Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1810.02117 · Physical Review A

Quantum tetrachotomous states: Superposition of four coherent states on a line in phase space

Namrata Shukla, Naeem Akhtar, Barry C. Sanders

Published Jun 10, 2019• 17 citations
arXivHTMLPDF

Abstract

The well-studied quantum optical Schr\"odinger cat state is a superposition of two distinguishable states, with quantum coherence between these macroscopically distinguishable states being of foundational and, in the context of quantum-information processing, practical use. We refer to these quantum-optical cat states as quantum dichotomous states, reflecting that the state is a superposition of two options, and we introduce the term quantum multichotomous state to refer to a superposition of multiple macroscopically distinguishable options. For a single degree of freedom, such as position, we construct the quantum multichotomous states as a superposition of Gaussian states on the position line in phase space. Using this nomenclature, a quantum tetrachotomous state (QTS) is a coherent superposition of four macroscopically distinguishable states. We define, analyze, and show how to create such states, and our focus on the QTSs is due to their exhibition of much richer phenomena than for the quantum dichotomous states. Our characterization of the QTS involves the Wigner function, its marginal distributions, and the photon-number distribution, and we discuss the QTS's approximate realization in a multiple-coupled-well system.

DOI
10.1103/physreva.99.063813

Related papers

Quantum statistical properties of phase-type three-headed Schrodinger cat state

Lin Dun-Qing, Zhu Ze-Qun, Wang Zu-Jian +1 more

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.

Acta Physica Sinica•Jan 1, 2017•0 citations

Study of decoherence of a superposition of macroscopic quantum states by means the consideration of a multimode state of a Schrödinger cat

Dmitriy V. Fastovets, Yu. I. Bogdanov, Nadezhda A. Bogdanova +1 more

Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state: an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems.

International Conference on Micro- and Nano-Electronics (ICMNE)•Jan 31, 2022•1 citation

Entanglement of macroscopically distinct states of light

Demid Sychev, Valeriy A. Novikov, Khurram K. Pirov +2 more

Schrödinger's famous Gedankenexperiment has inspired multiple generations of physicists to think about apparent paradoxes that arise when the logic of quantum physics is applied to macroscopic objects. The development of quantum technologies enabled us to produce physical analogues of Schrödinger's cats, such as superpositions of macroscopically distinct states as well as entangled states of microscopic and macroscopic entities. Here we take one step further and prepare an optical state which, in Schrödinger's language, is equivalent to a superposition of two cats, one of which is dead and the other alive, but it is not known in which state each individual cat is. Specifically, the alive and dead states are, respectively, the displaced single photon and displaced vacuum (coherent state), with the magnitude of displacement being on a scale of $10^8$ photons. These two states have significantly different photon statistics and are therefore macroscopically distinguishable.

arXiv:1811.01041•Optica•Nov 8, 2019•9 citations

Study of decoherence of a superposition of macroscopic quantum states by means the consideration of a multimode state of a Schrodinger cat

Fastovets, D. V., Yu. I. Bogdanov, Nadezhda A. Bogdanova +1 more

Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state - an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems.

arXiv:2201.03553•Jan 9, 2022•0 citations

Shaping of Coherent State Superpositions Via Deterministically Created SchröDinger-Cat States

Severin Daiss, Bastian Hacker, Stephan Welte +4 more

In 1935, Erwin Schrödinger put forward his famous gedanken experiment in which a microscopic degree of freedom - the emission state of a radioactive atom - and a macroscopic, classical state - the life or death of a cat - are entangled [1]. The resulting state is dubbed a Schrodinger-cat state. According to quantum theory, a measurement of the atom in a superposition basis projects the cat into a macroscopic superposition of being alive and dead. Here, we use coherent laser fields with different optical phases as a macroscopic model system and generate entangled Schrodinger-cat states in a deterministic way [3]. We reflect the coherent pulses from an optical cavity that contains a single strongly-coupled atom. This entangles the spin of the atom with the optical phase of the light field [2]. By manipulating and detecting the atomic state, we create a plethora of flying optical coherent-state superpositions. We reconstruct the corresponding Wigner functions (see Fig. 1) to demonstrate the tunability and non-classicality of the produced light state.

Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC)•Jun 1, 2019•0 citations

Phase properties of macroscopic superposition states

Edwin E. Hach, Christopher C. Gerry

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 Optics Journal of the European Optical Society Part B•Dec 1, 1993•11 citations
The well-studied quantum optical Schr"odinger cat state is a superposition of two distinguishable states, with quantum coherence between these macroscopically distinguishable states being of foundational and, in the context of quantum-information processing, practical use. We refer to these quantum-optical cat states as quantum dichotomous states, reflecting that the state is a superposition of two options, and we introduce the term quantum multichotomous state to refer to a superposition of multiple macroscopically distinguishable options. For a single degree of freedom, such as position, we construct the quantum multichotomous states as a superposition of Gaussian states on the position line in phase space. Using this nomenclature, a quantum tetrachotomous state (QTS) is a coherent superposition of four macroscopically distinguishable states. We define, analyze, and show how to create such states, and our focus on the QTSs is due to their exhibition of much richer phenomena than for the quantum dichotomous states. Our characterization of the QTS involves the Wigner function, its marginal distributions, and the photon-number distribution, and we discuss the QTS's approximate realization in a multiple-coupled-well system.
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.
Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state: an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems.
Schrödinger's famous Gedankenexperiment has inspired multiple generations of physicists to think about apparent paradoxes that arise when the logic of quantum physics is applied to macroscopic objects. The development of quantum technologies enabled us to produce physical analogues of Schrödinger's cats, such as superpositions of macroscopically distinct states as well as entangled states of microscopic and macroscopic entities. Here we take one step further and prepare an optical state which, in Schrödinger's language, is equivalent to a superposition of two cats, one of which is dead and the other alive, but it is not known in which state each individual cat is. Specifically, the alive and dead states are, respectively, the displaced single photon and displaced vacuum (coherent state), with the magnitude of displacement being on a scale of 10810^8108 photons. These two states have significantly different photon statistics and are therefore macroscopically distinguishable.
Quantum Schrodinger cat states are of great interest in quantum communications and quantum optics. These states are used in various scientific fields such as quantum computing, quantum error correction and high-precision measurements. The analysis of the Schrodinger cat states coherence is an important task for their complete practical application. Our developed approach makes it possible to estimate the coherence of the quantum Schrodinger cat state of arbitrary dimension, as well as to find the interference visibility of the state - an important optical characteristic. The obtained simple quantitative relationship between coherence and the Schmidt number, as well as the developed approach of reducing the multidimensional quantum cat state to a two-mode analog allow us to analyze macroscopic states formed by a large number of modes. Several explicit formulas for the reduced states that arise after measuring of some modes of the considered multimode system are obtained. The research results have significant application and can be used in the development of high-dimensional quantum information processing systems.
In 1935, Erwin Schrödinger put forward his famous gedanken experiment in which a microscopic degree of freedom - the emission state of a radioactive atom - and a macroscopic, classical state - the life or death of a cat - are entangled [1]. The resulting state is dubbed a Schrodinger-cat state. According to quantum theory, a measurement of the atom in a superposition basis projects the cat into a macroscopic superposition of being alive and dead. Here, we use coherent laser fields with different optical phases as a macroscopic model system and generate entangled Schrodinger-cat states in a deterministic way [3]. We reflect the coherent pulses from an optical cavity that contains a single strongly-coupled atom. This entangles the spin of the atom with the optical phase of the light field [2]. By manipulating and detecting the atomic state, we create a plethora of flying optical coherent-state superpositions. We reconstruct the corresponding Wigner functions (see Fig. 1) to demonstrate the tunability and non-classicality of the produced light state.
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.