{ "metadata": { "name": "", "signature": "sha256:51895f9982b1da68e92b74354f8e4ca0122adc96bdc8df04113fa2e184ed22bb" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Lecture 5 - Symbolic quantum mechanics using SymPsi - Optomechanics" ] }, { "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).\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.operatorordering import *" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Optomechanical system" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the standard Hamiltonian for an optomechanical system, including a classical driving signal on the optical mode:\n", "\n", "$$\n", "H = \\hbar\\omega_a a^\\dagger a + \\hbar \\omega_b b^\\dagger b - \\hbar g a^\\dagger a (b + b^\\dagger) + (A e^{-i\\omega_d t} + A^* e^{i\\omega_d t})(a + a^\\dagger)\n", "$$\n", "\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "omega_a, omega_b, g, A, Delta, t = symbols(\"omega_a, omega_b, g, A, Delta, t\", positive=True)\n", "Hsym, omega_d = symbols(\"H, omega_d\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "a, b = BosonOp(\"a\"), BosonOp(\"b\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "H0 = omega_a * Dagger(a) * a + omega_b * Dagger(b) * b - g * Dagger(a) * a * (b + Dagger(b))\n", "\n", "Hdrive = (A * exp(-I * omega_d * t) + conjugate(A) * exp(I * omega_d * t)) * (a + Dagger(a))\n", "\n", "H = H0 + Hdrive\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = - g {{a}^\\dagger} {a} \\left({{b}^\\dagger} + {b}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b} + \\left(A e^{i \\omega_{d} t} + A e^{- i \\omega_{d} t}\\right) \\left({{a}^\\dagger} + {a}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAl0AAAAcBAMAAACpNxf4AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nXv8dMCr3NjKackfzBvE2uP/PQ/4NzJtHcDU8tkEAAAAASUVORK5CYII=\n", "prompt_number": 5, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \u239b \u2148\u22c5\u03c9_d\u22c5t -\u2148\u22c5\u03c9_d\u22c5t\u239e \u239b \u2020 \n", "H = - g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b + \u239dA\u22c5\u212f + A\u22c5\u212f \u23a0\u22c5\u239da + \n", "\n", " \u239e\n", "a\u23a0" ] } ], "prompt_number": 5 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Linearized interaction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we apply the unitary transformation $U = e^{i \\omega_d a^\\dagger a t}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * Dagger(a) * a * omega_d * t)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\omega_{d} t {{a}^\\dagger} {a}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAWBAMAAABppzwEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFYWQP0vDQByG35j0TFPaBoybkIPi\n5lBoQLAON6vDCY7uLg6dBUHcRIeASgdBbnPTYqGDU0pHl27iVvATdHD3ck1yTWnsDfk975+7JAcU\nrklhkgU0IwUmy2up2odRznO4llUxY6atRfLdmdNZDLTuJpivkDZICHzG4cnlAbzWBzCRQkHsYs13\nK3K8AFYH3zAunCnAkEDcwHEXp9iCBRCGX1jMkW+hcoMCVcEPHjHChjwvIlewQ7MHCCSgKsZU7GGA\nbYx2udnb9Hlt6MZBBkB5TJ+fvh5YmdqwmtQWb+dqawbys9557dq6RVW8gvQF6d8fqUoCRiPgSsvH\nutsIU56bZ7xEU0l2gihlPUs38ISWy6hy1xou8+e8uryBFatOVxTkBcpK8/+WOYaX/XRBNdiPCpLE\n/gNa6jyqkqIGvwAAAABJRU5ErkJggg==\n", "prompt_number": 6, "text": [ " \u2020 \n", " \u2148\u22c5\u03c9_d\u22c5t\u22c5a \u22c5a\n", "\u212f " ] } ], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "H1 = hamiltonian_transformation(U, H, independent=True)\n", "\n", "H1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- g {{a}^\\dagger} {a} \\left({{b}^\\dagger} + {b}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b} - \\omega_{d} {{a}^\\dagger} {a} + \\left(A e^{i \\omega_{d} t} + A e^{- i \\omega_{d} t}\\right) \\left(e^{i \\omega_{d} t} {{a}^\\dagger} + e^{- i \\omega_{d} t} {a}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAsgAAAAcBAMAAACOtVRVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRAAAAABJRU5ErkJggg==\n", "prompt_number": 7, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \u2020 \u239b \u2148\u22c5\u03c9_d\u22c5t -\u2148\u22c5\u03c9_d\u22c5t\u239e\n", "- g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b - \u03c9_d\u22c5a \u22c5a + \u239dA\u22c5\u212f + A\u22c5\u212f \u23a0\n", "\n", " \u239b \u2148\u22c5\u03c9_d\u22c5t \u2020 -\u2148\u22c5\u03c9_d\u22c5t \u239e\n", "\u22c5\u239d\u212f \u22c5a + \u212f \u22c5a\u23a0" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now perform a rotating-wave approximation (RWA) by eliminating all terms that rotate with frequencies $2\\omega_d$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = drop_terms_containing(H1.expand(), [exp(-2*I*omega_d*t), exp(2*I*omega_d*t)])\n", "\n", "Eq(Symbol(\"H_{rwa}\"), H2)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H_{{rwa}} = A {{a}^\\dagger} + A {a} - g {{a}^\\dagger} {a} {{b}^\\dagger} - g {{a}^\\dagger} {a} {b} + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b} - \\omega_{d} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAikAAAAXBAMAAAA8UsLZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 8, "text": [ " \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \n", "H_{rwa} = A\u22c5a + A\u22c5a - g\u22c5a \u22c5a\u22c5b - g\u22c5a \u22c5a\u22c5b + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b - \u03c9_d\u22c5a \u22c5a" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Introduce the detuning $\\Delta = \\omega_a - \\omega_d$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = H2.subs(omega_a, Delta + omega_d).expand()\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A {{a}^\\dagger} + A {a} + \\Delta {{a}^\\dagger} {a} - g {{a}^\\dagger} {a} {{b}^\\dagger} - g {{a}^\\dagger} {a} {b} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAAXBAMAAAAM6K4JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n", "prompt_number": 9, "text": [ " \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \n", "A\u22c5a + A\u22c5a + \u0394\u22c5a \u22c5a - g\u22c5a \u22c5a\u22c5b - g\u22c5a \u22c5a\u22c5b + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To eliminate the coherent part of the state of the cavity mode we apply the unitary displacement operator $U = e^{\\alpha a^\\dagger - \\alpha^*a}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "alpha = symbols(\"alpha\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "UH = Dagger(a) * alpha - conjugate(alpha) * a\n", "U = exp(UH)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{\\alpha {{a}^\\dagger} - \\overline{\\alpha} {a}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAWBAMAAACPjvdAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDElEQVQoFWNgQAN8aHxsXCZsgqhizFv2oApg\n47FgE0QTY4fx+f+DAIyHQsPVoIiCOExGDowMjM7WCxiQ/KWswiAAFoMoL2PYzMQgvmC9AAPCPewJ\n3AYKYDGwGpYHDGfTGFwYlgNNhGgCkn4MPNcSwGJgIY4JDBkODJ0M1gwMXAwMnO+A4CGQy9YCEQOr\n4Q9gWC/A+JXhjSDcFAaGrwxcH5HEeBMY9BkZuri+HUBSc42B6YMAQozV+QyLJkO2pQmymhCbA04T\nIGKMyiYBSLqxMtMDWBWwSiAEWTsZRBYguFhZ3D3GB7FKIAnyb0Di4GDyK+CQQBLmAKoxQOJjYzJf\nYBAh6HcTmwPY9CKJAQDwgzkp4qt/twAAAABJRU5ErkJggg==\n", "prompt_number": 11, "text": [ " \u2020 _ \n", " \u03b1\u22c5a - \u03b1\u22c5a\n", "\u212f " ] } ], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = hamiltonian_transformation(U, H3, independent=True)\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A \\left(- \\alpha + {a}\\right) + A \\left(- \\overline{\\alpha} + {{a}^\\dagger}\\right) + \\Delta \\left(- \\overline{\\alpha} + {{a}^\\dagger}\\right) \\left(- \\alpha + {a}\\right) - g \\left(- \\overline{\\alpha} + {{a}^\\dagger}\\right) \\left(- \\alpha + {a}\\right) {{b}^\\dagger} - g \\left(- \\overline{\\alpha} + {{a}^\\dagger}\\right) \\left(- \\alpha + {a}\\right) {b} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAA8EAAAAbBAMAAACuK5DnAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 12, "text": [ " \u239b _ \u2020\u239e \u239b _ \u2020\u239e \u239b _ \u2020\u239e \u2020 \n", "A\u22c5(-\u03b1 + a) + A\u22c5\u239d- \u03b1 + a \u23a0 + \u0394\u22c5\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a) - g\u22c5\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a)\u22c5b -\n", "\n", " \u239b _ \u2020\u239e \u2020 \n", " g\u22c5\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a)\u22c5b + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now want to cancel out the drivng terms so we set $A - \\Delta \\alpha = 0$, i.e. $\\alpha = A/\\Delta$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = H4.expand().subs({A: alpha * Delta, conjugate(alpha): alpha}) \n", "\n", "H5 = collect(H5, [g * Dagger(a) * a, - alpha * g])\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\Delta \\alpha^{2} + \\Delta {{a}^\\dagger} {a} + \\alpha^{2} g \\left(- {{b}^\\dagger} - {b}\\right) + \\alpha g \\left({{a}^\\dagger} {{b}^\\dagger} + {{a}^\\dagger} {b} + {a} {{b}^\\dagger} + {a} {b}\\right) + g {{a}^\\dagger} {a} \\left(- {{b}^\\dagger} - {b}\\right) + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAtMAAAAcBAMAAABR+y5KAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n", "prompt_number": 13, "text": [ " 2 \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u2020 \u2020 \u2020 \u239e \u2020 \u239b\n", "- \u0394\u22c5\u03b1 + \u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239d- b - b\u23a0 + \u03b1\u22c5g\u22c5\u239da \u22c5b + a \u22c5b + a\u22c5b + a\u22c5b\u23a0 + g\u22c5a \u22c5a\u22c5\u239d\n", "\n", " \u2020 \u239e \u2020 \n", "- b - b\u23a0 + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Drop C-numbers from the Hamiltonian:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H6 = drop_c_number_terms(H5)\n", "\n", "H6" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\Delta {{a}^\\dagger} {a} + \\alpha^{2} g \\left(- {{b}^\\dagger} - {b}\\right) + \\alpha g \\left({{a}^\\dagger} {{b}^\\dagger} + {{a}^\\dagger} {b} + {a} {{b}^\\dagger} + {a} {b}\\right) + g {{a}^\\dagger} {a} \\left(- {{b}^\\dagger} - {b}\\right) + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAArEAAAAcBAMAAACE6foHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrkJggg==\n", "prompt_number": 14, "text": [ " \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u2020 \u2020 \u2020 \u239e \u2020 \u239b \u2020 \u239e\n", "\u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239d- b - b\u23a0 + \u03b1\u22c5g\u22c5\u239da \u22c5b + a \u22c5b + a\u22c5b + a\u22c5b\u23a0 + g\u22c5a \u22c5a\u22c5\u239d- b - b\u23a0\n", "\n", " \u2020 \n", " + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, if driving strength is large, so that $\\alpha \\gg 1$, we can drop the nonlinear interaction term, and we have an linear effective coupling:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H7 = H6.subs(g * Dagger(a) * a, 0)\n", "\n", "e = (a + Dagger(a)) * (b + Dagger(b))\n", "H7 = H7.subs(e.expand(), e)\n", "\n", "Eq(Hsym, H7)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\Delta {{a}^\\dagger} {a} + \\alpha^{2} g \\left(- {{b}^\\dagger} - {b}\\right) + \\alpha g \\left({{a}^\\dagger} + {a}\\right) \\left({{b}^\\dagger} + {b}\\right) + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAgcAAAAcBAMAAAAHAQF5AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrNlT26pYfru0B2dX/z/SsgJ/AQIk/fsDEU6tAAAAAElFTkSuQmCC\n", "prompt_number": 15, "text": [ " \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u239e \u239b \u2020 \u239e \u2020 \n", "H = \u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239d- b - b\u23a0 + \u03b1\u22c5g\u22c5\u239da + a\u23a0\u22c5\u239db + b\u23a0 + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This linearlized optomechanical Hamiltonian has at least two interesting regimes:" ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Red sideband" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The red sideband regime occurs when the detuning is $\\Delta = \\omega_b$. In this case, if we move to a frame rotating with the driving field, we obtain:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H1 = H7\n", "H1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\Delta {{a}^\\dagger} {a} + \\alpha^{2} g \\left(- {{b}^\\dagger} - {b}\\right) + \\alpha g \\left({{a}^\\dagger} + {a}\\right) \\left({{b}^\\dagger} + {b}\\right) + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAdgAAAAcBAMAAAAw+bO8AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 16, "text": [ " \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u239e \u239b \u2020 \u239e \u2020 \n", "\u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239d- b - b\u23a0 + \u03b1\u22c5g\u22c5\u239da + a\u23a0\u22c5\u239db + b\u23a0 + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * Dagger(a) * a * Delta * t)\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\Delta t {{a}^\\dagger} {a}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAWBAMAAABqEDIsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABMElEQVQoFX1PMUvDQBh910tDEtEqRBxz4B8I\nNA4Sh4BO6hDB/+Cgg7MiuFYcBAdHbxOztNjNKV07ZdSt4KLg0qGCU/vdpYUU2nvDd9/33vve3QFL\nMFjCz2gxa/TJk7kRiI/yKuOlNP0oZlWqCiS6Vov7oaZgveQuq5LurUDQ+TTl53Q7JnaHD4HTm0P4\nzXdgALD9WJbuGqWyENc0fYJdeWRMgC3Znl52Qqk14JnsI1iJR+ECOMALLWh8Aa/09gL2LZwH3gEk\n0IK6VoENpQr0/sE7m0G61lO5bITfDaUCbiHcfpZl37BC4cj2hWbvvL9cN2BvqTNWCO2utLuPx5o+\n341IZ9tRWroW17O0LhYrmq234EuDvnLf7BlkNOivJjSESQUc0kODhRfwjf+L9nLD+gQANT8e9ElQ\nlQAAAABJRU5ErkJggg==\n", "prompt_number": 17, "text": [ " \u2020 \n", " \u2148\u22c5\u0394\u22c5t\u22c5a \u22c5a\n", "\u212f " ] } ], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = hamiltonian_transformation(U, H1, independent=True)\n", "H2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\alpha^{2} g \\left(- {{b}^\\dagger} - {b}\\right) + \\alpha g \\left(e^{i \\Delta t} {{a}^\\dagger} + e^{- i \\Delta t} {a}\\right) \\left({{b}^\\dagger} + {b}\\right) + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAeUAAAAcBAMAAACt9Lo0AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAABJRU5ErkJggg==\n", "prompt_number": 18, "text": [ " 2 \u239b \u2020 \u239e \u239b \u2148\u22c5\u0394\u22c5t \u2020 -\u2148\u22c5\u0394\u22c5t \u239e \u239b \u2020 \u239e \u2020 \n", "\u03b1 \u22c5g\u22c5\u239d- b - b\u23a0 + \u03b1\u22c5g\u22c5\u239d\u212f \u22c5a + \u212f \u22c5a\u23a0\u22c5\u239db + b\u23a0 + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 18 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * Dagger(b) * b * omega_b * t)\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\omega_{b} t {{b}^\\dagger} {b}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAD4AAAAWBAMAAABqEDIsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABSklEQVQoFXWPP0vDQBiHfzXpmf7NgXUTeljd\nCy0I1iE4tksERwU3J9F+AKE4CFUHB6WDUDIUXKOCc8VRlGxSUCj0ExTp7ps0J7nGvpC89zy/N7k7\nYF4N5wWhF9Fcs6LEvoFao48jT9q0LVdAzsEakQUs0BOvIseA7AlNXsdToANMprkp/8pqYDT75o/v\nntbZT41jCBxXHnxDGxV5hto9u6D3AFqTzmsBG7xKTLXTwR5WoKNFMEG+nPUggBHeif0a4RavWEp6\nAGvBONBdOMA5GkGKxNjZxDPWU191aO6yyY3g5GO0p3nKE927zxtL7+9DL4tF/hH4ZnYYfv9o58/0\nS+TsLtiTk94Wge+tUkuUqvKayNEms3VoJ4V0rBTsK9HvyTYKTlTMrDNXlZcZpaDpKhgDU8SUIgzK\ny4pRQfNQ+LufGk2putX/T4fuF0tgPjdHfvJ7AAAAAElFTkSuQmCC\n", "prompt_number": 19, "text": [ " \u2020 \n", " \u2148\u22c5\u03c9_b\u22c5t\u22c5b \u22c5b\n", "\u212f " ] } ], "prompt_number": 