{ "metadata": { "name": "", "signature": "sha256:945973d0e876cbb291acfa62b69c6e726f1656c12bb0595d3a35cbf07432ed91" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Lecture 3 - Symbolic quantum mechanics using SymPsi - Resonators and cavities" ] }, { "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": [ "Introduction" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "In this notebook we will work with cavities and resonators. A single mode of a resonator can be modelled with a quantum harmonic oscillator. Here we look at the effect of classical driving fields, transformation to different rotating frames, and various types of coupling between different modes in a resonator." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Quantum Harmonic Oscillator" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "First we consider the Hamiltonian for a single harmonic oscillator, which describes a single mode of for example a cavity or a waveguide resonator:\n", "\n", "$$\n", "H = \\hbar \\omega_r a^\\dagger a\n", "$$\n", "\n", "To represent this Hamiltonian in sympy we create a symbol `omega_r` and an instance of the class `BosonOp`:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "omega_r = symbols(\"omega_r\", positive=True)\n", "Hsym = symbols(\"H\")\n", "a = BosonOp(\"a\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "H0 = omega_r * Dagger(a) * a\n", "\n", "Eq(Hsym, H0)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\omega_{r} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAGMAAAAWBAMAAAAm+j/UAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n", "prompt_number": 4, "text": [ " \u2020 \n", "H = \u03c9\u1d63\u22c5a \u22c5a" ] } ], "prompt_number": 4 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Classical drive signal" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "A classical driving field of frequency $\\omega_d$ and amplitude $A$ and phase $\\phi_0$ can be modelled by including an additional term in the Hamiltonian:\n", "\n", "$$\n", "H = \\hbar \\omega_r a^\\dagger a + A \\cos(\\omega_dt + \\phi_0) (a + a^\\dagger)\n", "$$\n", "\n", "where we have assumed that the driving field couples to the quadrature of the resonator $a + a^\\dagger$, which is the canonical case. It is convenient to rewrite the $\\cos$ factor\n", "\n", "$$\n", "H = \\hbar \\omega_r a^\\dagger a + (Ae^{-i\\omega_dt}+ A^*e^{i\\omega_dt})(a + a^\\dagger)\n", "$$\n", "\n", "where we have redefined $A \\rightarrow Ae^{-i\\phi_0}$. In Sympy we can represent this Hamiltonian as:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "omega_d, t = symbols(\"omega_d, t\")\n", "A = symbols(\"A\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 5 }, { "cell_type": "code", "collapsed": false, "input": [ "Hdrive = (A * exp(-I * omega_d * t) + conjugate(A) * exp(I * omega_d * t)) * (a + Dagger(a))\n", "\n", "Hdrive" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(A e^{- i \\omega_{d} t} + e^{i \\omega_{d} t} \\overline{A}\\right) \\left({{a}^\\dagger} + {a}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 6, "text": [ "\u239b -\u2148\u22c5\u03c9_d\u22c5t \u2148\u22c5\u03c9_d\u22c5t _\u239e \u239b \u2020 \u239e\n", "\u239dA\u22c5\u212f + \u212f \u22c5A\u23a0\u22c5\u239da + a\u23a0" ] } ], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "H = H0 + Hdrive\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\omega_{r} {{a}^\\dagger} {a} + \\left(A e^{- i \\omega_{d} t} + e^{i \\omega_{d} t} \\overline{A}\\right) \\left({{a}^\\dagger} + {a}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 7, "text": [ " \u2020 \u239b -\u2148\u22c5\u03c9_d\u22c5t \u2148\u22c5\u03c9_d\u22c5t _\u239e \u239b \u2020 \u239e\n", "H = \u03c9\u1d63\u22c5a \u22c5a + \u239dA\u22c5\u212f + \u212f \u22c5A\u23a0\u22c5\u239da + a\u23a0" ] } ], "prompt_number": 7 }, { "cell_type": "markdown", "metadata": {}, "source": [ "When working with Hamiltonians like this one, one common operation is to move to different rotating frames (by performing unitary transformations) where the Hamiltonian takes a simplier form.\n", "\n", "Here we want to transform this Hamiltonian to a rotating frame in which the drive term (and all other terms) are no longer explicitly depening on time `t`. We can accomplish this by performing the unitary transformation \n", "\n", "$$\n", "U = \\exp(i\\omega_d a^\\dagger a t)\n", "$$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * omega_d * t * Dagger(a) * a)\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": 8, "text": [ " \u2020 \n", " \u2148\u22c5\u03c9_d\u22c5t\u22c5a \u22c5a\n", "\u212f " ] } ], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "When doing a unitary basis transformation, the Hamiltonian is transformed as\n", "\n", "$$\n", "H \\rightarrow UHU^\\dagger -i U \\frac{d}{dt}U^\\dagger\n", "$$\n", "\n", "and we can carry out this transformation using the function `hamiltonian_transformation`:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = hamiltonian_transformation(U, H.expand())\n", "\n", "H2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A {{a}^\\dagger} + A e^{- 2 i \\omega_{d} t} {a} - \\omega_{d} {{a}^\\dagger} {a} + \\omega_{r} {{a}^\\dagger} {a} + e^{2 