Enhancing Sensitivity of an Atomic Interferometer to the Heisenberg Limit Using Increased Quantum Noise


Quantum metrology has been among the most vigorous branches of quantum technology. It involves using quantum effects to achieve better estimation of parameters of a physical sys- tem. Conventionally the system consists of an ensemble of N non-interacting atoms and the measurements are done on individual atomic states, such as a conventional Raman atomic interferometer (CRAIN) or a Raman-Ramsey atomic clock (RRAC). The measurement sen- sitivity of such a system is restricted by the standard quantum limit (SQL), which scales √ 3 as ∼ N (this is known as the shot-noise scaling). Introducing quantum entanglement in the system, it is possible to surpass the SQL, and a key goal in this context is to achieve the Heisenberg limit (HL), which scales as ∼ N (this is know as the Heisenberg scaling), √ representing an improvement by a factor of N. In this thesis, we propose a protocol that can enhance the measurement sensitivity of an atomic interferometer (or an atomic clock) to the HL, while at the same time make it substantially robust against excess noise present in the system. Specifically, the protocol employs critically tuned one-axis-twist (OAT) spin squeezing to generate maximially-entangled states among the atoms, known as Schr ̈odinger 4 cat (SC) states, in combination with the conventional detection (CD) scheme (measurements of individual atomic states). Since the interferometer makes use of Schr ̈odinger cat states, we name it as Schro ̈dinger cat atomic interferometer (SCAIN). A SCAIN relies on the collective behavior of the atomic ensemble, so as a first step, we investigate the behavior of an ensemble of N non-interacting, identical, two-level atoms, excited by the same laser field. Traditionally, the ensemble would be described using direct product states as the basis. In his seminal paper, R. H. Dicke proposed an alternative basis known as collective states, and showed that under ideal conditions, the dynamics of the system can be confined within the N + 1 sysmmetric collective states (also known as Dicke collective states), labeled as {|E0⟩ , |E1⟩ , . . . , |EN ⟩}, with all the other 2N − (N + 1) asym- metric collective states decoupled from the system. This simplifies greatly the descriptions of the ensemble. Furthermore, the collective state descriptions suggest a new detection scheme for the ensemble, named as collective state detection (CSD), in addition to the conventional detection scheme. The CSD can be applied to both an atomic interferometer, referred to as collective state atomic interferometer (COSAIN), and an atomic clock, referred to as collective state atomic clock (COSAC). We show that the fringe widths of a COSAIN and √ a COSAC are narrowed by a factor of CRAIN and RRAC, respectively, despite the fact that the measurement sensitivity remains the same at the SQL for both cases. With the model of collective state descriptions in place, we move to the key part of the protocol for realizing a SCAIN and review the concepts of spin squeezing. We first present the spin representation of the ensemble, where collective spins are defined for the system. We then introduce the coherent spin states (CSSs), which are direct product states of individual coherent states and which turn out to be equivalent to the Dicke collective N compared to their conventional counterparts, 5 states in this picture. Lastly based on CSSs, we define the spin squeezed states (SSSs) using the definitions proposed by Kitagawa and Ueda, and summarize the two approaches for generating squeezed spin states: one-axis-twist (OAT) and two-axis-counter-twist (TACT) spin squeezing. Before we get to the formal descriptions of the protocol for a SCAIN, we revisit the Sagnac effect that is essential for using the SCAIN for rotation sensing. We propose two alternative models for deriving and interpreting the Sagnac effect. The first one is based on Lorentz transformation of special relativity and can be generalized to an interferometer of an arbitrary shape, while the second one is a quantum-mechanical model which shows that the total effect can be split equally during each of the two dark zones of the interferometer. Finally, we present the detailed protocol for implementing a SCAIN. The core components of the protocol are the four pulses employed to implement the squeezing, rotation, inverse- rotation and unsqueezing operations, in addition to the usual π/2-dark-π-dark-π/2 pulse sequence of its conventional counterpart. The squeezing effect is controlled by the squeezing parameter μ, which indicates the length of interaction of the squeezing Hamiltonian. A squeezing pulse with μ = π/2 will split an initial CSS into equal superpositions of two extremal collective states. A rotation pulse is then applied to rotate the mean spin directions of those extremal collective states and align them with the z-axis, generating the SC state. After the second dark zone, an inverse-rotation pulse and an unsqueezing pulse are applied sequentially to undo the previous rotation and squeezing effects in order to extract the phase imprinted during the dark zones. One difficulty of this protocol is that the splitting of the CSS after the squeezing pulse depends on the parity of N, which requires rotation around different axes to produce the SC state. This leads to two equivalent versions of the protocol (protocol A and protocol B), depending on the axis around which the rotation pulse 6 is applied. At the end of the interferometer, the signal can be measured using either the collective state detection scheme, or the conventional detection scheme, where the former is referred to as CSD-SCAIN, while the latter as CD-SCAIN. For each detection scheme, we examine the signal fringes and the measurement sensitivity for different values of the squeezing parameter μ. We show that for both detection schemes, when μ = π/2, the signal fringes are narrowed by a factor of N for one of the two parities, and the measurement sensitivity reaches the HL for that same parity of N. When averaged over the two parities, √ the overall sensitivity is below the HL by a factor of 2. Despite the fact that both schemes can reach the HL, the CD-SCAIN can provide additional robustness against excess noise, due to the increase in the standard deviation of the signal. We compare and summarize the robustness against excess noise of different protocols proposed for atomic interferometers (and clocks) in the last chapter of the thesis. Lastly, we note that the core components of the protocol for SCAIN consisting of squeezing, rotation, inverse-rotation and unsqueezing operations can also be applied to atomic clocks and atomic accelerometers, which leads to similar results as SCAIN.

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