{ "metadata": { "name": "" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Reproduce: Johansson et al., PRA 87 043804 (2013)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "J. R. Johansson, http://jrjohansson.github.com, " ] }, { "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": [ "# Note: This notebook requires and experimental branch of sympy available at: https://github.com/jrjohansson/sympy\n", "from sympy.physics.quantum import *" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Introduction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this notebook I reproduce some results from Johansson et al., PRA 87, 043804 (2013), which is one of my own papers. Normally this would not qualify as a reproduced results, but in this case I will use a different method to obtain the same result. Here I will use SymPy and its quantum physics module to demonstrate how an analytical result can be obtained using symbolic operator manipulation, instead of calculations with pen and paper." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Here we are interested in evaluating the nonclassicality indicator [see, e.g., Miranowicz et al. Phys. Rev. A 82, 013824 (2010)]\n", "\n", "$$\n", "\\langle: f^\\dagger f :\\rangle\n", "$$\n", "\n", "which is positive for classical states, and where $f$ is a function of the field operator $b$. We will use the following choise of $f$, which is ideally suited for detecting the presence of nonclassial two-mode squeezing (nonclassical quadrature correlations between photons at difference frequencies):\n", "\n", "\n", "$$\n", "f = e^{i\\theta} b_- + e^{-i\\theta} b_-^\\dagger + i(e^{i\\theta} b_+ - e^{-i\\theta} b_+^\\dagger )\n", "$$\n", "\n", "where $b_\\pm$ are the field operators at frequencies $\\omega_\\pm = \\omega_d/2 \\pm \\delta\\omega$. Note that the structure of $f$ is $f = x_- - p_+$ where $x_\\pm$ and $p_\\pm$ are the $x$ and $p$ quadrature operators of modes with frequency $\\omega_\\pm$, which suggest that this form is suitable for two-mode quadrature squeezing where exactly this operator combination is squeezed below its vacuum level.\n", "\n", "Below we will evaluate $\\langle: f^\\dagger f :\\rangle$ for the state of the field $b$ that is produced by the dynamical Casimir effect when the boundary condition of the field is varied in time. The relation between the output field $b$ and the input field $a$ (that impinges on the time-varying boundary condition) is, in the perturbative regime, given as\n", "\n", "$$\n", "b_{\\pm} = -a_{\\pm} - i\\frac{\\delta\\!L_{\\rm eff}}{v}\\sqrt{\\omega_-\\omega_+} a_{\\mp}^\\dagger\n", "$$\n", "\n", "Here $\\frac{\\delta\\!L_{\\rm eff}}{v}\\sqrt{\\omega_-\\omega_+}$ is a small parameter in the perturbation calculation, which we for brievity will denote $x$ in the following (see PRA 87 043804 for details)." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We start by defining symbols and operators:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "theta = Symbol(\"theta\", real=True)\n", "x = Symbol(\"x\", real=True)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "dL, v, wp, wm = symbols(\"{\\delta\\!L}, v, omega_-, omega_+\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "code", "collapsed": false, "input": [ "# input field operators\n", "am = BosonOperator(\"a_-\")\n", "ap = BosonOperator(\"a_+\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "# output field operators\n", "bm = BosonOperator(\"b_-\")\n", "bp = BosonOperator(\"b_+\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "# relation between input field operators and output field operators\n", "bm_sub = -am - 1j * x * Dagger(ap)\n", "bp_sub = -ap - 1j * x * Dagger(am)\n", "\n", "bmd_sub = Dagger(bm_sub)\n", "bpd_sub = Dagger(bp_sub)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "$$\n", "f = e^{i\\theta} b_- + e^{-i\\theta} b_-^\\dagger + i(e^{i\\theta} b_+ - e^{-i\\theta} b_+^\\dagger )\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "f = exp(1j*theta) * bm + exp(-1j*theta) * Dagger(bm) + 1j * (exp(1j*theta) * bp - exp(-1j*theta) * Dagger(bp))\n", "\n", "f" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{- 