Macroscopic superpositions via nested interferometry: finite temperature and decoherence considerations

However, for a device of finite temperature, γ ≠ 0. Consider a mechanical resonator initially in a thermal state, a statistical mixture of coherent states:

$$\begin{equation} \hat{\rho}_{\mathrm{th}}=\frac{1}{\pi\bar{n}_{\mathrm{th}}}\int {\rm e}^{-|\gamma|^2/\bar{n}_{\mathrm{th}}} (\left| \gamma \rangle\langle \gamma \right|) {\rm d}^2\gamma, \end{equation} \tag{ 8 }$$

where $\bar {n}_{\mathrm {th}}$ is the average number of phonons:

$$\begin{equation} \bar{n}_{\mathrm{th}}\equiv \frac{1}{{\rm e}^{\hbar \omega_{\rm m}/k_{\mathrm{B}} T}-1}, \end{equation} \tag{ 9 }$$

and where k_B represents the Boltzmann constant. Note that $\bar {n}_{\mathrm {th}}$ is also the value of |γ|² averaged over the thermal distribution, equation (8). In this subsection we will deal only with mechanical states and will thus drop the m subscript.

For an initial coherent state, we will have created $| \psi _{\mathrm {ps}}(t) \rangle$ from equation (6), a superposition between a small early component with mechanical state $| \psi _{\mathrm {ps}}(t) \rangle$ and a large late component still in $\left | \gamma (t) \right >$.

To lowest order in κ, the overall probability of successful postselection for an initial coherent state will be

$$\begin{equation} \langle\psi_{\mathrm{ps}}(t) | \psi_{\mathrm{ps}}(t) \rangle \approx \textstyle\frac{1}{2}[ \kappa^2 (1-\cos \omega_{\mathrm{m}} t) + \textstyle\frac{1}{2}\kappa^2 |\gamma|^2 \sin^2 \omega_{\mathrm{m}} t ]. \end{equation} \tag{ 10 }$$

Note that $|\langle \gamma (t) | \psi _{\mathrm {ps}}(t) \rangle |^2\approx (1/4)\kappa ^2 |\gamma |^2 \sin ^2 \omega _{\mathrm {m}} t$, precisely the second term of equation (10). Thus the first term represents our signal, while the second term represents a background noise of dark port events due to finite temperature rather than successfully conveying a phonon to the device. Averaging equation (10) over the thermal distribution, equation (8), we arrive at

$$\begin{equation} \langle\langle \psi_{\mathrm{ps}}(t) | \psi_{\mathrm{ps}}(t) \rangle\rangle_{\mathrm{th}}\approx \left[ \kappa^2 \sin^2 \frac{\omega_{\mathrm{m}} t}{2} + \frac{1}{4}\kappa^2 \bar{n}_{\mathrm{th}} \sin^2 \omega_{\mathrm{m}} t \right]. \end{equation} \tag{ 11 }$$

So for the signal to be larger than the noise, we must have $\bar {n}_{\mathrm {th}} \ll 4\left [ \sin (\omega _{\mathrm {m}} t /2)/ \sin \omega _{\mathrm {m}} t\right ]^2= \sec ^2(\omega _{\mathrm {m}} t / 2)$. This implies that the nested interferometry proposal will only be successful if $\bar {n} \ll 1$, that is T ≪ ℏω_m/k_B. Thus, ground state cooling is essential for the success of this scheme. For a sideband-resolved device, this can be accomplished by driving the red (anti-Stokes) sideband of the cavity with a coherent beam [9–12].

The nested interferometry proposal [16] aims to use this amplification to create macroscopic superposition states, doing so by means of the extended optical setup pictured in figure 2. The postselection interferometer of figure 1 is nested in a larger interferometer with both an early and a late path.

