Nonclassical Properties of an Opto-Mechanical System Initially Prepared in N-Headed Cat State and Number State

Abstract

Non-classical quantum states have been applied more and more deeply in quantum communication and quantum computation. By using the time evolution theory and the nonlinearlity of opto-mechanical system, new quantum states can be prepared in the opto-mechanical system which is composed of an N-headed cat state of photonic mode and a number state of mechanical mode. The distribution of the Wigner function (WF) of the mechanical mode exhibits the superposition of several WF wave packets of displaced number states. And it is interesting to find that the parameter N affects the wave-packet numbers of displaced number states in phase space for mechanical mode and the shape of the WF wave packets for photonic mode. The nonclassical properties are investigated through the WF and the negative part volume of WF. An interesting result is that the nonclassicality increases (decreases) with the parameter N for mechanical (photonic) mode. And for the initial state of larger value of parameter N, the increment of the nonclassicality for mechanical (or photonic) mode is increasing more with parameter k (or α). Furthermore the parameter N also affects the entanglement degree between the photonic mode and the mechanical mode.

Appendix: Derivation of Time Evolution Operator (2)

The Hamiltonian of the opto-mechanical system is $H=\hbar \omega _{0}a^{\dagger }a+\hbar \omega _{m}b^{\dagger }b-\hbar g_{0}a^{\dagger }a\left (b^{\dagger }+b\right )$ , where $\hbar \omega _{0}a^{\dagger }a$ is the free Hamiltonian of light field, $\hbar \omega _{m}b^{\dagger }b$ is the Hamiltonian of mechanical oscillator and $\hbar g_{0}a^{\dagger }a\left (b^{\dagger }+b\right )$ is the interaction Hamiltonian between light field and mechanical oscillator. Let $H_{0}=\hbar \omega _{0}a^{\dagger }a$, $V=\hbar \omega _{m}b^{\dagger }b-\hbar g_{0}a^{\dagger }a\left (b^{\dagger }+b\right )$, then H = H₀ + V. And the time evolution operator is given by $U\left (t\right ) =\exp (-iH_{0}t/\hbar )=\exp \left [ -i\left (\omega _{0}a^{\dagger }a\right ) t\right ]$. According to the transformation formula between Schrödinger’s representation and interaction representation,

$$ H_{I}=i\hbar \frac{dU^{\dagger }}{dt}U+U^{\dagger }HU, $$

(16)

we obtain the Hamiltonian of the system in interaction representation [35]

$$ H_{I}=\hbar \omega_{m}b^{\dagger }b-\hbar g_{0}a^{\dagger }a\left( b^{\dagger }+b\right) . $$

(17)

Then we have the time evolution operator in interaction representation $U(t^{^{\prime }})=\exp \{-i[\omega _{m}b^{\dagger }b-g_{0}a^{\dagger }a\left (b^{\dagger }+b\right ) ]t^{^{\prime }}\}$. Let $t=\omega _{m}t^{^{\prime }}$ and g = g₀/ω_m, then U (t) = exp{−it[b^†b − ga^†a (b^† + b)]}. Using the formula e^{A + B} = e^Ae^Be^{−[A, B]/2} and [a, a^†] = [b, b^†] = 1, [a, b] = [a, b^†] = 0, we obtain the time evolution operator U (t)

$$ U\left( t\right) =\exp \left[ i\left( ga^{\dagger }a\right)^{2}t\right] \exp \left[ -it\left( b^{\dagger }-ga^{\dagger }a\right) \left( b-ga^{\dagger }a\right) \right] . $$

(18)

It is well known that $D_{b}\left (\alpha \right ) bD_{b}^{\dagger }\left (\alpha \right ) =b-\alpha$ where the displacement operator D_b (α) = exp [αb^†− α^∗b] acts on mechanical mode only, we can get $U\left (t\right ) =\exp \left [ i\left (ga^{\dagger }a\right )^{2}t\right ] D_{b}\left (ga^{\dagger }a\right ) \exp \left [ -itb^{\dagger }b\right ] D_{b}^{\dagger }\left (ga^{\dagger }a\right )$. Considering the normally ordering form (noted as ::) of operator exp [λb^†b] =: exp [(e^λ − 1) b^†b] :, we can further obtain

$$ U\left( t\right) =\exp \left[ i\left( ga^{\dagger }a\right)^{2}t\right] :\exp \left[ \eta \left( b^{\dagger }-ga^{\dagger }a\right) \left( b-ga^{\dagger }a\right) \right] :, $$

(19)

here η = 1 − e^−it. According to the Baker-Hausdorf formula $e^{\lambda A}Be^{-\lambda A}=B+\lambda \left [ A,B\right ] +\frac {\lambda ^{2}}{2!}\left [ A,\left [ A,B\right ] \right ] +{\cdots }$, we get rid of the sign of normally ordering and obtain

$$ U\left( t\right) =\exp \left[ i\left( ga^{\dagger }a\right)^{2}\left( t-\sin t\right) \right] \exp \left[ ga^{\dagger }a\left( \eta b^{\dagger }-\eta^{\ast }b\right) \right] \exp \left[ -it\left( b^{\dagger }b\right) \right] . $$

(20)