PT-SYMMETRIC INTERPRETATION of OPEN QUANTUM and CLASSICAL SYSTEMS

Transcription

1 PT-SYMMETRIC INTERPRETATION of OPEN QUANTUM and CLASSICAL SYSTEMS overview and examples Mariagiovanna Gianfreda Centro Fermi (Roma) and CNR-IFAC (Sesto Fiorentino, FI) - Open Quantum Systems: From atomic nuclei to ultracold atoms and quantum optics, ECT*- July 11, / 35

2 PT-symmetric Hamiltonians vs Hermitian Hamiltonians The Origin 2 / 35

3 PT- SYMMETRIC QM E XPERIMENTS IN PT- SYMMETRY PT-symmetric WGM G ENERALIZED B ATEMAN SYSTEM O THER APPLICATIONS 3 / 35

4 PT -symmetric quantum mechanics Is Dirac Hermiticity H = H a physical request? Dirac Hermiticity can be replaced by the physical and weaker condition of PT SYMMETRY P parity T time reversal 4 / 35

5 PT -symmetric quantum mechanics H = p 2 + x 2 (ix) ɛ (ɛ R) This class of Hamiltonians has PT symmetry REAL SPECTRUM! 1 1 Carl M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998) mathematical proof: P. Dorey, C. Dunning, R. Tateo, J. Phys. A: Math. Gen. 34, 5679 (2001) 5 / 35

6 PT-symmetry has been helpful to resolve long-standing problems: The Lee model A quantum field theory model that describes a three-particle interaction. Below a critical value, the Hamiltonian is non-hermitian and a state with negative norm appears (ghost state)... It has always been regarded as an unacceptable quantum theory because unitarity is violated... C. M. Bender, S. F. Brandt, J. H. Chen, and Q. Wang, Phys. Rev. D 71, (2005). The Pais-Uhlembeck model Fourth-order derivative oscillator model (generally associated with ghost states). Treated as PT -symmetric theory, the normalization of the wave functions takes place in complex stokes wedges, and the spectrum is real. Moreover, the C operator solves the problem of negative norm of the states. C. M. Bender and P. D. Mannheim, Phys. Rev. Lett. 100, (2008). Electromagnetic self-force problem The nonrelativistic classical motion of a charged particle suffers from the physical instabilities of runaway modes. If the particle interacts with a PT-symmetric partner, the runaway modes can be removed. C. M. Bender and M. Gianfreda, J. Phys A Math Theor 48, 34FT01 (2015). Other field-theory models... T. Curtright, E. Ivanov, L. Mezincescu, and P. L. Townsend, JHEP 0704:020 (2007); T. Curtright and A. Veitia, J. Math. Phys. 48, (2007); E. A. Ivanov and A. V. Smilga, JHEP 0707:36 (2007). 6 / 35

7 Experimental Realization of PT-Symmetric Systems 7 / 35

8 PT -symmetry in Optics The analogy between quantum mechanics and optics is based on the fact that they share the same mathematical formalism. The equation governing optical beam propagation is described the paraxial equation of diffraction, mathematically equivalent to the Schroödinger equation i φ z + 2 φ x 2 + V(x)φ = 0 where φ is proportional to the electric field envelope, z is the propagation distance and V(x) = n(x) = n R (x) + in I (x) is the complex refractive index distribution, that plays the role of an optical potential. The PT-symmetric condition imposes symmetric index guiding n R ( x) = n R (x) and antisymmetric gain/loss distribution n I ( x) = n I (x). Optical PT-symmetric systems can be realistically implemented through an inclusion of gain/loss regions in guided wave geometries a a C. Ruter et al, Nature Physics 24, 192 (2010) 8 / 35

9 Intuitive explanation of PT-phase transition Source and sink of equal and opposite powers: Two boxes together as a single system (source and sink equidistant from the origin) [ ] [ HL 0 re iθ 0 H = = 0 H G 0 re iθ ] i d ψ(t) = H(t) ψ(t) dt [ 1 ψ 1 (t) = 0 ] [ e ie 1 t 0, ψ 2 (t) = 1 ] e ie 2 t H L = E 1 = re iθ H G = E 2 = re iθ E 1,2 = r e ±iθ r > 0 0 < θ < π Im(E 1 ) > 0 Im(E 2 ) < 0 [ 0 1 H is PT -symmetric, where P = 1 0 ψ 1 (x) decays exponentially in time because there is a sink ψ 2 (x) grows exponentially in time ] because there is a source and T is complex conjugation. The eigenvectors ψ,12 (t) are NOT eigenvectors of PT and the system is not in equilibrium. 9 / 35

