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Zielińska JA, van der Laan F, Norrman A, Rimlinger M, Reimann R, Novotny L, Frimmer M. Controlling Optomechanical Libration with the Degree of Polarization. PHYSICAL REVIEW LETTERS 2023; 130:203603. [PMID: 37267539 DOI: 10.1103/physrevlett.130.203603] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2023] [Revised: 03/07/2023] [Accepted: 04/04/2023] [Indexed: 06/04/2023]
Abstract
Control of the potential energy and free evolution lie at the heart of levitodynamics as key requirements for sensing, wave function expansion, and mechanical squeezing protocols. Here, we experimentally demonstrate versatile control over the optical potential governing the libration motion of a levitated anisotropic nanoparticle. This control is achieved by introducing the degree of polarization as a new tool for rotational levitodynamics. We demonstrate thermally driven free rotation of a levitated anisotropic scatterer around its short axis and we use the rotational degrees of freedom to probe the local spin of a strongly focused laser beam.
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Affiliation(s)
- J A Zielińska
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
| | - F van der Laan
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
| | - A Norrman
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
- Center for Photonics Sciences, University of Eastern Finland, P.O. Box 111, FI-80101 Joensuu, Finland
| | - M Rimlinger
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
| | - R Reimann
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
- Quantum Research Center, Technology Innovation Institute, Abu Dhabi, United Arab Emirates
| | - L Novotny
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
| | - M Frimmer
- Photonics Laboratory, ETH Zürich, CH-8093 Zürich, Switzerland
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2
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Coelho SS, Queiroz L, Alves DT. Exact Solution of a Time-Dependent Quantum Harmonic Oscillator with Two Frequency Jumps via the Lewis-Riesenfeld Dynamical Invariant Method. ENTROPY (BASEL, SWITZERLAND) 2022; 24:1851. [PMID: 36554256 PMCID: PMC9778280 DOI: 10.3390/e24121851] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 11/30/2022] [Revised: 12/14/2022] [Accepted: 12/16/2022] [Indexed: 06/17/2023]
Abstract
Harmonic oscillators with multiple abrupt jumps in their frequencies have been investigated by several authors during the last decades. We investigate the dynamics of a quantum harmonic oscillator with initial frequency ω0, which undergoes a sudden jump to a frequency ω1 and, after a certain time interval, suddenly returns to its initial frequency. Using the Lewis−Riesenfeld method of dynamical invariants, we present expressions for the mean energy value, the mean number of excitations, and the transition probabilities, considering the initial state different from the fundamental. We show that the mean energy of the oscillator, after the jumps, is equal or greater than the one before the jumps, even when ω1<ω0. We also show that, for particular values of the time interval between the jumps, the oscillator returns to the same initial state.
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Affiliation(s)
- Stanley S. Coelho
- Faculdade de Física, Universidade Federal do Pará, Belém 66075-110, PA, Brazil
| | - Lucas Queiroz
- Faculdade de Física, Universidade Federal do Pará, Belém 66075-110, PA, Brazil
| | - Danilo T. Alves
- Faculdade de Física, Universidade Federal do Pará, Belém 66075-110, PA, Brazil
- Centro de Física, Universidade do Minho, 4710-057 Braga, Portugal
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Kustura K, Gonzalez-Ballestero C, Sommer ADLR, Meyer N, Quidant R, Romero-Isart O. Mechanical Squeezing via Unstable Dynamics in a Microcavity. PHYSICAL REVIEW LETTERS 2022; 128:143601. [PMID: 35476467 DOI: 10.1103/physrevlett.128.143601] [Citation(s) in RCA: 5] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2021] [Accepted: 03/10/2022] [Indexed: 06/14/2023]
Abstract
We theoretically show that strong mechanical quantum squeezing in a linear optomechanical system can be rapidly generated through the dynamical instability reached in the far red-detuned and ultrastrong coupling regime. We show that this mechanism, which harnesses unstable multimode quantum dynamics, is particularly suited to levitated optomechanics, and we argue for its feasibility for the case of a levitated nanoparticle coupled to a microcavity via coherent scattering. We predict that for submillimeter-sized cavities the particle motion, initially thermal and well above its ground state, becomes mechanically squeezed by tens of decibels on a microsecond timescale. Our results bring forth optical microcavities in the unresolved sideband regime as powerful mechanical squeezers for levitated nanoparticles, and hence as key tools for quantum-enhanced inertial and force sensing.
