Quantum and classical control of single photon states via a mechanical resonator

The classical beam splitter interaction between two optical modes a₁ and a₂ is defined by the unitary operator

$$\begin{eqnarray}&&{U}_{{\rm{BS}}}(\theta )={{\rm{e}}}^{-{\rm{i}}\theta ({a}_{1}^{\dagger }{a}_{2}+{a}_{1}{a}_{2}^{\dagger })},\end{eqnarray} \tag{ 1 }$$

under which the optical operators transform as

$$\begin{eqnarray}&&{U}_{{\rm{BS}}}^{\dagger }(\theta ){a}_{1}{U}_{{\rm{BS}}}(\theta )=\mathrm{cos}(\theta )\ {a}_{1}-{\rm{i}}\;\mathrm{sin}(\theta ){a}_{2},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{U}_{{\rm{BS}}}^{\dagger }(\theta ){a}_{2}{U}_{{\rm{BS}}}(\theta )=\mathrm{cos}(\theta ){a}_{2}-{\rm{i}}\;\mathrm{sin}(\theta ){a}_{1}.\end{eqnarray} \tag{ 3 }$$

Langford et al [26] introduced the concept of coherent photon conversion based on an ability to coherently control the exchange of photons between two optical modes. The Hamiltonian realized an optical three-wave mixing process in which a strong coherent pump field was used to create an enhanced nonlinearity in the nonlinear medium to convert a single photon into two single excitations in different frequencies. The defining Hamiltonian for the process is

$$\begin{eqnarray}&&{H}_{{\rm{I}}}={\hslash }g({a}_{1}^{\dagger }{a}_{2}{b}^{\dagger }+{a}_{1}{a}_{2}^{\dagger }b),\end{eqnarray} \tag{ 4 }$$

where a₁, a₂ are the annihilation operators for the bosonic modes we seek to control while b is the annihilation operator of the bosonic controller. In [26], the controller was taken to also be an optical mode but in this paper the controller will represent a mechanical degree of freedom. Our scheme uses the intrinsic nonlinearity of the optomechanical beam splitter and does not need a nonlinear crystal.

The Hamiltonian in equation (4) will, in general, dynamically entangle the optical and mechanical degrees of freedom depending on the initial states used. We will now describe a picture in which this entanglement can be viewed in terms of 'which-path' information in a kind of optical Stern–Gerlach device in which optical information is stored in the mechanical element.

We will assume that the optical modes begin in an eigenstate of the total photon number $N={a}_{1}^{\dagger }{a}_{1}+{a}_{2}^{\dagger }{a}_{2}$, while the mechanical element is prepared in an arbitrary state $| \phi {\rangle }_{b}$. It is convenient in this case to use the two mode Schwinger representation of ${\rm{SU}}(2)$ to write the joint state of the optical modes. Defining the generators of SU(2) as

$$\begin{eqnarray}&&{S}_{z}=\displaystyle \frac{1}{2}({a}_{2}^{\dagger }{a}_{2}-{a}_{1}^{\dagger }{a}_{1}),\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{S}_{x}=\displaystyle \frac{1}{2}({a}_{2}^{\dagger }{a}_{1}+{a}_{1}^{\dagger }{a}_{2}),\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&\quad {S}_{y}=-\displaystyle \frac{{\rm{i}}}{2}({a}_{2}^{\dagger }{a}_{1}-{a}_{1}^{\dagger }{a}_{2}),\end{eqnarray} \tag{ 7 }$$

with ${S}^{2}=(N/2+1)N/2$. We can then define the joint eigenstates of S² and S_z in terms of the tensor product photon number basis for modes a_k as $| s,m{\rangle }_{z}=| s-m{\rangle }_{1}\otimes | s+m{\rangle }_{2}$.

The interaction Hamiltonian equation (4) can then be written in the form

$$\begin{eqnarray}&&{H}_{{\rm{I}}}={\hslash }g({S}_{+}b+{S}_{-}{b}^{\dagger }),\end{eqnarray} \tag{ 8 }$$

where ${S}_{+}={S}_{-}^{\dagger }={a}_{2}^{\dagger }{a}_{1}$ or equivalently ${S}_{\pm }={S}_{x}\pm {\rm{i}}{S}_{y}$. If the initial state of the entire system is

$$\begin{eqnarray}&&| {\rm{\Psi }}(0)\rangle =| \psi (0){\rangle }_{{\rm{O}}}\otimes | \phi (0){\rangle }_{b},\end{eqnarray} \tag{ 9 }$$

the total state at time t > 0 can then be written as

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle =U(t)| {\rm{\Psi }}(0)\rangle =\displaystyle \sum _{m=-s}^{s}| s,m{\rangle }_{z}\otimes | {\phi }_{m}(t){\rangle }_{b},\end{eqnarray} \tag{ 10 }$$

with $U(t)={{\rm{e}}}^{-\tfrac{{\rm{i}}}{{\hslash }}{H}_{{\rm{I}}}t}$ and

$$\begin{eqnarray}&&| {\phi }_{m}(t){\rangle }_{b}={\text{}}_{z}\langle s,m| U(t)| \psi (0){\rangle }_{{\rm{O}}}\otimes | \phi (0){\rangle }_{b}.\end{eqnarray} \tag{ 11 }$$

Equation (10) is like a Stern–Gerlach device in which the mechanical controller is keeping track of 'which-path' information. We make this interpretation more explicit in section 2.2 below.

Clearly $| {\rm{\Psi }}(t)\rangle$ is an entangled state in general. Tracing out the state of the controller, we see that the state of the optical system is

$$\begin{eqnarray}&&{\rho }_{{\rm{O}}}(t)=\displaystyle \sum _{m,n=-s}^{s}{R}_{n,m}(t)| s,m{\rangle }_{z}\langle s,n| ,\end{eqnarray} \tag{ 12 }$$

where

$$\begin{eqnarray}&&{R}_{{nm}}(t)={\text{}}_{b}\langle {\phi }_{n}(t)| {\phi }_{m}(t){\rangle }_{b}.\end{eqnarray} \tag{ 13 }$$

In general, ρ_O(t) is a mixed state representing the degree of entanglement between the optical system and the controller; a correlation that encodes 'which-path' information if the states $| {\phi }_{m}(t){\rangle }_{b}$ are distinguishable.

We now consider the case in which the mechanical element is prepared in a coherent state, $| \beta {\rangle }_{b}$, with a large coherent amplitude β. We can use a canonical transformation $b\to \bar{b}+\beta ,$ to include this amplitude in the Hamiltonian while the initial mechanical state now becomes the ground state. The new interaction Hamiltonian is

$$\begin{eqnarray}&&{H}_{{\rm{I}}}={\hslash }\bar{g}({a}_{1}^{\dagger }{a}_{2}+{a}_{1}{a}_{2}^{\dagger })+{\hslash }g({a}_{1}^{\dagger }{a}_{2}{\bar{b}}^{\dagger }+{a}_{1}{a}_{2}^{\dagger }\bar{b}),\end{eqnarray} \tag{ 14 }$$

where $\bar{g}=g\beta$. For simplicity, and without loss of generality, we take β to be real.

As a coherent state with large amplitude is a semiclassical state, we expect that as $| \beta |$ becomes large, this model should reduce to the simple unitary model of equation (1). To see this, we fix $\bar{g}$ to be constant while letting $\beta \to \infty$. With this scaling we can regard $g=\displaystyle \frac{\bar{g}}{\beta }\ll 1$ as a perturbation parameter in the dynamics arising from equation (14). Terms arising to first and higher order terms in g describe optomechanical entanglement and a departure from the simple unitary coupling of the optical modes described by equation (1).

