PT-symmetric quantum mechanics

Transcription

1 PT-symmetric quantum mechanics Crab Lender Washing Nervy Tuitions Tokyo, Bed crème 2012

2 PT-symmetric quantum mechanics Carl Bender Washington University Kyoto, December 2012

3 Dirac Hermiticity H = H ( means transpose + complex conjugate) guarantees real energy and probability-conserving time evolution but is a mathematical axiom and not a physical axiom of quantum mechanics Dirac Hermiticity can be generalized...

4 The point of this talk: Replace Dirac Hermiticity by the physical and weaker condition of PT symmetry P = parity T = time reversal

5 Example: This Hamiltonian has PT symmetry!

6 H 2 3 p ix

7 A class of PT-symmetric Hamiltonians: CMB and S. Boettcher Physical Review Letters 80, 5243 (1998)

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9 Upside-down potential with real positive eigenvalues?!

10 Some of my work CMB and S. Boettcher, Physical Review Letters 80, 5243 (1998) CMB, D. Brody, H. Jones, Physical Review Letters 89, (2002) CMB, D. Brody, and H. Jones, Physical Review Letters 93, (2004) CMB, D. Brody, H. Jones, B. Meister, Physical Review Letters 98, (2007) CMB and P. Mannheim, Physical Review Letters 100, (2008) CMB, D. Hook, P. Meisinger, Q. Wang, Physical Review Letters 104, (2010) CMB and S. Klevansky, Physical Review Letters 105, (2010)

11 Some of my coauthors:

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15 PT papers ( ) K. Makris, R. El-Ganainy, D. Christodoulides, and Z. Musslimani, Phyical Review Letters 100, (2008) Z. Musslimani, K. Makris, R. El-Ganainy, and D. Christodoulides, Physical Review Letters 100, (2008) U. Günther and B. Samsonov, Physical Review Letters 101, (2008) E. Graefe, H. Korsch, and A. Niederle, Physical Review Letters 101, (2008) S. Klaiman, U. Günther, and N. Moiseyev, Physical Review Letters 101, (2008) CMB and P. Mannheim, Physical Review Letters 100, (2008) U. Jentschura, A. Surzhykov, and J. Zinn-Justin, Physical Review Letters 102, (2009) A. Mostafazadeh, Physical Review Letters 102, (2009) O. Bendix, R. Fleischmann, T. Kottos, and B. Shapiro, Physical Review Letters 103, (2009) S. Longhi, Physical Review Letters 103, (2009) A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, Physical Review Letters 103, (2009) H. Schomerus, Physical Review Letters 104, (2010) S. Longhi, Physical Review Letters 105, (2010) C. West, T. Kottos, T. Prosen, Physical Review Letters 104, (2010) S. Longhi, Physical Review Letters 105, (2010) T. Kottos, Nature Physics 6, 166 (2010) C. Ruter, K. Makris, R. El-Ganainy, D. Christodoulides, M. Segev, and D. Kip, Nature Physics 6, 192 (2010) CMB, D. Hook, P. Meisinger, Q. Wang, Physical Review Letters 104, (2010) CMB and S. Klevansky, Physical Review Letters 105, (2010)

16 PT papers ( ) Y. Chong, L. Ge, and A. Stone, Physical Review Letters 106, (2011) Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. Christodoulides, Physical Review Letters 106, (2011) P. Mannheim and J. O Brien, Physical Review Letters 106, (2011) L. Feng, M. Ayache, J. Huang, Y. Xu, M. Lu, Y. Chen, Y. Fainman, A. Scherer, Science 333, 729 (2011) S. Bittner, B. Dietz, U. Guenther, H. Harney, M. Miski-Oglu, A. Richter, F. Schaefer, Physical Review Letters 108, (2012) M. Liertzer, L. Ge, A. Cerjan, A. Stone, H. Tureci, and S. Rotter, Physical Review Letters 108, (2012) A. Zezyulin and V. V. Konotop, Physical Review Letters 108, (2012) H. Ramezani, D. Christodoulides, V. Kovanis, I. Vitebskiy, and T. Kottos, Physical Review Letters 109, (2012) A. Regensberger, C. Bersch, M.-A. Miri, G. Onishchukov, D. Christodoulides, Nature 488, 167 (2012) T. Prosen, Physical Review Letters 109, (2012) N. Chtchelkatchev, A. Golubov, T. Baturina, and V. Vinokur, Physical Review Letters 109, (2012) D. Brody and E.-M.. Graefe, Physical Review Letters 109, (2012)

