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arXiv:2209.07915 · Physical Review A

Hybrid approximation approach to the generation of atomic squeezing with quantum nondemolition measurements

Ebubechukwu O. Ilo-Okeke, Manikandan Kondappan, Ping Chen, Yuping Mao, Valentin Ivannikov, Tim Byrnes

Published May 8, 2023• 3 citations
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Abstract

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.

DOI
10.1103/physreva.107.052604

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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.
ABSTRACT: We theoretically study a scheme for generating squeezing between two Bose-Einstein condensates (BECs). The scheme involves placing two BECs in the path of a Mach-Zehnder interferometer, where the coherent light interacts with the atoms due to the ac Stark shift. The most relevant regime for producing squeezing is in the short time regime, defined as when the interaction time scales as the inverse square root of the atom number. In this regime it is possible to construct a very simple approximate theory for the overall effect of the scheme: amplitudes in the superposition between the two BECs with unequal spin eigenvalues are damped. We analyze the types of correlations, entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell correlations that are produced and show that the state is similar to a spin-EPR state. Using a two-pulse sequence the correlations can be dramatically improved, where the state further approaches a spin-EPR state.
We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin, and Polzik [Nature (London) 413, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein--Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find that this produces a state with characteristics similar to those of a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the characteristics of the two-mode squeezed state start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock-state probability distributions, Husimi QQQ distributions, and nonlocal entanglement between the ensembles. We compare and connect several quantities obtained by using the Holstein--Primakoff approach and our exact time evolution methods.
We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin and Polzik [Nature {\bf 413}, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein-Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find this produces a state with similar characteristics as a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the two-mode squeezed state characteristics start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock state probability distributions, Husimi QQQ distributions, and non-local entanglement between the ensembles. We compare and connect several quantities obtained using the Holstein-Primakoff approach and our exact time evolution methods.
Abstract We theoretically study a scheme for generating entanglement between two Bose–Einstein condensates (BECs). The scheme involves placing two BECs in the path of a Mach–Zehnder interferometer, where the coherent light interacts with the atoms due to a quantum nondemolition Hamiltonian. In contrast to standard approaches where a Holstein–Primakoff approximation is used, we use an exact wavefunction approach where the atoms can be initialized in an arbitrary state and the light–atom interaction times can be arbitrary. In the short time regime, it is possible to construct a very simple approximate theory for the overall effect of the scheme: amplitudes in the superposition between the two BECs with unequal spin eigenvalues are damped. We analyze the types of correlations, entanglement, Einstein–Podolsky–Rosen (EPR) steering, and Bell correlations that are produced and show that the state is similar to a spin-EPR state. Using a two-pulse sequence the correlations can be dramatically improved, where the state further approaches a spin-EPR state.
We study the robustness of quantum nondemolition (QND) measurement induced entanglement between Bose-Einstein Condensates (BECs). We consider an experimental scheme where two BECs are placed in the paths of a Mach-Zehnder interferometer, and a QND interaction creates entanglement between coherent light and the atoms. We analyze the two dominant channels of decoherence, atomic dephasing and photon loss on the entangled states produced by this scheme. We calculate the effect of dephasing on the variance and expectation values of the spin operators, entanglement, and correlation criteria. Our analysis does not use the Holstein-Primakoff approximation and is capable of modeling long light-atom interaction times, producing non-Gaussian states beyond the two-mode squeezed states. In the presence of dephasing, the entangled states are robust in the macroscopic limit as long as the dimensionless interaction time is less than $ 1/\sqrt{N}$, where $ N $ is the number of atoms in the BEC. For photon loss, the entangled states generated by long interaction times show a remarkable robustness that makes the scheme promising for various quantum information applications.