DOCSLIB.ORG

Quantum Entropy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Entropy Quantum Entropy Davi Geiger and Zvi M. Kedem Courant Institute of Mathematical Sciences New York University, New York, New York 10012 arXiv:2106.15375v1 [quant-ph] 29 Jun 2021 1 Abstract All the laws of particle physics are time-reversible. A time arrow emerges only when ensembles of classical particles are treated probabilistically, outside of physics laws, and the second law of thermodynamics is introduced. In quantum physics, despite its intrinsically probabilistic nature, no mechanism for a time arrow has been proposed. We propose an entropy for quantum physics, which may conduce to the emergence of a time arrow. The proposed entropy is a measure of randomness over the degrees of freedom of a quantum state. It is dimensionless, it is a relativistic scalar, it is invariant under coordinate transformation of position and momentum that maintain conjugate properties, and under CPT transformations; and its minimum is positive due to the uncertainty principle. I. INTRODUCTION Today’s classical and quantum physics laws are time-reversible and a time arrow emerges in physics only when a probabilistic behavior of ensembles of particles is considered. In contrast, no mechanism for a time arrow was proposed for quantum physics even though it introduces probability as intrinsic to the description of even a single-particle system. A probability P 0 is assigned to each value of 0 A ( ) ∈ for an observable A. For a finite A = 08 8=1,...,# , the Shannon entropy, H = { } # P 08 log P 08 , is a measure of information about A. − 8=1 ( ) 2 ( ) PExtending the concept of entropy to continuous variables, continuous distribu- tions, and to quantum mechanics (QM) has been challenging. For example, von Neumann’s entropy [17] requires the existence of classical statistics elements (mixed states)in order not to vanish, and consequentlyit assigns zero entropy to one-particle systems. Therefore, it is not possible to start with von Neumann’s entropy if one wants to assign an entropy that measures the randomness of a one particle state and then 2 extend it to multiple particles and a quantum field. There are two frameworks for quantum physics: QM and quantum field theory (QFT). A particle system is described in QM by a quantum state kC , which is a ray | i in Hilbert space, while in QFT such a state is described by an operator R acting on the vacuum state 0 . Our proposed entropy is applicable to both frameworks. | i In classical physics, Boltzmann entropy and Gibbs entropy, and their respective H-theorems [11], are formulated in the phase space, reflecting the degrees of freedomofasystem. InQM,thepositionandthemomentumareconjugate operators and their eigenstates allow to describe a phase space. In QFT the position is demoted to a variable, and the Fourier transform of the fields introduces a spatial frequency variable that together with position compose a quantum field phase space coordinate system. We require the entropy (i) to account for all degrees of freedom of a state, (ii) to be a measure of randomness of such a state, (iii) to be invariant under the applicable transformations, such as Lorentz transformations. The paper is organized as follows. In Section II, we propose an entropy measure of randomness of a quantum state. In Section III we prove its minimum. Then, we prove invariant properties under: continuous coordinate transformations in phase space in Section IV A, discrete CPT transformations in Section IV B, and special relativity in Section IV C. Section V concludes the paper. II. QUANTUM ENTROPY The entropy must account for both continuous and discrete degrees of freedom (dof). TheentropyforcontinuousdofsisdevelopedinsectionIIAanditisassociated with the space coordinates. Together with the spatial frequency (the conjugate variable to the spatial variable) they form the continuous phase space. The discrete dof is associated with the spin, which is an internal dof. The entropy for a spin 3 1 phase space is developed in section IIB for spins 0 and 2. The entropy of a state is then the sum of the entropy of the continuous dofs and the entropy of the spin. A. Entropy for the Continuous Degrees of Freedom To address the continuous coordinates degrees of freedom and the uncertainties associated with them, we associate with every QM state k the projection onto | iC the QM eigenstates of the conjugate operators rˆ and pˆ , i.e., r and p . Note that | i | i the continuous degrees of freedom are completely defined by one of the choices of continous coordinates, r or