The passage introduces the fundamental theory of trapped-ion quantum computing.
In Schrodinger picture, the Hamiltonian of the atom’s internal state and motional state is
H atom ( S ) = 1 2 ℏ ω 0 σ z = 1 2 ℏ ω 0 ( ∣ e ⟩ ⟨ e ∣ − ∣ g ⟩ ⟨ g ∣ ) = 1 2 ℏ ω 0 ( σ † σ − σ σ † ) H_\text{atom}^\mathrm{(S)} = \frac{1}{2} \hbar \omega_0 \sigma_z = \frac{1}{2} \hbar \omega_0 (\ket{e}\bra{e} - \ket{g}\bra{g}) = \frac{1}{2} \hbar\omega_0 (\sigma^\dagger\sigma - \sigma\sigma^\dagger)
H atom ( S ) = 2 1 ℏ ω 0 σ z = 2 1 ℏ ω 0 ( ∣ e ⟩ ⟨ e ∣ − ∣ g ⟩ ⟨ g ∣ ) = 2 1 ℏ ω 0 ( σ † σ − σ σ † )
H motion ( S ) = ℏ ν ( a † a + 1 2 ) H_\text{motion}^\mathrm{(S)} = \hbar \nu \left(a^\dagger a+\frac{1}{2} \right)
H motion ( S ) = ℏ ν ( a † a + 2 1 )
The quantized motional modes can be in any specific direction (i.e., a x , a y , a z a_x, a_y, a_z a x , a y , a z x x x x x x x x x a x † a x a_x^\dagger a_x a x † a x
The coupling Hamiltonian describing the interaction between the ion and the electromagnetic field is
H int ( S ) = 1 2 ℏ Ω 0 ( ∣ e ⟩ ⟨ g ∣ + ∣ g ⟩ ⟨ e ∣ ) [ e i ( k x ^ − ω t + ϕ ) + e − i ( k x ^ − ω t + ϕ ) ] H_\text{int}^\mathrm{(S)}=\frac{1}{2} \hbar \Omega_0 (\ket{e}\bra{g} + \ket{g}\bra{e}) \left[\mathrm{e}^{\mathrm{i}(k \hat{x} - \omega t +\phi)} + \mathrm{e}^{-\mathrm{i}(k \hat{x} - \omega t +\phi)}\right]
H int ( S ) = 2 1 ℏ Ω 0 ( ∣ e ⟩ ⟨ g ∣ + ∣ g ⟩ ⟨ e ∣ ) [ e i ( k x ^ − ω t + ϕ ) + e − i ( k x ^ − ω t + ϕ ) ]
where x ^ \hat{x} x ^ E = E 0 [ e i ( k x − ω t + ϕ ) + e − i ( k x − ω t + ϕ ) ] \boldsymbol{E} = \boldsymbol{E}_0 \left[\mathrm{e}^{\mathrm{i}(k x - \omega t +\phi)} + \mathrm{e}^{-\mathrm{i}(kx - \omega t +\phi)}\right] E = E 0 [ e i ( k x − ω t + ϕ ) + e − i ( k x − ω t + ϕ ) ] classical electromagnetic field running in the x x x y y y z z z
/* Important: The electromagnetic field is not quantized! */
Introducing the Lamb-Dicke parameter η = k ℏ 2 m ν \eta = k\sqrt{\frac{\hbar}{2m\nu}} η = k 2 m ν ℏ
k x ^ = η ( a + a † ) k\hat{x} = \eta \,(a+a^\dagger)
k x ^ = η ( a + a † )
and the interaction Hamiltonian
H int ( S ) = 1 2 ℏ Ω 0 ( σ † + σ ) { e i [ η ( a + a † ) − ω t + ϕ ] + e − i [ η ( a + a † ) − ω t + ϕ ] } H_\text{int}^\mathrm{(S)}=\frac{1}{2} \hbar \Omega_0 (\sigma^\dagger +\sigma)\left\{\mathrm{e}^{\mathrm{i}[\eta(a+a^\dagger) - \omega t +\phi]} + \mathrm{e}^{-\mathrm{i}[\eta(a+a^\dagger) - \omega t +\phi]}\right\}
H int ( S ) = 2 1 ℏ Ω 0 ( σ † + σ ) { e i [ η ( a + a † ) − ω t + ϕ ] + e − i [ η ( a + a † ) − ω t + ϕ ] }
Transforming the into the interaction picture, the Hamiltonian is
H int ( I ) = e i H 0 t H int ( S ) e − i H 0 t = 1 2 ℏ Ω 0 e i H A t ( σ † + σ ) e − i H A t ⊗ e i H M t { e i [ η ( a + a † ) − ω t + ϕ ] + e − i [ η ( a + a † ) − ω t + ϕ ] } e − i H M t \begin{aligned}
H_\text{int}^\mathrm{(I)} &= \mathrm{e}^{\mathrm{i}H_0 t} H_\text{int}^\mathrm{(S)} \mathrm{e}^{-\mathrm{i}H_0 t} \\
&= \frac{1}{2} \hbar \Omega_0 \,\mathrm{e}^{\mathrm{i}H_\text{A} t} (\sigma^\dagger +\sigma)\mathrm{e}^{-\mathrm{i}H_\text{A} t} \otimes \mathrm{e}^{\mathrm{i}H_\text{M} t} \left\{\mathrm{e}^{\mathrm{i}[\eta(a+a^\dagger) - \omega t +\phi]} + \mathrm{e}^{-\mathrm{i}[\eta(a+a^\dagger) - \omega t +\phi]}\right\} \mathrm{e}^{-\mathrm{i}H_\text{M} t}
\end{aligned}
H int ( I ) = e i H 0 t H int ( S ) e − i H 0 t = 2 1 ℏ Ω 0 e i H A t ( σ † + σ ) e − i H A t ⊗ e i H M t { e i [ η ( a + a † ) − ω t + ϕ ] + e − i [ η ( a + a † ) − ω t + ϕ ] } e − i H M t
Taking advantage of Baker-Campbell-Hausdorff formula, the Hamiltonian is
H int ( I ) = 1 2 ℏ Ω 0 ( σ † e i ω 0 t + σ e − i ω 0 t ) ⊗ { e i [ η ( a e − i ν t + a † e i ν t ) − ω t + ϕ ] + e − i [ η ( a e − i ν t + a † e i ν t ) − ω t + ϕ ] } H_\text{int}^\mathrm{(I)} = \frac{1}{2} \hbar \Omega_0 (\sigma^\dagger \mathrm{e}^{\mathrm{i} \omega_0 t} + \sigma \mathrm{e}^{-\mathrm{i} \omega_0 t}) \otimes \left\{\mathrm{e}^{\mathrm{i}[\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) - \omega t +\phi]} + \mathrm{e}^{-\mathrm{i}[\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) - \omega t +\phi]}\right\}
H int ( I ) = 2 1 ℏ Ω 0 ( σ † e i ω 0 t + σ e − i ω 0 t ) ⊗ { e i [ η ( a e − i ν t + a † e i ν t ) − ω t + ϕ ] + e − i [ η ( a e − i ν t + a † e i ν t ) − ω t + ϕ ] }
Due to micromotion, the Rabi frequency must be corrected. The effective Rabi frequency Ω \Omega Ω
