dissipative_time_crystal_harmonic_balance_no_LC.ipynb

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   "source": [
    "using DrWatson\n",
    "@quickactivate \"DrivenKerrResonator\"\n",
    "\n",
    "using HarmonicBalance\n",
    "using Symbolics"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "b280fe90",
   "metadata": {},
   "outputs": [
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      "text/plain": [
       "System of length 4\n",
       " 4 variables: αᵣ₁, αᵢ₁, αᵣ₂, αᵢ₂\n",
       " 5 parameters: Δ, κ, U, F, J\n",
       "\n",
       " -J*αᵢ₂ - 0.5*κ*αᵣ₁ + (-Δ + αᵢ₁^2 + αᵣ₁^2)*αᵢ₁\n",
       " -F*sqrt(U) + J*αᵣ₂ - 0.5*κ*αᵢ₁ - (-Δ + αᵢ₁^2 + αᵣ₁^2)*αᵣ₁\n",
       " -J*αᵢ₁ - 0.5*κ*αᵣ₂ + (-Δ + αᵢ₂^2 + αᵣ₂^2)*αᵢ₂\n",
       " J*αᵣ₁ - 0.5*κ*αᵢ₂ - (-Δ + αᵢ₂^2 + αᵣ₂^2)*αᵣ₂"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# defining variables\n",
    "@variables αᵣ₁::Real,αᵢ₁::Real,αᵣ₂::Real,αᵢ₂::Real\n",
    "@variables Δ::Real,κ::Real,U::Real, F::Real, J::Real\n",
    "@variables α₁::Complex{Real},α₂::Complex{Real}\n",
    "\n",
    "# accounting for the fact that we \"divide\" each complex equation in its real and imaginary part\n",
    "function replace_split(f,sub_rules)\n",
    "    exp = substitute(real(f),sub_rules) + 1.0im*substitute(imag(f),sub_rules)\n",
    "    [real(exp),imag(exp)]\n",
    "end\n",
    "\n",
    "function get_EOM_full()\n",
    "    K₁ = (-Δ + conj(α₁)*α₁ - 1.0im*κ/2)*α₁ + F*sqrt(U)\n",
    "    K₂ = (-Δ + conj(α₂)*α₂ - 1.0im*κ/2)*α₂ \n",
    "    \n",
    "    fψ₁ =   -1.0im*(K₁ - J*α₂) \n",
    "    fψ₂ =   -1.0im*(K₂ - J*α₁) \n",
    "    \n",
    "    sub_rules = Dict((real(α₁)=>αᵣ₁),(imag(α₁)=>αᵢ₁),(real(α₂)=>αᵣ₂),(imag(α₂)=>αᵢ₂))\n",
    "\n",
    "    variables = [αᵣ₁,αᵢ₁,αᵣ₂,αᵢ₂]\n",
    "    parameters = [Δ,κ,U,F,J]\n",
    "    system_f= reduce(vcat,[replace_split(f,sub_rules) for f in [fψ₁,fψ₂]])\n",
    "\n",
    "    equation_HC = HarmonicBalance.Problem(system_f,variables,parameters);\n",
    "    return equation_HC\n",
    "end\n",
    "\n",
    "\n",
    "MF_eqs = get_EOM_full()\n",
    "\n",
    "\n",
    "###Hot fix required for the eom field in the harmonic equations object to not be empty. \n",
    "###Will be corrected in future versions\n",
    "# declare dummy system: declare constant variables and two functions x(t) and y(t)\n",
    "@variables α, ω, ω0, t, η, x(t), y(t)\n",
    "\n",
    "# differential equations\n",
    "diff_eq = DifferentialEquation([d(x,t,2) + ω0^2*x + α*x^3 + η*d(x,t)*x^2 ~ 0,\n",
    "                                d(y,t,2) + ω0^2*y + α*y^3 + η*d(y,t)*y^2 ~ 0], [x,y])\n",
    "\n",
    "# specify the harmonic ansatz for x and y: x = u(T) cos(ωt) + v(T) sin(ωt)\n",
    "add_harmonic!(diff_eq, x, ω)\n",
    "add_harmonic!(diff_eq, y, ω)\n",
    "\n",
    "# implement ansatz to get harmonic equations\n",
    "harmonic_eq_dummy = get_harmonic_equations(diff_eq)\n",
    "\n",
    "MF_eqs.eom = harmonic_eq_dummy;\n",
    "MF_eqs.system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "id": "53391561",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\r\n",
      "\u001b[32mSolving for 40000 parameters... 100%|███████████████████| Time: 0:00:34\u001b[39m\r\n",
      "\u001b[34m  # parameters solved:  40000\u001b[39m\r\n",
      "\u001b[34m  # paths tracked:      360000\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "A steady state result for 40000 parameter points\n",
