Two-mode quantum interferometry with a single-mode Fock state and parity detection

Photon counting measurements are analyzed for obtaining a classical phase parameter in linear Mach Zehnder interferometer (MZI), by the use of phase estimation theories. The detailed analysis is made for four cases: a) Coherent states inserted into the interferometer. b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port. c) Coherent state inserted into one input port of the interferometer and squeezed-vacuum state in the other input port. d) Exchanging the first beam-splitter (BS1) of a MZI by a non-linear system which inserts a NOON state into the interferometer and by using photon counting for parity measurements. The properties of photon counting for obtaining minimal phase uncertainties for the above special cases and for the general case are discussed.

The Mach-Zehnder interferometer is a powerful device for detecting small phase shifts between two light beams. Simple input states, such as coherent states or single photons, can reach the standard quantum limit of phase estimation, while more complex states can be used to reach Heisenberg scaling; the latter, however, require challenging preparation and measurement strategies. The quest for highly sensitive phase estimation therefore calls for interferometers with nonlinear devices which would make the preparation of these complex states more efficient. Here, we show that the Heisenberg scaling can be recovered with simple input states (including Fock and coherent states) when the linear mirrors in the interferometer are replaced with controlled-swap gates and measurements on auxiliary qubits. These swap tests project the input Fock and coherent states onto NOON and entangled coherent states, respectively, and allow optimal or near-optimal measurements, leading to improved sensitivity to small phase shifts in one of the interferometer arms. We analyze auxiliary qubit errors in detail, showing that biasing the qubit towards phase flips offers a great advantage, and perform thorough numerical simulations of a possible implementation in circuit quantum electrodynamics with an auxiliary Kerr-cat qubit. Our results thus present a viable approach to phase estimation approaching Heisenberg-limited sensitivity and demonstrate potential advantages of using biased-noise qubits in quantum metrology. Published by the American Physical Society 2024

Parity-based estimation in an SU(1,1) interferometer with photon-subtracted squeezed vacuum states

We calculate the second-order moments, the Robertson-Schrödinger uncertainty product, and the Mandel factor for various superpositions of coherent phase states with opposite arguments, comparing the results with similar superpositions of the usual (Klauder-Glauber-Sudarshan) coherent states. We discover that the coordinate variance in the analog of even coherent states can show the most strong squeezing effect, close to the maximal possible squeezing for the given mean photon number. On the other hand, the Robertson-Schrödinger (RS) uncertainty product in superpositions of coherent phase states increases much slower (as function of the mean photon number) than in superpositions of the usual coherent states. A nontrivial behavior of the Mandel factor for small mean photon numbers is discovered in superpositions with unequal weights of two components. An exceptional nature of the even and odd superpositions is demonstrated.

A nonlinear phase shift is introduced to a Mach-Zehnder interferometer (MZI), and we present a scheme for enhancing the phase sensitivity. In our scheme, one input port of a standard MZI is injected with a coherent state and the other input port is injected with one mode of a two-mode squeezed-vacuum state. The final interference output of the MZI is detected with the method of active correlation output readout. Based on the optimal splitting ratio of beam splitters, the phase sensitivity can beat the standard quantum limit and approach the quantum Cram\'{e}r-Rao bound. The effects of photon loss on phase sensitivity are discussed. Our scheme can also provide some estimates for units of $\chi^{(3)}$, due to the relation between the nonlinear phase shift and the susceptibility $\chi^{(3)}$ of the Kerr medium.