We define $\theta =\bar{g}t$ as the effective beam splitter parameter that can be reached by unitary evolution; for example, a 50/50 beam splitter has $\theta =\pi /4$. Let the initial state of the optics and the mechanics be

$$\begin{eqnarray}&&| {\rm{\Psi }}(0)\rangle =| \psi (0){\rangle }_{{\rm{O}}}\otimes | \beta {\rangle }_{b},\end{eqnarray} \tag{ 15 }$$

where $| \psi (0){\rangle }_{{\rm{O}}}=| n{\rangle }_{1}\otimes | m{\rangle }_{2}$ is the initial state of the mode-1 and mode-2, taken as a product Fock state, and $| \beta {\rangle }_{b}$ is a coherent state. In terms of the SU(2) operators, $| \psi (0){\rangle }_{{\rm{O}}}$ is an eigenstate of ${\hat{S}}_{z}$. Therefore, equation (11) can be written as

$$\begin{eqnarray}&&| {\phi }_{m}(t){\rangle }_{b}=D(\beta ){\text{}}_{z}\langle s,m| \bar{U}(\theta )| \psi (0){\rangle }_{{\rm{O}}}\otimes | 0{\rangle }_{b},\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray}&&\bar{U}(\theta )={D}^{\dagger }(\beta ){{\rm{e}}}^{-{\rm{i}}{gt}({S}_{+}b+{S}_{-}{b}^{\dagger })}D(\beta )={{\rm{e}}}^{-2{\rm{i}}\theta {S}_{x}-{\rm{i}}{gt}({S}_{+}\bar{b}+{S}_{-}{\bar{b}}^{\dagger })},\end{eqnarray} \tag{ 17 }$$

and D(β) is the displacement operator with the property ${D}^{\dagger }(\beta ){bD}(\beta )=\bar{b}+\beta$. To second order in gt, we find that

$$\begin{eqnarray}| {\phi }_{m}(\theta )\rangle & = & {\text{}}_{z}\langle s,m| {U}_{{\rm{BS}}}| \psi (0){\rangle }_{{\rm{O}}}| \beta {\rangle }_{b}+\theta {gt}{\text{}}_{z}\langle s,m| {U}_{{\rm{BS}}}{S}_{z}| \psi (0){\rangle }_{{\rm{O}}}D(\beta )| 1{\rangle }_{b}\\ & & +\displaystyle \frac{{(\theta {gt})}^{2}}{2}{\text{}}_{z}\langle s,m| {U}_{{\rm{BS}}}{S}_{z}^{2}| \psi (0){\rangle }_{{\rm{O}}}(\sqrt{2}D(\beta )| 2{\rangle }_{b}-| \beta {\rangle }_{b})\;+\;\cdots ,\end{eqnarray} \tag{ 18 }$$

where U_BS is given by equation (1). Substituting this into equation (12), we then find that

$$\begin{eqnarray}&&{\rho }_{{\rm{O}}}(\theta )={U}_{{\rm{BS}}}(\theta )\left({\rho }_{{\rm{O}}}(0)-\displaystyle \frac{{(\theta {gt})}^{2}}{2}[{S}_{z},[{S}_{z},{\rho }_{{\rm{O}}}(0)]]+...\right){U}_{{\rm{BS}}}^{\dagger }(\theta )\end{eqnarray} \tag{ 19 }$$

with ${\rho }_{{\rm{O}}}(0)=| \psi (0){\rangle }_{{\rm{O}}}\langle \psi (0)|$. The state of the two optical cavities, after fixed interaction time such that $\theta =\bar{g}t={\rm{constant}}$, is thus given by a completely positive unital map of the initial state. We can now see that, up to second order correction to the action of a classical beam splitter, we achieve a completely positive map corresponding to a dephasing channel that might arise, for example, as a weak measurement of S_z. This is precisely what one would expect if the mechanics encoded 'which-path' information about the optical excitations. The classical control given by the beam splitter interaction U_BS is composed with a random relative phase shift between the two optical cavities. This leads to a weak suppression of the off-diagonal coherence terms in the S_z basis. This effect can be interpreted as an effective measurement of the relative occupation number of each cavity mode. The results of this measurement are hidden in the quantum correlations between the cavity modes and the mechanical degree of freedom and thus reflects residual entanglement between the controller and the optical system. This structure is typical of the way in which residual quantum entanglement in a semiclassical controller affects the ideal classical control transformation [35]. The dephasing channel described above is not the only decoherence phenomena that can happen (to second order in gt). However, this expression shows the point that once we go beyond zero order, the controller gets entangled with the optical system.

We now consider the opposite limit in which the mechanical element is a quantum controller for the optical states. Quantum control necessarily requires that the controller becomes entangled with the target state for appropriate states of the controller. For example, in a quantum CNOT gate, preparing the controller in a superposition of the two computational basis states with the target in one of the computational basis states, produces a Bell state for the combined system.

As in the previous section, the Schwinger representation allows one to see the kind of quantum control realized in this system. We now write the Hamiltonian given in equations (4) and (8), in terms of the (dimensionless) position and momentum operators for the mechanics

$$\begin{eqnarray}&&{H}_{{\rm{I}}}={\hslash }g({S}_{x}X-{S}_{y}Y),\end{eqnarray} \tag{ 20 }$$

where $X=b+{b}^{\dagger },Y=-{\rm{i}}(b-{b}^{\dagger })$. This Hamiltonian represents two orthogonal rotations of the pseudo-spin controlled by two non-commuting operators in the controller. This is the canonical example of quantum control considered in [43]: it is not possible to represent this kind of control using a measurement and feedback protocol from the controller to the optical subsystem.

To proceed, we need to fix the initial optical state. For practical reasons it is unlikely that we will be able to prepare states with N > 2 in the foreseeable future. The case N = 1 requires a single photon and encodes a qubit in the two optical modes [17]. In the next section, we will see that this can be done using a MZ interferometer set-up. The case N = 2 can be done by preparing a single photon in each optical mode and corresponds to encoding a qutrit into the optical system. In this case, the initial state of the optical system is $| 1{\rangle }_{1}\otimes | 1{\rangle }_{2}\equiv | 1,0{\rangle }_{z}$ in the Schwinger representation. In the next section, we will see that this suggests a HOM interferometer set-up to investigate quantum to classical control.

The connection to interferometry can be seen more clearly by writing equation (10) for the two cases N = 1, 2 with the initial optical states $| 0{\rangle }_{1}| 1{\rangle }_{2}$ and $| 1{\rangle }_{1}| 1{\rangle }_{2}$ respectively

$$\begin{eqnarray}&&| {{\rm{\Psi }}}_{1/2}(t)\rangle ={| 1/2,-\left.1/2\right\rangle }_{z}| {\phi }_{-1/2}(t){\rangle }_{b}+{| 1/2,\left.1/2\right\rangle }_{z}| {\phi }_{1/2}(t){\rangle }_{b},\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&| {{\rm{\Psi }}}_{1}(t)\rangle =| 1,-1{\rangle }_{z}| {\phi }_{-1}(t){\rangle }_{b}+| 1,0{\rangle }_{z}| {\phi }_{0}(t){\rangle }_{b}+| 1,1{\rangle }_{z}| {\phi }_{1}(t){\rangle }_{b}.\end{eqnarray} \tag{ 22 }$$

These two equations indicate that the mechanical element acts, in general, as quantum controller of a qubit (N = 1) or a qutrit (N = 2).