Ω = Ω 0 1 + q / 2 \Omega = \frac{\Omega_0}{1+q/2}
Ω = 1 + q / 2 Ω 0
where q q q
d 2 x d ξ 2 + [ a − 2 q cos ( 2 ξ ) ] x = 0 \frac{\mathrm{d}^2 x}{\mathrm{d}\xi^2}+[a-2q \cos(2\xi)]x=0
d ξ 2 d 2 x + [ a − 2 q cos ( 2 ξ ) ] x = 0
Typically, a ∼ 1 0 − 3 a \sim 10^{-3} a ∼ 1 0 − 3 q 2 ∼ 0.015 q^2 \sim 0.015 q 2 ∼ 0 . 0 1 5 β = a + q 2 / 2 ∼ 0.09 \beta = \sqrt{a+q^2/2}\sim 0.09 β = a + q 2 / 2 ∼ 0 . 0 9 ν = β ω RF \nu = \beta \omega_\text{RF} ν = β ω RF
Defining the detuning δ : = ω − ω 0 \delta := \omega - \omega_0 δ : = ω − ω 0 RWA (Rotating Wave Approximation):
H int ( I ) = 1 2 ℏ Ω [ σ † e i ϕ e − i δ t ⊗ e i η ( a e − i ν t + a † e i ν t ) + σ e − i ϕ e i δ t ⊗ e − i η ( a e − i ν t + a † e i ν t ) ] H_\text{int}^\mathrm{(I)} = \frac{1}{2} \hbar \Omega \left[\sigma^\dagger \mathrm{e}^{\mathrm{i}\phi}\mathrm{e}^{-\mathrm{i} \delta t} \otimes \mathrm{e}^{\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t})} + \sigma \mathrm{e}^{-\mathrm{i}\phi} \mathrm{e}^{\mathrm{i} \delta t} \otimes\mathrm{e}^{-\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t})} \right]
H int ( I ) = 2 1 ℏ Ω [ σ † e i ϕ e − i δ t ⊗ e i η ( a e − i ν t + a † e i ν t ) + σ e − i ϕ e i δ t ⊗ e − i η ( a e − i ν t + a † e i ν t ) ]
H int ( I ) = 1 2 ℏ Ω σ † e i ϕ e − i δ t exp [ i η ( a e − i ν t + a † e i ν t ) ] + h . c . H_\text{int}^\mathrm{(I)} = \frac{1}{2} \hbar \Omega \sigma^\dagger \mathrm{e}^{\mathrm{i}\phi}\mathrm{e}^{-\mathrm{i} \delta t} \exp \left[\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) \right] + \mathrm{h.c.}
H int ( I ) = 2 1 ℏ Ω σ † e i ϕ e − i δ t exp [ i η ( a e − i ν t + a † e i ν t ) ] + h . c .
The expression of the Hamiltonian can be further simplified in the Lamb-Dicke regime η ⟨ ( a + a † ) 2 ⟩ ≪ 1 \eta \sqrt{\langle (a+a^\dagger)^2 \rangle} \ll 1 η ⟨ ( a + a † ) 2 ⟩ ≪ 1
exp [ i η ( a e − i ν t + a † e i ν t ) ] = 1 + i η ( a e − i ν t + a † e i ν t ) − η 2 2 ( a e − i ν t + a † e i ν t ) 2 + ο ( η 3 ) \exp \left[\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) \right] = 1 + \mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) - \frac{\eta^2}{2} (a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t})^2 +\omicron(\eta^3)
exp [ i η ( a e − i ν t + a † e i ν t ) ] = 1 + i η ( a e − i ν t + a † e i ν t ) − 2 η 2 ( a e − i ν t + a † e i ν t ) 2 + ο ( η 3 )
H int ( I ) = 1 2 ℏ Ω σ † e i ϕ e − i δ t [ 1 + i η ( a e − i ν t + a † e i ν t ) ] + h . c . H_\text{int}^\mathrm{(I)} = \frac{1}{2} \hbar \Omega \sigma^\dagger \mathrm{e}^{\mathrm{i}\phi}\mathrm{e}^{-\mathrm{i} \delta t} \left[1+\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) \right] + \mathrm{h.c.}
H int ( I ) = 2 1 ℏ Ω σ † e i ϕ e − i δ t [ 1 + i η ( a e − i ν t + a † e i ν t ) ] + h . c .
By changing the detuning δ \delta δ
δ = 0 \delta = 0 δ = 0 H car ( I ) = ℏ Ω 2 ( σ † e i ϕ + σ e − i ϕ ) = ℏ Ω 2 σ † e i ϕ + h . c . H_\text{car}^\mathrm{(I)} = \frac{\hbar \Omega}{2} (\sigma^\dagger \mathrm{e}^{\mathrm{i}\phi} + \sigma \mathrm{e}^{-\mathrm{i}\phi}) = \frac{\hbar \Omega}{2} \sigma^\dagger \mathrm{e}^{\mathrm{i}\phi} + \mathrm{h.c.} H car ( I ) = 2 ℏ Ω ( σ † e i ϕ + σ e − i ϕ ) = 2 ℏ Ω σ † e i ϕ + h . c . δ = − ν \delta = -\nu δ = − ν red sideband, H rsb ( I ) = i η ℏ Ω 2 ( σ † a e i ϕ − σ a † e − i ϕ ) = i η ℏ Ω 2 σ † a e i ϕ + h . c . H_\text{rsb}^\mathrm{(I)} = \mathrm{i} \eta \frac{\hbar \Omega}{2} (\sigma^\dagger a \mathrm{e}^{\mathrm{i}\phi}- \sigma a^\dagger \mathrm{e}^{-\mathrm{i}\phi}) =\mathrm{i} \eta \frac{\hbar \Omega}{2} \sigma^\dagger a \mathrm{e}^{\mathrm{i}\phi}+ \mathrm{h.c.} H rsb ( I ) = i η 2 ℏ Ω ( σ † a e i ϕ − σ a † e − i ϕ ) = i η 2 ℏ Ω σ † a e i ϕ + h . c . δ = ν \delta = \nu δ = ν blue sideband, H bsb ( I ) = i η ℏ Ω 2 ( σ † a † e i ϕ − σ a e − i ϕ ) = i η ℏ Ω 2 σ † a † e i ϕ + h . c . H_\text{bsb}^\mathrm{(I)} = \mathrm{i} \eta \frac{\hbar \Omega}{2} (\sigma^\dagger a^\dagger \mathrm{e}^{\mathrm{i}\phi}- \sigma a \mathrm{e}^{-\mathrm{i}\phi}) =\mathrm{i} \eta \frac{\hbar \Omega}{2} \sigma^\dagger a^\dagger \mathrm{e}^{\mathrm{i}\phi}+ \mathrm{h.c.} H bsb ( I ) = i η 2 ℏ Ω ( σ † a † e i ϕ − σ a e − i ϕ ) = i η 2 ℏ Ω σ † a † e i ϕ + h . c .
/* We will set the interaction picture as default. The superscript (I) will be omitted from now on. */
To be continued.