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An experiment begins with the optomechanical device being cooled to its ground state by standard optomechanical cooling techniques [9, 10]. Single photons are input to the outer interferometer and are split into an early component and a late component. The late component enters the first delay line. The early component immediately enters the inner interferometer where it interacts with the device, and only the $\left | 1 \right >_{\mathrm {m}}$ component is passed through the dark port, entering a second equal length delay line. Conditional on the early component leaving the dark port, the unnormalized state will be

$$\begin{equation} \left| \psi(t) \right>\approx\frac{1}{\sqrt{2}}\left(\left| 1 \right>_{{\rm d}1}\left| 0 \right>_{\rm d2}\left| 0 \right>_{\mathrm{m}} + \frac{\alpha(t)}{2} \left| 0 \right>_{\rm d1}\left| 1 \right>_{\rm d2}\left| 1 \right>_{\mathrm{m}} \right), \end{equation} \tag{ 12 }$$

with d1 and d2 labeling the first and second delay lines respectively.

At this point, the late component is associated with mechanical component $\left | 0 \right >_{\mathrm {m}}$ while the early component is associated with $\left | 1 \right >_{\mathrm {m}}$. These components are left to evolve freely for the length of the delay lines, which can in principle be arbitrarily long. During this time they may experience decoherence from either traditional EID [17] or one of many proposed novel decoherence mechanisms [18–23].

Finally, the components exit the delay lines and the late component enters the interferometer. As before only the $\left | 1 \right >_{\mathrm {m}}$ component passes out of the dark port and we are left with both components in the $\left | 1 \right >_{\mathrm {m}}$ state, assuming no decoherence has occurred:

$$\begin{equation} \left| \psi(t) \right>\approx\frac{\alpha(t)}{2\sqrt{2}}\left({\rm e}^{{\rm i}\phi}\left| 1 \right>_{\rm s}\left| 0 \right>_{\rm d2}\left| 1 \right>_{\mathrm{m}} + \left| 0 \right>_{\rm s}\left| 1 \right>_{\rm d2}\left| 1 \right>_{\mathrm{m}} \right), \end{equation} \tag{ 13 }$$

with s representing the short path of the late photon through a variable phase shift ϕ prior to the final beam splitter. At this point both components are interfered and the experiment is repeated with ϕ varied to check for interference visibility, allowing us to measure whether decoherence has taken place.

This scheme has two advantages over previous schemes. Firstly, it allows weakly coupled devices to be placed in superpositions by a single photon. Secondly, in principle it allows observation of decoherence on an arbitrary time scale, as the delay lines can be varied. Previous schemes [2, 5, 8] were limited in the time scales by both the mechanical period of oscillation and the cavity lifetime. This would require new devices to measure at different time scales. Although it may be difficult to determine the cause of the decoherence beyond any doubt, it will be possible to vary the temperature and the characteristics of the device, such as mass, frequency, mechanical quality factor and optical finesse, allowing parameter dependence to be established.

Most devices proposed for ground state cooling [11–13] require that the device be optically cooled below the temperature T_env that the surrounding environment can reach by conventional cooling (there is one notable exception [27]). This is also true of the devices proposed in table 1.

In this situation, the mechanical resonator is modeled as coupled to an infinite bath of harmonic oscillators [8, 17]. In the limit of k_BT_env ≫ ℏω_m, mechanical quality factor Q_m ≫ 1, and a Markovian regime with no memory effects in the bath, the bath degrees of freedom can be eliminated and the system can be described by the master equation for the reduced density matrix $\hat {\rho }$ [8, 17, 28]:

$$\begin{equation} \frac{{\rm d}}{{\rm d} t}\hat{\rho}= \frac{{\rm i}}{\hbar}[\hat{\rho}, \hat{\mathcal{H}}_{\mathrm{ren}} ]-\frac{{\rm i} \gamma_{\mathrm{m}}}{\hbar}[\hat{x} , \{\hat{p} , \hat{\rho}\} ]-\frac{D}{\hbar^2}[\hat{x} , [\hat{x}, \hat{\rho} ] ], \end{equation} \tag{ 14 }$$

with $\hat {\mathcal {H}}_{\mathrm {ren}}$ the Hamiltonian from equation (1) renormalized by the interaction of the device and the bath, the damping coefficient γ_m = ω_m/Q_m, and the diffusion coefficient D = 2Mγ_mk_BT_env. The first term represents the unitary evolution of the system under the Hamiltonian from equation (1), while the second term represents the damping and the third term represents the diffusion. In the macroscopic regime the diffusion term proportional to D/ℏ² dominates equation (14) [8, 17]. Thus the resulting time scale for decoherence is