17 Review articles CMB, Contemporary Physics 46, 277 (2005) CMB, Reports on Progress in Physics 70, 947 (2007) P. Dorey, C. Dunning, and R. Tateo, Journal of Physics A 40, R205 (2007) A. Mostafazadeh, Int l Journal of Geometric Methods in Modern Physics 7, 1191 (2010)

18 Developments in PT Quantum Mechanics (Since official beginning in 1998) Over fifteen international conferences Over 1000 published papers About 135 posts to PT symmeter < in last 12 months (about 95 in previous 12 months) Lots of experimental results in last two years

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21 Rigorous proof of real eigenvalues Proof is difficult! Uses techniques from conformal field theory and statistical mechanics: (1) Bethe ansatz (2) Monodromy group (3) Baxter T-Q relation (4) Functional determinants ODE/IM Correspondence P. Dorey, C. Dunning, and R. Tateo

22 Region of broken PT symmetry PT Boundary Region of unbroken PT symmetry

23 n=3: n=2: CMB and D. Hook Phys. Rev. A 86, (2012)

24 Broken ParroT Unbroken ParroT

25 Broken PT symmetry in Paris

26 Hermitian Hamiltonians: BORING! Eigenvalues are always real nothing interesting happens

27 PT-symmetric Hamiltonians: ASTONISHING! Transition between parametric regions of broken and unbroken PT symmetry... Can be observed experimentally!

28 Intuitive explanation of PT transition

29 Classical harmonic oscillator Back and forth motion on the real axis: Turning point Turning point ( = 0)

30 Harmonic oscillator in complex plane Turning point Turning point ( = 0)

31 H 2 3 p ix (e = 1)

32 (e = 2)

33 p

34 Broken PT symmetry orbit not closed e< 0

35 Box 1: Loss Box 2: Gain

36 Two boxes together as a single system: This Hamiltonian is PT symmetric, where T is complex conjugation and

37 Couple boxes together with coupling strength s Eigenvalues become real if s is sufficiently large. Critical value given by:

38 Examining CLASSICAL limit of PT quantum mechanics provides intuitive explanation of the PT transition: Source antenna becomes infinitely strong as Sink antenna becomes infinitely strong as Time for classical particle to travel from source to sink:

39 Source and sink localized at + and - infinity

40 Complex eigenvalue problems and Stokes wedges At the quantum level:

41 Upside down potential

42 Step 1: Change path of integration

43 Step 1: Change path of integration

44 Step 2: Fourier transform

45 Step 3: Change dependent variable

46 Step 4: Rescale p

47 Result: A pair of exactly isospectral Hamiltonians CMB, D. C. Brody, J.-H. Chen, H. F. Jones, K. A. Milton, and M. C. Ogilvie Physical Review D 74, (2006) [arxiv: hep-th/ ]

48 Reflectionless potentials! Z. Ahmed, CMB, and M. V. Berry, J. Phys. A: Math. Gen. 38, L627 (2005) [arxiv: quant-ph/ ]

49 At a physical level, PT-symmetric systems are intermediate between closed and open systems. Hermitian H PT-symmetric H Non-Hermitian H

50 At a mathematical level, we are extending conventional classical mechanics and Hermitian quantum mechanics into the complex plane

51 Complex plane

52 The eigenvalues are real and positive, but is this quantum mechanics? Probabilistic interpretation?? Hilbert space with a positive metric?? Unitarity time evolution??

53 The Hamiltonian determines its own adjoint! Find the secret symmetry:

54 Unitarity With respect to the CPT adjoint the theory has UNITARY time evolution. Norms are strictly positive! Probability is conserved!