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Affiliation(s)
- Katja Kustura
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
- Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
| | - Carlos Gonzalez-Ballestero
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
- Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
| | - Andrés de Los Ríos Sommer
- Nanophotonic Systems Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland
- Quantum Center, ETH Zurich, 8083 Zurich, Switzerland
| | - Nadine Meyer
- Nanophotonic Systems Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland
- Quantum Center, ETH Zurich, 8083 Zurich, Switzerland
| | - Romain Quidant
- Nanophotonic Systems Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland
- Quantum Center, ETH Zurich, 8083 Zurich, Switzerland
| | - Oriol Romero-Isart
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
- Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
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4
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Energy and Magnetic Moment of a Quantum Charged Particle in Time-Dependent Magnetic and Electric Fields of Circular and Plane Solenoids. ENTROPY 2021; 23:e23121579. [PMID: 34945884 PMCID: PMC8700163 DOI: 10.3390/e23121579] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/16/2021] [Revised: 11/19/2021] [Accepted: 11/20/2021] [Indexed: 12/02/2022]
Abstract
We consider a quantum spinless nonrelativistic charged particle moving in the xy plane under the action of a time-dependent magnetic field, described by means of the linear vector potential A=B(t)−y(1+α),x(1−α)/2, with two fixed values of the gauge parameter α: α=0 (the circular gauge) and α=1 (the Landau gauge). While the magnetic field is the same in all the cases, the systems with different values of the gauge parameter are not equivalent for nonstationary magnetic fields due to different structures of induced electric fields, whose lines of force are circles for α=0 and straight lines for α=1. We derive general formulas for the time-dependent mean values of the energy and magnetic moment, as well as for their variances, for an arbitrary function B(t). They are expressed in terms of solutions to the classical equation of motion ε¨+ωα2(t)ε=0, with ω1=2ω0. Explicit results are found in the cases of the sudden jump of magnetic field, the parametric resonance, the adiabatic evolution, and for several specific functions B(t), when solutions can be expressed in terms of elementary or hypergeometric functions. These examples show that the evolution of the mentioned mean values can be rather different for the two gauges, if the evolution is not adiabatic. It appears that the adiabatic approximation fails when the magnetic field goes to zero. Moreover, the sudden jump approximation can fail in this case as well. The case of a slowly varying field changing its sign seems especially interesting. In all the cases, fluctuations of the magnetic moment are very strong, frequently exceeding the square of the mean value.
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Xin M, Leong WS, Chen Z, Wang Y, Lan SY. Rapid Quantum Squeezing by Jumping the Harmonic Oscillator Frequency. PHYSICAL REVIEW LETTERS 2021; 127:183602. [PMID: 34767425 DOI: 10.1103/physrevlett.127.183602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2021] [Accepted: 09/30/2021] [Indexed: 06/13/2023]
Abstract
Quantum sensing and quantum information processing use quantum advantages such as squeezed states that encode a quantity of interest with higher precision and generate quantum correlations to outperform classical methods. In harmonic oscillators, the rate of generating squeezing is set by a quantum speed limit. Therefore, the degree to which a quantum advantage can be used in practice is limited by the time needed to create the state relative to the rate of unavoidable decoherence. Alternatively, a sudden change of harmonic oscillator's frequency projects a ground state into a squeezed state which can circumvent the time constraint. Here, we create squeezed states of atomic motion by sudden changes of the harmonic oscillation frequency of atoms in an optical lattice. Building on this protocol, we demonstrate rapid quantum amplification of a displacement operator that could be used for detecting motion. Our results can speed up quantum gates and enable quantum sensing and quantum information processing in noisy environments.