10 Intuitive explanation of PT-phase transition What happens if we add a coupling term? ( re iθ g H = g re iθ ) Eigenvalues become real if the coupling g is sufficiently large! E ± = r cos θ ± g 2 r 2 sin 2 θ g critical = r 2 sin 2 (θ) PT -symmetric systems can be interpreted as nonisolated physical systems having balanced loss and gain they can be considered as intermediate between open systems and closed systems In the unbroken PT -symmetric region (equilibrium) they mimic a closed system in the broken PT -symmetric region they are no longer in equilibrium and mimic an open system 10 / 35

11 Experiments on PT -phase transition Phase transition between parametric regions of broken and unbroken PT symmetry can be observed experimentally! Electronic circuits: J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, Phys. Rev. A 84, (R) (2011). N. Bender et al., Phys. Rev. Lett. 110, (2013). Nuclear magnetic resonance : K. F. Zhao, M. Schaden, and Z. Wu, Phys. Rev. A 81, (2010). C. Zheng, L. Hao, and G. L. Long, Phil. Trans. R. Soc. A 371, (2013). Optics: A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, Phys. Rev. Lett. 103, (2009). C. E. Ruter, K. G. Makris, R. El-Ganainy, D. N. Christodoulides, M. Segev, and D. Kip, Nat. Phys. 6, (2010). L. Feng, M. Ayache, J. Huang, Y.-L. Xu, M. H. Lu, Y. F. Chen, Y. Fainman, and A. Scherer, Science 333, 729 (2011). A. Regensburger et al., Nature 488, 167 (2012). Metamaterials: L. Feng et al., Nature Mat. 12, 108 (2012). Microwave cavities: S. Bittner, B. Dietz, U. Gunther, H. L. Harney, M. Miski-Oglu, A. Richter, and F. Schafer, Phys. Rev. Lett. 108, (2012). Mechanical oscillators: C. M. Bender, B. Berntson, D. Parker, and E. Samuel, Am. J. Phys. 81, 173 (2013). Superconductors: N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, V. M. Vinokur, Phys. Rev. Lett. 109, (2012). 11 / 35

12 Experiments on PT -phase transition PT SYMMETRY IS IMPORTANT IN APPLIED PHYSICS In the last few years, considerable research effort has been invested in developing PT -symmetric artificial materials appropriately engineered to display properties NOT found in nature. IN OPTICAL SYSTEMS: Scattering processes in optical periodic structures with PT -symmetric refractive index. Because of the PT -symmetry breaking: Perfect transmission Unidirectional invisibility H. Hernandez-Coronadoa, D. Krejiky, P. Siegl, Phys. Lett. A (2011). The wave entering from the left goes through the sample entirely unaffected.the wave entering from the right experiences enhanced reflection. [Z. Lin, Phys. Rev. Lett. 106, (2011)] IN SOLID STATE PHYSICS: PT -symmetric superconducting wire: PT symmetry stabilizes superconductivity 12 / 35

13 PT -phase transition in optical resonators First demonstration of PT-symmetric breaking in optical resonator systems based on two coupled on-chip whispering-gallery-mode microtoroid silica resonator: 2 In a WGM resonator, light is confined by total internal reflection and circulates around the curved inner surface boundary of the resonator. The active resonator (gain) is a silica microtoroid doped with erbium ions. The passive resonator (loss) is also a silica microtoroid but without any dopant. The two microresonators are directly coupled and each one is coupled to a tapered fiber waveguide that couples light in and out. After it is separated form the pump, the output probe signal from the resonator system is monitored with a photodetector. The coupling strength between the microresonators is tuned to observe the PT-phase transition. 2 B. Peng, S.Ozdemir, F. Lei, F. Monifi, M. Gianfreda, G.Long, S. Fan, F. Nori, C. M. Bender, L. Yang, Nature Physics , (2014) 13 / 35

