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arXiv:1003.0157 · Physical Review A

Spin-squeezing and Dicke-state preparation by heterodyne measurement

Thomas Vanderbruggen, Simon Bernon, Andréa D Bertoldi, Arnaud Landragin, Philippe Bouyer

Published Jan 26, 2011• 23 citations
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Abstract

We investigate the quantum nondemolition (QND) measurement of an atomic population based on a heterodyne detection and show that the induced back-action allows for the preparation of both spin-squeezed and Dicke states. We use a wave-vector formalism to describe the stochastic process of the measurement and the associated atomic evolution. Analytical formulas of the atomic distribution momenta are derived in the weak-coupling regime both for short- and long-time behavior, and they are in good agreement with those obtained by a Monte Carlo simulation. The experimental implementation of the proposed heterodyne detection scheme is discussed. The role played in the squeezing process by the spontaneous emission is considered.

DOI
10.1103/physreva.83.013821

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We investigate the collapse of a coherent superposition of two-level atoms induced by a single-photon heterodyne quantum non-demolition (QND) measurement, and show that such a measurement process leads to the generation of spin-squeezed and Dicke states. We describe the stochastic process of the measurement and the associated atomic evolution in a wavevector formalism. Analytical formulas of the atomic distribution momenta are derived in the weak coupling regime for both short and long time behavior and are in good agreement with those from a Monte-Carlo simulation.
We propose a scheme for producing entangled spin-squeezed states of an atomic ensemble inside an optical cavity by backaction of a detuned quantum-nondemolition (QND) measurement. By illuminating the atoms with bichromatic light, an interaction Hamiltonian of the cross-Kerr effect between the cavity and atoms is generated to implement QND measurements. The feedback effect is obtained through mixing the phase and amplitude quadratures of the cavity field, due to the detuning of the optical cavity. Therefore the continuous nondemolition measurements are fed back to correct the quantum state of the atomic sample such that unconditional spin squeezing is produced without requiring the use of any external electronics.
A weak measurement approach is proposed to entangle and squeeze atoms. We show that even for very small coupling strength between light and atoms, one can achieve large squeezing unattainable with normal measurement-based squeezing. Post-selecting the photons interacting with spins via a weak off-resonant quantum nondemolition interaction (QND) in a state nearly orthogonal to its initial state can strengthen the entanglement among elementary spins and thus enhance spin squeezing. The squeezing process is probabilistic but the created state is unconditionally determined. Further control of the post-selection parameter and the detection process can transform the QND-like spin squeezing into the one-axis-twisting or even two-axis-twisting type.
We propose an experimentally accessible procedure for conditional preparation of highly nonclassical states of collective spin of an atomic ensemble. The quantum state engineering is based on a combination of quantum nondemolition interaction between atoms and light previously prepared in a non-Gaussian state using photon subtraction from squeezed vacuum beam, homodyne detection on the output light beam, and a coherent displacement of atomic state. The procedure is capable of nondeterministic preparation of a wide class of superpositions of atomic Dicke states. We present several techniques to optimize the performance of the protocol and maximize the trade off between fidelity of prepared state and success probability of the scheme.
We analyze a scheme that uses quantum nondemolition measurements to induce squeezing of a two-mode Bose-Einstein condensate in a double-well trap. In a previous paper [E. O. Ilo-Okeke, S. Sunami, C. J. Foot, and T. Byrnes, Phys. Rev. A 104, 053324 (2021)], we introduced a model to solve exactly the wave function for all atom-light interaction times. Here, we perform approximations for the short interaction time regime, which is relevant for producing squeezing. Our approach uses a Holstein-Primakoff approximation for the atoms while we treat the light variables exactly. It allows us to show that the measurement induces correlations within the condensate, which manifest in the state of the condensate as a superposition of even-parity states. In the long interaction time regime, our methods allow us to identify the mechanism for loss of squeezing correlation. We derive simple expressions for the variances of atomic spin variables conditioned on the measurement outcome. We find that the results agree with the exact solution in the short interaction time regime. Additionally, we show that the expressions are the sum of the variances of the atoms and the measurement. Beyond the short interaction time regime, our scheme agrees qualitatively with the exact solution for the spin variable that couples to light.
We examine how nondestructive measurements generate spin squeezing in an atomic Bose-Einstein condensate confined in a double-well trap. The condensate in each well is monitored using coherent light beams in a Mach-Zehnder configuration that interacts with the atoms through a quantum nondemolition Hamiltonian. We solve the dynamics of the light-atom system using an exact wave-function approach, in the presence of dephasing noise, which allows us to examine arbitrary interaction times and a general initial state. We find that monitoring the condensate at zero detection current and with identical coherent light beams minimizes the backaction of the measurement on the atoms. In the weak atom-light interaction regime, we find the mean spin direction is relatively unaffected, while the variance of the spins is squeezed along the axis coupled to the light. Additionally, squeezing persists in the presence of tunneling and dephasing noise. For long atom-light interaction time, we equally show our methods can be used to prepare non-Gaussian correlated spin states.