Operator Identities
For operators A A A B B B
e A B e − A = ∑ n = 0 ∞ 1 n ! [ A , [ A , ⋯ [ A ⏟ n , B ] ⋯ ] ] . e^A B e^{-A}
= \sum_{n=0}^{\infty}\frac{1}{n!}
\underbrace{[A,[A,\cdots[A}_{n},B]\cdots]] . e A B e − A = n = 0 ∑ ∞ n ! 1 n [ A , [ A , ⋯ [ A , B ] ⋯ ]] . Equivalently, define ad A ( B ) = [ A , B ] \operatorname{ad}_A(B)=[A,B] ad A ( B ) = [ A , B ]
e A B e − A = e ad A B . e^A B e^{-A}=e^{\operatorname{ad}_A}B . e A B e − A = e ad A B . For a simple oscillator with U = e − i ω a a † a t U=e^{-i\omega_a a^\dagger a t} U = e − i ω a a † a t
U † a U = e − i ω a t a . U^\dagger a U=e^{-i\omega_a t}a . U † a U = e − i ω a t a . Baker-Campbell-Hausdorff
If [ A , B ] [A,B] [ A , B ] A A A B B B
e A + B = e A e B e − 1 2 [ A , B ] . e^{A+B}=e^A e^B e^{-\frac12[A,B]} . e A + B = e A e B e − 2 1 [ A , B ] . For coherent-state displacement operators,
D α = e α a † − α ∗ a = e α a † e − α ∗ a e − ∣ α ∣ 2 / 2 , D_\alpha
= e^{\alpha a^\dagger-\alpha^*a}
= e^{\alpha a^\dagger}e^{-\alpha^*a}e^{-|\alpha|^2/2}, D α = e α a † − α ∗ a = e α a † e − α ∗ a e − ∣ α ∣ 2 /2 , D α D β = e ( α β ∗ − α ∗ β ) / 2 D α + β . D_\alpha D_\beta
= e^{(\alpha\beta^*-\alpha^*\beta)/2}D_{\alpha+\beta}. D α D β = e ( α β ∗ − α ∗ β ) /2 D α + β . Start from a time-independent Hamiltonian
H = H 0 + V , H=H_0+V, H = H 0 + V , where H 0 ∣ m ⟩ = E m ∣ m ⟩ H_0|m\rangle=E_m|m\rangle H 0 ∣ m ⟩ = E m ∣ m ⟩ V V V V V V H 0 H_0 H 0 ⟨ m ∣ V ∣ m ⟩ = 0 \langle m|V|m\rangle=0 ⟨ m ∣ V ∣ m ⟩ = 0
Define the dressed Hamiltonian by the unitary transformation
H ′ = e S H e − S . H'=e^SHe^{-S}. H ′ = e S H e − S . For small V V V S S S
H ′ = H + [ S , H ] + 1 2 [ S , [ S , H ] ] + ⋯ . H'=H+[S,H]+\frac12[S,[S,H]]+\cdots . H ′ = H + [ S , H ] + 2 1 [ S , [ S , H ]] + ⋯ . With S S S [ S , H 0 ] = − V [S, H_0] = -V [ S , H 0 ] = − V
H ′ = H 0 + 1 2 [ S , V ] + O ( V 3 ) , H'=H_0+\frac12[S,V]+O(V^3), H ′ = H 0 + 2 1 [ S , V ] + O ( V 3 ) , where, in the eigenbasis of H 0 H_0 H 0
⟨ m ∣ S ∣ m ′ ⟩ = ⟨ m ∣ V ∣ m ′ ⟩ E m − E m ′ , m ≠ m ′ , ⟨ m ∣ S ∣ m ⟩ = 0. \langle m|S|m'\rangle
