{ "metadata": { "name": "", "signature": "sha256:9bc90bf47bd622bb253206cc04ef151b8ee25119ca91ec4aafeeea80eda72674" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Lecture 4 - Symbolic quantum mechanics using SymPsi - Atom and cavity" ] }, { "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": [ "from sympy import *\n", "init_printing()" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 1 }, { "cell_type": "code", "collapsed": false, "input": [ "from sympsi import *\n", "from sympsi.boson import *\n", "from sympsi.pauli import *" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "The Jaynes-Cummings model" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The [Jaynes-Cummings model](http://en.wikipedia.org/wiki/Jaynes%E2%80%93Cummings_model) is one of the most elementary quantum mechanical models light-matter interaction. It describes a single two-level atom that interacts with a single harmonic-oscillator mode of a electromagnetic cavity.\n", "\n", "The Hamiltonian for a two-level system in its eigenbasis (see [Two-level systems](lecture-sympy-quantum-two-level-system.ipynb)) can be written as\n", "\n", "$$\n", "H = \\frac{1}{2}\\Omega \\sigma_z\n", "$$\n", "\n", "and the Hamiltonian of a quantum harmonic oscillator (see [Resonators and cavities](lecture-sympy-quantum-resonators.ipynb)) is\n", "\n", "$$\n", "H = \\hbar\\omega_r (a^\\dagger a + 1/2)\n", "$$\n", "\n", "The atom interacts with the electromagnetic field produced by the cavity mode $a + a^\\dagger$ through its dipole moment. The dipole-transition operators is $\\sigma_x$ (which cause a transition from the two dipole states of the atom). The combined atom-cavity Hamiltonian can therefore be written in the form\n", "\n", "$$\n", "H = \n", "\\hbar\\omega_r (a^\\dagger a + 1/2)\n", "+ \\frac{1}{2}\\hbar\\Omega\\sigma_z \n", "+\n", "\\hbar\n", "g\\sigma_x(a + a^\\dagger)\n", "$$\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "To obtain the Jaynes-Cumming Hamiltonian \n", "\n", "$$\n", "H = \n", "\\hbar\\omega_r (a^\\dagger a + 1/2)\n", "%-\\frac{1}{2}\\Delta\\sigma_x \n", "+ \\frac{1}{2}\\hbar\\Omega\\sigma_z \n", "+\n", "\\hbar\n", "g(\\sigma_+ a + \\sigma_- a^\\dagger)\n", "$$\n", "\n", "we also need to perform a rotating-wave approximation which simplifies the interaction part of the Hamiltonian. In the following we will begin with looking at how these two Hamiltonians are related.\n", "\n", "To represent the atom-cavity Hamiltonian in SymPy we creates an instances of the operator classes `BosonOp` and `SigmaX`, `SigmaY`, and `SigmaZ`, and use these to construct the Hamiltonian (we work in units where $\\hbar = 1$)." ] }, { "cell_type": "code", "collapsed": false, "input": [ "omega_r, Omega, g, Delta, t, x, Hsym = symbols(\"omega_r, Omega, g, Delta, t, x, H\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "sx, sy, sz, sm, sp = SigmaX(), SigmaY(), SigmaZ(), SigmaMinus(), SigmaPlus()\n", "a = BosonOp(\"a\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "H = omega_r * Dagger(a) * a + Omega/2 * sz + g * sx * (a + Dagger(a))\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\frac{\\Omega {\\sigma_z}}{2} + g {\\sigma_x} \\left({{a}^\\dagger} + {a}\\right) + \\omega_{r} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 5, "text": [ " \u03a9\u22c5False \u239b \u2020 \u239e \u2020 \n", "H = \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5False\u22c5\u239da + a\u23a0 + \u03c9\u1d63\u22c5a \u22c5a\n", " 2 " ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To simplify the interaction term we carry out two unitary transformations that corresponds to moving to the interaction picture:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * omega_r * t * Dagger(a) * a)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\omega_{r} t {{a}^\\dagger} {a}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"$$\\frac{\\Omega {\\sigma_z}}{2} + g e^{i \\omega_{r} t} {\\sigma_x} {{a}^\\dagger} + g e^{- i \\omega_{r} t} {\\sigma_x} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 7, "text": [ "\u03a9\u22c5False \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 -\u2148\u22c5\u03c9\u1d63\u22c5t \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5\u212f \u22c5False\u22c5a + g\u22c5\u212f \u22c5False\u22c5a\n", " 2 " ] } ], "prompt_number": 7 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * Omega * t * sp * sm)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\Omega t {\\sigma_+} {\\sigma_-}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega}{2} \\left(1 + {\\sigma_z}\\right) + \\frac{\\Omega {\\sigma_z}}{2} + g e^{- i \\Omega t - i \\omega_{r} t} {\\sigma_-} {a} + g e^{- i \\Omega t + i \\omega_{r} t} {\\sigma_-} {{a}^\\dagger} + g e^{i \\Omega t - i \\omega_{r} t} {\\sigma_+} {a} + g e^{i \\Omega t + i \\omega_{r} t} {\\sigma_+} {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 9, "text": [ " \u03a9\u22c5(1 + False) \u03a9\u22c5False -\u2148\u22c5\u03a9\u22c5t - \u2148\u22c5\u03c9\u1d63\u22c5t -\u2148\u22c5\u03a9\u22c5t + \u2148\u22c5\u03c9\u1d63\u22c5t \n", "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5\u212f \u22c5False\u22c5a + g\u22c5\u212f \u22c5Fa\n", " 2 2 \n", "\n", " \u2020 \u2148\u22c5\u03a9\u22c5t - \u2148\u22c5\u03c9\u1d63\u22c5t \u2148\u22c5\u03a9\u22c5t + \u2148\u22c5\u03c9\u1d63\u22c5t \u2020\n", "lse\u22c5a + g\u22c5\u212f \u22c5False\u22c5a + g\u22c5\u212f \u22c5False\u22c5a \n", " " ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We introduce the detuning parameter $\\Delta = \\Omega - \\omega_r$ and substitute into this expression" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# trick to simplify exponents\n", "def simplify_exp(e):\n", " if isinstance(e, exp):\n", " return exp(simplify(e.exp.expand()))\n", "\n", " if isinstance(e, (Add, Mul)):\n", " return type(e)(*(simplify_exp(arg) for arg in e.args)) \n", "\n", " return e" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = simplify_exp(H3).subs(-omega_r + Omega, Delta)\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega}{2} \\left(1 + {\\sigma_z}\\right) + \\frac{\\Omega {\\sigma_z}}{2} + g e^{i \\Delta t} {\\sigma_+} {a} + g e^{i t \\left(\\Omega + \\omega_{r}\\right)} {\\sigma_+} {{a}^\\dagger} + g e^{- i t \\left(\\Omega + \\omega_{r}\\right)} {\\sigma_-} {a} + g e^{- i \\Delta t} {\\sigma_-} {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 11, "text": [ " \u03a9\u22c5(1 + False) \u03a9\u22c5False \u2148\u22c5\u0394\u22c5t \u2148\u22c5t\u22c5(\u03a9 + \u03c9\u1d63) \u2020 -\n", "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5\u212f \u22c5False\u22c5a + g\u22c5\u212f \u22c5False\u22c5a + g\u22c5\u212f \n", " 2 2 \n", "\n", "\u2148\u22c5t\u22c5(\u03a9 + \u03c9\u1d63) -\u2148\u22c5\u0394\u22c5t \u2020\n", " \u22c5False\u22c5a + g\u22c5\u212f \u22c5False\u22c5a \n", " " ] } ], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, in the rotating-wave approximation we can drop the fast oscillating terms containing the factors $e^{\\pm i(\\Omega + \\omega_r)t}$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = drop_terms_containing(H4, [exp( I * (Omega + omega_r) * t),\n", " exp(-I * (Omega + omega_r) * t)])\n", "\n", "H5 = drop_c_number_terms(H5.expand())\n", "\n", "Eq(Hsym, H5)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = g e^{i \\Delta t} {\\sigma_+} {a} + g e^{- i \\Delta t} {\\sigma_-} {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 12, "text": [ " \u2148\u22c5\u0394\u22c5t -\u2148\u22c5\u0394\u22c5t \u2020\n", "H = g\u22c5\u212f \u22c5False\u22c5a + g\u22c5\u212f \u22c5False\u22c5a " ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the interaction term of in the Jaynes-Cumming model in the interaction picture. If we transform back to the Schr\u00f6dinger picture we have:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(-I * omega_r * t * Dagger(a) * a)\n", "H6 = hamiltonian_transformation(U, H5.expand())" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(-I * Omega * t * sp * sm)\n", "H7 = hamiltonian_transformation(U, H6.expand())" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 14 }, { "cell_type": "code", "collapsed": false, "input": [ "H8 = simplify_exp(H7).subs(Delta, Omega - omega_r)\n", "\n", "H8 = simplify_exp(powsimp(H8)).expand()\n", "\n", "H8 = drop_c_number_terms(H8)\n", "\n", "H = collect(H8, g)\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\frac{\\Omega {\\sigma_z}}{2} + g \\left({\\sigma_-} {{a}^\\dagger} + {\\sigma_+} {a}\\right) + \\omega_{r} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAUwAAAArBAMAAADhxqw6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n", "prompt_number": 15, "text": [ " \u03a9\u22c5False \u239b \u2020 \u239e \u2020 \n", "H = \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5\u239dFalse\u22c5a + False\u22c5a\u23a0 + \u03c9\u1d63\u22c5a \u22c5a\n", " 2 " ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the Jaynes-Cumming model give above, and we have now seen that it is obtained to the dipole interaction Hamiltonian through the rotating wave approximation." