{ "metadata": { "name": "", "signature": "sha256:ad0e3ea376d943ae08c5ab0150b58caf6e2d2166ac63b9aef3aee4f4374474d8" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Lecture 7 - Symbolic quantum mechanics using SymPsi - Semiclassical equations of motion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Author: J. R. Johansson (robert@riken.jp), [http://jrjohansson.github.io](http://jrjohansson.github.io), and Eunjong Kim.\n", "\n", "Status: Preliminary (work in progress)\n", "\n", "This notebook is part of a series of IPython notebooks on symbolic quantum mechanics computations using \n", "[SymPy](http://sympy.org) and [SymPsi](http://www.github.com/jrjohansson/sympsi). SymPsi is an experimental fork and extension of the [`sympy.physics.quantum`](http://docs.sympy.org/dev/modules/physics/quantum/) module in SymPy. The latest version of this notebook is available at [http://github.com/jrjohansson/sympy-quantum-notebooks](http://github.com/jrjohansson/sympy-quantum-notebooks), and the other notebooks in this lecture series are also indexed at [http://jrjohansson.github.io](http://jrjohansson.github.com).\n", "\n", "Requirements: A recent version of SymPy and the latest development version of SymPsi is required to execute this notebook. Instructions for how to install SymPsi is available [here](http://www.github.com/jrjohansson/sympsi).\n", "\n", "Disclaimer: The SymPsi module is still under active development and may change in behavior without notice, and the intention is to move some of its features to [`sympy.physics.quantum`](http://docs.sympy.org/dev/modules/physics/quantum/) when they matured and have been tested. However, these notebooks will be kept up-to-date the latest versions of SymPy and SymPsi." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Setup modules" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%matplotlib inline\n", "import matplotlib.pyplot as plt" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "code", "collapsed": false, "input": [ "import numpy as np" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "code", "collapsed": false, "input": [ "from sympy import *\n", "init_printing()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "from sympsi import *\n", "from sympsi.boson import *\n", "from sympsi.pauli import *\n", "from sympsi.operatorordering import *\n", "from sympsi.expectation import *\n", "from sympsi.operator import OperatorFunction" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Semiclassical equations of motion" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The dynamics of an open quantum system with a given Hamiltonian, $H$, and some interaction with an environment that acts on the system through the sytem operator $a$, and with rate $\\kappa$, can often be described with a Lindblad master equation for the dynamics of the system density matrix $\\rho$:\n", "\n", "$$\n", "\\frac{d}{dt}\\rho = -i[H, \\rho] + \\kappa \\mathcal{D}[a]\\rho,\n", "$$\n", "\n", "where the Lindblad superoperator $\\mathcal{D}$ is\n", "\n", "$$\n", "\\mathcal{D}[a]\\rho = a \\rho a^\\dagger -\\frac{1}{2}\\rho a^\\dagger a - \\frac{1}{2}a^\\dagger a \\rho.\n", "$$\n", "\n", "One common approach to solve for the dynamics of this system is to represent the system operators and the density operator as matrices, possibly in a truncated state space, and solve the matrix-valued ODE problem numerically.\n", "\n", "Another approach is to use the adjoint master equation for the system operators $X$:\n", "\n", "$$\n", "\\frac{d}{dt} X = i [H, X] + \\kappa \\mathcal{D}[a^\\dagger]X\n", "$$\n", "\n", "and then solve for dynamics of the expectation values of the relevant system operators. The advantage of this method is that the ODEs are no longer matrix-valued, unlike the ODE for the density matrix. However, from the density matrix we can calculate any same-time expectation values, but with explicit ODEs for expectation values we need to select in advance which operator's expectation values we want to generate equations for. \n", "\n", "We can easily generate an equation for the expectation value of a specific operator by multiplying the master equation for $\\rho$ from the left with an operator $X$, and then take the trace over the entire equation. Doing this we obtain:\n", "\n", "$$\n", "X\\frac{d}{dt}\\rho = -iX[H, \\rho] + \\kappa X\\mathcal{D}[a]\\rho\n", "$$\n", "\n", "and taking the trace:\n", "\n", "$$\n", "{\\rm Tr}\\left(X\\frac{d}{dt}\\rho\\right) = -i{\\rm Tr}\\left(X[H, \\rho]\\right) + \\kappa {\\rm Tr}\\left(X\\mathcal{D}[a]\\rho\\right)\n", "$$\n", "\n", "using the cyclic permutation properties of traces:\n", "\n", "$$\n", "\\frac{d}{dt}{\\rm Tr}\\left(X\\rho\\right) = -i{\\rm Tr}\\left([X, H]\\rho\\right) + \\kappa {\\rm Tr}\\left((\\mathcal{D}[a]X) \\rho\\right)\n", "$$\n", "\n", "we end up with an equation for the expectation value of the operator $X$:\n", "\n", "$$\n", "\\frac{d}{dt}\\langle X\\rangle \n", "= \n", "i\\langle [H, X] \\rangle + \\kappa \\langle \\mathcal{D}[a]X \\rangle\n", "$$\n", "\n", "Note that this is a C-number equation, and therefore not as complicated to solve as the master equation for the density matrix. However, the problem with this C-number equation is that the expressions $[H, X]$\n", " and $\\mathcal{D}[a]X$ in general will introduce dependencies on other system operators, so we obtain a system of coupled C-number equations. If this system of equations closes when a finite number of operators are included, then we can use this method to solve the dynamics of these expectation values exactly. If the system of equations do not close, which is often the case for coupled systems, then we can still use this method if we introduce some rule for truncating high-order operator expectation values (for example, by discarding high-order terms or by factoring them in expectation values of lower order). However, in this case the results are no longer exact, and is called a semi-classical equation of motion.\n", "\n", "With SymPsi we can automatically generate semiclassical equations of motion for operators in a system described by a given Hamiltonian and a set of collapse operators that describe its coupling to an environment." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Driven harmonic oscillator" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider a driven harmonic oscillator, which interaction with an bath at some temperature that corresponds to $N_{\\rm th}$ average photons. We begin by setting up symbolic variables for the problem parameters and the system operators in SymPsi: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "w, t, Nth, Ad, kappa = symbols(r\"\\omega, t, n_{th}, A_d, kappa\", positive=True)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "a = BosonOp(\"a\")\n", "rho = Operator(r\"\\rho\")\n", "rho_t = OperatorFunction(rho, t)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "H = w * Dagger(a) * a + Ad * (a + Dagger(a))\n", "\n", "Eq(Symbol(\"H\"), H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = A_{d} \\left({{a}^\\dagger} + {a}\\right) + \\omega {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAANwAAAAbBAMAAADlmfy+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", "prompt_number": 7, "text": [ " \u239b \u2020 \u239e \u2020 \n", "H = A_d\u22c5\u239da + a\u23a0 + \\omega\u22c5a \u22c5a" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The master equation for this system can be generated using the `master_equation` function:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "c_ops = [sqrt(kappa * (Nth + 1)) * a, sqrt(kappa * Nth) * Dagger(a)]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 8 }, { "cell_type": "code", "collapsed": false, "input": [ "me = master_equation(rho_t, t, H, c_ops)\n", "\n", "me" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\partial}{\\partial t} {{\\rho}(t)} = \\kappa n_{{th}} {{a}^\\dagger} {{\\rho}(t)} {a} - \\frac{\\kappa n_{{th}}}{2} {a} {{a}^\\dagger} {{\\rho}(t)} - \\frac{\\kappa n_{{th}}}{2} {{\\rho}(t)} {a} {{a}^\\dagger} - \\frac{\\kappa {{a}^\\dagger}}{2} \\left(n_{{th}} + 1\\right) {a} {{\\rho}(t)} + \\kappa \\left(n_{{th}} + 1\\right) {a} {{\\rho}(t)} {{a}^\\dagger} - \\frac{\\kappa {{\\rho}(t)}}{2} \\left(n_{{th}} + 1\\right) {{a}^\\dagger} {a} - i \\left[A_{d} \\left({{a}^\\dagger} + {a}\\right) + \\omega {{a}^\\dagger} {a},{{\\rho}(t)}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 9, "text": [ " \n", "\u2202 \u2020 \u03ba\u22c5n_{t\n", "\u2500\u2500(OperatorFunction(\\rho,t)) = \u03ba\u22c5n_{th}\u22c5a \u22c5OperatorFunction(\\rho,t)\u22c5a - \u2500\u2500\u2500\u2500\u2500\u2500\n", "\u2202t \n", "\n", " \u2020 \u2020 \n", "h}\u22c5a\u22c5a \u22c5OperatorFunction(\\rho,t) \u03ba\u22c5n_{th}\u22c5OperatorFunction(\\rho,t)\u22c5a\u22c5a \u03ba\u22c5\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\n", " 2 2 \n", "\n", " \u2020 \n", "(n_{th} + 1)\u22c5a \u22c5a\u22c5OperatorFunction(\\rho,t) \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u03ba\u22c5(n_{th} + 1)\u22c5a\u22c5OperatorFunction\n", " 2 \n", "\n", " \u2020 \n", " \u2020 \u03ba\u22c5(n_{th} + 1)\u22c5OperatorFunction(\\rho,t)\u22c5a \u22c5a \u23a1 \u239b \u2020 \u239e \n", "(\\rho,t)\u22c5a - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2148\u22c5\u23a3A_d\u22c5\u239da + a\u23a0 +\n", " 2 \n", "\n", " \n", " \u2020 \u23a4\n", " \\omega\u22c5a \u22c5a,OperatorFunction(\\rho,t)\u23a6\n", " " ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Equation for the system operators can be generated using the function `operator_master_equation`, and for the specific case of the cavity operator $a$ we obtain:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# first setup time-dependent operators\n", "a_t = OperatorFunction(a, t)\n", "a_to_a_t = {a: a_t, Dagger(a): Dagger(a_t)}\n", "H_t = H.subs(a_to_a_t)\n", "c_ops_t = [c.subs(a_to_a_t) for c in