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arXiv:2405.15704 · Physical Review A

Quantum nondemolition measurement operator with spontaneous emission

Ebubechukwu O. Ilo-Okeke, Tim Byrnes

Published Jun 25, 2024• 4 citations
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Abstract

We present a theory for quantum nondemolition (QND) measurements of an atomic ensemble in the presence of spontaneous emission. We derive the master equation that governs the evolution of the ground state of the atoms and the quantum state of light. Solving the master equation exactly without invoking the Holstein-Primakoff approximation and projecting out the quantum state of light, we derive a positive operator-valued measure that describes the QND measurement. We show that, at high spontaneous emission conditions, the QND measurement has a unique dominant state to which the measurement collapses. We additionally investigate the behavior of the QND measurement in the limiting case of strong atom-light interactions, where we show that the positive operator valued measure becomes a projection operator. We further analyze the effect of spontaneous emission noise on atomic state preparation. We find that it limits the width of the eigenvalue spectrum available to a quantum state in a linear superposition. This effect leads to state collapse on the dominant state. We generate various nonclassical states of the atom by tuning the atom-light interaction strength. We find that nonclassical states such as the Schr\"odinger-cat state, whose coherence spans the entire eigenvalue spectrum of the total spin operator ${J}_{z}$ for a given spin eigenvalue $J$, lose their coherence because spontaneous emission limits the accessibility of states farther away from the dominant state.

DOI
10.1103/physreva.109.063711

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We present a theory for quantum nondemolition (QND) measurements of an atomic ensemble in the presence of spontaneous emission. We derive the master equation that governs the evolution of the ground state of the atoms and the quantum state of light. Solving the master equation exactly without invoking the Holstein-Primakoff approximation and projecting out the quantum state of light, we derive a positive operator-valued measure that describes the QND measurement. We show that, at high spontaneous emission conditions, the QND measurement has a unique dominant state to which the measurement collapses. We additionally investigate the behavior of the QND measurement in the limiting case of strong atom-light interactions, where we show that the positive operator valued measure becomes a projection operator. We further analyze the effect of spontaneous emission noise on atomic state preparation. We find that it limits the width of the eigenvalue spectrum available to a quantum state in a linear superposition. This effect leads to state collapse on the dominant state. We generate various nonclassical states of the atom by tuning the atom-light interaction strength. We find that nonclassical states such as the Schr"odinger-cat state, whose coherence spans the entire eigenvalue spectrum of the total spin operator Jz{J}_{z}Jz​ for a given spin eigenvalue JJJ, lose their coherence because spontaneous emission limits the accessibility of states farther away from the dominant state.
We derive a measurement operator corresponding to a quantum nondemolition (QND) measurement of an atomic ensemble. The quantum measurement operator takes the form of a positive operator valued measure (POVM) and is valid for arbitrary interaction times, initial coherent state amplitudes, and final photon measurement outcomes. We analyze the dependence on various parameters and show that the effect of the QND measurement for short interaction times is to apply a Gaussian modulation of the initial state wavefunction. We derive approximate expressions for the POVM in various limits, such as the short interaction time regime and projective measurement limit. Several examples are shown, which show how spin squeezing and Schrodinger cat states can be generated using the measurement.
We develop a measurement operator formalism to handle quantum nondemolition (QND) measurement induced entanglement generation between two atomic gases. We first derive how the QND entangling scheme reduces to a positive operator valued measure (POVM), and consider its limiting case when it is a projection operator that collapses the state to a total spin projection state. We then analyze how a stroboscopic sequence of such projections made in the x and z basis evolves the initial wavefunction. We show that for this projection sequence there exists a unique basis where the state evolution can be simplified. This allows one to derive the exact state evolution in a highly efficient manner. Our formalism does not use the Holstein-Primakoff approximation as is conventionally done, and treats the spins of the atomic gases in an exact way.
We develop a measurement operator formalism to handle quantum nondemolition (QND) measurement induced entanglement generation between two atomic gases. We first derive how the QND entangling scheme reduces to a positive operator-valued measure, and consider its limiting case when it can be used to construct a projection operator that collapses the state to a total spin projection state. We then analyze how a stroboscopic sequence of such projections made in the xxx and zzz basis evolves the initial wave function. Such a sequence of QND projections can enhance the entanglement between the atomic ensembles and makes the state converge towards a highly entangled state. We show several mathematical identities which greatly simplify the state evolution in the projection sequence and allow one to derive the exact state in a highly efficient manner. Our formalism does not use the Holstein-Primakoff approximation as is conventionally done, and treats the spins of the atomic gases in an exact way.
We show that continuous quantum nondemolition (QND) measurement of an atomic ensemble is able to improve the precision of frequency estimation even in the presence of independent dephasing acting on each atom. We numerically simulate the dynamics of an ensemble with up to N=150 atoms initially prepared in a (classical) spin coherent state, and we show that, thanks to the spin squeezing dynamically generated by the measurement, the information obtainable from the continuous photocurrent scales superclassically with respect to the number of atoms N. We provide evidence that such superclassical scaling holds for different values of dephasing and monitoring efficiency. We moreover calculate the extra information obtainable via a final strong measurement on the conditional states generated during the dynamics and show that the corresponding ultimate limit is nearly achieved via a projective measurement of the spin-squeezed collective spin operator. We also briefly discuss the difference between our protocol and standard estimation schemes, where the state preparation time is neglected.
We study the effects of optical and atomic decoherence in entangled atomic ensembles produced via quantum nondemolition (QND) measurements. We examine potentially experimentally detrimental effects, such as optical phase diffusion, photon loss and gain, and atomic dephasing. For the optical decoherence channels, we use the technique of integration within ordered operators to obtain the associated Kraus operators. We analyze the effect of different decoherence channels on various quantities, such as the variances of the spin operators, entanglement and correlation criteria, logarithmic negativity, and the Bell–CHSH inequality. We generally find a smooth decay of correlations and entanglement in the presence of decoherence. In the short interaction time range, we find that various quantities show signals consistent with, and showing that entanglement exists under all three types of decoherence. Our results show that QND measurements are one of the most promising methods for entanglement generation between two Bose–Einstein condensates.
Schrödinger cat states, representing coherent superpositions of macroscopically distinguishable states, are indispensable nonclassical resources for continuous-variable quantum information processing. Existing generation protocols typically rely on strong nonlinear interactions, complicated control techniques, or engineered dissipation, posing challenges for experimental implementation. Here, we propose a simple measurement-based protocol for generating Schrödinger cat states in a cavity-QED system by combining coherent driving, dispersive atom--cavity interactions, and atomic postselection. The atom--cavity interaction establishes coherent correlations between the atomic and photonic degrees of freedom, while the subsequent atomic postselection projects the cavity field onto a non-Gaussian superposition state with pronounced Wigner negativity. Numerical simulations based on the Lindblad master equation show that the generated Schrödinger cat states remain robust against moderate cavity dissipation. Our results demonstrate that conditional atomic measurements provide an effective and experimentally accessible approach for preparing nonclassical cavity states without relying on strong optical nonlinearities or engineered dissipation.