k, as one of the projections can be recovered from the other one via a Fourier transform. However, due to the uncertainty principle, the uncertainties or randomness of the continuous coordinates of a particle are captured in the phase space described by k r,C = r kC , q p,C = p kC . By Born’s ( ( ) h | i ( ) h | i) 2 2 rule, dr r,C = k r,C and d? p,C = q˜ p,C are the probability densities of ( ) | ( )| ( ) | ( )| the phase space representation of the state. We define the continuous entropy of a particle to be 3 3 3 S = dr r,C d? p,C ln dr r,C d? p,C \ d r d p , −¾ ( ) ( ) ( ) ( ) 3 3 = dr r,C d: k,C ln dr r,C d: k,C d r d k , −¾ ( ) ( ) ( ( ) ( ) ) = Sr S: , (1) + 1 1 where k = p is the spatial frequency, d: k,C = d? p,C is the spatial frequency \ ( ) \3 ( ) probability density (so that the infinitesimal probability in a infinitesimal volume 3 is invariant under the change in variables), Sr = dr r,C ln dr r,C d r, and − ( ) ( ) ´ analogously for Sk. The proposed entropy is motivated by the work of Gibbs [11] and Jaynes [13]. The dimensionless phase space volume element is d3r d3k, and the dimensionless probability density in phase space is d r, k,C = dr r,C d: k,C . Thus the entropy ( ) ( ) ( ) 4 is dimensionless and invariant under changes of the units of measurements. A natural extension to an #-particle system in QM is 3 3 3 3 ( = d r1 d k1 ... d r# d k# dr r1,..., r# ,C dk k1,..., k# ,C −¾ ( ) ( ) ln dr r1,..., r# ,C d k1,..., k# ,C × ( ( ) k( )) 3 3 = d r1 ... d r# dr r1,..., r# ,C ln dr r1,..., r# ,C −¾ ¾ ( ) ( ) 3 3 d k1 ... d k# dk k1,..., k# ,C ln dp k1,..., k# ,C , −¾ ¾ ( ) ( ) 2 2 where dr r1,..., r# ,C = k r1,..., r# ,C and dk k1,..., k# ,C = q k1,..., k# ,C ( ) | ( )| ( ) | ( )| # are defined in QM via the projection of the state kC of # particles onto the po- sition r1 ... r# and the spatial frequency (momentum) k1 ... k# coordinate h | h | h | h | # systems. The state kC is defined in Fock spaces, the product of # Hilbert spaces, and requiring the construction of combinatorics to describe indistinguishable particles. In QFT, fields are described by the operators R r,C , where r,C become pa- ( ) ( ) rameters describing space-time, and Q k,C is the Fourier transform of R r,C . A ( ) ( ) representation used in QFT for a system of particles is based on Fock states with occupation number of the form =@1 ,=@2 ,,...,=@8 ,... , where =@8 is the number of particles in a QM Hilbert space state @8 . A special state is the vacuum sate 0 | i | i with no particles. One can create a Fock occupation state, or simply a Fock state, for sets of quantum numbers from the vaccum state and the creation operator as 1 = =@8 =@1 ,=@2 ,,...,=@8 ,...,=@ 0†@8 0 . = = ! ( ) | i Y8 1 @8 =p The number of particles of a Fock state is # 8=1 =@8 . Then, a QFT state is described in a Fock space as a linear superpositionP of Fock states, i.e., state = | i < U< =@1 ,=@2 ,,...,=@8 ,... , where < is an index for each possible configuration P 2 of a Fock state, U< ℂ,and1 = < U< . A general state in Fock space does not ∈ | | | i have a specific number of particlesP since it can be in a superposition of two Fock 5 states with different number of particles. The QFT operators act on the the states producing a phase space state given by R r,C state ,Q k,C state . We then ( ( )| i ( )| i) define the probability density function in position and momentum space as QFT 2 d r,C = R r,C state = state R † r,C R r,C state . r ( ) | ( )| i| h | ( ) ( )| i Analogously, we have dQFT k,C = Q k,C state 2 = state Q k,C Q k,C state . k ( ) | ( )| i| h | †( ) ( )| i One should not interpret these probability density distributionsin QFT as associated with the measurements of finding a particle in a given position or spatial frequency, but rather as distributions of the information about position and space frequency of the state of the field. The information and the uncertainties of the position and momentum of the field of particles in this state are captured by the distributions dQFT r,C and dQFT k,C . The QFT entropy of the position and momentum r ( ) k ( ) information is then given by = QFT QFT QFT QFT 3 3 S dr r,C dk k,C ln dr r,C dk k,C d r d k −¾ ( ) ( ) ( ) ( ) = Sr S: , + QFT QFT 3 where Sr = d r,C ln d r,C d r, and analogously for Sk. − r ( ) r ( ) In the rest´ of the paper the superscript QFT will generally be dropped as it will be clear whether we are using the representation of QM or of QFT. B. Entropy for the Spin Degrees of Freedom The dofs associated with the spin are captured by the vector or bispinor repre- sentation of the states in both frameworks, QM and QFT.
