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Quantum Optics Journal of the European Optical Society Part B

Phase properties of macroscopic superposition states

Edwin E. Hach, Christopher C. Gerry

Published Dec 1, 1993• 11 citations
Publisher

Abstract

Phase properties of states consisting of superpositions of the states mod a) and mod -a), coherent states of a single mode quantized electromagnetic field, are studied. These states are also known as macroscopic superposition states and 'Schrodinger cat' states. After first reviewing the squeezing and antibunching properties of these states, from the density operator and the phase states we derive a probability distribution for the phase. We also compare with this distributions obtained from the associated Q-function and the Wigner function.

DOI
10.1088/0954-8998/5/6/002

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Phase properties of states consisting of superpositions of the states mod a) and mod -a), coherent states of a single mode quantized electromagnetic field, are studied. These states are also known as macroscopic superposition states and 'Schrodinger cat' states. After first reviewing the squeezing and antibunching properties of these states, from the density operator and the phase states we derive a probability distribution for the phase. We also compare with this distributions obtained from the associated Q-function and the Wigner function.
Quantum superposition is a fundamental principle of quantum mechanics, which provides a crucial basis to observe phenomena beyond the predictions of classical physics. For example, a quantum entangled state can exhibit stronger correlation than classically possible one. In quantum state engineering, many new quantum states can be obtained from the superposition of many known states. In recent decades, the superposition of coherent states (CSs) with the same amplitude but two different phases has been a subject of great interest. This superposition state was often called Schrodinger cat state (here, we also name it 2-headed cat state (2HCS)), which becomes an important tool to study a lot of fundamental issues. Surprisingly, some studies have extended the quantum superposition to involving more than two component coherent states. In order to produce the superposition of three photons, people have considered the superposition of coherent states with three different phases (here, we also name it 3-headed cat state (3HCS)). Furthermore, in microwave cavity quantum electrodynamics of bang-bang quantum Zeno dynamics control, people have proposed the superposition of coherent states with four different phases (here, we also name it 4-headed cat state (4HCS)). In this paper, we make a detailed investigation on the quantum statistical properties of a phase-type 3HCS. These properties include photon number distribution, average photon number, sub-Poissionian distribution, squeezing effect, and Wigner function, etc. We derive their analytical expressions and make numerical simulations for these properties. The results are compared with the counterparts of the CS, the 2HCS and the 4HCS. The conclusions are obtained as follows. 1) The CS, the 2HCS, the 3HCS and the 4HCS have k, 2k, 3k and 4k photon number components, respectively (k is an integer); 2) small difference in average photon number among these quantum states in small-amplitude range can be observed, while their average photon numbers become almost equal in large-amplitude range; 3) the CS exhibits Poisson distribution, and the 2HCS, the 3HCS and the 4HCS exhibit super-Poisson distributions in most amplitude ranges, however, sub-Poisson distribution can be seen for the 3HCS and the 4HCS in some specific amplitude ranges; 4) except for the 2HCS that may have the squeezing property, no squeezing properties can be found in the CS, the 3HCS and the 4HCS; 5) negative values can exist in the Wigner functions for the 2HCS, the 3HCS and the 4HCS, while it is not found in the CS. Similar to the 2HCS and 4HCS, the Wigner function of the 3HCS has negative component, which implies its nonclassicality. Different from the 2HCS, the 3HCS exhibits sub-Poisson photon number distribution in a certain amplitude range, it is weaker than that of the 4HCS. At the same time, no squeezing is found in the 3 or 4HCS, which is another difference from the 2HCS.
We analyze the phase and squeezing properties of the cotangent states of the electromagnetic field. We calculate the phase distribution, phase variance, and number-phase uncertainty product for these states. Under appropriate conditions, the phase distribution develops oscillations resulting from the formation of states, reminiscent of macroscopic superpositions. In other cases the cotangent states are nearly minimum number-phase uncertainty states, emulating the coherent states. We also study the quadrature squeezing properties of the cotangent states and find that under a wide range of conditions they are highly squeezed.
In recent years, there has been an increased interest in the generation of superposition of coherent states with opposite phases, the so-called photonic Schrodinger-cat states. These experiments are very challenging and so far, cats involving small photon numbers only have been implemented. Here, we propose to consider two-mode squeezed states as examples of a Schrodinger-cat-like state. In particular, we are interested in several criteria aiming to identify quantum states that are macroscopic superpositions in a more general sense. We show how these criteria can be extended to continuous variable entangled states. We apply them to various squeezed states, argue that two-mode squeezed vacuum states belong to a class of general Schrodinger-cat states and compare the size of states obtained in several experiments. Our results not only promote two-mode squeezed states for exploring quantum effects at the macroscopic level but also provide direct measures to evaluate their usefulness for quantum metrology.
The Barut-Girardello (BG) coherent states (CS) representation is extended to the noncompact algebras and in (reducible) quadratic boson realizations. The BG CS take the form of multimode ordinary Schrödinger cat states. Macroscopic superpositions of CS ( canonical CS, n = 1, 2, ...) are pointed out which are overcomplete in the N -mode Hilbert space and the relation between the canonical CS and the BG-type CS representations is established. The sets of and BG CS and their discrete superpositions contain many states studied in quantum optics (even and odd N -mode CS, pair CS) and provide an approach to quadrature squeezing, alternative to that of intelligent states. New subsets of weakly and strongly nonclassical states are pointed out and their statistical properties (first- and second-order squeezing, photon number distributions) are discussed. For specific values of the angle parameters and small amplitude of the canonical CS components, these states approach multimode Fock states with one, two or three bosons/photons. It is shown that eigenstates of a squared non-Hermitian operator (generalized cat states) can exhibit squeezing of the quadratures of A .
We study the eigenstates of the square of the phase operator for a single-mode field, and show that they are given as superpositions of macroscopically distinguishable phase states, which we call Schrödinger phase cats, the phase states being eigenstates of the phase operator. Those solutions that are also eigenstates of the parity operator with even parity are shown to be very similar to the squeezed vacuum states over a range of relevant parameters. Thus, it is possible to consider some squeezed vacuum states as approximate Schrödinger phase-cat states. We discuss the connections to the phase states of the SU (1, 1) phase operators for various realizations and representations.