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arXiv:1612.07691 · Physical Review A

Collapse in ultracold Bose Josephson junctions

Marco Bilardello, Andrea Trombettoni, Angelo Bassi

Published Mar 29, 2017• 14 citations
arXivHTMLPDF

Abstract

We investigate how ultracold atoms in double-well potentials can be used to study and put bounds on models describing wave-function collapse. We refer in particular to the continuous spontaneous localization (CSL) model, which is the most well studied among dynamical reduction models. It modifies the Schr\"odinger equation in order to include the collapse of the wave function in its dynamics. We consider Bose Josephson junctions, where ultracold bosons are trapped in a double-well potential, since they can be experimentally controlled with high accuracy and are suited and used to study macroscopic quantum phenomena on a scale of microns, with a number of particles typically ranging from $\ensuremath{\sim}{10}^{2}--{10}^{3}$ to $\ensuremath{\sim}{10}^{5}--{10}^{6}$. We study the CSL dynamics of three atomic states showing macroscopic quantum coherence: the atomic coherent state, the superposition of two atomic coherent states, and the NOON state. We show that for the last two states, the suppression of quantum coherence induced by the CSL model increases exponentially with the number of atoms. We observe that in the case of optically trapped atoms, the spontaneous photon emission of the atoms induces a dynamics similar to the CSL one, and we conclude that magnetically trapped atoms may be more convenient to experimentally test the CSL model. Finally, we discuss decoherence effects in order to provide reasonable estimates on the bounds that it is (or will be) possible to obtain for the parameters of the CSL model in such class of experiments. As an example, we show that a NOON state with $N\ensuremath{\sim}{10}^{3}$ with a coherence time of $\ensuremath{\sim}1$ s can constrain the CSL parameters in a region where the other systems presently cannot.

DOI
10.1103/physreva.95.032134

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Collapse models solve the measurement problem by adding non-linear and stochastic terms to the usual Schroedinger dynamics. The resulting dynamics localizes in position macroscopic systems, while preserves quantum mechanical predictions for microscopic systems. \nTwo of the main consequences of this resulting dynamics are: the loss of any initial spatial coherence, with a decoherence rate which increases with the dimensions of the system (amplification mechanism); and a heating effect induced on the system that, for long times, leads either to a divergent or to a convergent value of the energy, depending on the particular collapse model considered. \nAn interesting class of system to experimentally test collapse models are cold atomic gases. Besides their very low temperatures (T = 10^(-11) - 10^(-7)K), cold atomic gases show mesoscopic quantum properties, i.e. involving a number of atoms of order of N = 10^(3)– 10^(11). These are all necessary ingredients to experimentally test collapse models. In this thesis, we use cold bosonic atomic gas to test one of the most studied and complete among collapse models, i.e. CSL, and also its non-Markovian and dissipative extensions. In particular, in this thesis we put bounds on CSL free-parameters by studying the diffusion effect induced by collapse noise on a free expanding gas, and we compare the theoretical results with experimental data known in literature. The bounds we put are among the strongest ever found non only for CSL model, but especially for its non-Markovian extension. \nIn this thesis we also specialize the CSL dynamics in Bose-Josephson junctions, and we focus on the localization mechanism induced by the noise on initially entangled states. In particular, we focus on decoherence induced by CSL on typical entangled states as atomic coherent states and NOON states. We find that in the latter ones, CSL effects become significant when the NOON state is composed by a number of atoms of 10^(4) or greater. Moreover, we compare CSL decoherence with environmental decoherence induced by typical realistic sources, as an external thermal cloud, imperfections in the trapping laser, and three body effects. We then find which conditions experiments must fulfill in order to test CSL in a Bose-Josephson junction in conditions never

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{The Continuous Spontaneous Localization (CSL) theory alters the Schrodinger equation. It describes wave function collapse as a dynamical process instead of an ill-defined postulate, thereby providing macroscopic uniqueness and solving the so-called measurement problem of standard quantum theory. CSL contains a parameter $\lambda$ giving the collapse rate of an isolated nucleon in a superposition of two spatially separated states and, more generally, characterizing the collapse time for any physical situation. CSL is experimentally testable, since it predicts some behavior different from that predicted by standard quantum theory. One example is the narrowing of wave functions, which results in energy imparted to particles. Here we consider energy given to trapped ultra-cold atoms. Since these are the coldest samples under experimental investigation, it is worth inquiring how they are affected by the CSL heating mechanism. We examine the CSL heating of a BEC in contact with its thermal cloud. Of course, other mechanisms also provide heat and also particle loss. From varied data on optically trapped cesium BEC's, we present an energy audit for known heating and loss mechanisms. The result provides an upper limit on CSL heating and thereby an upper limit on the parameter $\lambda$. We obtain $\lambda\lesssim 1(\pm1)\times 10^{-7}$sec$^{-1}$.}