$$\begin{equation} \tau_{\mathrm{EID}}\approx\frac{\hbar^2}{D(\Delta x)^2}=\frac{\hbar Q_{\mathrm{m}}}{k_{\mathrm{B}} T_{\mathrm{env}}}, \end{equation} \tag{ 15 }$$

with the superposition size Δx = x₀. It is helpful at this point to define an EID temperature [8]:

$$\begin{equation} T_{\mathrm{EID}}=\frac{\hbar \omega_{\mathrm{m}} Q_{\mathrm{m}}}{k_{\mathrm{B}}}. \end{equation} \tag{ 16 }$$

We note that the inverse of the decoherence time scale is τ⁻¹_EID = ω_m(T_env/T_EID). Thus for the EID to act on a time scale slower than the mechanical frequency it is necessary that T_env ≪ T_EID.

We will consider EID with a base temperature of T_env = 1 mK, obtainable with a dilution refrigerator. For this case, for the 300 kHz device, τ_EID ≈ 150 μs. For the 4.5 kHz device, τ_EID ≈ 15 ms.

Gravitationally induced decoherence, proposed independently by Diósi [18] and Penrose [19], is a type of decoherence caused by an object in the superposition's perturbation of spacetime. The time scale for such decoherence is

$$\begin{equation} \tau_{\mathrm{P}}=\hbar/\Delta_{\mathrm{P}}, \end{equation} \tag{ 17 }$$

with the gravitational self-energy Δ_P defined as follows:

$$\begin{equation} \Delta_{\mathrm{P}} = 4 \pi G \int\!\!\!\int \frac{(\rho_1(\vec{x})-\rho_2(\vec{x}))(\rho_1(\vec{y})-\rho_2(\vec{y}))}{|\vec{x}-\vec{y}|}{\rm d}^{3}x {\rm d}^{3}y, \end{equation} \tag{ 18 }$$

with $\rho _1(\vec {x})$ and $\rho _2(\vec {x})$ the mass distributions of the two superposed states.

There is considerable theoretical disagreement about the proper mass distribution to use for gravitationally induced decoherence [2, 8, 29–31]. Previous papers have used the zero point motion of the resonator itself, the nuclear radius of the nuclei making up the resonator, the zero point motion of the nuclei making up the resonator, and a completely homogeneous mass with no nuclear granularity. We will consider the decoherence rate for all of these mass distributions.

As in [8], we model the system as a disk composed of spheres representing nuclei. Since the largest component of the mass of a Ta₂O₅/SiO₂ dielectric mirror will be tantalum, we will make the simplifying assumption that the mirrors are composed of tantalum with nuclear mass m = 181 amu. The gravitational self-energy for one sphere is given by Δ⁰_P = 4π(E⁰_1,2 + E⁰_2,1 − E⁰_1,1 − E⁰_2,2), with $E_{m,n}^0= -G\int \!\!\!\int \rho _{m}(\vec {x})\rho _{n}(\vec {x})/|\vec {x}-\vec {y}|{\mathrm { d}}^3 x {\rm d}^3 y$. The spheres considered are far enough apart and displaced little enough that their most significant interaction is with themselves, and not neighboring spheres. This means that we can merely multiply by the number of nuclei, M/m, to get the total energy Δ_P = (M/m)Δ⁰_P = 4π(E_1,2 + E_2,1 − E_1,1 − E_2,2), with E_m,n = (M/m)E⁰_m,n.

For all cases, we will consider two spherical mass distributions with radii a equal to the size of the specific mass distribution that will be chosen, separated by $\Delta x=x_{0}=\sqrt {\hbar /(2M\omega _{\mathrm {m}})}$, the zero point motion of the resonator. Note that this is mathematically equivalent to the model of one sphere at x = 0 for $\left | 0 \right >_{\mathrm {m}}$, and two half-mass spheres at x = ± x₀ for $\left | 1 \right >_{\mathrm {m}}$.