i \\omega_{d} t} \\overline{A} {{a}^\\dagger} + \\overline{A} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 9, "text": [ " \u2020 -2\u22c5\u2148\u22c5\u03c9_d\u22c5t \u2020 \u2020 2\u22c5\u2148\u22c5\u03c9_d\u22c5t _ \u2020 _ \n", "A\u22c5a + A\u22c5\u212f \u22c5a - \u03c9_d\u22c5a \u22c5a + \u03c9\u1d63\u22c5a \u22c5a + \u212f \u22c5A\u22c5a + A\u22c5a" ] } ], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now let's introduce the detuning $\\Delta = \\omega_r - \\omega_d$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "Delta = symbols(\"Delta\", positive=True)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = collect(H2, Dagger(a) * a).subs(omega_r - omega_d, Delta)\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A {{a}^\\dagger} + A e^{- 2 i \\omega_{d} t} {a} + \\Delta {{a}^\\dagger} {a} + e^{2 i \\omega_{d} t} \\overline{A} {{a}^\\dagger} + \\overline{A} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 11, "text": [ " \u2020 -2\u22c5\u2148\u22c5\u03c9_d\u22c5t \u2020 2\u22c5\u2148\u22c5\u03c9_d\u22c5t _ \u2020 _ \n", "A\u22c5a + A\u22c5\u212f \u22c5a + \u0394\u22c5a \u22c5a + \u212f \u22c5A\u22c5a + A\u22c5a" ] } ], "prompt_number": 11 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Rotating-wave approximation (RWA)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we invoke the rotating-wave approximation, under which we assume that $\\omega_d$ is much larger than $\\Delta$, so that the we can neglect the two fast rotating terms which contains factors $\\exp(\\pm 2i \\omega_d t)$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = drop_terms_containing(H3, [exp( 2 * I * omega_d * t),\n", " exp(-2 * I * omega_d * t)])\n", "\n", "Eq(Hsym, H3)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = A {{a}^\\dagger} + \\Delta {{a}^\\dagger} {a} + \\overline{A} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 12, "text": [ " \u2020 \u2020 _ \n", "H = A\u22c5a + \u0394\u22c5a \u22c5a + A\u22c5a" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "This is a time-independent hamiltonian describing the resonator in a rotating frame, where fast rotating terms have been dropped. We can no apply a displacement transformation by applying the unitary transformation:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "alpha = symbols(\"alpha\")\n", "H = Dagger(a) * alpha - conjugate(alpha) * a\n", "U = exp(H)\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": 13, "text": [ " \u2020 _ \n", " \u03b1\u22c5a - \u03b1\u22c5a\n", "\u212f " ] } ], "prompt_number": 13 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = hamiltonian_transformation(U, H3)\n", "\n", "H4 = collect(H4.expand(), [Dagger(a)*a, a, Dagger(a)])\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- A \\overline{\\alpha} + \\Delta \\alpha \\overline{\\alpha} + \\Delta {{a}^\\dagger} {a} - \\alpha \\overline{A} + \\left(A - \\Delta \\alpha\\right) {{a}^\\dagger} + \\left(- \\Delta \\overline{\\alpha} + \\overline{A}\\right) {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAgwAAAAbBAMAAADi5Mo2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ny8g014L+N2HVt+NqG86VVov83+D+BIrQ+10kn3YEAAAAAElFTkSuQmCC\n", "prompt_number": 14, "text": [ " _ _ \u2020 _ \u2020 \u239b _ _\u239e \n", "- A\u22c5\u03b1 + \u0394\u22c5\u03b1\u22c5\u03b1 + \u0394\u22c5a \u22c5a - \u03b1\u22c5A + (A - \u0394\u22c5\u03b1)\u22c5a + \u239d- \u0394\u22c5\u03b1 + A\u23a0\u22c5a" ] } ], "prompt_number": 14 }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we choose $\\alpha = A/\\Delta$ and drop c-numbers from the hamiltonian, we obtain a particularly simple form:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = H4.subs(alpha, A/Delta)\n", "\n", "H5 = drop_c_number_terms(H5)\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\Delta {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAADAAAAATBAMAAAAkFJMsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZjJ2qxBEie9UmSLN\n3buAmmHLAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5UlEQVQYGWNgQAUsCah8OA+nBHsAXA0Kg+/c\niglIAgJwNit7KJzNwMD0GcFhUUCwGVj+IwzmQzYpub4ArgxFwiH+K1yCBUkHmwD7d4REApzJwMLA\n2g+yZEuxbgADlwKMxcCQyMBgD7SErZvhPQMD+wYYi4HBgYGBA2hJhADDOgYGZqBeCIuBUwDI/cXA\nsD6A4TIDAyvQJAgLaAUQAC3pY2D9BGIywFgzQWz7BazfGTg/gNQzwFgOIA7Hb9afDHwHZoPYUBbn\nqVVAsPYXw2EGWQeg44AAwuL9DwEBIebazgkgcQYwCwAqLj+PClXWAAAAAABJRU5ErkJggg==\n", "prompt_number": 15, "text": [ " \u2020 \n", "\u0394\u22c5a \u22c5a" ] } ], "prompt_number": 15 }, { "cell_type": "markdown", "metadata": {}, "source": [ "So a driven harmonic oscillator can be described by the hamiltonian of an undriven harmonic oscillator in a displaced frame." ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Optical parametric oscillator" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "An optical parametric oscillator (OPO) is a two-mode system with a particular nonlinear interaction. The Hamiltonian of the OPO is\n", "\n", "$$\n", "H = \\omega_r a^\\dagger a + \\omega_p b^\\dagger b + (\\kappa {a^\\dagger}^2 b + \\kappa^* a^2 b^\\dagger) + (A e^{i\\omega_pt} + A^* e^{-i\\omega_pt})(b + b^\\dagger)\n", "$$\n", "\n", "where the operators of the two modes are$a$ and $b$, and the nonlinear interaction strength is $\\kappa$, and where we have also included a classical drive field applied to mode $b$.