diff options
Diffstat (limited to 'mathemathica_fwm')
| -rw-r--r-- | mathemathica_fwm/RbXMDSSetup1.nb | 2972 |
1 files changed, 2972 insertions, 0 deletions
diff --git a/mathemathica_fwm/RbXMDSSetup1.nb b/mathemathica_fwm/RbXMDSSetup1.nb new file mode 100644 index 0000000..6733d40 --- /dev/null +++ b/mathemathica_fwm/RbXMDSSetup1.nb @@ -0,0 +1,2972 @@ +Notebook[{ + +Cell[CellGroupData[{ +Cell["Find evolution equations", "Section"], + +Cell["Load the package.", "Text"], + +Cell[BoxData[ + RowBox[{"<<", "AtomicDensityMatrix`"}]], "Input", + CellGroupingRules->{GroupTogetherGrouping, 10001.}, + CellID->2058623809], + +Cell["Use density matrix variables with explict time dependence.", "Text"], + +Cell[BoxData[ + RowBox[{"SetOptions", "[", + RowBox[{"DensityMatrix", ",", + RowBox[{"TimeDependence", "\[Rule]", "True"}]}], "]"}]], "Input"], + +Cell["\<\ +Pull quantum numbers and other basic info about the states from a database.\ +\>", "Text", + CellID->429217524], + +Cell[BoxData[ + RowBox[{"s12data", "=", + RowBox[{"AtomicData", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<s\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<S\>\"", ",", + FractionBox["1", "2"]}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{ + "Energy", ",", "J", ",", "L", ",", "S", ",", "NuclearSpin", ",", + "NaturalWidth", ",", "Parity"}], "}"}]}], "]"}]}]], "Input"], + +Cell[BoxData[ + RowBox[{"p12data", "=", + RowBox[{"Append", "[", + RowBox[{ + RowBox[{"AtomicData", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<p\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<P\>\"", ",", + FractionBox["1", "2"]}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"J", ",", "L", ",", "S", ",", "NuclearSpin", ",", "Parity"}], + "}"}]}], "]"}], ",", + RowBox[{ + RowBox[{"BranchingRatio", "[", "0", "]"}], "\[Rule]", "1"}]}], + "]"}]}]], "Input"], + +Cell[BoxData[ + RowBox[{"p32data", "=", + RowBox[{"Append", "[", + RowBox[{ + RowBox[{"AtomicData", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<p\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<P\>\"", ",", + FractionBox["3", "2"]}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"J", ",", "L", ",", "S", ",", "NuclearSpin", ",", "Parity"}], + "}"}]}], "]"}], ",", + RowBox[{ + RowBox[{"BranchingRatio", "[", "0", "]"}], "\[Rule]", "1"}]}], + "]"}]}]], "Input"], + +Cell["\<\ +Using the quantum numbers, create a list of all hyperfine and Zeeman \ +sublevels of the J states, which are labeled 0, 1, and 2 for reference. This \ +list will be passed to the functions from the ADM package that create the DM \ +evolution equations.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"system0", "=", + RowBox[{"Sublevels", "[", + RowBox[{"{", + RowBox[{ + RowBox[{"AtomicState", "[", + RowBox[{"0", ",", "s12data"}], "]"}], ",", + RowBox[{"AtomicState", "[", + RowBox[{"1", ",", "p12data"}], "]"}], ",", + RowBox[{"AtomicState", "[", + RowBox[{"2", ",", "p32data"}], "]"}]}], "}"}], "]"}]}], ";"}]], "Input", + CellID->433132487], + +Cell[TextData[{ + "For simplicity, we delete some excited states from the system, leaving only \ +", + Cell[BoxData[ + FormBox[ + RowBox[{ + RowBox[{ + SubscriptBox["P", + RowBox[{"1", "/", "2"}]], " ", "F"}], "=", "1"}], TraditionalForm]]], + " and ", + Cell[BoxData[ + FormBox[ + RowBox[{ + RowBox[{ + SubscriptBox["P", + RowBox[{"3", "/", "2"}]], " ", "F"}], "=", "2"}], TraditionalForm]]], + "." +}], "Text"], + +Cell[BoxData[ + RowBox[{"system", "=", + RowBox[{"DeleteStates", "[", + RowBox[{"system0", ",", + RowBox[{ + RowBox[{"(", + RowBox[{ + RowBox[{"Label", "\[Equal]", "1"}], "&&", + RowBox[{"F", "\[NotEqual]", "1"}]}], ")"}], "||", + RowBox[{"(", + RowBox[{ + RowBox[{"Label", "\[Equal]", "2"}], "&&", + RowBox[{"F", "\[NotEqual]", "2"}]}], ")"}]}]}], "]"}]}]], "Input"], + +Cell[TextData[{ + "Define the optical field with four frequencies, ", + Cell[BoxData[ + FormBox[ + StyleBox[ + SubscriptBox["\[Omega]", "1"], "InlineMath"], TraditionalForm]]], + ", ", + Cell[BoxData[ + FormBox[ + StyleBox[ + SubscriptBox["\[Omega]", "2"], "InlineMath"], TraditionalForm]]], + ", ", + Cell[BoxData[ + FormBox[ + StyleBox[ + SubscriptBox["\[Omega]", "3"], "InlineMath"], TraditionalForm]]], + ", and ", + Cell[BoxData[ + FormBox[ + StyleBox[ + SubscriptBox["\[Omega]", "4"], "InlineMath"], TraditionalForm]]], + ". Each field can have arbitrary ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["\[Sigma]", "+"], TraditionalForm]], + FormatType->"TraditionalForm"], + " and ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["\[Sigma]", "-"], TraditionalForm]], + FormatType->"TraditionalForm"], + " components, labeled ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", "1"}]], TraditionalForm]], + FormatType->"TraditionalForm"], + "and ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", + RowBox[{"-", "1"}]}]], TraditionalForm]], + FormatType->"TraditionalForm"], + "." +}], "Text", + CellID->133602844], + +Cell[BoxData[ + RowBox[{"SetOptions", "[", + RowBox[{"OpticalField", ",", + RowBox[{"PolarizationVector", "\[Rule]", + RowBox[{"{", + RowBox[{"1", ",", "0", ",", "0"}], "}"}]}], ",", + RowBox[{"CartesianCoordinates", "\[Rule]", + RowBox[{"{", + RowBox[{"x", ",", "y", ",", "z"}], "}"}]}]}], "]"}]], "Input"], + +Cell[TextData[{ + "Parameters for the OpticalField function are {frequency, wavenumber}, \ +{electric amplitude, phase}, {rotation angle (relative to \ +PolarizationVector), ellipticity}. Here ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", "q"}]], TraditionalForm]]], + " are the Rabi frequencies defined in terms of the dipole reduced matrix \ +elements." +}], "Text"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"rmes", "=", + RowBox[{"{", + RowBox[{ + RowBox[{"ReducedME", "[", + RowBox[{"0", ",", + RowBox[{"{", + RowBox[{"Dipole", ",", "1"}], "}"}], ",", "1"}], "]"}], ",", " ", + RowBox[{"ReducedME", "[", + RowBox[{"0", ",", + RowBox[{"{", + RowBox[{"Dipole", ",", "1"}], "}"}], ",", "1"}], "]"}], ",", " ", + RowBox[{"ReducedME", "[", + RowBox[{"0", ",", + RowBox[{"{", + RowBox[{"Dipole", ",", "1"}], "}"}], ",", "2"}], "]"}], ",", " ", + RowBox[{"ReducedME", "[", + RowBox[{"0", ",", + RowBox[{"{", + RowBox[{"Dipole", ",", "1"}], "}"}], ",", "2"}], "]"}]}], "}"}]}], + ";"}]], "Input"], + +Cell[BoxData[ + RowBox[{"contrafield", "=", + RowBox[{"Sum", "[", + RowBox[{ + RowBox[{ + FractionBox[ + SuperscriptBox["\[ExponentialE]", + RowBox[{"\[ImaginaryI]", " ", + RowBox[{"(", + RowBox[{ + RowBox[{"z", " ", + SubscriptBox["k", "i"]}], "+", + SubscriptBox["\[Phi]", "i"], "-", + RowBox[{"t", " ", + SubscriptBox["\[Omega]", "i"]}]}], ")"}]}]], + RowBox[{"rmes", "\[LeftDoubleBracket]", "i", "\[RightDoubleBracket]"}]], + " ", + RowBox[{"{", + RowBox[{ + RowBox[{"{", + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", "1"}]], "}"}], ",", + RowBox[{"{", "0", "}"}], ",", + RowBox[{"{", + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", + RowBox[{"-", "1"}]}]], "}"}]}], "}"}]}], ",", + RowBox[{"{", + RowBox[{"i", ",", "4"}], "}"}]}], "]"}]}]], "Input"], + +Cell[BoxData[ + RowBox[{"field", "=", + RowBox[{ + RowBox[{"ToCartesian", "[", "contrafield", "]"}], "//", + "Expand"}]}]], "Input"], + +Cell["\<\ +The Hamiltonian for the system subject to the optical field. Each field is \ +assumed to interact with only one transition\[LongDash]the replacement rule \ +(Cos[_]|Sin[_]) ReducedME[_,{Dipole,1},_]\[Rule]0 causes other terms to be \ +set to zero.