As the radius of the two spheres will be greater than x₀ regardless of the mass distribution used, there will always be significant overlap in the distributions. This will greatly complicate evaluation of equation (18). This has no effect on the self-energy terms but does affect the interaction terms. The 1/r potential between overlapping spheres has been evaluated previously [32]:

$$\begin{eqnarray} E_{1,2}= \left\{\begin{array}{@{}l@{\quad}l@{}} -GMm/\Delta x, & {\rm if} \ \Delta x > 2a, \\ -GMm \left[ \displaystyle \frac{12a^2-5\Delta x^2}{10a^3} - \displaystyle \frac{\Delta x^{5}-30\Delta x^{3} a^2}{160a^6}\right], & {\rm if} \ 0 \leqslant \Delta x \leqslant 2a. \end{array}\right. \end{eqnarray} \tag{ 19 }$$

For the E_1,1 and E_2,2 terms, we can just plug Δx = 0 into equation (19). This gives $E_{1,1}= E_{2,2}=-\frac {6GMm}{5a}$.

At this point, we will define the mass distributions to be considered in this paper.

3.2.1. Zero point motion of resonator

Zero point motion is defined as

$$\begin{equation} a=x_{0}=\sqrt{\frac{\hbar}{2M\omega_{\mathrm{m}}}}. \end{equation} \tag{ 20 }$$

For the 300 kHz device, a = 5.3 fm. For the 4.5 kHz device, a = 4.3 fm.

For this case, for the 300 kHz device, τ_P ≈ 3.5 ms. For the 4.5 kHz device, τ_P ≈ 28 μs. This type of decoherence might potentially be testable in the 4.5 kHz device, as it is faster than EID.

3.2.2. Radius of tantalum

The atomic nucleus has a size of approximately [33]

$$\begin{equation} a=r_0 A^{1/3}, \end{equation} \tag{ 21 }$$

with r₀ = 1.25 fm and A the atomic mass number. For tantalum, A = 181, so a ≈ 7 fm.

For this case, for the 300 kHz device, τ_P ≈ 7.1 ms. For the 4.5 kHz device, τ_P ≈ 100 μs. This type of decoherence might potentially be testable in the 4.5 kHz device, as it is faster than EID.

3.2.3. Zero point motion of nuclei

In the Debye model, the zero point motion of nuclei in a lattice is given (equation (12.3.10) in [34]):

$$\begin{equation} a=x_{0\mathrm{,nuc}}=\frac{3\hbar}{2\sqrt{k_{\mathrm{B}} \Theta_{\mathrm{D}} m}}. \end{equation} \tag{ 22 }$$

with Θ_D the Debye temperature and m the nuclear mass. The Debye temperature of tantalum is Θ_D = 240 K [35], and the nuclear mass m = 181 amu. Thus a ≈ 5 pm.

For this case, for the 300 kHz device, τ_P ≈ 1.8 × 10⁶ s. For the 4.5 kHz device, τ_P ≈ 28 × 10³ s. This type of decoherence would not be testable, as it is slower than EID in both devices.

3.2.4. Homogeneous mass

Continuous spontaneous localization (CSL) is a proposed position-localized decoherence mechanism in which a nonlinear stochastic classical field interacts with objects causing collapse of macroscopic superpositions. Proposed by Ghirardi et al [22, 23], the master equation and decay rate for position-localized decoherence have the following form [18–23, 26, 31]:

$$\begin{equation} \frac{{\rm d}}{{\rm d} t}\left< x \right|\hat{\rho}| x' \rangle =\frac{{\rm i}}{\hbar}\left< x \right|[\hat{\rho},\hat{\mathcal{H}}]| x' \rangle-\Gamma(x-x') \left< x \right| \hat{\rho} | x' \rangle, \end{equation} \tag{ 23 }$$

$$\begin{equation} \Gamma(x)\equiv\gamma \left[1- \exp \left(-\frac{x^2}{4a^2}\right)\right] \end{equation} \tag{ 24 }$$

$$\begin{equation} \approx\left\{\begin{array}{@{}l@{\quad}l@{}}\Lambda x^2, & \mathrm{if} \ x \ll 2a, \\ \gamma, & \mathrm{if} \ x \gg 2a,\\ \end{array}\right. \end{equation} \tag{ 25 }$$

with Γ(x) the decay rate, Λ = γ/(4a²) the localization parameter, γ the localization strength and a the localization distance. In all cases, the trampoline resonators considered are in the x ≪ 2a limit. For the single nucleon case, the CSL model [23] gives values a_CSL = 100 nm and γ⁰_CSL = 10⁻¹⁶ Hz based on phenomenological arguments.