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Affiliation(s)
- Mingjie Xin
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Wui Seng Leong
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Zilong Chen
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Yu Wang
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
| | - Shau-Yu Lan
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore
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Jaramillo JD, Deng J, Gong J. Quantum work fluctuations in connection with the Jarzynski equality. Phys Rev E 2018; 96:042119. [PMID: 29347528 DOI: 10.1103/physreve.96.042119] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/06/2017] [Indexed: 11/07/2022]
Abstract
A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change ΔF of a system at inverse temperature β from an ensemble average of nonequilibrium exponential work, i.e., 〈e^{-βW}〉=e^{-βΔF}. The number of experimental work values needed to reach a given accuracy of ΔF is determined by the variance of e^{-βW}, denoted var(e^{-βW}). We discover in this work that var(e^{-βW}) in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging var(e^{-βW}) at sufficiently low temperatures, markedly different from the classical behavior. The divergence of var(e^{-βW}) indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.
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Affiliation(s)
- Juan D Jaramillo
- Department of Physics, National University of Singapore, Singapore 117546
| | - Jiawen Deng
- NUS Graduate School for Integrative Science and Engineering, Singapore 117597
| | - Jiangbin Gong
- Department of Physics, National University of Singapore, Singapore 117546.,NUS Graduate School for Integrative Science and Engineering, Singapore 117597
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Silveri MP, Tuorila JA, Thuneberg EV, Paraoanu GS. Quantum systems under frequency modulation. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2017; 80:056002. [PMID: 28379844 DOI: 10.1088/1361-6633/aa5170] [Citation(s) in RCA: 31] [Impact Index Per Article: 4.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
We review the physical phenomena that arise when quantum mechanical energy levels are modulated in time. The dynamics resulting from changes in the transition frequency is a problem studied since the early days of quantum mechanics. It has been of constant interest both experimentally and theoretically since, with the simple two-state model providing an inexhaustible source of novel concepts. When the transition frequency of a quantum system is modulated, several phenomena can be observed, such as Landau-Zener-Stückelberg-Majorana interference, motional averaging and narrowing, and the formation of dressed states with the appearance of sidebands in the spectrum. Adiabatic changes result in the accumulation of geometric phases, which can be used to create topological states. In recent years, an exquisite experimental control in the time domain was gained through the parameters entering the Hamiltonian, and high-fidelity readout schemes allowed the state of the system to be monitored non-destructively. These developments were made in the field of quantum devices, especially in superconducting qubits, as a well as in atomic physics, in particular in ultracold gases. As a result of these advances, it became possible to demonstrate many of the fundamental effects that arise in a quantum system when its transition frequencies are modulated. The purpose of this review is to present some of these developments, from two-state atoms and harmonic oscillators to multilevel and many-particle systems.
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Affiliation(s)
- M P Silveri
- Department of Physics, University of Oulu, PO Box 3000, FI-90014, Finland. Department of Physics, Yale University, New Haven, CT 06520, United States of America
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Joshi C, Irish EK, Spiller TP. Qubit-flip-induced cavity mode squeezing in the strong dispersive regime of the quantum Rabi model. Sci Rep 2017; 7:45587. [PMID: 28358025 PMCID: PMC5372368 DOI: 10.1038/srep45587] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2016] [Accepted: 02/28/2017] [Indexed: 11/09/2022] Open
Abstract
Squeezed states of light are a set of nonclassical states in which the quantum fluctuations of one quadrature component are reduced below the standard quantum limit. With less noise than the best stabilised laser sources, squeezed light is a key resource in the field of quantum technologies and has already improved sensing capabilities in areas ranging from gravitational wave detection to biomedical applications. In this work we propose a novel technique for generating squeezed states of a confined light field strongly coupled to a two-level system, or qubit, in the dispersive regime. Utilising the dispersive energy shift caused by the interaction, control of the qubit state produces a time-dependent change in the frequency of the light field. An appropriately timed sequence of sudden frequency changes reduces the quantum noise fluctuations in one quadrature of the field well below the standard quantum limit. The degree of squeezing and the time of generation are directly controlled by the number of frequency shifts applied. Even in the presence of realistic noise and imperfections, our protocol promises to be capable of generating a useful degree of squeezing with present experimental capabilities.
