Is there a simple picture of what is the 3 dB limit in squeezing of standard parametric squeezing techniques?

This limit is mentioned in many papers e.g. the Google search for 3 dB limit in cavity squeezing.

Any suggestions as to where to find an elementary discussion of this limit?

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    $\begingroup$ It would be very helpful if this question could be more self-contained. As it stands, we must read a link to understand even what you're asking. $\endgroup$ – DanielSank Dec 4 '17 at 18:27

As often in continuous variable quantum physics, it is a case where the “limit” is not a hard one (like “no information faster than $c$” in relativity), but a value at which interesting things occurs, some approximations are no longer valid, and it makes an interesting benchmark for experimentalists.

After some googling, I fond this paper (PRL 111 207203) which explains this 3 dB “limit” in parametric amplification in its first page. I follow the beginning of their reasoning below.

One can look at parametric amplification in a purely classical way. Consider a damped harmonic oscillator of resonant frequency $ω$ and quality factor $Q$ excited at resonance by $f(t)$. Parametric amplification happens when one modulates the stiffness of the spring at frequency $2ω$ with a modulation depth $h$. (The stiffness is the main parameter of the oscillator, hence the name “parametric”.) We have the following differential equation: $$\ddot x + \frac{ω}{Q}\dot x + \omega^2 [1+h \sin 2ω t]x = f(t)$$ We now split $x$ and $f$ into in phase and out of phase quadratures : \begin{align} x(t)&=X(t)\cos ωt+ Y(t)\sin ωt\\ f(t)&=f_c(t)\cos \omega t + f_s(t) \sin ωt, \end{align} where $X$, $Y$, $f_c$ and $f_s$ are slowly varying since we are close to resonance. We then compute all the terms of the differential equation: \begin{align} \dot x&=[\dot X + ωY]\cos ωt+ [\dot Y - ω X]\sin ωt\\ \ddot x&=[\ddot X + 2ω\dot Y - ω^2 X]\cos ωt+ [\ddot Y - 2ω \dot X - ω^ 2 Y]\sin ωt\\ [\sin2\omega t]x &=\frac{X}{2}(\sin ω t + \sin 3 ωt)+\frac{Y}{2}(\cos ω t -\cos 3 ωt). \end{align} Neglecting the second order derivatives of slowly varying $X$ and $Y$ as well as out of resonance $3ωt$ terms, the differential equation becomes \begin{align} 2ω\dot Y - ω^2 X + \frac{ω}{Q}\dot X + \frac{ω^2}{Q}Y + ω^2 X + \frac{ω^2 h}{2}Y&=f_c\\ -2ω\dot X -ω^2 Y + \frac{ω}{Q}\dot Y - \frac{ω^2}{Q}X + ω^2 Y + \frac{ω^2 h}{2}X&= f_s \end{align} Which simplifies to \begin{align} \dot Y + \frac{\dot X}{2Q} + \frac{ω}{2Q}\left(1+\frac{h}{2/Q}\right)Y&=\frac{f_c}{2ω}\\ \dot X - \frac{\dot Y}{2Q} + \frac{ω}{2Q}\left(1-\frac{h}{2/Q}\right)X&= -\frac{ f_s}{2ω} \end{align} For high $Q$, we can neglect the terms $\frac{\dot X}{2Q}$ and $\frac{\dot X}{2Q}$, and the evolution of the two quadratures are decoupled. Let $h'=2/Q$. If $f_c$ and $f_s$ are constant, and $h<h'$, both quadratures $X$ and $Y$ tend to a steady state value: \begin{align} Y &→ \frac{1}{1+h/h'}\frac{Qf_c}{ω^2} \\ X &→ -\frac{1}{1-h/h'}\frac{Qf_s}{ω^2} \end{align} The quadrature are (de)amplified by a factor $\frac1{1±h/h'}$ compared to the case of null parametric modulation. When $h→h'$, the gain of the $Y$ quadrature tends to $\frac12$, leading to 3 dB of squeezing. But the gain of the $X$ quadrature $\frac{1}{1-h/h'}$ diverges leading to an infinite noise. Actually, when $h>h'$ the solutions of the differential equation for $X$ above become exponentially divergent, unstable even when $f=0$ and many approximations above become unjustified. In other words, funny stuff happen when one get close to 3 dB of parametric deamplification, even in a classical regime, and other techniques are needed to go beyond this 3 dB “limit”.