\ +\>", "Text", + CellID->462076121], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"H", "=", + RowBox[{ + RowBox[{"Expand", "@", + RowBox[{"Hamiltonian", "[", + RowBox[{"system", ",", + RowBox[{"ElectricField", "\[Rule]", "field"}], ",", + RowBox[{"MagneticField", "\[Rule]", + RowBox[{"{", + RowBox[{"0", ",", "0", ",", + RowBox[{"\[CapitalOmega]L", "/", "BohrMagneton"}]}], "}"}]}]}], + "]"}]}], "/.", " ", + RowBox[{ + RowBox[{ + RowBox[{"(", + RowBox[{ + RowBox[{"Cos", "[", "_", "]"}], "|", + RowBox[{"Sin", "[", "_", "]"}]}], ")"}], " ", + RowBox[{"ReducedME", "[", + RowBox[{"_", ",", + RowBox[{"{", + RowBox[{"Dipole", ",", "1"}], "}"}], ",", "_"}], "]"}]}], "\[Rule]", + "0"}]}]}], "]"}]], "Input", + CellID->494599775], + +Cell["\<\ +The level diagram for the system, showing optical couplings. Note that both \ +resonant and off-resonant (counter-rotating) couplings are shown, because we \ +have not yet performed the rotating-wave approximation.\ +\>", "Text", + CellID->358620443], + +Cell[BoxData[ + RowBox[{"LevelDiagram", "[", + RowBox[{"system", ",", + RowBox[{"H", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"Energy", "[", "0", "]"}], "\[Rule]", "0"}], ",", + RowBox[{ + RowBox[{"Energy", "[", "1", "]"}], "\[Rule]", "2"}], ",", + RowBox[{ + RowBox[{"Energy", "[", "2", "]"}], "\[Rule]", "4"}], ",", + RowBox[{ + RowBox[{"HyperfineA", "[", "_", "]"}], "\[Rule]", ".2"}], ",", + RowBox[{ + RowBox[{"HyperfineB", "[", "_", "]"}], "\[Rule]", ".1"}], ",", + RowBox[{"\[CapitalOmega]L", "\[Rule]", ".1"}]}], "}"}]}], ",", + RowBox[{"ParityOffset", "\[Rule]", "False"}]}], "]"}]], "Input", + CellID->167259034], + +Cell[TextData[{ + "Here we apply the rotating-wave approximation to the Hamiltonian. Here we \ +construct a list of frequency shifts for the list of Zeeman sublevels that \ +will have the effect of removing the optical frequencies from the \ +Hamiltonian. I.e., we hold ", + Cell[BoxData[ + FormBox[ + RowBox[{ + RowBox[{ + SubscriptBox["S", + RowBox[{"1", "/", "2"}]], "F"}], "=", "2"}], TraditionalForm]]], + " fixed, shift ", + Cell[BoxData[ + FormBox[ + SubscriptBox["P", + RowBox[{"1", "/", "2"}]], TraditionalForm]]], + " down by ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "1"], TraditionalForm]]], + ", shift ", + Cell[BoxData[ + FormBox[ + RowBox[{ + RowBox[{ + SubscriptBox["S", + RowBox[{"1", "/", "2"}]], " ", "F"}], "=", "1"}], TraditionalForm]]], + " down by ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["\[Omega]", "1"], "-", + SubscriptBox["\[Omega]", "2"]}], TraditionalForm]]], + ", and shift ", + Cell[BoxData[ + FormBox[ + SubscriptBox["P", + RowBox[{"3", "/", "2"}]], TraditionalForm]]], + " down by ", + Cell[BoxData[ + FormBox[ + RowBox[{ + SubscriptBox["\[Omega]", "1"], "-", + SubscriptBox["\[Omega]", "2"], "+", + SubscriptBox["\[Omega]", "3"]}], TraditionalForm]]], + "." +}], "Text", + CellID->577766068], + +Cell[BoxData[{ + RowBox[{ + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{"Label", "[", "#", "]"}], ",", + RowBox[{"F", "[", "#", "]"}]}], "}"}], "&"}], "/@", + "system"}], "\[IndentingNewLine]", + RowBox[{"shifts", "=", + RowBox[{"%", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"{", + RowBox[{"0", ",", "2"}], "}"}], "\[Rule]", "0"}], ",", + RowBox[{ + RowBox[{"{", + RowBox[{"1", ",", "_"}], "}"}], "\[Rule]", + SubscriptBox["\[Omega]", "1"]}], ",", + RowBox[{ + RowBox[{"{", + RowBox[{"0", ",", "1"}], "}"}], "\[Rule]", + RowBox[{ + SubscriptBox["\[Omega]", "1"], "-", + SubscriptBox["\[Omega]", "2"]}]}], ",", + RowBox[{ + RowBox[{"{", + RowBox[{"2", ",", "_"}], "}"}], "\[Rule]", + RowBox[{ + SubscriptBox["\[Omega]", "1"], "-", + SubscriptBox["\[Omega]", "2"], "+", + SubscriptBox["\[Omega]", "3"]}]}]}], "}"}]}]}]}], "Input"], + +Cell["\<\ +Make a unitary transformation matrix to implement the chosen frequency shifts.