In the N = 1 qubit case, it is easier to work in the dressed state basis rather than the tensor product basis as the dressed states are eigenstates of the Hamiltonian. These are defined by ${H}_{{\rm{I}}}| \pm ,\;n\rangle =\pm {\hslash }g\sqrt{n}| \pm ,\;n\rangle$, where

$$\begin{eqnarray}&&| +,\;n\rangle =1/\sqrt{2}(\left.\left|1/2\right.,n-1\right\rangle +\left.\left|-1/2\right.,n\right\rangle ),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&| -,\;n\rangle =1/\sqrt{2}(\left.\left|1/2\right.,n-1\right\rangle -\left.\left|-1/2\right.,n\right\rangle ),\end{eqnarray} \tag{ 24 }$$

where $| \pm 1/2,n\rangle =| 1/2,\pm 1/2\rangle \otimes | n{\rangle }_{b}$ with $| n{\rangle }_{b}$ a Fock state of the mechanical oscillator. If the initial state is written in the dressed-state basis as

$$\begin{eqnarray}&&| \psi (0){\rangle }_{{\rm{O}}}\otimes | \phi (0){\rangle }_{b}=\displaystyle \sum _{n}{c}_{n}^{+}| +,\;n\rangle +{c}_{n}^{-}| -,\;n\rangle ,\end{eqnarray} \tag{ 25 }$$

where ${c}_{n}^{\pm }$ are complex coefficients, the state at a later time is

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle =\displaystyle \sum _{n}{c}_{n}^{+}{{\rm{e}}}^{-{\rm{i}}g\sqrt{n}t}| +,\;n\rangle +{c}_{n}^{-}{{\rm{e}}}^{{\rm{i}}g\sqrt{n}t}| -,\;n\rangle ,\end{eqnarray} \tag{ 26 }$$

which is clearly a controlled rotation in the dressed state basis. For example, if the optical system is prepared in the state $| 1/2,-1/2\rangle$ while the mechanical controller is prepared with a single excitation, $| 1{\rangle }_{b}$, the state at a later time is

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle =-{\rm{i}}\;\mathrm{sin}({gt})| 1/2,1/2\rangle \otimes | 0{\rangle }_{b}+\mathrm{cos}({gt})| 1/2,-1/2\rangle \otimes | 1{\rangle }_{b},\end{eqnarray} \tag{ 27 }$$

an entangled state in general.

In the case N = 2, the qutrit case, the corresponding case for the initial state

$$\begin{eqnarray}&&| {\rm{\Psi }}(0)\rangle =| 1,0{\rangle }_{z}\otimes | 1{\rangle }_{b},\end{eqnarray} \tag{ 28 }$$

is

$$\begin{eqnarray}| {\rm{\Psi }}(t)\rangle & = & -{\rm{i}}\sqrt{\displaystyle \frac{1}{3}}\mathrm{sin}(\sqrt{6}{gt})| 1,1{\rangle }_{z}\otimes | 0{\rangle }_{b}+\mathrm{cos}(\sqrt{6}{gt})| 1,0{\rangle }_{z}\otimes | 1{\rangle }_{b}\\ & & -{\rm{i}}\sqrt{\displaystyle \frac{2}{3}}\mathrm{sin}(\sqrt{6}{gt})| 1,-1{\rangle }_{z}\otimes | 2{\rangle }_{b}.\end{eqnarray} \tag{ 29 }$$

If we write the state of the optical system in form of equation (12), we have

$$\begin{eqnarray}&&{\rho }_{{\rm{O}}}(t)={R}_{\mathrm{1,1}}(t)| 1,1{\rangle }_{z}\langle 1,1| +{R}_{\mathrm{0,0}}(t)| 1,0{\rangle }_{z}\langle 1,0| +{R}_{-1,-1}(t)| 1,-1{\rangle }_{z}\langle 1,-1| ,\end{eqnarray} \tag{ 30 }$$

where

$$\begin{eqnarray}&&{R}_{\mathrm{1,1}}(t)=\displaystyle \frac{1}{3}{\mathrm{sin}}^{2}(\sqrt{6}{gt}),\phantom{\rule{3mm}{0ex}}{R}_{\mathrm{0,0}}(t)={\mathrm{cos}}^{2}(\sqrt{6}{gt}),\phantom{\rule{3mm}{0ex}}{R}_{-1,-1}(t)=\displaystyle \frac{2}{3}{\mathrm{sin}}^{2}(\sqrt{6}{gt}).\end{eqnarray} \tag{ 31 }$$

We see that ${R}_{n,m}(t)=0$ for $n\ne m$ and there is a complete loss of coherence, due to the entanglement of the optical and mechanical systems, and perfect which-path information of the optical system is encoded in the mechanical object. Inspection of equation (29) indicates that this information is stored in the number of mechanical excitations.

If the optical system is prepared in the same initial state in each case N = 1, 2 but the mechanical controller is prepared in a superposition of ground and single excitation, $\tfrac{1}{\sqrt{2}}(| 0{\rangle }_{b}+| 1{\rangle }_{b})$, the state at time t for the case N = 1 becomes

$$\begin{eqnarray}&&| {\rm{\Psi }}(t)\rangle =\displaystyle \frac{1}{\sqrt{2}}(| 1/2,-1/2\rangle -{\rm{i}}\;\mathrm{sin}({gt})| 1/2,1/2\rangle )\otimes | 0{\rangle }_{b}+\displaystyle \frac{1}{\sqrt{2}}\mathrm{cos}({gt})| 1/2,-1/2\rangle \otimes | 1{\rangle }_{b},\end{eqnarray} \tag{ 32 }$$

and for the case N = 2 we have

$$\begin{eqnarray}| {\rm{\Psi }}(t)\rangle & = & \left(\displaystyle \frac{-{\rm{i}}}{\sqrt{6}}\mathrm{sin}(\sqrt{6}{gt})| 1,1{\rangle }_{z}+\displaystyle \frac{1}{\sqrt{2}}\mathrm{cos}(\sqrt{2}{gt})| 1,0{\rangle }_{z})\right)\otimes | 0{\rangle }_{b}\\ & & +\left(\displaystyle \frac{1}{\sqrt{2}}\mathrm{cos}(\sqrt{6}{gt})| 1,0{\rangle }_{z}-\displaystyle \frac{{\rm{i}}}{\sqrt{2}}\mathrm{sin}(\sqrt{2}{gt})| 1,-1{\rangle }_{z}\right)\otimes | 1{\rangle }_{b}\\ & & -\displaystyle \frac{{\rm{i}}}{\sqrt{9}}\mathrm{sin}(\sqrt{6}{gt})| 1,-1{\rangle }_{z}\otimes | 2{\rangle }_{b}.\end{eqnarray} \tag{ 33 }$$

Again we take the example for the case N = 2 to calculate the state of the optical system in form of equation (12) as

$$\begin{eqnarray}{\rho }_{{\rm{O}}}(t) & = & {R}_{\mathrm{1,1}}(t)| 1,1{\rangle }_{z}\langle 1,1| +{R}_{\mathrm{0,0}}(t)| 1,0{\rangle }_{z}\langle 1,0| +{R}_{-1,-1}(t)| 1,-1{\rangle }_{z}\langle 1,-1| \\ & & +{R}_{\mathrm{0,1}}(t)| 1,1{\rangle }_{z}\langle 1,0| +{R}_{\mathrm{1,0}}(t)| 1,0{\rangle }_{z}\langle 1,1| \\ & & +{R}_{0,-1}(t)| 1,-1{\rangle }_{z}\langle 1,0| +{R}_{-\mathrm{1,0}}(t)| 1,0{\rangle }_{z}\langle 1,-1| ,\end{eqnarray} \tag{ 34 }$$

where

$$\begin{eqnarray}&&{R}_{\mathrm{1,1}}(t)=\displaystyle \frac{1}{6}{\mathrm{sin}}^{2}(\sqrt{6}{gt}),\phantom{\rule{5mm}{0ex}}{R}_{\mathrm{0,0}}(t)=\displaystyle \frac{1}{2}({\mathrm{cos}}^{2}(\sqrt{6}{gt})+{\mathrm{cos}}^{2}(\sqrt{6}{gt})),\\ &&{R}_{-1,-1}(t)=\displaystyle \frac{1}{2}{\mathrm{sin}}^{2}(\sqrt{6}{gt})+\displaystyle \frac{1}{3}{\mathrm{sin}}^{2}(\sqrt{6}{gt}),\\ &&{R}_{\mathrm{0,1}}(t)={R}_{1,0}^{* }(t)=\displaystyle \frac{-{\rm{i}}}{2\sqrt{3}}\mathrm{sin}(\sqrt{6}{gt})\mathrm{cos}(\sqrt{2}{gt}),\\ &&{R}_{0,-1}(t)={R}_{-1,0}^{* }(t)=\displaystyle \frac{-{\rm{i}}}{2}\mathrm{sin}(\sqrt{2}{gt})\mathrm{cos}(\sqrt{6}{gt}).\end{eqnarray} \tag{ 35 }$$

Figure 1 shows the coefficients R_n,m ($n\ne m$) which measure the decoherence. It can be seen that these terms are not equal to zero at the same time thus there is always some coherence in the system which results in the reduction of the which-path information. These examples show that, by varying the state of the controller, the degree of entanglement between the optical system and the controller varies and so does the which-path information which affects the reduced state of the optical system.