       "\n",
       "Solution branches:   9\n",
       "   of which real:    7\n",
       "   of which stable:  6\n",
       "\n",
       "Classes: stable, physical, Hopf, binary_labels\n"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# parameters\n",
    "κᵣ = 1. #reference value for unitless parameters\n",
    "Uᵣ = 1.\n",
    "#under these assumptions, Ftilde = F and J/κ = J\n",
    "fixed = (κ=>κᵣ,Δ=>2*κᵣ,U=>Uᵣ)\n",
    "varied = (F=>LinRange(0,5,200),J => LinRange(0,2*κᵣ,200))\n",
    "\n",
    "# results\n",
    "result = get_steady_states(MF_eqs, varied, fixed)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "2300a6a4",
   "metadata": {},
   "source": [
    "## Phase diagrams"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8fb44073",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "id": "9aba97e4",
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = plot_phase_diagram(result,title=\"all\")\n",
    "p2 = plot_phase_diagram(result,class=\"stable\",title=\"stable\")\n",
    "p3 = plot_phase_diagram(result,not_class=\"stable\",title=\"unstable\")\n",
    "plot([p1,p2,p3]...,layout = (3,1),figsize = (24,20))"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "5914aaf2",
   "metadata": {},
   "source": [
    "## Replace RWA by harmonic balance around the driving frequency: The previous phase diagram can be recovered at a new effective detuning"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "id": "d0395d41",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A set of 4 harmonic equations\n",
       "Variables: u1(T), v1(T), u2(T), v2(T)\n",
       "Parameters: α, ω, γ, ω0, J, F, U\n",
       "\n",
       "Harmonic ansatz: \n",
       "x(t) = u1(T)*cos(ωt) + v1(T)*sin(ωt)\n",
       "y(t) = u2(T)*cos(ωt) + v2(T)*sin(ωt)\n",
       "\n",
       "Harmonic equations:\n",
       "\n",
       "γ*Differential(T)(u1(T)) + (ω0^2)*u1(T) + (3//4)*α*(u1(T)^3) + γ*ω*v1(T) + (2//1)*ω*Differential(T)(v1(T)) + (3//4)*α*(v1(T)^2)*u1(T) - J*u2(T) - (ω^2)*u1(T) ~ 0\n",
       "\n",
       "(ω0^2)*v1(T) + γ*Differential(T)(v1(T)) + (3//4)*α*(v1(T)^3) + (3//4)*α*(u1(T)^2)*v1(T) - J*v2(T) - (ω^2)*v1(T) - (2//1)*ω*Differential(T)(u1(T)) - γ*ω*u1(T) ~ F*sqrt(U)\n",
       "\n",
       "γ*Differential(T)(u2(T)) + (ω0^2)*u2(T) + (3//4)*α*(u2(T)^3) + γ*ω*v2(T) + (2//1)*ω*Differential(T)(v2(T)) + (3//4)*α*(v2(T)^2)*u2(T) - J*u1(T) - (ω^2)*u2(T) ~ 0\n",
       "\n",
       "γ*Differential(T)(v2(T)) + (ω0^2)*v2(T) + (3//4)*α*(v2(T)^3) + (3//4)*α*(u2(T)^2)*v2(T) - J*v1(T) - (2//1)*ω*Differential(T)(u2(T)) - (ω^2)*v2(T) - γ*ω*u2(T) ~ 0\n"
      ]
     },
     "execution_count": 205,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@variables α, ω, ω0, t, γ, x(t), y(t)\n",
    "\n",
    "# differential equations\n",
    "diffeq_beyond_RWA = DifferentialEquation([d(x,t,2) + γ*d(x,t,1) + ω0^2*x + α*x^3 - J*y ~ F*sqrt(U)*sin(ω*t),\n",
    "                                d(y,t,2) + γ*d(y,t,1) + ω0^2*y + α*y^3  - J*x ~ 0], [x,y])\n",
    "\n",
    "# specify the harmonic ansatz for x and y: x = u(T) cos(ωt) + v(T) sin(ωt)\n",
    "add_harmonic!(diffeq_beyond_RWA, x, ω)\n",
    "add_harmonic!(diffeq_beyond_RWA, y, ω)\n",
    "\n",
    "\n",
    "MF_eqs_beyond_RWA = get_harmonic_equations(diffeq_beyond_RWA)"
   ]
  },
  {
   "attachments": {},
   "cell_type": "markdown",
   "id": "c47eb552",