c_ops]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "# operator master equation for a\n", "ome_a = operator_master_equation(a_t, t, H_t, c_ops_t)\n", "\n", "Eq(ome_a.lhs, normal_ordered_form(ome_a.rhs.doit().expand()))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{d}{d t} {{{a}}(t)} = - i A_{d} - i \\omega {{{a}}(t)} - \\frac{\\kappa {{{a}}(t)}}{2}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAASAAAAAtBAMAAAD8RKvZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 11, "text": [ "d \u03ba\u22c5Operat\n", "\u2500\u2500(OperatorFunction(a,t)) = -\u2148\u22c5A_d - \u2148\u22c5\\omega\u22c5OperatorFunction(a,t) - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", "dt \n", "\n", "orFunction(a,t)\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " 2 " ] } ], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "# operator master equation for n = Dagger(a) * a\n", "ome_n = operator_master_equation(Dagger(a_t) * a_t, t, H_t, c_ops_t)\n", "\n", "Eq(ome_n.lhs, normal_ordered_form(ome_n.rhs.doit().expand()))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{d}{d t} {{{{a}^\\dagger}}(t)} {{{a}}(t)} + {{{{a}^\\dagger}}(t)} \\frac{d}{d t} {{{a}}(t)} = - i A_{d} {{{{a}^\\dagger}}(t)} + i A_{d} {{{a}}(t)} + \\kappa n_{{th}} - \\kappa {{{{a}^\\dagger}}(t)} {{{a}}(t)}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 12, "text": [ "d \u239b \u239b \u2020 \u239e\u239e \u239b \u2020 \u239e d \n", "\u2500\u2500\u239dOperatorFunction\u239da ,t\u23a0\u23a0\u22c5OperatorFunction(a,t) + OperatorFunction\u239da ,t\u23a0\u22c5\u2500\u2500(O\n", "dt dt \n", "\n", " \u239b \u2020 \u239e \n", "peratorFunction(a,t)) = - \u2148\u22c5A_d\u22c5OperatorFunction\u239da ,t\u23a0 + \u2148\u22c5A_d\u22c5OperatorFunctio\n", " \n", "\n", " \u239b \u2020 \u239e \n", "n(a,t) + \u03ba\u22c5n_{th} - \u03ba\u22c5OperatorFunction\u239da ,t\u23a0\u22c5OperatorFunction(a,t)\n", " " ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "From these operator equations we see that the equation for $a$ depends only on the operator $a$, while the equation for $n$ depends on $n$, $a$, $a^\\dagger$. So to solve the latter equation we therefore also have to generate an equation for $a^\\dagger$." ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "System of semiclassical equations" ] }, { "cell_type": "code", "collapsed": false, "input": [ "ops, op_eqm, sc_eqm, sc_ode, ofm, oim = semi_classical_eqm(H, c_ops)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "html_table([[Eq(Expectation(key), ofm[key]), sc_ode[key]] for key in operator_sort_by_order(sc_ode)])" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", "\n", "
$\\left\\langle {{a}^\\dagger} \\right\\rangle = \\operatorname{A_{0}}{\\left (t \\right )}$$\\frac{d}{d t} \\operatorname{A_{0}}{\\left (t \\right )} = i A_{d} + i \\omega \\operatorname{A_{0}}{\\left (t \\right )} - \\frac{\\kappa}{2} \\operatorname{A_{0}}{\\left (t \\right )}$
$\\left\\langle {a} \\right\\rangle = \\operatorname{A_{1}}{\\left (t \\right )}$$\\frac{d}{d t} \\operatorname{A_{1}}{\\left (t \\right )} = - i A_{d} - i \\omega \\operatorname{A_{1}}{\\left (t \\right )} - \\frac{\\kappa}{2} \\operatorname{A_{1}}{\\left (t \\right )}$
$\\left\\langle {{a}^\\dagger} {a} \\right\\rangle = \\operatorname{A_{2}}{\\left (t \\right )}$$\\frac{d}{d t} \\operatorname{A_{2}}{\\left (t \\right )} = - i A_{d} \\operatorname{A_{0}}{\\left (t \\right )} + i A_{d} \\operatorname{A_{1}}{\\left (t \\right )} + \\kappa n_{{th}} - \\kappa \\operatorname{A_{2}}{\\left (t \\right )}$
" ], "metadata": {}, "output_type": "pyout", "prompt_number": 14, "text": [ "" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Since this is a system of linear ODEs, we can write it on matrix form:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_eq, A, M, b = semi_classical_eqm_matrix_form(sc_ode, t, ofm)\n", "\n", "A_eq" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{d}{d t} \\left[\\begin{matrix}\\operatorname{A_{0}}{\\left (t \\right )}\\\\\\operatorname{A_{1}}{\\left (t \\right )}\\\\\\operatorname{A_{2}}{\\left (t \\right )}\\end{matrix}\\right] = \\left[\\begin{matrix}i A_{d}\\\\- i A_{d}\\\\\\kappa n_{{th}}\\end{matrix}\\right] + \\left[\\begin{matrix}i \\omega - \\frac{\\kappa}{2} & 0 & 0\\\\0 & - i \\omega - \\frac{\\kappa}{2} & 0\\\\- i A_{d} & i A_{d} & - \\kappa\\end{matrix}\\right] \\left[\\begin{matrix}\\operatorname{A_{0}}{\\left (t \\right )}\\\\\\operatorname{A_{1}}{\\left (t \\right )}\\\\\\operatorname{A_{2}}{\\left (t \\right )}\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 15, "text": [ " d \u23a1 \u03ba \u23a4 \n", "-\u2500\u2500\u239b\u23a1A\u2080(t)\u23a4\u239e = \u23a1 \u2148\u22c5A_d \u23a4 + \u23a2\u2148\u22c5\\omega - \u2500 0 0 \u23a5\u22c5\u23a1A\u2080(t)\u23a4\n", " dt\u239c\u23a2 \u23a5\u239f \u23a2 \u23a5 \u23a2 2 \u23a5 \u23a2 \u23a5\n", " \u239c\u23a2A\u2081(t)\u23a5\u239f \u23a2 -\u2148\u22c5A_d \u23a5 \u23a2 \u23a5 \u23a2A\u2081(t)\u23a5\n", " \u239c\u23a2 \u23a5\u239f \u23a2 \u23a5 \u23a2 \u03ba \u23a5 \u23a2 \u23a5\n", " \u239d\u23a3A\u2082(t)\u23a6\u23a0 \u23a3\u03ba\u22c5n_{th}\u23a6 \u23a2 0 -\u2148\u22c5\\omega - \u2500 0 \u23a5 \u23a3A\u2082(t)\u23a6\n", " \u23a2 2 \u23a5 \n", " \u23a2 \u23a5 \n", " \u23a3 -\u2148\u22c5A_d \u2148\u22c5A_d -\u03ba\u23a6 " ] } ], "prompt_number": 15 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Steadystate" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can solve for the steadystate by setting the left-hand-side of the ODE to zero, and solve the linear system of equations:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol = M.LUsolve(-b)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 16 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The solution for the three system operators are:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol[ops.index(Dagger(a)*a)]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{1}{\\kappa} \\left(\\frac{A_{d}^{2}}{i \\omega - \\frac{\\kappa}{2}} + \\frac{A_{d}^{2}}{- i \\omega - \\frac{\\kappa}{2}} - \\kappa n_{{th}}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 17, "text": [ " \u239b 2 2 \u239e \n", " \u239c A_d A_d \u239f \n", "-\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u03ba\u22c5n_{th}\u239f \n", " \u239c \u03ba \u03ba \u239f \n", " \u239c\u2148\u22c5\\omega - \u2500 -\u2148\u22c5\\omega - \u2500 \u239f \n", " \u239d 2 2 \u23a0 \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u03ba " ] } ], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol[ops.index(a)]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{i A_{d}}{- i \\omega - \\frac{\\kappa}{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAxBAMAAACG8II6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n5YmzeRAAAAAASUVORK5CYII=\n", "prompt_number": 18, "text": [ " \u2148\u22c5A_d \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u03ba\n", "-\u2148\u22c5\\omega - \u2500\n", " 2" ] } ], "prompt_number": 18 }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol[ops.index(Dagger(a))]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{i A_{d}}{i \\omega - \\frac{\\kappa}{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAxBAMAAACG8II6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nB7gFo9KCJdeQ5psAAAAASUVORK5CYII=\n", "prompt_number": 19, "text": [ " -\u2148\u22c5A_d \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u03ba\n", "\u2148\u22c5\\omega - \u2500\n", " 2" ] } ], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can also solve for the steadystate directly from the ODE by settings its right-hand-side to zero, and using the SymPy `solve` function: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "solve([eq.rhs for eq in sc_ode.values()], list(ofm.values()))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left \\{ \\operatorname{A_{0}}{\\left (t \\right )} : - \\frac{2 i A_{d}}{2 i \\omega - \\kappa}, \\quad \\operatorname{A_{1}}{\\left (t \\right )} : - \\frac{2 i A_{d}}{2 i \\omega + \\kappa}, \\quad \\operatorname{A_{2}}{\\left (t \\right )} : \\frac{1}{4 \\omega^{2} + \\kappa^{2}} \\left(4 A_{d}^{2} + 4 \\omega^{2} n_{{th}} + \\kappa^{2} n_{{th}}\\right)\\right \\}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 20, "text": [ "\u23a7 2 2 \n", "\u23aa -2\u22c5\u2148\u22c5A_d -2\u22c5\u2148\u22c5A_d 4\u22c5A_d + 4\u22c5\\omega \u22c5n_{th\n", "\u23a8A\u2080(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, A\u2081(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, A\u2082(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", "\u23aa 2\u22c5\u2148\u22c5\\omega - \u03ba 2\u22c5\u2148\u22c5\\omega + \u03ba 2 \n", "\u23a9 4\u22c5\\omega + \n", "\n", " 2 \u23ab\n", "} + \u03ba \u22c5n_{th}\u23aa\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23ac\n", " 2 \u23aa\n", "\u03ba \u23ad" ] } ], "prompt_number": 20 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Solve in the ODEs" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "For systems with a small number of dependent operators we can solve the resulting system of ODEs directly:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sols = dsolve(list(sc_ode.values())); sols" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left [ \\operatorname{A_{0}}{\\left (t \\right )} = C_{1} e^{t \\left(i \\omega - \\frac{\\kappa}{2}\\right)}, \\quad \\operatorname{A_{1}}{\\left (t \\right )} = C_{2} e^{t \\left(- i \\omega - \\frac{\\kappa}{2}\\right)}, \\quad \\operatorname{A_{2}}{\\left (t \\right )} = - \\frac{i A_{d} C_{1} e^{t \\left(i \\omega - \\frac{\\kappa}{2}\\right)}}{i \\omega + \\frac{\\kappa}{2}} + \\frac{i A_{d} C_{2} e^{t \\left(- i \\omega - \\frac{\\kappa}{2}\\right)}}{- i \\omega + \\frac{\\kappa}{2}} + \\frac{C_{3}}{e^{\\kappa t}}\\right ]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 21, "text": [ "\u23a1 \n", "\u23a2 \u239b \u03ba\u239e \u239b \u03ba\u239e \n", "\u23a2 t\u22c5\u239c\u2148\u22c5\\omega - \u2500\u239f t\u22c5\u239c-\u2148\u22c5\\omega - \u2500\u239f \n", "\u23a2 \u239d 2\u23a0 \u239d 2\u23a0 \u2148\u22c5A_d\u22c5\n", "\u23a2A\u2080(t) = C\u2081\u22c5\u212f , A\u2081(t) = C\u2082\u22c5\u212f , A\u2082(t) = - \u2500\u2500\u2500\u2500\u2500\u2500\n", "\u23a2 \n", "\u23a2 \n", "\u23a3 \n", "\n", " \u239b \u03ba\u239e \u239b \u03ba\u239e \u23a4\n", " t\u22c5\u239c\u2148\u22c5\\omega - \u2500\u239f t\u22c5\u239c-\u2148\u22c5\\omega - \u2500\u239f \u23a5\n", " \u239d 2\u23a0 \u239d 2\u23a0 \u23a5\n", "C\u2081\u22c5\u212f \u2148\u22c5A_d\u22c5C\u2082\u22c5\u212f -\u03ba\u22c5t\u23a5\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + C\u2083\u22c5\u212f \u23a5\n", " \u03ba \u03ba \u23a5\n", " \u2148\u22c5\\omega + \u2500 -\u2148\u22c5\\omega + \u2500 \u23a5\n", " 2 2 \u23a6" ] } ], "prompt_number": 21 }, { "cell_type": "code", "collapsed": false, "input": [ "# hack\n", "tt = [s for s in sols[0].rhs.free_symbols if s.name == 't'][0]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We also need o specify the initial conditions: Here the initial conditions are $\\langle a(0) \\rangle = \\langle a^\\dagger(0) \\rangle = 2$ and $\\langle a^\\dagger(0)a(0) \\rangle = 4$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "ics = {ofm[Dagger(a)].subs(tt, 0): 2, \n", " ofm[a].subs(tt, 0): 2,\n", " ofm[Dagger(a)*a].subs(tt, 0): 4}; ics" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left \\{ \\operatorname{A_{0}}{\\left (0 \\right )} : 2, \\quad \\operatorname{A_{1}}{\\left (0 \\right )} : 2, \\quad \\operatorname{A_{2}}{\\left (0 \\right )} : 4\\right \\}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 23, "text": [ "{A\u2080(0): 2, A\u2081(0): 2, A\u2082(0): 4}" ] } ], "prompt_number": 23 }, { "cell_type": "code", "collapsed": false, "input": [ "constants = set(sum([[s for s in sol.free_symbols if (str(s)[0] == 'C')] for sol in sols], [])); constants" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left\\{C_{1}, C_{2}, C_{3}\\right\\}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAG0AAAAVBAMAAAC+p33JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ngo7PSpzzZOL/sdvLrT4CtZh07irrd/4AAAAASUVORK5CYII=\n", "prompt_number": 24, "text": [ "set([C\u2081, C\u2082, C\u2083])" ] } ], "prompt_number": 24 }, { "cell_type": "code", "collapsed": false, "input": [ "C_sols = solve([sol.subs(tt, 0).subs(ics) for sol in sols], constants); C_sols" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left \\{ C_{1} : 2, \\quad C_{2} : 2, \\quad C_{3} : \\frac{1}{4 \\omega^{2} + \\kappa^{2}} \\left(16 A_{d} \\omega + 16 \\omega^{2} + 4 \\kappa^{2}\\right)\\right \\}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 25, "text": [ "\u23a7 \u239b 2 2\u239e\u23ab\n", "\u23aa 4\u22c5\u239d4\u22c5A_d\u22c5\\omega + 4\u22c5\\omega + \u03ba \u23a0\u23aa\n", "\u23a8C\u2081: 2, C\u2082: 2, C\u2083: \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23ac\n", "\u23aa 2 2 \u23aa\n", "\u23a9 4\u22c5\\omega + \u03ba \u23ad" ] } ], "prompt_number": 25 }, { "cell_type": "code", "collapsed": false, "input": [ "sols_with_ics = [sol.subs(C_sols) for sol in sols]; sols_with_ics" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left [ \\operatorname{A_{0}}{\\left (t \\right )} = 2 e^{t \\left(i \\omega - \\frac{\\kappa}{2}\\right)}, \\quad \\operatorname{A_{1}}{\\left (t \\right )} = 2 e^{t \\left(- i \\omega - \\frac{\\kappa}{2}\\right)}, \\quad \\operatorname{A_{2}}{\\left (t \\right )} = - \\frac{2 i A_{d} e^{t \\left(i \\omega - \\frac{\\kappa}{2}\\right)}}{i \\omega + \\frac{\\kappa}{2}} + \\frac{2 i A_{d} e^{t \\left(- i \\omega - \\frac{\\kappa}{2}\\right)}}{- i \\omega + \\frac{\\kappa}{2}} + \\frac{16 A_{d} \\omega + 16 \\omega^{2} + 4 \\kappa^{2}}{\\left(4 \\omega^{2} + \\kappa^{2}\\right) e^{\\kappa t}}\\right ]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 26, "text": [ "\u23a1 \n", "\u23a2 \u239b \u03ba\u239e \u239b \u03ba\u239e \n", "\u23a2 t\u22c5\u239c\u2148\u22c5\\omega - \u2500\u239f t\u22c5\u239c-\u2148\u22c5\\omega - \u2500\u239f \n", "\u23a2 \u239d 2\u23a0 \u239d 2\u23a0 2\u22c5\u2148\u22c5A_d\u22c5\n", "\u23a2A\u2080(t) = 2\u22c5\u212f , A\u2081(t) = 2\u22c5\u212f , A\u2082(t) = - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", "\u23a2 \n", "\u23a2 \u2148\n", "\u23a3 \n", "\n", " \u239b \u03ba\u239e \u239b \u03ba\u239e \n", " t\u22c5\u239c\u2148\u22c5\\omega - \u2500\u239f t\u22c5\u239c-\u2148\u22c5\\omega - \u2500\u239f \n", " \u239d 2\u23a0 \u239d 2\u23a0 \u239b 2 \n", "\u212f 2\u22c5\u2148\u22c5A_d\u22c5\u212f 4\u22c5\u239d4\u22c5A_d\u22c5\\omega + 4\u22c5\\omega +\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u03ba \u03ba 2 2 \n", "\u22c5\\omega + \u2500 -\u2148\u22c5\\omega + \u2500 4\u22c5\\omega + \u03ba \n", " 2 2 \n", "\n", " \u23a4\n", " \u23a5\n", " 2\u239e -\u03ba\u22c5t\u23a5\n", " \u03ba \u23a0\u22c5\u212f \u23a5\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n", " \u23a5\n", " \u23a5\n", " \u23a6" ] } ], "prompt_number": 