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Quantum Entanglement and Networking with Spin-Optomechanics

VA Montenegro Tobar

Published Oct 28, 2015• 0 citations
OpenAlex

Abstract

Non-relativistic quantum mechanics have proven to be a significant framework to understand the non-classical behaviour of light and the microcosmos. Perhaps, one of the first technological revolutions within quantum theory came with the invention of the transistor, whereby a purely quantum mechanical description was required. Currently, another outstanding revolution is taking place in a crossroad where information science meets quantum mechanics (this being the quantum information field). Such an area of work contemplates both the fascinating theoretical aspect of quantum correlations, as well as implementations towards quantum tasks performed by a universal quantum computer; tasks that cannot be realised (or they are hard to implement) within the classical domain. This Thesis is devoted to study the dynamics of quantum entanglement in spin-optomechanics systems. In particular, we explore the quantum stabilization of quantum entanglement, a quantum concentration scheme in opto-mechanics and an interfacing of matter and light towards quantum networking applications. Additionally, we also investigate theoretical aspects of quantum correlations within thermal environments, as well as the topical area of quantum sudden transitions. In Chapter 1, we provide a brief summary of quantum information and of the quantum optics framework to cover elementary concepts and techniques used subsequently in this work. Subsequently, in Chapter 2 we present the stabilization of quantum entanglement in a non-linear qubit-oscillator system. The inclusion of a modest nonlinearity gives three results, i) the loss of periodicity of the system, ii) the occurrence of quadrature squeezing appearing for a short time, and iii) the quantum entanglement reaches higher values in contrast to the case without non-linearity. In Chapter 3, a technique to concentrate/distill a two-mode vacuum state in optomechanics via unsharp measurements is presented. Here, one of the optical modes is injected into a cavity at first, and thereafter, it is nonlinearly coupled to a mechanical oscillator. Afterwards, the position of the oscillator is measured using pulsed optomechanics and homodyne detection. The results show that this measurement can conditionally increase the initial entanglement. Next, in Chapter 4, stimulated by optomechanical transducers and quantum networking, a light-matter system is constructed where a qubit is coupled to a cavity mode mediated through a mechanical oscillator. The qubit-oscillator conditionally displaced Hamiltonian and the oscillator-cavity radiation-pressure interaction generate a maximal qubit-cavity entanglement. Additionally, we consider the case in which the cavity mode is coupled to a waveguide, numerical calculations show a promising qubit-fibre entanglement under a weak matter-light coupling. For the quantum network case, we coupled a generic qubit in the first node to a second qubit-cavity distant Jaynes-Cummings system coupled through an optical fibre, where qubit-qubit correlations can be achieved in the quantum open systems scenario. In Chapter 5, we study the evolution of an open quantum system within the Born-Markov microscopic master equation (MME). Essentially, two distant two-level atoms are trapped in fibre-coupled cavities. Under the approximation of one-excitation allowed in the atom-cavity-fibre basis, we can obtain quantum correlations induced by thermal fluctuations from the environments. Lastly, in Chapter 6, we bring together previously elements explored in this Thesis. The system is a hybrid atomic-mechanical system formed from two remote qubits interacting with individual harmonic oscillators. This system, as in Chapter 4, explores interesting applications in quantum networking schemes. The two qubits are initially prepared in a Bell-diagonal state, and consequently the two-qubit correlations exhibit few interesting effects such as freezing, sudden changes and revivals in the evolution of the quantum entropic discord. To conclude, I summarize my findings in Chapter 7.