The first link I'm shown in the Google search you linked to is the paper

Squeezing of the mechanical motion and beating 3 dB limit using dispersive optomechanical interactions. U. Satya Sainadh & M. Anil Kumar. J. Mod. Opt. 64, 1121 (2016), arXiv:1604.02541,

where a pdf search for "3 dB" search leads to the line

helpful in breaking the 3 dB limit that is paramount for ultra sensitive precision measurements and some quantum information applications [43],

where reference [43] is

Quantum information with continuous variables. S. L. Braunstein and P. van Loock. Rev. Mod. Phys. 77, 513 (2005), arXiv:quant-ph/0410100,

which itself reads, in p. 545,

The fidelity boundary in Eq. (175) is exceeded for any squeezing in the EPR channel, whereas fulfillment of the teleportation criteria of Ralph and Lam (1998) requires more than 3 dB squeezing,

referring in turn to

Teleportation with Bright Squeezed Light. T. C. Ralph and P. K. Lam. Phys. Rev. Lett. 81, 5668 (1998), eprint.

This doesn't seem to mention any hard 3 dB limit, but the papers that cite it do: in particular e.g.

Quantum teleportation criteria for continuous variables. P. Grangier and F. Grosshans (2000). arXiv:quant-ph/0009079

has all the details.

Squeezing is a process which refers to situations with a quantum system with two canonically conjugate variables $q$ and $p$ that, by the Heisenberg Uncertainty principle, must have nonzero uncertainties $\Delta q$ and $\Delta p$ obeying $$ \Delta q\,\Delta p \geq \frac12 \hbar. $$ A squeezed state is normally understood in a normalization where both $q$ and $p$ have the same dimension (normally optical field quadratures), so in the most symmetric configuration the system is driven by the hamiltonian $H=\frac12 (q^2 + p^2)$, and the ground state has symmetric uncertainties $$ \Delta q= \sqrt{\frac12 \hbar} = \Delta p. $$ A squeezed state is a state where one of those two uncertainties smaller than that limit (while driving the other one up by a corresponding amount). The degree of squeezing is often given by quoting the factor $$ s = \frac{\Delta q^2}{\hbar/2} $$ by which the limit-beating variance is smaller than the symmetric limit; this is then often quoted, as is quite normal for ratios, in decibels. Thus, in this context, the 3 dB limit corresponds to $s<1/2$, and the $3$ is obtained as $$ -\log_{10}(s)=-\log_{10}(1/2) \approx 0.3. $$

  • $\begingroup$ Thanks for the answer @emilio-pisanty. I understand that 3 dB is equivalent to reducing the variance by half. I realize my question was unnecessarily vague. I want to understand more precisely what is the " 3 dB limit of standard parametric squeezing techniques" alluded in the abstract of Quantum nondemolition measurement of mechanical squeezed state beyond the 3 dB limit. $\endgroup$ – Nicolás Quesada Dec 4 '17 at 20:12
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    $\begingroup$ @NicolásQuesada : I suggest you should update your question itself to include the words " 3 dB limit of standard parametric squeezing techniques": I was about to write an answer about the “3 dB limit in CV quantum information”, giving more historical details to Emilio Pisanty’s answer $\endgroup$ – Frédéric Grosshans Dec 5 '17 at 11:01
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    $\begingroup$ @EmilioPisanty: A few year after arXiv:quant-ph/0009079 , I would not speak about a hard limit at 3 dB. Until around 2002, may researchers in quantum optics thought that there was an “obvious” 3 dB limit in CV quantum information, both in squeezing and losses, but none proved it. These “limits” are now mainly thought as convenient benchmarks. $\endgroup$ – Frédéric Grosshans Dec 5 '17 at 11:08
  • $\begingroup$ @Frédéric Thanks for the added context. By 'limit' here I mostly meant benchmark, but that did not get on the text as clearly as I'd've liked. Given how poorly phrased the OP was, I'm not sure I'm going to dedicate any more effort to this thread, though. $\endgroup$ – Emilio Pisanty Dec 5 '17 at 11:11
  • $\begingroup$ @Frédéric Good idea. I have updated the title. $\endgroup$ – Nicolás Quesada Dec 5 '17 at 15:49

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