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{}, "outputs": [], "prompt_number": 3 }, { "cell_type": "code", "collapsed": false, "input": [ "a = BosonOp(\"a\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 4 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Transformations with $U = \\exp(i \\omega_0 a^\\dagger a t)$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "A = I * omega_0 * Dagger(a) * a * t\n", "U = exp(A)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\omega_{0} t {{a}^\\dagger} {a}}$$" ], "metadata": {}, "output_type": "pyout", "png": 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a$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "e = bch_expansion(A, Dagger(a) * a, n).doit()\n", "\n", "normal_ordered_form(e)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$${{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAATBAMAAACaQstxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjElEQVQYGWNggAANKA2nMATM4VIQxro1c1FF\nCswLUAUYVNH4DGsJCuiiq0ByB6OyazoDw1GgCgiLoVOA6wMDgyFQAMJin83As4CBwZ2BAcri+c7A\n5sDAAHQXlMW5gKE+AKiegQHKqjdguA+SZ2CAsuovMKzjFQALQFh8DtxJzCA+A5TFrmIq5AEWgLAA\nNNUfGy2S/ZoAAAAASUVORK5CYII=\n", "prompt_number": 6, "text": [ " \u2020 \n", "a \u22c5a" ] } ], "prompt_number": 6 }, { "cell_type": "code", "collapsed": false, "input": [ "unitary_transformation_auto(U, Dagger(a) * a)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$${{a}^\\dagger} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAB8AAAATBAMAAACaQstxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIquJdjLdEETvu2aZ\nVM0GsGrEAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAjElEQVQYGWNggAANKA2nMATM4VIQxro1c1FF\nCswLUAUYVNH4DGsJCuiiq0ByB6OyazoDw1GgCgiLoVOA6wMDgyFQAMJin83As4CBwZ2BAcri+c7A\n5sDAAHQXlMW5gKE+AKiegQHKqjdguA+SZ2CAsuovMKzjFQALQFh8DtxJzCA+A5TFrmIq5AEWgLAA\nNNUfGy2S/ZoAAAAASUVORK5CYII=\n", "prompt_number": 7, "text": [ " \u2020 \n", "a \u22c5a" ] } ], "prompt_number": 7 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "$U a U^\\dagger = a e^{-i\\omega_0t}$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "unitary_transformation_auto(U, a)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{- i \\omega_{0} t} {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"$$\\left(\\frac{i \\omega_{0}^{7}}{5040} t^{7} - \\frac{\\omega_{0}^{6} t^{6}}{720} - \\frac{i \\omega_{0}^{5}}{120} t^{5} + \\frac{\\omega_{0}^{4} t^{4}}{24} + \\frac{i \\omega_{0}^{3}}{6} t^{3} - \\frac{\\omega_{0}^{2} t^{2}}{2} - i \\omega_{0} t + 1\\right) {a}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"$U a^\\dagger U^\\dagger = a^\\dagger e^{i\\omega_0t}$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "unitary_transformation_auto(U, Dagger(a))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{i \\omega_{0} t} {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAADwAAAAUBAMAAAAjLUMaAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOklEQVQoFXWQPUgDMRiG3/R+2lp7d2DdxLsi\nXcThxOJQHTqLwwkOgu66OosgbsUOHZQOgmbrWloo6KQ4unQTt4IuDgU7uJvk8nkn1gzJ871Pvtwl\ngBpGPV7/mWeiRBQ58YAgvfoeVVN1myzKmuwN2C3gWZa7p1s6Hb83Ysr4XkFQh62Ihldt4ZX1Z3ba\nOMACTMdbBL5I457oDVd4wpyLnGefUYgPTWzCaxigsocsN7rz5Ennh8HN9ctlXWozDEjfUXcvcs7N\nC6jD+5y0vDdbqv48mfo1Ka1e7RiQNzyMrEAmcrDtSgxHkfMJiMJqoMTjLJlZE+YIWAUKzbXHJNdk\nTpANAfEqbvePBHIj+FzlbjBF+wFuZa/YKLQ45/fwWxjnI5kZQ5QUpDcUw9mTTBxUNx/SRjHrL++v\nS/oGEZo3m34usEEAAAAASUVORK5CYII=\n", "prompt_number": 11, "text": [ " \u2148\u22c5\u03c9\u2080\u22c5t \u2020\n", "\u212f \u22c5a " ] } ], "prompt_number": 11 }, { "cell_type": "code", "collapsed": false, "input": [ "e = bch_expansion(A, Dagger(a), n).doit()\n", "\n", "e = normal_ordered_form(expand(e))\n", "\n", "e = collect(e, Dagger(a))\n", "\n", "e" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(- \\frac{i \\omega_{0}^{7}}{5040} t^{7} - \\frac{\\omega_{0}^{6} t^{6}}{720} + \\frac{i \\omega_{0}^{5}}{120} t^{5} + \\frac{\\omega_{0}^{4} t^{4}}{24} - \\frac{i \\omega_{0}^{3}}{6} t^{3} - \\frac{\\omega_{0}^{2} t^{2}}{2} + i \\omega_{0} t + 1\\right) {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 12, "text": [ "\u239b 7 7 6 6 5 5 4 4 3 3 2 2 \u239e \n", "\u239c \u2148\u22c5\u03c9\u2080 \u22c5t \u03c9\u2080 \u22c5t \u2148\u22c5\u03c9\u2080 \u22c5t \u03c9\u2080 \u22c5t \u2148\u22c5\u03c9\u2080 \u22c5t \u03c9\u2080 \u22c5t \u239f \u2020\n", "\u239c- \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500 - \u2500\u2500\u2500\u2500\u2500\u2500 + \u2148\u22c5\u03c9\u2080\u22c5t + 1\u239f\u22c5a \n", "\u239d 5040 720 120 24 6 2 \u23a0 " ] } ], "prompt_number": 12 }, { "cell_type": "code", "collapsed": false, "input": [ "e == (exp(I * omega_0 * t).series(t, 0, n=n).removeO()) * Dagger(a)" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "pyout", "prompt_number": 13, "text": [ "True" ] } ], "prompt_number": 13 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Transformations with $U = D(\\alpha) = e^{\\alpha a^\\dagger - \\alpha^* a}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "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": 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{a}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEEAAAAOBAMAAACC12zPAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMomZdkQi71S7\nZqsQsTlgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAxElEQVQYGWNgwAIksYihCm1H5WLh4VDBmp4B\nUwxTwfm0xBAmxsDAacEgNWE2hA9T0TWB5wNCBVcAA09CIYoKxrUMbAcYGISMQUCFIV+AgU0hAEUF\n20cG5gKICIhcw8DAulQAyAhSUrJSUlIDsrgOMMg7AGkIYPwKVPEbyoG6Qz6AYT8DSBME/AWq+ABl\nw1RsYDjEPQGmgCGPgeHaB0cIF6qCqYDXjhWugIE19i5DagGKCsbnIfMqECqQWbDwQBZDZfcicwGO\ndid+zGJgXgAAAABJRU5ErkJggg==\n", "prompt_number": 15, "text": [ "-\u03b1 + a" ] } ], "prompt_number": 15 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "$U a^\\dagger U^\\dagger = a^\\dagger - \\alpha^*$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "unitary_transformation_auto(U, Dagger(a))" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\overline{\\alpha} + {{a}^\\dagger}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEgAAAAVBAMAAAAX07VdAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEM3dMomZdkQi71S7\nZqsQsTlgAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA/klEQVQoFWNgIAheEFQBVECUokAiTDp0bDVh\nVQKBAoQVMTyBq+H/DwYf4AIIxmkEE5nFOAGZh0MRdwGyotfIHAQbVRFyOLGmZ8CUwRRxPi0xZGBI\ngQkzMHBaMEhNmA3hwxR1TeAB+iQSoYgrgIEnoRBFEeNaBrYDDAxAQSFjEFBhyBdgYFMIQFHE9pGB\nGegFAYggiFzDwMC6FMRnVVJSXqSklMDAwHWAQd4BKAIHjF+B0r+hXKib5AMY9iObw8DwF6joA5qi\nDQyHuCfAzQEy8hgYrn1whIhATWIq4LVjRVbDwBp7lyG1AEUR4/OQeRUoipA5sHBCFsNgc1xAFQIA\nUyQ3bsv+LmkAAAAASUVORK5CYII=\n", "prompt_number": 16, "text": [ " _ \u2020\n", "- \u03b1 + a " ] } ], "prompt_number": 16 }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "$U a^\\dagger a U^\\dagger = a^\\dagger a - \\alpha a^\\dagger - \\alpha^* a + |\\alpha|^2$" ] }, { "cell_type": "code", "collapsed": false, "input": [ "unitary_transformation_auto(U, Dagger(a) * a)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\left(- \\overline{\\alpha} + {{a}^\\dagger}\\right) \\left(- \\alpha + {a}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAAbBAMAAAAXJakbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nA4YInHLT+PvZAAAAAElFTkSuQmCC\n", "prompt_number": 17, "text": [ "\u239b _ \u2020\u239e \n", "\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a)" ] } ], "prompt_number": 17 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Optomechanical system" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Consider the standard Hamiltonian for an optomechanical syste, including a classical driving signal on the optical mode:\n", "\n", "$$\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", "$$\n", "\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "n = 6\n", "omega_a, omega_b, g, A, Delta = symbols(\"omega_a, omega_b, g, A, Delta\", positive=True)\n", "Hsym, omega_d = symbols(\"H, omega_d\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 33 }, { "cell_type": "code", "collapsed": false, "input": [ "a, b = BosonOp(\"a\"), BosonOp(\"b\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 34 }, { "cell_type": "code", "collapsed": false, "input": [ "H = omega_a * Dagger(a) * a + omega_b * Dagger(b) * b + g * Dagger(a) * a * (b + Dagger(b)) \\\n", " + (A * exp(-I * omega_d * t) + conjugate(A) * exp(I * omega_d * t)) * (a + Dagger(a))\n", "\n", "Eq(Hsym, H)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$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": "iVBORw0KGgoAAAANSUhEUgAAAkwAAAAcBAMAAABhW/0uAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZqvNmRDdRHYyiVS7\nIu/EmopNAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAH+ElEQVRoBeVXa4hc5Rl+ZnbOzn13FLT+aHGy\n/uiqoEs2kUoVp1Ql2hqHEC1qdBY1ooh2lFqLheaYbjUa0EHFX7U7pUZi62VAKSLKjrdfXnYQf6mY\nFcFLohIT18TEkL7vd//mnDl70T/SF7Ln/Z73+Z73cr5z5gT4Mdv0MotfLn958j+PoaebMeASoKXt\ni8sYI/6Igw21nMUAdzH+0oobIA6v6LycT6xiOEjB4rH7bFh5XkYbVbkt4HiNurNYgkv8UNO0bnxx\nhqbpA647XPzvclGIq+pWlxjvx+6LUL2MNqpyW8