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Affiliation(s)
- Chaitanya Joshi
- Department of Physics and York Centre for Quantum Technologies, University of York, Heslington, York, YO10 5DD, UK
| | - Elinor K Irish
- Physics and Astronomy, University of Southampton, Highfield, Southampton, SO17 1BJ, UK
| | - Timothy P Spiller
- Department of Physics and York Centre for Quantum Technologies, University of York, Heslington, York, YO10 5DD, UK
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Rashid M, Tufarelli T, Bateman J, Vovrosh J, Hempston D, Kim MS, Ulbricht H. Experimental Realization of a Thermal Squeezed State of Levitated Optomechanics. PHYSICAL REVIEW LETTERS 2016; 117:273601. [PMID: 28084746 DOI: 10.1103/physrevlett.117.273601] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2016] [Indexed: 06/06/2023]
Abstract
We experimentally squeeze the thermal motional state of an optically levitated nanosphere by fast switching between two trapping frequencies. The measured phase-space distribution of the center of mass of our particle shows the typical shape of a squeezed thermal state, from which we infer up to 2.7 dB of squeezing along one motional direction. In these experiments the average thermal occupancy is high and, even after squeezing, the motional state remains in the remit of classical statistical mechanics. Nevertheless, we argue that the manipulation scheme described here could be used to achieve squeezing in the quantum regime if preceded by cooling of the levitated mechanical oscillator. Additionally, a higher degree of squeezing could, in principle, be achieved by repeating the frequency-switching protocol multiple times.
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Affiliation(s)
- Muddassar Rashid
- Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - Tommaso Tufarelli
- School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, United Kingdom
| | - James Bateman
- Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
- Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom
| | - Jamie Vovrosh
- Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - David Hempston
- Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
| | - M S Kim
- QOLS, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
| | - Hendrik Ulbricht
- Department of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, United Kingdom
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10
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Manzano G, Galve F, Zambrini R, Parrondo JMR. Entropy production and thermodynamic power of the squeezed thermal reservoir. Phys Rev E 2016; 93:052120. [PMID: 27300843 DOI: 10.1103/physreve.93.052120] [Citation(s) in RCA: 21] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/29/2016] [Indexed: 06/06/2023]
Abstract
We analyze the entropy production and the maximal extractable work from a squeezed thermal reservoir. The nonequilibrium quantum nature of the reservoir induces an entropy transfer with a coherent contribution while modifying its thermal part, allowing work extraction from a single reservoir, as well as great improvements in power and efficiency for quantum heat engines. Introducing a modified quantum Otto cycle, our approach fully characterizes operational regimes forbidden in the standard case, such as refrigeration and work extraction at the same time, accompanied by efficiencies equal to unity.