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A coherent state with a very small amplitude, $\beta \ll 1$, can also be approximated as an asymmetric superposition of $| 0\rangle$ and $| 1\rangle$

$$\begin{eqnarray}&&| \beta \rangle \simeq \displaystyle \frac{| 0\rangle +\beta | 1\rangle }{\sqrt{1+| \beta {| }^{2}}}.\end{eqnarray} \tag{ 36 }$$

Therefore, for a mechanical system prepared in coherent state with small amplitude there is less entanglement between the controller and the optical system than there is for the mechanical number state $| 0{\rangle }_{b}\;\mathrm{or}\;| 1{\rangle }_{b}$, resulting in less decoherence and less which-path information stored in the mechanics. In section 5, we consider the visibility of the interference pattern, in a MZ interferometer for the qubit case and in a HOM interferometer for the optical qutrit, as a measure of this which-path information.

As we demonstrated in section 2.2, the semiclassical limit is obtained when the mechanics is prepared in a coherent state with large coherent amplitude and the coupling constant, g, is small, while the effective coupling, $\bar{g}=\beta g$, is constant. We now consider this limit for the case of cavities driven by external single photon sources. We can then compute the visibility of one and two-photon interferometry and how it depends on dephasing corrections that appear in equation (19) due to entanglement between the optical and mechanical sub-systems.

Assume that each cavity is driven by single photon pulse states with wavepacket envelope functions $\xi (t),\eta (t)$ for the input to cavity a₁ and a₂, respectively, and that the coupling between the cavities is given by the first term in the Hamiltonian in equation (14), the semiclassical approximation. Note that we will assume that the carrier frequency of each single photon pulse is resonant with the respective cavity into which it is injected. We do not explicitly see the carrier frequencies here as we are already working in an interaction picture. We further assume the symmetric case for which ${\kappa }_{1}={\kappa }_{2}=\kappa$. In the semiclassical regime, we use the quantum Langevin equations for the optical fields [47, 48]

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{a}_{1}(t)}{{\rm{d}}t} & = & -{\rm{i}}\bar{g}{a}_{2}(t)-\displaystyle \frac{\kappa }{2}{a}_{1}(t)+\sqrt{\kappa }{a}_{1,{\rm{in}}}(t),\\ \displaystyle \frac{{\rm{d}}{a}_{2}(t)}{{\rm{d}}t} & = & -{\rm{i}}\bar{g}{a}_{1}(t)-\displaystyle \frac{\kappa }{2}{a}_{2}(t)+\sqrt{\kappa }{a}_{2,{\rm{in}}}(t),\end{eqnarray} \tag{ 50 }$$

in which the amplitude functions for single photon state inputs in ${a}_{1,{\rm{in}}}(t)$ and ${a}_{2,{\rm{in}}}(t)$ are respectively given by $\xi (t)$ and $\eta (t)$. The solution to these linear equations is

$$\begin{eqnarray}{a}_{1}(t) & = & \sqrt{\kappa }\left[A(t){\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime }(C({t}^{\prime }){a}_{1,{\rm{in}}}({t}^{\prime })+D({t}^{\prime }){a}_{2,{\rm{in}}}({t}^{\prime }))\right.\\ & & +\left.B(t){\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime }(D({t}^{\prime }){a}_{1,{\rm{in}}}({t}^{\prime })+C({t}^{\prime }){a}_{2,{\rm{in}}}({t}^{\prime }))\right],\\ {a}_{2}(t) & = & \sqrt{\kappa }\left[B(t){\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime }(C({t}^{\prime }){a}_{1,{\rm{in}}}({t}^{\prime })+D({t}^{\prime }){a}_{2,{\rm{in}}}({t}^{\prime }))\right.\\ & & +\left.A(t){\displaystyle \int }_{0}^{t}{\rm{d}}{t}^{\prime }(D({t}^{\prime }){a}_{1,{\rm{in}}}({t}^{\prime })+C({t}^{\prime }){a}_{2,{\rm{in}}}({t}^{\prime }))\right],\end{eqnarray} \tag{ 51 }$$

where $A(t)={{\rm{e}}}^{-\kappa t/2}\mathrm{cos}(\bar{g}t)$, $B(t)=-{\mathrm{ie}}^{-\kappa t/2}\mathrm{sin}(\bar{g}t)$, $C(t)={{\rm{e}}}^{\kappa t/2}\mathrm{cos}(\bar{g}t)$ and $D(t)={\mathrm{ie}}^{\kappa t/2}\mathrm{sin}(\bar{g}t)$.

We can now derive the effective transmission and reflection coefficients when only one photon is incident on the system. These are defined by

$$\begin{eqnarray}R & = & {\displaystyle \int }_{0}^{\infty }\langle {a}_{1,{\rm{out}}}^{\dagger }{a}_{1,{\rm{out}}}{\rangle }_{t}{\rm{d}}t,\\ T & = & {\displaystyle \int }_{0}^{\infty }\langle {a}_{2,{\rm{out}}}^{\dagger }{a}_{2,{\rm{out}}}{\rangle }_{t}{\rm{d}}t.\end{eqnarray} \tag{ 52 }$$

For the single photon with pulse shape $\xi (t)=\sqrt{\gamma }{{\rm{e}}}^{-\gamma t/2}$, resulting from the decay of a photon from a cavity, these are given by

$$\begin{eqnarray}T & = & \displaystyle \frac{8\kappa {\bar{g}}^{2}(\gamma +2\kappa )}{(4{\bar{g}}^{2}+{\kappa }^{2})(4{\bar{g}}^{2}+{(\gamma +\kappa )}^{2})},\\ R & = & 1-\displaystyle \frac{8\kappa {\bar{g}}^{2}(\gamma +2\kappa )}{(4{\bar{g}}^{2}+{\kappa }^{2})(4{\bar{g}}^{2}+{(\gamma +\kappa )}^{2})},\end{eqnarray} \tag{ 53 }$$

where γ is the single photon bandwidth. Figure 2 shows the transmission, T, as a function of κ and $\bar{g}$ in units of γ. We can use this figure to configure the optomechanical system as one port in an optical interferometer. Of the two branches in the figure, the lower branch is of experimental relevance as it enables us to operate with smaller values of $\bar{g}$.

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We insert a controlled beam splitter of the type described before by Hamiltonian (4) into the output beam splitter of a MZ interferometer, see figure 3. We inject a single photon with an exponentially decaying shape, $\xi (t)=\sqrt{\gamma }{{\rm{e}}}^{-\tfrac{1}{2}\gamma t}$, into the interferometer through the port containing cavity-1. Note that the carrier frequency of this pulse needs to be set equal to the resonance frequency of cavity-1. We further use dimensionless units assuming the cavity damping rate κ = 1 in numerical simulations.