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In a nutshell, a cavity optomechanical system consists of a mechanical oscillator inside an optical cavity. Cavity optomechanics studies the interplay between the light-matter radiation-pressure interaction and the change in the cavity frequency induced by mechanical motion. By leveraging the tools of quantum optics and quantum measurement, cavity quantum optomechanics enables the engineering of nonclassical quantum states of bulk matter. In this thesis, I present protocols for engineering such nonclassical quantum states in Part II. First, I introduce a scheme to ‘grow’ a quantum kitten state of a mechanical oscillator into a macroscopic Schrödinger cat state using a series of nonlinear pulsed optomechanical interactions. Next, I detail our proposed schemes to generate and verify entanglement between two mechanical oscillators. Moving on from pulsed optomechanics, I describe the theory behind an experiment that achieved multi-phonon subtraction from a mechanical oscillator in a continuously driven Brillouin optomechanical system. In the final chapter of Part II, I provide a theoretical preview of an experiment that investigates the behaviour of the mechanical oscillator when no light is detected at the output. In Part III, I introduce a new framework for nonlinear cavity quantum optomechanics that incorporates the intrinsic nonlinearity of the radiation-pressure interaction, the nonlinearity from the cavity enhancement, and higher-order mechanical position terms in the expansion of the cavity frequency. I then propose a scheme for measuring mechanical position using general-dyne detection, applicable to pulsed and continuous measurements. This predicts a new regime of operation, the “stochastic Gaussian” regime. Finally, I discuss a protocol for deterministic mechanical squeezing that utilizes the cavity response.

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This thesis concentrates on generating and measuring non-classical states of mechanical oscillators by coupling them to atomic and molecular quantum systems. We start with a discussion of what novel physics can be explored by mechanical systems operating in the quantum regime. We then discuss one technique making it a possibility- cavity optomechanics, particularly optomechanical cooling. We investigate the limits of optomechanical cooling and review how the coupling of mechanical oscillators to external heat baths limits the minimum attainable phonon occupation number. As a possible alternative for circumventing clamping losses, we consider an all-optical approach where the mechanical element (in this case, a Bragg mirror) is suspended via optical forces, and discuss some limitations of this approach. We explore several schemes aimed at the generation of quantum states in mechanical oscillators. We consider specifically two examples: one in which the mechanical oscillator is coupled to polar molecules via dipole-dipole interaction, and another where it is magnetically coupled to a Bose condensate. The first example emphasizes that such an interaction can generate parametric squeezing and entanglement. The second scheme demonstrates that the back action of BEC spin measurements can be used to generate quantum states of motion of a mechanical oscillator. We then discuss possible methods for measuring the entire density matrix of a mechanical oscillator. The first method achieves the tomographic reconstruction of the mechanical Wigner function by coupling it simultaneously to a classical optical oscillator and a qubit. The second approach involves a state transfer scheme between momentum excitations of a bose-condensate in a cavity and a moving mirror of the cavity that is entirely mediated by the light field. We conclude with a discussion of the broader implications of this work, and some future research directions.

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Quantum measurement and control in single-photon opto-mechanics