26 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's insert numerical values for the system parameters so we can plot the solution:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "values = {w: 1.0, Ad: 0.0, kappa: 0.1, Nth: 0.0}" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 27 }, { "cell_type": "code", "collapsed": false, "input": [ "sols_funcs = [sol.rhs.subs(values) for sol in sols_with_ics]; sols_funcs" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left [ 2 e^{t \\left(-0.05 + 1.0 i\\right)}, \\quad 2 e^{t \\left(-0.05 - 1.0 i\\right)}, \\quad \\frac{4}{e^{0.1 t}}\\right ]$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 28, "text": [ "\u23a1 t\u22c5(-0.05 + 1.0\u22c5\u2148) t\u22c5(-0.05 - 1.0\u22c5\u2148) -0.1\u22c5t\u23a4\n", "\u23a32\u22c5\u212f , 2\u22c5\u212f , 4\u22c5\u212f \u23a6" ] } ], "prompt_number": 28 }, { "cell_type": "code", "collapsed": false, "input": [ "times = np.linspace(0, 50, 500)\n", "\n", "y_funcs = [lambdify([tt], sol_func, 'numpy') for sol_func in sols_funcs]\n", "\n", "fig, axes = plt.subplots(len(y_funcs), 1, figsize=(12, 6))\n", "\n", "for n, y_func in enumerate(y_funcs):\n", " axes[n].plot(times, np.real(y_func(times)), 'r')\n", " \n", "axes[2].set_ylim(0, 5);" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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66xg0izAoPn2ad9duu43v0eXL8/+kbNnwPgdzEAuY3XnlFaBkSQYCwWLpUuCl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\nJiRkWCVDROaB3frS6qaqvyR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azgz/9BNzlX3hlltY5nD8eAaIOd3GjWysU6UKFzp36QLs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\nI5z7LEmSxoRmDsyHA78EFlHKygGclZk/7XaNgblZZMJdd3WF51tvhSl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"text": [ "" ] } ], "prompt_number": 29 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Driven dissipative two-level system" ] }, { "cell_type": "code", "collapsed": false, "input": [ "sx, sy, sz, sm, sp = SigmaX(), SigmaY(), SigmaZ(), SigmaMinus(), SigmaPlus()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 30 }, { "cell_type": "code", "collapsed": false, "input": [ "Omega, gamma_0, N, t = symbols(\"\\Omega, \\gamma_0, N, t\", positive=True)\n", "\n", "values = {Omega: 1.0, gamma_0: 0.5, N: 1.75}" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 31 }, { "cell_type": "code", "collapsed": false, "input": [ "H = -Omega/2 * sx\n", "H" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega {\\sigma_x}}{2}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAADgAAAArBAMAAADf6lYYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 32, "text": [ "-\\Omega\u22c5False \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " 2 " ] } ], "prompt_number": 32 }, { "cell_type": "code", "collapsed": false, "input": [ "c_ops = [sqrt(gamma_0 * (N + 1)) * pauli_represent_x_y(sm), \n", " sqrt(gamma_0 * N) * pauli_represent_x_y(sp)]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 33 }, { "cell_type": "code", "collapsed": false, "input": [ "ops, op_eqm, sc_eqm, sc_ode, ofm, oim = semi_classical_eqm(H, c_ops)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 34 }, { "cell_type": "code", "collapsed": false, "input": [ "html_table([[Eq(Expectation(key), ofm[key]), sc_ode[key]] for key in operator_sort_by_order(sc_ode)])" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "\n", "\n", "
$\\left\\langle {\\sigma_x} \\right\\rangle = \\operatorname{A_{0}}{\\left (t \\right )}$$\\frac{d}{d t} \\operatorname{A_{0}}{\\left (t \\right )} = - N \\gamma_{0} \\operatorname{A_{0}}{\\left (t \\right )} - \\frac{\\gamma_{0}}{2} \\operatorname{A_{0}}{\\left (t \\right )}$
$\\left\\langle {\\sigma_y} \\right\\rangle = \\operatorname{A_{1}}{\\left (t \\right )}$$\\frac{d}{d t} \\operatorname{A_{1}}{\\left (t \\right )} = - N \\gamma_{0} \\operatorname{A_{1}}{\\left (t \\right )} + \\Omega \\operatorname{A_{2}}{\\left (t \\right )} - \\frac{\\gamma_{0}}{2} \\operatorname{A_{1}}{\\left (t \\right )}$
$\\left\\langle {\\sigma_z} \\right\\rangle = \\operatorname{A_{2}}{\\left (t \\right )}$$\\frac{d}{d t} \\operatorname{A_{2}}{\\left (t \\right )} = - 2 N \\gamma_{0} \\operatorname{A_{2}}{\\left (t \\right )} - \\Omega \\operatorname{A_{1}}{\\left (t \\right )} - \\gamma_{0} \\operatorname{A_{2}}{\\left (t \\right )} - \\gamma_{0}$