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The initial input state to the optomechanical beam splitter after passing the first beam splitter is

$$\begin{eqnarray}&&| \psi (0)\rangle =\displaystyle \frac{1}{\sqrt{2}}({{\rm{e}}}^{{\rm{i}}\phi }| {1}_{{a}_{1}},{0}_{{a}_{2}}\rangle +| {0}_{{a}_{1}},{1}_{{a}_{2}}\rangle ).\end{eqnarray} \tag{ 54 }$$

We use the solutions to the Langevin equations (50) given in relations (51) to calculate the detection probability in t to t + dt at the upper detector D₁

$$\begin{eqnarray}&&{P}_{u}(t\;:t+{\rm{d}}t)=\langle {a}_{1,{\rm{out}}}^{\dagger }(t){a}_{1,{\rm{out}}}(t)\rangle {\rm{d}}t.\end{eqnarray} \tag{ 55 }$$

This probability versus detection time and phase shift is plotted in figure 4. For $\kappa t\leqslant 1$ the decrease in P_u is due to the transmission of a photon which has not interacted strongly with the mechanics. This is evident because the decay of P_u follows $\xi (t)\propto {{\rm{e}}}^{-\gamma t/2}$. In figure 4, it appears that the maximum visibility of the fringes occurs at $\kappa t\approx 3$, but this can be deceiving. For this reason we use P_u to calculate the visibility of the interference pattern at each detection time which is given by

$$\begin{eqnarray}&&v(t)=\displaystyle \frac{{P}_{u}^{{\rm{max}}}(t)-{P}_{u}^{{\rm{min}}}(t)}{{P}_{u}^{{\rm{max}}}(t)+{P}_{u}^{{\rm{min}}}(t)},\end{eqnarray} \tag{ 56 }$$

where ${P}_{u}^{{\rm{max}}}$ (${P}_{u}^{{\rm{min}}}$) is maximized (minimized) over the phase shift ϕ.

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For the fully quantum mechanical description of the system, we use the unconditional Fock state master equation method [49, 50] which for a system having two input modes is

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\rho }_{m,n;p,q}(t) & = & -{\rm{i}}[H,{\rho }_{m,n;p,q}]+({ \mathcal L }[{L}_{1}]+{ \mathcal L }[{L}_{2}]){\rho }_{m,n;p,q}\\ & & +\sqrt{m}\xi (t)[{\rho }_{m-1,n;p,q},{L}_{1}^{\dagger }]+\sqrt{p}\eta (t)[{\rho }_{m,n;p-1,q},{L}_{2}^{\dagger }]\\ & & +\sqrt{n}{\xi }^{* }(t)[{L}_{1},{\rho }_{m,n-1;p,q}]+\sqrt{q}{\eta }^{* }(t)[{L}_{2},{\rho }_{m,n;p,q-1}],\end{eqnarray} \tag{ 57 }$$

where H is the Hamiltonian given in the equation (4), ${L}_{i}=\sqrt{{\kappa }_{i}}{a}_{i}$ and the superoperator ${ \mathcal L }$ is defined by

$$\begin{eqnarray}&&{ \mathcal L }[L]\rho =L\rho {L}^{\dagger }-\displaystyle \frac{1}{2}({L}^{\dagger }L\rho +\rho {L}^{\dagger }L).\end{eqnarray} \tag{ 58 }$$

The dynamics is reduced to solving the hierarchy of equations for the operators ${\rho }_{m,n;p,q}$. These act on the joint Hilbert space of the system and the input fields. The subscripts m, n refer to the Fock basis for the input to cavity-1 described by the wave packet $\xi (t)$ while p, q refer to the Fock basis of the input to cavity-2 described by the wave packet η(t). As each input has, at most, one photon, the indices are restricted to the values 1,0. For example, if we had a single photon input at each cavity we would need to solve for dρ_1,1;1,1 which couples all the way down to dρ_0,0;0,0 in the hierarchy of coupled differential equations.

After the photon passes through the first beam splitter and the phase shifter, the initial state of the input field incident on the controlled beam splitter is given by equation (54) which is not a pure Fock state. Therefore, the initial sate of the field has the form

$$\begin{eqnarray}&&{\rho }_{{\rm{field}}}(0)=\displaystyle \sum _{m,n,p,q=0}^{\infty }{c}_{m,n;p,q}| {n}_{\xi };{q}_{\eta }\rangle \langle {m}_{\xi };{p}_{\eta }| ,\end{eqnarray} \tag{ 59 }$$

The initial conditions are ${c}_{\mathrm{1,1};\mathrm{0,0}}={c}_{\mathrm{0,0};\mathrm{1,1}}=\tfrac{1}{2}$, ${c}_{\mathrm{0,1};\mathrm{1,0}}=\tfrac{1}{2}{{\rm{e}}}^{-{\rm{i}}\theta }$, ${c}_{\mathrm{1,0};\mathrm{0,1}}=\tfrac{1}{2}{{\rm{e}}}^{{\rm{i}}\theta }$ and all other coefficients are zero. The initial total state is given by equation (59), as [49]

$$\begin{eqnarray}&&{\rho }_{{\rm{system}}}(t)=\displaystyle \sum _{m,n,p,q=0}^{\infty }{c}_{m,n;p,q}^{* }{\rho }_{m,n;p,q}(t).\end{eqnarray} \tag{ 60 }$$

Therefore, we solve the hierarchy of differential equations produced by master equation (57) for $\xi (t)=\eta (t)=\sqrt{\gamma }{{\rm{e}}}^{-\tfrac{1}{2}\gamma t}$. We need the solutions for ${\rho }_{\mathrm{1,1};\mathrm{0,0}}(t),{\rho }_{\mathrm{0,0};\mathrm{1,1}}(t),{\rho }_{\mathrm{0,1};\mathrm{1,0}}(t)$ and ${\rho }_{\mathrm{1,0};\mathrm{0,1}}(t)$ to calculate the dynamical state of the optomechanical system given by equation (60). When we use the Fock state master equation approach, a different phase convention to the input–output relation (46) is used (see section five of [48]) such that we have

$$\begin{eqnarray}&&{a}_{j,{\rm{out}}}(t)=\sqrt{{\kappa }_{j}}{a}_{j}(t)+{a}_{j,{\rm{in}}}(t).\end{eqnarray} \tag{ 61 }$$

To calculate the detection probability at the top detector, D₁, which is defined in equation (95), one also needs the action of the operators a_{1, in}(t) and ${a}_{2,{\rm{in}}}(t)$ on two mode Fock states

$$\begin{eqnarray}{a}_{1,{\rm{in}}}(t)| {n}_{\xi };{q}_{\eta }\rangle & = & \xi (t)\sqrt{n}| n-{1}_{\xi };{q}_{\eta }\rangle ,\\ {a}_{2,{\rm{in}}}(t)| {n}_{\xi };{q}_{\eta }\rangle & = & \eta (t)\sqrt{q}| {n}_{\xi };q-{1}_{\eta }\rangle .\end{eqnarray} \tag{ 62 }$$

We then use equation (56) to calculate the visibility of the interference pattern for different values of the coherent state amplitude shown in figure 5. This figure demonstrates that as β increases, the corrections due to the first and higher order terms in g, which was discussed in section 2, become negligible and when β is large enough, we recover a semiclassical interaction without any entanglement between the optical and mechanical degrees of freedom.

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In practice, photo detector operates with a finite integration time. Therefore, it is more convenient to integrate over time to calculate the detection probability at D₁ as

$$\begin{eqnarray}&&{P}_{u}={\int }_{0}^{\infty }\langle {a}_{1,{\rm{out}}}^{\dagger }(t){a}_{1,{\rm{out}}}(t)\rangle {\rm{d}}t,\end{eqnarray} \tag{ 63 }$$

which gives a detection probability and an interference visibility independent of the detection time as one would expect in an experiment. This visibility is plotted in figure 6 versus coherent state amplitude prepared in the mechanics. This figure shows that by enhancing β, the visibility saturates to the value obtained in the semiclassical limit. The interference visibility of a MZ interferometer can be used as a sign to show the transition from quantum control to classical control that is obtained for a certain value of β.

