Decoherence and degradation of squeezed states in quantum filter cavities

The squeezer is represented by the operator $\mathbf{S}(\sigma,\phi)$, given by

which describes squeezing by $\mathrm{e}^{-\sigma}$ at angle $\phi$ and anti-squeezing by $\mathrm{e}^{\sigma}$ at $\phi+\pi/2$. Conventionally, squeezing magnitudes are expressed in decibels (dB), with $\sigma_{dB}=\sigma\times 20\log_{10}e$.

In general, all optical losses outside of the filter cavity are frequency independent or the frequency dependence is so small that it can be neglected. Examples of optical losses are residual transmissions of steering mirrors, scattering, absorption and imperfections in polarisation optics. The last of these is likely to dominate the frequency-independent losses incurred in the passage of the squeezed field to the readout, therefore these losses are represented in Figure 1 as occurring at the optical isolator.

Since there are no non-linear elements in our system between the squeezer and the interferometer (i.e. nothing which mixes upper and lower audio sidebands) we can combine all of the input losses together into a single frequency-independent “injection loss”, $\Lambda^{2}_{\rm inj}$, which represents the total power loss outside of the filter cavity and before the readout (this work does not consider any losses within the interferometer itself).

Amalgamating the losses with the action of the squeezer, we arrive at the two-photon transfer matrix which takes $v_{1}$ to the filter cavity¹¹1Our model treats all injection losses as occurring before the filter cavity. This approach is valid as the action of the filter cavity on coherent vacuum states is null.,

where the attenuation due to $\Lambda^{2}_{\rm inj}$ is described by the transfer coefficient $\tau_{\rm inj}=\tau(\Lambda_{\rm inj})=\sqrt{1-\Lambda^{2}_{\rm inj}}$.

Reflection from a filter cavity is a linear process which can easily be described in the one-photon, and therefore two-photon, formalisms, as in equation (A9) of Evans et al. (2013). However, the approach therein does not permit one to explore the consequences of filter cavity imperfections analytically, with resulting loss of physical insight. Here we revisit this equation and, by making appropriate approximations, construct a closed-form expression for the action of a filter cavity in the two-photon formalism.

For a given signal sideband frequency $\Omega$, the complex reflectivity, $r_{\rm fc}(\Omega)$, of a filter cavity, using the same notation as Evans et al. (2013), is given by

where $r_{\rm in}$ is the amplitude reflectivity of the input mirror and $r_{\rm rt}$ is the cavity’s round-trip amplitude reflectivity. For a cavity of length $L_{\rm fc}$ and resonant frequency ${\omega_{\rm fc}}$, the round-trip phase $\Phi(\Omega)$ is defined as

where $\Delta{\omega_{\rm fc}}={\omega_{\rm fc}}-\omega_{0}$ is the cavity detuning with respect to the carrier frequency $\omega_{0}$ and $c$ is the speed of light.

For a high-finesse cavity near to resonance, we can make the approximations

where $\Lambda_{\rm rt}^{2}$ accounts for the power lost during one round-trip in the cavity (not including input mirror transmission).

Under these approximations, and neglecting terms of order 1 or greater in $\Lambda_{\rm rt}^{2}$, $t_{\rm in}^{2}$ and $\Phi(\Omega)$, (3) can be rewritten as ²²2Note that the $\epsilon$ defined here is similar to that in equation (94) of Kimble et al. (2001) and that $\epsilon\to 1$ for an optimally coupled cavity.

and the cavity half-width-half-maximum-power linewidth is defined as

As noted by previous authors Khalili (2010), for a given cavity half-width ${\gamma_{\rm fc}}$, the filter cavity performance is determined entirely by the loss per unit length $\Lambda_{\rm rt}^{2}/L_{\rm fc}$.

To investigate the effect the filter cavity has on a squeezed field we must convert its response, (7), into the two-photon picture. This is done with the one-photon to two-photon conversion matrix (see Evans et al. (2013) and section A.3),

yielding the transfer matrix

where $r_{\pm}=r_{\rm fc}(\pm\Omega)$.

To cast this expression in a more instructive form, we require several sum and difference quantities based on $r_{\rm fc}(\Omega)$. In terms of $\epsilon$ and $\xi(\Omega)$, the complex phase and magnitude of $r_{\rm fc}(\Omega)$ are given by

Whence we define

where the subscripts $p$ and $m$ are used to denote the sum and difference of the phases and magnitudes.