14 PT -phase transition in optical resonators: the model ȧ 1 = i ω 1 a 1 (γ 1 + γ c )a 1 i κ a 2 γ c a in ȧ 2 = i ω 2 a 2 γ 2 a 2 i κ a 1 a out = a in + γ c a 1 For ω 2 = ω 1 = ω 0, after the substitution a i (t) = A i (t)e i ω t we obtain the frequency of the supermodes ω ± = ω 0 i 4 (γ 1 + γ 2 + γ c ) ± 1 16κ 4 2 (γ 1 + γ c γ 2 ) 2 T = A out 2 A in 14 / 35

15 PT -phase transition in optical resonators: the model E = BROKEN REGION Complex conjugate frequencies D= UNBROKEN REGION Two dinstinct frequencies (REAL in the exact-pt-symmetric configuration) 15 / 35

16 PT -phase transition in optical resonators: the model Non reciprocal effects: light is transmitted in one direction but blocked in the other direction. Nonreciprocity is based on nonlinear effects (kerr and termal nonlinearity). Nonlinearity is enhanced only in the broken region! Work in progress at IFAC-CNR: Mathematical modeling of the nonlinear regime 16 / 35

17 PT -symmetric optical resonators: Quantum Model ȧ 1 = i ω 1 a 1 (γ c + γ 1 )a 1 i κ a 2 γ c a in ȧ 2 = i ω 2 a 2 γ 2 a 2 i κ a 1 a out = a in + γ c a 1 d a 1 dt d a 2 dt = i [a 1, H] = i [a 2, H] H = (ω 1 i γ 1 i γ c ) a 1 a 1 + (ω 2 i γ 2 ) a 2 a 2 + κ(a 1 a 2 + a 1 a 2) i γ c (a 1 a in + a 1 a in ) H. A. Haus, Waves and Fields in Optoelectronics, Prentice-Hall, New Jersey, / 35

18 PT -symmetric optical resonators: Quantum Model Neglect γ c. For ω 1 = ω 2 = ω 0 and balanced loss-gain γ 1 = γ 2 = γ H = (ω 0 i γ) a 1 a 1 + (ω 0 + i γ) a 2 a 2 + κ(a 1 a 2 + a 1 a 2) can be diagonalized as ( H = ω 0 + ) (Ā ) ( κ 2 γ 2 A ω 0 ) ) κ 2 γ ( B 2 B Ā A! [A, Ā] = [B, B] = 1 κ 2 γ 2 > 0 UNBROKEN PT-SYMMETRY! MG and G. Landonfi, work in progress. R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, (2005). 18 / 35

19 My experiment at IFAC Ultra-high Q 3D-WGMR (Three Dimensional Whispering Gallery Mode Resonators) will replace the toroidal cavities for testing the PT-symmetric two coupled oscillators system. (a) Active microspherical resonator made in commercial Er3+ doped glass 3 and (b) passive silica microsphere made directly from the tip of an optical fiber. We expect good flexibility for setting the coupling strengths that needs to be finely adjusted 4. Big challenge: to balance exactly loss and gain in order to observe an exact PT-phase transition (not possible with integrated micro-toroids). Experiment conducted at IFAC-CNR with the collaboration of Dr. Gualtiero Nunzi Conti, coordinator of the Research Unit Material and Devices for Photonics (a) (b) 3 D. Ristic, et al., Journal of Luminescence (2015) doi:/ /j.jlumin D. Farnesi, et al., invited paper, SPIE 9133, (2014); D. Farnesi, et al., Phys. Rev. Lett. 112, (2014). 19 / 35

20 PT- SYMMETRIC QM E XPERIMENTS IN PT- SYMMETRY PT-symmetric WGM G ENERALIZED B ATEMAN SYSTEM O THER APPLICATIONS Generalized (coupled) Bateman s system 2 x + ω x + γ x = 0 y + ω 2 y γ y = 0 H = pq + γ(yq xp) + (ω 2 γ 2 )xy H. Bateman, Phys. Rev. 38, 815 (1931) C. M. Bender and MG, Phys. Rev. A, 88, (2013). 20 / 35

21 PT- SYMMETRIC QM E XPERIMENTS IN PT- SYMMETRY PT-symmetric WGM G ENERALIZED B ATEMAN SYSTEM O THER APPLICATIONS Generalized (coupled) Bateman s system 2 x + ω x + γ x = 0 y + ω 2 y γ y = 0 2 x + ω x + γ x = y y + ω 2 y γ y = x H = pq + γ(yq xp) + (ω 2 γ 2 )xy+ (x2 + y2 ) 2 H. Bateman, Phys. Rev. 38, 815 (1931) C. M. Bender and MG, Phys. Rev. A, 88, (2013). 20 / 35