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We investigate how ultracold atoms in double-well potentials can be used to study and put bounds on models describing wave-function collapse. We refer in particular to the continuous spontaneous localization (CSL) model, which is the most well studied among dynamical reduction models. It modifies the Schr"odinger equation in order to include the collapse of the wave function in its dynamics. We consider Bose Josephson junctions, where ultracold bosons are trapped in a double-well potential, since they can be experimentally controlled with high accuracy and are suited and used to study macroscopic quantum phenomena on a scale of microns, with a number of particles typically ranging from \ensuremath∼102−−103\ensuremath{\sim}{10}^{2}--{10}^{3}\ensuremath∼102−−103 to \ensuremath∼105−−106\ensuremath{\sim}{10}^{5}--{10}^{6}\ensuremath∼105−−106. We study the CSL dynamics of three atomic states showing macroscopic quantum coherence: the atomic coherent state, the superposition of two atomic coherent states, and the NOON state. We show that for the last two states, the suppression of quantum coherence induced by the CSL model increases exponentially with the number of atoms. We observe that in the case of optically trapped atoms, the spontaneous photon emission of the atoms induces a dynamics similar to the CSL one, and we conclude that magnetically trapped atoms may be more convenient to experimentally test the CSL model. Finally, we discuss decoherence effects in order to provide reasonable estimates on the bounds that it is (or will be) possible to obtain for the parameters of the CSL model in such class of experiments. As an example, we show that a NOON state with N\ensuremath∼103N\ensuremath{\sim}{10}^{3}N\ensuremath∼103 with a coherence time of \ensuremath∼1\ensuremath{\sim}1\ensuremath∼1 s can constrain the CSL parameters in a region where the other systems presently cannot.
Collapse models solve the measurement problem by adding non-linear and stochastic terms to the usual Schroedinger dynamics. The resulting dynamics localizes in position macroscopic systems, while preserves quantum mechanical predictions for microscopic systems. \nTwo of the main consequences of this resulting dynamics are: the loss of any initial spatial coherence, with a decoherence rate which increases with the dimensions of the system (amplification mechanism); and a heating effect induced on the system that, for long times, leads either to a divergent or to a convergent value of the energy, depending on the particular collapse model considered. \nAn interesting class of system to experimentally test collapse models are cold atomic gases. Besides their very low temperatures (T = 10^(-11) - 10^(-7)K), cold atomic gases show mesoscopic quantum properties, i.e. involving a number of atoms of order of N = 10^(3)– 10^(11). These are all necessary ingredients to experimentally test collapse models. In this thesis, we use cold bosonic atomic gas to test one of the most studied and complete among collapse models, i.e. CSL, and also its non-Markovian and dissipative extensions. In particular, in this thesis we put bounds on CSL free-parameters by studying the diffusion effect induced by collapse noise on a free expanding gas, and we compare the theoretical results with experimental data known in literature. The bounds we put are among the strongest ever found non only for CSL model, but especially for its non-Markovian extension. \nIn this thesis we also specialize the CSL dynamics in Bose-Josephson junctions, and we focus on the localization mechanism induced by the noise on initially entangled states. In particular, we focus on decoherence induced by CSL on typical entangled states as atomic coherent states and NOON states. We find that in the latter ones, CSL effects become significant when the NOON state is composed by a number of atoms of 10^(4) or greater. Moreover, we compare CSL decoherence with environmental decoherence induced by typical realistic sources, as an external thermal cloud, imperfections in the trapping laser, and three body effects. We then find which conditions experiments must fulfill in order to test CSL in a Bose-Josephson junction in conditions never
The continuous spontaneous localization (CSL) theory alters the Schr\"odinger equation. It describes wave-function collapse as a dynamical process instead of an ill-defined postulate, thereby providing macroscopic uniqueness and solving the so-called measurement problem of standard quantum theory. CSL contains a parameter $\ensuremath{\lambda}$ giving the collapse rate of an isolated nucleon in a superposition of two spatially separated states and, more generally, characterizing the collapse time for any physical situation. CSL is experimentally testable, since it predicts some behavior different from that predicted by standard quantum theory. One example is the narrowing of wave functions, which results in energy imparted to particles. Here we consider energy given to trapped ultracold atoms. Since these are the coldest samples under experimental investigation, it is worth inquiring how they are affected by the CSL heating mechanism. We examine the CSL heating of a Bose-Einstein condensate (BEC) in contact with its thermal cloud. Of course, other mechanisms also provide heat and also particle loss. From varied data on optically trapped cesium BECs, we present an energy audit for known heating and loss mechanisms. The result provides an upper limit on CSL heating and thereby an upper limit on the parameter $\ensuremath{\lambda}$. We obtain $\ensuremath{\lambda}\ensuremath{\lesssim}1(\ifmmode\pm\else\textpm\fi{}1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ ${\mathrm{s}}^{\ensuremath{-}1}$.