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Quantum opto-mechanics is a fast moving, broad research field which aims to investigate and control the interaction of light and the quantized motion of mechanical objects and the coupling between them. This allows one to engineer the mechanical quantum state by controlling and measuring the electromagnetic field, and conversely, to control the dynamical evolution of the optical field via its interaction with a mechanical system. Mechanical resonators are accessible in many different shapes and sizes, from kilogram scales in experiments that test gravitational theories to micro- and nano-scales in resolved sideband regime, where the frequency of the mechanical resonator is larger than the bandwidth of the optical cavity. In this regime, it is possible to cool the mechanical object to the ground state which allows a high degree of control and the ability to prepare interesting non-classical states, states that could aid in investigating the quantum dynamics of macroscopic objects. The field of quantum opto-mechanics is a promising testbed for quantum control technologies, such as fundamental tests of quantum mechanics, ultra-sensitive quantum sensors for precise measurements, phonon lasing, mechanical memories with long lifetime and quantum information processing. Furthermore, experiments are progressing toward strong opto-mechanical coupling regimes with single photons. This creates a platform that can take advantage of the nonlinear optical interactions to generate highly non-classical states in such light-matter interfaces. Photonic systems working in the single photon regime will be promising for low power applications and also for implementations of quantum information and computation. Motivated by these applications, this thesis consists of theoretical studies in the context of single photonics. This includes investigating non-classical mechanical state generation and mechanically controlling the dynamical evolution of optical fields in the strong coupling regime single photon opto-mechanics and furthermore, proposing a quantum sensor using single photonics. In this regard, we employ three important formalisms to treat and measure open quantum systems with non-Gaussian input fields: the conditional and unconditional cascaded master equations, Fock-state master equations and input-output Langevin equations. In the context of opto-mechanics, the system to be investigated is composed of two optical cavities with a coupling modulated via a mechanical resonator. In one case, a sequence of single photons are irreversibly sent to one of the optical cavities. After photons go through the opto-mechanical interaction, photon counting measurements are performed on the cavity output, conditionally steering the mechanical state to a highly non-classical Fock state. Numerical calculations are performed to predict the experimental parameters needed to generate a mechanical Fock state before the mechanical object is thermalized. In another case, the same opto-mechanical scheme is employed to conversely modify the state of the optical mode by engineering the state of the mechanical degree of freedom. In this case, it is shown that the opto-mechanical system operates as an effective beam splitter for the input single photons, with a transmission that can be controlled by tuning the number of excitations in the bulk mechanical resonator. When the mechanical resonator has a small coherent amplitude it acts as a quantum control, entangling the optical and mechanical degrees of freedom. As the coherent amplitude of the resonator increases, it acts as a classical control for the input photons. Finally, we propose a novel multi-purpose sensor architecture that can be used for force, refractive index and possibly local temperature detection. In this scheme, two coupled cavities behave as an effective beam splitter. This sensor is based on fourth order interference (the Hong-Ou-Mandel effect) and requires a sequence of single photon pulses and consequently has low pulse power. Changes in the parameter to be measured induce variations in the effective beam splitter reflectivity and result in changes to the visibility of interference. We propose that this generic scheme could be implemented with coupled L3 photonic crystal cavities and find that this system, which only relies on photon coincidence detection and does not need any spectral resolution, can estimate forces as small as 10-7 Newtons and can measure one part per million change in refractive index using a very low input power of 10-10 W.

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In this thesis we have focused on the study of various systems which are presently widely studied in different areas of quantum optics and quantum information sciences. These, for example, include the coupled system of photonic waveguides which are known to be highly efficient in manipulating the flow of light. The Hamiltonian describing the evolution of field mode in coupled waveguides is effectively identical to the well-known tight binding Hamiltonian used in solid state physics. The advantage of waveguide system is the possibility to control various interactions by design and their low decoherence rate. The excellent stability offered by coupled waveguides has led to the observation of many key coherent effects such as quantum walk, Bloch oscillation, and discrete Talbot effect. For example, Bloch oscillations have been investigated in coupled waveguides using coherent beam of light. We wanted to inquire whether coherent phenomena such as Bloch oscillations can be possible with incoherent single photon sources. We discovered that Bloch oscillations are indeed possible with single photons provided we prepare single photons in a W state. Moreover, coupled waveguides also find applications in the field of quantum information processing. Since entanglement plays a prominent role in all these applications, it is important to understand the entanglement dynamics in these structures. We considered the case of squeezed input in one of the waveguide and showed that one can generate entanglement between the waveguide modes. We further continued our work on the entanglement generation in coupled waveguides by incorporating the effect of loss in the waveguide structure for the squeezed and photon number input states. We considered relevant experimental parameters and showed that waveguide structures are reasonably robust against the effect of loss. Another system which has attracted a great deal of interest is the optomechanical system. We consider an optomechanical system where an optical cavity mode is coupled to the square of the position of a mechanical oscillator. The optomechanical system can then be regarded as a quantum optical spring, i.e., a spring whose spring constant depends on the quantum state of another system. In particular, we consider the situation where the field inside the cavity is in a coherent state and the oscillator is prepared in its ground state. The quantized nature of the field produces new features in the optomechanical system.