55 Example: 2 x 2 Non-Hermitian matrix PT-symmetric Hamiltonian where

56 Overview of talk so far:

57 PT symmetric systems are being observed experimentally!

58 Laboratory observation of PT transition using optical wave guides A. Guo, G. Salamo, D. Duchesne, R. Morandotti, M. Volatier- Ravat, V. Aimez, G. Siviloglou, and D. Christodoulides, Physical Review Letters 103, (2009) C. Ruter, K. Makris, R. El-Ganainy, D. Christodoulides, M. Segev, and D. Kip, Nature Physics 6, 192 (2010)

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61 The observed PT transition

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65 Another experiment...

66 Yet another...

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75 APS: Spotlighting exceptional research J. Schindler et al., Phys. Rev. A (2011) Experimental study of active LRC circuits with PT symmetries Joseph Schindler, Ang Li, Mei C. Zheng, F. M. Ellis, and Tsampikos Kottos Phys. Rev. A 84, (2011) Published October 13, 2011 Everyone learns in a first course on quantum mechanics that the result of a measurement cannot be a complex number, so the quantum mechanical operator that corresponds to a measurement must be Hermitian. However, certain classes of complex Hamiltonians that are not Hermitian can still have real eigenvalues. The key property of these Hamiltonians is that they are parity-time (PT) symmetric, that is, they are invariant under a mirror reflection and complex conjugation (which is equivalent to time reversal). Hamiltonians that have PT symmetry have been used to describe the depinning of vortex flux lines in type-ii superconductors and optical effects that involve a complex index of refraction, but there has never been a simple physical system where the effects of PT symmetry can be clearly understood and explored. Now, Joseph Schindler and colleagues at Wesleyan University in Connecticut have devised a simple LRC electrical circuit that displays directly the effects of PT symmetry. The key components are a pair of coupled resonant circuits, one with active gain and the other with an equivalent amount of loss. Schindler et al. explore the eigenfrequencies of this system as a function of the gain/loss parameter that controls the degree of amplification and attenuation of the system. For a critical value of this parameter, the eigenfrequencies undergo a spontaneous phase transition from real to complex values, while the eigenstates coalesce and acquire a definite chirality (handedness). This simple electronic analog to a quantum Hamiltonian could be a useful reference point for studying more complex applications. Gordon W. F. Drake

76 PT-symmetric system of coupled pendula Best way to have loss and gain: Set a=0 Remove r (0 < r < 1) of the energy of the x pendulum and transfer it to the y pendulum.

77 CMB, B. Berntson, D. Parker, E. Samuel, American Journal of Physics (in press) [arxiv: math-ph/ ]

78 Magnets off Theory: Unbroken PT, Rabi oscillations (pendula in equilibrium) (r=0) Experiment:

79 Unbroken PT region Theory: Weak magnets, Rabi oscillations (pendula in equilibrium) (r=0.01) Experiment:

80 Broken PT region Theory: Strong magnets, no Rabi oscillations (pendula out of equilibrium) (r=0.3) Experiment:

81 PT quantum mechanics is fun! You can re-visit things you already know about traditional Hermitian quantum mechanics.

82 Three examples: 1. Ghost Busting: PT-Symmetric Interpretation of the Lee Model CMB, S. Brandt, J.-H. Chen, and Q. Wang Phys. Rev. D 71, (2005) [arxiv: hep-th/ ] 2. No-ghost Theorem for the Fourth-Order Derivative Pais-Uhlenbeck Oscillator Model CMB and P. Mannheim Phys. Rev. Lett. 100, (2008) [arxiv: hep-th/ ] 3. Resolution of Ambiguity in the Double-Scaling Limit CMB, M. Moshe, and S. Sarkar [arxiv: hep-th/ ]

83 Possible future applications: 1. PT Higgs model: theory is asymptotically free, stable, conformally invariant, and has 2. PT QED like a theory of magnetic charge, asymptotically free, opposite Coulomb force 3. PT gravity has a repulsive force 4. PT Dirac equation allows for massless neutrinos to undergo oscillations

84 THE END!