19 }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = hamiltonian_transformation(U, H2, independent=True)\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\alpha^{2} g \\left(- e^{i \\omega_{b} t} {{b}^\\dagger} - e^{- i \\omega_{b} t} {b}\\right) + \\alpha g \\left(e^{i \\Delta t} {{a}^\\dagger} + e^{- i \\Delta t} {a}\\right) \\left(e^{i \\omega_{b} t} {{b}^\\dagger} + e^{- i \\omega_{b} t} {b}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAlcAAAAcBAMAAAC+FYcxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6lJZtAfXZW4AEMk3APof4s3Af+y7OsR45vSHAAAAAElFTkSuQmCC\n", "prompt_number": 20, "text": [ " 2 \u239b \u2148\u22c5\u03c9_b\u22c5t \u2020 -\u2148\u22c5\u03c9_b\u22c5t \u239e \u239b \u2148\u22c5\u0394\u22c5t \u2020 -\u2148\u22c5\u0394\u22c5t \u239e \u239b \u2148\u22c5\u03c9_b\u22c5t \u2020 \n", "\u03b1 \u22c5g\u22c5\u239d- \u212f \u22c5b - \u212f \u22c5b\u23a0 + \u03b1\u22c5g\u22c5\u239d\u212f \u22c5a + \u212f \u22c5a\u23a0\u22c5\u239d\u212f \u22c5b \n", "\n", " -\u2148\u22c5\u03c9_b\u22c5t \u239e\n", "+ \u212f \u22c5b\u23a0" ] } ], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we substitute $\\Delta = \\omega_b$ we get:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = H3.expand().subs(Delta, omega_b)\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\alpha^{2} g e^{i \\omega_{b} t} {{b}^\\dagger} - \\alpha^{2} g e^{- i \\omega_{b} t} {b} + \\alpha g e^{2 i \\omega_{b} t} {{a}^\\dagger} {{b}^\\dagger} + \\alpha g {{a}^\\dagger} {b} + \\alpha g {a} {{b}^\\dagger} + \\alpha g e^{- 2 i \\omega_{b} t} {a} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAosAAAAXBAMAAACR/xg1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 21, "text": [ " 2 \u2148\u22c5\u03c9_b\u22c5t \u2020 2 -\u2148\u22c5\u03c9_b\u22c5t 2\u22c5\u2148\u22c5\u03c9_b\u22c5t \u2020 \u2020 \u2020 \n", "- \u03b1 \u22c5g\u22c5\u212f \u22c5b - \u03b1 \u22c5g\u22c5\u212f \u22c5b + \u03b1\u22c5g\u22c5\u212f \u22c5a \u22c5b + \u03b1\u22c5g\u22c5a \u22c5b + \u03b1\u22c5g\u22c5\n", "\n", " \u2020 -2\u22c5\u2148\u22c5\u03c9_b\u22c5t \n", "a\u22c5b + \u03b1\u22c5g\u22c5\u212f \u22c5a\u22c5b" ] } ], "prompt_number": 21 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we can do a rotating-wave approximation and get rid of terms rotating at angular frequencies $2\\omega_b$, and then transform back to the original frame:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = drop_terms_containing(H4, [exp(+2 * I * omega_b * t), exp(-2 * I * omega_b * t)])\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\alpha^{2} g e^{i \\omega_{b} t} {{b}^\\dagger} - \\alpha^{2} g e^{- i \\omega_{b} t} {b} + \\alpha g {{a}^\\dagger} {b} + \\alpha g {a} {{b}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAX8AAAAYBAMAAADg7OACAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nDwAAAABJRU5ErkJggg==\n", "prompt_number": 22, "text": [ " 2 \u2148\u22c5\u03c9_b\u22c5t \u2020 2 -\u2148\u22c5\u03c9_b\u22c5t \u2020 \u2020\n", "- \u03b1 \u22c5g\u22c5\u212f \u22c5b - \u03b1 \u22c5g\u22c5\u212f \u22c5b + \u03b1\u22c5g\u22c5a \u22c5b + \u03b1\u22c5g\u22c5a\u22c5b " ] } ], "prompt_number": 22 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(-I * Dagger(a) * a * omega_b * t) # Delta = omega_b\n", "H6 = hamiltonian_transformation(U, H5, independent=True)\n", "U = exp(-I * Dagger(b) * b * omega_b * t)\n", "H7 = hamiltonian_transformation(U, H6, independent=True)\n", "H7 = collect(H7, [alpha**2, g])\n", "H7" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\alpha^{2} \\left(- g {{b}^\\dagger} - g {b}\\right) + \\alpha \\left(g {{a}^\\dagger} {b} + g {a} {{b}^\\dagger}\\right) + \\omega_{b} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAcIAAAAcBAMAAAAAdaKdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 23, "text": [ " 2 \u239b \u2020 \u239e \u239b \u2020 \u2020\u239e \u2020 \u2020 \n", "\u03b1 \u22c5\u239d- g\u22c5b - g\u22c5b\u23a0 + \u03b1\u22c5\u239dg\u22c5a \u22c5b + g\u22c5a\u22c5b \u23a0 + \u03c9_b\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 23 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now the interaction term is $a^\\dagger b + ab^\\dagger$, which is a swapping interaction that can be used for state transfer or energy transfer in for example cooling application (side-band cooling)." ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Blue sideband" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "If, instead, we choose a driving frequency such that $\\Delta = -\\omega_b$, we obtain:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = H3.expand().subs(Delta, -omega_b)\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\alpha^{2} g e^{i \\omega_{b} t} {{b}^\\dagger} - \\alpha^{2} g e^{- i \\omega_{b} t} {b} + \\alpha g e^{2 i \\omega_{b} t} {a} {{b}^\\dagger} + \\alpha g {{a}^\\dagger} {{b}^\\dagger} + \\alpha g {a} {b} + \\alpha g e^{- 2 i \\omega_{b} t} {{a}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAosAAAAXBAMAAACR/xg1AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nWqe7PHqGfkM3Z01Am4DUa/0LdJ2d5fRSxl8AAAAASUVORK5CYII=\n", "prompt_number": 24, "text": [ " 2 \u2148\u22c5\u03c9_b\u22c5t \u2020 2 -\u2148\u22c5\u03c9_b\u22c5t 2\u22c5\u2148\u22c5\u03c9_b\u22c5t \u2020 \u2020 \u2020 \n", "- \u03b1 \u22c5g\u22c5\u212f \u22c5b - \u03b1 \u22c5g\u22c5\u212f \u22c5b + \u03b1\u22c5g\u22c5\u212f \u22c5a\u22c5b + \u03b1\u22c5g\u22c5a \u22c5b + \u03b1\u22c5g\u22c5\n", "\n", " -2\u22c5\u2148\u22c5\u03c9_b\u22c5t \u2020 \n", "a\u22c5b + \u03b1\u22c5g\u22c5\u212f \u22c5a \u22c5b" ] } ], "prompt_number": 24 }, { "cell_type": "markdown", "metadata": {}, "source": [ "As before, we do a rotating-wave approximation to get rid of fast oscillating terms:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = drop_terms_containing(H4, [exp(+2 * I * omega_b * t), exp(-2 * I * omega_b * t)])\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\alpha^{2} g e^{i \\omega_{b} t} {{b}^\\dagger} - \\alpha^{2} g e^{- i \\omega_{b} t} {b} + \\alpha g {{a}^\\dagger} {{b}^\\dagger} + \\alpha g {a} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAYAAAAAYBAMAAAABqzcWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n", "prompt_number": 25, "text": [ " 2 \u2148\u22c5\u03c9_b\u22c5t \u2020 2 -\u2148\u22c5\u03c9_b\u22c5t \u2020 \u2020 \n", "- \u03b1 \u22c5g\u22c5\u212f \u22c5b - \u03b1 \u22c5g\u22c5\u212f \u22c5b + \u03b1\u22c5g\u22c5a \u22c5b + \u03b1\u22c5g\u22c5a\u22c5b" ] } ], "prompt_number": 25 }, { "cell_type": "markdown", "metadata": {}, "source": [ "and moving back to the original frame results in:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp( I * Dagger(a) * a * omega_b * t) # Delta = -omega_b\n", "H6 = hamiltonian_transformation(U, H5, independent=True)\n", "U = exp(-I * Dagger(b) * b * omega_b * t)\n", "H7 = hamiltonian_transformation(U, H6, independent=True)\n", "H7 = collect(H7, [alpha**2, g])\n", "H7" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\alpha^{2} \\left(- g {{b}^\\dagger} - g {b}\\right) + \\alpha \\left(g {{a}^\\dagger} {{b}^\\dagger} + g {a} {b}\\right) - \\omega_{b} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAcIAAAAcBAMAAAAAdaKdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nb060VX2y5onSpES1D/L/UJcd+BtHz8qSrd3SQgAAAABJRU5ErkJggg==\n", "prompt_number": 26, "text": [ " 2 \u239b \u2020 \u239e \u239b \u2020 \u2020 \u239e \u2020 \u2020 \n", "\u03b1 \u22c5\u239d- g\u22c5b - g\u22c5b\u23a0 + \u03b1\u22c5\u239dg\u22c5a \u22c5b + g\u22c5a\u22c5b\u23a0 - \u03c9_b\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 26 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here, instead of a swap interaction we have obtained an interaction on the form $a^\\dagger b^\\dagger + a b$, which is the parametric amplification Hamiltonian. It can be used to parametrically amplify the states of the optical and mechanical modes, and generated interesting nonclassical states like Schrodinger-cat states." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Nonlinear regime: effective Kerr nonlinearity" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In the regime where $g \\sim \\omega_b$, the effect of the coupling to the mechanical mode $b$ on the optical mode $a$ is an effective Kerr-nonlinearity, i.e., a term on the form $(a^\\dagger a)^2$ in the Hamiltonian. To see this we can perform the so-called polariton defined by the unitary\n", "\n", "$$\n", "U = \\exp\\left(-\\frac{g}{\\omega_b} a^\\dagger a (b^\\dagger - b)\\right)\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H0" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- g {{a}^\\dagger} {a} \\left({{b}^\\dagger} + {b}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 27, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \n", "- g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 27 }, { "cell_type": "code", "collapsed": false, "input": [ "x = symbols(\"x\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 28 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(- x * Dagger(a) * a * (Dagger(b) - b))\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{- x {{a}^\\dagger} {a} \\left({{b}^\\dagger} - {b}\\right)}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 29 }, { "cell_type": "code", "collapsed": false, "input": [ "H1 = H0.subs(A, 0)\n", "\n", "H1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- g {{a}^\\dagger} {a} \\left({{b}^\\dagger} + {b}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 30, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \n", "- g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 30 }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = hamiltonian_transformation(U, H1, independent=True, expansion_search=False, N=2)\n", "\n", "H3 = normal_ordered_form(H2.expand(), independent=True)\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- 2 g x^{3} {{a}^\\dagger} {a} + 2 g x^{3} {{a}^\\dagger} {a} \\left({b}\\right)^{2} - 4 g x^{3} {{a}^\\dagger} {{b}^\\dagger} {a} {b} + 2 g x^{3} {{a}^\\dagger} \\left({{b}^\\dagger}\\right)^{2} {a} - 2 g x^{3} \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} + 2 g x^{3} \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} \\left({b}\\right)^{2} - 4 g x^{3} \\left({{a}^\\dagger}\\right)^{2} {{b}^\\dagger} \\left({a}\\right)^{2} {b} + 2 g x^{3} \\left({{a}^\\dagger}\\right)^{2} \\left({{b}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} + g x^{2} {{a}^\\dagger} {a} {b} + g x^{2} {{a}^\\dagger} {a} \\left({b}\\right)^{3} - 3 g x^{2} {{a}^\\dagger} {{b}^\\dagger} {a} - g x^{2} {{a}^\\dagger} {{b}^\\dagger} {a} \\left({b}\\right)^{2} - g x^{2} {{a}^\\dagger} \\left({{b}^\\dagger}\\right)^{2} {a} {b} + g x^{2} {{a}^\\dagger} \\left({{b}^\\dagger}\\right)^{3} {a} - 2 g x {{a}^\\dagger} {a} - 2 g x \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} - g {{a}^\\dagger} {a} {b} - g {{a}^\\dagger} {{b}^\\dagger} {a} + \\omega_{a} x^{2} {{a}^\\dagger} {a} - \\omega_{a} x^{2} {{a}^\\dagger} {a} \\left({b}\\right)^{2} + 2 \\omega_{a} x^{2} {{a}^\\dagger} {{b}^\\dagger} {a} {b} - \\omega_{a} x^{2} {{a}^\\dagger} \\left({{b}^\\dagger}\\right)^{2} {a} + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} x^{2} {{a}^\\dagger} {a} + \\omega_{b} x^{2} \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} + \\omega_{b} x {{a}^\\dagger} {a} {b} + \\omega_{b} x {{a}^\\dagger} {{b}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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VZwY+PBt+Lpto23nv6vxm2zqsvYbXB2v8nr32cM/sdafVc25LDcNAnpCbyTR0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\naMswZ2iRAAAAAElFTkSuQmCC\n", "prompt_number": 31, "text": [ " 2 \n", " 3 \u2020 3 \u2020 2 3 \u2020 \u2020 3 \u2020 \u239b \u2020\u239e 3\n", "- 2\u22c5g\u22c5x \u22c5a \u22c5a + 2\u22c5g\u22c5x \u22c5a \u22c5a\u22c5b - 4\u22c5g\u22c5x \u22c5a \u22c5b \u22c5a\u22c5b + 2\u22c5g\u22c5x \u22c5a \u22c5\u239db \u23a0 \u22c5a - 2\u22c5g\u22c5x \n", "\n", " 2 2 2 2 2 \n", " \u239b \u2020\u239e 2 3 \u239b \u2020\u239e 2 2 3 \u239b \u2020\u239e \u2020 2 3 \u239b \u2020\u239e \u239b \u2020\u239e 2 \n", "\u22c5\u239da \u23a0 \u22c5a + 2\u22c5g\u22c5x \u22c5\u239da \u23a0 \u22c5a \u22c5b - 4\u22c5g\u22c5x \u22c5\u239da \u23a0 \u22c5b \u22c5a \u22c5b + 2\u22c5g\u22c5x \u22c5\u239da \u23a0 \u22c5\u239db \u23a0 \u22c5a \n", "\n", " \n", " 2 \u2020 2 \u2020 3 2 \u2020 \u2020 2 \u2020 \u2020 2 2 \u2020 \u239b \u2020\u239e\n", "+ g\u22c5x \u22c5a \u22c5a\u22c5b + g\u22c5x \u22c5a \u22c5a\u22c5b - 3\u22c5g\u22c5x \u22c5a \u22c5b \u22c5a - g\u22c5x \u22c5a \u22c5b \u22c5a\u22c5b - g\u22c5x \u22c5a \u22c5\u239db \u23a0\n", "\n", "2 3 2 \n", " 2 \u2020 \u239b \u2020\u239e \u2020 \u239b \u2020\u239e 2 \u2020 \u2020 \u2020 \n", " \u22c5a\u22c5b + g\u22c5x \u22c5a \u22c5\u239db \u23a0 \u22c5a - 2\u22c5g\u22c5x\u22c5a \u22c5a - 2\u22c5g\u22c5x\u22c5\u239da \u23a0 \u22c5a - g\u22c5a \u22c5a\u22c5b - g\u22c5a \u22c5b \u22c5a +\n", "\n", " 2 \n", " 2 \u2020 2 \u2020 2 2 \u2020 \u2020 2 \u2020 \u239b \u2020\u239e \u2020 \n", " \u03c9\u2090\u22c5x \u22c5a \u22c5a - \u03c9\u2090\u22c5x \u22c5a \u22c5a\u22c5b + 2\u22c5\u03c9\u2090\u22c5x \u22c5a \u22c5b \u22c5a\u22c5b - \u03c9\u2090\u22c5x \u22c5a \u22c5\u239db \u23a0 \u22c5a + \u03c9\u2090\u22c5a \u22c5a +\n", "\n", " 2 \n", " 2 \u2020 2 \u239b \u2020\u239e 2 \u2020 \u2020 \u2020 \u2020 \n", " \u03c9_b\u22c5x \u22c5a \u22c5a + \u03c9_b\u22c5x \u22c5\u239da \u23a0 \u22c5a + \u03c9_b\u22c5x\u22c5a \u22c5a\u22c5b + \u03c9_b\u22c5x\u22c5a \u22c5b \u22c5a + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 31 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = H3.subs({x**2: 0, x**3: 0, x: g/omega_b}) # neglect higher order terms\n", "\n", "H4 = collect(H4, Dagger(a)*a)\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\frac{2 g^{2}}{\\omega_{b}} \\left({{a}^\\dagger}\\right)^{2} \\left({a}\\right)^{2} + \\omega_{b} {{b}^\\dagger} {b} + \\left(- \\frac{2 g^{2}}{\\omega_{b}} + \\omega_{a}\\right) {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 32, "text": [ " 2 \n", " 2 \u239b \u2020\u239e 2 \u239b 2 \u239e \n", " 2\u22c5g \u22c5\u239da \u23a0 \u22c5a \u2020 \u239c 2\u22c5g \u239f \u2020 \n", "- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u03c9_b\u22c5b \u22c5b + \u239c- \u2500\u2500\u2500\u2500 + \u03c9\u2090\u239f\u22c5a \u22c5a\n", " \u03c9_b \u239d \u03c9_b \u23a0 " ] } ], "prompt_number": 32 }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this Hamiltonian, the mechanical and optical mode is effectively decoupled, but the influence of the mechanical mode on the optical mode is described by the induced Kerr nonlinearity for the optical mode." ] }, { "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
Thu Oct 09 16:16:53 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|}{Thu Oct 09 16:16:53 2014 JST} \\\\ \\hline\n", "\\end{tabular}\n" ], "metadata": {}, "output_type": "pyout", "prompt_number": 33, "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", "Thu Oct 09 16:16:53 2014 JST" ] } ], "prompt_number": 33 } ], "metadata": {} } ] }