An atom interferometric gyroscope (AIG) made with an uncorrelated ensemble of N two-level atoms, rotating at a rate ΩG about an axis normal to the area Θ accrues a phase ϕ = 2ωCΘΩG/c2 due to the Sagnac effect. Here ωC is the Compton frequency of the atoms used, and c is the speed of light in vacuum. The rotation sensitivity of such an AIG is restricted by the standard quantum limit (SQL), δΩG = c2/2ωCΘ√N. This is a direct consequence of the Heisenberg uncertainty principle. Introducing entanglement in the system can enhance this precision upto the fundamental Heisenberg limit, ∆ϕ = c2/2ωCΘN. Essentially, this can be interpreted as an AIG with a single particle of Compton frequency NωC. Motivated by this, we explore the use of large number of particles treated as a single entity, first without entanglement in a collective state atomic interferometer (COSAIN), and then under spin squeezing in a Schrödinger cat atomic interferometer (SCAIN). As a first step towards achieving this goal, we investigate the behavior of an ensemble of N non-interacting, identical atoms, excited by a laser with a wavelength of λ. In his seminal paper, R. H. Dicke showed that such an ensemble evolves into a superposition of N + 1 symmetric states, {|E0⟩,|E1⟩,...,|EN⟩} under conditions that each atom experiences identical motion induced Doppler shift, and Rabi frequency. We show that when these ideal conditions are breached, i.e., the i-th atom experiences Rabi frequency Ωi, and Doppler shift δi, the ensemble evolves into a superposition of N + 1 symmetric, as well as 2N − (N + 1) asymmetric states. For a large value of N , the number of asymmetric states is far greater than that of the symmetric states. It is important to understand the behavior of all the collective states under various non-idealities for the realization of a COSAIN, as well as a collective state atomic clock (COSAC) - a device based on similar principles. In this thesis, we show first how to formulate the properties of all the collective states under various non-idealities, and use this formulation to understand the dynamics thereof. We show that the collective states corresponding to the absorption of a given number of photons can be visualized as an abstract, multi-dimensional rotation in the Hilbert space spanned by the ordered product states of individual atoms. 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. In particular, we show 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. Based on the model of collective states developed thus far, we describe the COSAIN with the signal fringe as a function of ϕ, and therefore, ΩG narrowed by √N compared to a conventional Raman atomic interferometer (CRAIN). This effect arises from the interferences among the collective states, and is a manifestation of interference at 2πωC = 1040 Hz, de Broglie wavelength of 4.5 × 10−15 m, for N = 106 and v = 1 m/s. The population of the collective state of interest is detected by a null measurement scheme, in which an event corresponding to detection of zero photons corresponds to the system being in that particular collective state. The signal is detected by collecting fluorescence through stimulated Raman scattering of Stokes photons, which are emitted predominantly against the direction of the probe beam, for a high enough resonant optical density. The sensitivity of the ideal COSAIN is found to be given by the SQL. However, when detection efficiency and collection efficiency are taken into account, the detection scheme of the COSAIN increases the quantum efficiency of detection significantly in comparison to a typical CRAIN employing fluorescence detection, yielding a net improvement in stability by as much as a factor of 10. We discuss how the inhomogeneities arising from the nonuniformity in experimental parameters affect the COSAIN signal. We also describe an alternate experimental scheme to enhance resonant optical density in a COSAIN by using cross-linearly polarized counter-propagating Raman beams. Finally, we explore the application of spin squeezing echo to surpass the SQL, and achieve Heisenberg scaling of rotation sensitivity in an AIG. We first review the spin representation of two-level atoms, and Coherent Spin States (CSS) which are equivalent of the Dicke collective states in this picture. The quantum fluctuations of a CSS are isotropic in the plane orthogonal to the direction of the mean spin. The application of spin squeezing to correlate the individual spins via a nonlinear interaction suppresses the quantum fluctuations along an orthogonal axis (to the mean spin) while inflating that along the third axis, generating Squeezed Spin States. We describe the SCAIN, which is a COSAIN with Heisenberg scaling of phase sensitivity, enhanced by the application of squeezing echo. Explicitly, we employ what is known as the one axis twisting (OAT) spin squeezing on an initial CSS followed by a perturbation, at the end of which we seek to reverse the squeezing by switching the sign of the nonlinear interaction. In practice, this encourages increased interference between states with higher contrast in Compton frequency, resulting in the narrowing of signal fringe width by a factor ∼ N. The parameter of squeezing µ, which indicates length of interaction, dictates the signal fringe width, and therefore, δΩG. However, this technique inherently depends on whether N is even or odd. For large ensembles, it is virtually impossible to determine this a priori. The protocol we describe here eliminates this complexity. At µ = π, the well known Greenberger–Horne–Zeilinger (GHZ) states are obtained for even values of N. These states are pure cat states of an equal superposition of |E0⟩ and |EN⟩. The signal generated from the echo protocol in this regime is narrowed by a factor of N. On the contrary, odd values of N produce a null signal. In an actual experiment, the total signal is averaged over multiple runs, and leads to a signal halved in amplitude than what is expected for even N. The rotation sensitivity of the SCAIN in this regime is given by the Heisenberg limit. For values of µ < π, both odd and even N generate almost identical signal fringes. We show that the SCAIN can attain a rotation sensitivity lower than the Heisenberg limit by a factor of e−1/3 for interaction times significantly less than µ = π.

Last modified

• 02/21/2018

Creator

• Resham Sarkar

DOI

• https://doi.org/10.21985/N2QD6F

Subject

• Physics and Astronomy

Keyword

• Schrödinger cat states
• Atomic Interferometer
• Heisenberg limit
• Collective States

Date created

• 2016-01-01

Resource type

• Dissertation

Rights statement

• In Copyright