of the spectrum of a non-Hermitian Hamiltonian, J. of Math. Phys., 43, 205 (2002).

[6] C. M. Bender, D. C. Brody and H. F. Jones, Must a Hamiltonianbe Hermitian?, Am. J. Phys., 71 (11), 1095-1102 (2003).

[7] C. M. Bender, B. K. Berntson, D. Parker and E. Samuel, Obser-vation of PT phase transition in a simple mechanical system,Am. J. Phys., 81 (3), 173-179 (2013).

[8] A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou and D. N. Christodoulides,Observation of PT-Symmetry Breaking in Complex Optical Po-tentials, Phys. Rev. Lett., 103, No. 093902 (2009).

[9] C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N.Christodoulides, M. Segev and D. Kip, Observation of Parity-Time Symmetry in Optics, Nat. Phys., 6, 192-195 (2010).

[10] L. Chang, X. Jiang,S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G.Wang and M. Xiao, Parity-Time Symmetry and Variable Opti-cal Isolation in Active-Passive-Coupled Microresonators, Nat.Photonics, 8, 524-529 (2014).

[11] M. Wimmer, A. Regensburger, M. A. Miri, C. Bersch, D. N.Christodoulides and U. Peschel, Observation of Optical Soli-tons in PT-Symmetric Lattices, Nat. Commun., 6, 7782 (2015).

[12] P. Peng, W. Cao, C. Shen, W. Qu, J. Wen, L. Jiang and Y. Xiao,Anti-Parity-Time Symmetry with Flying Atoms, Nat. Phys. 121139-1145 (2016).

[13] Y. Jiang, Y. Mei, Y. Zuo, Y. Zhai, J. Li, J. Wen and S. Du,Anti-Parity-Time Symmetric Optical Four-Wave Mixing in ColdAtoms, Phys. Rev. Lett. 123, 193604 (2019).

[14] Y. Choi, C. Hahn, J. W. Yoon and S.H. Song, Observation of anAnti-PT-Symmetric Exceptional Point and Energy-DifferenceConserving Dynamics in Electrical Circuit Resonators, Nat.Commun. 9, 2182 (2018).

[15] X. L. Zhang, T. Jiang and C.T. Chan, Dynamically En-circling an Exceptional Point in Anti-Parity-Time SymmetricSystems: Asymmetric Mode Switching for Symmetry-BrokenModes, Light: Sci. Appl. 8, 88 (2019)

[16] C. F. Li and G. C. Guo, Experimental Simulation of Anti-Parity-Time Symmetric Lorentz Dynamics, Optica 6, 67 (2019).

[17] J. Zhao, Y. Liu, L. Wu, C. K. Duan, Y. X. Liu, and J. Du, Obser-vation of Anti-PT-Symmetry Phase Transition in the Magnon-Cavity-Magnon Coupled System, Phys. Rev. Appl. 13, No.014053 (2020).

[18] Ying Li, Yu-Gui Peng, Lei Han, Mohammad-Ali Miri, Wei Li,Meng Xiao, Xue-Feng Zhu, Jianlin Zhao, Andrea Alu, ShanhuiFan and Cheng-Wei Qiu, Anti–parity-time symmetry in diffusivesystems, Science 364, 6436, 170-173 (2019)

[19] J. Doppler, A. A. Mailybaev, J. Bohm, A. G. U. Kuhl and, F.Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Dy-namically encircling an exceptional point for asymmetric modeswitching, Nature 537, 76–79 (2016).

[20] H. Xu, D. Mason, L. Jiang, and J. G. E. Harris, Topologicalenergy transfer in an optomechanical system with exceptionalpoints, Nature 537, 80–83 (2016).

[21] Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D.N. Christodoulides, Unidirectional invisibility induced by PT-symmetric periodic structures, Phys. Rev. Lett. 106, 213901(2011).

[22] B. Peng, S. K. Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G.L. Long, S. Fan, F. Nori, C. M. Bender, and L. Yang, Parity-time-symmetric whispering-gallery microcavities, Nat. Phys. 10(2014).

[23] L. Feng, Y.-L. Xu, W. S. Fegadolli, M.-H. Lu, J. E. Oliveira, V.R. Almeida, Y.-F. Chen, and A. Scherer, Experimental demon-stration of a unidirectional reflectionless parity-time metamate-rial at optical frequencies, Nat. Mater. 12, 108–113 (2013).

[24] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Science 346,972–975 (2014).

[25] H. Hodaei, M.-A. Miri, M. Heinrich, D. N. Christodoulides,and M. Khajavikhan, Parity-time–symmetric microring lasers,Science 346, 975–978 (2014).

[26] Y. Sun, W. Tan, H.Q. Li, J. Li and H. Chen, Experimentaldemonstration of a coherent perfect absorber with PT phasetransition, Phys. Rev. Lett. 121 073901 (2018)