DOCSLIB.ORG

Phase-Space Formulation of Quantum Mechanics and Quantum State

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Phase-Space Formulation of Quantum Mechanics and Quantum State formulation of quantum mechanics and quantum state reconstruction for View metadata,Phase-space citation and similar papers at core.ac.uk brought to you by CORE ysical systems with Lie-group symmetries ph provided by CERN Document Server ? ? C. Brif and A. Mann Department of Physics, Technion { Israel Institute of Technology, Haifa 32000, Israel We present a detailed discussion of a general theory of phase-space distributions, intro duced recently by the authors [J. Phys. A 31, L9 1998]. This theory provides a uni ed phase-space for- mulation of quantum mechanics for physical systems p ossessing Lie-group symmetries. The concept of generalized coherent states and the metho d of harmonic analysis are used to construct explicitly a family of phase-space functions which are p ostulated to satisfy the Stratonovich-Weyl corresp on- dence with a generalized traciality condition. The symb ol calculus for the phase-space functions is given by means of the generalized twisted pro duct. The phase-space formalism is used to study the problem of the reconstruction of quantum states. In particular, we consider the reconstruction metho d based on measurements of displaced pro jectors, which comprises a numb er of recently pro- p osed quantum-optical schemes and is also related to the standard metho ds of signal pro cessing. A general group-theoretic description of this metho d is develop ed using the technique of harmonic expansions on the phase space. 03.65.Bz, 03.65.Fd I. INTRODUCTION P function [?,?] is asso ciated with the normal order- ing and the Husimi Q function [?] with the antinormal y ordering of a and a . Moreover, a whole family of s- The phase-space formulation of quantum mechanics parameterized functions can be de ned on the complex has a long history. In 1932 Wigner [?] intro duced his plane which is equivalent to the q -p at phase space. The famous function which has found numerous applications index s is related to the corresp onding ordering pro cedure y in many areas of physics and electronics. In 1949 Moyal of a and a ; the values +1, 0, and 1of s corresp ond to [?] discovered that the Weyl corresp ondence rule [?] can the P , W , and Q functions, resp ectively. These phase- be inverted by the Wigner transform from an op erator space functions are referred to as quasiprobability distri- on the Hilb ert space to a function on the phase space. butions QPDs, as they play in the Moyal formulation As a result, the quantum exp ectation value of an op- of quantum mechanics a role similar to that of genuine erator can b e represented by the statistical-likeaverage probability distributions in classical statistical mechan- of the corresp onding phase-space function with the sta- ics. Various QPDs has b een extensively used in many tistical density given by the Wigner function asso ciated quantum-optical applications [?,?]. Most recently, there with the density matrix of the quantum state. In this is great interest in the s-parameterized distributions b e- way quantum mechanics can b e formally represented as cause of their role in mo dern schemes for measuring the a statistical theory on classical phase space. It should quantum state of the radiation eld [?]. b e emphasized that this phase-space formalism do es not The mathematical framework and the conceptual back- replace quantum mechanics by a classical or semiclassical ground of the Moyal quantization have b een essentially theory. In fact, the phase-space formulation of quantum enlarged and generalized in two imp ortant pap ers by mechanics also known as the Moyal quantization is in Bayen et al. [?]. Sp eci cally,itwas shown that noncom- principle equivalenttoconventional formulations due to mutative deformations of the algebra of classical phase- Heisenb erg, Schrodinger, and Feynman. However, the space functions de ned by the ordinary multiplication formal resemblance of quantum mechanics in the Moyal give rise to op erator algebras of quantum mechanics. formulation to classical statistical mechanics can yield This fact means that intro ducing noncommutative sym- deep er understanding of di erences between the quan- b ol calculus based on the so-called twisted pro duct also tum and classical theories. Extensive lists of the litera- known as the star or Moyal pro duct, one obtains a com- ture on this sub ject can b e found in reviews and b o oks pletely autonomous reformulation of quantum mechanics [?,?,?,?,?,?]. in terms of phase-space functions instead of Hilb ert-space states and op erators. This program of \quantization by The ideas of Moyal were further develop ed in the late deformation" has b een develop ed in a numberofworks sixties in the works of Cahill and Glaub er [?] and Agar- [?,?,?,?,?]