Two superimposed laser fields at positive and negative detuning ± ν \pm \nu ± ν
H bic = i η ℏ Ω 2 ( σ † a e i ϕ r − σ a † e − i ϕ r + σ † a † e i ϕ b − σ a e − i ϕ b ) H_\text{bic} = \mathrm{i} \eta \frac{\hbar \Omega}{2} (\sigma^\dagger a \mathrm{e}^{\mathrm{i}\phi_r}- \sigma a^\dagger \mathrm{e}^{-\mathrm{i}\phi_r} + \sigma^\dagger a^\dagger \mathrm{e}^{\mathrm{i}\phi_b}- \sigma a \mathrm{e}^{-\mathrm{i}\phi_b})
H bic = i η 2 ℏ Ω ( σ † a e i ϕ r − σ a † e − i ϕ r + σ † a † e i ϕ b − σ a e − i ϕ b )
Note: We only consider the superposition of the first sideband Hamiltonians in the expression above. In fact, the complete interaction Hamiltonian should be
H int = H car + H rsb + H bsb = H car + H bic H_\text{int} = H_\text{car} + H_\text{rsb} + H_\text{bsb} = H_\text{car}+H_\text{bic}
H int = H car + H rsb + H bsb = H car + H bic
H int = ℏ Ω cos ( ν t + ϕ − ) ( σ † e i ϕ + + σ e − i ϕ + ) + i η ℏ Ω 2 ( σ † a e i ϕ r − σ a † e − i ϕ r + σ † a † e i ϕ b − σ a e − i ϕ b ) H_\text{int} = \hbar\Omega \cos (\nu t + \phi_-) (\sigma^\dagger \mathrm{e}^{\mathrm{i}\phi_+} + \sigma \mathrm{e}^{-\mathrm{i}\phi_+}) +\mathrm{i} \eta \frac{\hbar \Omega}{2} (\sigma^\dagger a \mathrm{e}^{\mathrm{i}\phi_r}- \sigma a^\dagger \mathrm{e}^{-\mathrm{i}\phi_r} + \sigma^\dagger a^\dagger \mathrm{e}^{\mathrm{i}\phi_b}- \sigma a \mathrm{e}^{-\mathrm{i}\phi_b})
H int = ℏ Ω cos ( ν t + ϕ − ) ( σ † e i ϕ + + σ e − i ϕ + ) + i η 2 ℏ Ω ( σ † a e i ϕ r − σ a † e − i ϕ r + σ † a † e i ϕ b − σ a e − i ϕ b )
Though carrier resonance term H car H_\text{car} H car ν \nu ν Ω \Omega Ω H bic H_\text{bic} H bic η Ω \eta \Omega η Ω H int ≈ H bic H_\text{int} \approx H_\text{bic} H int ≈ H bic
Utilizing the relationship between ladder operators and Pauli operators
σ † = σ x + i σ y σ = σ x − i σ y X = 1 2 ( a + a † ) P = 1 2 i ( a − a † ) \begin{gathered}
\sigma^\dagger = \sigma_x + \mathrm{i} \sigma_y \qquad \sigma = \sigma_x - \mathrm{i} \sigma_y \\
X = \frac{1}{2} (a+a^\dagger) \qquad P = \frac{1}{2\mathrm{i}}(a-a^\dagger)
\end{gathered}
σ † = σ x + i σ y σ = σ x − i σ y X = 2 1 ( a + a † ) P = 2 i 1 ( a − a † )
and defining the following parameters
ϕ + = ϕ b + ϕ r 2 ϕ − = ϕ b − ϕ r 2 \phi_+ = \frac{\phi_b + \phi_r}{2} \qquad \phi_- = \frac{\phi_b - \phi_r}{2}
ϕ + = 2 ϕ b + ϕ r ϕ − = 2 ϕ b − ϕ r
the Hamiltonian of bichromatic light field could take these alternative forms:
H bic = − η ℏ Ω [ ( a + a † ) cos ϕ − − i ( a − a † ) sin ϕ − ] ( σ x sin ϕ + + σ y cos ϕ + ) H_\text{bic} = - \eta \hbar \Omega \left[(a+a^\dagger) \cos \phi_- -\mathrm{i} (a-a^\dagger)\sin\phi_-\right] (\sigma_x \sin\phi_+ +\sigma_y\cos\phi_+)
H bic = − η ℏ Ω [ ( a + a † ) cos ϕ − − i ( a − a † ) sin ϕ − ] ( σ x sin ϕ + + σ y cos ϕ + )
H bic = − 2 η ℏ Ω ( X cos ϕ − + P sin ϕ − ) ( σ x sin ϕ + + σ y cos ϕ + ) H_\text{bic} = -2\eta\hbar\Omega (X \cos\phi_- + P\sin \phi_-) (\sigma_x \sin\phi_+ +\sigma_y\cos\phi_+)
H bic = − 2 η ℏ Ω ( X cos ϕ − + P sin ϕ − ) ( σ x sin ϕ + + σ y cos ϕ + )
H bic = − η ℏ Ω ( a e − i ϕ − + a † e i ϕ − ) σ π / 2 − ϕ + H_\text{bic} = -\eta\hbar\Omega \left(a \mathrm{e}^{-\mathrm{i}\phi_-} + a^\dagger \mathrm{e}^{\mathrm{i}\phi_-}\right) \sigma_{\pi/2-\phi_+}
H bic = − η ℏ Ω ( a e − i ϕ − + a † e i ϕ − ) σ π / 2 − ϕ +
where σ ϕ = σ x cos ϕ + σ y sin ϕ \sigma_\phi =\sigma_x \cos\phi+\sigma_y \sin\phi σ ϕ = σ x cos ϕ + σ y sin ϕ ∣ + ⟩ ϕ \ket{+}_\phi ∣ + ⟩ ϕ ∣ − ⟩ ϕ \ket{-}_\phi ∣ − ⟩ ϕ ( cos ϕ , sin ϕ , 0 ) (\cos\phi,\sin\phi,0) ( cos ϕ , sin ϕ , 0 ) ( − cos ϕ , − sin ϕ , 0 ) (-\cos\phi,-\sin\phi,0) ( − cos ϕ , − sin ϕ , 0 )
The evolution operator (in the interaction picture) is
U I = exp ( − i ℏ H bic t ) = exp [ i η Ω t ( a e − i ϕ − + a † e i ϕ − ) σ π / 2 − ϕ + ] = D ( α σ π / 2 − ϕ + ) U^\mathrm{I} = \exp\left(-\frac{\mathrm{i}}{\hbar}H_\text{bic}t\right) = \exp\left[\mathrm{i}\eta\Omega t (a \mathrm{e}^{-\mathrm{i}\phi_-} + a^\dagger \mathrm{e}^{\mathrm{i}\phi_-}) \sigma_{\pi/2-\phi_+}\right] = D(\alpha \sigma_{\pi/2-\phi_+} )
U I = exp ( − ℏ i H bic t ) = exp [ i η Ω t ( a e − i ϕ − + a † e i ϕ − ) σ π / 2 − ϕ + ] = D ( α σ π / 2 − ϕ + )
