Bishop, RF

Davidson, NJ

Quick, RM

van der Walt, DM

English

The Rabi Hamiltonian, which describes the interaction of a single mode of electromagnetic radiation with a two-level system, is one of the fundamental models of quantum optics. It is also of wider interest as it provides a. generic model for the interaction of bosons and fermions. To allow for a systematic analysis of the strong-coupling behaviour, we have applied the coupled cluster method (CCM) to the Rabi Hamiltonian to calculate its spectrum. We find strong evidence for the existence of a somewhat subtle quantum phase transition

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The University of Manchester ResearchQuantum optics meets quantum many-body theoryDocument VersionAccepted author manuscriptLink to publication record in Manchester Research ExplorerCitation for published version (APA):Davidson, NJ., Quick, RM., Bishop, RF., & van der Walt, DM. (1998). Quantum optics meets quantum many-bodytheory: Coupled cluster studies of the Rabi Hamiltonian. In D. Neilson, & RF. Bishop (Eds.), Recent Progress inMany-Body Theories: Proceedings of the 9th International Conference, Sydney, Australia, 21-25 July 1997 (pp.163-166). (Series on Advances in Quantum Many-Body Theory; Vol. 1). World Scientific Publishing Co. Pte. Ltd.http://personalpages.manchester.ac.uk/staff/raymond.bishop/RFB_papers/[143] SerAQMBT_1(1998)163Published in:Recent Progress in Many-Body TheoriesCiting this paperPlease note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscriptor Proof version this may differ from the final Published version. If citing, it is advised that you check and use thepublisher's definitive version.General rightsCopyright and moral rights for the publications made accessible in the Research Explorer are retained by theauthors and/or other copyright owners and it is a condition of accessing publications that users recognise andabide by the legal requirements associated with these rights.Takedown policyIf you believe that this document breaches copyright please refer to the University of Manchester’s TakedownProcedures [http://man.ac.uk/04Y6Bo] or contact uml.scholarlycommunications@manchester.ac.uk providingrelevant details, so we can investigate your claim.Download date:09. Jun. 2022Series on Advances in Quantum Many-Body Theory - Vol. 1The Proceedings of the 9th International ConferenceRECENT PROGRESS INMANY-BODY THEORIESSydney,Australia July 21 - 25, 1997EditorsDavid NeilsonThe University of Neto South Wales,Sydney, AustraliaRaymond F. BishopUMIST, Manchester, UK'b World ScientificIII Singapore· New Jersey» London· Hong KongPublished byWorld Scientific Publishing Co. Pte. Ltd.POBox 128, Farrer Road, Singapore 912805USA office: Suite 1B, 1060 Main Street, River Edge, NJ 07661UK office: 57 Shelton Street, Covent Garden, London WC2H 9HEBritish Library Cataloguing-In-PubHcatioD DataA catalogue record for this book is available from the British Library.RECENT PROGRESS IN MANY-BODY THEORIFSCopyright © 1998 by World Scientific Publishing Co. Pte. Ltd.AI/ rights reserved. This book. or parts thereof, may not be reproduced in any form or by any TTUUUIS, electronic ormechanical, including photocopying, recording or any information storage and retrieval system now known or tobe invented, without written permission from the Publisher.For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center,Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required fromthe publisher.ISBN 981-02-3369-8Printed in Singapore by Uto-Print163QUANTUM OPTICS MEETS QUANTUM MANY-BODY THEORY:COUPLED CLUSTER STUDIES OF THE RABI HAMILTONIANN. J. DAVIDSON, R. M. QUICKDepartment of Physics, University of Pretoria, Pretoria 0002, South AfricaE-mail: davidson@scientia.up.ac.zaR. F. BISHOPDepartment of Physics, University of Manchester Institute of Science and Technology(UMIST), POBox 88, Manchester M60 1QD, United KingdomD. M. VAN DER WALTDepartment of Physics, VISTA University (Mamelodi Campus), Private Bag X 1911,Silverton 0127, South AfricaThe Rabi Hamiltonian, which describes the interaction of a single mode of electromag-netic radiation with a two-level system, is one of the fundamental models of quantumoptics. It is also of wider interest as it provides a. generic model for the interactionof bosons and fermior.s. To allow for a systematic analysis of the strong-coupling be-haviour, we have applied the coupled cluster method (CCM) to the Rabi Hamiltonian tocalculate its spectrum. We find strong evidence for the existence of a somewhat subtlequantum phase transition.1 IntroductionThe