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Physical Review A

Potential barrier mimicking frequent location measurements in quantum Zeno dynamics

Miguel A. Porras, Isabel Gonzalo, Alfredo Luis

Published Apr 7, 2016• 3 citations
Publisher

Abstract

We show that quantum Zeno dynamics can be mimicked by the isolated evolution of an unobserved system in an effective potential. Monitoring frequently whether a particle remains in a region of space leads to the same wave-packet dynamics as placing the region on top of a potential barrier and letting the particle evolve on its own, without external couplings. We focus on very frequent but not continuous observation so that the particle abandons the initial region with some finite probability. The height of the barrier relative to the surroundings for a high frequency $\ensuremath{\nu}$ of the observations being mimicked is found numerically to be $h\ensuremath{\nu}/2$, where $h$ is Planck's constant.

DOI
10.1103/physreva.93.040101

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We show that quantum Zeno dynamics can be mimicked by the isolated evolution of an unobserved system in an effective potential. Monitoring frequently whether a particle remains in a region of space leads to the same wave-packet dynamics as placing the region on top of a potential barrier and letting the particle evolve on its own, without external couplings. We focus on very frequent but not continuous observation so that the particle abandons the initial region with some finite probability. The height of the barrier relative to the surroundings for a high frequency \ensuremathν\ensuremath{\nu}\ensuremathν of the observations being mimicked is found numerically to be h\ensuremathν/2h\ensuremath{\nu}/2h\ensuremathν/2, where hhh is Planck's constant.
Abstract. If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected “quantum Zeno subspace ” is called “quantum Zeno dynamics: ” for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.
If frequent measurements ascertain whether a quantum system is still in its initial state, transitions to other states are hindered and the quantum Zeno effect takes place. However, in its broader formulation, the quantum Zeno effect does not necessarily freeze everything. On the contrary, for frequent projections onto a multidimensional subspace, the system can evolve away from its initial state, although it remains in the subspace defined by the measurement. The continuing time evolution within the projected "quantum Zeno subspace" is called "quantum Zeno dynamics:" for instance, if the measurements ascertain whether a quantum particle is in a given spatial region, the evolution is unitary and the generator of the Zeno dynamics is the Hamiltonian with hard-wall (Dirichlet) boundary conditions. We discuss the physical and mathematical aspects of this evolution, highlighting the open mathematical problems. We then analyze some alternative strategies to obtain a Zeno dynamics and show that they are physically equivalent.
Although the quantum Zeno effect takes its name from Zeno's arrow paradox, the effect of frequently observing the position of a freely moving particle on its motion has not been analyzed in detail in the frame of standard quantum mechanics. We study the evolution of a moving free particle while monitoring whether it lingers in a given region of space, and explain the dependence of the lingering probability on the frequency of the measurements and the initial momentum of the particle. Stopping the particle entails the emergence of Schr\"odinger cat states during the observed evolution, closely connected to the high-order diffraction modes in Fabry-P\'erot optical resonators.
The evolution of a quantum system undergoing very frequent measurements takes place in a subspace of the total Hilbert space (quantum Zeno effect). The dynamical properties A quantum system, prepared in a state that does not belong to an eigenvalue of its total Hamiltonian, starts to evolve quadratically in time [1, 2]. This behavior leads to the so-called quantum Zeno phenomenon: by performing frequent measurements on the system, in order to check whether it is still in its initial state, one can “slow
According to quantum Zeno dynamics (QZD), the evolution of a quantum system can be restricted to a subspace of its Hilbert space by frequent measurements. A crucial question in QZD of a particle's position is: how short the time interval between successive measurements should be, in order to confine the particle in its initial spatial region? To address this question, we consider a toy model with two ions initially known to be, for simplicity, in a one-dimensional spatial region. By simulating the evolution of this two-body quantum system, we estimate the measurement frequency needed to keep the ions within their initial confined region at a desired confidence level. Two key parameters we employ in our calculations are the Zeno time and the leakage probability of the quantum system. The measurement frequencies are calculated and compared when ions are located initially at different spatial regions. For our simulation, we introduce the Python code 2IonQZD.
Quantum tunneling is a fundamental quantum-mechanical effect involved in plenty of physical phenomena. Its control would impact these phenomena and the technologies based on them. We show that the quantum tunneling probability through a potential barrier can be increased to approach unity in a quantum Zeno dynamics undergone by the tunneling particle in which the direction of the momentum is frequently monitored. We first model the measurements of the momentum direction as selective von Neumann projections, and then as nonselective, direction-sensitive interactions of the particle with probe particles. Nonselective measurements are more efficient than selective measurements in enhancing the quantum tunneling probability.