where α = i η Ω t e i ϕ − \alpha = \mathrm{i}\eta\Omega t \,\mathrm{e}^{\mathrm{i}\phi_-} α = i η Ω t e i ϕ − /* The displacement operator is D ( α ) = exp ( α a † − α ⋆ a ) D(\alpha) = \exp(\alpha a^\dagger-\alpha^\star a) D ( α ) = exp ( α a † − α ⋆ a )
Two superimposed laser fields at positive and negative detuning ± ( ν + Δ ) \pm (\nu+\Delta) ± ( ν + Δ )
H bic = i η ℏ Ω 2 ( σ † a e i Δ t + i ϕ r − σ a † e − i Δ t − i ϕ r + σ † a † e − i Δ t + i ϕ b − σ a e i Δ t − i ϕ b ) H_\text{bic} = \mathrm{i} \eta \frac{\hbar \Omega}{2} (\sigma^\dagger a \mathrm{e}^{\mathrm{i} \Delta t +\mathrm{i}\phi_r}- \sigma a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t -\mathrm{i}\phi_r} + \sigma^\dagger a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t+\mathrm{i}\phi_b}- \sigma a \mathrm{e}^{\mathrm{i}\Delta t-\mathrm{i}\phi_b})
H bic = i η 2 ℏ Ω ( σ † a e i Δ t + i ϕ r − σ a † e − i Δ t − i ϕ r + σ † a † e − i Δ t + i ϕ b − σ a e i Δ t − i ϕ b )
which is equivalent to a transformation with ϕ b → ϕ b − Δ t \phi_b \to \phi_b - \Delta t ϕ b → ϕ b − Δ t ϕ r → ϕ r + Δ t \phi_r \to \phi_r +\Delta t ϕ r → ϕ r + Δ t ϕ + , ϕ − \phi_+, \phi_- ϕ + , ϕ −
ϕ + = ϕ b + ϕ r 2 → ϕ + ϕ − = ϕ b − ϕ r 2 → ϕ − − Δ t \phi_+ = \frac{\phi_b + \phi_r}{2}\to \phi_+ \qquad \phi_- = \frac{\phi_b - \phi_r}{2} \to \phi_- - \Delta t
ϕ + = 2 ϕ b + ϕ r → ϕ + ϕ − = 2 ϕ b − ϕ r → ϕ − − Δ t
The interaction Hamiltonian takes the form
H bic = − η ℏ Ω ( a e i Δ t − i ϕ − + a † e − i Δ t + i ϕ − ) σ π / 2 − ϕ + H_\text{bic} = -\eta\hbar\Omega \left(a \mathrm{e}^{\mathrm{i}\Delta t-\mathrm{i}\phi_-} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t+\mathrm{i}\phi_-}\right) \sigma_{\pi/2-\phi_+}
H bic = − η ℏ Ω ( a e i Δ t − i ϕ − + a † e − i Δ t + i ϕ − ) σ π / 2 − ϕ +
where σ ϕ = σ x cos ϕ + σ y sin ϕ \sigma_\phi =\sigma_x \cos\phi+\sigma_y \sin\phi σ ϕ = σ x cos ϕ + σ y sin ϕ ∣ + ⟩ ϕ \ket{+}_\phi ∣ + ⟩ ϕ ∣ − ⟩ ϕ \ket{-}_\phi ∣ − ⟩ ϕ ( cos ϕ , sin ϕ , 0 ) (\cos\phi,\sin\phi,0) ( cos ϕ , sin ϕ , 0 ) ( − cos ϕ , − sin ϕ , 0 ) (-\cos\phi,-\sin\phi,0) ( − cos ϕ , − sin ϕ , 0 )
Since the Hamiltonian is time-dependent, we must use the Magnus expansion to calculate the evolution operator
U I = exp ( − i ℏ ∫ 0 t H int d t − 1 2 ℏ 2 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] + ⋯ ) U^\mathrm{I} = \exp\left(-\frac{\mathrm{i}}{\hbar} \int_0^t H_\text{int}\,\mathrm{d}t -\frac{1}{2\hbar^2} \int_0^t \mathrm{d}t_1 \int_0^{t_1} \mathrm{d}t_2 \, [H_\text{int}(t_1),H_\text{int}(t_2)] +\cdots\right)
U I = exp ( − ℏ i ∫ 0 t H int d t − 2 ℏ 2 1 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] + ⋯ )
Now, we must fix ϕ + \phi_+ ϕ + ± π / 2 \pm \pi/2 ± π / 2 σ π / 2 − ϕ + = σ x \sigma_{\pi/2-\phi_+} = \sigma_x σ π / 2 − ϕ + = σ x ϕ + \phi_+ ϕ + σ π / 2 − ϕ + = σ y \sigma_{\pi/2-\phi_+} = \sigma_y σ π / 2 − ϕ + = σ y σ ϕ \sigma_\phi σ ϕ σ x , σ y , σ z \sigma_x,\sigma_y,\sigma_z σ x , σ y , σ z [ H int ( t 1 ) , H int ( t 2 ) ] [H_\text{int}(t_1),H_\text{int}(t_2)] [ H int ( t 1 ) , H int ( t 2 ) ]
We choose σ x \sigma_x σ x
exp ( − i ℏ ∫ 0 t H int d t ) = exp ( η Ω Δ [ a † e i ϕ − ( 1 − e − i Δ t ) − a e − i ϕ − ( 1 − e i Δ t ) ] σ x ) \exp\left(-\frac{\mathrm{i}}{\hbar} \int_0^t H_\text{int}\,\mathrm{d}t\right) = \exp \left( \frac{\eta\Omega}{\Delta} \left[a^\dagger \mathrm{e}^{\mathrm{i}\phi_-} (1-\mathrm{e}^{-\mathrm{i}\Delta t})-a \mathrm{e}^{-\mathrm{i}\phi_-}(1-\mathrm{e}^{\mathrm{i}\Delta t})\right] \sigma_x\right)
exp ( − ℏ i ∫ 0 t H int d t ) = exp ( Δ η Ω [ a † e i ϕ − ( 1 − e − i Δ t ) − a e − i ϕ − ( 1 − e i Δ t ) ] σ x )
exp ( − 1 2 ℏ 2 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] ) = exp [ − i ( η Ω Δ ) 2 ( Δ t − sin Δ t ) ] \exp\left(-\frac{1}{2\hbar^2} \int_0^t \mathrm{d}t_1 \int_0^{t_1} \mathrm{d}t_2 \, [H_\text{int}(t_1),H_\text{int}(t_2)]\right) = \exp \left[ -\mathrm{i}\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t)\right]
exp ( − 2 ℏ 2 1 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] ) = exp [ − i ( Δ η Ω ) 2 ( Δ t − sin Δ t ) ]
Using Baker-Campbell-Hausdorff formula e A + B = e A e B e − [ A , B ] / 2 \mathrm{e}^{A+B} = \mathrm{e}^A \mathrm{e}^B \mathrm{e}^{-[A,B]/2} e A + B = e A e B e − [ A , B ] / 2 e − [ A , B ] / 2 \mathrm{e}^{-[A,B]/2} e − [ A , B ] / 2