1.0 i \\theta} {{b_-}^\\dagger} + e^{1.0 i \\theta} {b_-} + 1.0 i \\left(- e^{- 1.0 i \\theta} {{b_+}^\\dagger} + e^{1.0 i \\theta} {b_+}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 8, "text": [ " -1.0\u22c5\u2148\u22c5\u03b8 1.0\u22c5\u2148\u22c5\u03b8 \u239b -1.0\u22c5\n", "\u212f \u22c5BosonOperator(b\u208b,0) + \u212f \u22c5BosonOperator(b\u208b,1) + 1.0\u22c5\u2148\u22c5\u239d- \u212f \n", "\n", "\u2148\u22c5\u03b8 1.0\u22c5\u2148\u22c5\u03b8 \u239e\n", " \u22c5BosonOperator(b\u208a,0) + \u212f \u22c5BosonOperator(b\u208a,1)\u23a0" ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we construct $f^\\dagger f$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf = Dagger(f) * f" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "code", "collapsed": false, "input": [ "fdf.expand()" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- 1.0 i e^{- 2.0 i \\theta} {{b_+}^\\dagger} {{b_-}^\\dagger} - 1.0 e^{- 2.0 i \\theta} \\left({{b_+}^\\dagger}\\right)^{2} - 1.0 i e^{- 2.0 i \\theta} {{b_-}^\\dagger} {{b_+}^\\dagger} + e^{- 2.0 i \\theta} \\left({{b_-}^\\dagger}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {b_+} {b_-} - 1.0 e^{2.0 i \\theta} \\left({b_+}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {b_-} {b_+} + e^{2.0 i \\theta} \\left({b_-}\\right)^{2} + 1.0 {{b_+}^\\dagger} {b_+} - 1.0 i {{b_+}^\\dagger} {b_-} + 1.0 {b_+} {{b_+}^\\dagger} + 1.0 i {b_+} {{b_-}^\\dagger} + 1.0 i {{b_-}^\\dagger} {b_+} + {{b_-}^\\dagger} {b_-} - 1.0 i {b_-} {{b_+}^\\dagger} + {b_-} {{b_-}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 10, "text": [ " -2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 \n", "- 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208a,0)\u22c5BosonOperator(b\u208b,0) - 1.0\u22c5\u212f \u22c5Boso\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "nOperator(b\u208a,0) - 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208b,0)\u22c5BosonOperator(b\u208a,0) + \u212f\n", "\n", "-2.0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", " \u22c5BosonOperator(b\u208b,0) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208a,1)\u22c5BosonOperat\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", "or(b\u208b,1) - 1.0\u22c5\u212f \u22c5BosonOperator(b\u208a,1) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208b\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2 \n", ",1)\u22c5BosonOperator(b\u208a,1) + \u212f \u22c5BosonOperator(b\u208b,1) + 1.0\u22c5BosonOperator(b\u208a\n", "\n", " \n", ",0)\u22c5BosonOperator(b\u208a,1) - 1.0\u22c5\u2148\u22c5BosonOperator(b\u208a,0)\u22c5BosonOperator(b\u208b,1) + 1.0\u22c5\n", "\n", " \n", "BosonOperator(b\u208a,1)\u22c5BosonOperator(b\u208a,0) + 1.0\u22c5\u2148\u22c5BosonOperator(b\u208a,1)\u22c5BosonOpera\n", "\n", " \n", "tor(b\u208b,0) + 1.0\u22c5\u2148\u22c5BosonOperator(b\u208b,0)\u22c5BosonOperator(b\u208a,1) + BosonOperator(b\u208b,0\n", "\n", " \n", ")\u22c5BosonOperator(b\u208b,1) - 1.0\u22c5\u2148\u22c5BosonOperator(b\u208b,1)\u22c5BosonOperator(b\u208a,0) + BosonO\n", "\n", " \n", "perator(b\u208b,1)\u22c5BosonOperator(b\u208b,0)" ] } ], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we calculate the normal order expression $:f^\\dagger f:$, with respect to the operators in the output field ($b_\\pm$)" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_normal = normal_order(fdf.expand())" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_normal" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- 1.0 i e^{- 2.0 i \\theta} {{b_+}^\\dagger} {{b_-}^\\dagger} - 1.0 e^{- 2.0 i \\theta} \\left({{b_+}^\\dagger}\\right)^{2} - 1.0 i e^{- 2.0 i \\theta} {{b_-}^\\dagger} {{b_+}^\\dagger} + e^{- 2.0 i \\theta} \\left({{b_-}^\\dagger}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {b_+} {b_-} - 1.0 e^{2.0 i \\theta} \\left({b_+}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {b_-} {b_+} + e^{2.0 i \\theta} \\left({b_-}\\right)^{2} + 2.0 {{b_+}^\\dagger} {b_+} - 2.0 i {{b_+}^\\dagger} {b_-} + 2.0 i {{b_-}^\\dagger} {b_+} + 2 {{b_-}^\\dagger} {b_-}$$" ], "metadata": {}, "output_type": "pyout", "png": 