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Macroscopic quantum effects based on Kerr nonlinearities

Tian Wang

Published Jan 1, 2014• 0 citations
Publisher

Abstract

Glorious victories have been achieved when quantum theory (QT) is applied to microscopic systems. However, although there might be good reasons for us to believe that QT applies at the macroscopic level as well, to give a definite answer ”yes” there is still a long journey. If it does apply, a direct result is that it predicts highly counter-intuitive macroscopic quantum superpositions and entanglements, which we never experience in our daily lives. In this thesis we assume that QT applies to the macroscopic level, and try to find out why we never really observe macroscopic quantum effects. The thesis contains two projects, aiming at two reasons for the above problem. In the first project (Chapter 3 and 4), we show that the required resolution to observe macroscopic quantum effects increases with the size of the system, when both outcome precision and control precision are taken into account. This means that for really large quantum effects we need a very good measuring resolution to observe them, while what we usually do are coarse-grained measurements, whose resolutions are much lower. In the second project (Chapter 5), we try to deal with decoherence, another obstacle preventing us from observing macroscopic quantum effects. We propose to create and detect strong entanglement of micro-macro and macro-macro beams of photons with very weak cross Kerr nonlinearities that are obtainable by current technology. We analyze the entanglement under environmental decoherence with various methods, and show that strong entanglement can still be created and detected under decoherence. We hope that the above results will help to push the boundary of the realm of QT towards the macroscopic level.

DOI
10.11575/prism/26808

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Recent advances in quantum technologies enabled us to make large quantum states \nand pushed towards examining quantum theory at the macroscopic level. However observation \nof quantum e ects at a macroscopic level still remains a demanding task. In this \nthesis we try to address one of the challenges and propose and explore some new solutions. \nOne of the obstacles for observation of macroscopic quantum e ects is the sensitivity \nto the measurement resolution. For many di erent cases, it has been observed that the \nprecision requirement for measuring quantum e ects increases with the system size. We \nformalize this as a conjecture that for observation of macroscopic quantum e ects, either \nthe outcome precision or the control precision of the measurements has to increase with \nsystem size. This indicates that the complexity of macroscopic quantum measurement \nincreases with the system size and sheds some lights on the quantum-to-classical transition \nat the macroscopic level. \nWe also introduce a technique to go around the sensitivity problem for observation \nof micro-macro entanglement. We propose that using a unitary deampli cation process, \none can bring the system back to the microscopic level where the measurements are less \ndemanding and quantum e ects are easier to verify. As the unitary processes do not change \nthe entanglement, this serves as a veri cation tool for micro-macro entanglement. \nWe also explored the connection between quantum e ects and thermodynamics of \nmacroscopic quantum systems for two speci c cases. For one, we investigated the e ect of \nentanglement in composite bosons and Bose-Einstein condensation. We showed that as the \nstate of the composite boson approaches a maximally entangled state, the condensation \nrate also approaches one. \nThe other case we considered was heat-bath algorithmic cooling. We found the cooling \nlimit of this class of thermodynamic transformations and showed that it decreases \nexponentially with the number of qubits. \nWe also developed an entropic version of Mermin's inequality. Here the idea is to \ndevelop a tool to reveal the entanglement in many-body quantum systems based on the \nentropy of the measurement outcomes. We introduce a new inequality that holds for locally \nrealistic models, yet can be violated with quantum measurements. One of the nice features \nof this inequality is that it can be violated maximally with quantum measurements. This \nresembles the GHZ paradox but for entropies of the measurement outcomes.