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The scheme for our system working as a beam splitter inside a HOM interferometer is shown in figure 7. We send one photon to each of the modes a₁ and a₂. The single photons are specified by the same amplitude function except one is time shifted with respect to the other

$$\begin{eqnarray}\xi (t) & = & \sqrt{\gamma }{{\rm{e}}}^{-\tfrac{1}{2}\gamma t},\\ \eta (t) & = & \sqrt{\gamma }{{\rm{e}}}^{-\tfrac{1}{2}\gamma (t-\tau )},\end{eqnarray} \tag{ 64 }$$

and, as before, we assume that they have carrier frequencies resort with their respective cavities. We can then use differential equations (50) together with the input–output relation (46) to analytically calculate the probability of the joint photon counting at D₁ and D₂ in the semiclassical regime [7] which is defined as

$$\begin{eqnarray}&&{G}^{(2)}(\tau )=\displaystyle \frac{{\int }_{0}^{\infty }{\int }_{0}^{\infty }\langle {a}_{1,{\rm{out}}}^{\dagger }(t){a}_{2,{\rm{out}}}^{\dagger }(t^{\prime} ){a}_{2,{\rm{out}}}(t^{\prime} ){a}_{1,{\rm{out}}}(t)\rangle {\rm{d}}t{\rm{d}}t^{\prime} }{{\int }_{0}^{\infty }\langle {a}_{1,{\rm{out}}}^{\dagger }{a}_{1,{\rm{out}}}(t)\rangle {\rm{d}}t{\int }_{0}^{\infty }\langle {a}_{2,{\rm{out}}}^{\dagger }{a}_{2,{\rm{out}}}(t)\rangle {\rm{d}}t}.\end{eqnarray} \tag{ 65 }$$

In the quantum regime, one way to calculate this two-time correlation function is to use the quantum regression theorem, for which one needs to solve the unconditional master equation. However, for this two-mode, two-input photon case, this becomes complicated. Another way is to numerically simulate a HOM experiment using the stochastic theory of quantum jumps. We choose the latter approach in this work, the details for which will be given later in this section. Before that, we start with a simpler calculation to give some physical insight into testing the quantum to classical control by employing a HOM interferometer.

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We consider the physically idealistic case of coincidence detection at a specific detection time t by calculating the coincidence detection rate $C(t,\tau )={\kappa }_{1}{\kappa }_{2}\langle {a}_{1}{(t)}^{\dagger }{a}_{1}(t){a}_{2}{(t)}^{\dagger }{a}_{2}(t)\rangle$. This expectation value can be computed using the Fock state master equation (57) as

$$\begin{eqnarray}&&C(t,\tau )={\kappa }_{1}{\kappa }_{2}{\rm{Tr}}[{a}_{1}^{\dagger }{a}_{1}{a}_{2}^{\dagger }{a}_{2}{\rho }_{\mathrm{1,1};\mathrm{1,1}}(t)].\end{eqnarray} \tag{ 66 }$$

This coincidence rate is plotted in figure 8(a) versus detection time and the time shift between the input photons for the semiclassical regime. However, we get qualitatively the same plot for the quantum regime with a HOM dip forming around the τ = 0 point. The HOM dip can be clearly seen in this figure at fixed detection times. We choose $\kappa t=4.7$ for which the HOM visibility in the semiclassical regime is 1 and then plot the coincidence rate versus the time shift between the input photons for different values of coherent state amplitude, β, changing from a quantum regime, β = 1, to very strong amplitudes, as can be seen in figure 8(b). One feature we observe in this figure is the asymmetry in the HOM dip in the quantum regime, which arises from the asymmetry in interaction Hamiltonian given in equation (4). By increasing β, this asymmetry is gradually removed since as we increase β, there is a gradual transition to the semiclassical regime with a symmetric interaction Hamiltonian between the optical modes. This figure also suggests that the change in the HOM dip can be used as a measure of the transition from the quantum control regime to the classical control regime.

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We calculate the visibility of the HOM interference pattern as

$$\begin{eqnarray}&&v(t)=\displaystyle \frac{{C}^{{\rm{max}}}(t,{\tau }_{{\rm{negative}}})-C(t,0)}{{C}^{{\rm{max}}}(t,{\tau }_{{\rm{negative}}})+C(t,0)},\end{eqnarray} \tag{ 67 }$$

which gives the worst case visibility for each curve. Figure 8(c) shows HOM visibility versus the detection time. For the chosen parameter regimes of γ and $\bar{g}$ in this figure, maximum visibility occurs at κt = 4.7, the detection time that we choose to plot visibility versus coherent state amplitude prepared in the mechanics in figure 8(d). This figure also shows that HOM visibility can be an indicator to test the transition form the quantum control to the classical control in which the visibility saturates to a maximum value.

In the next step, we perform the calculation using the more realistic definition of the joint detection probability given in equation (65). As discussed earlier, we perform a Monte-Carlo simulation using the stochastic version of the Fock state master equation to simulate the HOM interference which shows the ratio of coincident photo-detections over the total number of measurements versus the time shift, (τ). We need to model the conditional evolution of the system. Following the same procedure as introduced for unconditional Fock state master equation [49], we derive the conditional master equation (for more details see [51]) describing the dynamics of the system given vacuum detection in both modes up to time t

$$\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{{\rm{d}}t}{\tilde{\rho }}_{m,n;p,q}^{({0}_{1},{0}_{2})}(t) & = & -{\rm{i}}[H,{\rho }_{m,n;p,q}]-\displaystyle \frac{1}{2}\{{L}_{1}^{\dagger }{L}_{1},{\rho }_{m,n;p,q}\}-\displaystyle \frac{1}{2}\{{L}_{2}^{\dagger }{L}_{2},{\rho }_{m,n;p,q}\}\\ & & -\sqrt{m}\xi (t){L}_{1}^{\dagger }{\rho }_{m-1,n;p,q}-\sqrt{p}\eta (t){L}_{2}^{\dagger }{\rho }_{m,n;p-1,q}\\ & & -\sqrt{n}{\xi }^{* }(t){\rho }_{m,n-1;p,q}{L}_{1}-\sqrt{q}{\eta }^{* }(t){\rho }_{m,n;p,q-1}{L}_{2}\\ & & -\sqrt{{mn}}| \xi (t){| }^{2}{\rho }_{m-1,n-1;p,q}-\sqrt{{pq}}| \eta (t){| }^{2}{\rho }_{m,n;p-1,q-1},\end{eqnarray} \tag{ 68 }$$

where ${\tilde{\rho }}_{m,n;p,q}^{({0}_{1},{0}_{2})}(t)$ is the conditional un-normalized state of the system in which n_i in the superscript $({n}_{1},{n}_{2})$ is the number of counts at detector D_i. The top level generalized density operator ${\tilde{\rho }}_{1,1;1,1}^{({0}_{1},{0}_{2})}(t)$ is the physical state of the system and is used to calculate the normalization factor: ${\rm{Tr}}[{\tilde{\rho }}_{1,1;1,1}^{({0}_{1},{0}_{2})}(t+{\rm{d}}t)]$, which is in fact the probability for a vacuum detection occurring in the time interval $(t,t+{\rm{d}}t]$

$$\begin{eqnarray}{P}^{({0}_{1},{0}_{2})}(t) & = & 1-{\rm{d}}t(| \xi (t){| }^{2}{\rm{Tr}}[{\rho }_{\mathrm{0,0};\mathrm{1,1}}(t)]-| \eta (t){| }^{2}{\rm{Tr}}[{\rho }_{\mathrm{1,1};\mathrm{0,0}}(t)]\\ & & -{\rm{Tr}}[{L}_{1}^{\dagger }{L}_{1}{\rho }_{\mathrm{1,1};\mathrm{1,1}}]-{\rm{Tr}}[{L}_{2}^{\dagger }{L}_{2}{\rho }_{\mathrm{1,1};\mathrm{1,1}}]\\ & & -\xi (t){\rm{Tr}}[{L}_{1}^{\dagger }{\rho }_{\mathrm{0,1};\mathrm{1,1}}]-{\xi }^{* }(t){\rm{Tr}}[{L}_{1}{\rho }_{\mathrm{1,0};\mathrm{1,1}}]\\ & & -\eta (t){\rm{Tr}}[{L}_{2}^{\dagger }{\rho }_{\mathrm{1,1};\mathrm{0,1}}]-{\eta }^{* }(t){\rm{Tr}}[{L}_{2}{\rho }_{\mathrm{1,1};\mathrm{1,0}}]).\end{eqnarray} \tag{ 69 }$$

