While the uncertainty principle for linear position and linear momentum, and more recently for angular position and angular momentum, is well established, its radial equivalent has so far eluded researchers. Here we exploit the logarithmic radial position, ln r , and hyperbolic momentum, P H , to formulate a rigorous uncertainty principle for the radial degree of freedom of transverse light modes. We show that the product of their uncertainties is bounded by Planck’s constant, Δ ln r · Δ P H ≥ ℏ / 2 , and identify a set of radial intelligent states that satisfy the equality. We illustrate the radial uncertainty principle for a variety of intelligent states, by preparing transverse light modes with suitable radial profiles. We use eigenmode projection to measure the corresponding hyperbolic momenta, confirming the minimum uncertainty bound. Optical systems are most naturally described in terms of cylindrical coordinates, and our radial uncertainty relation provides the missing piece in characterizing optical quantum measurements, providing a new platform for the fundamental tests and applications of quantum optics.While the uncertainty principle for linear position and linear momentum, and more recently for angular position and angular momentum, is well established, its radial equivalent has so far eluded researchers. Here we exploit the logarithmic radial position, ln r , and hyperbolic momentum, P H , to formulate a rigorous uncertainty principle for the radial degree of freedom of transverse light modes. We show that the product of their uncertainties is bounded by Planck’s constant, Δ ln r · Δ P H ≥ ℏ / 2 , and identify a set of radial intelligent states that satisfy the equality. We illustrate the radial uncertainty principle for a variety of intelligent states, by preparing transverse light modes with suitable radial profiles. We use eigenmode projection to measure the corresponding hyperbolic momenta, confirming the minimum uncertainty bound. Optical systems are most naturally described in terms of cylindrical coordinates, and our radial uncertainty relation provides the missing piece in characterizing optical quantum measurements, providing a new platform for the fundamental tests and applications of quantum optics.

Photonics ResearchSep 1, 20226 citations

Minimum Uncertainty States

Pankaj Sharan

Lecture notes in physicsJan 1, 20230 citations

Viscous control of minimum uncertainty state in hydrodynamics

Tomoi Koide C

Abstract A minimum uncertainty state for position and momentum of a fluid element is obtained. We consider a general fluid described by the Navier–Stokes–Korteweg (NSK) equation, which reproduces the behaviors of a standard viscous fluid, a fluid with the capillary action and a quantum fluid, with the proper choice of parameters. When the parameters of the NSK equation is adjusted to reproduce Madelung’s hydrodynamic representation of the Schrödinger equation, the uncertainty relation of a fluid element reproduces the Kennard and the Robertson–Schrödinger inequalities in quantum mechanics. The derived minimum uncertainty state is the generalization of the coherent state and its uncertainty is given by a function of the shear viscosity. The viscous uncertainty can be smaller than the inviscid minimum value when the shear viscosity is smaller than a critical value which is similar in magnitude to the Kovtun–Son–Starinets bound. This uncertainty reflects the information of the fluctuating microscopic degrees of freedom in the fluid and will modify the standard hydrodynamic scenario, for example, in heavy-ion collisions.Abstract A minimum uncertainty state for position and momentum of a fluid element is obtained. We consider a general fluid described by the Navier–Stokes–Korteweg (NSK) equation, which reproduces the behaviors of a standard viscous fluid, a fluid with the capillary action and a quantum fluid, with the proper choice of parameters. When the parameters of the NSK equation is adjusted to reproduce Madelung’s hydrodynamic representation of the Schrödinger equation, the uncertainty relation of a fluid element reproduces the Kennard and the Robertson–Schrödinger inequalities in quantum mechanics. The derived minimum uncertainty state is the generalization of the coherent state and its uncertainty is given by a function of the shear viscosity. The viscous uncertainty can be smaller than the inviscid minimum value when the shear viscosity is smaller than a critical value which is similar in magnitude to the Kovtun–Son–Starinets bound. This uncertainty reflects the information of the fluctuating microscopic degrees of freedom in the fluid and will modify the standard hydrodynamic scenario, for example, in heavy-ion collisions.

arXiv:2104.11777Journal of Statistical Mechanics Theory and ExperimentFeb 1, 20223 citations

Systems with Infinite Number of Degrees of Freedom

И. Д. Феранчук, A. A. Ivanov, Van-Hoang Le +1 more

Lecture notes in physicsNov 9, 20141 citation

Gaussian states

Alexander Semenovich Holevo

Consider a quantum degree of freedom, e.g., an oscillator, described by the canonical observables q = Q, p = ℏP. The ground state |0)(0| is the minimum-uncertainty state in which Q and P have zero mean values. The state 5.1.1 $\left| {\left. {\overline P ,\overline Q } \right)\left( {\overline Q ,\overline P } \right.} \right| = W_{\overline Q ,\hbar \overline P } \left| {\left. 0 \right)\left( 0 \right.} \right|W_{\overline Q ,\hbar \overline P }^*$ (where to simplify notations we put $$ \left| {\left. {\overline P ,\overline Q } \right)} \right. \equiv \left| {\left. {\overline P ,\overline Q ;{\hbar \mathord{\left/ {\vphantom {\hbar {2\omega }}} \right. \kern-\nulldelimiterspace} {2\omega }}} \right)} \right. $$ ω being the oscillator frequency) can be regarded as the result of an external influence onto the object in ground state, shifting the mean values of the canonical observables but leaving their uncertainties unchanged.Consider a quantum degree of freedom, e.g., an oscillator, described by the canonical observables q = Q, p = ℏP. The ground state |0)(0| is the minimum-uncertainty state in which Q and P have zero mean values. The state 5.1.1 $$ \left| {\left. {\overline P ,\overline Q } \right)\left( {\overline Q ,\overline P } \right.} \right| = W_{\overline Q ,\hbar \overline P } \left| {\left. 0 \right)\left( 0 \right.} \right|W_{\overline Q ,\hbar \overline P }^* $$ (where to simplify notations we put $$ \left| {\left. {\overline P ,\overline Q } \right)} \right. \equiv \left| {\left. {\overline P ,\overline Q ;{\hbar \mathord{\left/ {\vphantom {\hbar {2\omega }}} \right. \kern-\nulldelimiterspace} {2\omega }}} \right)} \right. $$ ω being the oscillator frequency) can be regarded as the result of an external influence onto the object in ground state, shifting the mean values of the canonical observables but leaving their uncertainties unchanged.

Edizioni della Normale eBooksJan 1, 20112 citations

\left| {\left. {\overline P ,\overline Q } \right)\left( {\overline Q ,\overline P } \right.} \right| = W_{\overline Q ,\hbar \overline P } \left| {\left. 0 \right)\left( 0 \right.} \right|W_{\overline Q ,\hbar \overline P }^*

(where to simplify notations we put $$ \left| {\left. {\overline P ,\overline Q } \right)} \right. \equiv \left| {\left. {\overline P ,\overline Q ;{\hbar \mathord{\left/ {\vphantom {\hbar {2\omega }}} \right. \kern-\nulldelimiterspace} {2\omega }}} \right)} \right. $$ ω being the oscillator frequency) can be regarded as the result of an external influence onto the object in ground state, shifting the mean values of the canonical observables but leaving their uncertainties unchanged.