prototypical model for the interaction of a single mode of a quantised elec-·tromagnetic field with a two-level system, popular in quantum optics 1,2 and morerecently in studies of quantum chaos 3 is the Rabi Hamiltonian, given by(1)where a", e+ and a" are Pauli matrices, and b and bt are bosonic annihilation andcreation operators respectively, obeying the commutation relation [b, bt] = 1. Thereis a conserved parity Il associated with the Hamiltonian (1),(2)Although the Rabi Hamiltonian (1) does not appear to be integrable, (and see,however, Ref. 4), it clearly becomes so if the rotating wave approximation (RWA) ismade, leading to the Jaynes-Cummings model,' In the RWA the terms g(bt u++bu-)are neglected, so that [HRWA' N] = 0 and the model is exactly soluble by a series of2 x 2 diagonalisations.Historically, most calculations based on (1) have made use of the RWA forapplications where g/w « 1, and of diagonalisation in large but finite vectorspaces for 9/ w '" 1. In the present work, we report on the results obtained from theapplication of the coupled cluster method (CCM) to the Hamiltonian of Eq. (1).The CCM is well known in the quantum many-body theory community, and thereader is referred to the literature (see, e.g., Ref. 5) for a detailed discussion of the164CCM. For the formalism for the application of the CCM to the Rabi Hamiltonian,the reader is referred to previous work~2 Ground State Results: Normal Coupled Cluster Method (NCCM)We turn nowto our results for the ground-state energy for the Hamiltonian (1) withWo > o. For simplicity,wewillonly quote results for the case of a scaled Hamiltonianat resonance (w = Wo = 1). Since the Rabi Hamiltonian is not integrable, someapproximation has to be made. The simplest is the SUB-N approximation, inwhich the CCM correlation operator is truncated so as include at most N-bodyexcitations relative to the chosen model state~We find that the ground-state energy shows evidence for spontaneous breakingof the parity symmetry. For each N, the solution with positive parity terminatesat a finite value of 9 which we indicate by gtJ• It is possible to find solutions abovethis coupling with mixed symmetry; these solutions reproduce the expected large-gbehaviour, but appear to be rather unreliable for 9 < 2.In Figure 1 we show the ground-state energy Eg as a function of the coupling9 for several SUB-N approximations, together with configuration-interaction (CI)method results from a diagonalisation of (1) in a basis of 100positive parity states.For 9 < gf:J, the agreement between the results of the CCM and CI calculations isvery good. The termination points of the CCM solutions with positive parity areclearly visible. Note that there is a difference in the nature of the termination ofthe positive parity solutions depending on whether !f is odd or even!' In the samereference, an attempt is made to determine critical indices for a possible parity-breaking transition using the coherent anomaly method (CAM) of Suzuki: Theresults of this analysis suggest that there may be an essential singularity in theground state energy at 9 ~ 0.665.Similar behaviour is certainly known to exist in some comparable exactly in-tegrable models, for example at the isotropic Heisenberg point for a 1D chain ofspin-1/2 atoms interacting via the XXZ Hamiltonian. The CCM has been appliedto this model in both 1D and 2D, and equally strong evidence for similar criticalbehaviour has been observed in various truncation schemesfThe CI and CCM results appear to be contradictory, in that the CI calculationsgive no indication of a possible phase transition and always yield a positive-parityground state. However, the application of a unitary transformation to the Hamil-tonian yields states which alwayshave mixed parity, so it is by no means clear thatthe CI results are reliable in the prediction of a possible phase transition. Moredetailed investigations of the CI results are in progress.3 Excited States (NCCM)The CCM formalism also permits the calculation of the spectrum of a givenHamil-tonian. The standard approach is that of Emrich; in which the excited-state ket isparameterised by means of an excitation operator acting on the CCM ground-stateketo A problem with this approach is that another approximation, namely in theexcitation operator, is necessary to enable calculations to be performed.