Multimode microwave circuit optomechanics as a platform to study coupled quantum harmonic oscillators

As children, whispering into the ear of a friend in the presence of others allows us to pass a secret without interception, and forms one of the simplest attempts at secret communications we can employ. However, sending secret messages becomes deeply nontrivial over long distances. A solution for two parties to communicate securely is to encrypt and decrypt a message with two identical strings of bits, one for each party. In this case, the security of the encrypted message is provable and does not rely on assumptions on computational power. Quantum theory provides a clear solution for the initial distribution of these identical bit strings through Quantum Key Distribution. However, once long distances are involved, the corresponding loss involved in direct transmission ruins the effectiveness of quantum key distribution by reducing the effective rate exponentially with the distance. To circumvent the losses involved in direct transmission, quantum repeater architectures have been proposed. We present our contributions towards three aspects of quantum repeater systems in this thesis. We ensure conditions for implementing quantum repeaters with atomic ensembles, explore the option of optomechanical systems for implementing quantum repeaters and verify the success of completed quantum repeater protocols. In the first part of this thesis, we show how we can ensure conditions for the successful implementation of quantum repeater systems with atomic ensembles. These quantum repeater systems are formed with 1-dimensional networks, where the nodes are made up of quantum memories connected by means of single photons. This requires memories that are highly efficient. Also, if quantum repeater systems are implemented with hybrid resources, tunable photon waveforms will be desirable. We propose a protocol to implement quantum memories with atomic ensembles using a clear recipe to optimise the efficiency. We also demonstrate that a cold ensemble of Rubidium-87 can act as an efficient tunable source of single photons, along with flexibility in the produced temporal shapes. Next, we show how we can explore alternative options for the nodes of quantum repeater systems. We focus on optomechanical oscillators, and recognise that they can also be used as quantum memories. We present a witness to certify that this memory successfully operates in the quantum regime. Finally, we focus on the verification of successfully implemented quantum repeater protocols. This verification will be essential for certifying that quantum repeater systems operate as instructed. We use only local homodyne measurements to witness the success of the network, and find that the witness is robust to loss. We thus present distinct contributions towards three important aspects of quantum repeater systems. As far as a full-fledged quantum repeater system might seem to be right now, we have faith that our work brings the field of quantum-enabled secure communications forward.

Sum sideband generation in a generic optomechanical system is discussed in the parameter configuration of optomechanically induced transparency. The nonlinear terms of the optomechanical dynamics are taken account and the features of the sum sideband generation are identified based on the analytical treatment. The nonlinear optomechanical interactions between cavity fields and the mechanical oscillation, which emerging as a new frontier in cavity optomechanics, are responsible for the generation of the frequency components at the sum sideband. We analyze in detail the influences of some parameters, including the pump power of the control field and the frequencies of the probe fields, on the sum sideband generation. The results clearly indicate that sum sideband generation can be significantly enhanced via achieving the matching conditions. The effect of sum sideband generation may be accessible in experiments and have potential application for achieving high precision measurement and on-chip manipulation of light propagation.

Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. The radiation-pressure force can be used for cooling atomic motion. Laser cooling was experimentally realized in the 1980s and has since become an extraordinarily important technique. Many applications have been enabled by laser cooling, including optical atomic clocks, precision measurements of the gravitational field, and systematic studies of quantum many-body physics in trapped clouds of atoms. Since the advent of both optical microcavities and advanced nanofabrication techniques, the optomechanical interaction could be studied in mesoscopic mechannical resonator. In 2005, Kippenberg et al. discovered that optical microtoroid resonators with their high optical finesse at the same time contain mechanical modes and thus are able to display optomechanical effects, in particular, radiation-pressure-induced self-oscillations. During the last decade, cavity optomechanics has advanced rapidly and optomechanical coupling has been reported in numerous novel systems. Furthermore, a number of remarkable optomechanical phenomena have been demonstrated in experiments, such as amplifying and cooling of phonon modes, optomechanically induced transparency, strong coupling between mechanical oscillator and optical cavity, optical wavelength conversion, conversion between optical light and microwave, optomechanical light storage and optomechanically induced non-reciprocity. In future, both in the field of precision sensing, classical and quantum information processing, cavity optomechanics will play an increasingly important role.

We report on numerical analysis of opto-mechanical (OM) cavities, with a full description of the rigorous numerical scheme involved, with emphasis on the mechanical stress induced by EM fields. In the future, OM structures are likely to be integrated in planar technology as pumped phonon sources and detectors, as well as phonon processing components (waveguides, splitters, memories) to process information by means of phonons. The solution of the equations governing the light behavior in opto-mechanical cavities is formally addressed by the Transformation Optics (TO) method. Such kind of calculation will help to engineer the coupling and the synchronization of GHz to tens of GHz coherent phonon sources.