Dx/uv4S3GJb8rVuvHFGVqy7Cuv7nEIa4VfOmt9\nxwGVW6hEMR+J3+dzAD+jjcrcZv2nKv4MpKoMvHPNBXxJtH6+KVfpDihO0o49uqbO/wZmqO9wYpkQ\nmz4GgsIlPj94a4qAzT4YXUX29VOEDmeUgl44E4plfkKhOxZQBfJTwPnALxXoXh50F0CEr8rNhJI3\nqDhJ20+kbyQx/u/JDryLRraa1umWA7I7VKU/i/+URPbxXteEDmcUjhsB5+bAXomWr5gqV4IKasAf\ngCMS9P6e462ifFWu0o1pSu4XtOxBuiNfe4J9i9ed9aPkH6Z/pY4Cr1TXUQZ2qQU0qtfmavYJJOiZ\ngHaEDmcUDqNGi3OTHb8gryiFxV4qDELgn8jOQ5ojmfp9XWLFN8hei/JVuUrXacoRIQVBK3wL5Ljz\ngeaOiV5TWS7TtPuc2raFSxquqJVG1dJezD4BFUMbUZ7Q4YzCYdBoUW6y8rP64DQuy7XSzUIvte0m\n5Kpqmo7kexf7dyHCV+VKXZI2xTkiBAsan+GSvkFcR8ROtUieiIUpWqc7CtRNPMbrTLcPVUt7MfsE\n5JcjIKHDGYXDkM7AuclyuQPiCjxTKL5+0o3n4nnMIxO2JepIhvc3FVNeInxZrtIljinOESFY0Ea7\ndMCnaOVbcef2Lsqn3EPvL/6KeXvnw3W6pObp3TA+3YH5xNBNrN72G2qBwsI02qXVB6pYISH2KTFA\nl1MiWvkTsVPocEbhMKS1ODfZbcWv6O/l2y7ChifxVnt9FWfmu8iOt0TUSNKpr89W6Mdm53RPRqJ8\nWa7QtcUxWdcl+xe0LavHxuZqQulfk2xrhP8EdtXwl15pL3ABpfwCcwxnJoDGWnpfZlqCZZoI9tdH\ne/J55IBqLdUkvxEyoiR4nxaz5fyE4sFBZkkdyigdhvSYODfhzTwRszfhXV4J21zYrl0rCfwOMyHw\nx165baPCM3zx+hA9yZJMU3pMsn9Be5oKv7/Tp4TiAr1Dgw/FgXsSuKqC3zKl1AVm6ziEQk/tUE0w\nv408veeEKTTDi4IUlxK8T4vZnmrM+092khChQxmlw7geE+emAaFMvzcfdAIapbI7TviVdq0k0OQb\nXd6DjS0bFZ7hy3JZ1xYnGGpMqn9B41/NubqIOn9GupjpZRYwHAJ/Ay6u43qODlWBXyM4hJzeoQcy\ngUYbZf3aUOjtwBRSNBkyKcH7tJjpKdvEMPAz0E8OHRnSoYzS4Y16TJwbeGBs7DvgtBPu1AUw6Jg+\nCMiPjZ3eRfrDbcc5Ud+V5bKuLU4wlIjqX9AoKfjTybeZCu4CzarRAqigzxGIgzJcA9YgexiBoNNT\nerN8Sodq2NJBWfxiWjREvoaCVJYSvE+JFSYnz/hocpIUc8BDQJs/EaUOZZQOrBY4N58RfATsYzdq\nVpKPHUYmMFqNsgwiy2VdWxydflOX6p9p4rPpkNz52HVsV/PixB5e5fs6y0Pi41Pcyx3y5L8CTdmY\nutejFb4j/aephlxHvUSMhBUzpykDbEe5zZ+IUofUjeOfpiKFbgY99YPMnKYO/ZLP85tgsJnTZIuT\nZCWi+mea+GzSz4pVnOWvyEYHrxR7BAaHUZp4kRwe0wF+ko3pMTXpQUaeGhWm0DcxU8PZCNZtbRkJ\nK2bGVOrR9+1QL3twa31U6JCGcfwxvUchek3sFkdQ5vL/6jGVaUz0rTPSBkKfYVeyXOrJFieDekyy\nf6YN7439bGr00kcwFKaulo/ML3B8tU0S6S6wis6ZNTWQdKVUo6Opj5lCn3p0z76n2ri0NVOHloDj\nqXIKu9fPnrIbuc25mtQheeOYMXHuYIpCcz38FcFF5MWYHtOlFCsdQW4CG/lGx5osl3VtcYKoRFT/\nRNtwzr4wM/dd2C+THf/3FILxx9++W0Q23PvIeJO83DywcWvFYauBBOtWEag+bcxrd9P+1n2fAufi\nYUBLuJ4qJ5g7M/dlSP+ZSE1IHZIyjhkT5Q7m9tfxwtFrkVt3jFOD6yrJTUfXojj3TQ3Td7rVukxd\nLvdkixMMXZfs33Tl75YrPh0RM8fFRNSY5DrTVbiHYg+2Gr7nqHIUNtLMVL2wXGitaO4YsvkyjIv1\nY7LcGF2/LtNV/36ghQafnojJT2UHvsXx6SlVKw8NjuClS1ya8bPenR6tj3RMyDpGK5LbcqznS1o8\nzlPlRnV9EdNVRKOwgJcjIAMXxqIabNS1510/yx/w5uEF7WK4/lO7iPGSc8dsWAxS5S6mO6ArUi+u\nmo67scA/ElMfGx+9dfv0UsaUv7Adv1+hybkTt8YHVbmL6Q7oKl5Toun46aktNyRt/d6x5NwrkFfl\nLqa7gq6CiYRyUu2E4PcPJeZegbwudxFdTVtWhmcT2O8nxH6IUFLuFeibcpN1DW05KYoJT915yxFa\nATcp9wrkTLnJuoa2ghT/X1v+B1saQvdT5NZYAAAAAElFTkSuQmCC\n", "prompt_number": 35, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \u239b \u2148\u22c5\u03c9_d\u22c5t -\u2148\u22c5\u03c9_d\u22c5t\u239e \u239b \u2020 \u239e\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 + a\u23a0" ] } ], "prompt_number": 35 }, { "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}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEMAAAAWBAMAAABppzwEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABOUlEQVQoFYWQP0vDQByG35j0TFPaBoybkIPi\n5lBoQLAON6vDCY7uLg6dBUHcRIeASgdBbnPTYqGDU0pHl27iVvATdHD3ck1yTWnsDfk975+7JAcU\nrklhkgU0IwUmy2up2odRznO4llUxY6atRfLdmdNZDLTuJpivkDZICHzG4cnlAbzWBzCRQkHsYs13\nK3K8AFYH3zAunCnAkEDcwHEXp9iCBRCGX1jMkW+hcoMCVcEPHjHChjwvIlewQ7MHCCSgKsZU7GGA\nbYx2udnb9Hlt6MZBBkB5TJ+fvh5YmdqwmtQWb+dqawbys9557dq6RVW8gvQF6d8fqUoCRiPgSsvH\nutsIU56bZ7xEU0l2gihlPUs38ISWy6hy1xou8+e8uryBFatOVxTkBcpK8/+WOYaX/XRBNdiPCpLE\n/gNa6jyqkqIGvwAAAABJRU5ErkJggg==\n", "prompt_number": 36, "text": [ " \u2020 \n", " \u2148\u22c5\u03c9_d\u22c5t\u22c5a \u22c5a\n", "\u212f " ] } ], "prompt_number": 36 }, { "cell_type": "code", "collapsed": false, "input": [ "unitary_transformation_auto(U, H, independent=True)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$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(e^{i \\omega_{d} t} {{a}^\\dagger} + e^{- i \\omega_{d} t} {a}\\right)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 37, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \u239b \u2148\u22c5\u03c9_d\u22c5t -\u2148\u22c5\u03c9_d\u22c5t\u239e \u239b \u2148\u22c5\u03c9_d\u22c5t \u2020\n", "g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b + \u239dA\u22c5\u212f + A\u22c5\u212f \u23a0\u22c5\u239d\u212f \u22c5a \n", "\n", " -\u2148\u22c5\u03c9_d\u22c5t \u239e\n", " + \u212f \u22c5a\u23a0" ] } ], "prompt_number": 37 }, { "cell_type": "code", "collapsed": false, "input": [ "H1 = hamiltonian_transformation_auto(U, H, independent=True)\n", "\n", "H1" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$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)$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 38, "text": [ " \u2020 \u239b \u2020 \u239e \u2020 \u2020 \u2020 \u239b \u2148\u22c5\u03c9_d\u22c5t -\u2148\u22c5\u03c9_d\u22c5t\u239e \u239b\n", "g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b - \u03c9_d\u22c5a \u22c5a + \u239dA\u22c5\u212f + A\u22c5\u212f \u23a0\u22c5\u239d\n", "\n", " \u2148\u22c5\u03c9_d\u22c5t \u2020 -\u2148\u22c5\u03c9_d\u22c5t \u239e\n", "\u212f \u22c5a + \u212f \u22c5a\u23a0" ] } ], "prompt_number": 38 }, { "cell_type": "markdown", "metadata": {}, "source": [ "We can now perform a rotating-wave approximation (RWA) by eliminating all terms that rotate with frequencies $2\\omega_d$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H2 = drop_terms_containing(H1.expand(), [exp(-2*I*omega_d*t), exp(2*I*omega_d*t)])\n", "\n", "Eq(Symbol(\"H_{rwa}\"), H2)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$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}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 39, "text": [ " \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \n", "H_{rwa} = A\u22c5a + A\u22c5a + g\u22c5a \u22c5a\u22c5b + g\u22c5a \u22c5a\u22c5b + \u03c9\u2090\u22c5a \u22c5a + \u03c9_b\u22c5b \u22c5b - \u03c9_d\u22c5a \u22c5a" ] } ], "prompt_number": 39 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now we introduce the detuning $\\Delta = \\omega_a - \\omega_d$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H3 = H2.subs(omega_a, Delta + omega_d).expand()\n", "\n", "H3" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$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}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAYkAAAAXBAMAAAAM6K4JAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n", "prompt_number": 40, "text": [ " \u2020 \u2020 \u2020 \u2020 \u2020 \u2020 \n", "A\u22c5a + A\u22c5a + \u0394\u22c5a \u22c5a + g\u22c5a \u22c5a\u22c5b + g\u22c5a \u22c5a\u22c5b + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 40 }, { "cell_type": "markdown", "metadata": {}, "source": [ "To eliminate the coherent part of the state of the cavity mode we apply the unitary displacement operator $U = e^{\\alpha a^\\dagger - \\alpha^*a}$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "alpha = symbols(\"alpha\")" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 41 }, { "cell_type": "code", "collapsed": false, "input": [ "H = Dagger(a) * alpha - conjugate(alpha) * a\n", "U = exp(H)\n", "\n", "U" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$e^{\\alpha {{a}^\\dagger} - \\overline{\\alpha} {a}}$$" ], "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAEYAAAAWBAMAAACPjvdAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEIl2mSJE3e9UMqtm\nzbsXyEShAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDElEQVQoFWNgQAN8aHxsXCZsgqhizFv2oApg\n47FgE0QTY4fx+f+DAIyHQsPVoIiCOExGDowMjM7WCxiQ/KWswiAAFoMoL2PYzMQgvmC9AAPCPewJ\n3AYKYDGwGpYHDGfTGFwYlgNNhGgCkn4MPNcSwGJgIY4JDBkODJ0M1gwMXAwMnO+A4CGQy9YCEQOr\n4Q9gWC/A+JXhjSDcFAaGrwxcH5HEeBMY9BkZuri+HUBSc42B6YMAQozV+QyLJkO2pQmymhCbA04T\nIGKMyiYBSLqxMtMDWBWwSiAEWTsZRBYguFhZ3D3GB7FKIAnyb0Di4GDyK+CQQBLmAKoxQOJjYzJf\nYBAh6HcTmwPY9CKJAQDwgzkp4qt/twAAAABJRU5ErkJggg==\n", "prompt_number": 42, "text": [ " \u2020 _ \n", " \u03b1\u22c5a - \u03b1\u22c5a\n", "\u212f " ] } ], "prompt_number": 42 }, { "cell_type": "code", "collapsed": false, "input": [ "H4 = hamiltonian_transformation_auto(U, H3, independent=True)\n", "\n", "H4" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$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}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 43, "text": [ " \u239b _ \u2020\u239e \u239b _ \u2020\u239e \u239b _ \u2020\u239e \u2020 \n", "A\u22c5(-\u03b1 + a) + A\u22c5\u239d- \u03b1 + a \u23a0 + \u0394\u22c5\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a) + g\u22c5\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a)\u22c5b +\n", "\n", " \u239b _ \u2020\u239e \u2020 \n", " g\u22c5\u239d- \u03b1 + a \u23a0\u22c5(-\u03b1 + a)\u22c5b + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 43 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now want to cancel out the drivng terms so we set $A - \\Delta \\alpha = 0$, i.e. $\\alpha = A/\\Delta$:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H5 = H4.expand().subs({A: alpha * Delta, conjugate(alpha): alpha}) \n", "\n", "H5 = collect(H5, [g * Dagger(a) * a, - alpha * g])\n", "\n", "H5" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$- \\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}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 44, "text": [ " 2 \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u2020 \u2020 \u2020 \u239e \u2020 \u239b\n", "- \u0394\u22c5\u03b1 + \u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239db + b\u23a0 + \u03b1\u22c5g\u22c5\u239d- a \u22c5b - a \u22c5b - a\u22c5b - a\u22c5b\u23a0 + g\u22c5a \u22c5a\u22c5\u239d\n", "\n", " \u2020 \u239e \u2020 \n", "b + b\u23a0 + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 44 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Drop C-numbers from the hamiltonian:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H6 = drop_c_number_terms(H5)\n", "\n", "H6" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$\\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}$$" ], "metadata": {}, "output_type": "pyout", "png": 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"prompt_number": 45, "text": [ " \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u2020 \u2020 \u2020 \u239e \u2020 \u239b \u2020 \u239e \n", "\u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239db + b\u23a0 + \u03b1\u22c5g\u22c5\u239d- a \u22c5b - a \u22c5b - a\u22c5b - a\u22c5b\u23a0 + g\u22c5a \u22c5a\u22c5\u239db + b\u23a0 +\n", "\n", " \u2020 \n", " \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 45 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Now, if driving strength is large, so that $\\alpha \\gg 1$, we can drop the nonlinear interaction term, and we have an linear effective coupling:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "H7 = drop_terms_containing(H6, [g * Dagger(a) * a]) # XXX not working\n", "\n", "Eq(Hsym, H7)" ], "language": "python", "metadata": {}, "outputs": [ { "latex": [ "$$H = \\Delta {{a}^\\dagger} {a} + \\alpha^{2} g \\left({{b}^\\dagger} + {b}\\right) + \\alpha g \\left(- {{a}^\\dagger} {{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": 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"prompt_number": 46, "text": [ " \u2020 2 \u239b \u2020 \u239e \u239b \u2020 \u2020 \u2020 \u2020 \u239e \u2020 \u239b \u2020 \n", "H = \u0394\u22c5a \u22c5a + \u03b1 \u22c5g\u22c5\u239db + b\u23a0 + \u03b1\u22c5g\u22c5\u239d- a \u22c5b - a \u22c5b - a\u22c5b - a\u22c5b\u23a0 + g\u22c5a \u22c5a\u22c5\u239db + \n", "\n", " \u239e \u2020 \n", "b\u23a0 + \u03c9_b\u22c5b \u22c5b" ] } ], "prompt_number": 46 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Versions" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%reload_ext version_information\n", "\n", "%version_information sympy" ], "language": "python", "metadata": {}, "outputs": [ { "html": [ "
Software | Version |
---|---|
Python | 3.4.0 (default, Apr 11 2014, 13:05:11) [GCC 4.8.2] |
IPython | 3.0.0-dev |
OS | posix [linux] |
sympy | 0.7.5-git |
Tue Jun 10 17:25:19 2014 JST |