. wal and Wolf [?]. As mentioned, the Wigner function is related to the Weyl symmetric ordering of the p osi- For a long time applications of the Moyal formulation tion and momentum op erators q and p or, equivalently, were restricted to description of systems like a spinless of the b osonic annihilation and creation op erators a and non-relativistic quantum particle or a mo de of the quan- y a . However, there exist other p ossibilities of ordering. In tized radiation eld mo deled by a quantum harmonic particular, it was shown [?] that the Glaub er-Sudarshan oscillator, i.e., to the case of the at phase space. There- 1 fore, an imp ortant problem is the generalization of the This description can b e useful not only for measurements standard Moyal quantization for quantum systems p os- of quantum states but also in the eld of signal pro cess- sessing an intrinsic group of symmetries, with the phase ing. space b eing a homogeneous manifold on which the group of transformations acts transitively [?,?]. It has b een recently understo o d that this problem can b e solved us- I I. BASICS OF MOYAL QUANTIZATION ing the Stratonovich-Weyl SW corresp ondence. The idea of the SW corresp ondence is that the linear bijec- A. Generalized coherent states and the de nition of tive mapping b etween op erators on the Hilb ert space and quantum phase space functions on the phase space can be implemented by a kernel which satis es a number of physically sensible p os- Given a sp eci c physical system, the rst thing one tulates, with covariance and traciality b eing the two most needs to do for the Moyal quantization i.e., for con- imp ortant ones. This idea rst app eared in a pap er by structing phase-space functions is to determine what is Stratonovich[?] in 1956, but it was almost forgotten for the related phase space. This can often b e done by anal- decades. The SW corresp ondence, that has b een restated ogy with the corresp onding classical problem, thereby some years ago by Gracia-Bond a and V arilly [?,?], has providing a direct route for the quantum-classical corre- given a new impulse to the phase-space formulation of sp ondence. From the technical p oint of view, the phase quantum theory. The SW metho d of the Moyal quan- space can b e conveniently determined using the concept tization has b een applied to a numb er of imp ortant sit- of coherent states [?]. The coherent-state approach is not uations: a nonrelativistic free particle with spin, using just a convenient mathematical to ol, but it also helps to the extended Galilei group [?]; a relativistic free particle understand howphysical prop erties of the system are re- with spin, using the Poincar e group [?]; the spin, using ected by the geometrical structure of the related phase the SU2 group [?]; compact semisimple Lie groups [?]; space. It is p ossible to say that the concept of coher- one- and two-dimensional kinematical groups [?,?,?,?]; ent states constitutes a bridge b etween the Moyal phase- the two-dimensional Euclidean group [?,?]; systems of space quantization and the Berezin geometric quantiza- identical quantum particles [?]. For a review of basic tion [?]. results see Ref. [?]. Let G b e a Lie group connected and simply connected, Notwithstanding the success of the SW metho d in the with nite dimension n, which is the dynamical symme- Moyal quantization of many imp ortantphysical systems, try group of a given quantum system. Let T be a uni- the theory su ered from a serious problem. Sp eci cally, tary irreducible representation of G acting on the Hilb ert it was the absence of a simple and e ective metho d for space H . By cho osing a xed normalized reference state the construction of the SW kernel which should imple- j i 2 H , one can de ne the system of coherent states 0 ment the mapping b etween Hilb ert-space op erators and fj ig: g phase-space functions. The construction pro cedures for the SW kernels, considered during the last decade see, j i = T g j i; g 2 G: 2.1 g 0 e.g., Ref. [?], did not guarantee that the kernel will sat- isfy all the SW p ostulates. Only very recently a general The isotropy subgroup H G consists of all the group algorithm for constructing the SW kernel for quantum elements h that leave the reference state invariantupto systems p ossessing Lie-group symmetries was prop osed a phase factor, [?]. It has b een shown that the constructed kernel ex- plicitly satis es all the desired prop erties the SW p ostu- ih ih T hj i = e j i; je j =1; h 2 H: 2.2 0 0 lates and that in the particular cases of the Heisenb erg- Weyl group and SU2 our general expression reduces to For every element g 2 G, there is a decomp osition of g the known results.