-0.4 ~-'--'----'----r------'-0.6-0.8Eg -1-1.2-1.4-1.6 '------'-_-'----'-_---L---lo 0.3 0.6 0.9 1.2 1.59Figure 1: The ground---state energy Eg as afunction of the coupling 9 for the indicatedCCM SUB-N approximations for the casew = wo = 1. Also shown ace the results ofa CI calculation in a basis of 100 positive-parity states.165E12 ~~I---r-I~I---r-I~10 ....,;~~~~~~~f~~~:~:~°F====~~~~ ~~~~~,~~~~~~-8 '------'-_-'----'-_---L---lo 0.1 0.2 0.3 0.4 0.59Figure 2: The spectrum of the Rabi Hamil-tonian, as determined from lineae repsonsetheory, for the CCM SUB-6 approximation(w = wo = 1). The ground-state energyis shown as the solid line, with the excitedstates shown as dots. Missing regions inthe excited states indicates complex ener-gies. Note the presence of the spurious(negative-energy) excitations.An alternative method which avoids the need to introduce another truncationis to use the theory of linear responser? Here a dynamical matrix is determined, theeigenvaluesof which are the excitation energies. At the SUB-N level of approxima-tion, the matrix is of dimension 4N; however, half of the eigenvalues are spuriousdue to symmetries in the matrix.Figure 2 shows the full spectrum (spurious states included) for the SUB-6approximation for the Rabi Hamiltonian at resonance. The bunching of the statesnear 9£6) is noteworthy, and may provide additional support for the possibility of aparity-breaking phase transition in this system.4 Ground State Results: Extended Coupled Cluster Method (ECCM)Although the NCCM has been very successfully applied to a wide variety of phys-ical systems, its one major failing is its apparent inability to traverse regions ofparameter space near phase transitions. However, a modified version of the CCMansatz, which makes use of a double similarity transform of the Hamiltonian ratherthan the single such transform of the NCCM, has been very successful in correctingthis shortcoming. This approach is called the extended coupled cluster methcd.P166We have applied the ECCM to the Rabi Hamiltonian to calculate the ground-state energy. Apart from some difficulty in the closeproximity of the NCCM criticalpoint, our initial ECCM results are very close to those of the CI. However, theeigenket below the transition region can easily be shown to be of positive parity,while that above the transition is definitely of mixed parity. This result is verydifferent to the conclusions which would be drawn from the CI calculation.5 Conclusions and Future WorkIn contrast to the perturbative or semiclassical treatments of previous work on theRabi Hamiltonian, wehave applied the CCM to the calculation of its spectrum. Forthe weak-coupling range of parameters typical of optical applications to date themethod convergesextremely rapidly to yield very accurate results. However, sincequantum optics experiments are nowadays being performed with ever-increasingatom-field couplings, it is of considerable interest to explore the full Rabi Hamilto-nian with a non-perturbative approach such as the CCM. One outcome has beenthat for stronger couplings than have hitherto been explored.experimentally, ourresults provide strong evidence for a subtle quantum phase transition. Future workincludes higher-order ECCM calculations, and the determination of a suitable orderparameter for the transition. In the longer term, the application of the CCM toother Hamiltonians of relevance in quantum optics, such as multi-level systems, isenvisaged.AcknowledgmentsR.F.B. acknowledgesthe financial support of the EPSRC, Great Britain. N.J.D.,and R.M.Q. acknowledgethe financial support of the FRD, South Africa.References1. E. T. Jaynes and F. W. Cummings, Proc. IEEE 51, 89 (1963).2. P. W. Milonni and S. Singh, Adv. At. MoL Opt. Phys. 28, 75 (1991); B. W.Shore and P. L. Knight, J. Mod. Opt. 40, 1195 (1993).3. P. W. Milonni, J. R. Ackerhalt, and H. W. Galbraith, Phys. Rev. Lett. 50,966 (1983);M. KuS, Phys. Rev. Lett. 54,1343 (1985).4. H. G. Reik, P. Lais, M. E. Stiitzle, and M. Doucha, J. Phys. A 20, 6327(1987).5. R. F. Bishop, Theor. Chim. Acta. 80,95 (1991).6. R. F. Bishop, N. J. Davidson, R. M. Quick, and D. M. van der Walt, Phys.Rev. A 54, R4657 (1996).7. M. Suzuki, J. Phys. Soc. Jpn. 55,4205 (1986).8. R. F. Bishop, J. B. Parkinson, and Y. Xian, Phys. Rev. B 44, 9425 (1991);R. F. Bishop, R. G. Hale, and Y. Xian, Phys. Rev. Lett. 73,3157 (1994).9. K. Emrich, Nacl. Phys. A 351, 379 (1981).10. J. S. Arponen, R. F. Bishop, and E. Pajanne, Phys. Rev. A 36, 2539 (1987).11. J. Arponen, Ann. Phys. (NY) 151, 311 (1983).

Quantum optics meets quantum many-body theory:Coupled cluster studies of the Rabi Hamiltonian

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Quantum optics meets quantum many-body theory:Coupled cluster studies of the Rabi Hamiltonian

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