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Affiliation(s)
- Gonzalo Manzano
- Departamento de Física Atómica, Molecular y Nuclear and GISC, Universidad Complutense Madrid, 28040 Madrid, Spain
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Fernando Galve
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Roberta Zambrini
- Instituto de Física Interdisciplinar y Sistemas Complejos IFISC (CSIC-UIB), Campus Universitat Illes Balears, E-07122 Palma de Mallorca, Spain
| | - Juan M R Parrondo
- Departamento de Física Atómica, Molecular y Nuclear and GISC, Universidad Complutense Madrid, 28040 Madrid, Spain
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Roßnagel J, Abah O, Schmidt-Kaler F, Singer K, Lutz E. Nanoscale heat engine beyond the Carnot limit. PHYSICAL REVIEW LETTERS 2014; 112:030602. [PMID: 24484127 DOI: 10.1103/physrevlett.112.030602] [Citation(s) in RCA: 184] [Impact Index Per Article: 18.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2013] [Indexed: 06/03/2023]
Abstract
We consider a quantum Otto cycle for a time-dependent harmonic oscillator coupled to a squeezed thermal reservoir. We show that the efficiency at maximum power increases with the degree of squeezing, surpassing the standard Carnot limit and approaching unity exponentially for large squeezing parameters. We further propose an experimental scheme to implement such a model system by using a single trapped ion in a linear Paul trap with special geometry. Our analytical investigations are supported by Monte Carlo simulations that demonstrate the feasibility of our proposal. For realistic trap parameters, an increase of the efficiency at maximum power of up to a factor of 4 is reached, largely exceeding the Carnot bound.
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Affiliation(s)
- J Roßnagel
- Quantum, Institut für Physik, Universität Mainz, D-55128 Mainz, Germany
| | - O Abah
- Institute for Theoretical Physics, University of Erlangen-Nürnberg, D-91058 Erlangen, Germany
| | - F Schmidt-Kaler
- Quantum, Institut für Physik, Universität Mainz, D-55128 Mainz, Germany
| | - K Singer
- Quantum, Institut für Physik, Universität Mainz, D-55128 Mainz, Germany
| | - E Lutz
- Institute for Theoretical Physics, University of Erlangen-Nürnberg, D-91058 Erlangen, Germany
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Abstract
Studying mechanical resonators via radiation pressure offers a rich avenue for the exploration of quantum mechanical behavior in a macroscopic regime. However, quantum state preparation and especially quantum state reconstruction of mechanical oscillators remains a significant challenge. Here we propose a scheme to realize quantum state tomography, squeezing, and state purification of a mechanical resonator using short optical pulses. The scheme presented allows observation of mechanical quantum features despite preparation from a thermal state and is shown to be experimentally feasible using optical microcavities. Our framework thus provides a promising means to explore the quantum nature of massive mechanical oscillators and can be applied to other systems such as trapped ions.
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Zagoskin AM, Il'ichev E, McCutcheon MW, Young JF, Nori F. Controlled generation of squeezed states of microwave radiation in a superconducting resonant circuit. PHYSICAL REVIEW LETTERS 2008; 101:253602. [PMID: 19113707 DOI: 10.1103/physrevlett.101.253602] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2008] [Indexed: 05/27/2023]
Abstract
Superconducting oscillators have been successfully used for quantum control and readout devices in conjunction with superconducting qubits. Also, squeezed states can improve the accuracy of measurements to subquantum, or at least subthermal, levels. Here, we show theoretically how to produce squeezed states of microwave radiation in a superconducting oscillator with tunable parameters. Its resonance frequency can be changed by controlling an rf SQUID inductively coupled to the oscillator. By repeatedly shifting the resonance frequency between any two values, it is possible to produce squeezed and subthermal states of the electromagnetic field in the (0.1-10) GHz range, even when the relative frequency change is small. We propose experimental protocols for the verification of squeezed state generation, and for their use to improve the readout fidelity when such oscillators serve as quantum transducers.
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Affiliation(s)
- A M Zagoskin
- Department of Physics, Loughborough University, Loughborough, Leics LE11 3TU, United Kingdom
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Averbukh I, Sherman B, Kurizki G. Enhanced squeezing by periodic frequency modulation under parametric instability conditions. PHYSICAL REVIEW A 1994; 50:5301-5308. [PMID: 9911531 DOI: 10.1103/physreva.50.5301] [Citation(s) in RCA: 18] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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15
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Kiss T, Janszky J, Adam P. Time evolution of harmonic oscillators with time-dependent parameters: A step-function approximation. PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1994; 49:4935-4942. [PMID: 9910814 DOI: 10.1103/physreva.49.4935] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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