Load more

Recommended publications
  • Squeezed and Entangled States of a Single Spin
    SQUEEZED AND ENTANGLED STATES OF A SINGLE SPIN Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky Faculty of Science, Bilkent University Bilkent, Ankara, 06800 Turkey e-mail: [email protected] (Received 23 December 2007; accepted 1 March 2007) Abstract We show correspondence between the notions of spin squeez- ing and spin entanglement. We propose a new measure of spin squeezing. We consider a number of physical examples. Concepts of Physics, Vol. IV, No. 3 (2007) 441 DOI: 10.2478/v10005-007-0020-0 Barı¸s Oztop,¨ Alexander A. Klyachko and Alexander S. Shumovsky It is well known that the concept of squeezed states [1] was orig- inated from the famous work by N.N. Bogoliubov [2] on the super- fluidity of liquid He4 and canonical transformations. Initially, it was developed for the Bose-fields. Later on, it has been extended on spin systems as well. The two main objectives of the present paper are on the one hand to show that the single spin s 1 can be prepared in a squeezed state and on the other hand to demonstrate≥ one-to-one correspondence between the notions of spin squeezing and entanglement. The results are illustrated by physical examples. Spin-coherent states | Historically, the notion of spin-coherent states had been introduced [3] before the notion of spin-squeezed states. In a sense, it just reflected the idea of Glauber [4] about creation of Bose-field coherent states from the vacuum by means of the displacement operator + α field = D(α) vac ;D(α) = exp(αa α∗a); (1) j i j i − where α C is an arbitrary complex parameter and a+; a are the Boson creation2 and annihilation operators.
    [Show full text]
  • Lecture 5 6.Pdf
    Lecture 6. Quantum operators and quantum states Physical meaning of operators, states, uncertainty relations, mean values. States: Fock, coherent, mixed states. Examples of such states. In quantum mechanics, physical observables (coordinate, momentum, angular momentum, energy,…) are described using operators, their eigenvalues and eigenstates. A physical system can be in one of possible quantum states. A state can be pure or mixed. We will start from the operators, but before we will introduce the so-called Dirac’s notation. 1. Dirac’s notation. A pure state is described by a state vector: . This is so-called Dirac’s notation (a ‘ket’ vector; there is also a ‘bra’ vector, the Hermitian conjugated one, ( ) ( * )T .) Originally, in quantum mechanics they spoke of wavefunctions, or probability amplitudes to have a certain observable x , (x) . But any function (x) can be viewed as a vector (Fig.1): (x) {(x1 );(x2 );....(xn )} This was a very smart idea by Paul Dirac, to represent x wavefunctions by vectors and to use operator algebra. Then, an operator acts on a state vector and a new vector emerges: Aˆ . Fig.1 An operator can be represented as a matrix if some basis of vectors is used. Two vectors can be multiplied in two different ways; inner product (scalar product) gives a number, K, 1 (state vectors are normalized), but outer product gives a matrix, or an operator: Oˆ, ˆ (a projector): ˆ K , ˆ 2 ˆ . The mean value of an operator is in quantum optics calculated by ‘sandwiching’ this operator between a state (ket) and its conjugate: A Aˆ .
    [Show full text]
  • Cavity State Manipulation Using Photon-Number Selective Phase Gates
    week ending PRL 115, 137002 (2015) PHYSICAL REVIEW LETTERS 25 SEPTEMBER 2015 Cavity State Manipulation Using Photon-Number Selective Phase Gates Reinier W. Heeres, Brian Vlastakis, Eric Holland, Stefan Krastanov, Victor V. Albert, Luigi Frunzio, Liang Jiang, and Robert J. Schoelkopf Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA (Received 27 February 2015; published 22 September 2015) The large available Hilbert space and high coherence of cavity resonators make these systems an interesting resource for storing encoded quantum bits. To perform a quantum gate on this encoded information, however, complex nonlinear operations must be applied to the many levels of the oscillator simultaneously. In this work, we introduce the selective number-dependent arbitrary phase (SNAP) gate, which imparts a different phase to each Fock-state component using an off-resonantly coupled qubit. We show that the SNAP gate allows control over the quantum phases by correcting the unwanted phase evolution due to the Kerr effect. Furthermore, by combining the SNAP gate with oscillator displacements, we create a one-photon Fock state with high fidelity. Using just these two controls, one can construct arbitrary unitary operations, offering a scalable route to performing logical manipulations on oscillator- encoded qubits. DOI: 10.1103/PhysRevLett.115.137002 PACS numbers: 85.25.Hv, 03.67.Ac, 85.25.Cp Traditional quantum information processing schemes It has been shown that universal control is possible using rely on coupling a large number of two-level systems a single nonlinear term in addition to linear controls [11], (qubits) to solve quantum problems [1].