The conditional state of the system given that one photon is detected at D₁ in the time interval $(t,t+{\rm{d}}t]$, should be updated as

$$\begin{eqnarray}{\tilde{\rho }}_{m,n;p,q}^{({1}_{1},{0}_{2})}(t+{\rm{d}}t) & = & {\rm{d}}t({L}_{1}{\rho }_{m,n;p,q}(t){L}_{1}^{\dagger }+\sqrt{{mn}}| \xi (t){| }^{2}{\rho }_{m-1,n-1;p,q}\\ & & +\sqrt{m}\xi (t){\rho }_{m-1,n;p,q}{L}_{1}^{\dagger }+\sqrt{n}\xi {(t)}^{* }{L}_{1}{\rho }_{m,n-1;p,q}),\end{eqnarray} \tag{ 70 }$$

and for a count occurring at D₂ in t to t + dt we have

$$\begin{eqnarray}{\tilde{\rho }}_{m,n;p,q}^{({0}_{1},{1}_{2})}(t+{\rm{d}}t) & = & {\rm{d}}t({L}_{2}{\rho }_{m,n;p,q}(t){L}_{2}^{\dagger }+\sqrt{{pq}}| \eta (t){| }^{2}{\rho }_{m,n;p-1,q-1}\\ & & +\sqrt{p}\eta (t){\rho }_{m,n;p-1,q}{L}_{2}^{\dagger }+\sqrt{q}\eta {(t)}^{* }{L}_{2}{\rho }_{m,n;p,q-1}),\end{eqnarray} \tag{ 71 }$$

The associated normalization with the states given in equations (70) and (71) gives the probability for a photo-detection occurring in the time interval $(t,t+{\rm{d}}t]$ at detectors D₁ and D₂, respectively

$$\begin{eqnarray}{P}^{({1}_{1},{0}_{2})}(t) & = & {\rm{d}}t({\rm{Tr}}[{L}_{1}^{\dagger }{L}_{1}{\rho }_{\mathrm{1,1};\mathrm{1,1}}(t)]+| \xi (t){| }^{2}{\rho }_{\mathrm{0,0};\mathrm{1,1}}\\ & & +\xi (t){\rm{Tr}}[{L}_{1}^{\dagger }{\rho }_{\mathrm{0,1};\mathrm{1,1}}(t)]+{\xi }^{* }(t){\rm{Tr}}[{L}_{1}{\rho }_{\mathrm{1,0};\mathrm{1,1}}(t)]),\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}{P}^{({0}_{1},{1}_{2})}(t) & = & {\rm{d}}t({\rm{Tr}}[{L}_{2}^{\dagger }{L}_{2}{\rho }_{\mathrm{1,1};\mathrm{1,1}}(t)]+| \eta (t){| }^{2}{\rho }_{\mathrm{1,1};\mathrm{0,0}}\\ & & +\eta (t){\rm{Tr}}[{L}_{2}^{\dagger }{\rho }_{\mathrm{1,1};\mathrm{0,1}}(t)]+{\eta }^{* }(t){\rm{Tr}}[{L}_{2}{\rho }_{\mathrm{1,1};\mathrm{1,0}}(t)]).\end{eqnarray} \tag{ 73 }$$

We perform the two-jump Monte-Carlo simulation in four steps as follows: (1) start with the optomechanical initial state $| \psi (0)\rangle =| 0{\rangle }_{1}| 0{\rangle }_{2}| \beta {\rangle }_{b}$ and inject two single photons, with a time shift τ, to the cavity inputs. (2) Generate a random number, rand, in the range 0–1. If ${P}^{({0}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)\gt \mathrm{rand}$, no jump occurs and the normalized state of the system at the end of the interval should be updated as

$$\begin{eqnarray}&&{\rho }_{m,n;p,q}(t+{\rm{d}}t)=\displaystyle \frac{{\tilde{\rho }}_{m,n;p,q}^{({0}_{1},{0}_{2})}(t+{\rm{d}}t)}{{P}^{({0}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)}.\end{eqnarray} \tag{ 74 }$$

If ${P}^{({0}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)\lt \mathrm{rand}$, a jump occurs and we choose a second random number rand_J to decide if the jump occurs in mode one or in mode two.

$$\begin{eqnarray}&&{\rm{If}}\displaystyle \frac{{P}^{({1}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)}{1-{P}^{({0}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)}\gt {\mathrm{rand}}_{J}\longrightarrow {\rho }_{m,n;p,q}(t+{\rm{d}}t)=\displaystyle \frac{{\tilde{\rho }}_{m,n;p,q}^{({1}_{1},{0}_{2})}(t+{\rm{d}}t)}{{P}^{({1}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)},\end{eqnarray} \tag{ 75 }$$

$$\begin{eqnarray}&&{\rm{If}}\displaystyle \frac{{P}^{({1}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)}{1-{P}^{({0}_{1},{0}_{2})}(t\;:t+{\rm{d}}t)}\lt {\mathrm{rand}}_{J}\longrightarrow {\rho }_{m,n;p,q}(t+{\rm{d}}t)=\displaystyle \frac{{\tilde{\rho }}_{m,n;p,q}^{({0}_{1},{1}_{2})}(t+{\rm{d}}t)}{{P}^{({0}_{1},{1}_{2})}(t\;:t+{\rm{d}}t)}.\end{eqnarray} \tag{ 76 }$$

(3) We repeat step (2) until we have detected both photons. (4) We repeat steps (1)–(3) for a large number of trajectories.

Figure 9 shows the probability of having one count at each of detectors D₁ and D₂. The solid line shows the analytical results for the semiclassical regime as a solution to equation (65) [7]. Blue squares show the numerical results for the semiclassical case obtained by using Monte-Carlo simulation. For each τ, we performed 1200 trajectories. For other values of β in the quantum regime, we only compute the joint detection probability at τ = 0 since we are limited by our computation resources. However, this plot clearly shows the trend we expect to see; a decrease in the HOM dip with increasing mechanical coherent state amplitude, as we observed in the previous figures.

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In the semiclassical limit, there is an offset from zero in the HOM dip for totally indistinguishable photons. Considering the fact that with the chosen parameters the effective beam splitter is performing at T = 0.4, we also give a full description of the behavior of the effective beam splitter in the semiclassical regime implemented in both MZ and HOM interferometers in appendix B. The analysis fully include all the phenomena involved in the visibility reduction in this effective beam splitter which are not involved in a conventional beam splitter interaction.

