Leonard Susskind

Bio: Leonard Susskind is an academic researcher from Cornell University. The author has contributed to research in topics: Gluon & Ladder operator. The author has an hindex of 2, co-authored 2 publications receiving 716 citations.

Quantum mechanical phase and time operator

Abstract: The phase operator for an oscillator is shown not to exist. It is replaced by a pair of non-commuting sin and cos operators which can be used to define uncertainty relations for phase and number. The relation between phase and angle operators is carefully discussed. The possibility of using a phase variable as a quantum clock is demonstrated and the states for which the clock is most accurate are constructed.

Asymptotic freedom and the "absence" of vector-gluon exchange in wide-angle hadronic collisions

Abstract: The naive, pointlike parton model of Berman, Bjorken, and Kogut is generalized to scale-invariant and asymptotically free field theories. The asymptotically free field generalization is studied in detail. Although such theories contain vector fields, single vector-gluon exchange contributes insignificantly to wide-angle hadronic collisions. This follows from (1) the smallness of the invariant charge at small distances and (2) the breakdown of naive scaling in these theories. These effects should explain the apparent absence of vector exchange in inclusive and exclusive hadronic collisions at large momentum transfers observed at Fermilab and at the CERN ISR.

Quantum metrology

Abstract: We point out a general framework that encompasses most cases in which quantum effects enable an increase in precision when estimating a parameter (quantum metrology) The typical quantum precision-enhancement is of the order of the square root of the number of times the system is sampled We prove that this is optimal and we point out the different strategies (classical and quantum) that permit to attain this bound

Phase and Angle Variables in Quantum Mechanics

Abstract: The quantum-mechanical description of phase and angle variables is reviewed, with emphasis on the proper mathematical description of these coordinates. The relations among the operators and state vectors under consideration are clarified in the context of the Heisenberg uncertainty relations. The familiar case of the azimuthal angle variable $\ensuremath{\phi}$ and its "conjugate" angular momentum ${L}_{z}$ is discussed. Various pitfalls associated with the periodicity problem are avoided by employing periodic variables ($sin\ensuremath{\phi}$ and $cos\ensuremath{\phi}$ to describe the phase variable. Well-defined uncertainty relations are derived and discussed. A detailed analysis of the three-dimensional harmonic oscillator excited in coherent states is given. A detailed analysis of the simple harmonic oscillator is given. The usual assumption that a (Hermitian) phase operator $\ensuremath{\varphi}$ (conjugate to the number operator $N$) exists is shown to be erroneous. However, cosine and sine operators $C$ and $S$ exist and are the appr\'opriate phase variables. A Poisson bracket argument using action-angle (rather $J$, $cos\ensuremath{\varphi}$, $sin\ensuremath{\varphi}$) variables is used to deduce $C$ and $S$. The spectra and eigenfunctions of these operators are investigated, along with the important "phase-difference" periodic variables. The properties of the oscillator variables in the various types of states are analyzed with special attention to the uncertainty relations and the transition to the classical limit. The utility of coherent states as a basis for the description of the evolution of the density matrix is emphasized. In this basis it is easy to identify the classical Liouville equation in action-angle variables along with quantum-mechanical "corrections." Mention is made of possible physical applications to superfluid systems.

'Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years

Abstract: Seventy five years ago, three remarkable papers by Schr¨ odinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called ‘nonclassical states’ (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the ‘standard citation package’ used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and ‘nonclassical’ states commenced. The 1990s are described in a more ‘squeezed’ manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.

Phase properties of the quantized single-mode electromagnetic field.

Abstract: The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.

On the Hermitian Optical Phase Operator

Abstract: It has long been believed that no Hermitian optical phase operator exists. However, such an operator can be constructed from the phase states. We demonstrate that its properties are precisely in accord with the results of semiclassical and phenomenological approaches when such approximate methods are valid. We find that the number-phase commutator differs from that originally postulated by Dirac. This difference allows the consistent use of the commutator for inherently quantum states. It also leads to the correct periodic phase behaviour of the Poisson bracket in the classical regime.