    [Show full text]
  • Research Archive
    http://dx.doi.org/10.1108/IJBM-05-2016-0075 View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Hertfordshire Research Archive Research Archive Citation for published version: Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel, ‘Wigner's representation of quantum mechanics in integral form and its applications’, Physical Review A, Vol. 95, 022127, published 27 February 2017. DOI: https://doi.org/10.1103/PhysRevA.95.022127 Document Version: This is the Accepted Manuscript version. The version in the University of Hertfordshire Research Archive may differ from the final published version. Users should always cite the published version of record. Copyright and Reuse: ©2017 American Physical Society. This manuscript version is made available under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Enquiries If you believe this document infringes copyright, please contact the Research & Scholarly Communications Team at [email protected] Wigner's representation of quantum mechanics in integral form and its applications Dimitris Kakofengitis, Maxime Oliva, and Ole Steuernagel School of Physics, Astronomy and Mathematics, University of Hertfordshire, Hatfield, AL10 9AB, UK (Dated: February 28, 2017) We consider quantum phase-space dynamics using Wigner's representation of quantum mechanics. We stress the usefulness of the integral form for the description of Wigner's phase-space current J as an alternative to the popular Moyal bracket. The integral form brings out the symmetries between momentum and position representations of quantum mechanics, is numerically stable, and allows us to perform some calculations using elementary integrals instead of Groenewold star products.
    [Show full text]
  • 4 Representations of the Electromagnetic Field
    4 Representations of the Electromagnetic Field 4.1 Basics A quantum mechanical state is completely described by its density matrix ½. The density matrix ½ can be expanded in di®erent basises fjÃig: X ­ ¯ ¯ ½ = Dij jÃii Ãj for a discrete set of basis states (158) i;j ¯ ® ¯ The c-number matrix Dij = hÃij ½ Ãj is a representation of the density operator. One example is the number state or Fock state representation of the density ma- trix. X ½nm = Pnm jni hmj (159) n;m The diagonal elements in the Fock representation give the probability to ¯nd exactly n photons in the ¯eld! A trivial example is the density matrix of a Fock State jki: ½ = jki hkj and ½nm = ±nk±km (160) Another example is the density matrix of a thermal state of a free single mode ¯eld: + exp[¡~!a a=kbT ] ½ = + (161) T rfexp[¡~!a a=kbT ]g In the Fock representation the matrix is: X ½ = exp[¡~!n=kbT ][1 ¡ exp(¡~!=kbT )] jni hnj (162) n X hnin = jni hnj (163) (1 + hni)n+1 n with + ¡1 hni = T r(a a½) = [exp(~!=kbT ) ¡ 1] (164) 37 This leads to the well-known Bose-Einstein distribution: hnin ½ = hnj ½ jni = P = (165) nn n (1 + hni)n+1 4.2 Glauber-Sudarshan or P-representation The P-representation is an expansion in a coherent state basis jf®gi: Z ½ = P (®; ®¤) j®i h®j d2® (166) A trivial example is again the P-representation of a coherent state j®i, which is simply the delta function ±(® ¡ a0). The P-representation of a thermal state is a Gaussian: 1 2 P (®) = e¡j®j =n (167) ¼n Again it follows: Z ¤ 2 2 Pn = hnj ½ jni = P (®; ® ) jhnj®ij d ® (168) Z 2n 1 2 j®j 2 = e¡j®j =n e¡j®j d2® (169) ¼n n! The last integral can be evaluated and gives again the thermal distribution: hnin P = (170) n (1 + hni)n+1 4.3 Optical Equivalence Theorem Why is the P-representation useful? ² j®i corresponds to a classical state.
    [Show full text]
  • Generation and Detection of Fock-States of the Radiation Field
    Generation and Detection of Fock-States of the Radiation Field Herbert Walther Max-Planck-Institut für Quantenoptik and Sektion Physik der Universität München, 85748 Garching, Germany Reprint requests to Prof. H. W.; Fax: +49/89-32905-710; E-mail: [email protected] Z. Naturforsch. 56 a, 117-123 (2001); received January 12, 2001 Presented at the 3rd Workshop on Mysteries, Puzzles and Paradoxes in Quantum Mechanics, Gargnano, Italy, September 1 7 -2 3 , 2000. In this paper we give a survey of our experiments performed with the micromaser on the generation of Fock states. Three methods can be used for this purpose: the trapping states leading to Fock states in a continuous wave operation, state reduction of a pulsed pumping beam, and finally using a pulsed pumping beam to produce Fock states on demand where trapping states stabilize the photon number. Key words: Quantum Optics; Cavity Quantum Electrodynamics; Nonclassical States; Fock States; One-Atom-Maser. I. Introduction highly excited Rydberg atoms interact with a single mode of a superconducting cavity which can have a The quantum treatment of the radiation field uses quality factor as high as 4 x 1010, leading to a photon the number of photons in a particular mode to char­ lifetime in the cavity of 0.3 s. The steady-state field acterize the quantum states. In the ideal case the generated in the cavity has already been the object modes are defined by the boundary conditions of a of detailed studies of the sub-Poissonian statistical cavity giving a discrete set of eigen-frequencies.