\n", "\n", "To model this system in SymPy we create two instances of `BosonOp`, one for each mode, and construct the Hamiltonian:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "a, b = BosonOp(\"a\"), BosonOp(\"b\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 16 }, { "cell_type": "code", "collapsed": false, "input": [ "kappa, omega_p, omega_a, omega_b, t = symbols(\"kappa, omega_p, omega_a, omega_b, t\")\n", "Delta_a, Delta_b = symbols(\"Delta_a, Delta_b\", positive=True)" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 17 }, { "cell_type": "code", "collapsed": false, "input": [ "Hdrive = (A * exp(-I * omega_p * t) + conjugate(A) * exp(I * omega_p * t)) * (b + Dagger(b))\n", "H1 = omega_a * Dagger(a) * a + omega_b * Dagger(b) * b + kappa * (a ** 2 * Dagger(b) + Dagger(a) ** 2 * b) + Hdrive\n", "\n", "Eq(Hsym, H1)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\kappa \\left(\\left({{a}^\\dagger}\\right)^{2} {b} + \\left({a}\\right)^{2} {{b}^\\dagger}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b} + \\left(A e^{- i \\omega_{p} t} + e^{i \\omega_{p} t} \\overline{A}\\right) \\left({{b}^\\dagger} + {b}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAosAAAAmBAMAAABe2cw9AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n", "prompt_number": 18, "text": [ " \u239b 2 \u239e \n", " \u239c\u239b \u2020\u239e 2 \u2020\u239f \u2020 \u2020 \u239b -\u2148\u22c5\u03c9_p\u22c5t \u2148\u22c5\u03c9_p\u22c5t _\u239e \u239b \u2020 \n", "H = \u03ba\u22c5\u239d\u239da \u23a0 \u22c5b + a \u22c5b \u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b + \u239dA\u22c5\u212f + \u212f \u22c5A\u23a0\u22c5\u239db \n", "\n", " \n", " \u239e\n", "+ b\u23a0" ] } ], "prompt_number": 18 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We first move to a frame rotating with frequency $\\omega_p$ with respect to mode $b$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * omega_p * t * Dagger(b) * b)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\omega_{p} t {{b}^\\dagger} {b}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEAAAAAWBAMAAACCkIcHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFXWPMUvDUBSFz0uaatLYFOwoGBR3\nsQGh7RDo4OASJxd3BREr/gFxEIoOHRQHQTIIXVXEyaHqKEIGQXAqdHPLP/C9G++r0vQM+c459/LI\nBSapPGnAvcGGaIb/4l4C8+EJkPyVE7GT/IIRAgXpFMclUsx0gSk5URyX3YYnH1QLiqRiA0W5/KaC\ntfYR79fuoK5QJBnzlZI0PQry3dVKQP+gSNq4xBbmqKN3h3iHOlMx0xAXeMUshR3gBOtwZFAkiTSu\n4xFLbuuli4aIUnSoZ8JO/Ourz/PQXhFttAwcuANaYELcR+Xjwinw7STYreNmgeYZxWLA1wJHRj8b\n/fluR5bP0TpsstW0OqjGnFyf3Yils9qzTpsDbbXxbrXNN56f3+t2Wi4s65RjzATV0Zk5Cwia/bya\nux8GwzxDC6YuAwAAAABJRU5ErkJggg==\n", "prompt_number": 19, "text": [ " \u2020 \n", " \u2148\u22c5\u03c9_p\u22c5t\u22c5b \u22c5b\n", "\u212f " ] } ], "prompt_number": 19 }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = hamiltonian_transformation(U, H1.expand(), independent=True)\n", "\n", "H2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A {{b}^\\dagger} + A e^{- 2 i \\omega_{p} t} {b} + \\kappa e^{i \\omega_{p} t} \\left({a}\\right)^{2} {{b}^\\dagger} + \\kappa e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} {b} + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b} - \\omega_{p} {{b}^\\dagger} {b} + e^{2 i \\omega_{p} t} \\overline{A} {{b}^\\dagger} + \\overline{A} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 20, "text": [ " 2 \n", " \u2020 -2\u22c5\u2148\u22c5\u03c9_p\u22c5t \u2148\u22c5\u03c9_p\u22c5t 2 \u2020 -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u2020 \n", "A\u22c5b + A\u22c5\u212f \u22c5b + \u03ba\u22c5\u212f \u22c5a \u22c5b + \u03ba\u22c5\u212f \u22c5\u239da \u23a0 \u22c5b + \u03c9\u2090\u22c5a \u22c5a + \u03c9_\n", "\n", " \n", " \u2020 \u2020 2\u22c5\u2148\u22c5\u03c9_p\u22c5t _ \u2020 _ \n", "b\u22c5b \u22c5b - \u03c9_p\u22c5b \u22c5b + \u212f \u22c5A\u22c5b + A\u22c5b" ] } ], "prompt_number": 20 }, { "cell_type": "markdown", "metadata": {}, "source": [ "and we can perform the rotating wave approximation and drop the fast rotating terms $\\exp{\\left(\\pm i \\omega_p t\\right)}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = drop_terms_containing(H2, [exp(2 * I * omega_p * t),\n", " exp(-2 * I * omega_p * t)])\n", "\n", "H3 = collect(H3, kappa)\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A {{b}^\\dagger} + \\kappa \\left(e^{i \\omega_{p} t} \\left({a}\\right)^{2} {{b}^\\dagger} + e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} {b}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\omega_{b} {{b}^\\dagger} {b} - \\omega_{p} {{b}^\\dagger} {b} + \\overline{A} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 21, "text": [ " \u239b 2 \u239e \n", " \u2020 \u239c \u2148\u22c5\u03c9_p\u22c5t 2 \u2020 -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u239f \u2020 \u2020 \u2020 \n", "A\u22c5b + \u03ba\u22c5\u239d\u212f \u22c5a \u22c5b + \u212f \u22c5\u239da \u23a0 \u22c5b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b - \u03c9_p\u22c5b \u22c5b \n", "\n", " \n", " _ \n", "+ A\u22c5b" ] } ], "prompt_number": 21 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Introduce