{The Continuous Spontaneous Localization (CSL) theory alters the Schrodinger equation. It describes wave function collapse as a dynamical process instead of an ill-defined postulate, thereby providing macroscopic uniqueness and solving the so-called measurement problem of standard quantum theory. CSL contains a parameter λ\lambdaλ giving the collapse rate of an isolated nucleon in a superposition of two spatially separated states and, more generally, characterizing the collapse time for any physical situation. CSL is experimentally testable, since it predicts some behavior different from that predicted by standard quantum theory. One example is the narrowing of wave functions, which results in energy imparted to particles. Here we consider energy given to trapped ultra-cold atoms. Since these are the coldest samples under experimental investigation, it is worth inquiring how they are affected by the CSL heating mechanism. We examine the CSL heating of a BEC in contact with its thermal cloud. Of course, other mechanisms also provide heat and also particle loss. From varied data on optically trapped cesium BEC's, we present an energy audit for known heating and loss mechanisms. The result provides an upper limit on CSL heating and thereby an upper limit on the parameter λ\lambdaλ. We obtain λ≲1(±1)×10−7\lambda\lesssim 1(\pm1)\times 10^{-7}λ≲1(±1)×10−7sec−1^{-1}−1.}
Collapse models describe phenomenologically the quantum-to-classical transition by adding suitable nonlinear and stochastic terms to the Schr\"odinger equation, thus (slightly) modifying the dynamics of quantum systems. Experimental bounds on the collapse parameters have been derived from various experiments involving a plethora of different systems, from single atoms to gravitational wave detectors. Here, we give a comprehensive treatment of the continuous spontaneous localization (CSL) model, the most studied among collapse models, for Fermi liquids. We consider both the white and non-white noise case. Application to various astrophysical sources is presented.
Quantum mechanics exhibits a broad collection of theoretical results in complete agreement with experimental evidence. Besides showing unquestionable success in the description of isolated systems, it can be also successfully used to characterize non-isolated quantum system. In such a case, phenomena like dissipation, diffusion or decoherence. The theory of open quantum systems provides the framework where such features can be conveniently explained. \nThis thesis is about decoherence and collapse models. Albeit conceptually they are very far from each other, they both belong the framework of open quantum systems. Indeed, they consider systems interacting with an external entity: an environment for decoherence models, a noise for collapse models. Although the external influence has different origin, they can be described by similar dynamical equations and, to confirm or falsify them, the same experimental tests can be performed. \nThe quantum Brownian motion model can be considered the paradigm of decoherence models. We provide an exact and analytic equation for the time evolution of the operators and we show that the corresponding equation for the states is equivalent to well-known results in the literature. Our derivation allows to compute the time evolution of physically relevant quantities in a much easier way than previous formulations. Moreover, we are not bound to compute the time evolution of the state of the system, which in general is a complicated task. The explicit dependence on the initial state appears only in the initial expectation values and not in the dynamics. This makes possible the derivation of expectation values also for nontrivial states. \nAnother decoherence model we considered is a recently proposed model, based on the mass-energy equivalence. The model describes a decoherence source acting universally on every system whose superposition is extended on positions experiencing different gravitational potentials. We studied the conditions under which this mechanism becomes the dominant decoherence effect in typical interferometric experiments. We show that current experiments are off by several orders of magnitude. New ideas are needed to achieve the necessary requirements. \nThe second part of this thesis concerns collapse models, in particular the Continuous Spontaneous Localization (CSL) model. We focus on experimental tests that can probe it. In this respect, experiments can be grouped in two classes: interferometric tests and non-interferometric ones. The first class includes those experiments, which directly create and detect quantum superpositions of the center of mass of massive systems. The strongest bounds on the CSL parameters come from the second class of non-interferometric experiments, which are sensitive to small position displacements and detect CSL-induced diffusion in position. \nWe investigate how we can benefit from the non-interferometric perspective given by optomechanical setups, which have reached high sensitivities as force and position sensors. Three examples are considered.\nFirst, we compute the upper bounds on the CSL parameters, which can be inferred by the gravitational wave detectors LIGO, LISA Pathfinder and AURIGA.\nSecond, we report new results from an experiment based on a high-quality cantilever cooled to millikelvin temperature. High accuracy measurements of the cantilever thermal fluctuations reveal a nonthermal force noise of unknown origin. This excess noise is compatible with the CSL heating predicted by Adler. Several physical mechanisms able to explain the observed noise have been ruled out.\nThird, we propose an unattempted non-interferometric test aimed to investigate the still unexplored region of the CSL parameter space. Our proposal relies on torsional degrees of freedom rather than the usual vibrational ones. We believe that the test that has been put forward here, will eventually probe the unexplored CSL parameter space.
We investigate an attractive atomic Bose-Einstein condensate (BEC) trapped by a double-well potential in the axial direction and by a harmonic potential in the transverse directions. We obtain numerically a quantum phase diagram which includes all the three relevant phases of the system: Josephson, spontaneous symmetry breaking (SSB), and collapse. We consider also the coherent dynamics of the BEC and calculate the frequency of population-imbalance mode in the Josephson phase and in the SSB phase up to the collapse. We show that these phases can be observed by using ultracold vapors of 7Li^{7}\mathrm{Li}7Li atoms in a magneto-optical trap.