Mach-Zehnder Interferometer (MZI) is used to measure the phase shift (Δφ) between two light beams related to relative length difference (ΔL) of MZI's two arms. The phase shift of the interferometer output is represented by output Free Spectral Range (FSR). Thus the FSR can be configured by setting its length difference of the two arms, to vary the phase shift. In this paper, the FSR of two couplers MZI are compared with three couplers MZI with two configurations based on the two couplers MZI's length difference. Length difference of the two couplers MZI is utilized for both of the length differences for the first configuration of three couplers MZI. As for the second configuration, each of the three couplers MZI length differences is half of the two couplers MZI length difference. The two couplers MZI are designed to have FSR at 10 nm, which can be obtained by using 0.166 mm as the length difference. 3 dB couplers are used for all of the configurations. Only one input port is utilized and the outputs are simulated in a normalized graph form. The MZIs simulation shows that the FSR of three couplers MZI first configuration is 5 nm, half of the FSR of two couplers MZI, while the FSR of three couplers MZI second configuration is 10 nm, the same as the FSR of two couplers MZI.

Conventional Glauber coherent states (CS) can be defined in several equivalent ways, e.g., by displacing the vacuum or, explicitly, by their infinite Poissonian expansion in Fock states. It is well known that these definitions become inequivalent if applied to finite $d$-level systems (qudits). We present a comparative Wigner-function description of the qudit CS defined (i) by the action of the truncated displacement operator on the vacuum and (ii) by the Poissonian expansion in Fock states of the Glauber CS truncated at $(d\ensuremath{-}1)$-photon Fock state. These states can be generated from a classical light by its optical truncation using nonlinear and linear quantum scissors devices, respectively. We show a surprising effect that a macroscopically distinguishable superposition of two qudit CS (according to both definitions) can be generated with high fidelity by displacing the vacuum in the qudit Hilbert space. If the qudit dimension $d$ is even (odd), then the superposition state contains Fock states with only odd (even) photon numbers, which can be referred to as the odd (even) qudit CS or Schr\"odinger's cat state. This phenomenon can be interpreted as an interference of a single CS with its reflection from the highest-energy Fock state of the Hilbert space, as clearly seen via phase-space interference of the Wigner function. We also analyze nonclassical properties of the qudit CS including their photon-number statistics and nonclassical volume of the Wigner function, which is a quantitative parameter of nonclassicality (quantumness) of states. Finally, we study optical tomograms, which can be directly measured in the homodyne detection of the analyzed qudit cat states and enable the complete reconstructions of their Wigner functions.

We theoretically study the phase sensitivity of an SU(1,1) interferometer with a thermal state and a squeezed vacuum state as inputs and parity detection as the measurement. We find that the phase sensitivity can beat the shot-noise limit and approaches the Heisenberg limit, with increasing input photon number, in an ideal situation. We also consider the effect of various noises, including photon loss, dark counts, and thermal photon noise. Our results show that the phase sensitivity is below the shot-noise limit with photon loss and dark counts, but cannot beat the shot-noise limit in the presence of thermal noise.

We present explicit evaluations of quantum speed limit times pertinent to the Markovian dynamics of an open continuous-variable system. Specifically, we consider the standard setting of a cavity mode of the quantum radiation field weakly coupled to a thermal bosonic reservoir. The evolution of the field state is ruled by the quantum optical master equation, which is known to have an exact analytic solution. Starting from a pure input state, we employ two indicators of how different the initial and evolved states are, namely, the fidelity of evolution and the Hilbert-Schmidt distance of evolution. The former was introduced by del Campo {\em et al.} who derived a time-independent speed limit for the evolution of a Markovian open system. We evaluate it for this field-reservoir setting, with an arbitrary input pure state of the field mode. The resultant formula is then specialized to the coherent and Fock states. On the other hand, we exploit an alternative approach that employs both indicators of evolution mentioned above. Their rates of change have the same upper bound, and consequently provide a unique time-dependent quantum speed limit. It turns out that the associate quantum speed limit time built with the Hilbert-Schmidt metric is tighter than the fidelity-based one. As apposite applications, we investigate the damping of the coherent and Fock states by using the characteristic functions of the corresponding evolved states. General expressions of both the fidelity and the Hilbert-Schmidt distance of evolution are obtained and analyzed for these two classes of input states. In the case of a coherent state, we derive accurate formulas for their common speed limit and the pair of associate limit times.