We inject a single photon with an exponentially decaying shape into the interferometer already described in figure 3. The visibility of the interference pattern is given by the relation

$$\begin{eqnarray}&&v=\displaystyle \frac{{P}_{u}^{{\rm{max}}}-{P}_{u}^{{\rm{min}}}}{{P}_{u}^{{\rm{max}}}+{P}_{u}^{{\rm{min}}}},\end{eqnarray} \tag{ 92 }$$

where

$$\begin{eqnarray}&&{P}_{u}={\int }_{0}^{\infty }\langle {a}_{1,{\rm{out}}}^{\dagger }{a}_{1,{\rm{out}}}{\rangle }_{t}{\rm{d}}t,\end{eqnarray} \tag{ 93 }$$

is the probability to detect a single photon at any time in the upper detector D₁. The input state incident on the effective beam splitter (second beam splitter shown in figure 3) after passing the conventional 50/50 beam splitter (first beam splitter shown in figure 3) is

$$\begin{eqnarray}&&| \psi (0)\rangle =\displaystyle \frac{1}{\sqrt{2}}({{\rm{e}}}^{{\rm{i}}\phi }| {1}_{{a}_{1}},{0}_{{a}_{2}}\rangle +| {0}_{{a}_{1}},{1}_{{a}_{2}}\rangle ).\end{eqnarray} \tag{ 94 }$$

Therefore, P_u becomes

$$\begin{eqnarray}&&{P}_{u}=\displaystyle \frac{4\kappa g(4{g}^{2}-\kappa (\kappa +\gamma ))\mathrm{sin}(\phi )}{(4{g}^{2}+{\kappa }^{2})(4{g}^{2}+{(\kappa +\gamma )}^{2})}+\displaystyle \frac{1}{2}.\end{eqnarray} \tag{ 95 }$$

This relation gives the typical interference pattern for a MZ interferometer shown in figure B1 (a). The interference visibility is given by

$$\begin{eqnarray}&&v=\left|\displaystyle \frac{8\kappa g(4{g}^{2}-\kappa (\kappa +\gamma ))}{(4{g}^{2}+{\kappa }^{2})(4{g}^{2}+{(\kappa +\gamma )}^{2})}\right|.\end{eqnarray} \tag{ 96 }$$

Figure B1(b) shows the MZ visibility for different values of $\kappa /\gamma$ and $\bar{g}/\gamma$.

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For a MZ interferometer in which the first beam splitter is a conventional 50/50 beam splitter and the second beam splitter is a conventional beam splitter with reflectivity/transmissivity ${\boldsymbol{ \mathcal R }}/{\boldsymbol{ \mathcal T }}$, the visibility of the interference is $2\sqrt{{\boldsymbol{ \mathcal R }}{\boldsymbol{ \mathcal T }}}$. For ${\boldsymbol{ \mathcal R }}=0.5$, the visibility is one. The red dashed line in figure B1(b) shows the parameter regime where according to figure 2, the transmission is 0.5. In this case, to compare the cavity beam splitter with a conventional 50/50 beam splitter, we need to achieve a visibility as close as possible to 1. This figure shows that to achieve a visibility greater than 0.9, in the case of T = 0.5, one needs to work in regimes where $\kappa /\gamma \gg 1$. However, one can see that for values of T other than T = 0.5, the overlap of the corresponding transmission contour given in figure 2 with the expected visibility value can be achieved in regimes where $\kappa /\gamma \simeq 1$ or $\kappa /\gamma \lt 1$.

In the next section, we use the effective beam splitter in a HOM interferometer which demonstrates a fully quantum phenomena.

The scheme for a HOM interferometer implemented with the cavity beam splitter is shown in figure 7. We send one photon into each of the cavities a₁ and a₂. The single photons are specified by the same amplitude function but one of the input photons is time shifted with respect to the other

$$\begin{eqnarray}\xi (t) & = & \sqrt{\gamma }{{\rm{e}}}^{-\tfrac{1}{2}\gamma t},\\ \eta (t) & = & \sqrt{\gamma }{{\rm{e}}}^{-\tfrac{1}{2}\gamma (t-\tau )}.\end{eqnarray} \tag{ 97 }$$

The joint photon counting probability is given by equation (65). For the initial state $| \psi (0)\rangle =| {1}_{{a}_{1},\xi }{1}_{{a}_{2},\eta }\rangle$ this joint detection probability becomes [7]

$$\begin{eqnarray}&&{G}^{(2)}(\delta \tau )=\displaystyle \frac{{{\rm{e}}}^{-\tfrac{3}{2}\delta \tau (\kappa +\gamma )}}{A}\left(B{{\rm{e}}}^{\tfrac{3}{2}\delta \tau (\kappa +\gamma )}+C{{\rm{e}}}^{-\tfrac{1}{2}\delta \tau (3\kappa +\gamma )}+D{{\rm{e}}}^{\tfrac{1}{2}\delta \tau (\kappa +3\gamma )}+E{{\rm{e}}}^{\delta \tau (\kappa +\gamma )}\right),\end{eqnarray} \tag{ 98 }$$

where

$$\begin{eqnarray*}&&A={(4{g}^{2}+{\kappa }^{2})}^{2}{(16{g}^{4}+{({\gamma }^{2}-{\kappa }^{2})}^{2}+8{g}^{2}({\gamma }^{2}+{\kappa }^{2}))}^{2},\end{eqnarray*}$$

$$\begin{eqnarray*}B & = & {(4{g}^{2}+{(\gamma -\kappa )}^{2})}^{2}(256{g}^{8}+{\kappa }^{4}{(\gamma +\kappa )}^{4}+8{g}^{2}({\gamma }^{2}-2{\kappa }^{2})(16{g}^{4}+{\kappa }^{2}{(\gamma +\kappa )}^{2})\\ & & +16{g}^{4}({\gamma }^{4}+2{\gamma }^{2}{\kappa }^{2}+20\gamma {\kappa }^{3}+22{\kappa }^{4})),\end{eqnarray*}$$

$$\begin{eqnarray*}&&C=-32{g}^{2}{\kappa }^{2}{(4{g}^{2}+{\gamma }^{2}-{\kappa }^{2})}^{2}{(4{g}^{2}+{\kappa }^{2})}^{2},\end{eqnarray*}$$

$$\begin{eqnarray*}&&D=-32{g}^{2}{\gamma }^{2}{\kappa }^{2}{F}^{2},\end{eqnarray*}$$

$$\begin{eqnarray*}&&E=-64{g}^{2}\gamma {\kappa }^{2}(4{g}^{2}+{\gamma }^{2}-{\kappa }^{2})(4{g}^{2}+{\kappa }^{2})F,\end{eqnarray*}$$

and

$$\begin{eqnarray*}&&F=\kappa (-12{g}^{2}-{\gamma }^{2}+{\kappa }^{2})\mathrm{cos}(g\delta \tau )+2g(4{g}^{2}+{\gamma }^{2}-3{\kappa }^{2})\mathrm{sin}(g\delta \tau ).\end{eqnarray*}$$

Figure B2 (a) shows the HOM interference pattern for some arbitrary parameters κ, and g. For τ = 0, where input photons are indistinguishable, quantum interference results in photon bunching and we see the HOM dip. The visibility is defined as

$$\begin{eqnarray}&&{v}_{{\rm{HOM}}}=\displaystyle \frac{{G}^{(2)}(\delta \tau \to \infty )-{G}^{(2)}(0)}{{G}^{(2)}(\delta \tau \to \infty )+{G}^{(2)}(0)}.\end{eqnarray} \tag{ 99 }$$

Figure B2(b) shows HOM visibility for different values of κ/γ and $\bar{g}/\gamma$. We see that compared to what we had in the case of a MZ interferometer, HOM visibility is more sensitive to changes in $\bar{g}$ and κ. Moreover, to work in regime where R = 0.5 and visibility >0.9, we need a larger κ/γ compared to those needed in MZ interferometer case. We also need to work in stronger coupling regimes. The red dashed line shows the parameters for which the transmission of the effective beam splitter T = 0.5.

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