22 PT- SYMMETRIC QM E XPERIMENTS IN PT- SYMMETRY PT-symmetric WGM G ENERALIZED B ATEMAN SYSTEM O THER APPLICATIONS Generalized (coupled) Bateman s system 2 x + ω x + γ x = 0 2 x + ω x + γ x = y y + ω 2 y γ y = 0 y + ω 2 y γ y = x H = pq + γ(yq xp) + (ω 2 γ 2 )xy+ (x2 + y2 ) 2 P: x y, y x, p q, q p T : p q, q p, i i H. Bateman, Phys. Rev. 38, 815 (1931) C. M. Bender and MG, Phys. Rev. A, 88, (2013). 20 / 35

23 PT- SYMMETRIC QM E XPERIMENTS IN PT- SYMMETRY PT-symmetric WGM G ENERALIZED B ATEMAN SYSTEM O THER APPLICATIONS Generalized (coupled) Bateman s system 2 x + ω x + γ x = 0 2 x + ω x + γ x = y y + ω 2 y γ y = 0 y + ω 2 y γ y = x H = pq + γ(yq xp) + (ω 2 γ 2 )xy+ (x2 + y2 ) 2 P: x y, y x, p q, q p T : p q, q p, i i H. Bateman, Phys. Rev. 38, 815 (1931) C. M. Bender and MG, Phys. Rev. A, 88, (2013). 20 / 35

24 Classical solutions: x(t) = e iλt ɛ < 2γ ω 2 ɛ 2 the energy flowing into the y resonator cannot transfer fast enough to the x resonator, where the energy is flowing out. The system cannot be in equilibrium. ɛ > 2γ ω 2 ɛ 2 All of the energy flowing into the y resonator can transfer to the x resonator and the entire system can attain equilibrium. 21 / 35

25 SECOND TRANSITION If ɛ > ω 2 the PT symmetry is broken again! ɛ < 2γ ω 2 ɛ 2 2γ ω 2 ɛ 2 < ɛ < ω 2 ɛ > ω 2 first broken region unbroken region second broken region Could this transition be observed in a classical systems? WHAT ABOUT THE QUANTUM MODEL? 22 / 35

26 ... Quantum model: Eigenfunctions [ x y iγ(y y x x) + (ω 2 γ 2 )xy + ɛ(x 2 + y 2 )/2] = E m,nψ m,n(x, y) P 0,0 P 1,0 P 1,1 P 2,0 P 2,1 P 2,2 P 3,0 P 3,1 P 3,2 P 3, ψ m,n(x, y) = e (2axy+bx2 +cy 2 )/2 Pm,n b = c = ɛ/(2iγ + 2a) a is solution to the quartic equation: a 4 +(2γ 2 ω 2 )a 2 +ɛ 2 /4 γ 2 ω 2 +γ 4 = 0. The orthogonal polynomials P m,n satisfy three terms recursion relations in terms of = bc γ 2 : (iγ )x + cy n( iγ) P n+1,0 = P n,0 + c c(a ) P n 1,0, The operators x and y are lowering and rising operators: ( + iγ)x + cy P n+1,n+1 = c n( + iγ) P n,n c(2a + ) P n 1,n 1. + iγ xp n,0 = n P n 1,0, yp n,0 = np n 1,0, xp n,n = n + iγ P n 1,n 1, yp n,n = np n 1,n 1. c c Ladder operators: L = (y + b 1 x) y + (b 2 y x) x + b 3 2 y + b 4 2 x + b 5 x y R = (x + f 1 y) x + (f 2 x y) y + f 3 2 x + f 4 2 y + f 5 x y (iγ + ) (iγ ) LP m,n = 2n P m,n 1, RP m,n = 2(m n) P m,n 1, bc bc [R, L]P m,n = 4m 2 bc Pm,n 23 / 35