    [Show full text]
  • Weyl Quantization and Wigner Distributions on Phase Space
    faculteit Wiskunde en Natuurwetenschappen Weyl quantization and Wigner distributions on phase space Bachelor thesis in Physics and Mathematics June 2014 Student: R.S. Wezeman Supervisor in Physics: Prof. dr. D. Boer Supervisor in Mathematics: Prof. dr H. Waalkens Abstract This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.
    [Show full text]
  • Quantum Correlations of Few Dipolar Bosons in a Double-Well Trap
    J Low Temp Phys manuscript No. (will be inserted by the editor) Quantum correlations of few dipolar bosons in a double-well trap Michele Pizzardo1, Giovanni Mazzarella1;2, and Luca Salasnich1;2;3 October 15, 2018 Abstract We consider N interacting dipolar bosonic atoms at zero temper- ature in a double-well potential. This system is described by the two-space- mode extended Bose-Hubbard (EBH) Hamiltonian which includes (in addition to the familiar BH terms) the nearest-neighbor interaction, correlated hopping and bosonic-pair hopping. For systems with N = 2 and N = 3 particles we calculate analytically both the ground state and the Fisher information, the coherence visibility, and the entanglement entropy that characterize the corre- lations of the lowest energy state. The structure of the ground state crucially depends on the correlated hopping Kc. On one hand we find that this process makes possible the occurrence of Schr¨odinger-catstates even if the onsite in- teratomic attraction is not strong enough to guarantee the formation of such states. On the other hand, in the presence of a strong onsite attraction, suffi- ciently large values of jKcj destroys the cat-like state in favor of a delocalized atomic coherent state. Keywords Ultracold gases, trapped gases, dipolar bosonic gases, quantum tunneling, quantum correlations PACS 03.75.Lm,03.75.Hh,67.85.-d 1 Introduction Ultracold and interacting dilute alkali-metal vapors trapped by one-dimensional double-well potentials [1] offer the opportunity to study the formation of 1 Dipartimento di Fisica e Astronomia "Galileo Galilei", Universit`adi Padova, Via F.
    [Show full text]
  • Phase-Space Description of the Coherent State Dynamics in a Small One-Dimensional System
    Open Phys. 2016; 14:354–359 Research Article Open Access Urszula Kaczor, Bogusław Klimas, Dominik Szydłowski, Maciej Wołoszyn, and Bartłomiej J. Spisak* Phase-space description of the coherent state dynamics in a small one-dimensional system DOI 10.1515/phys-2016-0036 space variables. In this case, the product of any functions Received Jun 21, 2016; accepted Jul 28, 2016 on the phase space is noncommutative. The classical alge- bra of observables (commutative) is recovered in the limit Abstract: The Wigner-Moyal approach is applied to investi- of the reduced Planck constant approaching zero ( ! 0). gate the dynamics of the Gaussian wave packet moving in a ~ Hence the presented approach is called the phase space double-well potential in the ‘Mexican hat’ form. Quantum quantum mechanics, or sometimes is referred to as the de- trajectories in the phase space are computed for different formation quantization [6–8]. Currently, the deformation kinetic energies of the initial wave packet in the Wigner theory has found applications in many fields of modern form. The results are compared with the classical trajecto- physics, such as quantum gravity, string and M-theory, nu- ries. Some additional information on the dynamics of the clear physics, quantum optics, condensed matter physics, wave packet in the phase space is extracted from the analy- and quantum field theory. sis of the cross-correlation of the Wigner distribution func- In the present contribution, we examine the dynam- tion with itself at different points in time. ics of the initially coherent state in the double-well poten- Keywords: Wigner distribution, wave packet, quantum tial using the phase space formulation of the quantum me- trajectory chanics.