Following [31, 36], the value of the localization parameter Λ_CSL can be shown to be

$$\begin{equation} \Lambda_{\mathrm{CSL}}=\frac{M^2}{m_0^2} \frac{\gamma_{\mathrm{CSL}}^0}{4 a_{\mathrm{CSL}}^{2}} f(R,b,a), \end{equation} \tag{ 26 }$$

with M the resonator mass, m₀ the nucleon mass, R the radius of the sphere and f(R,b,a) a parameter depending on the geometry of the device. Disk geometry was considered in [36]. For motion perpendicular to the disk face f is evaluated (see [36], section 5.2, appendix A and equation (A.11)):

$$\begin{equation} \fl f(R,b,a)=4\left(\frac{2a}{R}\right)^4\left( \frac{2a}{b} \right)^2 [1-\mathrm{e}^{-b^2/4a^2}]\int_{0}^{R/2a}x\, {\rm d} x \int_{0}^{R/2a}x' {\rm d} x' \mathrm{e}^{-(x^2+x'^2)}I_{0}(2xx'), \end{equation} \tag{ 27 }$$

with R the disk radius, b the disk thickness, I₀(x) the n = 0 modified Bessel function of the first kind, and a the localization distance (for CSL, a_CSL = 100 nm). In the (R/2a)² ≫ 1 and (b/2a)² ≫ 1 limits, applicable in this case, f ≈ (2a/R)²(2a/b)².

Thus, for the 300 kHz device, using a thickness of ∼5 μm and a radius of ∼4 μm (values consistent with the proposed finesse and mass), we obtain a decoherence time of order τ_CSL = 10⁷ s. For the 4.5 kHz device, using a thickness of ∼5 μm and a radius of ∼40 μm, we obtain a decoherence time of order τ_CSL = 1.5 × 10⁵ s. This type of decoherence would not be testable, as it is slower than EID in both devices.

It has been proposed that quantum gravity might cause a form of position-localized decoherence due to coupling of the system to spacetime foam. This was first proposed by Ellis et al [20] and subsequently elaborated [21, 26] with others. Notably, this model is phenomenologically equivalent to the CSL model with altered values for the constants [31]: a_QG = ℏm_P/2 cm²₀ with $m_{\mathrm {P}}=\sqrt {\hbar c/G}$ the Planck mass, and γ⁰_QG = 4a²_QGc⁴m⁶₀/ℏ³m³_P. This gives us

$$\begin{equation} \Lambda_{\mathrm{QG}} = \frac{M^2}{m_0^2}\frac{\gamma_{\mathrm{QG}}^0}{4a_{\mathrm{QG}}^2}f(R,b,a)= \frac{c^4 M^2 m_0^4}{\hbar^3 m_{\rm P}^3}f(R,b,a), \end{equation} \tag{ 28 }$$

with f(R,b,a) as in equation (27). However, since R ≪ a_QG and b ≪ a_QG, we can set f to 1 [36]:

$$\begin{equation} \Lambda_{\mathrm{QG}} \approx \frac{c^4 M^2 m_0^4}{\hbar^3 m_{\mathrm{P}}^3}. \end{equation} \tag{ 29 }$$

Thus, for the 300 kHz device, using a thickness of ∼5 μm and a radius of ∼4 μm, we get a decoherence time of order τ_QG = 7.1 s. For the 4.5 kHz device, using a thickness of ∼5 μm and a radius of ∼40 μm, we get a decoherence time of order τ_QG = 1.1 ms. This type of decoherence might potentially be testable in the 4.5 kHz device, as it is faster than EID.