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Lydia A. Kanari-Naish, Jack Clarke, Sofia Qvarfort +1 more

Abstract Cavity quantum optomechanics has emerged as a new platform for quantum science and technology with applications ranging from quantum-information processing to tests of the foundations of physics. Of crucial importance for optomechanics is the generation and verification of non-Gaussian states of motion and a key outstanding challenge is the observation of a canonical two-mode Schrödinger-cat state in the displacement of two mechanical oscillators. In this work, we introduce a pulsed approach that utilizes the nonlinearity of the radiation–pressure interaction combined with photon-counting measurements to generate this entangled non-Gaussian mechanical state, and, importantly, describe a protocol using subsequent pulsed interactions to verify the non-Gaussian entanglement generated. Our pulsed verification protocol allows quadrature moments of the two mechanical oscillators to be measured up to any finite order providing a toolset for experimental characterisation of bipartite mechanical quantum states and allowing a broad range of inseparability criteria to be evaluated. Key experimental factors, such as optical loss and open-system dynamics, are carefully analyzed and we show that the scheme is feasible with only minor improvements to current experiments that operate outside the resolved-sideband regime. Our scheme provides a new avenue for quantum experiments with entangled mechanical oscillators and offers significant potential for further research and development that utilizes such non-Gaussian states for quantum-information and sensing applications, and for studying the quantum-to-classical transition.

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Enabling entanglement distillation via optomechanics

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Quantum networking based on optical Gaussian states, although promising in terms of scalability, is hindered by the fact that their entanglement cannot be distilled via Gaussian operations. We show that optomechanics, and particularly the possibility to measure the mechanical degree of freedom in an integrable system, can address this problem. Here, one of the optical modes of a two-mode squeezed vacuum is injected into a single-sided Fabry-P\'erot cavity and nonlinearly coupled to a mechanical oscillator. Afterwards, the position of the oscillator is measured using pulsed optomechanics and homodyne detection. We show that this measurement can supply non-Gaussian entangled states frequently enough to enable scalable entanglement distillation. Moreover, it can conditionally increase the initial entanglement under an optimal radiation-pressure interaction strength, which corresponds to an effective unsharp measurement of the photon number inside the cavity. We show how the resulting entanglement enhancement can be verified by using a standard teleportation procedure.