27 Quantum model: Eigenvalues E m,n = (m + 1)a + (n m/2) (m = 0, 1, 2..., n = 0, 1, 2... m) a = 2ω 2 4γ ω 4 ɛ 2 Look at the Stokes wedges in the complex plane in which the eigenfunctions ψ m,n vanish: e (2axy+bx2 +cy 2 )/2 e (bu2 +Rv 2 )/2 u = x+a/b y, v = i y, R = a 2 /b c (a, Re(b), Re(R)) > 0 in the unbroken region Eigenspectrum bounded below Other approach: diagonalization ) ) H = λ 1 (āa λ 2 2 ( bb + 1, λ 2 1,2 = 1 2γ ± 4γ 4 4γ 2 + ɛ 2 [a, ā] = 1, [b, b] = 1, ā a 24 / 35

28 Norm of the states Does H define a physical acceptable quantum theory? We must define an inner product whose associated norm is positive definite and conserved in time... One natural choice could be: (f, g) = dx(pt f )g, (ψ n, ψ m) = ( 1) n δ n,m. This norm is conserved in time but it is not positive definite! One half of the energy eigenstates have norm +1 and one half have norm 1. The theory is defined in an Hilbert space endowed with indefinite metrics! IN STANDARD QUANTUM MECHANICS THE NORM OF THE STATES CARRIES A PROBABILISTIC INTERPRETATION 25 / 35

29 Norm of the states A linear operator C can be constructed, it represents an hidden symmetry of the PT -symmetric Hamiltonian. In terms of C, an inner product with a positive norm can be defined: ψ ζ CPT = dx ψ CPT (x)ζ(x) ψ CPT (x) = dy C(x, y)ψ (y) With respect to the CPT-adjoint: Norms are strictly positive The Hamiltonian determines its own adjoint: replace by CPT The theory has unitary time evolution (norm is preserved in time) Probability is conserved 5 5 C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett. 89, (2002); C. M. Bender, Rept. Prog. Phys. 70, 947 (2007). 26 / 35

30 Norm of the states PROBLEM 1) The C operator is defined only in the unbroker region...what happens in the broken region? PROBLEM 2)The C operator is defined only for SYMMETRIC Hamiltonians under the PT -scalar product 27 / 35

31 Norm of the states PROBLEM 1) The C operator is defined only in the unbroker region...what happens in the broken region? PROBLEM 2)The C operator is defined only for SYMMETRIC Hamiltonians under the PT -scalar product USE BIORTHOGONAL BASIS! Biorthogonal basis formalism doesn t need the introduction of any additional operator, and it can be used in both broken and unbroken region h ψ m = E m ψ m, h ψ m = E m ψ m, ψ m ψ n = δ m,n 27 / 35

32 Three resonators ẍ(t) + ω 2 x(t) + γẋ(t) = ɛ 1 y(t) + ɛ 2 z(t) ÿ(t) + ω 2 y(t) = ɛ 1 (x(t) + z(t)) z(t) + ω 2 z(t) γ ż(t) = ɛ 1 y(t) + ɛ 2 x(t). C. M. Bender M.G. and S. P. Klevansky, Phys. Rev. A 90, (2014). 28 / 35

33 PT-symmetric models: other applications 29 / 35

34 Nonlinear PT-symmetric models The Hamiltonian model of a PT -symmetric oscillator with loss and gain is a playground! ẍ + ω 2 x + γ ẋ = ɛ y, ÿ + ω 2 y γ ẏ = ɛ x....it is inspiring the investigation of more general nonlinear systems composed by coupled oscillators with loss and gain... Some papers: J. Cuevas, P.G. Kevrekidis, A. Saxena, and A. Khare, Phys. Rev. A 88, (2013); D. A. Zezyulin and V. V. Konotop, J. Phys. A: Math. Theor. 46, (2013); I. V. Barashenkov and M. Gianfreda, Fast Track Comm. J. Phys. A: Math. and Theor. 47, (2014); J. Cuevas, A. Khare, P. G. Kevrekidis, H. Xu, and A. Saxena, Int. J. Theor. Phys, (2014); I. V. Baeashenkov, Phys. Rev. A 90, (2014); R. Banerjee and P. Mukherjee, arxiv: , (2014); S. Karthiga, V.K. Chandrasekar, M. Senthilvelan and M. Lakshmanan, arxiv: (2014). 30 / 35