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Dispersive regime" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the dispersive regime, where the two-level system is detuned from the cavity by much more than the interaction strength, $\\Delta \\gg g$, an effective Hamiltonian can be dervied which describes the Stark shift of the two-level system (which depends on the number of photons in the cavity) and the frequency shift of the cavity (which depend on the state of the two-level system).\n", "\n", "This effective Hamiltonian, which is correct up to second order in the small paramter $g/\\Delta$, is obtained by performing the unitary transformation\n", "\n", "$$\n", "U = e^{\\frac{g}{\\Delta}(a \\sigma_- - a^\\dagger \\sigma_+)}\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp((x * (a * sp - Dagger(a) * sm)).expand())\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{- x {{a}^\\dagger} {\\sigma_-} + x {a} {\\sigma_+}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAHoAAAAWBAMAAAD9QZX2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQJalN2JvkBhAX/5ic7c3t725bBn4BS7YUCdoxJvMAAAAAElFTkSuQmCC\n", "prompt_number": 16, "text": [ " \u2020 \n", " - x\u22c5a \u22c5False + x\u22c5a\u22c5False\n", "\u212f " ] } ], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "#H1 = unitary_transformation(U, H, allinone=True, expansion_search=False, N=3).expand()\n", "#H1 = qsimplify(H1)\n", "#H1" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "H1 = hamiltonian_transformation(U, H, expansion_search=False, N=3).expand()\n", "\n", "H1 = qsimplify(H1)\n", "\n", "H1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega x^{2}}{2} - \\Omega x^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega x^{2}}{2} {\\sigma_z} - \\Omega x {{a}^\\dagger} {\\sigma_-} - \\Omega x {a} {\\sigma_+} + \\frac{\\Omega {\\sigma_z}}{2} + g x^{4} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{3 g}{4} x^{4} {{a}^\\dagger} {\\sigma_-} - g x^{4} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{5 g}{4} x^{4} {a} {\\sigma_+} - 2 g x^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g x^{2} {{a}^\\dagger} {\\sigma_-} - 2 g x^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g x^{2} {a} {\\sigma_+} + g x + 2 g x {{a}^\\dagger} {a} {\\sigma_z} + g x {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\frac{\\omega_{r} x^{4}}{4} {{a}^\\dagger} {a} - \\frac{\\omega_{r} x^{3}}{2} {{a}^\\dagger} {\\sigma_-} - \\frac{\\omega_{r} x^{3}}{2} {a} {\\sigma_+} + \\frac{\\omega_{r} x^{2}}{2} + \\omega_{r} x^{2} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\omega_{r} x^{2}}{2} {\\sigma_z} + \\omega_{r} x {{a}^\\dagger} {\\sigma_-} + \\omega_{r} x {a} {\\sigma_+} + \\omega_{r} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 18, "text": [ " 2 2 \n", " \u03a9\u22c5x 2 \u2020 \u03a9\u22c5x \u22c5False \u2020 \u03a9\u22c5False \n", "- \u2500\u2500\u2500\u2500 - \u03a9\u22c5x \u22c5a \u22c5a\u22c5False - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u03a9\u22c5x\u22c5a \u22c5False - \u03a9\u22c5x\u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 +\n", " 2 2 2 \n", "\n", " 4 \u2020 2 4 \n", " 4 \u2020 2 3\u22c5g\u22c5x \u22c5a \u22c5False 4 \u239b \u2020\u239e 5\u22c5g\u22c5x \u22c5a\u22c5False \n", " g\u22c5x \u22c5a \u22c5a \u22c5False - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - g\u22c5x \u22c5\u239da \u23a0 \u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - 2\u22c5\n", " 4 4 \n", "\n", " 2 \n", " 2 \u2020 2 2 \u2020 2 \u239b \u2020\u239e 2 \n", "g\u22c5x \u22c5a \u22c5a \u22c5False - 2\u22c5g\u22c5x \u22c5a \u22c5False - 2\u22c5g\u22c5x \u22c5\u239da \u23a0 \u22c5a\u22c5False - 2\u22c5g\u22c5x \u22c5a\u22c5False + g\n", " \n", "\n", " 4 \u2020 \n", " \u2020 \u2020 \u03c9\u1d63\u22c5x \u22c5a \u22c5a \u03c9\u1d63\u22c5x\n", "\u22c5x + 2\u22c5g\u22c5x\u22c5a \u22c5a\u22c5False + g\u22c5x\u22c5False + g\u22c5a \u22c5False + g\u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\n", " 4 \n", "\n", "3 \u2020 3 2 2 \n", " \u22c5a \u22c5False \u03c9\u1d63\u22c5x \u22c5a\u22c5False \u03c9\u1d63\u22c5x 2 \u2020 \u03c9\u1d63\u22c5x \u22c5False \u2020 \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 + \u03c9\u1d63\u22c5x \u22c5a \u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u03c9\u1d63\u22c5x\u22c5a \u22c5\n", " 2 2 2 2 \n", "\n", " \n", " \u2020 \n", "False + \u03c9\u1d63\u22c5x\u22c5a\u22c5False + \u03c9\u1d63\u22c5a \u22c5a\n", " " ] } ], "prompt_number": 18 }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = drop_terms_containing(H1.expand(), [x**3, x**4])\n", "\n", "H2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega x^{2}}{2} - \\Omega x^{2} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega x^{2}}{2} {\\sigma_z} - \\Omega x {{a}^\\dagger} {\\sigma_-} - \\Omega x {a} {\\sigma_+} + \\frac{\\Omega {\\sigma_z}}{2} - 2 g x^{2} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - 2 g x^{2} {{a}^\\dagger} {\\sigma_-} - 2 g x^{2} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - 2 g x^{2} {a} {\\sigma_+} + g x + 2 g x {{a}^\\dagger} {a} {\\sigma_z} + g x {\\sigma_z} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\frac{\\omega_{r} x^{2}}{2} + \\omega_{r} x^{2} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{\\omega_{r} x^{2}}{2} {\\sigma_z} + \\omega_{r} x {{a}^\\dagger} {\\sigma_-} + \\omega_{r} x {a} {\\sigma_+} + \\omega_{r} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 19, "text": [ " 2 2 \n", " \u03a9\u22c5x 2 \u2020 \u03a9\u22c5x \u22c5False \u2020 \u03a9\u22c5False \n", "- \u2500\u2500\u2500\u2500 - \u03a9\u22c5x \u22c5a \u22c5a\u22c5False - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u03a9\u22c5x\u22c5a \u22c5False - \u03a9\u22c5x\u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 -\n", " 2 2 2 \n", "\n", " 2 \n", " 2 \u2020 2 2 \u2020 2 \u239b \u2020\u239e 2 \n", " 2\u22c5g\u22c5x \u22c5a \u22c5a \u22c5False - 2\u22c5g\u22c5x \u22c5a \u22c5False - 2\u22c5g\u22c5x \u22c5\u239da \u23a0 \u22c5a\u22c5False - 2\u22c5g\u22c5x \u22c5a\u22c5False \n", " \n", "\n", " 2 \n", " \u2020 \u2020 \u03c9\u1d63\u22c5x 2 \n", "+ g\u22c5x + 2\u22c5g\u22c5x\u22c5a \u22c5a\u22c5False + g\u22c5x\u22c5False + g\u22c5a \u22c5False + g\u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500 + \u03c9\u1d63\u22c5x \u22c5\n", " 2 \n", "\n", " 2 \n", " \u2020 \u03c9\u1d63\u22c5x \u22c5False \u2020 \u2020 \n", "a \u22c5a\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u03c9\u1d63\u22c5x\u22c5a \u22c5False + \u03c9\u1d63\u22c5x\u22c5a\u22c5False + \u03c9\u1d63\u22c5a \u22c5a\n", " 2 " ] } ], "prompt_number": 19 }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = H2.subs(x, g/Delta)\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\Omega {\\sigma_z}}{2} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\omega_{r} {{a}^\\dagger} {a} - \\frac{\\Omega g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g}{\\Delta} {a} {\\sigma_+} + \\frac{g^{2}}{\\Delta} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega_{r}}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega_{r}}{\\Delta} {a} {\\sigma_+} - \\frac{\\Omega g^{2}}{2 \\Delta^{2}} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+} + \\frac{g^{2} \\omega_{r}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 20, "text": [ " \n", " \u2020 2 \n", "\u03a9\u22c5False \u2020 \u2020 \u03a9\u22c5g\u22c5a \u22c5False \u03a9\u22c5g\u22c5a\u22c5False g \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5a \u22c5False + g\u22c5a\u22c5False + \u03c9\u1d63\u22c5a \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 +\n", " 2 \u0394 \u0394 \u0394 \n", " \n", "\n", " \n", " 2 \u2020 2 \u2020 2 2 \u2020 \n", " 2\u22c5g \u22c5a \u22c5a\u22c5False g \u22c5False g\u22c5\u03c9\u1d63\u22c5a \u22c5False g\u22c5\u03c9\u1d63\u22c5a\u22c5False \u03a9\u22c5g \u03a9\u22c5g \u22c5a \u22c5a\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\n", " \u0394 \u0394 \u0394 \u0394 2 2 \n", " 2\u22c5\u0394 \u0394 \n", "\n", " 2 \n", " 2 3 \u2020 2 3 \u2020 3 \u239b \u2020\u239e \n", "False \u03a9\u22c5g \u22c5False 2\u22c5g \u22c5a \u22c5a \u22c5False 2\u22c5g \u22c5a \u22c5False 2\u22c5g \u22c5\u239da \u23a0 \u22c5a\u22c5False 2\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\n", " 2 2 2 2 \n", " 2\u22c5\u0394 \u0394 \u0394 \u0394 \n", "\n", " \n", " 3 2 2 \u2020 2 \n", "\u22c5g \u22c5a\u22c5False g \u22c5\u03c9\u1d63 g \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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", " 2 2 2 2 \n", " \u0394 2\u22c5\u0394 \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 20 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = drop_c_number_terms(H3)\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\Omega {\\sigma_z}}{2} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\omega_{r} {{a}^\\dagger} {a} - \\frac{\\Omega