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2eq8ngADex5HQjhvam4CcaRURqAIxTGaFrX83+cr/+UtLr7i28gLFCDWPoq6lpaVJObda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\n5y1n36U/vpkAAAAASUVORK5CYII=\n", "prompt_number": 12, "text": [ " -2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 \n", "- 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208a,0)\u22c5BosonOperator(b\u208b,0) - 1.0\u22c5\u212f \u22c5Boso\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "nOperator(b\u208a,0) - 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208b,0)\u22c5BosonOperator(b\u208a,0) + \u212f\n", "\n", "-2.0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", " \u22c5BosonOperator(b\u208b,0) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208a,1)\u22c5BosonOperat\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", "or(b\u208b,1) - 1.0\u22c5\u212f \u22c5BosonOperator(b\u208a,1) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(b\u208b\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2 \n", ",1)\u22c5BosonOperator(b\u208a,1) + \u212f \u22c5BosonOperator(b\u208b,1) + 2.0\u22c5BosonOperator(b\u208a\n", "\n", " \n", ",0)\u22c5BosonOperator(b\u208a,1) - 2.0\u22c5\u2148\u22c5BosonOperator(b\u208a,0)\u22c5BosonOperator(b\u208b,1) + 2.0\u22c5\n", "\n", " \n", "\u2148\u22c5BosonOperator(b\u208b,0)\u22c5BosonOperator(b\u208a,1) + 2\u22c5BosonOperator(b\u208b,0)\u22c5BosonOperato\n", "\n", " \n", "r(b\u208b,1)" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now substitute the output field operators with the input field operators:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_1 = fdf_normal.expand().subs({bm: bm_sub, Dagger(bm): bmd_sub, bp: bp_sub, Dagger(bp): bpd_sub}).expand()\n", "\n", "fdf_1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$1.0 i x^{2} e^{- 2.0 i \\theta} {a_+} {a_-} - 1.0 x^{2} e^{- 2.0 i \\theta} \\left({a_+}\\right)^{2} + 1.0 i x^{2} e^{- 2.0 i \\theta} {a_-} {a_+} + 1.0 x^{2} e^{- 2.0 i \\theta} \\left({a_-}\\right)^{2} - 1.0 i x^{2} e^{2.0 i \\theta} {{a_+}^\\dagger} {{a_-}^\\dagger} - 1.0 x^{2} e^{2.0 i \\theta} \\left({{a_+}^\\dagger}\\right)^{2} - 1.0 i x^{2} e^{2.0 i \\theta} {{a_-}^\\dagger} {{a_+}^\\dagger} + 1.0 x^{2} e^{2.0 i \\theta} \\left({{a_-}^\\dagger}\\right)^{2} + 2.0 x^{2} {a_+} {{a_+}^\\dagger} + 2.0 i x^{2} {a_+} {{a_-}^\\dagger} - 2.0 i x^{2} {a_-} {{a_+}^\\dagger} + 2.0 x^{2} {a_-} {{a_-}^\\dagger} - 1.0 x e^{- 2.0 i \\theta} {{a_+}^\\dagger} {a_+} + 1.0 i x e^{- 2.0 i \\theta} {{a_+}^\\dagger} {a_-} - 1.0 x e^{- 2.0 i \\theta} {a_+} {{a_+}^\\dagger} - 1.0 i x e^{- 2.0 i \\theta} {a_+} {{a_-}^\\dagger} - 1.0 i x e^{- 2.0 i \\theta} {{a_-}^\\dagger} {a_+} - 1.0 x e^{- 2.0 i \\theta} {{a_-}^\\dagger} {a_-} + 1.0 i x e^{- 2.0 i \\theta} {a_-} {{a_+}^\\dagger} - 1.0 x e^{- 2.0 i \\theta} {a_-} {{a_-}^\\dagger} - 1.0 x e^{2.0 i \\theta} {{a_+}^\\dagger} {a_+} + 1.0 i x e^{2.0 i \\theta} {{a_+}^\\dagger} {a_-} - 1.0 x e^{2.0 i \\theta} {a_+} {{a_+}^\\dagger} - 1.0 i x e^{2.0 i \\theta} {a_+} {{a_-}^\\dagger} - 1.0 i x e^{2.0 i \\theta} {{a_-}^\\dagger} {a_+} - 1.0 x e^{2.0 i \\theta} {{a_-}^\\dagger} {a_-} + 1.0 i x e^{2.0 i \\theta} {a_-} {{a_+}^\\dagger} - 1.0 x e^{2.0 i \\theta} {a_-} {{a_-}^\\dagger} + 2.0 i x {{a_+}^\\dagger} {{a_-}^\\dagger} + 2.0 x \\left({{a_+}^\\dagger}\\right)^{2} - 2.0 i x {a_+} {a_-} + 2.0 x \\left({a_+}\\right)^{2} + 2.0 i x {{a_-}^\\dagger} {{a_+}^\\dagger} - 2.0 x \\left({{a_-}^\\dagger}\\right)^{2} - 2.0 i x {a_-} {a_+} - 2.0 x \\left({a_-}\\right)^{2} - 1.0 i e^{- 2.0 i \\theta} {{a_+}^\\dagger} {{a_-}^\\dagger} - 1.0 e^{- 2.0 i \\theta} \\left({{a_+}^\\dagger}\\right)^{2} - 1.0 i e^{- 2.0 i \\theta} {{a_-}^\\dagger} {{a_+}^\\dagger} + e^{- 2.0 i \\theta} \\left({{a_-}^\\dagger}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {a_+} {a_-} - 1.0 e^{2.0 i \\theta} \\left({a_+}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {a_-} {a_+} + e^{2.0 i \\theta} \\left({a_-}\\right)^{2} + 2.0 {{a_+}^\\dagger} {a_+} - 2.0 i {{a_+}^\\dagger} {a_-} + 2.0 i {{a_-}^\\dagger} {a_+} + 2 {{a_-}^\\dagger} {a_-}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 13, "text": [ " 2 -2.0\u22c5\u2148\u22c5\u03b8 