The transfer matrix of the filter cavity can then be expressed in a form which clearly shows the effect of intra-cavity loss,

where $\mathbf{I}$ is the 2$\times$2 identity matrix.

The first term in this expression, marked “lossless”, consists of a rotation operation and an overall phase which are identical to the rotation and phase provided by a lossless filter cavity Kimble et al. (2001).

The second,“lossy”, term goes to unity for a lossless filter cavity ($\rho_{p}=1$ and $\rho_{m}=0$). However, in the presence of losses, this term mixes the quadratures of the squeezed state, corrupting “squeezing” with “anti-squeezing”. We emphasise that this effect is not decoherence, as we have not yet introduced the vacuum fluctuations which enter as a consequence of the filter cavity losses, but rather a coherent dephasing of the squeezed quadratures which cannot be undone by rotation of the state. This dephasing is a direct result of different reflection magnitudes experienced by the upper and lower audio sidebands (i.e. $\rho_{m}\neq 0$). The ramifications of this effect on the measured noise are presented in section II.5.

Additionally, by combining (13) and (II.2), we are now able to write an explicit expression for the squeezed quadrature rotation, $\alpha_{p}$, produced by the filter cavity,

which holds for typical filter cavity parameters, $\epsilon\ll 1\Rightarrow\Lambda_{\rm rt}^{2}\ll t_{\rm in}^{2}$. In particular, for a lossless filter cavity ($\epsilon=0$),

consistent with the expression for $\alpha_{p}$ which can be deduced from (88) of Kimble et al. (2001) (note that the referenced equation is missing factor of 2, as reported in Harms et al. (2003)).

A quantum filter cavity modifies the phase of the squeezed state which is coupled into its resonant mode. In a laboratory context, free-space optics are used to perform this coupling, maximising the spatial overlap between the cavity mode and the incident beam. This process is known as “mode-matching” and the result is inevitably imperfect. In the case of quantum filter cavities, imperfect mode-matching results in both a source of loss and in a path by which the squeezed state can bypass the filter cavity. In this section we develop a model describing how imperfect filter cavity mode-matching affects a squeezed state. Furthermore, we also include the effects of loss arising from mode-mismatch between the squeezed field and the beam, known as the “local oscillator” (LO), used to detect it.

The previously stated filter cavity reflectivity $r_{\rm fc}$ applies only to a field perfectly mode-matched to the cavity fundamental mode. In order to incorporate mode-mismatch, we express the LO and the beam from the squeezed light source in an orthonormal basis of spatial modes $U_{n}$ (e.g. Hermite-Gauss or Laguerre-Gauss modes) such that

where $a_{n}$ and $b_{n}$ are complex coefficients. We further choose this basis such that $U_{0}$ is the filter cavity fundamental mode. For $a_{0}=1$ the beam from the squeezed light source is perfectly matched to the filter cavity mode. Similarly, $b_{0}=1$ indicates that the local oscillator beam has perfectly spatial overlap with the filter cavity mode.

Since the filter cavity is held near the resonance of the fundamental mode, we assume that all other modes ($U_{n}$ with $n>0$) are far from resonance, with $\xi\gg 1$ and $r_{\rm fc}\simeq 1$. Thus, the squeezed beam after reflection from the filter cavity is given by

The fundamental mode’s amplitude and phase are modified by the filter cavity, whereas those of the other modes remain unchanged since these modes are not resonant and the filter cavity acts like a simple mirror.

The spatial overlap integral of the reflected field $U_{\rm rfc}$ and the local oscillator $U_{\rm lo}$ is

where $t_{00}=a_{0}b_{0}^{*}{\rm~and~}t_{\rm mm}=\sum_{n=1}^{\infty}a_{n}b_{n}^{*}.$ Note that $t_{\rm mm}$ represents the overlap between the mismatched part of the beam from the squeezed light source and the mismatched LO. The squeezed field which follows this path essentially bypasses the filter cavity, and thereby experiences no frequency dependent rotation. It may, however, acquire a frequency independent rotation with respect to the field which couples into the filter cavity, as can be seen from the two-photon mode-mismatch matrix

The addition of this coupling path results in a frequency dependent rotation error with respect to the rotation expected from a perfectly mode matched filter cavity. For modest amounts of mode-mismatch (less than $10\%$), this error can be corrected by a small change in the filter cavity detuning.

