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arXiv:2110.09043 · Physical Review A

Stroboscopic quantum nondemolition measurements for enhanced entanglement generation between atomic ensembles

Manish Chaudhary, Yuping Mao, Manikandan Kondappan, Amiel S. P. Paz, Valentin Ivannikov, Tim Byrnes

Published Feb 28, 2022• 7 citations
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Abstract

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 $x$ and $z$ 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.

DOI
10.1103/physreva.105.022443

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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 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 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 scheme to prepare a macroscopic maximally entangled state between two atomic ensembles using adaptive quantum nondemolition (QND) measurements. The quantum state of the system is evolved using a sequence of QND measurements followed by adaptive unitaries such that the desired measurement outcome is obtained with asymptotically unit probability. This procedure is repeated in zzz- and xxx-spin bases alternately such that the state converges deterministically towards the maximally entangled state. Up to a local spin-basis rotation, the maximally entangled state has zero total spin angular momentum, i.e., it is a singlet state. Our protocol does not perform postselection and works beyond the Holstein-Primakoff regime for the atomic spin degrees of freedom, producing genuine macroscopic entanglement.
We develop a scheme to prepare a macroscopic maximally entangled state (MMES) between two atomic ensembles using adaptive quantum nondemolition (QND) measurements. The quantum state of the system is evolved using a sequence of QND measurements followed by adaptive unitaries, such that the desired measurement outcome is obtained with asymptotically unit probability. This procedure is repeated in z and x spin basis alternately such that the state converges deterministically towards the maximally entangled state. Up to a local spin-basis rotation, the maximally entangled state has zero total spin angular momentum, i.e. it is a singlet state. Our protocol does not perform postselection and works beyond the Holstein-Primakoff regime for the atomic spin degrees of freedom, producing genuine macroscopic entanglement.