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "1.0\u22c5\u2148\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208b,1) - 1.0\u22c5x \u22c5\u212f \u22c5\n", "\n", " 2 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "BosonOperator(a\u208a,1) + 1.0\u22c5\u2148\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 2 2 2.0\u22c5\u2148\u22c5\u03b8 \n", ",1) + 1.0\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208b,1) - 1.0\u22c5\u2148\u22c5x \u22c5\u212f \u22c5BosonOperator(\n", "\n", " 2 2.0\u22c5\u2148\u22c5\u03b8 2 2 2\n", "a\u208a,0)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208a,0) - 1.0\u22c5\u2148\u22c5x \u22c5\u212f \n", "\n", ".0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", " \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,0) + 1.0\u22c5x \u22c5\u212f \u22c5BosonOperator\n", "\n", " 2 2 2 \n", "(a\u208b,0) + 2.0\u22c5x \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208a,0) + 2.0\u22c5\u2148\u22c5x \u22c5BosonOpera\n", "\n", " 2 \n", "tor(a\u208a,1)\u22c5BosonOperator(a\u208b,0) - 2.0\u22c5\u2148\u22c5x \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a,\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "0) + 2.0\u22c5x \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5x\u22c5\u212f \u22c5BosonOpe\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "rator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) + 1.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5BosonO\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "perator(a\u208b,1) - 1.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208a,0) - 1.0\u22c5\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 \n", "\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5Boso\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "nOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,1) - 1.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5Boso\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "nOperator(a\u208b,1) + 1.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a,0) - \n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 \n", "1.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5x\u22c5\u212f \u22c5Boson\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", "Operator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) + 1.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5Boso\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", "nOperator(a\u208b,1) - 1.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208a,0) - 1.0\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 \n", "\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5Boson\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", "Operator(a\u208b,0)\u22c5BosonOperator(a\u208a,1) - 1.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5BosonO\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", "perator(a\u208b,1) + 1.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a,0) - 1.0\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", "\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208b,0) + 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(a\u208a\n", "\n", " 2 \n", ",0)\u22c5BosonOperator(a\u208b,0) + 2.0\u22c5x\u22c5BosonOperator(a\u208a,0) - 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(a\n", "\n", " 2 \n", "\u208a,1)\u22c5BosonOperator(a\u208b,1) + 2.0\u22c5x\u22c5BosonOperator(a\u208a,1) + 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(\n", "\n", " 2 \n", "a\u208b,0)\u22c5BosonOperator(a\u208a,0) - 2.0\u22c5x\u22c5BosonOperator(a\u208b,0) - 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "(a\u208b,1)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5x\u22c5BosonOperator(a\u208b,1) - 1.0\u22c5\u2148\u22c5\u212f \u22c5Boso\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 2 \n", "nOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5\u212f \u22c5BosonOperator(a\u208a,0) - 1.0\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 \n", "\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,0) + \u212f \u22c5BosonOperator\n", "\n", " 2 2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\n", "(a\u208b,0) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208b,1) - 