35 Nonlinear PT-symmetric dimer I. Barashenkov and M.G., J. Phys. A: Math. and Theor. 47, (2014) ẍ + x + 2γẋ = ɛy + x 3 + 3xy 2, ÿ + y 2γẏ = ɛx + y 3 + 3yx 2. H = pq γ(xp yq) + (1 γ 2 )xy ɛ(x2 + y 2 ) + xy 3 + x 3 y Linearization around equilibrium (small amplitude solutions) gives the Nonlinear Shrödinger Dimer that may describe two waveguides with loss and gain 6 : iψ 1 + ψ 2 + ( ψ ψ 2 2 )ψ 1 + ψ2 2 ψ 1 = iγψ 1 iψ 2 + ψ 1 + ( ψ ψ 1 2 )ψ 2 + ψ1 2 ψ 2 = iγψ 2 (ψ 1, ψ 2 ) complex light beam amplitudes (e.g. in coupled optical wave guides) γ > 0 gain-loss rate τ distance along the parallel cores P 2 = ψ 2 2, P 1 = ψ 1 2 powers carried by the active and lossy channel Nonlinearity softens the PT -symmetry transition: stable periodic orbits with big coupling ɛ persist for an arbitrarily large value of the gain-loss coefficient γ. 6 S. Jensen, IEEE J. Quantum Electron. 18, 1580 (1982); A. W. Snyder and Y. Chen, Opt. Lett. 14, 517 (1989); Y. Chen, A. W. Snyder and D. N. Payne, IEEE J. Quantum Electron. 28, 239 (1992). 31 / 35

36 PT -symmetric dynamical systems: chaos PT-symmetric Lotka Volterra Systems A Predator-Prey (Lotka-Volterra) model x 1 = x 1 x 1 y 1 cx 2 1 y 1 = y 1 + x 1 y 1 32 / 35

37 PT -symmetric dynamical systems: chaos PT-symmetric Lotka Volterra Systems A Predator-Prey (Lotka-Volterra) model and its PT-symmetric counterpart x 1 = x 1 x 1 y 1 cx 2 1 y 1 = y 1 + x 1 y 1 x 2 = x 2 + x 2 y 2 + cx 2 2 y 2 = y 2 x 2 y 2 32 / 35

38 PT-symmetric Lotka Volterra Systems A Predator-Prey (Lotka-Volterra) model and its PT-symmetric counterpart coupled together x 1 = x 1 x 1 y 1 cx 2 1 +gx 1x 2 y 1 = y 1 + x 1 y 1 +fy 1 y 2 x 2 = x 2 + x 2 y 2 + cx 2 2 gx 1x 2 y 2 = y 2 x 2 y 2 fy 1 y 2 33 / 35

39 PT-symmetric Lotka Volterra Systems A Predator-Prey (Lotka-Volterra) model and its PT-symmetric counterpart coupled together x 1 = x 1 x 1 y 1 cx 2 1 +gx 1x 2 y 1 = y 1 + x 1 y 1 +fy 1 y 2 x 2 = x 2 + x 2 y 2 + cx 2 2 gx 1x 2 y 2 = y 2 x 2 y 2 fy 1 y 2 PT-SYMMETRIC BROKEN REGION 33 / 35

40 PT-symmetric Lotka Volterra Systems A Predator-Prey (Lotka-Volterra) model and its PT-symmetric counterpart coupled together x 1 = x 1 x 1 y 1 cx 2 1 +gx 1x 2 y 1 = y 1 + x 1 y 1 +fy 1 y 2 x 2 = x 2 + x 2 y 2 + cx 2 2 gx 1x 2 y 2 = y 2 x 2 y 2 fy 1 y 2 PT-SYMMETRIC BROKEN REGION PT-SYMMETRIC UNBROKEN REGION 33 / 35

41 A PT-symmetric model of the immune response (x 1, y 1 ), (x 2, y 2 ) denote the concentration of two systems of antibody-antigen 7 P : Antigen Antibody (x 1 y 2, y 1 x 2 ) If only (x 1, y 1 ) is present, the antigens grow in concentration If only (x 2, y 2 ) is present, the concentration of antigens is under control If the host suffers from the antigen y 1 and a new system (x 2, y 2 ) is injected, the host can be cured, or it develops a chronic disease 7 C. M. Bender, A. Ghatak and M.G,J. Phys. A: Math. Theor. 50, (2016). Work based on: G. I. Bell, Math. Biosci. 16, 291 (1973) 34 / 35

42 35 / 35