The Laser Interferometric Gravitational-Wave Observatory (LIGO) is designed to detect the Gravitational Waves (GW) predicted by Albert Einstein’s general theory of relativity. The advanced LIGO project is ongoing an upgrade to increase the detection sensitivity by more than a factor of 10, which will make the events detection a routine occurrence. In addition to using higher power lasers, heavier test mass, and better isolation systems, several new designs and techniques are proposed in the long-term upgrade, such as modifying the optics configuration to reduce the quantum noise, active noise cancellation of the Newtonian noise, optimizing the coating structure, and employing non-Guassian laser beams etc. In the first part of my thesis (Chapters 2 and 3), I apply statistical mechanics and elastostatics to the LIGO coated mirrors, and study the thermal fluctuations that dominate advanced LIGO’s most sensitive frequency band from 40 Hz to 200 Hz. In particular, in Chapter 2, I study the so-called coating Brownian noise, fluctuations of mirrors coated with multiple layers of dielectrics due to internal friction. Assuming coating materials to be isotropic and homogeneous, I calculate the cross spectra of Brownian fluctuations in the bulk and shear strains of the coating layers, as well as fluctuations in the height of the coating-substrate interface. The additional phase shifting and back-scattering caused by photo elastic effects are also considered for the first time. In Chapter 3, I study whether it is realistic to adopt higher-order Laguerre-Gauss modes in LIGO, in order to mitigate the effect of mirror thermal noise. We investigate the effect on the detector’s contrast defect caused by the mode degeneracy. With both analytical calculation and numerical simulation, we show that with this approach, the detector’s susceptibility to mirror figure errors is reduced greatly compared to using the nondegenerate modes, therefore making it unacceptable for LIGO requirements. For the future GW detectors, with much lower noises and higher sensitivity, this might be used to investigate the quantum behaviors of macroscopic mechanical objects. In recent years the linear optomechanical systems with cavity modes coupling to a mechanical oscillator have been studied extensively. In the second part of my thesis (Chapter 4), I study the interaction between a single photon and a high-finesse cavity with a movable mirror, in the so-called strong coupling regime, where the recoil of the photon can cause significant change in the momentum of the mirror. The results are applied to analyze the case with a Fabry-Perot cavity. We also present that with engineering the photon wave function, it is possible to prepare the oscillator into an arbitrary quantum state.

Mechanical resonators are used in a wide variety of technical applications, from precision time keeping and sensing, to the delay and filtering of microwave signals in mobile communication systems. Critical to many of these applications is the ability of a mechanical object to store vibrational energy at a well defined frequency of oscillation and with minimal damping. Energy damping can occur through acoustic radiation into the resonator support structure, or through impurities and defects in the resonator material, and is highly dependent on the temperature of operation due to the inherent anharmonic motion of atoms within solid-state materials. Here, we present optical measurements down to milliKelvin temperatures of the acoustic mode properties of a crystalline silicon nanobeam cavity incorporating a three-dimensional phononic bandgap support structure for acoustic confinement. Utilizing pulsed laser light to excite a co-localized optical mode of the optomechanical crystal (OMC) device, we are able to measure the dynamics of the internal cavity acoustic modes which are coupled to the light field via radiation pressure. These measurements represent an almost ideal scenario in which the ringdown occurs free of any additional mechanical or probe field contact, and where elastic scattering or radiation of the acoustic field does not lead to energy damping due to the full bandgap shield. The resulting ringdown measurements for the fundamental 5 GHz acoustic mode of the cavity show an exponential increase in phonon lifetime versus phononic shield period number, which at a bath temperature of 35 milliKelvin saturates above six periods to a value as long as 1.5 seconds. This ultra-long lifetime, corresponding to an effective phonon propagation length of several kilometers, is found at the lowest temperatures to be consistent with damping from non-resonant tunneling states whose energy lies below the acoustic shield phononic bandgap, and which are most likely present in the amorphous etch-damaged region of the silicon surface. Other, more rapid forms of damping such as resonant tunneling state damping or three-phonon scattering are suppressed due to the phononic bandgap shield and the reduced density of phonon states in the effectively one-dimensional nanobeam geometry. Prospects for newapplications of ultra-coherent nanoscale mechanical resonators include tests of various collapse models of quantum mechanics, or, if appropriately integrated with microwave superconducting quantum circuits, as miniature quantum memory or processing units with potentially many-orders of magnitude longer coherence time than their electromagnetic counterparts.

We present a scheme to realize quantum thermal transistor effects in a continuous-variable electromechanical system including two microwave cavities and one mechanical oscillator. The thermal noise fluxes between the quantum system and its baths are evaluated by quantum master equation. It is shown that the thermal noise flux at one microwave cavity as an emitter can be dissipated into the other as a collector by combining the heating Stokes and cooling anti-Stokes processes. The indirect energy transfers between the two microwave modes can be significantly amplified by small energy changes at the mechanical oscillator as the base. The extremely high amplification depends sensitively on the detunings of the two microwave modes, which provides a new tool for precision measurements. This study opens the door for constructing quantum thermal transistors using various continuous-variable systems and is well accessible based on current experimental techniques.

We demonstrate all-optical control of the spontaneous-emission rate within the emission lifetime in copper-doped-silicon nanobeam optomechanical cavities via mechanical oscillation driven by repetitive laser pulses, representing the first experimental realization of optomechanical cavity QED systems.

