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arXiv:quant-ph/0306205 · Physical Review A

Spin squeezing via atom-cavity field coupling

Claudiu Genes, P. R. Berman, Alberto G. Rojo

Published Oct 8, 2003• 36 citations
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Abstract

Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, ${\ensuremath{\alpha}}^{2},$ initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing $\ensuremath{\alpha},$ even though the field more closely resembles a classical field for which no spin squeezing could be achieved.

DOI
10.1103/physreva.68.043809

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Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, \ensuremathα2,{\ensuremath{\alpha}}^{2},\ensuremathα2, initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing \ensuremathα,\ensuremath{\alpha},\ensuremathα, even though the field more closely resembles a classical field for which no spin squeezing could be achieved.
The interaction of classical and quantized electromagnetic fields with an ensemble of atoms in an optical cavity is considered. Four fields drive a double-lambda level scheme in the atoms, consisting of a pair of lambda systems sharing the same set of lower levels. Two of the fields produce maximum coherence, rho12 = -1/2, between the ground state sublevels 1 and 2. This pumping scheme involves equal intensity fields that are resonant with both the one and two-photon transitions of the lambda system. There is no steady-state absorption of these fields, implying that the fields induce a type electromagnetically-induced transparency (EIT) in the medium. An additional pair of fields interacting with the second lambda system, combined with the EIT fields, leads to squeezing of the atom spin associated with the ground state sublevels. Our method involves a new mechanism for creating steady-state spin squeezing using an optical cavity. As the cooperativity parameter C is increased, the optimal squeezing varies as C^{-1/3}. For experimentally accessible values of C, squeezing as large as 90% can be achieved.
We study interaction between a pair of indistinguishable two-level atoms and the single-mode cavity field. It is supposed that the pair of two-level atoms is laser cooled and trapped into the ground vibrational state, in which vibrational quantum number ⟨nv⟩=0\langle n_v\rangle =0⟨nv​⟩=0 . Two Jaynes–Cummings models are investigated. One is the Jaynes–Cummings model with intensity-dependent coupling and the another is the two-photon Jaynes–Cummings model of a pair of indistinguishable two-level atoms. It should be noted that in the present model, at initial moment t=0t=0t=0 , in intensity-dependent Jaynes–Cummings model the cavity field is prepared in Holstein–Primakoff SU(1,1) CS, while in two-photon Jaynes–Cummings model it is prepared in the squeezed vacuum state. Moreover, at initial moment t=0t=0t=0 , pair of atoms is supposed to be in the first excited state ∣e1⟩\vert e_1\rangle ∣e1​⟩ in both models. By using exact analytical solutions for state-vectors of the coupled atom-field systems amplitude-squared squeezing of the quantized cavity field is examined as a function of the ∣ξ∣\vert \xi \vert ∣ξ∣ parameter. In this situation, in both models higher-order squeezing has the tendency towards oscillations, but exact periodicity of these oscillations is violated by the analogy with the second-order squeezing.
The k-photon Jaynes-Cummings model is studied with respect to the properties of the radiation mode. In particular, the generation of coherence and squeezing in the one- and two-photon model is examined for the case that the field is initially in the vacuum state and the atom is prepared in a coherent superposition of states. Coherence is found in the one-photon model, and squeezing appears in both the one- and the two-photon model, the maximum noise reduction being 25% and 45%, respectively. In the one-photon model coherent squeezed radiation is generated, whereas the two-photon model yields a squeezed vacuum. For an initial atomic preparation close to the ground state, this squeezed vacuum may be interpreted as the result of two-photon interaction between the quantized cavity field and a classical atomic current.
Using the well-known solution for a two-level atom interacting with a single mode of radiation (Jaynes-Cummings model), we reanalyze the possibility of squeezing. We found that if only the initial coherent state has a mean photon number exceeding 10 (for a resonant case) significant squeezing can be achieved. Its strength can be arbitrarily large for increasing average photon number. Finally, we solve our model in the situation when the em field is damped and analyze the impact of damping on both squeezing and revival phenomena.
By utilizing our previous operator solution [17], we have investigated the squeezing in the radiation field of the Tavis-Cummings model (collective N ≫ 1 two-level atoms interacting with a resonant single cavity quantized mode). With field and atoms initially in coherent field state strong or weak and atomic coherent state (of few excited atoms), periodic time-dependent squeezing in the field and the macroscopic polarization is expressed in terms of Jacobian elliptic functions of the first kind. The statistical investigations are carried out for the quasiprobability distribution functions (Wigner function and Q function). The distribution function of the field quadrature has a variance less (greater) than that for a coherent state if this quadrature is squeezed (unsqueezed).
Squeezing of a coherent light field in a cavity by repeated interaction with atoms is studied in the resonance Jaynes-Cummings model. The least quadrature variance \ensuremath{\delta} is calculated for different values of N (mean photon number), t (interaction interval for each time), and K (number of times of interaction). In the one-photon-transition case, \ensuremath{\delta} could be reduced by this process to a certain extent for suitably chosen t. The squeezing becomes worse if K is too large. \ensuremath{\delta} as a function of t and K (for fixed N) has a minimum, which becomes smaller when N increases. The situation of the two-photon-transition case is similar, the only new feature is that \ensuremath{\delta}(K,t) varies with t periodically when N is large. With a favorably chosen value of t (half of the period), analytical formulas for \ensuremath{\delta}(N,K) are obtained in the large-N approximation. These formulas show definitely, both for atoms initially in the upper or lower level, the least quadrature variance \ensuremath{\delta} goes down when K increases up to a quite large value (for atoms in the upper level it is of order N3/2{\mathit{N}}^{3/2}N3/2, while for atoms in the lower level it is of order N) and the achievable minimum value of ${\mathrm{\ensuremath{\delta}}}^{2}$ is of order 1/ \ensuremath{\surd}N . Moreover, the light fields so obtained are nearly in pure states with minimum uncertainty product ${\mathrm{\ensuremath{\delta}}}_{1}$${\mathrm{\ensuremath{\delta}}}_{2}$=1/4.