   "metadata": {},
   "source": [
    "Observation: Harmonic Balance equations and regular RWA equations differ in the sign of J. Probably is just the fact the quadratures are rotated in both cases"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "id": "c102f767",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "System of length 4\n",
       " 4 variables: αᵣ₁, αᵢ₁, αᵣ₂, αᵢ₂\n",
       " 5 parameters: Δ, κ, U, F, J\n",
       "\n",
       " -J*αᵢ₂ - 0.5*κ*αᵣ₁ + (-Δ + αᵢ₁^2 + αᵣ₁^2)*αᵢ₁\n",
       " -F*sqrt(U) + J*αᵣ₂ - 0.5*κ*αᵢ₁ - (-Δ + αᵢ₁^2 + αᵣ₁^2)*αᵣ₁\n",
       " -J*αᵢ₁ - 0.5*κ*αᵣ₂ + (-Δ + αᵢ₂^2 + αᵣ₂^2)*αᵢ₂\n",
       " J*αᵣ₁ - 0.5*κ*αᵢ₂ - (-Δ + αᵢ₂^2 + αᵣ₂^2)*αᵣ₂"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result.problem.system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 207,
   "id": "7b9af7bc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "System of length 4\n",
       " 4 variables: u1, v1, u2, v2\n",
       " 6 parameters: ω, γ, ω0, J, F, U\n",
       "\n",
       " -J*u2 + u1*v1^2 - u1*ω^2 + u1*ω0^2 + v1*γ*ω + u1^3\n",
       " -F*sqrt(U) - J*v2 + u1^2*v1 - v1*ω^2 + v1*ω0^2 - u1*γ*ω + v1^3\n",
       " -J*u1 + u2*v2^2 - u2*ω^2 + u2*ω0^2 + v2*γ*ω + u2^3\n",
       " -J*v1 + u2^2*v2 - v2*ω^2 + v2*ω0^2 - u2*γ*ω + v2^3"
      ]
     },
     "execution_count": 207,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "result_beyond_RWA.problem.system"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 210,
   "id": "9eb2ce32",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\r\n",
      "\u001b[32mSolving for 40000 parameters... 100%|███████████████████| Time: 0:00:41\u001b[39m\r\n",
      "\u001b[34m  # parameters solved:  40000\u001b[39m\r\n",
      "\u001b[34m  # paths tracked:      360000\u001b[39m\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "A steady state result for 40000 parameter points\n",
       "\n",
       "Solution branches:   9\n",
       "   of which real:    7\n",
       "   of which stable:  7\n",
       "\n",
       "Classes: stable, physical, Hopf, binary_labels\n"
      ]
     },
     "execution_count": 210,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# parameters\n",
    "κᵣ = 1. #reference value for unitless parameters\n",
    "Uᵣ = 1.\n",
    "ωᵣ = sqrt(2*κᵣ) #the effective detuning parameter is (ω0^2-ω^2)/ω in the equations \n",
    "#under these assumptions, Ftilde = F and J/κ = J\n",
    "fixed = (γ=>κᵣ/(2*ωᵣ),ω=>ωᵣ,ω0=>0.,α=>(4/3),U=>1)\n",
    "varied = (F=>LinRange(0.0,5,200),J => LinRange(0.0,2.,200))\n",
    "\n",
    "# results\n",
    "result_beyond_RWA = get_steady_states(MF_eqs_beyond_RWA, varied, fixed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "id": "1ade9307",
   "metadata": {},
   "outputs": [
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       "</svg>\n"
      ]
     },
     "execution_count": 211,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = plot_phase_diagram(result_beyond_RWA ,title=\"all\")\n",
    "p2 = plot_phase_diagram(result_beyond_RWA ,class=\"stable\",title=\"stable\")\n",
    "p3 = plot_phase_diagram(result_beyond_RWA ,not_class=\"stable\",title=\"unstable\")\n",
    "plot([p1,p2,p3]...,layout = (3,1),figsize = (24,20))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "6db3a14f",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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