U I = exp ( η Ω Δ [ a † e i ϕ − ( 1 − e − i Δ t ) − a e − i ϕ − ( 1 − e i Δ t ) ] σ x ) exp [ − i ( η Ω Δ ) 2 ( Δ t − sin Δ t ) ] U^\mathrm{I} = \exp \left( \frac{\eta\Omega}{\Delta} \left[a^\dagger \mathrm{e}^{\mathrm{i}\phi_-} (1-\mathrm{e}^{-\mathrm{i}\Delta t})-a \mathrm{e}^{-\mathrm{i}\phi_-}(1-\mathrm{e}^{\mathrm{i}\Delta t})\right] \sigma_x\right) \exp \left[ -\mathrm{i}\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t)\right]
U I = exp ( Δ η Ω [ a † e i ϕ − ( 1 − e − i Δ t ) − a e − i ϕ − ( 1 − e i Δ t ) ] σ x ) exp [ − i ( Δ η Ω ) 2 ( Δ t − sin Δ t ) ]
The evolution operator can be decomposed into a displacement operator and a global phase
U I = D [ α ( t ) σ x ] e − i Φ ( t ) U^I = D[\alpha(t) \sigma_x] \, \mathrm{e}^{-\mathrm{i}\Phi(t)}
U I = D [ α ( t ) σ x ] e − i Φ ( t )
α ( t ) = η Ω Δ ( 1 − e − i Δ t ) e i ϕ − Φ ( t ) = ( η Ω Δ ) 2 ( Δ t − sin Δ t ) \alpha(t)= \frac{\eta\Omega}{\Delta} (1-\mathrm{e}^{-\mathrm{i}\Delta t}) \mathrm{e}^{\mathrm{i}\phi_-} \qquad \Phi(t) =\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t)
α ( t ) = Δ η Ω ( 1 − e − i Δ t ) e i ϕ − Φ ( t ) = ( Δ η Ω ) 2 ( Δ t − sin Δ t )
For the displacement parameter, η Ω / Δ \eta \Omega / \Delta η Ω / Δ e i ϕ − \mathrm{e}^{\mathrm{i}\phi_-} e i ϕ − 2 π / Δ 2\pi / \Delta 2 π / Δ
For the global phase, at time t = 2 π N / Δ t = 2\pi N/\Delta t = 2 π N / Δ 2 π N ( η Ω Δ ) 2 2\pi N \left(\frac{\eta\Omega}{\Delta}\right)^2 2 π N ( Δ η Ω ) 2
The Pauli operator acting on the internal state space makes the displacement spin-dependent. For ∣ + ⟩ x \ket{+}_x ∣ + ⟩ x α ( t ) \alpha(t) α ( t ) ∣ − ⟩ x \ket{-}_x ∣ − ⟩ x − α ( t ) -\alpha(t) − α ( t )
On-resonant bichromatic field
H bic ∼ η ℏ Ω ( a + a † ) σ ϕ H_\text{bic} \sim \eta\hbar\Omega (a+a^\dagger) \sigma_\phi
H bic ∼ η ℏ Ω ( a + a † ) σ ϕ
U I ∼ D ( α ( t ) σ ϕ ) , ∣ α ∣ ∼ η Ω t U^\mathrm{I} \sim D(\alpha(t)\sigma_\phi), \, |\alpha| \sim\eta\Omega t
U I ∼ D ( α ( t ) σ ϕ ) , ∣ α ∣ ∼ η Ω t
Off-resonant bichromatic field
H bic ∼ η ℏ Ω ( a e i Δ t + a † e − i Δ t ) σ ϕ H_\text{bic} \sim \eta\hbar\Omega (a \mathrm{e}^{\mathrm{i}\Delta t} +a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t}) \sigma_\phi
H bic ∼ η ℏ Ω ( a e i Δ t + a † e − i Δ t ) σ ϕ
U I ∼ D [ α ( t ) σ ϕ ] e − i Φ ( t ) U^\mathrm{I} \sim D[\alpha(t)\sigma_\phi] \, \mathrm{e}^{-\mathrm{i}\Phi(t)}
U I ∼ D [ α ( t ) σ ϕ ] e − i Φ ( t )
∣ α ∣ ∼ η Ω Δ ( 1 − e − i Δ t ) Φ ( t ) = ( η Ω Δ ) 2 ( Δ t − sin Δ t ) |\alpha| \sim \frac{\eta\Omega}{\Delta} (1-\mathrm{e}^{-\mathrm{i}\Delta t}) \qquad \Phi(t) =\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t)
∣ α ∣ ∼ Δ η Ω ( 1 − e − i Δ t ) Φ ( t ) = ( Δ η Ω ) 2 ( Δ t − sin Δ t )
When considering the second sideband, the Taylor expansion of exp [ i η ( a e − i ν t + a † e i ν t ) ] \exp \left[\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) \right] exp [ i η ( a e − i ν t + a † e i ν t ) ]
exp [ i η ( a e − i ν t + a † e i ν t ) ] = 1 + i η ( a e − i ν t + a † e i ν t ) − η 2 2 ( a e − i ν t + a † e i ν t ) 2 + o ( η 3 ) \exp \left[\mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) \right]= 1 + \mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) - \frac{\eta^2}{2} (a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t})^2 +o(\eta^3)
exp [ i η ( a e − i ν t + a † e i ν t ) ] = 1 + i η ( a e − i ν t + a † e i ν t ) − 2 η 2 ( a e − i ν t + a † e i ν t ) 2 + o ( η 3 )
H int ( I ) = 1 2 ℏ Ω σ † e i ϕ e − i δ t [ 1 − η 2 2 ( 2 a † a + 1 ) + i η ( a e − i ν t + a † e i ν t ) − η 2 2 ( a 2 e − i 2 ν t + a † 2 e i 2 ν t ) ] + h . c . H_\text{int}^\mathrm{(I)} = \frac{1}{2} \hbar \Omega \sigma^\dagger \mathrm{e}^{\mathrm{i}\phi}\mathrm{e}^{-\mathrm{i} \delta t} \left[1-\frac{\eta^2}{2}(2a^\dagger a+1) + \mathrm{i}\eta(a \mathrm{e}^{-\mathrm{i} \nu t}+a^\dagger \mathrm{e}^{\mathrm{i} \nu t}) - \frac{\eta^2}{2} (a^2 \mathrm{e}^{-\mathrm{i} 2\nu t}+a^{\dagger 2} \mathrm{e}^{\mathrm{i} 2\nu t})\right] + \mathrm{h.c.}
H int ( I ) = 2 1 ℏ Ω σ † e i ϕ e − i δ t [ 1 − 2 η 2 ( 2 a † a + 1 ) + i η ( a e − i ν t + a † e i ν t ) − 2 η 2 ( a 2 e − i 2 ν t + a † 2 e i 2 ν t ) ] + h . c .