A universal practical formula is given for calculating an integral which includes two confluent hypergeometric functions, power and exponential functions; then by means of this formula, the expressions of the radial matrix elements for a relativistic harmonic oscillator are given.

Accessing the full quantum nature of a macroscopic mechanical oscillator first requires elimination of its classical, thermal motion. The flourishing field of cavity optomechanics provides a nearly ideal architecture for both preparation and detection of mechanical motion at the quantum level. We realize a microwave cavity optomechanical system by coupling the motion of an aluminum membrane to the resonance frequency of a superconducting circuit [1]. By exciting the microwave circuit below its resonance frequency, we damp and cool the membrane motion with radiation pressure forces, analogous to laser cooling of the motion of trapped ions. The microwave excitation serves not only to cool, but also to monitor the displacement of the membrane. A nearly shot-noise limited, Josephson parametric amplifier is used to detect the mechanical sidebands of this microwave excitation and quantify the thermal motion as it is cooled with radiation pressure forces to its quantum ground state [2].

Diverse exceptional points (EPs) are theoretically studied in an experimentally feasible magno‐optomechanics consisting of an optomechanical subsystem coupled to a magnomechanical subsystem via physically direct contact. By adiabatically eliminating both the cavity and the Kittel mode, dissipative and parity‐time symmetric exceptional points can be observed. When only the cavity mode is eliminated, a second (third)‐order pseudo‐Hermitian EP emerges for nondegenerate (degenerate) mechanical modes. The distinct dynamical behavior of two mechanical modes around these EPs are further studied. The proposal provides a promising way to engineer diverse EPs and quantify non‐Hermitian phase transition with exceptional dynamical behavior in magno‐optomechanics.

In the present paper we enlarge the Max Planck law E = h(Ω 1 + Ω 2 ) for non-Hermitian operators and have examined the behavior of PT and CPT symmetries. For a quantum theory based on a non-Hermitian Hamiltonian, the spectrum in the general case is complex, and the time-evolution is non-unitary. We study also the case of non-commutative and non-Hermitian deformed harmonic oscillator and the behaviour of CPT symmetry. For further research, there appears the main problem of the connection between noncommutative geometry for non-Hermitian operators just as the conservation or not of PT and CPT symmetries.

(1990). The Geometry of Harmonic Oscillators. The American Mathematical Monthly: Vol. 97, No. 6, pp. 457-465.

In this paper we discuss the extension to exponential splitting methods with respect to time-dependent operators. We concentrate on the Suzuki’s method, which incorporates ideas to the time-ordered exponential of [3,11,12,34]. We formulate the methods with respect to higher order by using kernels for an extrapolation scheme. The advantages include more accurate and less computational intensive schemes to special time-dependent harmonic oscillator problems. The benefits of the higher order kernels are given on different numerical examples.

In this paper, the physical nature of quantum usual and self-friction (SF) harmonic oscillators is presented. The procedure for studying these harmonic oscillators is identical; therefore, we can benefit from the theory of the usual harmonic oscillator. To study the SF harmonic oscillator, using analytical formulae for the $$L^{{(p_{l}^{ * } )}}$$ -SF Laguerre polynomials ( $$L^{{(p_{l}^{ * } )}}$$ -SFLPs) and $$L^{{(\alpha^{*} )}}$$ -modified SFLPs ( $$L^{{(\alpha^{*} )}}$$ -MSFLPs) in standard convention, the $$V^{{(p_{l}^{ * } )}}$$ -SF potentials ( $$V^{{(p_{l}^{ * } )}}$$ -SFPs), $$V^{{(\alpha^{*} )}}$$ -modified SFPs ( $$V^{{(\alpha^{*} )}}$$ -MSFPs), $$F^{{(p_{l}^{ * } )}}$$ -SF forces ( $$F^{{(p_{l}^{ * } )}}$$ -SFFs) and $$F^{{(\alpha^{*} )}}$$ -modified SFFs ( $$F^{{(\alpha^{*} )}}$$ -MSFFs) are investigated, where $$p_{l}^{ * } = 2l + 2 - \alpha^{*}$$ and $$\,\alpha^{*}$$ is the integer ( $$\alpha^{*} = \alpha$$ , $$\, - \infty < \alpha \le 2)$$ or non-integer ( $$\alpha^{*} \ne \alpha$$ , $$\, - \infty < \alpha < 3)$$ SF quantum number. We note that the potentials ( $$V^{{(p_{l}^{ * } )}}$$ -SFPs and $$V^{{(\alpha^{*} )}}$$ -MSFPs), and forces ( $$F^{{(p_{l}^{ * } )}}$$ -SFFs and $$F^{{(\alpha^{*} )}}$$ -MSFFs), respectively, are independent functions. It is shown that the numerical values of these independent functions are the same, i.e., $$V_{num}^{{(p_{l}^{ * } )}} = V_{num}^{{(\alpha^{*} )}}$$ and $$F_{num}^{{(p_{l}^{ * } )}} = F_{num}^{{(\alpha^{*} )}}$$ . The dependence of the SF harmonic oscillator as a function of the distance is analyzed. The presented relationships are valid for arbitrary values of parameters.