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"transmat", "=", + RowBox[{"MatrixExp", "[", + RowBox[{ + RowBox[{ + RowBox[{"-", "\[ImaginaryI]"}], " ", "t", " ", + RowBox[{"DiagonalMatrix", "[", "shifts", "]"}]}], "+", + RowBox[{"\[ImaginaryI]", " ", "z", " ", + RowBox[{"DiagonalMatrix", "[", + RowBox[{"shifts", "/.", + RowBox[{"\[Omega]", "\[Rule]", "k"}]}], "]"}]}]}], "]"}]}], + "]"}]], "Input"], + +Cell[TextData[{ + "Write ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "4"], TraditionalForm]]], + " and ", + Cell[BoxData[ + FormBox[ + SubscriptBox["k", "4"], TraditionalForm]]], + " in terms of the non-degenerate detuning and phase-mismatch parameters \ +\[Delta]\[Omega] and \[Delta]k, which we set here to zero for simplicity. We \ +then apply the transform matrix to the Hamiltonian and set off resonant terms \ +oscillating at harmonics of ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "1"], TraditionalForm]]], + ", ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "2"], TraditionalForm]]], + ", and ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "3"], TraditionalForm]]], + " to zero." +}], "Text"], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"H1", "=", + RowBox[{"RotatingWaveApproximation", "[", + RowBox[{"system", ",", + RowBox[{ + RowBox[{"H", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + SubscriptBox["\[Omega]", "4"], "\[Rule]", + RowBox[{ + SubscriptBox["\[Omega]", "1"], "-", + SubscriptBox["\[Omega]", "2"], "+", + SubscriptBox["\[Omega]", "3"], "-", "\[Delta]\[Omega]"}]}], ",", + RowBox[{ + SubscriptBox["k", "4"], "\[Rule]", + RowBox[{ + SubscriptBox["k", "1"], "-", + SubscriptBox["k", "2"], "+", + SubscriptBox["k", "3"], "-", "\[Delta]k"}]}]}], "}"}]}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{"\[Delta]\[Omega]", "\[Rule]", "0"}], ",", + RowBox[{"\[Delta]k", "\[Rule]", "0"}]}], "}"}]}], ",", + RowBox[{"{", + RowBox[{ + SubscriptBox["\[Omega]", "1"], ",", + SubscriptBox["\[Omega]", "2"], ",", + SubscriptBox["\[Omega]", "3"]}], "}"}], ",", + RowBox[{"TransformMatrix", "\[Rule]", "transmat"}]}], "]"}]}], + "]"}]], "Input"], + +Cell[TextData[{ + "There are remaining fast-oscillating terms in the Hamiltonian at the \ +difference frequency between ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "1"], TraditionalForm]]], + " and ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "2"], TraditionalForm]]], + ", due to the fact that each optical field can interact with both \ +ground-state hyperfine levels. We set these terms to zero, which is \ +equivalent to assuming that each field interacts with only the ", + Cell[BoxData[ + FormBox[ + RowBox[{"F", "=", "1"}], TraditionalForm]]], + " or the ", + Cell[BoxData[ + FormBox[ + RowBox[{"F", "=", "2"}], TraditionalForm]]], + " ground-state hyperfine sublevel. We also absorb the complex phases into \ +the \[CapitalOmega]\[CloseCurlyQuote]s. (There\[CloseCurlyQuote]s a little \ +bit of a cheat here, since the \[CapitalOmega]\[CloseCurlyQuote]s should have \ +already been distinguished from the ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["\[CapitalOmega]", "*"], TraditionalForm]], + FormatType->"TraditionalForm"], + "\[CloseCurlyQuote]s, but the sign of the \[Phi] phase keeps track of the \ +distinction.)" +}], "Text"], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"H2", "=", + RowBox[{"H1", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + SuperscriptBox["\[ExponentialE]", + RowBox[{"\[Phi]_", "-", + RowBox[{"\[ImaginaryI]", " ", "t", " ", + SubscriptBox["\[Omega]", "1"]}], "+", + RowBox[{"\[ImaginaryI]", " ", "t", " ", + SubscriptBox["\[Omega]", "2"]}]}]], "\[Rule]", "0"}], ",", + RowBox[{ + SuperscriptBox["\[ExponentialE]", + RowBox[{"\[Phi]_", "+", + RowBox[{"\[ImaginaryI]", " ", "t", " ", + SubscriptBox["\[Omega]", "1"]}], "-", + RowBox[{"\[ImaginaryI]", " ", "t", " ", + SubscriptBox["\[Omega]", "2"]}]}]], "\[Rule]", "0"}], ",", + RowBox[{ + RowBox[{ + SuperscriptBox["\[ExponentialE]", + RowBox[{"\[ImaginaryI]", " ", + SubscriptBox["\[Phi]", "i_"]}]], " ", + SubscriptBox["\[CapitalOmega]", + RowBox[{"i_", ",", "q_"}]]}], "\[Rule]", + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", "q"}]]}], ",", + RowBox[{ + RowBox[{ + SuperscriptBox["\[ExponentialE]", + RowBox[{ + RowBox[{"-", "\[ImaginaryI]"}], " ", + SubscriptBox["\[Phi]", "i_"]}]], " ", + SubscriptBox["\[CapitalOmega]", + RowBox[{"i_", ",", "q_"}]]}], "\[Rule]", + RowBox[{"Conjugate", "[", + SubscriptBox["\[CapitalOmega]", + RowBox[{"i", ",", "q"}]], "]"}]}]}], "}"}]}]}], "]"}]], "Input"], + +Cell["\<\ +Find the frequencies at which each field is assumed to be resonant with its F\ +\[Rule]F\[CloseCurlyQuote] transition.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"\[Omega]1res", "=", + RowBox[{ + RowBox[{ + RowBox[{"Hamiltonian", "[", + RowBox[{"{", + RowBox[{"SelectState", "[", + RowBox[{"system", ",", " ", + RowBox[{ + RowBox[{"Label", "\[Equal]", "1"}], "&&", + RowBox[{"F", "\[Equal]", "1"}]}]}], "]"}], "}"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}], "-", + RowBox[{ + RowBox[{"Hamiltonian", "[", + RowBox[{"{", + RowBox[{"SelectState", "[", + RowBox[{"system", ",", " ", + RowBox[{ + RowBox[{"Label", "\[Equal]", "0"}], "&&", + RowBox[{"F", "\[Equal]", "2"}]}]}], "]"}], "}"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}]}]}]], "Input"], + +Cell[BoxData[ + RowBox[{"\[Omega]2res", "=", + RowBox[{ + RowBox[{ + RowBox[{"Hamiltonian", "[", + RowBox[{"{", + RowBox[{"SelectState", "[", + RowBox[{"system", ",", " ", + RowBox[{ + RowBox[{"Label", "\[Equal]", "1"}], "&&", + RowBox[{"F", "\[Equal]", "1"}]}]}], "]"}], "}"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}], "-", + RowBox[{ + RowBox[{"Hamiltonian", "[", + RowBox[{"{", + RowBox[{"SelectState", "[", + RowBox[{"system", ",", " ", + RowBox[{ + RowBox[{"Label", "\[Equal]", "0"}], "&&", + RowBox[{"F", "\[Equal]", "1"}]}]}], "]"}], "}"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}]}]}]], "Input"], + +Cell[BoxData[ + RowBox[{"\[Omega]3res", "=", + RowBox[{ + RowBox[{ + RowBox[{"Hamiltonian", "[", + RowBox[{"{", + RowBox[{"SelectState", "[", + RowBox[{"system", ",", " ", + RowBox[{ + RowBox[{"Label", "\[Equal]", "2"}], "&&", + RowBox[{"F", "\[Equal]", "2"}]}]}], "]"}], "}"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}], "-", + RowBox[{ + RowBox[{"Hamiltonian", "[", + RowBox[{"{", + RowBox[{"SelectState", "[", + RowBox[{"system", ",", " ", + RowBox[{ + RowBox[{"Label", "\[Equal]", "0"}], "&&", + RowBox[{"F", "\[Equal]", "1"}]}]}], "]"}], "}"}], "]"}], + "\[LeftDoubleBracket]", + RowBox[{"1", ",", "1"}], "\[RightDoubleBracket]"}]}]}]], "Input"], + +Cell[TextData[{ + "Rewrite the frequencies ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "1"], TraditionalForm]]], + ", ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "2"], TraditionalForm]]], + ", and ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Omega]", "3"], TraditionalForm]]], + " in terms of a detuning ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Delta]", "1"], TraditionalForm]]], + ", ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Delta]", "2"], TraditionalForm]]], + ", or ", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Delta]", "3"], TraditionalForm]]], + " from resonance with the appropriate transition between ground-state and \ +excited state hyperfine levels, and subtract a constant term off of the \ +diagonal to simplify the appearance." +}], "Text"], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"Hrwa", "=", + RowBox[{ + RowBox[{"(", + RowBox[{"H2", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + SubscriptBox["\[Omega]", "1"], "\[Rule]", + RowBox[{"\[Omega]1res", "+", + SubscriptBox["\[Delta]", "1"]}]}], ",", + RowBox[{ + SubscriptBox["\[Omega]", "2"], "\[Rule]", + RowBox[{"\[Omega]2res", "+", + SubscriptBox["\[Delta]", "2"]}]}], ",", + RowBox[{ + SubscriptBox["\[Omega]", "3"], "\[Rule]", + RowBox[{"\[Omega]3res", "+", + SubscriptBox["\[Delta]", "3"]}]}]}], "}"}]}], ")"}], "-", + RowBox[{ + FractionBox[ + RowBox[{"3", " ", + RowBox[{"HyperfineA", "[", "0", "]"}]}], "4"], + RowBox[{"IdentityMatrix", "[", + RowBox[{"Length", "[", "system", "]"}], "]"}]}]}]}], "]"}]], "Input"], + +Cell["\<\ +The level diagram showing resonant (co-rotating) optical couplings. \ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"diagram", "=", + RowBox[{"LevelDiagram", "[", + RowBox[{"system", ",", + RowBox[{ + RowBox[{"Hrwa", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + SubscriptBox["\[Delta]", "1"], "\[Rule]", + RowBox[{"-", "\[Omega]1res"}]}], ",", + RowBox[{ + SubscriptBox["\[Delta]", "2"], "\[Rule]", + RowBox[{"-", "\[Omega]2res"}]}], ",", + RowBox[{ + SubscriptBox["\[Delta]", "3"], "\[Rule]", + RowBox[{"-", 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", + Cell[BoxData[ + FormBox[ + SubscriptBox["\[Gamma]", "t"], TraditionalForm]]], + " is the transit rate." +}], "Text", + CellID->610306692], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"relax", "=", + RowBox[{ + RowBox[{"IntrinsicRelaxation", "[", "system", "]"}], "+", + RowBox[{"TransitRelaxation", "[", + RowBox[{"system", ",", "\[Gamma]t"}], "]"}]}]}], "]"}]], "Input", + CellID->645617687], + +Cell[TextData[{ + Cell[BoxData[ + ButtonBox["OpticalRepopulation", + BaseStyle->"Link", + ButtonData->"paclet:AtomicDensityMatrix/ref/OpticalRepopulation"]]], + " and ", + Cell[BoxData[ + ButtonBox["TransitRepopulation", + BaseStyle->"Link", + ButtonData->"paclet:AtomicDensityMatrix/ref/TransitRepopulation"]]], + " supply the matrices describing repopulation of the ground state due to \ +spontaneous decay and atomic transit." +}], "Text", + CellID->854192725], + +Cell[BoxData[ + RowBox[{"MatrixForm", "[", + RowBox[{"repop", "=", + RowBox[{ + RowBox[{"OpticalRepopulation", "[", "system", "]"}], "+", + RowBox[{"TransitRepopulation", "[", + RowBox[{"system", ",", "\[Gamma]t"}], "]"}]}]}], "]"}]], "Input", + CellID->465762594], + +Cell["Here are the evolution equations.", "Text", + CellID->314466782], + +Cell[BoxData[ + RowBox[{"TableForm", "[", + RowBox[{"eqs0", "=", + RowBox[{"LiouvilleEquation", "[", + RowBox[{"system", ",", "Hrwa", ",", "relax", ",", "repop"}], "]"}]}], + "]"}]], "Input"], + +Cell["\<\ +Here we pull useful numerical atomic data for Rb out of the database. These \ +numbers are in omega units, so that the unit \[OpenCurlyDoubleQuote]Hertz\ +\[CloseCurlyDoubleQuote] actually corresponds to rad/s.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"atomicdata", "=", + RowBox[{"Join", "[", + RowBox[{ + RowBox[{"AtomicData", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<s\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<S\>\"", ",", + FractionBox["1", "2"]}], "}"}]}], "}"}], ",", + RowBox[{"{", "HyperfineA", "}"}], ",", + RowBox[{"Label", "\[Rule]", "0"}]}], "]"}], ",", + RowBox[{"AtomicData", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<p\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<P\>\"", ",", + FractionBox["1", "2"]}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"HyperfineA", ",", "NaturalWidth"}], "}"}], ",", + RowBox[{"Label", "\[Rule]", "1"}]}], "]"}], ",", + RowBox[{"AtomicData", "[", + RowBox[{ + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<p\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<P\>\"", ",", + FractionBox["3", "2"]}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"HyperfineA", ",", "HyperfineB", ",", "NaturalWidth"}], "}"}], + ",", + RowBox[{"Label", "\[Rule]", "2"}]}], "]"}], ",", + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"Wavelength", "[", "1", "]"}], "->", + RowBox[{"1", "/", + RowBox[{"Wavenumber", "[", + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<p\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<P\>\"", ",", + FractionBox["1", "2"]}], "}"}]}], "}"}], "]"}]}]}], ",", + RowBox[{ + RowBox[{"Wavelength", "[", "2", "]"}], "->", + RowBox[{"1", "/", + RowBox[{"Wavenumber", "[", + RowBox[{"{", + RowBox[{"\"\<Rb\>\"", ",", "87", ",", + RowBox[{"{", + RowBox[{"\"\<Kr\>\"", ",", + RowBox[{"{", + RowBox[{"5", ",", "\"\<p\>\""}], "}"}]}], "}"}], ",", + RowBox[{"{", + RowBox[{"2", ",", "\"\<P\>\"", ",", + FractionBox["3", "2"]}], "}"}]}], "}"}], "]"}]}]}]}], "}"}]}], + "]"}]}]], "Input"], + +Cell[TextData[{ + "Here we find DM elements that are always identically zero, so we can remove \ +them from the evolution equations. We put in sample values for all of the \ +parameters, use the values for the atomic data from above, and set time \ +derivatives to zero with ", + Cell[BoxData[ + RowBox[{ + RowBox[{ + SuperscriptBox[ + SubscriptBox["\[Rho]", + RowBox[{"s1_", ",", "s2_"}]], "\[Prime]", + MultilineFunction->None], "[", "t", "]"}], "\[Rule]", "0"}]]], + " to find the steady state. " +}], "Text"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"sol", "=", + RowBox[{ + RowBox[{"NSolve", "[", + RowBox[{ + RowBox[{"Evaluate", "[", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"eqs0", "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + SubscriptBox["\[Phi]", "_"], "\[Rule]", "1."}], ",", + RowBox[{"\[Delta]k", "\[Rule]", "0"}], ",", + RowBox[{"\[Delta]\[Omega]", "\[Rule]", "0"}], ",", + RowBox[{ + SubscriptBox["\[CapitalOmega]", + RowBox[{"_", ",", "1"}]], "\[Rule]", + SuperscriptBox["10", "11"]}], ",", + RowBox[{ + SubscriptBox["\[CapitalOmega]", + RowBox[{"_", ",", + RowBox[{"-", "1"}]}]], "\[Rule]", + RowBox[{"3.", " ", + SuperscriptBox["10", "11"]}]}], ",", + RowBox[{ + SubscriptBox["\[Delta]", "_"], "\[Rule]", + RowBox[{"1.", " ", + SuperscriptBox["10", "7"]}]}], ",", + RowBox[{"\[Gamma]t", "\[Rule]", "10."}], ",", + RowBox[{"\[CapitalOmega]L", "\[Rule]", ".1"}]}], "}"}]}], "/.", + "atomicdata"}], "/.", + RowBox[{"Hertz", "\[Rule]", "1"}]}], "/.", + RowBox[{"Mega", "\[Rule]", "1"}]}], "/.", + RowBox[{ + RowBox[{ + SuperscriptBox[ + SubscriptBox["\[Rho]", + RowBox[{"s1_", ",", "s2_"}]], "\[Prime]", + MultilineFunction->None], "[", "t", "]"}], "\[Rule]", "0"}]}], + "]"}], ",", + RowBox[{"DMVariables", "[", "system", "]"}]}], "]"}], "[", + RowBox[{"[", "1", "]"}], "]"}]}], ";"}]], "Input"], + +Cell["\<\ +Find the position in the solution list of all of the DM elements that are \ +zero.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"delpos", "=", + RowBox[{"Position", "[", + RowBox[{"sol", ",", + RowBox[{"_", "\[Rule]", + RowBox[{"0.", "+", + RowBox[{"0.", " ", "\[ImaginaryI]"}]}]}]}], "]"}]}]], "Input"], + +Cell["\<\ +Find the list of DM elements that correspond to these zero positions.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"delvars", "=", + RowBox[{"Extract", "[", + RowBox[{ + RowBox[{"DMVariables", "[", "system", "]"}], ",", "delpos"}], + "]"}]}]], "Input"], + +Cell["Create a list of rules to set these DM elements to zero.", "Text"], + +Cell[BoxData[ + RowBox[{ + RowBox[{"delreps", "=", + RowBox[{"(", + RowBox[{ + RowBox[{ + RowBox[{"(", + RowBox[{"#", "\[Rule]", "0"}], ")"}], "&"}], "/@", "delvars"}], + ")"}]}], ";"}]], "Input"], + +Cell["\<\ +Find the list of variables corresponding to nonzero DM elements.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"vars", "=", + RowBox[{"Delete", "[", + RowBox[{ + RowBox[{"DMVariables", "[", "system", "]"}], ",", "delpos"}], + "]"}]}]], "Input"], + +Cell["\<\ +Remove all of the zero elements from the evolution equations.