g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g}{\\Delta} {a} {\\sigma_+} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega_{r}}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega_{r}}{\\Delta} {a} {\\sigma_+} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 21, "text": [ " \n", " \u2020 2\n", "\u03a9\u22c5False \u2020 \u2020 \u03a9\u22c5g\u22c5a \u22c5False \u03a9\u22c5g\u22c5a\u22c5False 2\u22c5g \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5a \u22c5False + g\u22c5a\u22c5False + \u03c9\u1d63\u22c5a \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\n", " 2 \u0394 \u0394 \n", " \n", "\n", " \n", " \u2020 2 \u2020 2 \u2020 2\n", "\u22c5a \u22c5a\u22c5False g \u22c5False g\u22c5\u03c9\u1d63\u22c5a \u22c5False g\u22c5\u03c9\u1d63\u22c5a\u22c5False \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \u03a9\u22c5g \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\n", " \u0394 \u0394 \u0394 \u0394 2 \n", " \u0394 2\n", "\n", " 2 \n", " 3 \u2020 2 3 \u2020 3 \u239b \u2020\u239e 3 \n", "\u22c5False 2\u22c5g \u22c5a \u22c5a \u22c5False 2\u22c5g \u22c5a \u22c5False 2\u22c5g \u22c5\u239da \u23a0 \u22c5a\u22c5False 2\u22c5g \u22c5a\u22c5False \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 \n", " 2 2 2 2 2 \n", "\u22c5\u0394 \u0394 \u0394 \u0394 \u0394 \n", "\n", " \n", " 2 \u2020 2 \n", " g \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 \n", " \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 21 }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = collect(H4, [Dagger(a) * a, sz])\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\left(\\frac{\\Omega}{2} + \\frac{g^{2}}{\\Delta} - \\frac{\\Omega g^{2}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega_{r}}{2 \\Delta^{2}}\\right) {\\sigma_z} + {{a}^\\dagger} {a} \\left(\\omega_{r} + \\frac{2 g^{2}}{\\Delta} {\\sigma_z} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {\\sigma_z} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {\\sigma_z}\\right) - \\frac{\\Omega g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g}{\\Delta} {a} {\\sigma_+} + \\frac{g \\omega_{r}}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega_{r}}{\\Delta} {a} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 22, "text": [ " \n", " \u239b 2 2 2 \u239e \u239b 2 \n", " \u2020 \u239c\u03a9 g \u03a9\u22c5g g \u22c5\u03c9\u1d63\u239f \u2020 \u239c 2\u22c5g \u22c5False\n", "g\u22c5a \u22c5False + g\u22c5a\u22c5False + \u239c\u2500 + \u2500\u2500 - \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u239f\u22c5False + a \u22c5a\u22c5\u239c\u03c9\u1d63 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\n", " \u239c2 \u0394 2 2\u239f \u239c \u0394 \n", " \u239d 2\u22c5\u0394 2\u22c5\u0394 \u23a0 \u239d \n", "\n", " \n", " 2 2 \u239e \u2020 \u2020 \n", " \u03a9\u22c5g \u22c5False g \u22c5\u03c9\u1d63\u22c5False\u239f \u03a9\u22c5g\u22c5a \u22c5False \u03a9\u22c5g\u22c5a\u22c5False g\u22c5\u03c9\u1d63\u22c5a \u22c5False g\u22c5\n", " - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\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 \u239f \u0394 \u0394 \u0394 \n", " \u0394 \u0394 \u23a0 \n", "\n", " 2 \n", " 3 \u2020 2 3 \u2020 3 \u239b \u2020\u239e 3 \n", "\u03c9\u1d63\u22c5a\u22c5False 2\u22c5g \u22c5a \u22c5a \u22c5False 2\u22c5g \u22c5a \u22c5False 2\u22c5g \u22c5\u239da \u23a0 \u22c5a\u22c5False 2\u22c5g \u22c5a\u22c5Fa\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\n", " \u0394 2 2 2 2 \n", " \u0394 \u0394 \u0394 \u0394 \n", "\n", " \n", " \n", "lse\n", "\u2500\u2500\u2500\n", " \n", " " ] } ], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now move to a frame co-rotating with the qubit and oscillator frequencies: " ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5.expand()" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\Omega {\\sigma_z}}{2} + g {{a}^\\dagger} {\\sigma_-} + g {a} {\\sigma_+} + \\omega_{r} {{a}^\\dagger} {a} - \\frac{\\Omega g}{\\Delta} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g}{\\Delta} {a} {\\sigma_+} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega_{r}}{\\Delta} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega_{r}}{\\Delta} {a} {\\sigma_+} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} {a} {\\sigma_+} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 23, "text": [ " \n", " \u2020 2\n", "\u03a9\u22c5False \u2020 \u2020 \u03a9\u22c5g\u22c5a \u22c5False \u03a9\u22c5g\u22c5a\u22c5False 2\u22c5g \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5a \u22c5False + g\u22c5a\u22c5False + \u03c9\u1d63\u22c5a \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\n", " 2 \u0394 \u0394 \n", " \n", "\n", " \n", " \u2020 2 \u2020 2 \u2020 2\n", "\u22c5a \u22c5a\u22c5False g \u22c5False g\u22c5\u03c9\u1d63\u22c5a \u22c5False g\u22c5\u03c9\u1d63\u22c5a\u22c5False \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \u03a9\u22c5g \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\n", " \u0394 \u0394 \u0394 \u0394 2 \n", " \u0394 2\n", "\n", " 2 \n", " 3 \u2020 2 3 \u2020 3 \u239b \u2020\u239e 3 \n", "\u22c5False 2\u22c5g \u22c5a \u22c5a \u22c5False 2\u22c5g \u22c5a \u22c5False 2\u22c5g \u22c5\u239da \u23a0 \u22c5a\u22c5False 2\u22c5g \u22c5a\u22c5False \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 \n", " 2 2 2 2 2 \n", "\u22c5\u0394 \u0394 \u0394 \u0394 \u0394 \n", "\n", " \n", " 2 \u2020 2 \n", " g \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 \n", " \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 23 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * omega_r * t * Dagger(a) * a)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 24 }, { "cell_type": "code", "collapsed": false, "input": [ "H6 = hamiltonian_transformation(U, H5.expand()); H6" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\Omega {\\sigma_z}}{2} + g e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} + g e^{- i \\omega_{r} t} {a} {\\sigma_+} - \\frac{\\Omega g}{\\Delta} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g}{\\Delta} e^{- i \\omega_{r} t} {a} {\\sigma_+} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega_{r}}{\\Delta} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} + \\frac{g \\omega_{r}}{\\Delta} e^{- i \\omega_{r} t} {a} {\\sigma_+} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\omega_{r} t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\omega_{r} t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\omega_{r} t} {a} {\\sigma_+} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 25, "text": [ " \n", " \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 \n", "\u03a9\u22c5False \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 -\u2148\u22c5\u03c9\u1d63\u22c5t \u03a9\u22c5g\u22c5\u212f \u22c5a \u22c5False \u03a9\u22c5g\n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5\u212f \u22c5a \u22c5False + g\u22c5\u212f \u22c5a\u22c5False - \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 \u0394 \n", " \n", "\n", " \n", " -\u2148\u22c5\u03c9\u1d63\u22c5t 2 \u2020 2 \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 \n", "\u22c5\u212f \u22c5a\u22c5False 2\u22c5g \u22c5a \u22c5a\u22c5False g \u22c5False g\u22c5\u03c9\u1d63\u22c5\u212f \u22c5a \u22c5False g\u22c5\u03c9\u1d63\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\n", " \u0394 \u0394 \u0394 \u0394 \n", " \n", "\n", " \n", " -\u2148\u22c5\u03c9\u1d63\u22c5t 2 \u2020 2 3 \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 3\n", "\u212f \u22c5a\u22c5False \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \u03a9\u22c5g \u22c5False 2\u22c5g \u22c5\u212f \u22c5a \u22c5False 2\u22c5g \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\n", " \u0394 2 2 2 \n", " \u0394 2\u22c5\u0394 \u0394 \n", "\n", " 2 \n", " \u2148\u22c5\u03c9\u1d63\u22c5t \u239b \u2020\u239e 3 -\u2148\u22c5\u03c9\u1d63\u22c5t \u2020 2 3 -\u2148\u22c5\u03c9\u1d63\u22c5t \n", "\u22c5\u212f \u22c5\u239da \u23a0 \u22c5a\u22c5False 2\u22c5g \u22c5\u212f \u22c5a \u22c5a \u22c5False 2\u22c5g \u22c5\u212f \u22c5a\u22c5False g\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\n", " 2 2 2 \n", " \u0394 \u0394 \u0394 \n", "\n", " \n", "2 \u2020 2 \n", " \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 \n", " \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 25 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * Omega * t * Dagger(sm) * sm)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 26 }, { "cell_type": "code", "collapsed": false, "input": [ "H7 = hamiltonian_transformation(U, H6.expand()); H7" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\Omega {\\sigma_+} {\\sigma_-} + \\frac{\\Omega {\\sigma_z}}{2} + g e^{i \\Omega t} e^{- i \\omega_{r} t} {a} {\\sigma_+} + g e^{- i \\Omega t} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g}{\\Delta} e^{i \\Omega t} e^{- i \\omega_{r} t} {a} {\\sigma_+} - \\frac{\\Omega g}{\\Delta} e^{- i \\Omega t} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} + \\frac{g \\omega_{r}}{\\Delta} e^{i \\Omega t} e^{- i \\omega_{r} t} {a} {\\sigma_+} + \\frac{g \\omega_{r}}{\\Delta} e^{- i \\Omega t} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\Omega t} e^{- i \\omega_{r} t} {{a}^\\dagger} \\left({a}\\right)^{2} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{i \\Omega t} e^{- i \\omega_{r} t} {a} {\\sigma_+} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\Omega t} e^{i \\omega_{r} t} {{a}^\\dagger} {\\sigma_-} - \\frac{2 g^{3}}{\\Delta^{2}} e^{- i \\Omega t} e^{i \\omega_{r} t} \\left({{a}^\\dagger}\\right)^{2} {a} {\\sigma_-} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 27, "text": [ " \n", " \n", " \u03a9\u22c5False \u2148\u22c5\u03a9\u22c5t -\u2148\u22c5\u03c9\u1d63\u22c5t -\u2148\u22c5\u03a9\u22c5t \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 \n", "-\u03a9\u22c5False\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + g\u22c5\u212f \u22c5\u212f \u22c5a\u22c5False + g\u22c5\u212f \u22c5\u212f \u22c5a \u22c5Fa\n", " 2 \n", " \n", "\n", " \n", " \u2148\u22c5\u03a9\u22c5t -\u2148\u22c5\u03c9\u1d63\u22c5t -\u2148\u22c5\u03a9\u22c5t \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 2 \u2020 \n", " \u03a9\u22c5g\u22c5\u212f \u22c5\u212f \u22c5a\u22c5False \u03a9\u22c5g\u22c5\u212f \u22c5\u212f \u22c5a \u22c5False 2\u22c5g \u22c5a \u22c5a\u22c5F\n", "lse - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\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", " \u0394 \u0394 \u0394 \n", " \n", "\n", " \n", " 2 \u2148\u22c5\u03a9\u22c5t -\u2148\u22c5\u03c9\u1d63\u22c5t -\u2148\u22c5\u03a9\u22c5t \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 \n", "alse g \u22c5False g\u22c5\u03c9\u1d63\u22c5\u212f \u22c5\u212f \u22c5a\u22c5False g\u22c5\u03c9\u1d63\u22c5\u212f \u22c5\u212f \u22c5a \u22c5False\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\n", " \u0394 \u0394 \u0394 \n", " \n", "\n", " \n", " 2 \u2020 2 3 \u2148\u22c5\u03a9\u22c5t -\u2148\u22c5\u03c9\u1d63\u22c5t \u2020 2 3 \u2148\u22c5\u03a9\n", " \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \u03a9\u22c5g \u22c5False 2\u22c5g \u22c5\u212f \u22c5\u212f \u22c5a \u22c5a \u22c5False 2\u22c5g \u22c5\u212f \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\n", " 2 2 2 \n", " \u0394 2\u22c5\u0394 \u0394 \n", "\n", " \n", "\u22c5t -\u2148\u22c5\u03c9\u1d63\u22c5t 3 -\u2148\u22c5\u03a9\u22c5t \u2148\u22c5\u03c9\u1d63\u22c5t \u2020 3 -\u2148\u22c5\u03a9\u22c5t \u2148\u22c5\u03c9\u1d63\u22c5t \u239b \u2020\n", " \u22c5\u212f \u22c5a\u22c5False 2\u22c5g \u22c5\u212f \u22c5\u212f \u22c5a \u22c5False 2\u22c5g \u22c5\u212f \u22c5\u212f \u22c5\u239da \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 2 \n", " \u0394 \u0394 \u0394 \n", "\n", " 2 \n", "\u239e 2 \u2020 2 \n", "\u23a0 \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 \n", " \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 27 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, since we are in the dispersive regime $|\\Omega-\\omega_r| \\gg g$, we can do a rotating-wave approximation and drop all the fast rotating terms in the Hamiltonian above:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H8 = drop_terms_containing(H7, [exp(I * omega_r * t), exp(-I * omega_r * t),\n", " exp(I * Omega * t), exp(-I * Omega * t)])\n", "\n", "H8" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\Omega {\\sigma_+} {\\sigma_-} + \\frac{\\Omega {\\sigma_z}}{2} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 28, "text": [ " 2 \u2020 2 2 \u2020 2\n", " \u03a9\u22c5False 2\u22c5g \u22c5a \u22c5a\u22c5False g \u22c5False \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \u03a9\u22c5g \n", "-\u03a9\u22c5False\u22c5False + \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\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 \u0394 \u0394 2 \n", " \u0394 2\n", "\n", " 2 \u2020 2 \n", "\u22c5False g \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 2 \n", "\u22c5\u0394 \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 28 }, { "cell_type": "code", "collapsed": false, "input": [ "H9 = qsimplify(H8)\n", "\n", "H9 = collect(H9, [Dagger(a) * a, sz])\n", "\n", "H9" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega}{2} + \\left(\\frac{g^{2}}{\\Delta} - \\frac{\\Omega g^{2}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega_{r}}{2 \\Delta^{2}}\\right) {\\sigma_z} + {{a}^\\dagger} {a} \\left(\\frac{2 g^{2}}{\\Delta} {\\sigma_z} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {\\sigma_z} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {\\sigma_z}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 29, "text": [ " \u239b 2 