a new variable for the detuning $\\Delta_b = \\omega_b - \\omega_p$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = H3.subs(omega_b, Delta_b + omega_p).expand()\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A {{b}^\\dagger} + \\Delta_{b} {{b}^\\dagger} {b} + \\kappa e^{i \\omega_{p} t} \\left({a}\\right)^{2} {{b}^\\dagger} + \\kappa e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} {b} + \\omega_{a} {{a}^\\dagger} {a} + \\overline{A} {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 22, "text": [ " 2 \n", " \u2020 \u2020 \u2148\u22c5\u03c9_p\u22c5t 2 \u2020 -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u2020 _ \n", "A\u22c5b + \u0394_b\u22c5b \u22c5b + \u03ba\u22c5\u212f \u22c5a \u22c5b + \u03ba\u22c5\u212f \u22c5\u239da \u23a0 \u22c5b + \u03c9\u2090\u22c5a \u22c5a + A\u22c5b" ] } ], "prompt_number": 22 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Next we want to displace the mode $b$ so that the drive terms are eliminated:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "beta = symbols(\"beta\")\n", "H = Dagger(b) * beta - conjugate(beta) * b\n", "U = exp(H)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{\\beta {{b}^\\dagger} - \\overline{\\beta} {b}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAWBAMAAACGZVc6AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABO0lEQVQoFX2OP0vEMByG32uuvfasNuB9gPhn\nua3YisOJlFvPoeDkdoPgdGg/gotw4lAQnTuI8/kNzsFRKTg5CBV3QcRZI2mSFou/IXmfNw9JgPoU\nEr3v35FU2Vkl1yMJ59YLMBjNdR/sYpIrXEGCdU6RamDQDjUUuzFO8MxPj7VxD5suppINOD6+6gbD\nBbxYGvuTHVifA4pCNjCvngocBbdlcQkjJQmJKv9wYvSxRcPSYCDRku/mYOoOAwjxhoey8NFm9rg9\nQwY473xeQYAhTjESRsuHnXnUVj/n9TJwhg9MhWFF3O/QR0FiPeDXInELQWRvI0V3yASJdTPoA9er\nHFprYczf/DO+ag5jk/1rmFP0shvlq9Ady7hwHtzJ3Lx7s+Zetx7TuTnZ3NDfbnJIjl7cdKC7cHuu\noSH9AE2POxGdPZzsAAAAAElFTkSuQmCC\n", "prompt_number": 23, "text": [ " \u2020 _ \n", " \u03b2\u22c5b - \u03b2\u22c5b\n", "\u212f " ] } ], "prompt_number": 23 }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = hamiltonian_transformation(U, H4.expand(), independent=True)\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$A \\left(- \\overline{\\beta} + {{b}^\\dagger}\\right) + \\Delta_{b} \\left(- \\overline{\\beta} + {{b}^\\dagger}\\right) \\left(- \\beta + {b}\\right) + \\kappa e^{i \\omega_{p} t} \\left({a}\\right)^{2} \\left(- \\overline{\\beta} + {{b}^\\dagger}\\right) + \\kappa e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} \\left(- \\beta + {b}\\right) + \\omega_{a} {{a}^\\dagger} {a} + \\overline{A} \\left(- \\beta + {b}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 24, "text": [ " \n", " \u239b _ \u2020\u239e \u239b _ \u2020\u239e \u2148\u22c5\u03c9_p\u22c5t 2 \u239b _ \u2020\u239e -\u2148\u22c5\u03c9_p\u22c5\n", "A\u22c5\u239d- \u03b2 + b \u23a0 + \u0394_b\u22c5\u239d- \u03b2 + b \u23a0\u22c5(-\u03b2 + b) + \u03ba\u22c5\u212f \u22c5a \u22c5\u239d- \u03b2 + b \u23a0 + \u03ba\u22c5\u212f \n", "\n", " 2 \n", "t \u239b \u2020\u239e \u2020 _ \n", " \u22c5\u239da \u23a0 \u22c5(-\u03b2 + b) + \u03c9\u2090\u22c5a \u22c5a + A\u22c5(-\u03b2 + b)" ] } ], "prompt_number": 24 }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = collect(H5.expand(), [Dagger(a) * a, Dagger(b) * b, Dagger(a) ** 2 * b, a ** 2 * Dagger(b), b, Dagger(b)])\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- A \\overline{\\beta} + \\Delta_{b} \\beta \\overline{\\beta} + \\Delta_{b} {{b}^\\dagger} {b} - \\beta \\kappa e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} - \\beta \\overline{A} - \\kappa e^{i \\omega_{p} t} \\overline{\\beta} \\left({a}\\right)^{2} + \\kappa e^{i \\omega_{p} t} \\left({a}\\right)^{2} {{b}^\\dagger} + \\kappa e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} {b} + \\omega_{a} {{a}^\\dagger} {a} + \\left(A - \\Delta_{b} \\beta\\right) {{b}^\\dagger} + \\left(- \\Delta_{b} \\overline{\\beta} + \\overline{A}\\right) {b}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 25, "text": [ " 2 \n", " _ _ \u2020 -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e _ \u2148\u22c5\u03c9_p\u22c5t _ 2 \n", "- A\u22c5\u03b2 + \u0394_b\u22c5\u03b2\u22c5\u03b2 + \u0394_b\u22c5b \u22c5b - \u03b2\u22c5\u03ba\u22c5\u212f \u22c5\u239da \u23a0 - \u03b2\u22c5A - \u03ba\u22c5\u212f \u22c5\u03b2\u22c5a + \u03ba\u22c5\u212f\n", "\n", " 2 \n", "\u2148\u22c5\u03c9_p\u22c5t 2 \u2020 -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u2020 \u2020 \u239b _ _\u239e\n", " \u22c5a \u22c5b + \u03ba\u22c5\u212f \u22c5\u239da \u23a0 \u22c5b + \u03c9\u2090\u22c5a \u22c5a + (A - \u0394_b\u22c5\u03b2)\u22c5b + \u239d- \u0394_b\u22c5\u03b2 + A\u23a0\n", "\n", " \n", " \n", "\u22c5b" ] } ], "prompt_number": 25 }, { "cell_type": "markdown", "metadata": {}, "source": [ "The choice $\\beta = A/\\Delta_b$ eliminates the drive terms. After dropping c numbers we have:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H6 = H5.subs(beta, A/Delta_b)\n", "\n", "H6 = drop_c_number_terms(H6)\n", "\n", "H6 = collect(H6, [exp( I * omega_p * t) * kappa * a ** 2,\n", " exp(-I * omega_p * t) * kappa * Dagger(a) ** 2])\n", "\n", "H6" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\Delta_{b} {{b}^\\dagger} {b} + \\kappa e^{i \\omega_{p} t} \\left({a}\\right)^{2} \\left({{b}^\\dagger} - \\frac{\\overline{A}}{\\Delta_{b}}\\right) + \\kappa e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} \\left(- \\frac{A}{\\Delta_{b}} + {b}\\right) + \\omega_{a} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 26, "text": [ " \u239b _ \u239e 2 \n", " \u2020 \u2148\u22c5\u03c9_p\u22c5t 2 \u239c \u2020 A \u239f -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u239b A \u239e \u2020 \n", "\u0394_b\u22c5b \u22c5b + \u03ba\u22c5\u212f \u22c5a \u22c5\u239cb - \u2500\u2500\u2500\u239f + \u03ba\u22c5\u212f \u22c5\u239da \u23a0 \u22c5\u239c- \u2500\u2500\u2500 + b\u239f + \u03c9\u2090\u22c5a \u22c5a\n", " \u239d \u0394_b\u23a0 \u239d \u0394_b \u23a0 " ] } ], "prompt_number": 26 }, { "cell_type": "markdown", "metadata": {}, "source": [ "If we now assume that the dynamics of the mode $b$ is dominated by the classical drive field, we can neglect the $b$ operators in the hamiltonian (in this displaced frame, which describes deviations from the dynamics induced by the classical driving field):" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H7 = drop_terms_containing(H6.expand(), [b, Dagger(b)])\n", "\n", "H7 = collect(H7, kappa / Delta_b)\n", "\n", "H7" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\omega_{a} {{a}^\\dagger} {a} + \\frac{\\kappa}{\\Delta_{b}} \\left(- A e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} - e^{i \\omega_{p} t} \\overline{A} \\left({a}\\right)^{2}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 27, "text": [ " \u239b 2 \u239e\n", " \u239c -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u2148\u22c5\u03c9_p\u22c5t _ 2\u239f\n", " \u2020 \u03ba\u22c5\u239d- A\u22c5\u212f \u22c5\u239da \u23a0 - \u212f \u22c5A\u22c5a \u23a0\n", "\u03c9\u2090\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\n", " \u0394_b " ] } ], "prompt_number": 27 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To simpify the notation we redefine $- \\kappa A / \\Delta_b \\rightarrow \\kappa$, and assume that $A$ is real:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H = omega_a * Dagger(a) * a + kappa * (a ** 2 * exp(I * omega_p * t) + Dagger(a) ** 2 * exp(-I * omega_p * t))\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\kappa \\left(e^{i \\omega_{p} t} \\left({a}\\right)^{2} + e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2}\\right) + \\omega_{a} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAXsAAAAnBAMAAAAcK/UAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqvNmRDdRHYyiVS7\nIu/EmopNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGbUlEQVRYCc1XbYhUVRh+ZnbuzufdmcSihGjQ\nH5n1Y6gtCgrnh4qF2LSoFZUzCYWxGZs/LAhs3DZSgxwQ/JXtVBobsTaUhZDmmAWK2U5SQhGu+seP\nLdq0zW/sfe/5uPfO3hlmd2ajF/ae533e93nP2XPOPecO8D+yyKL5UzGaG6aiqEfNQcxClwffHBVK\nNadvWL0K/ZVgyztbm2loAM0v/SkM96C3oc4aTzLONJbLS9+s7c2gLdFsEbfen5Z+SAHpVzW89FWU\nw+1z4NpwJWCO1Q5PJtKv9k7YOboHxpWyln4cq4itClBbrb1dxSJpQjOVV7v199SOVUc6qwn2Da85\noqVvxJQ2rAZBw89bwun81NNlUZ6PCQzfvOZVIZrzYGnpGzGlfVclbwdWMw4WgwWgLa/4mm2ksXli\nfSAnqwR7ESwB09g1F3RSR1VmLX0Vp9xXk3gN8CXJ19p7yWHb/+0IIgkCm7/aTePyHWeynsUeXEwD\nacw68jIvks34CW6x3DahD3OvZLfww1p6Bh62fQxJIJzjkNQG8uyQZbbToJ8j8Of16/Q0LzBZz4zI\nUjs87Xpnhv9sxoWGC9L9+Uv8iMcQsNx4xWp+kzE/JYmll0RVYz6VMxNGAmnmpXZY93gHkY6T6Swn\n1TXuTtt5Qv9orxrcpokD2IOb8YTlrxWsWn+k1NLrbAmi35N9h1g+WvHljTyzUrtNZgCHCA1rr4Hr\nI1ays4MXaVX/tv0qtED5xlhhA27HoG/hjBJ6jQrxoaIKblRLz8QzirXb7JOhgr8nUpm+7hOpxYiO\n8vDbE9rdpJBHHRFyDj9yicZxRUnGtS8qJpoq7v7svp3p6EYjh4URpjv4f7BsuQLc7nI6An8eiR6a\n1T0Pu4rLpDZsH713Uk6grEV7VVWPOiLJX9LJaBsFYnYtOyCQXhdjTiV2NnASOBhOYfUGjq7RyfJ1\nFL5Ht12D+KG4OIlYYbPUOg4YvrZCx3WtfjU4jzoiyXnux8v0NuW0WIHZ+OkiY+thDPSpKQGOW6cc\nx3j7mnM20KnhOjRVty4V59N64X0BEEgREOKHCQXtCcwmZIqqUyb/WI8gjwxsySBQUFLg9btnzhxK\nW8EP72HrZGxc8H1xkNu/2FtTMYvcspkr1wkA7COwthIbpcM6pThqVbdOlQwfXZ+QKFYmIMSDhMK0\nhaXF8xLIOr4e8rOCDP6BIZqsipICnxaATSWp0E30EhelwfLwzREsoSxhvqJC6KZ/74S1bdVNaoVk\nty6V1MzQ2rakFr9JpOO0jxdlkqwTYDciRrg8gUW00zKqX2AuBYcyUqGbwIqBCjthfqn9J9bfqCNH\nyxqeolduDO15ShvVpJ59l0qETX0QoD2txdy5aR8ecYpYJof/CpCDzxoNHs1gJf3bWgpcpVQ++t3W\nnvBZt1+YZz+edAeVR8PvKCNbcA6fNuBLYgPWUgk1z74UM+Gc/TQTdp08wmlYJx3wOwyxy5TUOvYv\nswD46Hm2FQzpsj1r0n9sbR69nBxxGG2ebBF7kUEw56DlrNVSiUwevhQzEba/DLVO1kkjVMKvMBb2\nFozLiI7S3NtS69gf/5lBl+1p2mIwLlJuR1F90rLSNnp1syXsj1Y8X91aKqH3l7WYCcfJE0+IDHUE\nHEZ/Gg9hWaE/Y1xBLLWHw7JftI96Hvt3Ad3WgvHChFJYQisxzj7gz1vfCsoLpB1BOWu1VCKTD3op\nZsJxDQyXRIYa/o5tI+d2FDGPvxbvx03JIoeltGvuuXxg6GqeKacdom9wPhFwkh9969SMsKeNri1j\n9sdH3qJqPZrUr24tlcjk6ZZiJgJlQdOzX9WS0/D0+cI7p0EfGb1A18ats62wQ6qFXmC/Fym59ooK\nHlOAW9mtk/LA9N47rC2vHf2jzVXHuIZ9S3VOw2BHnUxfWgUHFOD2ZadTEz/iimQz2tU9uuucCV9I\n6JyGgZ4LL8VhSRopr2h97j1X2PoRJ5hvXAHtrH67bxLD118guo4DqJ8rsYqDbBA6PxuBF2wV3zQt\ns1iyTilTzsetdXJqhVwr5ivqNMcNoLnJA8eJNvkinsqdDlYtI1GRSWxER6UqSDfFFFm0ZBeeb8MO\nuaI20xR6oyn1xMVfT1xST9FRqBdtfezZ1pZ0/RJpbWmvai3v7oBXL1PGHau0uPSy/3L3GN0tHj3M\nVa2uWKderLXnDvf0S53uWh16fEIF/wWkYpOqdtuusAAAAABJRU5ErkJggg==\n", "prompt_number": 28, "text": [ " \u239b 2\u239e \n", " \u239c \u2148\u22c5\u03c9_p\u22c5t 2 -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e \u239f \u2020 \n", "H = \u03ba\u22c5\u239d\u212f \u22c5a + \u212f \u22c5\u239da \u23a0 \u23a0 + \u03c9\u2090\u22c5a \u22c5a" ] } ], "prompt_number": 