arXiv:1503.04462•Physical Review A•Oct 14, 2019•8 citations
Non-relativistic quantum mechanics have proven to be a significant framework to understand the non-classical behaviour of light and the microcosmos. Perhaps, one of the first technological revolutions within quantum theory came with the invention of the transistor, whereby a purely quantum mechanical description was required. Currently, another outstanding revolution is taking place in a crossroad where information science meets quantum mechanics (this being the quantum information field). Such an area of work contemplates both the fascinating theoretical aspect of quantum correlations, as well as implementations towards quantum tasks performed by a universal quantum computer; tasks that cannot be realised (or they are hard to implement) within the classical domain. This Thesis is devoted to study the dynamics of quantum entanglement in spin-optomechanics systems. In particular, we explore the quantum stabilization of quantum entanglement, a quantum concentration scheme in opto-mechanics and an interfacing of matter and light towards quantum networking applications. Additionally, we also investigate theoretical aspects of quantum correlations within thermal environments, as well as the topical area of quantum sudden transitions. In Chapter 1, we provide a brief summary of quantum information and of the quantum optics framework to cover elementary concepts and techniques used subsequently in this work. Subsequently, in Chapter 2 we present the stabilization of quantum entanglement in a non-linear qubit-oscillator system. The inclusion of a modest nonlinearity gives three results, i) the loss of periodicity of the system, ii) the occurrence of quadrature squeezing appearing for a short time, and iii) the quantum entanglement reaches higher values in contrast to the case without non-linearity. In Chapter 3, a technique to concentrate/distill a two-mode vacuum state in optomechanics via unsharp measurements is presented. Here, one of the optical modes is injected into a cavity at first, and thereafter, it is nonlinearly coupled to a mechanical oscillator. Afterwards, the position of the oscillator is measured using pulsed optomechanics and homodyne detection. The results show that this measurement can conditionally increase the initial entanglement. Next, in Chapter 4, stimulated by optomechanical transducers and quantum networking, a light-matter system is constructed where a qubit is coupled to a cavity mode mediated through a mechanical oscillator. The qubit-oscillator conditionally displaced Hamiltonian and the oscillator-cavity radiation-pressure interaction generate a maximal qubit-cavity entanglement. Additionally, we consider the case in which the cavity mode is coupled to a waveguide, numerical calculations show a promising qubit-fibre entanglement under a weak matter-light coupling. For the quantum network case, we coupled a generic qubit in the first node to a second qubit-cavity distant Jaynes-Cummings system coupled through an optical fibre, where qubit-qubit correlations can be achieved in the quantum open systems scenario. In Chapter 5, we study the evolution of an open quantum system within the Born-Markov microscopic master equation (MME). Essentially, two distant two-level atoms are trapped in fibre-coupled cavities. Under the approximation of one-excitation allowed in the atom-cavity-fibre basis, we can obtain quantum correlations induced by thermal fluctuations from the environments. Lastly, in Chapter 6, we bring together previously elements explored in this Thesis. The system is a hybrid atomic-mechanical system formed from two remote qubits interacting with individual harmonic oscillators. This system, as in Chapter 4, explores interesting applications in quantum networking schemes. The two qubits are initially prepared in a Bell-diagonal state, and consequently the two-qubit correlations exhibit few interesting effects such as freezing, sudden changes and revivals in the evolution of the quantum entropic discord. To conclude, I summarize my findings in Chapter 7.
In a nutshell, a cavity optomechanical system consists of a mechanical oscillator inside an optical cavity. Cavity optomechanics studies the interplay between the light-matter radiation-pressure interaction and the change in the cavity frequency induced by mechanical motion. By leveraging the tools of quantum optics and quantum measurement, cavity quantum optomechanics enables the engineering of nonclassical quantum states of bulk matter. In this thesis, I present protocols for engineering such nonclassical quantum states in Part II. First, I introduce a scheme to ‘grow’ a quantum kitten state of a mechanical oscillator into a macroscopic Schrödinger cat state using a series of nonlinear pulsed optomechanical interactions. Next, I detail our proposed schemes to generate and verify entanglement between two mechanical oscillators. Moving on from pulsed optomechanics, I describe the theory behind an experiment that achieved multi-phonon subtraction from a mechanical oscillator in a continuously driven Brillouin optomechanical system. In the final chapter of Part II, I provide a theoretical preview of an experiment that investigates the behaviour of the mechanical oscillator when no light is detected at the output. In Part III, I introduce a new framework for nonlinear cavity quantum optomechanics that incorporates the intrinsic nonlinearity of the radiation-pressure interaction, the nonlinearity from the cavity enhancement, and higher-order mechanical position terms in the expansion of the cavity frequency. I then propose a scheme for measuring mechanical position using general-dyne detection, applicable to pulsed and continuous measurements. This predicts a new regime of operation, the “stochastic Gaussian” regime. Finally, I discuss a protocol for deterministic mechanical squeezing that utilizes the cavity response.