Load more

Recommended publications
  • 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]
  • 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]
  • 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]
  • 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]
  • Molecular Interferometers: Effects of Pauli Principle on Entangled- Enhanced Precision Measurements
    PAPER • OPEN ACCESS Molecular interferometers: effects of Pauli principle on entangled- enhanced precision measurements To cite this article: P Alexander Bouvrie et al 2019 New J. Phys. 21 123011 View the article online for updates and enhancements. This content was downloaded from IP address 186.13.115.241 on 12/12/2019 at 14:37 New J. Phys. 21 (2019) 123011 https://doi.org/10.1088/1367-2630/ab5b2f PAPER Molecular interferometers: effects of Pauli principle on entangled- OPEN ACCESS enhanced precision measurements RECEIVED 15 August 2019 P Alexander Bouvrie1,5 , Ana P Majtey2,3 , Francisco Figueiredo1 and Itzhak Roditi1,4 REVISED 1 Centro Brasileiro de Pesquisas Físicas, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil 6 November 2019 2 Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Av. Medina Allende s/n, Ciudad Universitaria, ACCEPTED FOR PUBLICATION X5000HUA Córdoba, Argentina 25 November 2019 3 Consejo de Investigaciones Científicas y Técnicas de la República Argentina, Av. Rivadavia 1917, C1033AAJ, Ciudad Autónoma de PUBLISHED Buenos Aires, Argentina 12 December 2019 4 Institute for Theoretical Physics, ETH Zurich 8093, Switzerland 5 Author to whom any correspondence should be addressed. Original content from this E-mail: [email protected] work may be used under the terms of the Creative Commons Attribution 3.0 Keywords: molecular BEC, fermion interference, quantum metrology licence. Any further distribution of this work must maintain attribution to the Abstract author(s) and the title of Feshbach molecules forming a Bose–Einstein condensate (BEC) behave as non-ideal bosonic particles the work, journal citation and DOI.
    [Show full text]
  • Experimental Preparation and Measurement of Quantum States Of
    journal of modern optics, 1997, vol. 44, no. 11/12, 2485± 2505 Expe rim e n tal pre paration an d m e asu re m e n t of qu an tu m state s of m otion of a trappe d atom ² D. LEIBFRIED, D. M. MEEKHOF, C. MONROE, B. E. KING, W. M. IT ANO and D. J. WINELAND Time and Frequency Division, National Institute of Standards and Technology, 325 Broadway, Boulder, CO 80303-3328, USA ( Received 5 November 1996) Abstrac t. We report the creation and full determination of several quantum states of motion of a 9Be+ ion bound in a RF (Paul) trap. The states are coherently prepared from an ion which has been initially laser cooled to the zero-point of motion. We create states having both classical and non-classical character including thermal, number, coherent, squeezed, and `SchroÈ dinger cat’ states. The motional quantum state is fully reconstructed using two novel schemes that determine the density matrix in the number state basis or the Wigner function. Our techniques allow well controlled experiments decoher- ence and related phenomena on the quantum± classical borderline. 1. In trod u c tion The ability to create and completely characterize a variety of fundamental quantum states has long been sought after in the laboratory since it brings to the forefront issues involving the relationship between quantum and classical physics. Since most theoretical proposals to achieve these goals were put forward in the ® eld of quantum optics, it might seem surprising that some of the ® rst experiments succeeding in both respects were realized on the motion of a trapped atom.
    [Show full text]
  • Coherent States.Pdf
    4 II. COHERENT STATES A. Derivation of Coherent States The number states studied in the previous section are mathematically very simple to study, but very difficult to realize in the laboratory. Coherent states α, as we will now study, are quasi-classical states produced by lasers. The mathematics of coherent states are motivated by a simple relation, namely: a^ α = α α : (16) j i j i This equation tells us that coherent states α are eigenstates of the destruction operator a^. j i A simple physical meaning is that measurement of a small portion of a coherent state does not change the coherent state as long as the field is strong (quasi-classical). For example, a small portion of a coherent state (laser) can be sampled without affecting the unsampled portion of the coherent state. This is not the case when number states are used. The decomposition of the coherent state in terms of the number states can be found from the relation in eqn. (14). Assume, α c n ; = X n (17) j i n j i which implies a α c pn n : ^ = X n 1 (18) j i n j − i Substituting these results into eqn. (16) yields c pn n c α n X n 1 = X n (19) n j − i j i which brings about the recurrence relation: cn+1pn + 1 = cnα: (20) Hence, α α2 αn cn = cn−1 = cn−1 = c0: (21) pn (n)(n 1) pn! q − Therefore, fixing the c0 coefficient and using the normalization condition, c 2 X n = 1 (22) n j j 5 uniquely determines all of the coefficients.
