We present a protocol for an atomic interferometer that reaches the Heisenberg Limit (HL), within a factor of ∼ √ 2, via collective state detection and critical tuning of one-axis twist spin squeezing. It generates a Schrödinger cat (SC) state, as a superposition of two extremal collective states. When this SC interferometer is used as a gyroscope, the interference occurs at an ultrahigh Compton frequency, corresponding to a mesoscopic single object with a mass of N m, where N is the number of particles in the ensemble, and m is the mass of each particle. For 87 Rb atoms, with N = 10 6 , for example, the intereference would occur at a Compton frequency of ∼ 2 × 10 31 Hz. Under this scheme, the signal is found to depend critically on the parity of N . We present two variants of the protocol. Under Protocol A, the fringes are narrowed by a factor of N for one parity, while for the other parity the signal is zero. Under Protocol B, the fringes are narrowed by a factor of N for one parity, and by a factor of √ N for the other parity. Both protocols can be modified in a manner that reverses the behavior of the signals for the two parities. Over repeated measurements under which the probability of being even or odd is equal, the averaged sensitivity is smaller than the HL by a factor of ∼ √ 2 for both versions of the protocol. We describe an experimental scheme for realizing such an atomic interferometer, and discuss potential limitations due to experimental constraints imposed by the current state of the art, for both collective state detection and oneaxis-twist squeezing. We show that when the SC interferometer is configured as an accelerometer, the effective two-photon wave vector is enhanced by a factor of N , leading to the same degree of enhancement in sensitivity. We also show that such a mesoscopic single object can be used to increase the effective base frequency of an atomic clock by a factor of N , with a sensitivity that is equivalent to the HL, within a factor of ∼ √ 2.
We investigate the behavior of an ensemble of N non-interacting, identical atoms, excited by a laser. In general, the i-th atom sees a Rabi frequency Ω i , an initial position dependent laser phase φ i , and a motion induced Doppler shift of δ i . When Ω i or δ i is distinct for each atom, the system evolves into a superposition of 2 N intercoupled states, of which there are N + 1 symmetric and (2 N − (N + 1)) asymmetric collective states. For a collective state atomic interferometer (COSAIN) we recently proposed, it is important to understand the behavior of all the collective states under various conditions. In this paper, we show how to formulate the properties of these states under various non-idealities, and use this formulation to understand the dynamics thereof.We also consider the effect of treating the center of mass degree of freedom of the atoms quantum mechanically on the description of the collective states, illustrating that it is indeed possible to construct a generalized collective state, as needed for the COSAIN, when each atom is assumed to be in a localized wave packet. The analysis presented in this paper is important for understanding the dynamics of the COSAIN, and will help advance the analysis and optimization of spin squeezing in the presence of practically unavoidable non-idealities as well as in the domain where the center of mass motion of the atoms is quantized.
We describe a collective state atomic clock (COSAC) with Ramsey fringes narrowed by a factor of √ N compared to a conventional clock -N being the number of non-interacting atoms -without violating the uncertainty relation. This narrowing is explained as being due to interferences among the collective states, representing an effective √ N fold increase in the clock frequency, without entanglement. We discuss the experimental inhomogeneities that affect the signal and show that experimental parameters can be adjusted to produce a near ideal signal. The detection process collects fluorescence through stimulated Raman scattering of Stokes photons, which emits photons predominantly in the direction of the probe beam for a high enough optical density. By using a null measurement scheme, in which detection of zero photons corresponds to the system being in a single collective state, we detect the population in a collective state of interest. The quantum and classical noise of the ideal COSAC is still limited by the standard quantum limit and performs only as well as the conventional clock. However, when detection efficiency and collection efficiency are taken into account, the detection scheme of the COSAC increases the quantum efficiency of detection significantly in comparison to a typical conventional clock employing fluorescence detection, yielding a net improvement in stability by as much as a factor of 10.
In a conventional atomic interferometer employing N atoms, the phase sensitivity is at the standard quantum limit: 1/ √ N . Using spin-squeezing, the sensitivity can be increased, either by lowering the quantum noise or via phase amplification, or a combination thereof. Here, we show how to increase the sensitivity, to the Heisenberg limit of 1/N , while increasing the quantum noise by √ N , thereby suppressing by the same factor the effect of excess noise. The proposed protocol makes use of a Schrödinger Cat state representing a mesoscopic superposition of two collective states of N atoms, behaving as a single entity with an N -fold increase in Compton frequency. The resulting N -fold phase magnification is revealed by using atomic state detection instead of collective state detection. We also show how to realize an atomic clock based on such a Schrödinger Cat state, with an N -fold increase in the effective transition frequency. We also discuss potential experimental constraints for implementing this scheme, using one axis twist squeezing employing the cavity feedback scheme, and show that the effects of cavity decay and spontaneous emission are highly suppressed due to the N -fold phase magnification. We find that even for a modest value of the cavity cooperativity parameter that should be readily accessible experimentally, the maximum improvement in sensitivity is very close to the ideal limit, for as many as ten million atoms.PACS numbers: 06.30. Gv, 03.75.Dg, 37.25.+k Using the fact that Ȯ = tr(ρÔ) for any operatorÔ, it
In a conventional atomic interferometer employing N atoms, the phase sensitivity is at the standard quantum limit: 1 / N . Under usual spin squeezing, the sensitivity is increased by lowering the quantum noise. It is also possible to increase the sensitivity by leaving the quantum noise unchanged while producing phase amplification. Here we show how to increase the sensitivity, to the Heisenberg limit of 1 / N , while increasing the quantum noise by N and amplifying the phase by a factor of N . Because of the enhancement of the quantum noise and the large phase magnification, the effect of excess noise is highly suppressed. The protocol uses a Schrödinger cat state representing a maximally entangled superposition of two collective states of N atoms. The phase magnification occurs when we use either atomic state detection or collective state detection; however, the robustness against excess noise occurs only when atomic state detection is employed. We show that for one version of the protocol, the signal amplitude is N when N is even, and is vanishingly small when N is odd, for both types of detection. We also show how the protocol can be modified to reverse the nature of the signal for odd versus even values of N . Thus, for a situation where the probability of N being even or odd is equal, the net sensitivity is within a factor of 2 of the Heisenberg limit. Finally, we discuss potential experimental constraints for implementing this scheme via one-axis-twist squeezing employing the cavity feedback scheme, and show that the effects of cavity decay and spontaneous emission are highly suppressed because of the increased quantum noise and the large phase magnification inherent to the protocol. As a result, we find that the maximum improvement in sensitivity can be close to the ideal limit for as many as 10 million atoms.
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