1.0\u22c5\u212f \n", "\n", "\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", " \u22c5BosonOperator(a\u208a,1) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a,1\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2 \n", ") + \u212f \u22c5BosonOperator(a\u208b,1) + 2.0\u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1\n", "\n", " \n", ") - 2.0\u22c5\u2148\u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,1) + 2.0\u22c5\u2148\u22c5BosonOperator(a\u208b,0)\u22c5\n", "\n", " \n", "BosonOperator(a\u208a,1) + 2\u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208b,1)" ] } ], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Write all input operators in normal ordered form:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_2 = normal_ordered_form(fdf_1.expand(), independent=True)\n", "\n", "fdf_2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$1.0 i x^{2} e^{- 2.0 i \\theta} {a_+} {a_-} - 1.0 x^{2} e^{- 2.0 i \\theta} \\left({a_+}\\right)^{2} + 1.0 i x^{2} e^{- 2.0 i \\theta} {a_-} {a_+} + 1.0 x^{2} e^{- 2.0 i \\theta} \\left({a_-}\\right)^{2} - 1.0 i x^{2} e^{2.0 i \\theta} {{a_+}^\\dagger} {{a_-}^\\dagger} - 1.0 x^{2} e^{2.0 i \\theta} \\left({{a_+}^\\dagger}\\right)^{2} - 1.0 i x^{2} e^{2.0 i \\theta} {{a_-}^\\dagger} {{a_+}^\\dagger} + 1.0 x^{2} e^{2.0 i \\theta} \\left({{a_-}^\\dagger}\\right)^{2} + 4.0 x^{2} + 2.0 x^{2} {{a_+}^\\dagger} {a_+} - 2.0 i x^{2} {{a_+}^\\dagger} {a_-} + 2.0 i x^{2} {{a_-}^\\dagger} {a_+} + 2.0 x^{2} {{a_-}^\\dagger} {a_-} - 2.0 x e^{- 2.0 i \\theta} - 2.0 x e^{- 2.0 i \\theta} {{a_+}^\\dagger} {a_+} + 2.0 i x e^{- 2.0 i \\theta} {{a_+}^\\dagger} {a_-} - 2.0 i x e^{- 2.0 i \\theta} {{a_-}^\\dagger} {a_+} - 2.0 x e^{- 2.0 i \\theta} {{a_-}^\\dagger} {a_-} - 2.0 x e^{2.0 i \\theta} - 2.0 x e^{2.0 i \\theta} {{a_+}^\\dagger} {a_+} + 2.0 i x e^{2.0 i \\theta} {{a_+}^\\dagger} {a_-} - 2.0 i x e^{2.0 i \\theta} {{a_-}^\\dagger} {a_+} - 2.0 x e^{2.0 i \\theta} {{a_-}^\\dagger} {a_-} + 2.0 i x {{a_+}^\\dagger} {{a_-}^\\dagger} + 2.0 x \\left({{a_+}^\\dagger}\\right)^{2} - 2.0 i x {a_+} {a_-} + 2.0 x \\left({a_+}\\right)^{2} + 2.0 i x {{a_-}^\\dagger} {{a_+}^\\dagger} - 2.0 x \\left({{a_-}^\\dagger}\\right)^{2} - 2.0 i x {a_-} {a_+} - 2.0 x \\left({a_-}\\right)^{2} - 1.0 i e^{- 2.0 i \\theta} {{a_+}^\\dagger} {{a_-}^\\dagger} - 1.0 e^{- 2.0 i \\theta} \\left({{a_+}^\\dagger}\\right)^{2} - 1.0 i e^{- 2.0 i \\theta} {{a_-}^\\dagger} {{a_+}^\\dagger} + e^{- 2.0 i \\theta} \\left({{a_-}^\\dagger}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {a_+} {a_-} - 1.0 e^{2.0 i \\theta} \\left({a_+}\\right)^{2} + 1.0 i e^{2.0 i \\theta} {a_-} {a_+} + e^{2.0 i \\theta} \\left({a_-}\\right)^{2} + 2.0 {{a_+}^\\dagger} {a_+} - 2.0 i {{a_+}^\\dagger} {a_-} + 2.0 i {{a_-}^\\dagger} {a_+} + 2 {{a_-}^\\dagger} {a_-}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 14, "text": [ " 2 -2.0\u22c5\u2148\u22c5\u03b8 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "1.0\u22c5\u2148\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208b,1) - 1.0\u22c5x \u22c5\u212f \u22c5\n", "\n", " 2 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "BosonOperator(a\u208a,1) + 1.0\u22c5\u2148\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 2 2 2.0\u22c5\u2148\u22c5\u03b8 \n", ",1) + 1.0\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208b,1) - 1.0\u22c5\u2148\u22c5x \u22c5\u212f \u22c5BosonOperator(\n", "\n", " 2 2.0\u22c5\u2148\u22c5\u03b8 2 2 2\n", "a\u208a,0)\u22c5BosonOperator(a\u208b,0) - 1.0\u22c5x \u22c5\u212f \u22c5BosonOperator(a\u208a,0) - 1.0\u22c5\u2148\u22c5x \u22c5\u212f \n", "\n", ".0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", " \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,0) + 1.0\u22c5x \u22c5\u212f \u22c5BosonOperator\n", "\n", " 2 2 2 2 \n", "(a\u208b,0) + 4.0\u22c5x + 2.0\u22c5x \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5\u2148\u22c5x \u22c5B\n", "\n", " 2 \n", "osonOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,1) + 2.0\u22c5\u2148\u22c5x \u22c5BosonOperator(a\u208b,0)\u22c5BosonOpe\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8\n", "rator(a\u208a,1) + 2.0\u22c5x \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208b,1) - 2.0\u22c5x\u22c5\u212f \n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8\n", " - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) + 2.0\u22c5\u2148\u22c5x\u22c5\u212f \n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "\u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,1) - 2.