    [Show full text]
  • P. A. M. Dirac and the Maverick Mathematician
    Journal & Proceedings of the Royal Society of New South Wales, vol. 150, part 2, 2017, pp. 188–194. ISSN 0035-9173/17/020188-07 P. A. M. Dirac and the Maverick Mathematician Ann Moyal Emeritus Fellow, ANU, Canberra, Australia Email: [email protected] Abstract Historian of science Ann Moyal recounts the story of a singular correspondence between the great British physicist, P. A. M. Dirac, at Cambridge, and J. E. Moyal, then a scientist from outside academia working at the de Havilland Aircraft Company in Britain (later an academic in Australia), on the ques- tion of a statistical basis for quantum mechanics. A David and Goliath saga, it marks a paradigmatic study in the history of quantum physics. A. M. Dirac (1902–1984) is a pre- neering at the Institut d’Electrotechnique in Peminent name in scientific history. In Grenoble, enrolling subsequently at the Ecole 1962 it was my privilege to acquire a set Supérieure d’Electricité in Paris. Trained as of the letters he exchanged with the then a civil engineer, Moyal worked for a period young mathematician, José Enrique Moyal in Tel Aviv but returned to Paris in 1937, (1910–1998), for the Basser Library of the where his exposure to such foundation works Australian Academy of Science, inaugurated as Georges Darmois’s Statistique Mathéma- as a centre for the archives of the history of tique and A. N. Kolmogorov’s Foundations Australian science. This is the only manu- of the Theory of Probability introduced him script correspondence of Dirac (known to to a knowledge of pioneering European stud- colleagues as a very reluctant correspondent) ies of stochastic processes.
    [Show full text]
  • Collège De France Abroad Lectures Quantum Information with Real Or Artificial Atoms and Photons in Cavities
    Collège de France abroad Lectures Quantum information with real or artificial atoms and photons in cavities Lecture 6: Circuit QED experiments synthesing arbitrary states of quantum oscillators. Introduction to State synthesis and reconstruction in Circuit QED We describe in this last lecture Circuit QED experiments in which quantum harmonic oscillator states are manipulated and non-classical states are synthesised and reconstructed by coupling rf resonators to superconducting qubits. We start by the description of deterministic Fock state preparation (&VI-A), and we then generalize to the synthesis of arbitrary resonator states (§VI-B). We finally describe an experiment in which entangled states of fields pertaining to two rf resonators have been generated and reconstructed (§VI-C). We compare these experiments to Cavity QED and ion trap studies. VI-A Deterministic preparation of Fock states in Circuit QED M.Hofheinz et al, Nature 454, 310 (2008) Generating Fock states in a superconducting resonator Microphotograph showing the phase qubit (left) and the coplanar resonator at 6.56 GHz (right) coupled by a capacitor. The qubit and the resonator are separately coupled to their own rf source. The phase qubit is tuned around the resonator frequency by using the flux bias. Qubit detection is achieved by a SQUID. Figure montrant avec un code couleur (code sur l’échelle de droite) la probabilité de mesurer le qubit dans son état excité en fonction de la fréquence d’excitation (en ordonnée) et du désaccord du qubit par rapport au résonateur (en abscisse). Le spectre micro-onde obtenu montre l’anticroisement à résonance, la distance minimale des niveaux mesurant la fréquence de Rabi du vide !/2"= 36 MHz.
    [Show full text]
  • Operators and States
    Operators and States University Press Scholarship Online Oxford Scholarship Online Methods in Theoretical Quantum Optics Stephen Barnett and Paul Radmore Print publication date: 2002 Print ISBN-13: 9780198563617 Published to Oxford Scholarship Online: January 2010 DOI: 10.1093/acprof:oso/9780198563617.001.0001 Operators and States STEPHEN M. BARNETT PAUL M. RADMORE DOI:10.1093/acprof:oso/9780198563617.003.0003 Abstract and Keywords This chapter is concerned with providing the necessary rules governing the manipulation of the operators and the properties of the states to enable us top model and describe some physical systems. Atom and field operators for spin, angular momentum, the harmonic oscillator or single mode field and for continuous fields are introduced. Techniques are derived for treating functions of operators and ordering theorems for manipulating these. Important states of the electromagnetic field and their properties are presented. These include the number states, thermal states, coherent states and squeezed states. The coherent and squeezed states are generated by the actions of the Glauber displacement operator and the squeezing operator. Angular momentum coherent states are described and applied to the coherent evolution of a two-state atom and to the action of a beam- splitter. Page 1 of 63 PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter
    [Show full text]