Jeff Tollaksen

A time-symmetric formulation of quantum mechanics

Quantum mechanics allows one to independently select both the initial and final states of a single system. Such pre- and postselection reveals novel effects that challenge our ideas about what time is and how it flows.

Quantum interference experiments, modular variables and weak measurements

We address the problem of interference using the Heisenberg picture and highlight some new aspects through the use of pre-selection, post-selection, weak measurements and modular variables. We present a physical explanation for the different behaviors of a single particle when the distant slit is open or closed; instead of having a quantum wave that passes through all slits, we have a localized particle with non-local interactions with the other slit(s). We introduce a Gedanken experiment to measure this non-local exchange. While the Heisenberg and Schrodinger pictures are equivalent formulations of quantum mechanics, nevertheless, the results discussed here support a new approach to quantum mechanics which has lead to new insights, new intuitions, new experiments and even the possibility of new devices that were missing from the old perspective.

Some mathematical properties of superoscillations

In this paper, we give a possible mathematical setting for superoscillations. We define the set of superoscillation in terms of the uniform convergence of functions on such a set and study the problem of the approximation of a function by superoscillating functions.

Observation of a quantum Cheshire Cat in a matter-wave interferometer experiment

From its very beginning, quantum theory has been revealing extraordinary and counter-intuitive phenomena, such as wave-particle duality, Schrödinger cats and quantum non-locality. Another paradoxical phenomenon found within the framework of quantum mechanics is the ‘quantum Cheshire Cat’: if a quantum system is subject to a certain pre- and postselection, it can behave as if a particle and its property are spatially separated. It has been suggested to employ weak measurements in order to explore the Cheshire Cat’s nature. Here we report an experiment in which we send neutrons through a perfect silicon crystal interferometer and perform weak measurements to probe the location of the particle and its magnetic moment. The experimental results suggest that the system behaves as if the neutrons go through one beam path, while their magnetic moment travels along the other.

Multiple-time states and multiple-time measurements in quantum mechanics

We discuss experimental situations that consist of multiple preparation and measurement stages. This leads us to an alternative approach to quantum mechanics. In particular, we introduce the idea of multitime quantum states which are the appropriate tools for describing these experimental situations. We also describe multitime measurements and discuss their relation to multitime states. A consequence of our formalism is to put states and operators on an equal footing. Finally we discuss the implications of our approach to quantum mechanics for the problem of the flow of time.

Quantum violation of the pigeonhole principle and the nature of quantum correlations

Significance We show that quantum mechanics violates one of the fundamental principles of nature: If you put three particles in two boxes, necessarily two particles will end up in the same box. We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box, a seemingly impossible and absurd effect. This is only one of a host of related quantum effects which we discovered and which point to a very interesting structure of quantum mechanics that was hitherto unnoticed and has major implications for our understanding of nature. It requires us to revisit some of the most basic notions of quantum physics––the notions of separability, of correlations, and of interactions. The pigeonhole principle: “If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole,” is an obvious yet fundamental principle of nature as it captures the very essence of counting. Here however we show that in quantum mechanics this is not true! We find instances when three quantum particles are put in two boxes, yet no two particles are in the same box. Furthermore, we show that the above “quantum pigeonhole principle” is only one of a host of related quantum effects, and points to a very interesting structure of quantum mechanics that was hitherto unnoticed. Our results shed new light on the very notions of separability and correlations in quantum mechanics and on the nature of interactions. It also presents a new role for entanglement, complementary to the usual one. Finally, interferometric experiments that illustrate our effects are proposed.

Quantum Theory: A Two-Time Success Story

Superoscillatory functions vary faster than their fastest Fourier component. Here they are employed to give an alternative description and explicit recipe for creating endfire arrays with supergain, that is antennas with radiation patterns concentrated in an arbitrarily narrow angular range and of arbitrary form. Two examples are radiation patterns described by sinc and Gaussian functions. [Editor’s note: for a video of the talk given by Prof. Berry (titled ‘Weak Value Probabilities’) at the Aharonov-80 conference in 2012 at Chapman University, see quantum.chapman.edu/talk-6.] Dedicated to Yakir Aharonov on his 80th birthday: still quick, still deep, still subtle.

Superoscillation phenomena in SO(3)

We prove a new asymptotic formula for the Wigner d-functions, as a consequence of the discovery of superoscillatory behaviour in SO(3).

Evolution of superoscillatory data

Weak measurements and the theory of weak values have a very deep meaning in quantum mechanics, and new phenomena associated with them has recently been observed experimentally. This theory has also directly led to the notion of superoscillating sequences of functions. In this paper we consider Cauchy problems with superoscillatory initial conditions (in particular, the Cauchy problem for the Schrodinger equation and some of its variations), and we give conditions under which the superoscillations persist in time. Our work is based on results from the theory of formal solutions to Cauchy problems and on the study of the specific growth of superoscillatory sequences, when regarded as functions of a complex variable. There are two main aims of this paper: one is to explain the mathematical tools that are necessary to study superoscillations, also repeating a few results that we have already proved in other papers in order to clarify the strategy. The second aim is to show that our technique applies to a large class of problems, showing under which conditions the superoscillatory phenomenon persists. Finally, we point out that our strategy can be applied also to non-constant coefficients differential equations as the quantum harmonic oscillator.

Robust weak measurements on finite samples

A new weak measurement procedure is introduced for finite samples which yields accurate weak values that are outside the range of eigenvalues and which do not require an exponentially rare ensemble. This procedure provides a unique advantage in the amplification of small nonrandom signals by minimizing uncertainties in determining the weak value and by minimizing sample size. This procedure can also extend the strength of the coupling between the system and measuring device to a new regime.