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "0)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208b,\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 \n", "1) - 2.0\u22c5x\u22c5\u212f - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) +\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 \n", " 2.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,1) - 2.0\u22c5\u2148\u22c5x\u22c5\u212f \u22c5B\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", "osonOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5Bo\n", "\n", " \n", "sonOperator(a\u208b,1) + 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,0) + 2.0\u22c5x\u22c5Bo\n", "\n", " 2 \n", "sonOperator(a\u208a,0) - 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(a\u208a,1)\u22c5BosonOperator(a\u208b,1) + 2.0\u22c5x\u22c5B\n", "\n", " 2 \n", "osonOperator(a\u208a,1) + 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,0) - 2.0\u22c5x\u22c5\n", "\n", " 2 \n", "BosonOperator(a\u208b,0) - 2.0\u22c5\u2148\u22c5x\u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5x\n", "\n", " 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", "\u22c5BosonOperator(a\u208b,1) - 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208b,0\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 2 -2.0\u22c5\u2148\u22c5\u03b8 \n", ") - 1.0\u22c5\u212f \u22c5BosonOperator(a\u208a,0) - 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5B\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 2 2.0\u22c5\u2148\u22c5\u03b8 \n", "osonOperator(a\u208a,0) + \u212f \u22c5BosonOperator(a\u208b,0) + 1.0\u22c5\u2148\u22c5\u212f \u22c5BosonOper\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2 2.\n", "ator(a\u208a,1)\u22c5BosonOperator(a\u208b,1) - 1.0\u22c5\u212f \u22c5BosonOperator(a\u208a,1) + 1.0\u22c5\u2148\u22c5\u212f \n", "\n", "0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 2 \n", " \u22c5BosonOperator(a\u208b,1)\u22c5BosonOperator(a\u208a,1) + \u212f \u22c5BosonOperator(a\u208b,1) \n", "\n", " \n", "+ 2.0\u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5\u2148\u22c5BosonOperator(a\u208a,0)\u22c5Boso\n", "\n", " \n", "nOperator(a\u208b,1) + 2.0\u22c5\u2148\u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208a,1) + 2\u22c5BosonOpera\n", "\n", " \n", "tor(a\u208b,0)\u22c5BosonOperator(a\u208b,1)" ] } ], "prompt_number": 14 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Assume thermal input field" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Elimitate operators that do not contribute to expectation values assuming thermal input field. The only operators that contribute are $a_-^\\dagger a_-$ and $a_+^\\dagger a_+$, and expressions such as for example $\\langle a_-a_-\\rangle = 0$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# assume thermal input: first remove nonthermal operator combinations\n", "fdf_3 = fdf_2.subs({am**2: 0, ap**2: 0,\n", " ap*am: 0, am*ap: 0, \n", " Dagger(ap) * am: 0, Dagger(am) * ap: 0, \n", " Dagger(ap) * Dagger(am): 0, Dagger(am) * Dagger(ap): 0, \n", " Dagger(ap)**2: 0, \n", " Dagger(am)**2: 0})\n", "\n", "fdf_3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$4.0 x^{2} + 2.0 x^{2} {{a_+}^\\dagger} {a_+} + 2.0 x^{2} {{a_-}^\\dagger} {a_-} - 2.0 x e^{- 2.0 i \\theta} - 2.0 x e^{- 2.0 i \\theta} {{a_+}^\\dagger} {a_+} - 2.0 x e^{- 2.0 i \\theta} {{a_-}^\\dagger} {a_-} - 2.0 x e^{2.0 i \\theta} - 2.0 x e^{2.0 i \\theta} {{a_+}^\\dagger} {a_+} - 2.0 x e^{2.0 i \\theta} {{a_-}^\\dagger} {a_-} + 2.0 {{a_+}^\\dagger} {a_+} + 2 {{a_-}^\\dagger} {a_-}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 15, "text": [ " 2 2 2 \n", "4.0\u22c5x + 2.0\u22c5x \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1) + 2.0\u22c5x \u22c5BosonOperator\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 \n", "(a\u208b,0)\u22c5BosonOperator(a\u208b,1) - 2.0\u22c5x\u22c5\u212f - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\n", "\n", " -2.0\u22c5\u2148\u22c5\u03b8 \n", "\u208a,0)\u22c5BosonOperator(a\u208a,1) - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 \n", "\u208b,1) - 2.0\u22c5x\u22c5\u212f - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208a,0)\u22c5BosonOperator(a\u208a,1)\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 \n", " - 2.0\u22c5x\u22c5\u212f \u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208b,1) + 2.0\u22c5BosonOperator(\n", "\n", " \n", "a\u208a,0)\u22c5BosonOperator(a\u208a,1) + 2\u22c5BosonOperator(a\u208b,0)\u22c5BosonOperator(a\u208b,1)" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now replace the operators that do contibute with their expectation values:\n", "\n", "$$\n", "\\langle a_-^\\dagger a_-\\rangle = n_{\\rm th}^-\n", "$$\n", "\n", "$$\n", "\\langle a_+^\\dagger a_+\\rangle = n_{\\rm th}^+\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "n_th_m, n_th_p = symbols(\"{n^-_{th}}, {n^+_{th}}\", positive=True)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "# thermal operator combinations \n", "fdf_4 = fdf_3.subs({Dagger(am) * am: n_th_m, Dagger(ap) * ap: n_th_p})\n", "\n", "fdf_4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$2.0 x^{2} {n^{+}_{{th}}} + 2.0 x^{2} {n^{-}_{{th}}} + 4.0 x^{2} - 2.0 x {n^{+}_{{th}}} e^{- 2.0 i \\theta} - 2.0 x {n^{+}_{{th}}} e^{2.0 i \\theta} - 2.0 x {n^{-}_{{th}}} e^{- 2.0 i \\theta} - 2.0 x {n^{-}_{{th}}} e^{2.0 i \\theta} - 2.0 x e^{- 2.0 i \\theta} - 2.0 x e^{2.0 i \\theta} + 2.0 {n^{+}_{{th}}} + 2 {n^{-}_{{th}}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 17, "text": [ " 2 2 2 -2.0\u22c5\u2148\u22c5\u03b8\n", "2.0\u22c5x \u22c5{n_{th}}__+ + 2.0\u22c5x \u22c5{n_{th}}__- + 4.0\u22c5x - 2.0\u22c5x\u22c5{n_{th}}__+\u22c5\u212f \n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 \n", " - 2.0\u22c5x\u22c5{n_{th}}__+\u22c5\u212f - 2.0\u22c5x\u22c5{n_{th}}__-\u22c5\u212f - 2.0\u22c5x\u22c5{n_{th}}__\n", "\n", " 2.0\u22c5\u2148\u22c5\u03b8 -2.0\u22c5\u2148\u22c5\u03b8 2.0\u22c5\u2148\u22c5\u03b8 \n", "-\u22c5\u212f - 2.0\u22c5x\u22c5\u212f - 2.0\u22c5x\u22c5\u212f + 2.0\u22c5{n_{th}}__+ + 2\u22c5{n_{th}}__\n", "\n", " \n", "-" ] } ], "prompt_number": 17 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Replace $e^{\\pm i\\theta}$ with trigonometric expressions, and simplify the expression:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_5 = fdf_4.collect(x).replace(exp(2j*theta), cos(2*theta) + 1j * sin(2*theta)).\\\n", " replace(exp(-2j*theta), cos(2*theta) - 1j * sin(2*theta))\n", " \n", "fdf_6 = fdf_5.simplify().collect(cos(2*theta)).collect(x).simplify()\n", "\n", "fdf_6" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$x^{2} \\left(2.0 {n^{+}_{{th}}} + 2.0 {n^{-}_{{th}}} + 4.0\\right) - 4.0 x \\left({n^{+}_{{th}}} + {n^{-}_{{th}}} + 1\\right) \\cos{\\left (2 \\theta \\right )} + 2.0 {n^{+}_{{th}}} + 2.0 {n^{-}_{{th}}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 18, "text": [ " 2 \n", "x \u22c5(2.0\u22c5{n_{th}}__+ + 2.0\u22c5{n_{th}}__- + 4.0) - 4.0\u22c5x\u22c5({n_{th}}__+ + {n_{th}}__\n", "\n", " \n", "- + 1)\u22c5cos(2\u22c5\u03b8) + 2.0\u22c5{n_{th}}__+ + 2.0\u22c5{n_{th}}__-" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Drop the $x^2$ term because our relation between the output field and input field is only valid to first order in $x$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_7 = fdf_6.subs(x**2, 0)\n", "\n", "fdf_7" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- 4.0 x \\left({n^{+}_{{th}}} + {n^{-}_{{th}}} + 1\\right) \\cos{\\left (2 \\theta \\right )} + 2.0 {n^{+}_{{th}}} + 2.0 {n^{-}_{{th}}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 19, "text": [ "-4.0\u22c5x\u22c5({n_{th}}__+ + {n_{th}}__- + 1)\u22c5cos(2\u22c5\u03b8) + 2.0\u22c5{n_{th}}__+ + 2.0\u22c5{n_{th\n", "}}__-" ] } ], "prompt_number": 19 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Substitute the small parameter $x$ with the physical parameters it represents:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_8 = fdf_7.subs(x**2, 0).subs(x, dL/v * sqrt(wp * wm))\n", "\n", "fdf_8" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$2.0 {n^{+}_{{th}}} + 2.0 {n^{-}_{{th}}} - \\frac{4.0 {\\delta\\!L}}{v} \\sqrt{\\omega_{+} \\omega_{-}} \\left({n^{+}_{{th}}} + {n^{-}_{{th}}} + 1\\right) \\cos{\\left (2 \\theta \\right )}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 20, "text": [ " _______ \n", " 4.0\u22c5{\\delta\\!L}\u22c5\u2572\u2571 \u03c9\u208a\u22c5\u03c9\u208b \u22c5({n_{th}}__+ + {\n", "2.0\u22c5{n_{th}}__+ + 2.0\u22c5{n_{th}}__- - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\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", " v \n", "\n", " \n", "n_{th}}__- + 1)\u22c5cos(2\u22c5\u03b8)\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\n", " " ] } ], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is equation 14 in Phys. Rev. A 87, 043804 (2013).\n", "\n", "The interesting regime is when this expression is negative, because it indicates that there are nonclassical correlations between the quadrature at the different frequencies $\\omega_\\pm$. It is clear that $\\theta = 0$ minimizes the expression, and that for finite temperature the expression is initially positive (the prefactor to the other term is small). We can find the transition point from positive to negative by solving $\\langle:f^\\dagger f:\\rangle = 0$ for $x = \\frac{\\delta\\!L_{\\rm eff}}{v}\\sqrt{\\omega_-\\omega_+}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# switch back to using x as small paramter, and set theta = 0\n", "fdf_9 = fdf_8.subs(dL/v * sqrt(wp*wm), x).subs(theta, 0)\n", "\n", "fdf_9" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- 4.0 x \\left({n^{+}_{{th}}} + {n^{-}_{{th}}} + 1\\right) + 2.0 {n^{+}_{{th}}} + 2.0 {n^{-}_{{th}}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 21, "text": [ "-4.0\u22c5x\u22c5({n_{th}}__+ + {n_{th}}__- + 1) + 2.0\u22c5{n_{th}}__+ + 2.0\u22c5{n_{th}}__-" ] } ], "prompt_number": 21 }, { "cell_type": "code", "collapsed": false, "input": [ "fdf_neg = solve(fdf_9, x)[0]\n", "\n", "fdf_neg" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{0.5 {n^{+}_{{th}}} + 0.5 {n^{-}_{{th}}}}{{n^{+}_{{th}}} + {n^{-}_{{th}}} + 1.0}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 22, "text": [ "0.5\u22c5({n_{th}}__+ + {n_{th}}__-)\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\n", "{n_{th}}__+ + {n_{th}}__- + 1.0" ] } ], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Which in the low tempeature case ($n_{\\rm th}^+ + n_{\\rm th}^- \\ll 1$) simply gives\n", "\n", "$$\n", "\\frac{\\delta\\!L_{\\rm eff}}{v}\\sqrt{\\omega_-\\omega_+} > \\sim (n_{\\rm th}^+ + n_{\\rm th}^-)/2\n", "$$\n", "\n", "as a condition for when the output field becomes nonclassical." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Versions" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py\n", "%reload_ext version_information\n", "\n", "%version_information sympy" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "
SoftwareVersion
Python3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1]
IPython1.1.0
OSposix [linux]
sympy0.7.4.1-git
Wed Dec 18 14:05:32 2013 KST
" ], "json": [ "{ \"Software versions\" : [{ \"module\" : \"Python\", \"version\" : \"3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1]\" }, { \"module\" : \"IPython\", \"version\" : \"1.1.0\" }, { \"module\" : \"OS\", \"version\" : \"posix [linux]\" }, { \"module\" : \"sympy\", \"version\" : \"0.7.4.1-git\" } ] }" ], "latex": [ "\\begin{tabular}{|l|l|}\\hline\n", "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n", "Python & 3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1] \\\\ \\hline\n", "IPython & 1.1.0 \\\\ \\hline\n", "OS & posix [linux] \\\\ \\hline\n", "sympy & 0.7.4.1-git \\\\ \\hline\n", "\\hline \\multicolumn{2}{|l|}{Wed Dec 18 14:05:32 2013 KST} \\\\ \\hline\n", "\\end{tabular}\n" ], "metadata": {}, "output_type": "pyout", "prompt_number": 23, "text": [ "Software versions\n", "Python 3.3.2+ (default, Oct 9 2013, 14:50:09) [GCC 4.8.1]\n", "IPython 1.1.0\n", "OS posix [linux]\n", "sympy 0.7.4.1-git\n", "\n", "Wed Dec 18 14:05:32 2013 KST" ] } ], "prompt_number": 23 } ], "metadata": {} } ] }