This thesis concentrates on generating and measuring non-classical states of mechanical oscillators by coupling them to atomic and molecular quantum systems. We start with a discussion of what novel physics can be explored by mechanical systems operating in the quantum regime. We then discuss one technique making it a possibility- cavity optomechanics, particularly optomechanical cooling. We investigate the limits of optomechanical cooling and review how the coupling of mechanical oscillators to external heat baths limits the minimum attainable phonon occupation number. As a possible alternative for circumventing clamping losses, we consider an all-optical approach where the mechanical element (in this case, a Bragg mirror) is suspended via optical forces, and discuss some limitations of this approach. We explore several schemes aimed at the generation of quantum states in mechanical oscillators. We consider specifically two examples: one in which the mechanical oscillator is coupled to polar molecules via dipole-dipole interaction, and another where it is magnetically coupled to a Bose condensate. The first example emphasizes that such an interaction can generate parametric squeezing and entanglement. The second scheme demonstrates that the back action of BEC spin measurements can be used to generate quantum states of motion of a mechanical oscillator. We then discuss possible methods for measuring the entire density matrix of a mechanical oscillator. The first method achieves the tomographic reconstruction of the mechanical Wigner function by coupling it simultaneously to a classical optical oscillator and a qubit. The second approach involves a state transfer scheme between momentum excitations of a bose-condensate in a cavity and a moving mirror of the cavity that is entirely mediated by the light field. We conclude with a discussion of the broader implications of this work, and some future research directions.
Quantum opto-mechanics is a fast moving, broad research field which aims to investigate and control the interaction of light and the quantized motion of mechanical objects and the coupling between them. This allows one to engineer the mechanical quantum state by controlling and measuring the electromagnetic field, and conversely, to control the dynamical evolution of the optical field via its interaction with a mechanical system. Mechanical resonators are accessible in many different shapes and sizes, from kilogram scales in experiments that test gravitational theories to micro- and nano-scales in resolved sideband regime, where the frequency of the mechanical resonator is larger than the bandwidth of the optical cavity. In this regime, it is possible to cool the mechanical object to the ground state which allows a high degree of control and the ability to prepare interesting non-classical states, states that could aid in investigating the quantum dynamics of macroscopic objects. The field of quantum opto-mechanics is a promising testbed for quantum control technologies, such as fundamental tests of quantum mechanics, ultra-sensitive quantum sensors for precise measurements, phonon lasing, mechanical memories with long lifetime and quantum information processing. Furthermore, experiments are progressing toward strong opto-mechanical coupling regimes with single photons. This creates a platform that can take advantage of the nonlinear optical interactions to generate highly non-classical states in such light-matter interfaces. Photonic systems working in the single photon regime will be promising for low power applications and also for implementations of quantum information and computation. Motivated by these applications, this thesis consists of theoretical studies in the context of single photonics. This includes investigating non-classical mechanical state generation and mechanically controlling the dynamical evolution of optical fields in the strong coupling regime single photon opto-mechanics and furthermore, proposing a quantum sensor using single photonics. In this regard, we employ three important formalisms to treat and measure open quantum systems with non-Gaussian input fields: the conditional and unconditional cascaded master equations, Fock-state master equations and input-output Langevin equations. In the context of opto-mechanics, the system to be investigated is composed of two optical cavities with a coupling modulated via a mechanical resonator. In one case, a sequence of single photons are irreversibly sent to one of the optical cavities. After photons go through the opto-mechanical interaction, photon counting measurements are performed on the cavity output, conditionally steering the mechanical state to a highly non-classical Fock state. Numerical calculations are performed to predict the experimental parameters needed to generate a mechanical Fock state before the mechanical object is thermalized. In another case, the same opto-mechanical scheme is employed to conversely modify the state of the optical mode by engineering the state of the mechanical degree of freedom. In this case, it is shown that the opto-mechanical system operates as an effective beam splitter for the input single photons, with a transmission that can be controlled by tuning the number of excitations in the bulk mechanical resonator. When the mechanical resonator has a small coherent amplitude it acts as a quantum control, entangling the optical and mechanical degrees of freedom. As the coherent amplitude of the resonator increases, it acts as a classical control for the input photons. Finally, we propose a novel multi-purpose sensor architecture that can be used for force, refractive index and possibly local temperature detection. In this scheme, two coupled cavities behave as an effective beam splitter. This sensor is based on fourth order interference (the Hong-Ou-Mandel effect) and requires a sequence of single photon pulses and consequently has low pulse power. Changes in the parameter to be measured induce variations in the effective beam splitter reflectivity and result in changes to the visibility of interference. We propose that this generic scheme could be implemented with coupled L3 photonic crystal cavities and find that this system, which only relies on photon coincidence detection and does not need any spectral resolution, can estimate forces as small as 10-7 Newtons and can measure one part per million change in refractive index using a very low input power of 10-10 W.