" ], "metadata": {}, "output_type": "pyout", "prompt_number": 35, "text": [ "" ] } ], "prompt_number": 35 }, { "cell_type": "code", "collapsed": false, "input": [ "A_eq, A, M, b = semi_classical_eqm_matrix_form(sc_ode, t, ofm)\n", "\n", "A_eq" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{d}{d t} \\left[\\begin{matrix}\\operatorname{A_{0}}{\\left (t \\right )}\\\\\\operatorname{A_{1}}{\\left (t \\right )}\\\\\\operatorname{A_{2}}{\\left (t \\right )}\\end{matrix}\\right] = \\left[\\begin{matrix}0\\\\0\\\\- \\gamma_{0}\\end{matrix}\\right] + \\left[\\begin{matrix}- N \\gamma_{0} - \\frac{\\gamma_{0}}{2} & 0 & 0\\\\0 & - N \\gamma_{0} - \\frac{\\gamma_{0}}{2} & \\Omega\\\\0 & - \\Omega & - 2 N \\gamma_{0} - \\gamma_{0}\\end{matrix}\\right] \\left[\\begin{matrix}\\operatorname{A_{0}}{\\left (t \\right )}\\\\\\operatorname{A_{1}}{\\left (t \\right )}\\\\\\operatorname{A_{2}}{\\left (t \\right )}\\end{matrix}\\right]$$" ], "metadata": {}, "output_type": 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"prompt_number": 36, "text": [ " d \u23a1 \\gamma\u2080 \n", "-\u2500\u2500\u239b\u23a1A\u2080(t)\u23a4\u239e = \u23a1 0 \u23a4 + \u23a2-N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 0 \n", " dt\u239c\u23a2 \u23a5\u239f \u23a2 \u23a5 \u23a2 2 \n", " \u239c\u23a2A\u2081(t)\u23a5\u239f \u23a2 0 \u23a5 \u23a2 \n", " \u239c\u23a2 \u23a5\u239f \u23a2 \u23a5 \u23a2 \\gamma\u2080 \n", " \u239d\u23a3A\u2082(t)\u23a6\u23a0 \u23a3-\\gamma\u2080\u23a6 \u23a2 0 -N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n", " \u23a2 2 \n", " \u23a2 \n", " \u23a3 0 -\\Omega -2\u22c5N\u22c5\n", "\n", " \u23a4 \n", " 0 \u23a5\u22c5\u23a1A\u2080(t)\u23a4\n", " \u23a5 \u23a2 \u23a5\n", " \u23a5 \u23a2A\u2081(t)\u23a5\n", " \u23a5 \u23a2 \u23a5\n", " \\Omega \u23a5 \u23a3A\u2082(t)\u23a6\n", " \u23a5 \n", " \u23a5 \n", "\\gamma\u2080 - \\gamma\u2080\u23a6 " ] } ], "prompt_number": 36 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Steadystate" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol = M.LUsolve(-b)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 37 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The steadystate expectation value of $\\sigma_x$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol[ops.index(sx)]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$0$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEJmJdjLNVN0iZu+7\nq0QgoRR7AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAVklEQVQIHWNgEDJRZWBgSGeQmMDAtYGBOYGB\n5wID+0cG/gsMfN8Z5BUY+L4wzDdgYP0MJeUNQCL8Cgzs3xk4DjBwfWRg2cDAlMDA0M4gHcDAIOxy\nlQEA9FISlFfRJtkAAAAASUVORK5CYII=\n", "prompt_number": 38, "text": [ "0" ] } ], "prompt_number": 38 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The steadystate expectation value of $\\sigma_y$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol[ops.index(sy)]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega \\gamma_{0}}{\\left(- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}\\right) \\left(- 2 N \\gamma_{0} + \\frac{\\Omega^{2}}{- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}} - \\gamma_{0}\\right)}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 39, "text": [ " -\\Omega\u22c5\\gamma\u2080 \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u239b 2 \u239e\n", "\u239b \\gamma\u2080\u239e \u239c \\Omega \u239f\n", "\u239c-N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u239c-2\u22c5N\u22c5\\gamma\u2080 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \\gamma\u2080\u239f\n", "\u239d 2 \u23a0 \u239c \\gamma\u2080 \u239f\n", " \u239c -N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u239f\n", " \u239d 2 \u23a0" ] } ], "prompt_number": 39 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The steadystate expectation value of $\\sigma_z$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A_sol[ops.index(sz)]" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\gamma_{0}}{- 2 N \\gamma_{0} + \\frac{\\Omega^{2}}{- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}} - \\gamma_{0}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 40, "text": [ " \\gamma\u2080 \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " 2 \n", " \\Omega \n", "-2\u22c5N\u22c5\\gamma\u2080 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \\gamma\u2080\n", " \\gamma\u2080 \n", " -N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \n", " 2 " ] } ], "prompt_number": 40 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Steadystate of $\\sigma_+$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "pauli_represent_x_y(sp).subs({sx: A_sol[ops.index(sx)], sy: A_sol[ops.index(sy)]})" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{i \\Omega \\gamma_{0}}{2 \\left(- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}\\right) \\left(- 2 N \\gamma_{0} + \\frac{\\Omega^{2}}{- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}} - \\gamma_{0}\\right)}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 41, "text": [ " -\u2148\u22c5\\Omega\u22c5\\gamma\u2080 \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u239b 2 \u239e\n", " \u239b \\gamma\u2080\u239e \u239c \\Omega \u239f\n", "2\u22c5\u239c-N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u239c-2\u22c5N\u22c5\\gamma\u2080 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \\gamma\u2080\u239f\n", " \u239d 2 \u23a0 \u239c \\gamma\u2080 \u239f\n", " \u239c -N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u239f\n", " \u239d 2 \u23a0" ] } ], "prompt_number": 41 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Steadystate of $\\sigma_-$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "pauli_represent_x_y(sm).subs({sx: A_sol[ops.index(sx)], sy: A_sol[ops.index(sy)]})" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{i \\Omega \\gamma_{0}}{2 \\left(- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}\\right) \\left(- 2 N \\gamma_{0} + \\frac{\\Omega^{2}}{- N \\gamma_{0} - \\frac{\\gamma_{0}}{2}} - \\gamma_{0}\\right)}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 42, "text": [ " \u2148\u22c5\\Omega\u22c5\\gamma\u2080 \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u239b 2 \u239e\n", " \u239b \\gamma\u2080\u239e \u239c \\Omega \u239f\n", "2\u22c5\u239c-N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\u22c5\u239c-2\u22c5N\u22c5\\gamma\u2080 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \\gamma\u2080\u239f\n", " \u239d 2 \u23a0 \u239c \\gamma\u2080 \u239f\n", " \u239c -N\u22c5\\gamma\u2080 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u239f\n", " \u239d 2 \u23a0" ] } ], "prompt_number": 42 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Alternatively we can also use the SymPy `solve` function to find the steadystate solutions:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "solve([eq.rhs for eq in sc_ode.values()], list(ofm.values()))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left \\{ \\operatorname{A_{0}}{\\left (t \\right )} : 0, \\quad \\operatorname{A_{1}}{\\left (t \\right )} : - \\frac{2 \\Omega \\gamma_{0}}{2 \\Omega^{2} + \\gamma_{0}^{2} \\left(2 N + 1\\right)^{2}}, \\quad \\operatorname{A_{2}}{\\left (t \\right )} : - \\frac{\\gamma_{0}^{2} \\left(2 N + 1\\right)}{2 \\Omega^{2} + \\gamma_{0}^{2} \\left(2 N + 1\\right)^{2}}\\right \\}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 43, "text": [ "\u23a7 2 \n", "\u23aa -2\u22c5\\Omega\u22c5\\gamma\u2080 -\\gamma\u2080 \u22c5(2\u22c5N\n", "\u23a8A\u2080(t): 0, A\u2081(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, A\u2082(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", "\u23aa 2 2 2 2 2\n", "\u23a9 2\u22c5\\Omega + \\gamma\u2080 \u22c5(2\u22c5N + 1) 2\u22c5\\Omega + \\gamma\u2080 \n", "\n", " \u23ab\n", " + 1) \u23aa\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23ac\n", " 2\u23aa\n", "\u22c5(2\u22c5N + 1) \u23ad" ] } ], "prompt_number": 43 }, { "cell_type": "markdown", "metadata": {}, "source": [ "At zero temperature:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "solve([eq.subs(N, 0).rhs for eq in sc_ode.values()], list(ofm.values()))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left \\{ \\operatorname{A_{0}}{\\left (t \\right )} : 0, \\quad \\operatorname{A_{1}}{\\left (t \\right )} : - \\frac{2 \\Omega \\gamma_{0}}{2 \\Omega^{2} + \\gamma_{0}^{2}}, \\quad \\operatorname{A_{2}}{\\left (t \\right )} : - \\frac{\\gamma_{0}^{2}}{2 \\Omega^{2} + \\gamma_{0}^{2}}\\right \\}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 44, "text": [ "\u23a7 2 \u23ab\n", "\u23aa -2\u22c5\\Omega\u22c5\\gamma\u2080 -\\gamma\u2080 \u23aa\n", "\u23a8A\u2080(t): 0, A\u2081(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500, A\u2082(t): \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23ac\n", "\u23aa 2 2 2 2\u23aa\n", "\u23a9 2\u22c5\\Omega + \\gamma\u2080 2\u22c5\\Omega + \\gamma\u2080 \u23ad" ] } ], "prompt_number": 44 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Versions" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%reload_ext version_information\n", "\n", "%version_information sympy, sympsi" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "
SoftwareVersion
Python3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]
IPython2.3.0
OSDarwin 13.4.0 x86_64 i386 64bit
sympy0.7.5-git
sympsi0.1.0.dev-0c6e514
Sun Oct 12 21:42:08 2014 JST
" ], "json": [ "{\"Software versions\": [{\"version\": \"3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]\", \"module\": \"Python\"}, {\"version\": \"2.3.0\", \"module\": \"IPython\"}, {\"version\": \"Darwin 13.4.0 x86_64 i386 64bit\", \"module\": \"OS\"}, {\"version\": \"0.7.5-git\", \"module\": \"sympy\"}, {\"version\": \"0.1.0.dev-0c6e514\", \"module\": \"sympsi\"}]}" ], "latex": [ "\\begin{tabular}{|l|l|}\\hline\n", "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n", "Python & 3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)] \\\\ \\hline\n", "IPython & 2.3.0 \\\\ \\hline\n", "OS & Darwin 13.4.0 x86\\letterunderscore{}64 i386 64bit \\\\ \\hline\n", "sympy & 0.7.5-git \\\\ \\hline\n", "sympsi & 0.1.0.dev-0c6e514 \\\\ \\hline\n", "\\hline \\multicolumn{2}{|l|}{Sun Oct 12 21:42:08 2014 JST} \\\\ \\hline\n", "\\end{tabular}\n" ], "metadata": {}, "output_type": "pyout", "prompt_number": 45, "text": [ "Software versions\n", "Python 3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)]\n", "IPython 2.3.0\n", "OS Darwin 13.4.0 x86_64 i386 64bit\n", "sympy 0.7.5-git\n", "sympsi 0.1.0.dev-0c6e514\n", "Sun Oct 12 21:42:08 2014 JST" ] } ], "prompt_number": 45 } ], "metadata": {} } ] }