diff --git a/README.md b/README.md
index 41c56c1..7e766bd 100644
--- a/README.md
+++ b/README.md
@@ -14,6 +14,7 @@ It combines a symbolic layer for the SLH formalism based on [SecondQuantizedAlge
 ## Development status
 
 [![Tests](https://github.com/qojulia/QuantumInputOutput.jl/actions/workflows/Tests.yml/badge.svg)](https://github.com/qojulia/QuantumInputOutput.jl/actions/workflows/Tests.yml)
+[![codecov](https://codecov.io/gh/qojulia/QuantumInputOutput.jl/graph/badge.svg)](https://codecov.io/gh/qojulia/QuantumInputOutput.jl)
 [![Documentation dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://qojulia.github.io/QuantumInputOutput.jl/dev/)
 
 Note that **QuantumInputOutput.jl** is still at an early stage of development.
diff --git a/TODO.md b/TODO.md
index c7f70f2..2fd2aa9 100644
--- a/TODO.md
+++ b/TODO.md
@@ -11,21 +11,16 @@
 - [ ] create more tests (cov +80%)
   - [x] simple full example
   - [ ] QC.jl: comparison cavity drive with full quantum (example)
-  - [ ] translate.jl
-  - [ ] utils.jl
-  - [ ] codecov
+  - [x] translate.jl
+  - [x] utils.jl
+  - [x] codecov
   - [ ] interaction picture: test general A(t)
   - [ ] cumulants: comparison example 04-1 and 04-2
-  - [ ] u_Gaussian, v_Gaussian; slower?
 
 - [ ] more functionalities
   - [ ] pulse delay 
   - [ ] better method for u_eff and v_eff (Victor)
 
-- [] docu
-  - [x] u_eff and v_eff theory 
-  - [ ] development status
-
 ## Additional TODOs
 
 - [x] u_eff and v_eff equation (PRA 2020, appendix) theory, API, implementation
@@ -90,4 +85,8 @@
 
 - [x] Formatter/SpellCheck
 - [x] JET
-- [x] Aqua
\ No newline at end of file
+- [x] Aqua
+  
+- [x] docu
+  - [x] u_eff and v_eff theory 
+  - [x] development status
diff --git a/test/runtests.jl b/test/runtests.jl
index 0a14d45..2f6c82f 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -3,6 +3,7 @@ names = [
     "test_SLH.jl",
     "test_feedback.jl",
     "test_translate.jl",
+    "test_correlations.jl",
     "test_example_cavity_scattering.jl",
     "test_compare_example_05_1_05_2.jl",
     "test_interaction_picture.jl",
diff --git a/test/test_SLH.jl b/test/test_SLH.jl
index 2ee292d..9e5ea20 100644
--- a/test/test_SLH.jl
+++ b/test/test_SLH.jl
@@ -4,7 +4,7 @@ using QuantumOptics
 using SymbolicUtils
 using FunctionWrappers: FunctionWrapper
 using StaticArrays
-using Test 
+using Test
 
 @testset "SLH" begin
     hu1 = FockSpace(:u1)
diff --git a/test/test_correlations.jl b/test/test_correlations.jl
new file mode 100644
index 0000000..0d364e2
--- /dev/null
+++ b/test/test_correlations.jl
@@ -0,0 +1,47 @@
+using QuantumInputOutput
+using QuantumOptics
+using LinearAlgebra
+using Test
+
+@testset "correlations_single_tls_decay" begin
+    γ = 1.0
+    ba = NLevelBasis(2)
+    σm = transition(ba, 1, 2)
+    ψ0 = nlevelstate(ba, 2) # initially excited
+
+    H = 0.0 * one(ba)
+    J = [sqrt(γ) * σm]
+    Ls = J[1]
+
+    T_end = 10 / γ
+    T = collect(0.0:0.01:T_end)
+    ΔT = T[2] - T[1]
+
+    function f_const(t, ρ)
+        return H, J, dagger.(J)
+    end
+
+    _, ρt_dyn = timeevolution.master_dynamic(T, ψ0, f_const)
+    g1_dyn = correlation_matrix(T, ρt_dyn, f_const, Ls)
+
+    _, ρt_static = timeevolution.master(T, ψ0, H, J)
+    g1_static = correlation_matrix(T, ρt_static, H, J, Ls)
+
+    @test maximum(abs.(g1_dyn .- g1_static)) < 1e-8
+
+    F = eigen(Hermitian((g1_static + g1_static') / 2))
+    n_modes = real.(F.values) * ΔT
+
+    expected_n = 1 - exp(-γ * T_end)
+    @test abs(n_modes[end] - expected_n) < 1e-2
+    @test abs(n_modes[end-1]) < 1e-2
+
+    mode_num = F.vectors[:, end] / sqrt(ΔT)
+    mode_exp = sqrt.(γ) .* exp.(-0.5 * γ .* T)
+    mode_exp ./= sqrt(sum(abs2.(mode_exp)) * ΔT)
+
+    overlap = sum(conj.(mode_exp) .* mode_num) * ΔT
+    mode_num_aligned = mode_num * exp(-1im * angle(overlap))
+    overlap_aligned = abs(sum(conj.(mode_exp) .* mode_num_aligned) * ΔT)
+    @test overlap_aligned > 0.99
+end
diff --git a/test/test_translate.jl b/test/test_translate.jl
index 8d90dd2..c2dc5f4 100644
--- a/test/test_translate.jl
+++ b/test/test_translate.jl
@@ -136,6 +136,39 @@ using Test
         @test sum(abs.((F_multi(0.4) - expected).data)) < 1e-8
     end
 
+    @testset "translate_qo vector overload" begin
+        ops = [a, Δ, a * E]
+        translated_ops = translate_qo(
+            ops,
+            bc1;
+            parameter = dict_p1,
+            time_parameter = dict_p_t2,
+            operators = ops_dict,
+        )
+        @test length(translated_ops) == length(ops)
+        @test all(op -> op isa Function, translated_ops)
+        @test isequal(translated_ops[1](0.3), a_QO)
+        @test isequal(translated_ops[2](0.3), one(bc1) * Δn)
+        @test isequal(translated_ops[3](0.3), a_QO * E_t(0.3))
+    end
+
+    @testset "translate_qo SLH overload" begin
+        G_sym = SLH(1, [sqrt(κ_R) * a * E], Δ * a' * a)
+        H_QO, L_QO = translate_qo(
+            G_sym,
+            bc1;
+            parameter = dict_p1,
+            time_parameter = dict_p_t2,
+            operators = ops_dict,
+        )
+
+        @test H_QO isa Function
+        @test length(L_QO) == 1
+        @test L_QO[1] isa Function
+        @test sum(abs.((H_QO(0.4) - dense(Δn * dagger(a_QO) * a_QO)).data)) < 1e-8
+        @test sum(abs.((L_QO[1](0.4) - dense(sqrt(κ_Rn) * a_QO * E_t(0.4))).data)) < 1e-8
+    end
+
     @testset "substitute_operators_qmul" begin
         h2 = FockSpace(:h2)
         a = Destroy(h2, :a, 1)
@@ -151,6 +184,8 @@ using Test
 
         @test isequal(simplify(y - (a_1*a_1*c1 + a_1*c3)), 0)
         @test isequal(simplify(y2 - (a_2*a_2*c1 + a_2*c3)), 0)
+
+        @test substitute_operators(5, dict_sub) == 5
     end
 
     @testset "substitute_operators adjoint in args_nc" begin