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Quantum engineering with Kerr media: Schrödinger cat and kitten generation, and photon blockade

Małgorzata Paprzycka

Abstract

The main subject of this PhD thesis is the application of a Kerr medium to generate nonclassical states, like Schrödinger cat states and photon-number truncated states via photon blockade. The dissertation is composed of three main parts. In the first part we studied the problem of the generation of discrete superpositions of coherent states (referred to as the Schrödinger cat and kitten states) in course of evolution of m-photon anharmonic oscillator. We obtained exact analytical formulae for the superposition coefficients with an arbitrary number of components. We showed that, in contrast to the two-photon process, the superposition components for m>2 enter the superposition with different probability making the superposition less symmetrical. We applied the phase distribution function P(θ) obtained from the Pegg-Barnett Hermitian phase formalism to show explicity the symmetry of the superpositions. The polar plots of this function clearly reflect such symmetry showing the number of components (if the states are well separated), their probabilities and phases. The phase distributions calculated numerically from the exact quantum state of the field, for the evolution times Ʈ=2π M/N (where M and N are coprime integers), supported our predictions based on the analitycal formulae for the superposition coefficients. We used the Wigner and Husimi functions to demonstrate the characteristic feature of generating state. These quasiprobability functions confirmed our earlier conclusions. In particular, the Wigner function showed explicity the nonclassicality of our states because in certain areas of the phase space it had negative values. In the second part we analized a photon blockade process. Photon blockade, in analogy to Coulomb’s or phonon blockades, is a phenomenon when a single photon in a nonlinear cavity blocks the transmission of a second photon. This effect can occur in Kerr-type systems driven by a laser due to strong nonlinear photon-photon interactions. We predicted the occurrence of higher-order photon blockades where the transmission of more than two photons is effectively blocked by single- and two-photon states. This photon blockade can be achieved by tuning the frequency of the laser driving field to be equal to the sum of the Kerr nonlinearity and the cavity resonance frequency. We refer to this phenomenon as two-photon blockade or two-photon state truncation via nonlinear scissors, and can also be interpreted as photon-induced tunneling. We also show that, for a driving-field frequency fulfilling another resonance condition and for higher strengths of the driving field, even a three-photon blockade can occur but less clearly than in the case of single- and two-photon blockades. We demonstrate how various photon blockades can be identified by analyzing photon-number correlations, coherence and entropic properties, the Wigner functions, and spectra of squeezing. We show that two- and three-photon blockades can, in principle, be observed in various cavity and circuit quantum electrodynamical systems for which the standard single-photon blockade was observed without the need of using higher-order driving interactions or Kerr media exhibiting higher-order nonlinear susceptibility. In the last part we discussed the generation of qudit coherent states (CS), in particular, the qudit Schrödinger cat states. Conventional Glauber 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 presented 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 − 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 showed 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 Schrödinger’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 analyzed 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 studied 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 hope that the results of this thesis can stimulate further interest in finding applications of the Kerr effect in quantum information processing (including quantum teleportation) with qudits and quantum engineering.
Record ID
UAM37b492cc610d4b8d9876ccbbfb66a684
Diploma type
Doctor of Philosophy
Author
Title in Polish
Inżynieria kwantowa z wykorzystaniem ośrodków kerrowskich: generacja kotów i kociąt Schrödingera oraz blokada fotonowa
Title in English
Quantum engineering with Kerr media: Schrödinger cat and kitten generation, and photon blockade
Language
pol (pl) Polish
Certifying Unit
Faculty of Physics (SNŚ/WyF/FoP)
Discipline
physics / (physical sciences domain) / (physical sciences)
Scientific discipline (2.0)
6.6 physical sciences
Status
Finished
Defense Date
29-10-2015
Title date
29-10-2015
Supervisor
URL
http://hdl.handle.net/10593/13975 Opening in a new tab
Keywords in English
Kerr medium, Schrödinger cat, photon blockade, qudit
Abstract in English
The main subject of this PhD thesis is the application of a Kerr medium to generate nonclassical states, like Schrödinger cat states and photon-number truncated states via photon blockade. The dissertation is composed of three main parts. In the first part we studied the problem of the generation of discrete superpositions of coherent states (referred to as the Schrödinger cat and kitten states) in course of evolution of m-photon anharmonic oscillator. We obtained exact analytical formulae for the superposition coefficients with an arbitrary number of components. We showed that, in contrast to the two-photon process, the superposition components for m>2 enter the superposition with different probability making the superposition less symmetrical. We applied the phase distribution function P(θ) obtained from the Pegg-Barnett Hermitian phase formalism to show explicity the symmetry of the superpositions. The polar plots of this function clearly reflect such symmetry showing the number of components (if the states are well separated), their probabilities and phases. The phase distributions calculated numerically from the exact quantum state of the field, for the evolution times Ʈ=2π M/N (where M and N are coprime integers), supported our predictions based on the analitycal formulae for the superposition coefficients. We used the Wigner and Husimi functions to demonstrate the characteristic feature of generating state. These quasiprobability functions confirmed our earlier conclusions. In particular, the Wigner function showed explicity the nonclassicality of our states because in certain areas of the phase space it had negative values. In the second part we analized a photon blockade process. Photon blockade, in analogy to Coulomb’s or phonon blockades, is a phenomenon when a single photon in a nonlinear cavity blocks the transmission of a second photon. This effect can occur in Kerr-type systems driven by a laser due to strong nonlinear photon-photon interactions. We predicted the occurrence of higher-order photon blockades where the transmission of more than two photons is effectively blocked by single- and two-photon states. This photon blockade can be achieved by tuning the frequency of the laser driving field to be equal to the sum of the Kerr nonlinearity and the cavity resonance frequency. We refer to this phenomenon as two-photon blockade or two-photon state truncation via nonlinear scissors, and can also be interpreted as photon-induced tunneling. We also show that, for a driving-field frequency fulfilling another resonance condition and for higher strengths of the driving field, even a three-photon blockade can occur but less clearly than in the case of single- and two-photon blockades. We demonstrate how various photon blockades can be identified by analyzing photon-number correlations, coherence and entropic properties, the Wigner functions, and spectra of squeezing. We show that two- and three-photon blockades can, in principle, be observed in various cavity and circuit quantum electrodynamical systems for which the standard single-photon blockade was observed without the need of using higher-order driving interactions or Kerr media exhibiting higher-order nonlinear susceptibility. In the last part we discussed the generation of qudit coherent states (CS), in particular, the qudit Schrödinger cat states. Conventional Glauber 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 presented 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 − 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 showed 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 Schrödinger’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 analyzed 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 studied 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 hope that the results of this thesis can stimulate further interest in finding applications of the Kerr effect in quantum information processing (including quantum teleportation) with qudits and quantum engineering.

Uniform Resource Identifier
https://researchportal.amu.edu.pl/info/phd/UAM37b492cc610d4b8d9876ccbbfb66a684/
URN
urn:amu-prod:UAM37b492cc610d4b8d9876ccbbfb66a684

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