=\frac{\langle m|V|m'\rangle}{E_m-E_{m'}},
\qquad m\ne m',
\qquad \langle m|S|m\rangle=0. ⟨ m ∣ S ∣ m ′ ⟩ = E m − E m ′ ⟨ m ∣ V ∣ m ′ ⟩ , m = m ′ , ⟨ m ∣ S ∣ m ⟩ = 0. Interaction Frame
Start from a lab-frame Hamiltonian
H ( t ) = H 0 ( t ) + H I ( t ) . H(t)=H_0(t)+H_I(t). H ( t ) = H 0 ( t ) + H I ( t ) . The static or unperturbed evolution is
U 0 ( t ) = T exp [ − i ℏ ∫ 0 t H 0 ( t ′ ) d t ′ ] . U_0(t)=T\exp\left[-\frac{i}{\hbar}\int_0^tH_0(t')\,dt'\right]. U 0 ( t ) = T exp [ − ℏ i ∫ 0 t H 0 ( t ′ ) d t ′ ] . Writing U ( t ) = U 0 ( t ) U I ( t ) U(t)=U_0(t)U_I(t) U ( t ) = U 0 ( t ) U I ( t )
U ˙ I ( t ) = − i ℏ H I ( I ) ( t ) U I ( t ) , \dot U_I(t)=-\frac{i}{\hbar}H_I^{(I)}(t)U_I(t), U ˙ I ( t ) = − ℏ i H I ( I ) ( t ) U I ( t ) , where
H I ( I ) ( t ) = U 0 † ( t ) ( H ( t ) − H 0 ( t ) ) U 0 ( t ) = U 0 † ( t ) H I ( t ) U 0 ( t ) . H_I^{(I)}(t)
=U_0^\dagger(t)\bigl(H(t)-H_0(t)\bigr)U_0(t)
=U_0^\dagger(t)H_I(t)U_0(t). H I ( I ) ( t ) = U 0 † ( t ) ( H ( t ) − H 0 ( t ) ) U 0 ( t ) = U 0 † ( t ) H I ( t ) U 0 ( t ) . Time-Dependent Expansions
Magnus Expansion
For a time-ordered exponential,
T exp ( ∫ 0 t H ( t 1 ) d t 1 ) = exp [ ∫ 0 t H ( t 1 ) d t 1 + 1 2 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H ( t 1 ) , H ( t 2 ) ] + O ( H 3 ) ] . T\exp\left(\int_0^t H(t_1)\,dt_1\right)
= \exp\left[
\int_0^t H(t_1)\,dt_1
+\frac12\int_0^t dt_1\int_0^{t_1}dt_2\,[H(t_1),H(t_2)]
+O(H^3)
\right]. T exp ( ∫ 0 t H ( t 1 ) d t 1 ) = exp [ ∫ 0 t H ( t 1 ) d t 1 + 2 1 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H ( t 1 ) , H ( t 2 )] + O ( H 3 ) ] . For Schrödinger evolution, insert the usual factor − i / ℏ -i/\hbar − i /ℏ
Dyson Expansion
The notes only flag Dyson expansion and effective Hamiltonians. A common starting point is
U ( t ) = T exp [ − i ℏ ∫ 0 t H ( t 1 ) d t 1 ] , U(t)=T\exp\left[-\frac{i}{\hbar}\int_0^t H(t_1)\,dt_1\right], U ( t ) = T exp [ − ℏ i ∫ 0 t H ( t 1 ) d t 1 ] , expanded as a time-ordered series before collecting terms into an effective Hamiltonian.