In this thesis we have focused on the study of various systems which are presently widely studied in different areas of quantum optics and quantum information sciences. These, for example, include the coupled system of photonic waveguides which are known to be highly efficient in manipulating the flow of light. The Hamiltonian describing the evolution of field mode in coupled waveguides is effectively identical to the well-known tight binding Hamiltonian used in solid state physics. The advantage of waveguide system is the possibility to control various interactions by design and their low decoherence rate. The excellent stability offered by coupled waveguides has led to the observation of many key coherent effects such as quantum walk, Bloch oscillation, and discrete Talbot effect. For example, Bloch oscillations have been investigated in coupled waveguides using coherent beam of light. We wanted to inquire whether coherent phenomena such as Bloch oscillations can be possible with incoherent single photon sources. We discovered that Bloch oscillations are indeed possible with single photons provided we prepare single photons in a W state. Moreover, coupled waveguides also find applications in the field of quantum information processing. Since entanglement plays a prominent role in all these applications, it is important to understand the entanglement dynamics in these structures. We considered the case of squeezed input in one of the waveguide and showed that one can generate entanglement between the waveguide modes. We further continued our work on the entanglement generation in coupled waveguides by incorporating the effect of loss in the waveguide structure for the squeezed and photon number input states. We considered relevant experimental parameters and showed that waveguide structures are reasonably robust against the effect of loss. Another system which has attracted a great deal of interest is the optomechanical system. We consider an optomechanical system where an optical cavity mode is coupled to the square of the position of a mechanical oscillator. The optomechanical system can then be regarded as a quantum optical spring, i.e., a spring whose spring constant depends on the quantum state of another system. In particular, we consider the situation where the field inside the cavity is in a coherent state and the oscillator is prepared in its ground state. The quantized nature of the field produces new features in the optomechanical system.
Abstract Cavity quantum optomechanics has emerged as a new platform for quantum science and technology with applications ranging from quantum-information processing to tests of the foundations of physics. Of crucial importance for optomechanics is the generation and verification of non-Gaussian states of motion and a key outstanding challenge is the observation of a canonical two-mode Schrödinger-cat state in the displacement of two mechanical oscillators. In this work, we introduce a pulsed approach that utilizes the nonlinearity of the radiation–pressure interaction combined with photon-counting measurements to generate this entangled non-Gaussian mechanical state, and, importantly, describe a protocol using subsequent pulsed interactions to verify the non-Gaussian entanglement generated. Our pulsed verification protocol allows quadrature moments of the two mechanical oscillators to be measured up to any finite order providing a toolset for experimental characterisation of bipartite mechanical quantum states and allowing a broad range of inseparability criteria to be evaluated. Key experimental factors, such as optical loss and open-system dynamics, are carefully analyzed and we show that the scheme is feasible with only minor improvements to current experiments that operate outside the resolved-sideband regime. Our scheme provides a new avenue for quantum experiments with entangled mechanical oscillators and offers significant potential for further research and development that utilizes such non-Gaussian states for quantum-information and sensing applications, and for studying the quantum-to-classical transition.
Quantum networking based on optical Gaussian states, although promising in terms of scalability, is hindered by the fact that their entanglement cannot be distilled via Gaussian operations. We show that optomechanics, and particularly the possibility to measure the mechanical degree of freedom in an integrable system, can address this problem. Here, one of the optical modes of a two-mode squeezed vacuum is injected into a single-sided Fabry-P\'erot cavity and nonlinearly coupled to a mechanical oscillator. Afterwards, the position of the oscillator is measured using pulsed optomechanics and homodyne detection. We show that this measurement can supply non-Gaussian entangled states frequently enough to enable scalable entanglement distillation. Moreover, it can conditionally increase the initial entanglement under an optimal radiation-pressure interaction strength, which corresponds to an effective unsharp measurement of the photon number inside the cavity. We show how the resulting entanglement enhancement can be verified by using a standard teleportation procedure.