By changing the detuning δ \delta δ
δ = − 2 ν \delta = -2\nu δ = − 2 ν red sideband, H rsb2 = − 1 4 η 2 ℏ Ω ( σ † a 2 e i ϕ + σ a † 2 e − i ϕ ) = − 1 4 η 2 ℏ Ω σ † a 2 e i ϕ + h . c . H_\text{rsb2} = -\frac{1}{4} \eta^2 \hbar \Omega (\sigma^\dagger a^2 \mathrm{e}^{\mathrm{i}\phi} + \sigma a^{\dagger 2} \mathrm{e}^{-\mathrm{i}\phi}) =-\frac{1}{4} \eta^2 \hbar \Omega \sigma^\dagger a^2 \mathrm{e}^{\mathrm{i}\phi} + \mathrm{h.c.} H rsb2 = − 4 1 η 2 ℏ Ω ( σ † a 2 e i ϕ + σ a † 2 e − i ϕ ) = − 4 1 η 2 ℏ Ω σ † a 2 e i ϕ + h . c . δ = 2 ν \delta = 2\nu δ = 2 ν blue sideband, H bsb2 = − 1 4 η 2 ℏ Ω ( σ † a † 2 e i ϕ + σ a 2 e − i ϕ ) = − 1 4 η 2 ℏ Ω σ † a † 2 e i ϕ + h . c . H_\text{bsb2} = -\frac{1}{4} \eta^2 \hbar \Omega (\sigma^\dagger a^{\dagger 2} \mathrm{e}^{\mathrm{i}\phi} + \sigma a^2 \mathrm{e}^{-\mathrm{i}\phi}) =-\frac{1}{4} \eta^2 \hbar \Omega \sigma^\dagger a^{\dagger 2} \mathrm{e}^{\mathrm{i}\phi} + \mathrm{h.c.} H bsb2 = − 4 1 η 2 ℏ Ω ( σ † a † 2 e i ϕ + σ a 2 e − i ϕ ) = − 4 1 η 2 ℏ Ω σ † a † 2 e i ϕ + h . c .
The bichromatic field constituted by two second sideband light are
H bic2 = − 1 4 η 2 ℏ Ω ( σ † a 2 e i ϕ r + σ a † 2 e − i ϕ r + σ † a † 2 e i ϕ b + σ a 2 e − i ϕ b ) H_\text{bic2} = -\frac{1}{4} \eta^2 \hbar \Omega (\sigma^\dagger a^2 \mathrm{e}^{\mathrm{i}\phi_r} + \sigma a^{\dagger 2} \mathrm{e}^{-\mathrm{i}\phi_r} + \sigma^\dagger a^{\dagger 2} \mathrm{e}^{\mathrm{i}\phi_b} + \sigma a^2 \mathrm{e}^{-\mathrm{i}\phi_b})
H bic2 = − 4 1 η 2 ℏ Ω ( σ † a 2 e i ϕ r + σ a † 2 e − i ϕ r + σ † a † 2 e i ϕ b + σ a 2 e − i ϕ b )
Still defining ϕ + = ( ϕ b + ϕ r ) / 2 , ϕ − = ( ϕ b − ϕ r ) / 2 \phi_+ = (\phi_b + \phi_r)/2,\phi_- = (\phi_b - \phi_r)/2 ϕ + = ( ϕ b + ϕ r ) / 2 , ϕ − = ( ϕ b − ϕ r ) / 2
H bic2 = − 1 2 η 2 ℏ Ω ( a 2 e − i ϕ − + a † 2 e i ϕ − ) ( σ x cos ϕ + − σ y sin ϕ + ) H_\text{bic2} = -\frac{1}{2} \eta^2 \hbar \Omega (a^2 \mathrm{e}^{-\mathrm{i}\phi_-} + a^{\dagger 2} \mathrm{e}^{\mathrm{i}\phi_-})(\sigma_x \cos\phi_+ -\sigma_y \sin\phi_+)
H bic2 = − 2 1 η 2 ℏ Ω ( a 2 e − i ϕ − + a † 2 e i ϕ − ) ( σ x cos ϕ + − σ y sin ϕ + )
The evolution operator is thus
U I = exp ( − i ℏ H bic2 t ) = exp [ i 2 η 2 Ω t ( a 2 e − i ϕ − + a † 2 e i ϕ − ) σ − ϕ + ] = S ( ξ σ − ϕ + ) U^\mathrm{I} = \exp\left(-\frac{\mathrm{i}}{\hbar}H_\text{bic2}t\right) = \exp\left[\frac{\mathrm{i}}{2} \eta^2 \Omega t (a^2 \mathrm{e}^{-\mathrm{i}\phi_-} + a^{\dagger 2} \mathrm{e}^{\mathrm{i}\phi_-}) \sigma_{-\phi_+}\right] = S(\xi \sigma_{-\phi_+} )
U I = exp ( − ℏ i H bic2 t ) = exp [ 2 i η 2 Ω t ( a 2 e − i ϕ − + a † 2 e i ϕ − ) σ − ϕ + ] = S ( ξ σ − ϕ + )
where S S S
S ( ξ ) = exp ( 1 2 ξ ∗ a 2 − 1 2 ξ a † 2 ) , ξ = − i 2 η 2 Ω t e i ϕ − S(\xi)=\exp \left(\frac{1}{2}{\xi^* a^2 -\frac{1}{2} \xi a^{\dagger 2}}\right) ,\quad \xi = -\frac{\mathrm{i}}{2}\eta^2 \Omega t \mathrm{e}^{\mathrm{i}\phi_-}
S ( ξ ) = exp ( 2 1 ξ ∗ a 2 − 2 1 ξ a † 2 ) , ξ = − 2 i η 2 Ω t e i ϕ −
New scheme include 4 beams of light, 2 pairs of bichromatic field.