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"TableForm", "[", + RowBox[{"eqs", "=", + RowBox[{"Delete", "[", + RowBox[{ + RowBox[{"eqs0", "/.", "delreps"}], ",", "delpos"}], "]"}]}], + "]"}]], "Input"], + +Cell["Initial conditions for the time-dependent case.", "Text"], + +Cell[BoxData[ + RowBox[{"inits", "=", + RowBox[{"Delete", "[", + RowBox[{ + RowBox[{"InitialConditions", "[", + RowBox[{"system", ",", + RowBox[{"TransitRepopulation", "[", + RowBox[{"system", ",", "1"}], "]"}], ",", "0"}], "]"}], ",", "delpos"}] + , "]"}]}]], "Input"], + +Cell[TextData[{ + "Find equations for the steady state by setting time derivatives to zero. We \ +also substitute in numerical values for natural widths in units of ", + Cell[BoxData[ + FormBox[ + SuperscriptBox["10", "6"], TraditionalForm]]], + " rad/s." +}], "Text"], + +Cell[BoxData[ + RowBox[{"steadyeqs", "=", + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{ + RowBox[{"eqs", "/.", + RowBox[{ + SubscriptBox["\[Phi]", "_"], "\[Rule]", "0"}]}], "/.", "atomicdata"}], + "/.", + RowBox[{"Hertz", "\[Rule]", "1"}]}], "/.", + RowBox[{"Mega", "\[Rule]", "1"}]}], "/.", + RowBox[{ + RowBox[{ + SuperscriptBox[ + SubscriptBox["\[Rho]", + RowBox[{"s1_", ",", "s2_"}]], "\[Prime]", + MultilineFunction->None], "[", "t", "]"}], "\[Rule]", + "0"}]}]}]], "Input"], + +Cell[TextData[{ + "Here we find expressions for the change of the spherical components of the \ +probe light amplitude upon propagation through a thin slice of medium in \ +terms of the density matrix elements. Since the probe light does not interact \ +with the D2 transition, we set DM elements involving the ", + Cell[BoxData[ + FormBox[ + SubscriptBox["P", + RowBox[{"3", "/", "2"}]], TraditionalForm]]], + " state to zero. We also set the vanishing DM elements found above to zero." +}], "Text"], + +Cell["\<\ +Polarization components for the probe light.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"pc", "=", + RowBox[{"Simplify", "[", + RowBox[{ + RowBox[{ + RowBox[{"PolarizationComponents", "[", + RowBox[{"system", ",", + RowBox[{"Energy", "[", "1", "]"}], ",", "E0"}], "]"}], "/.", + RowBox[{ + RowBox[{ + RowBox[{"DMElementPattern", "[", "]"}], "/;", + RowBox[{ + RowBox[{"Label2", "\[Equal]", "2"}], "||", + RowBox[{"F1", "\[Equal]", "2"}]}]}], "\[Rule]", "0"}]}], "/.", + "delreps"}], "]"}]}]], "Input"], + +Cell["\<\ +Differential change of the spherical components.\ +\>", "Text"], + +Cell[BoxData[{ + RowBox[{ + RowBox[{ + RowBox[{"{", + RowBox[{ + RowBox[{ + RowBox[{"EpRe", "'"}], "+", + RowBox[{"\[ImaginaryI]", " ", + RowBox[{"EpIm", "'"}]}]}], ",", + RowBox[{ + RowBox[{"EmRe", "'"}], "+", + RowBox[{"\[ImaginaryI]", " ", + RowBox[{"EmIm", "'"}]}]}]}], "}"}], "/.", + RowBox[{"{", + RowBox[{ + RowBox[{ + 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"/.", + "params"}], "/.", + RowBox[{"\[Omega]", "\[Rule]", + RowBox[{"Energy", "[", "1", "]"}]}]}]}]}], "Input"], + +Cell["\<\ +Divide change of the spherical components by the linear absorption per unit \ +length to find change per absorption length.\ +\>", "Text"], + +Cell[BoxData[ + RowBox[{"obs", "=", + RowBox[{ + RowBox[{"dsigma0", "/", "linabs"}], "//", "Chop"}]}]], "Input"], + +Cell["\<\ +Relate absorption and phase shift to changes in the real and imaginary parts \ +of the spherical field components\ +\>", "Text"], + +Cell[BoxData[{ + RowBox[{"{", + RowBox[{ + RowBox[{"{", + RowBox[{"(", + RowBox[{ + SubscriptBox["\[CapitalOmega]", + RowBox[{"1", ",", "Re"}]], "+", + RowBox[{"\[ImaginaryI]", " ", + SubscriptBox["\[CapitalOmega]", + RowBox[{"1", ",", "Im"}]]}]}], ")"}], "}"}], ",", + RowBox[{"{", "0", "}"}], ",", + RowBox[{"{", + RowBox[{"(", + RowBox[{ + SubscriptBox["\[CapitalOmega]", + RowBox[{ + RowBox[{"-", "1"}], ",", "Re"}]], "+", + RowBox[{"\[ImaginaryI]", " ", + 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