Phase estimation has a wide range of applications. Over the years, several strategies have been studied to improve precision in phase estimation. These strategies include using exotic quantum states to quantum detection schemes. This dissertation summarizes my effort in improving the precision of phase estimation with a linear and nonlinear interferometer. Chapter 1 introduces quantum optics and quantum metrology. I introduce all relevant quantum states of light used. We also look into tools and terminologies of quantum metrology such as Fisher information, shot-noise limit, Heisenberg limit, etc., along with examples of phase estimation with a Mach-Zehnder interferometer. In Chapter 2, I discuss multiple phase estimation using a multimode interferometer. Building upon previous work, our scheme consists of a multimode interferometer with single-photon inputs. By using a quantum Fisher information analysis, we show that our scheme gives a constant improvement over other schemes. We also show that our scheme with photon-number-resolving detection approaches the quantum Cram\'er-Rao bound. Moreover, we also consider the probabilistic nature of photon emission at the input, and we study its effect on phase sensitivity. I discuss phase estimation with SU(1,1) interferometer in Chapter 3. We look at phase sensitivity in this interferometer with different input states. Namely, we consider two different phase estimation scheme, one using thermal and squeezed states, and others using coherent and displaced squeezed states with parity and on-off as a detection scheme. We also look into the effect of photon loss inside the interferometer. In Chapter 4, we revisit phase estimation in SU(1,1) interferometer from the perspective of quantum Fisher information. I discuss in detail a longstanding confusion regarding the use of quantum Fisher information in SU(1,1) interferometer. We show that phase averaging or quantum Fisher information matrix method is needed in general for calculating the phase sensitivity which resolves inconsistencies reported in previously published articles.

We present a method for directly obtaining the parity of a Gaussian state of light without recourse to photon-number counting. The scheme uses only a simple balanced homodyne technique and intensity correlation. Thus interferometric schemes utilizing coherent or squeezed light and parity detection may be practically implemented for an arbitrary photon flux. Specifically, we investigate a two-mode, squeezed light, Mach–Zehnder interferometer and show how the parity of the output state may be obtained. We also show that the detection may be described independently of the parity operator and that this ‘parity-by-proxy’ measurement has the same signal as traditional parity.

We show that Fock states and Schrodinger-cat states can be conditionally generated by adding multiple photons to a squeezed vacuum state. To demonstrate this, we mainly investigate the photon-number distributions and the Wigner function of the multiple-photon-added squeezed vacuum state (MPASVS). We find that for small squeezing parameter r, the MPASVS approaches to the Fock state |m⟩, and the fidelity between the m-photon PASVS and the Fock state |m⟩ is close to 1. In terms of the fidelity between the MPASVS and the Schrodinger-cat state, as well as the Wigner function of the MPASVS, we show that the MPASVS is basically a Schrodinger-cat state with high fidelities and large amplitudes for the appropriate squeezing parameter.

Classicality and anticlassicality measures of pure and mixed quantum states

When a quantum-optical Fredkin gate is embedded into a Mach-Zehnder interferometer, state reduction techniques permit the generation of maximally entangled states of the radiation field when Fock states are input to the device. These states exactly reach the Heisenberg limit of phase sensitivity. We investigate the consequences of injecting more general states, and particularly coherent states, into the apparatus. Applications to interferometry and photolithography are discussed.

Strong coupling between a two-level system (TLS) and bosonic modes produces dramatic quantum optics effects. We consider a one-dimensional continuum of bosons coupled to a single localized TLS, a system which may be realized in a variety of plasmonic, photonic, or electronic contexts. We present the exact many-body scattering eigenstate obtained by imposing open boundary conditions. Multi-photon bound states appear in the scattering of two or more photons due to the coupling between the photons and the TLS. Such bound states are shown to have a large effect on scattering of both Fock and coherent state wavepackets, especially in the intermediate coupling strength regime. We compare the statistics of the transmitted light with a coherent state having the same mean photon number: as the interaction strength increases, the one-photon probability is suppressed rapidly, and the two- and three-photon probabilities are greatly enhanced due to the many-body bound states. This results in non-Poissonian light.