A bichromatic field with detuning ± ( ν + Δ ) \pm (\nu+\Delta) ± ( ν + Δ ) H bic ± ( ν + Δ ) = − η ℏ Ω ( a e i Δ t − i ϕ − + a † e − i Δ t + i ϕ − ) σ π / 2 − ϕ + H_\text{bic}^{\pm (\nu+\Delta)} = -\eta\hbar\Omega \left(a \mathrm{e}^{\mathrm{i}\Delta t-\mathrm{i}\phi_-} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t+\mathrm{i}\phi_-}\right) \sigma_{\pi/2-\phi_+}
H bic ± ( ν + Δ ) = − η ℏ Ω ( a e i Δ t − i ϕ − + a † e − i Δ t + i ϕ − ) σ π / 2 − ϕ +
Given ϕ − = 0 , ϕ + = ± π / 2 \phi_- = 0, \phi_+ = \pm \pi/2 ϕ − = 0 , ϕ + = ± π / 2 H bic ± ( ν + Δ ) = − η ℏ Ω ( a e i Δ t + a † e − i Δ t ) σ x H_\text{bic}^{\pm (\nu+\Delta)} = -\eta\hbar\Omega \left(a \mathrm{e}^{\mathrm{i}\Delta t} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t}\right) \sigma_{x}
H bic ± ( ν + Δ ) = − η ℏ Ω ( a e i Δ t + a † e − i Δ t ) σ x
A bichromatic field with detuning ± ( ν − Δ ) \pm (\nu-\Delta) ± ( ν − Δ ) H bic ± ( ν − Δ ) = − η ℏ Ω ( a e − i Δ t − i ϕ − + a † e i Δ t + i ϕ − ) σ π / 2 − ϕ + H_\text{bic}^{\pm (\nu-\Delta)} = -\eta\hbar\Omega \left(a \mathrm{e}^{-\mathrm{i}\Delta t-\mathrm{i}\phi_-} + a^\dagger \mathrm{e}^{\mathrm{i}\Delta t+\mathrm{i}\phi_-}\right) \sigma_{\pi/2-\phi_+}
H bic ± ( ν − Δ ) = − η ℏ Ω ( a e − i Δ t − i ϕ − + a † e i Δ t + i ϕ − ) σ π / 2 − ϕ +
Given ϕ − → ϕ eff \phi_- \to \phi_\text{eff} ϕ − → ϕ eff ϕ + = 0 \phi_+ = 0 ϕ + = 0 H bic ± ( ν − Δ ) = − η ℏ Ω ( a e − i Δ t − i ϕ eff + a † e i Δ t + i ϕ eff ) σ y H_\text{bic}^{\pm (\nu-\Delta)} = -\eta\hbar\Omega \left(a \mathrm{e}^{-\mathrm{i}\Delta t-\mathrm{i}\phi_\text{eff}} + a^\dagger \mathrm{e}^{\mathrm{i}\Delta t+\mathrm{i}\phi_\text{eff}}\right) \sigma_y
H bic ± ( ν − Δ ) = − η ℏ Ω ( a e − i Δ t − i ϕ eff + a † e i Δ t + i ϕ eff ) σ y
The total interaction Hamiltonian is
H int = − η ℏ Ω [ ( a e i Δ t + a † e − i Δ t ) σ x + ( a e − i Δ t − i ϕ eff + a † e i Δ t + i ϕ eff ) σ y ] H_\text{int} = -\eta\hbar\Omega \left[ (a \mathrm{e}^{\mathrm{i}\Delta t} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t}) \sigma_{x} + (a \mathrm{e}^{-\mathrm{i}\Delta t-\mathrm{i}\phi_\text{eff}} + a^\dagger \mathrm{e}^{\mathrm{i}\Delta t+\mathrm{i}\phi_\text{eff}}) \sigma_y\right]
H int = − η ℏ Ω [ ( a e i Δ t + a † e − i Δ t ) σ x + ( a e − i Δ t − i ϕ eff + a † e i Δ t + i ϕ eff ) σ y ]
The first order term still behaves like a displacement operator
exp ( − i ℏ ∫ 0 t H int d t ) ∼ D [ α 1 ( t ) σ x + α 2 ( t ) σ y ] \exp\left(-\frac{\mathrm{i}}{\hbar} \int_0^t H_\text{int}\,\mathrm{d}t\right) \sim D[\alpha_1(t)\sigma_x+ \alpha_2(t)\sigma_y]
exp ( − ℏ i ∫ 0 t H int d t ) ∼ D [ α 1 ( t ) σ x + α 2 ( t ) σ y ]
where ∣ α 1 ∣ ∼ η Ω Δ ( 1 − e − i Δ t ) , ∣ α 2 ∣ ∼ η Ω Δ ( 1 − e i Δ t ) |\alpha_1| \sim \frac{\eta\Omega}{\Delta} (1-\mathrm{e}^{-\mathrm{i}\Delta t}),\, |\alpha_2| \sim \frac{\eta\Omega}{\Delta} (1-\mathrm{e}^{\mathrm{i}\Delta t}) ∣ α 1 ∣ ∼ Δ η Ω ( 1 − e − i Δ t ) , ∣ α 2 ∣ ∼ Δ η Ω ( 1 − e i Δ t ) t = 2 π N / Δ t = 2\pi N / \Delta t = 2 π N / Δ
The second order term, however, differs a lot. When calculating the commutator [ H int ( t 1 ) , H int ( t 2 ) ] [H_\text{int}(t_1),H_\text{int}(t_2)] [ H int ( t 1 ) , H int ( t 2 ) ]
[ ( a e i Δ t 1 + a † e − i Δ t 1 ) σ x , ( a e i Δ t 2 + a † e − i Δ t 2 ) σ x ] [(a \mathrm{e}^{\mathrm{i}\Delta t_1} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t_1}) \sigma_{x}, (a \mathrm{e}^{\mathrm{i}\Delta t_2} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t_2}) \sigma_{x}] [ ( a e i Δ t 1 + a † e − i Δ t 1 ) σ x , ( a e i Δ t 2 + a † e − i Δ t 2 ) σ x ] Φ 1 ( t ) = ( η Ω Δ ) 2 ( Δ t − sin Δ t ) \Phi_1(t) =\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t) Φ 1 ( t ) = ( Δ η Ω ) 2 ( Δ t − sin Δ t ) [ ( a e i Δ t 1 + a † e − i Δ t 1 ) σ y , ( a e i Δ t 2 + a † e − i Δ t 2 ) σ y ] [(a \mathrm{e}^{\mathrm{i}\Delta t_1} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t_1}) \sigma_{y}, (a \mathrm{e}^{\mathrm{i}\Delta t_2} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t_2}) \sigma_{y}] [ ( a e i Δ t 1 + a † e − i Δ t 1 ) σ y , ( a e i Δ t 2 + a † e − i Δ t 2 ) σ y ] Φ 2 ( t ) = ( η Ω Δ ) 2 ( Δ t − sin Δ t ) \Phi_2(t) =\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t) Φ 2 ( t ) = ( Δ η Ω ) 2 ( Δ t − sin Δ t ) Cross-terms of [ σ x , σ y ] [\sigma_x,\sigma_y] [ σ x , σ y ] exp [ i η 2 Ω 2 Δ t − sin Δ t Δ 2 ( a 2 e − i ϕ eff + a † 2 e i ϕ eff ) σ z ] \exp [\mathrm{i} \eta^2 \Omega^2 \frac{\Delta t - \sin \Delta t}{\Delta^2} (a^2 \mathrm{e}^{-\mathrm{i}\phi_\text{eff}} + a^{\dagger 2} \mathrm{e}^{\mathrm{i}\phi_\text{eff}}) \sigma_z] exp [ i η 2 Ω 2 Δ 2 Δ t − s i n Δ t ( a 2 e − i ϕ eff + a † 2 e i ϕ eff ) σ z ]