The magnitude of the mode-mismatch is constrained by $t_{00}$ such that

while the phase is in general unconstrained. The $\left<U_{\rm lo}|U_{\rm rfc}\right>$ overlap is maximised when $t_{\rm mm}$ is real and positive and minimised when it is real and negative.

Experimentally, the quantities which one can easily measure are the squeezed field/filter cavity power mode-coupling, $a_{0}^{2}$, and the squeezed field/local oscillator power mode-coupling, $c_{0}^{2}$, say. From these values one can determine $b_{0}$, the overlap between the LO and filter cavity modes, in the following way,

where $\phi_{\mathrm{mm}}$ captures the ambiguity in the $t_{\mathrm{mm}}$ phase. The parameters of interest for noise propagation are then easily determined,

Note that the second equality in (25) is not universally true. The magnitude of the second term (the expression multiplying the exponential) can be smaller than that given, depending on the unknown character of the mode-mismatches. However this choice, an upper bound, allows one to explore the full range of $b_{0}$ values necessary to constrain the mode-mismatch-induced noise.

The non-linear action of radiation pressure in an interferometer affects any vacuum field incident upon it. In our analysis, we include an idealised lossless interferometer to illustrate this phenomenon. Operated on resonance, such an interferometer may be described by the transfer matrix

as reported in Buonanno and Chen (2001). Here, $\mathcal{K}$ characterises the coupling of amplitude fluctuations introduced at the interferometer’s dark port to phase fluctuations exiting the same port and takes the form

where $\gamma_{\rm ifo}$ is the interferometer signal-bandwidth and ${\Omega_{\rm SQL}}$ is a characteristic frequency, dependent on the particular interferometer configuration, which approximates the frequency at which the interferometer quantum noise equals the Standard Quantum Limit (i.e. where radiation pressure noise intersects shot noise Kimble et al. (2001)).

For a conventional interferometer without a signal recycling mirror, like the power-recycled Michelson interferometer described in Kimble et al. (2001),

where $P_{\rm arm}$ is the laser power stored inside the interferometer arm cavities, $\omega_{0}$ is the frequency of the carrier field, $L_{\rm arm}$ is the arm cavity length, $m$ is the mass of each test mass mirror, $T_{\rm arm}$ is the power transmissivity of the arm cavity input mirrors and approximations are valid provided arm cavity finesse is high.

For a dual-recycled interferometer, operating with a tuned signal-recycling cavity of length $L_{\rm src}$, it can be shown that, for $\Omega\ll c/L_{\rm src}$,

where $t_{\rm sr}$ and $r_{\rm sr}$ are the amplitude transmissivity and reflectivity of the signal recycling mirror. Given the Advanced LIGO parameters reported in Table 1,

confirming that the effect of signal recycling in Advanced LIGO is to increase the interferometer’s bandwidth whilst reducing the frequency at which its quantum noise reaches the SQL. For such an interferometer, in which $\gamma_{\rm ifo}\gg{\Omega_{\rm SQL}}$, $\mathcal{K}$ may be approximated by $({\Omega_{\rm SQL}}/\Omega)^{2}$ in the region of interest (where $\mathcal{K}$ is order unity or larger).

While (28) is very simple, greater appreciation of the action of the interferometer can be gained by noting that $\mathbf{T_{ifo}}$ can be recast in terms of the previously defined squeeze and rotation operators as

with

The role of the filter cavity is to rotate the input squeezed quadrature as a function of frequency such that it is always aligned with the signal quadrature at the output of the interferometer, even in the presence of rotation by $\theta_{\mathrm{ifo}}$ and the effective rotation caused by squeezing at angle $\phi_{\mathrm{ifo}}$. The required filter cavity rotation is given by

We now combine the intermediate results of previous sections to compute the quantum noise observed in the interferometer readout. Three vacuum fields make contributions to this noise: $v_{1}$ which passes through the squeezer, $v_{2}$ which enters before the interferometer but does not pass through the squeezer and $v_{3}$ which enters after the interferometer. We formulate transfer matrices for each of these fields in turn before providing, in (43), a final expression for the measured noise.