28 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To eliminate the time-dependence in the interaction term we move to a rotating frame using the unitary transformation:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(I * omega_d * t * Dagger(a) * a)\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": 29, "text": [ " \u2020 \n", " \u2148\u22c5\u03c9_d\u22c5t\u22c5a \u22c5a\n", "\u212f " ] } ], "prompt_number": 29 }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = hamiltonian_transformation(U, H)\n", "\n", "H2" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\kappa \\left(e^{2 i \\omega_{d} t} e^{- i \\omega_{p} t} \\left({{a}^\\dagger}\\right)^{2} + e^{- 2 i \\omega_{d} t} e^{i \\omega_{p} t} \\left({a}\\right)^{2}\\right) + \\omega_{a} {{a}^\\dagger} {a} - \\omega_{d} {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAgwAAAAnBAMAAACRXA3gAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAq3YimRBEZlS7Mond\n782N2PdbAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAIDklEQVRoBc1YDYhcVxX+5ufN35vZ3S5S1GB5\nCakJidZXqIjYsI/YIq0NTjT+tmEmIiq2hWm0Cv6ObYUGa3e1lcaC7YMWpFDdcbOhIkS2CaGlaFkV\n29hFHbRCDGHZGjeWFozn3J/37pt3Z7PNvoEe2HvP+c653ztz3v17C4xI5h9sjoh55LTlTmaPcP1c\nPzOyBNHehDUK46rsSOt+7Vx2bCZTqW9aI9BLf8qOtNEunM+OTTO5N1wPbNbWiPr9YZbEldUs2STX\nEVwBjGW3dK0ZvkWhouZbrSECzAfDfbGn7sX6oMb7Zz4YRMnOB9QMlxOY9VEYQX2NR9Y0vaj5Y4Zn\nLDQMoOYb5hlDJzXX1va8VtK92D+tLFYwJngBU23gshgYpsVpDIuQ+OVp91igMFFz098afvY5qna1\nthzAz/dYLQfc2iW1f2qWRLgVnKZUZoenowk2UIYp/ZJlzTUj9fcZ+oBaWZDAEwp3KcUDrN+KLyko\n3aX2T82SCLWCxylkzEvE2QxOYz1yNB10UwxNNz87gS/SK51g7Jrnb4xdQGkzSl3gDsYKN70cco9/\nihbVf+3pwu3Qwjl9748lZG0rq1YWK2gSuAFZ5UUTsukiDZtjAHvkwVMDCPCHGDmOo6ugEtQWAKrO\n6dhDmttq5qh7UoBjXdEVPdHBcT9OykmgceHCioSsbd2LWYoLUCxW6hoVleXr3HyFm4sfxTINjl1b\nmkebgwGF1yLEDQqfWih0nA4C4HNA8iL03vvwPnwSRRE+7otuStPlQrLXOGVENDCPmKUUQLEYIMUp\n8P1qCDOXeiVqsaSg4Z1IY7g78rw90rRSXtEa17zqVfyy53jAUygtKs+hn5A08SPcjYP4qABvl67o\nWKh2CZiSoKVVDLx/Riy0hhSLAdJYBarlBvSBX9x7iOtNl4eLiEjjIjHsPpyKobmphGve+kw+zLVd\nv7zlBPITH9Mu7p3VcAcux5Hy7m90sdnxGYuWmHh+vcOYEhmgLdnT/qlZcPBdvmaxUed7euhO4G8X\nLrD1Hg0l8tIg95dehvqM5uGa3+9WDl/x7HX4PhZR9KJUOKTS7x367tl7gspOZwG7XYaiKwdyXTKL\nERUZFY+apPD+GbH0GlAsVuqGr8fu1wqmNXZXBA0oIo0BzGa+NQWOBxrimu89gp/29kzgTG0GpclQ\nu7h3NvnVpeLvgYdrfRzYwVC8d+faZOYXGVRiKQPvn5plLHyCjljBYqW+TfNE+yhdHLoKHF6GdjRs\nTSW+Pk3imn9z6LjH7c1bPshdJCfd7VIfdBC6yCejkGKfuqvnnmyiGJJWUncqUo3ZYGEgd7051+Uw\nOHNb9Us2qXnXKWzacZJOqIDDWFr6uboMMwTuarPLTEMkJEHZVqkrvCB1yXkjGSLMOV/+3sPsEdyl\nE/iADFPtrQc/JLSUgwiPb9OhVcqj9Ec8SrnyT6nFh05cBgsDDy89tVX+qNv8Qk/zGdQPEXa7X12h\nKdcnVYh8Y6SqMpTbpLc8asw0ZEICVM1XqXf4nX8kVJxH9IDKa6SyTHE2u7oOFygtFkc5yhpjE8D+\nDm6g9dCksYmDveJJNguD+ZjCKewLNWBQP0uZvyh2m/hWOa6frMpQ5HFuV4yO05CaAFUTcP8r+mtp\nzit13sWn53z2ygX3joPb+HekxeK4diYKqwdU4SaOU8qMFV6NPPFssDAYUci9uOXOyDaoaQ4XV1H3\naI6taH+0jakyfB5YQFn+jDgNqcE5+1uS50N6723UgS8Ty/2ak36sDKt3yucF/WyHuleEammGOkQs\nz4Y/w9FrIZ4N7vLy2b8uLwcUtTYDxonCJlSGxgxaYboMv1le/t3y8ss0yEMtgDiyzDSMhCRxHngc\n6JExrzlJl2FTIZYKPpliUbwuB6TboQ4RSmVwXkdlRcwFSvicQaAXxdoMiCa6MZRVWhStHqbRRGmB\nTCFRrJoNAfJd2tOc3ZvDOI1Y08No7WwHbUD7tpzWnDR9Zd7foY8JsaDFFrNEq1uPSvRDHSIqNwPn\nVVT7d8sx1pNibQY06C15iUdKg7bIVhePVHyU+xKhQ62jNFWGBzAb4B+4JZxtxmnEmh5W9Wv/wZjv\nPEfXHMVJZZB5v40KLuaTKPFVcJIHpqYY6hABfFH4O745QT+FJb5GkKFnw9oMyPexzxejk80v+cO6\n/DSlWAy0Z6qrNFWGY/OnXjnWw3X8uRenEWsq2l3aM71piXhqi5qTPDLsMF1eruS4xgQ1+d1fYz0t\nQx0ilF//3p2PTbblQOstcm0G+h7b1kk/FqDrkzP566vfSdVQ7LSba02V4dP/C99N38en+L+1cRqx\npnidR8/k/+Kh0c3TGpac5BkIq/ZV9KV0tJEZkvi/iJ4Nhv+NqPVoiuyKhk03larKIC3nHB7iz/yL\nScuv3qkJ0rHuYhpbN/LhRGTin3alTsL3Ro1yoEfMaQXHtPYFrYj+pdr59TyrEc7+LDEuYWzoH+o/\nT1DdkbA2aDygxjv9iOiHkZZQDmxXF9IEmjJKk3fJm3HKw4DzXyu8PjDXNeOeM42N6vrfLlU/YtpI\nphHJMGVIjYeFJ3DjVdFB0Uv4NmgU1ETn25+UxLVEg5n139oI0z3GYP3+DChT1e1nSjdARjeKS5eK\nMfj6S6dZ18jGejbCdTHZgvJ9G/omxH4w2pxeGi19ZuzPZMZkJfq2FX3TgeX+aFOqeqPlz4h9V3xy\nZsSYpHEyPe+T3NlZDn16j1auHXGdM8m+OtJzglMshJkkOlqST4yC/v+acgBXN5/2gQAAAABJRU5E\nrkJggg==\n", "prompt_number": 30, "text": [ " \u239b 2 \u239e \n", " \u239c 2\u22c5\u2148\u22c5\u03c9_d\u22c5t -\u2148\u22c5\u03c9_p\u22c5t \u239b \u2020\u239e -2\u22c5\u2148\u22c5\u03c9_d\u22c5t \u2148\u22c5\u03c9_p\u22c5t 2\u239f \u2020 \u2020 \n", "\u03ba\u22c5\u239d\u212f \u22c5\u212f \u22c5\u239da \u23a0 + \u212f \u22c5\u212f \u22c5a \u23a0 + \u03c9\u2090\u22c5a \u22c5a - \u03c9_d\u22c5a \u22c5a" ] } ], "prompt_number": 30 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now consider the case $\\omega_p = 2 \\omega_d$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = H2.subs(omega_d, omega_p/2)\n", "\n", "H3 = collect(H3, Dagger(a) * a)\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\kappa \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\left(\\omega_{a} - \\frac{\\omega_{p}}{2}\\right) {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAATUAAAAnBAMAAACcWMITAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nF89pb3H9cb/mAAAAAElFTkSuQmCC\n", "prompt_number": 31, "text": [ " \u239b 2 \u239e \n", " \u239c\u239b \u2020\u239e 2\u239f \u239b \u03c9_p\u239e \u2020 \n", "\u03ba\u22c5\u239d\u239da \u23a0 + a \u23a0 + \u239c\u03c9\u2090 - \u2500\u2500\u2500\u239f\u22c5a \u22c5a\n", " \u239d 2 \u23a0 " ] } ], "prompt_number": 31 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Introduce $\\Delta_a = \\omega_a - \\omega_p / 2$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = H3.subs(omega_a, Delta_a + omega_p/2)\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\Delta_{a} {{a}^\\dagger} {a} + \\kappa \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 32, "text": [ " \u239b 2 \u239e\n", " \u2020 \u239c\u239b \u2020\u239e 2\u239f\n", "\u0394\u2090\u22c5a \u22c5a + \u03ba\u22c5\u239d\u239da \u23a0 + a \u23a0" ] } ], "prompt_number": 32 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can diagonalize this Hamiltonian by introducing the squeezing transformation:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "chi = symbols(\"chi\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 33 }, { "cell_type": "code", "collapsed": false, "input": [ "U = exp(chi/2 * a **2 - chi/2 * Dagger(a) ** 2)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{- \\frac{\\chi \\left({{a}^\\dagger}\\right)^{2}}{2} + \\frac{\\chi \\left({a}\\right)^{2}}{2}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAIEAAAAkBAMAAABcX0Q+AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 34, "text": [ " 2 \n", " \u239b \u2020\u239e 2\n", " \u03c7\u22c5\u239da \u23a0 \u03c7\u22c5a \n", " - \u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\n", " 2 2 \n", "\u212f " ] } ], "prompt_number": 34 }, { "cell_type": "markdown", "metadata": {}, "source": [ "which tranforms the operators $a$ and $a^\\dagger$ according to the well-known relations:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "hamiltonian_transformation(U, a)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\sinh{\\left (\\chi \\right )} {{a}^\\dagger} + \\cosh{\\left (\\chi \\right )} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 35, "text": [ " \u2020 \n", "sinh(\u03c7)\u22c5a + cosh(\u03c7)\u22c5a" ] } ], "prompt_number": 35 }, { "cell_type": "code", "collapsed": false, "input": [ "hamiltonian_transformation(U, Dagger(a))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\sinh{\\left (\\chi \\right )} {a} + \\cosh{\\left (\\chi \\right )} {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 36, "text": [ " \u2020\n", "sinh(\u03c7)\u22c5a + cosh(\u03c7)\u22c5a " ] } ], "prompt_number": 36 }, { "cell_type": "markdown", "metadata": {}, "source": [ "and in this frame the Hamiltonian takes the form:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = hamiltonian_transformation(U, H4)\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\Delta_{a} \\left(\\sinh{\\left (\\chi \\right )} {a} + \\cosh{\\left (\\chi \\right )} {{a}^\\dagger}\\right) \\left(\\sinh{\\left (\\chi \\right )} {{a}^\\dagger} + \\cosh{\\left (\\chi \\right )} {a}\\right) + \\kappa \\left(\\left(\\sinh{\\left (\\chi \\right )} {{a}^\\dagger} + \\cosh{\\left (\\chi \\right )} {a}\\right)^{2} + \\left(\\sinh{\\left (\\chi \\right )} {a} + \\cosh{\\left (\\chi \\right )} {{a}^\\dagger}\\right)^{2}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 37, "text": [ " \u239b \n", " \u239b \u2020\u239e \u239b \u2020 \u239e \u239c\u239b \u2020 \n", "\u0394\u2090\u22c5\u239dsinh(\u03c7)\u22c5a + cosh(\u03c7)\u22c5a \u23a0\u22c5\u239dsinh(\u03c7)\u22c5a + cosh(\u03c7)\u22c5a\u23a0 + \u03ba\u22c5\u239d\u239dsinh(\u03c7)\u22c5a + cosh(\u03c7\n", "\n", " 2 2\u239e\n", " \u239e \u239b \u2020\u239e \u239f\n", ")\u22c5a\u23a0 + \u239dsinh(\u03c7)\u22c5a + cosh(\u03c7)\u22c5a \u23a0 \u23a0" ] } ], "prompt_number": 37 }, { "cell_type": "code", "collapsed": false, "input": [ "H6 = normal_ordered_form(H5.expand(), independent=True)\n", "\n", "H6 = drop_c_number_terms(H6)\n", "\n", "H6 = collect(H6, [Dagger(a) * a, Dagger(a)**2, a**2])\n", "\n", "H6" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(\\Delta_{a} \\sinh^{2}{\\left (\\chi \\right )} + \\Delta_{a} \\cosh^{2}{\\left (\\chi \\right )} + 4 \\kappa \\sinh{\\left (\\chi \\right )} \\cosh{\\left (\\chi \\right )}\\right) {{a}^\\dagger} {a} + \\left(\\Delta_{a} \\sinh{\\left (\\chi \\right )} \\cosh{\\left (\\chi \\right )} + \\kappa \\sinh^{2}{\\left (\\chi \\right )} + \\kappa \\cosh^{2}{\\left (\\chi \\right )}\\right) \\left({{a}^\\dagger}\\right)^{2} + \\left(\\Delta_{a} \\sinh{\\left (\\chi \\right )} \\cosh{\\left (\\chi \\right )} + \\kappa \\sinh^{2}{\\left (\\chi \\right )} + \\kappa \\cosh^{2}{\\left (\\chi \\right )}\\right) \\left({a}\\right)^{2}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 38, "text": [ " \n", "\u239b 2 2 \u239e \u2020 \u239b \n", "\u239d\u0394\u2090\u22c5sinh (\u03c7) + \u0394\u2090\u22c5cosh (\u03c7) + 4\u22c5\u03ba\u22c5sinh(\u03c7)\u22c5cosh(\u03c7)\u23a0\u22c5a \u22c5a + \u239d\u0394\u2090\u22c5sinh(\u03c7)\u22c5cosh(\u03c7) +\n", "\n", " 2 \n", " 2 2 \u239e \u239b \u2020\u239e \u239b 2 2 \n", " \u03ba\u22c5sinh (\u03c7) + \u03ba\u22c5cosh (\u03c7)\u23a0\u22c5\u239da \u23a0 + \u239d\u0394\u2090\u22c5sinh(\u03c7)\u22c5cosh(\u03c7) + \u03ba\u22c5sinh (\u03c7) + \u03ba\u22c5cosh (\u03c7\n", "\n", " \n", " \u239e 2\n", ")\u23a0\u22c5a " ] } ], "prompt_number": 38 }, { "cell_type": "code", "collapsed": false, "input": [ "H7 = collect(H6, H6.args[1].args[0])\n", "\n", "H7" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(\\Delta_{a} \\sinh^{2}{\\left (\\chi \\right )} + \\Delta_{a} \\cosh^{2}{\\left (\\chi \\right )} + 4 \\kappa \\sinh{\\left (\\chi \\right )} \\cosh{\\left (\\chi \\right )}\\right) {{a}^\\dagger} {a} + \\left(\\Delta_{a} \\sinh{\\left (\\chi \\right )} \\cosh{\\left (\\chi \\right )} + \\kappa \\sinh^{2}{\\left (\\chi \\right )} + \\kappa \\cosh^{2}{\\left (\\chi \\right )}\\right) \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 39, "text": [ " \n", "\u239b 2 2 \u239e \u2020 \u239b \n", "\u239d\u0394\u2090\u22c5sinh (\u03c7) + \u0394\u2090\u22c5cosh (\u03c7) + 4\u22c5\u03ba\u22c5sinh(\u03c7)\u22c5cosh(\u03c7)\u23a0\u22c5a \u22c5a + \u239d\u0394\u2090\u22c5sinh(\u03c7)\u22c5cosh(\u03c7) +\n", "\n", " \u239b 2 \u239e\n", " 2 2 \u239e \u239c\u239b \u2020\u239e 2\u239f\n", " \u03ba\u22c5sinh (\u03c7) + \u03ba\u22c5cosh (\u03c7)\u23a0\u22c5\u239d\u239da \u23a0 + a \u23a0" ] } ], "prompt_number": 39 }, { "cell_type": "code", "collapsed": false, "input": [ "# Trick to simplify the coefficients for the quantum operators\n", "H8 = Add(*(simplify(arg.args[0]) * Mul(*(arg.args[1:])) for arg in H7.args))\n", "\n", "H8" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(\\frac{\\Delta_{a}}{2} \\sinh{\\left (2 \\chi \\right )} + \\kappa \\cosh{\\left (2 \\chi \\right )}\\right) \\left(\\left({{a}^\\dagger}\\right)^{2} + \\left({a}\\right)^{2}\\right) + \\left(\\Delta_{a} \\cosh{\\left (2 \\chi \\right )} + 2 \\kappa \\sinh{\\left (2 \\chi \\right )}\\right) {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 40, "text": [ " \u239b 2 \u239e \n", "\u239b\u0394\u2090\u22c5sinh(2\u22c5\u03c7) \u239e \u239c\u239b \u2020\u239e 2\u239f \u2020 \n", "\u239c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u03ba\u22c5cosh(2\u22c5\u03c7)\u239f\u22c5\u239d\u239da \u23a0 + a \u23a0 + (\u0394\u2090\u22c5cosh(2\u22c5\u03c7) + 2\u22c5\u03ba\u22c5sinh(2\u22c5\u03c7))\u22c5a \u22c5\n", "\u239d 2 \u23a0 \n", "\n", " \n", " \n", "a\n", " " ] } ], "prompt_number": 40 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now if we choose $\\chi$ such that the coefficient of $a^2 + {a^\\dagger}^2$ is zero:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "chi_eq = H8.args[0].args[0]" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 41 }, { "cell_type": "code", "collapsed": false, "input": [ "Eq(chi_eq, 0)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\frac{\\Delta_{a}}{2} \\sinh{\\left (2 \\chi \\right )} + \\kappa \\cosh{\\left (2 \\chi \\right )} = 0$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 42, "text": [ "\u0394\u2090\u22c5sinh(2\u22c5\u03c7) \n", "\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u03ba\u22c5cosh(2\u22c5\u03c7) = 0\n", " 2 " ] } ], "prompt_number": 42 }, { "cell_type": "code", "collapsed": false, "input": [ "chi_sol = simplify(solve(chi_eq, chi)[3])\n", "\n", "Eq(chi, chi_sol)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\chi = \\log{\\left (\\sqrt[4]{\\frac{\\Delta_{a} - 2 \\kappa}{\\Delta_{a} + 2 \\kappa}} \\right )}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 43, "text": [ " \u239b __________\u239e\n", " \u239c \u2571 \u0394\u2090 - 2\u22c5\u03ba \u239f\n", "\u03c7 = log\u239c4 \u2571 \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 \u239f\n", " \u239d\u2572\u2571 \u0394\u2090 + 2\u22c5\u03ba \u23a0" ] } ], "prompt_number": 43 }, { "cell_type": "markdown", "metadata": {}, "source": [ "we obtain a diagonal Hamiltonian:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H9 = H8.subs(chi_eq.args[0], -chi_eq.args[1])\n", "\n", "H9" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(\\Delta_{a} \\cosh{\\left (2 \\chi \\right )} + 2 \\kappa \\sinh{\\left (2 \\chi \\right )}\\right) {{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 44, "text": [ " \u2020 \n", "(\u0394\u2090\u22c5cosh(2\u22c5\u03c7) + 2\u22c5\u03ba\u22c5sinh(2\u22c5\u03c7))\u22c5a \u22c5a" ] } ], "prompt_number": 44 }, { "cell_type": "markdown", "metadata": {}, "source": [ "with frequency:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "Eq(Symbol(\"omega\"), H9.args[0])" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\omega = \\Delta_{a} \\cosh{\\left (2 \\chi \\right )} + 2 \\kappa \\sinh{\\left (2 \\chi \\right )}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 45, "text": [ "\u03c9 = \u0394\u2090\u22c5cosh(2\u22c5\u03c7) + 2\u22c5\u03ba\u22c5sinh(2\u22c5\u03c7)" ] } ], "prompt_number": 45 }, { "cell_type": "heading", "level": 3, "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 16:01:18 2014 JST
" ], "json": [ "{\"Software versions\": [{\"module\": \"Python\", \"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\": \"IPython\", \"version\": \"2.3.0\"}, {\"module\": \"OS\", \"version\": \"Darwin 13.4.0 x86_64 i386 64bit\"}, {\"module\": \"sympy\", \"version\": \"0.7.5-git\"}, {\"module\": \"sympsi\", \"version\": \"0.1.0.dev-0c6e514\"}]}" ], "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 16:01:18 2014 JST} \\\\ \\hline\n", "\\end{tabular}\n" ], "metadata": {}, "output_type": "pyout", "prompt_number": 46, "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 16:01:18 2014 JST" ] } ], "prompt_number": 46 } ], "metadata": {} } ] }