    [Show full text]
  • Quantum Trajectories for Propagating Fock States
    PHYSICAL REVIEW A 96, 023819 (2017) Quantum trajectories for propagating Fock states Ben Q. Baragiola1,2,* and Joshua Combes2,3,4,5,† 1Centre for Engineered Quantum Systems, Macquarie University, Sydney, New South Wales 2109, Australia 2Center for Quantum Information and Control, University of New Mexico, Albuquerque, New Mexico 87131, USA 3Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada 4Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 5Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia (Received 9 April 2017; published 9 August 2017) We derive quantum trajectories (also known as stochastic master equations) that describe an arbitrary quantum system probed by a propagating wave packet of light prepared in a continuous-mode Fock state. We consider three detection schemes of the output light: photon counting, homodyne detection, and heterodyne detection. We generalize to input field states in superpositions and mixtures of Fock states and illustrate our formalism with several examples. DOI: 10.1103/PhysRevA.96.023819 I. INTRODUCTION previous time t t, or (ii) the photon has not yet arrived at the system and can be found with certainty in the remaining, future Propagating Fock states, wave packets with a definite input field t >t. The origin of system-field entanglement is number of photons, are well suited for the role of relaying in- the superposition of these options. This is a departure from formation between nodes of a quantum computing device [1]. the typically considered situation for open quantum systems In the optical and microwave domains, single-photon fields are where the input field at time t is uncorrelated both with the routinely produced and manipulated, with ongoing progress to- system and with the field at all other times.
    [Show full text]
  • Why Ladder Operators Are Useful
    Why ladder operators are useful Ryan M. Wilson (Dated: February 13, 2015) I. HARMONIC OSCILLATOR Let's consider a 1D harmonic oscillator with the Hamiltonian p2 1 H = − + m!2x2: (1) 2m 2 This yields eigenstates φn(x) with energies En = ~!(n + 1=2). We saw that we can recast this Hamiltonian in terms of \ladder" operators, y H = ~!(^a a^ + 1=2); (2) wherea ^y is a \raising" operator anda ^ is a \lowering" operator. When acting on a real-space basis of harmonic y oscillator eigenstates,a ^ φn = φn+1, andaφ ^ n = φn−1. II. TWO PARTICLES What if we have two particles in a 1D harmonic oscillator? Then our Hamiltonian is H = H1 + H2, where p2 1 H = − i + m!2x2: (3) i 2m 2 i In general, our wave function will have to be written as (x1; x2). Our Schrodinger equation is H (x1; x2) = E (x1; x2) X p2 1 − i + m!2x2 (x ; x ) = E (x ; x ): (4) 2m 2 i 1 2 1 2 i=1;2 Can we solve this by writing the separable form (x1; x2) = 1(x1) 2(x2)? Let's try. p2 p2 1 1 − 1 − 2 + m!2x2 + m!2x2 (x ) (x ) = E (x ) (x ) 2m 2m 2 1 2 2 1 1 2 2 1 1 2 2 p2 p2 1 1 − (x ) 1 (x ) − (x ) 2 (x ) + (x ) m!2x2 (x ) + (x ) m!2x2 (x ) = E (x ) (x ) 2 2 2m 1 1 1 1 2m 2 2 2 2 2 1 1 1 1 1 2 2 2 2 1 1 2 2 2 2 1 p1 1 p2 1 2 2 1 2 2 − 1(x1) − 2(x2) + m! x1 + m! x2 = E = Ep1 + Ep2; (5) 1(x1) 2m 2(x2) 2m 2 2 so this is a separable differential equation, and 1(x1) and 2(x2) are harmonic oscillator eigenstates with energies Ep1 and Ep2, respectively! A.
    [Show full text]
  • Arxiv:Quant-Ph/0108024V2 10 Jun 2002
    Preprint SB/F/01-289 On the Squeezed Number States and their Phase Space Representations L. Albano 1, D.F.Mundarain 1 and J. Stephany1,2 1Universidad Sim´on Bol´ıvar,Departamento de F´ısica,Apartado Postal 89000, Caracas 1080-A, Venezuela. 2Abdus Salam International Centre for Theoretical Physics, Strada Costiera, 11, 34014 Trieste, Italy. e-mail: lalbano@fis.usb.ve, dmundara@fis.usb.ve, [email protected] Abstract We compute the photon number distribution, the Q(α) distribution function and the wave functions in the momentum and position representation for a single mode squeezed number state using generating functions which allow to obtain any matrix element in the squeezed number state representation from the matrix elements in the squeezed coherent state representation. For highly squeezed number states we discuss the previously unnoted oscillations which appear in the Q(α) function. We also note that these oscillations can be related to the arXiv:quant-ph/0108024v2 10 Jun 2002 photon-number distribution oscillations and to the momentum representation of the wave function. UNIVERSIDAD SIMON BOLIVAR 1 Introduction The study and experimental detection of squeezed states [1, 2, 3, 4] and other non classical states of optical systems, is an interesting issue, from both the fundamental and the technological point of view. On one hand, disentangling the properties of intrinsically quantum states of light enhances our understanding of the behavior and interactions of photons, and improves our knowledge of Quantum Electrodynamics. On the other hand, the use of light states with reduced quantum fluctuations in one of the conjugate quadratures is at the heart of attractive proposals for the detection of very weak signals[5, 6].
    [Show full text]