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Glorious victories have been achieved when quantum theory (QT) is applied to microscopic systems. However, although there might be good reasons for us to believe that QT applies at the macroscopic level as well, to give a definite answer ”yes” there is still a long journey. If it does apply, a direct result is that it predicts highly counter-intuitive macroscopic quantum superpositions and entanglements, which we never experience in our daily lives. In this thesis we assume that QT applies to the macroscopic level, and try to find out why we never really observe macroscopic quantum effects. The thesis contains two projects, aiming at two reasons for the above problem. In the first project (Chapter 3 and 4), we show that the required resolution to observe macroscopic quantum effects increases with the size of the system, when both outcome precision and control precision are taken into account. This means that for really large quantum effects we need a very good measuring resolution to observe them, while what we usually do are coarse-grained measurements, whose resolutions are much lower. In the second project (Chapter 5), we try to deal with decoherence, another obstacle preventing us from observing macroscopic quantum effects. We propose to create and detect strong entanglement of micro-macro and macro-macro beams of photons with very weak cross Kerr nonlinearities that are obtainable by current technology. We analyze the entanglement under environmental decoherence with various methods, and show that strong entanglement can still be created and detected under decoherence. We hope that the above results will help to push the boundary of the realm of QT towards the macroscopic level.
It has recently been conjectured that detecting quantum effects such as superposition or entanglement for macroscopic systems always requires high measurement precision. Analyzing an apparent counter-example involving macroscopic coherent states and Kerr non-linearities, we find that while measurements with coarse outcomes can be sufficient, the phase control precision of the necessary non-linear operations has to increase with the size of the system. This suggests a refined conjecture that either the {\it outcome precision} or the {\it control precision} of the measurements has to increase with system size.
In this thesis, I studied two different approaches which are promising for creating macroscopic quantum superpositions. The first approach, which is the main focus of this thesis, is based on optomechanical systems. The basic challenge to realize quantum effects in optomechanical systems, is considered to be the weak optomechanical coupling rate compared to the decoherence rates of the system caused by the cavity and mechanical damping. To enhance the optomechanical coupling it has been suggested to pump the cavity with coherent laser light \cite{Vitali07}. By this method the strong coupling regime in which the optomechanical coupling rate exceeds all decoherence rates has been demonstrated very recently \cite{Verhagen12,Groblacher09}. It is also known that the optomechanical systems may exhibit nonlinear behaviours such as bistability. After introducing the basic ideas of optomechanical systems in chapter 1, in chapter 2 we studied the connection between optomechanical entanglement and bistability. In chapter 3, we studied the drawback of using a laser to enhance the interaction. We studied the effect of the laser phase noise, as an additional decoherence channel, on the optomechanical entanglement. We have shown that the system is very sensitive to this new source of decoherence and the sensitivity is maximum in the bistable threshold. We discussed that our results are independent of how we model the laser phase noise. In chapter 4, we studied single photon optomechanics. In this regime, we proposed a new scheme to enhance the optomechanical coupling based on post selection. Our scheme is potentially promising to create a superposition of the massive oscillator in different motional states. We also discussed the potential application of our scheme to test some decoherence models. The second approach that we consider in this thesis is based on amplifying single photons with high gain to create micro-macro superpositions . In chapter 5, we discussed the De Martini scheme. We then present our scheme for creating optical micro-macro superposition and discussed its experimental realization under realistic conditions when losses are taken into account.
It has recently been conjectured that detecting quantum effects such as superposition or entanglement for macroscopic systems always requires high measurement precision. Analyzing an apparent counterexample involving macroscopic coherent states and Kerr or higher-order nonlinearities, we find that while measurements with coarse outcomes can be sufficient, the phase control precision of the necessary nonlinear operations has to increase with the size of the system. This suggests a refined conjecture that either the outcome precision or the control precision of the measurements has to increase with system size.
Entangled coherent states, which can in principle be created using strong Kerr non-linearities, allow the violation of Bell inequalities for very coarse-grained measurements. This seems to contradict a recent conjecture that observing quantum effects in macroscopic systems generally requires very precise measurements. However, here we show that both the creation of the required states and the required measurements rely on being able to control the phase of the necessary Kerr-nonlinearity based unitary operations with extreme precision. This lends support to the idea that there is a general principle that makes macroscopic quantum effects difficult to observe, even in the absence of decoherence.
Recent advances in quantum technologies enabled us to make large quantum states \nand pushed towards examining quantum theory at the macroscopic level. However observation \nof quantum e ects at a macroscopic level still remains a demanding task. In this \nthesis we try to address one of the challenges and propose and explore some new solutions. \nOne of the obstacles for observation of macroscopic quantum e ects is the sensitivity \nto the measurement resolution. For many di erent cases, it has been observed that the \nprecision requirement for measuring quantum e ects increases with the system size. We \nformalize this as a conjecture that for observation of macroscopic quantum e ects, either \nthe outcome precision or the control precision of the measurements has to increase with \nsystem size. This indicates that the complexity of macroscopic quantum measurement \nincreases with the system size and sheds some lights on the quantum-to-classical transition \nat the macroscopic level. \nWe also introduce a technique to go around the sensitivity problem for observation \nof micro-macro entanglement. We propose that using a unitary deampli cation process, \none can bring the system back to the microscopic level where the measurements are less \ndemanding and quantum e ects are easier to verify. As the unitary processes do not change \nthe entanglement, this serves as a veri cation tool for micro-macro entanglement. \nWe also explored the connection between quantum e ects and thermodynamics of \nmacroscopic quantum systems for two speci c cases. For one, we investigated the e ect of \nentanglement in composite bosons and Bose-Einstein condensation. We showed that as the \nstate of the composite boson approaches a maximally entangled state, the condensation \nrate also approaches one. \nThe other case we considered was heat-bath algorithmic cooling. We found the cooling \nlimit of this class of thermodynamic transformations and showed that it decreases \nexponentially with the number of qubits. \nWe also developed an entropic version of Mermin's inequality. Here the idea is to \ndevelop a tool to reveal the entanglement in many-body quantum systems based on the \nentropy of the measurement outcomes. We introduce a new inequality that holds for locally \nrealistic models, yet can be violated with quantum measurements. One of the nice features \nof this inequality is that it can be violated maximally with quantum measurements. This \nresembles the GHZ paradox but for entropies of the measurement outcomes.
We study the entanglement generated by a weak cross-Kerr nonlinearity between two initial coherent states, one of which has an amplitude close to the single-photon level, while the other one is macroscopic. We show that strong micro-macro entanglement is possible for weak phase shifts by choosing the amplitude of the macroscopic beam sufficiently large. We analyze the effects of loss and discuss possible experimental demonstrations of the micro-macro entanglement based on homodyne tomography and an entanglement witness. We also discuss the macroscopicity of the generated entanglement from the point of view of a recently proposed measure.