so the second-order term is
exp ( − 1 2 ℏ 2 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] ) ∼ S [ ξ ( t ) σ z ] e − i Φ ( t ) \exp\left(-\frac{1}{2\hbar^2} \int_0^t \mathrm{d}t_1 \int_0^{t_1} \mathrm{d}t_2 \, [H_\text{int}(t_1),H_\text{int}(t_2)]\right) \sim S[\xi(t) \sigma_z]\mathrm{e}^{-\mathrm{i}\Phi(t)}
exp ( − 2 ℏ 2 1 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] ) ∼ S [ ξ ( t ) σ z ] e − i Φ ( t )
where ∣ ξ ∣ ∼ ( η Ω Δ ) 2 ( Δ t − sin Δ t ) |\xi| \sim \left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t) ∣ ξ ∣ ∼ ( Δ η Ω ) 2 ( Δ t − sin Δ t ) t = 2 π N / Δ t = 2\pi N / \Delta t = 2 π N / Δ ∣ ξ ∣ ∼ η 2 Ω 2 Δ t |\xi| \sim \frac{\eta^2\Omega^2}{\Delta}t ∣ ξ ∣ ∼ Δ η 2 Ω 2 t
Using Baker-Campbell-Hausdorff formula e A + B = e A e B e − [ A , B ] / 2 \mathrm{e}^{A+B} = \mathrm{e}^A \mathrm{e}^B \mathrm{e}^{-[A,B]/2} e A + B = e A e B e − [ A , B ] / 2 e − [ A , B ] / 2 \mathrm{e}^{-[A,B]/2} e − [ A , B ] / 2 t = 2 π N / Δ t = 2\pi N / \Delta t = 2 π N / Δ [ A , B ] [A,B] [ A , B ] η Ω \eta \Omega η Ω
Old scheme
H int ∼ η 2 ℏ Ω ( a 2 + a † 2 ) H_\text{int} \sim \eta^2 \hbar\Omega(a^2 +a^{\dagger2})
H int ∼ η 2 ℏ Ω ( a 2 + a † 2 )
U I ∼ S [ ξ ( t ) σ ϕ ] , ∣ ξ ∣ ∼ η 2 Ω t U^{\mathrm{I}} \sim S[\xi(t) \sigma_\phi],\quad |\xi| \sim \eta^2 \Omega t
U I ∼ S [ ξ ( t ) σ ϕ ] , ∣ ξ ∣ ∼ η 2 Ω t
New scheme
H int ∼ η ℏ Ω [ ( a e i Δ t + a † e − i Δ t ) σ x + ( a e − i Δ t + a † e i Δ t ) σ y ] H_\text{int} \sim \eta \hbar\Omega \left[ (a \mathrm{e}^{\mathrm{i}\Delta t} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t}) \sigma_{x} + (a \mathrm{e}^{-\mathrm{i}\Delta t} + a^\dagger \mathrm{e}^{\mathrm{i}\Delta t}) \sigma_y\right]
H int ∼ η ℏ Ω [ ( a e i Δ t + a † e − i Δ t ) σ x + ( a e − i Δ t + a † e i Δ t ) σ y ]
U I ∼ D [ α 1 ( t ) σ x + α 2 ( t ) σ y ] S [ ξ ( t ) σ z ] e − i Φ ( t ) U^{\mathrm{I}} \sim D[\alpha_1(t)\sigma_x+ \alpha_2(t)\sigma_y] S[\xi(t) \sigma_z]\mathrm{e}^{-\mathrm{i}\Phi(t)}
U I ∼ D [ α 1 ( t ) σ x + α 2 ( t ) σ y ] S [ ξ ( t ) σ z ] e − i Φ ( t )
∣ α ∣ ∼ η Ω Δ ( 1 − e − i Δ t ) ∣ ξ ∣ ∼ ( η Ω Δ ) 2 ( Δ t − sin Δ t ) Φ ( t ) ∼ ( η Ω Δ ) 2 ( Δ t − sin Δ t ) |\alpha| \sim \frac{\eta\Omega}{\Delta} (1-\mathrm{e}^{-\mathrm{i}\Delta t}) \qquad |\xi| \sim \left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t) \qquad \Phi(t) \sim\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t)
∣ α ∣ ∼ Δ η Ω ( 1 − e − i Δ t ) ∣ ξ ∣ ∼ ( Δ η Ω ) 2 ( Δ t − sin Δ t ) Φ ( t ) ∼ ( Δ η Ω ) 2 ( Δ t − sin Δ t )
Question 1: Could we achieve the squeezing operation with two on-resonant first sideband beams and take ϕ = π / 4 \phi = \pi /4 ϕ = π / 4
H ∼ η ℏ Ω ( a + a † ) ( σ x + σ y ) H \sim\eta\hbar\Omega (a+a^\dagger) (\sigma_x + \sigma_y)
H ∼ η ℏ Ω ( a + a † ) ( σ x + σ y )
Answer: No. On-resonant fields are time-dependent. [ H ( t 1 ) , H ( t 2 ) ] [H(t_1),H(t_2)] [ H ( t 1 ) , H ( t 2 ) ] U I = exp ( − i ℏ H t ) U^\mathrm{I} = \exp(-\frac{\mathrm{i}}{\hbar}Ht) U I = exp ( − ℏ i H t )
Question 2: Could we achieve the squeezing operation with two off-resonant first sideband beams and take ϕ = π / 4 \phi = \pi/4 ϕ = π / 4
Answer: No. Taking a bichromatic field with detuning ± ( ν + Δ ) \pm (\nu+\Delta) ± ( ν + Δ )
H ∼ η ℏ Ω ( a e i Δ t + a † e − i Δ t ) ( σ x + σ y ) H \sim \eta\hbar\Omega \left(a \mathrm{e}^{\mathrm{i}\Delta t} + a^\dagger \mathrm{e}^{-\mathrm{i}\Delta t}\right)(\sigma_x+\sigma_y)
H ∼ η ℏ Ω ( a e i Δ t + a † e − i Δ t ) ( σ x + σ y )
The first-order term is
exp ( − i ℏ ∫ 0 t H int d t ) = exp ( η Ω Δ [ a † ( 1 − e − i Δ t ) − a ( 1 − e i Δ t ) ] [ σ x + σ y ] ) ∼ D [ α ( t ) ( σ x + σ y ) ] \exp\left(-\frac{\mathrm{i}}{\hbar} \int_0^t H_\text{int}\,\mathrm{d}t\right) = \exp \left( \frac{\eta\Omega}{\Delta} \left[a^\dagger (1-\mathrm{e}^{-\mathrm{i}\Delta t})-a (1-\mathrm{e}^{\mathrm{i}\Delta t})\right] [\sigma_x+\sigma_y]\right) \sim D[\alpha(t) (\sigma_x+\sigma_y)]
exp ( − ℏ i ∫ 0 t H int d t ) = exp ( Δ η Ω [ a † ( 1 − e − i Δ t ) − a ( 1 − e i Δ t ) ] [ σ x + σ y ] ) ∼ D [ α ( t ) ( σ x + σ y ) ]
To avoid redundant displacement, we must select t = 2 π N / Δ t = 2\pi N / \Delta t = 2 π N / Δ
The second-order term is
exp ( − 1 2 ℏ 2 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] ) = exp [ − 2 i ( η Ω Δ ) 2 ( Δ t − sin Δ t ) ] \exp\left(-\frac{1}{2\hbar^2} \int_0^t \mathrm{d}t_1 \int_0^{t_1} \mathrm{d}t_2 \, [H_\text{int}(t_1),H_\text{int}(t_2)]\right) = \exp \left[ -2\mathrm{i}\left(\frac{\eta\Omega}{\Delta}\right)^2 (\Delta t - \sin \Delta t)\right]
exp ( − 2 ℏ 2 1 ∫ 0 t d t 1 ∫ 0 t 1 d t 2 [ H int ( t 1 ) , H int ( t 2 ) ] ) = exp [ − 2 i ( Δ η Ω ) 2 ( Δ t − sin Δ t ) ]
No a 2 a^2 a 2 a † 2 a^{\dagger2} a † 2