Converting the result of (22) into a two-photon transfer matrix and including losses in the injection and readout paths, via $\mathbf{T_{inj}}$ and $\tau_{\mathrm{ro}}$ respectively (see (2) and (42)), we arrive at the full expression describing the transfer of vacuum field $v_{1}$ through the squeezer, filter cavity and interferometer to the detection point,

We now consider the vacuum field $v_{2}$, which accounts for all fluctuations coupled into the beam due to injection losses, losses inside the filter cavity itself and imperfect mode-matching. The audio-sideband transmission coefficient from the squeezer to the interferometer is

In the two-photon picture, the average of the upper and lower sideband losses gives the source term for the $v_{2}$ vacuum fluctuations, so that

Finally, frequency independent losses $\Lambda^{2}_{\rm ro}$ between the interferometer and the readout introduce a second source of attenuation of the squeezed state and accompanying vacuum fluctuations $v_{3}$, a process described by the following transfer matrix and transmission coefficient

These losses cannot be added to the injection losses mentioned above since they are separated by the non-linear effects of the interferometer. Explicitly, losses before and after the interferometer are not equivalent.

The single-sided power spectrum of the quantum noise at the interferometer readout is then given by

where the local oscillator field $\mathbf{\bar{b}}_{\zeta}=A_{\rm LO}\begin{pmatrix}\sin\zeta&\cos\zeta\end{pmatrix}$, with amplitude $A_{\rm LO}$, determines the readout quadrature. All mathematical operations are as defined in Evans et al. (2013) and $\zeta$ is defined such that $N(\zeta=0)$ is the noise in the quadrature containing the interferometer signal.

We now investigate (43) more closely, providing analytical expressions for the contribution of each term. To improve readability, we normalise all noise powers with respect to shot noise (see appendix  A.2). This action is denoted through the use of an additional circumflex, i.e. $\widehat{N}$ rather than $N$.

In addition to optical losses and mode-mismatch, a further cause of squeezing degradation is phase noise, also referred to as “squeezed quadrature fluctuations” Dwyer (2013). In this section we develop a means of quantifying the impact of this important degradation mechanism.

Assuming some parameter $X$ in $\mathbf{T_{1}}$ or $\mathbf{T_{2}}$ has small, Gaussian-distributed fluctuations with variance $\delta X^{2}$, the average readout noise is given by

Extending this approach to multiple incoherent noise parameters $X_{n}$ yields

where the parameters not explicitly listed as arguments to $\widehat{N}$, including $\zeta$, are assumed to take on their mean values.

While (61) is sufficient to evaluate $\widehat{N}_{\mathrm{avg}}$ for any collection of phase noise sources, we choose to follow the same approach adopted in the treatment of optical losses, considering two classes of squeezed quadrature fluctuations: extra-cavity fluctuations which are frequency independent and intra-cavity fluctuations which are frequency dependent.

Examples of frequency-independent phase noise sources include length fluctuations in the squeezed field injection path and instabilities in the relative phase of the local oscillator or the radio-frequency sidebands which co-propagate with the squeezed field. Such frequency-independent noise may be represented by variations, $\delta\zeta$, in the homodyne readout angle $\zeta$.

Frequency-dependent phase noise is caused by variability in the filter cavity detuning $\Delta{\omega_{\rm fc}}$ (see (4)). This detuning noise results from filter cavity length noise $\delta L_{\rm fc}$, driven by seismic excitation of the cavity mirrors or sensor noise associated with the filter cavity length control loop, according to

Detuning noise gives rise to frequency-dependent phase noise through the properties of the filter cavity resonance. For example, the dependence of $\mathbf{T_{fc}}$ on $\Delta{\omega_{\rm fc}}$ is weak for $\Omega\gg\Delta{\omega_{\rm fc}}$, i.e. for frequencies far from resonance, and stronger for $\Omega\simeq\Delta{\omega_{\rm fc}}$, i.e. for frequencies close to resonance.

General analytic expressions for $\widehat{N}_{\mathrm{avg}}$ as a function of $\delta\zeta$ and $\delta L_{\rm fc}$ are neither concise nor especially edifying. Therefore, in the following section, we apply (61) numerically to illustrate the impact of phase noise in a typical advanced gravitational-wave detector.