Jacobi-Anger Expansion
The Bessel-function expansions are
e i z cos θ = ∑ n = − ∞ ∞ i n J n ( z ) e i n θ , e^{iz\cos\theta}
=\sum_{n=-\infty}^{\infty}i^nJ_n(z)e^{in\theta}, e i z c o s θ = n = − ∞ ∑ ∞ i n J n ( z ) e in θ , e i z sin θ = ∑ n = − ∞ ∞ J n ( z ) e i n θ . e^{iz\sin\theta}
=\sum_{n=-\infty}^{\infty}J_n(z)e^{in\theta}. e i z s i n θ = n = − ∞ ∑ ∞ J n ( z ) e in θ . Here J n J_n J n n n n
J − n ( z ) = ( − 1 ) n J n ( z ) , J n ( − z ) = ( − 1 ) n J n ( z ) . J_{-n}(z)=(-1)^nJ_n(z),\qquad
J_n(-z)=(-1)^nJ_n(z). J − n ( z ) = ( − 1 ) n J n ( z ) , J n ( − z ) = ( − 1 ) n J n ( z ) . In trigonometric form,
e i z cos θ = J 0 ( z ) + 2 ∑ n = 1 ∞ i n J n ( z ) cos ( n θ ) . e^{iz\cos\theta}
=J_0(z)+2\sum_{n=1}^{\infty}i^nJ_n(z)\cos(n\theta). e i z c o s θ = J 0 ( z ) + 2 n = 1 ∑ ∞ i n J n ( z ) cos ( n θ ) . Supplementary Proof Notes
Adjoint Action
Let
O ( λ ) = e λ A B e − λ A . O(\lambda)=e^{\lambda A}Be^{-\lambda A}. O ( λ ) = e λ A B e − λ A . Then
d O d λ = A O ( λ ) − O ( λ ) A = [ A , O ( λ ) ] . \frac{dO}{d\lambda}=AO(\lambda)-O(\lambda)A=[A,O(\lambda)]. d λ d O = A O ( λ ) − O ( λ ) A = [ A , O ( λ )] . The nested-commutator series and O ( λ ) O(\lambda) O ( λ ) O ( 0 ) = B O(0)=B O ( 0 ) = B
BCH For Displacements
With
A = α a † , B = − α ∗ a , A=\alpha a^\dagger,\qquad B=-\alpha^*a, A = α a † , B = − α ∗ a , the commutator [ A , B ] = − ∣ α ∣ 2 [A,B]=-|\alpha|^2 [ A , B ] = − ∣ α ∣ 2 D α D β D_\alpha D_\beta D α D β
Schrieffer-Wolff Expansion
Substitute H = H 0 + V H=H_0+V H = H 0 + V
H ′ = H 0 + V + [ S , H 0 ] + [ S , V ] + 1 2 [ S , [ S , H 0 ] ] + 1 2 [ S , [ S , V ] ] + ⋯ . H'=H_0+V+[S,H_0]+[S,V]
+\frac12[S,[S,H_0]]+\frac12[S,[S,V]]+\cdots . H ′ = H 0 + V + [ S , H 0 ] + [ S , V ] + 2 1 [ S , [ S , H 0 ]] + 2 1 [ S , [ S , V ]] + ⋯ . Since S = O ( V ) S=O(V) S = O ( V ) [ S , [ S , V ] ] [S,[S,V]] [ S , [ S , V ]] O ( V 3 ) O(V^3) O ( V 3 ) S S S
V + [ S , H 0 ] = 0. V+[S,H_0]=0. V + [ S , H 0 ] = 0. Then [ S , [ S , H 0 ] ] = − [ S , V ] [S,[S,H_0]]=-[S,V] [ S , [ S , H 0 ]] = − [ S , V ]
H ′ = H 0 + 1 2 [ S , V ] + O ( V 3 ) . H'=H_0+\frac12[S,V]+O(V^3). H ′ = H 0 + 2 1 [ S , V ] + O ( V 3 ) . In the eigenbasis of H 0 H_0 H 0
[ S , H 0 ] m m ′ = ( E m ′ − E m ) S m m ′ , [S,H_0]_{mm'}=(E_{m'}-E_m)S_{mm'}, [ S , H 0 ] m m ′ = ( E m ′ − E m ) S m m ′ , so the first-order cancellation condition gives
S m m ′ = V m m ′ E m − E m ′ , m ≠ m ′ . S_{mm'}=\frac{V_{mm'}}{E_m-E_{m'}},\qquad m\ne m'. S m m ′ = E m − E m ′ V m m ′ , m = m ′ . For Hermitian V V V S † = − S S^\dagger=-S S † = − S e S e^S e S