2 2 \u239e \u239b 2 2 2 \u239e\n", " \u03a9 \u239cg \u03a9\u22c5g g \u22c5\u03c9\u1d63\u239f \u2020 \u239c2\u22c5g \u22c5False \u03a9\u22c5g \u22c5False g \u22c5\u03c9\u1d63\u22c5False\u239f\n", "- \u2500 + \u239c\u2500\u2500 - \u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u239f\u22c5False + a \u22c5a\u22c5\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\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n", " 2 \u239c\u0394 2 2\u239f \u239c \u0394 2 2 \u239f\n", " \u239d 2\u22c5\u0394 2\u22c5\u0394 \u23a0 \u239d \u0394 \u0394 \u23a0" ] } ], "prompt_number": 29 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now move back to the lab frame:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(-I * omega_r * t * Dagger(a) * a)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 30 }, { "cell_type": "code", "collapsed": false, "input": [ "H10 = hamiltonian_transformation(U, H9.expand()); H10" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega}{2} + \\omega_{r} {{a}^\\dagger} {a} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 31, "text": [ " 2 \u2020 2 2 \u2020 2 2\n", " \u03a9 \u2020 2\u22c5g \u22c5a \u22c5a\u22c5False g \u22c5False \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \u03a9\u22c5g \u22c5False g \n", "- \u2500 + \u03c9\u1d63\u22c5a \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\u2500\u2500\u2500\u2500 + \u2500\u2500\n", " 2 \u0394 \u0394 2 2 \n", " \u0394 2\u22c5\u0394 \n", "\n", " \u2020 2 \n", "\u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 \n", " \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 31 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(-I * Omega * t * Dagger(sm) * sm)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 32 }, { "cell_type": "code", "collapsed": false, "input": [ "H11 = hamiltonian_transformation(U, H10.expand()); H11" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{\\Omega}{2} + \\Omega {\\sigma_+} {\\sigma_-} + \\omega_{r} {{a}^\\dagger} {a} + \\frac{2 g^{2}}{\\Delta} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} {\\sigma_z}}{\\Delta} - \\frac{\\Omega g^{2}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} - \\frac{\\Omega g^{2} {\\sigma_z}}{2 \\Delta^{2}} + \\frac{g^{2} \\omega_{r}}{\\Delta^{2}} {{a}^\\dagger} {a} {\\sigma_z} + \\frac{g^{2} \\omega_{r} {\\sigma_z}}{2 \\Delta^{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 33, "text": [ " 2 \u2020 2 2 \u2020 \n", " \u03a9 \u2020 2\u22c5g \u22c5a \u22c5a\u22c5False g \u22c5False \u03a9\u22c5g \u22c5a \u22c5a\u22c5False \n", "- \u2500 + \u03a9\u22c5False\u22c5False + \u03c9\u1d63\u22c5a \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 -\n", " 2 \u0394 \u0394 2 \n", " \u0394 \n", "\n", " 2 2 \u2020 2 \n", " \u03a9\u22c5g \u22c5False g \u22c5\u03c9\u1d63\u22c5a \u22c5a\u22c5False g \u22c5\u03c9\u1d63\u22c5False\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\n", " 2 2 2 \n", " 2\u22c5\u0394 \u0394 2\u22c5\u0394 " ] } ], "prompt_number": 33 }, { "cell_type": "code", "collapsed": false, "input": [ "H12 = qsimplify(H11)\n", "\n", "H12 = collect(H12, [Dagger(a) * a, sz])\n", "\n", "H12 = H12.subs(omega_r, Omega-Delta).expand().collect([Dagger(a)*a, sz]).subs(Omega-Delta,omega_r)\n", "\n", "Eq(Hsym, H12)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\left(\\frac{\\Omega}{2} + \\frac{g^{2}}{2 \\Delta}\\right) {\\sigma_z} + {{a}^\\dagger} {a} \\left(\\omega_{r} + \\frac{g^{2} {\\sigma_z}}{\\Delta}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 34, "text": [ " \u239b 2\u239e \u239b 2 \u239e\n", " \u239c\u03a9 g \u239f \u2020 \u239c g \u22c5False\u239f\n", "H = \u239c\u2500 + \u2500\u2500\u2500\u239f\u22c5False + a \u22c5a\u22c5\u239c\u03c9\u1d63 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u239f\n", " \u239d2 2\u22c5\u0394\u23a0 \u239d \u0394 \u23a0" ] } ], "prompt_number": 34 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is the Hamiltonian of the Jaynes-Cummings model in the the dispersive regime. It can be interpreted as the resonator having a qubit-state-dependent frequency shift, or alternatively that the qubit is feeling a resonator-photon-number dependent Stark-shift." ] }, { "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": [ "
Software | Version |
---|---|
Python | 3.4.1 (default, Sep 20 2014, 19:44:17) [GCC 4.2.1 Compatible Apple LLVM 5.1 (clang-503.0.40)] |
IPython | 2.1.0 |
OS | Darwin 13.4.0 x86_64 i386 64bit |
sympy | 0.7.5-git |
sympsi | 0.1.0.dev-9060485 |
Thu Oct 09 16:25:51 2014 JST |