Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators

TL;DR

This paper introduces a broad class of quantum states, including generic coherent and squeezed contractive states, that maintain or reduce uncertainty widths over time, enhancing the detection of small disturbances like gravitational waves.

Contribution

The paper generalizes existing quantum states to include states with arbitrary uncertainty products and time-dependent widths, useful for quantum monitoring applications.

Findings

Generic coherent states with time-independent width

Squeezed contractive states with decreasing width over time

States with arbitrarily large uncertainty products

Abstract

Monitoring photon quadratures and free masses are useful tools to detect small disturbances such as gravitational waves.Here we report a large class of states for photon quadratures and free masses potentially useful for this purpose: (1)'generic coherent states' (GCS) of photons, whose width is independent of time and uncertainty product $\sigma (x) \sigma (p) $ is arbitrarily large (a generalization of the minimum uncertainty Schr\"odinger coherent states and (2) `squeezed generic contractive states' (SGCS) for photons and free masses (a generalization of the Yuen states \cite{Yuen}) whose width decreases with time ,uncertainty product is arbitrarily large, and the covariance squared $ <\{\Delta \hat{x}, \Delta \hat{p} \}>^2 $ has an arbitrary value within the allowed range $(0,4 \sigma ^2 (x) \sigma ^2 (p) -1\>)$. Dedicated to the 125th birth anniversary of S. N. Bose.

Equations

$${\sigma }^2(X(t))\ge {\sigma }^2(X(0))+(t^2/m^2){\sigma }^2(P(0))$$

$$\ge 2(t/m)\sigma (X(0))\sigma (P(0))\ge \hslash t/m,$$

$$<p|\psi (t)>=(\pi \alpha {)}^{-1/4}exp[-\frac{(p-\beta {)}^2}{2\alpha }-it\frac{p^2}{2m}],$$

$${\sigma }^2(P(t))=\frac{\alpha }{2},{\sigma }^2(X(t))={\hslash }^2\frac{1+(\alpha t/(m\hslash ){)}^2}{2\alpha },$$

$$A(t)\equiv exp(iHt/\hslash )Aexp(-iHt/\hslash ),$$

$${\sigma }^2(A(t))\equiv <\psi (0)|(\Delta A(t){)}^2|\psi (0)>,$$

$$\Delta A(t)\equiv A(t)-<A(t)>,$$

$$<A(t)>\equiv <\psi (0)|A(t)|\psi (0)>$$

$$\Delta X(t)=\Delta X(0)+(t/m)\Delta P(0),$$

$${\sigma }^2(X(t))={\sigma }^2(X(0))+(t^2/m^2){\sigma }^2(P(0))+$$

$$(t/m)<\psi (0)|\{\Delta X(0),\Delta P(0)\}|\psi (0)>.$$

$$<\psi (0)|\{\Delta X(0),\Delta P(0)\}|\psi (0)>+i\hslash$$

$$=2<\psi (0)|\Delta X(0)\Delta P(0)|\psi (0)>.$$

$$\displaystyle\big{(}<{\psi(0)}|\Delta X(0)\Delta P(0)+\Delta P(0)\Delta X(0)|{\psi(0)}>\big{)}^{2}$$

$$\le 4{\sigma }^2(X(0)){\sigma }^2(P(0))-{\hslash }^2,$$

$$|{\sigma }^2(X(t))-{\sigma }^2(X(0))-(t/m{)}^2{\sigma }^2(P(0))|$$

$$\le (t/m)\sqrt{4{\sigma }^2(X(0)){\sigma }^2(P(0))-{\hslash }^2}.$$

$$\Delta P(0)|\psi (0)>=i\lambda \Delta X(0)|\psi (0)>,$$

$$\displaystyle<X^{\prime}|{\psi(0)}>=\big{(}\frac{Re\lambda}{\pi\hbar}\big{)}^{1/4}$$

$$\displaystyle\times exp\big{(}\frac{i<P(0)>X^{\prime}}{\hbar}-\frac{\lambda(X^{\prime}-<X(0)>)^{2}}{2\hbar}\big{)},$$

$$|Im\lambda |=\frac{1}{2{\sigma }^2(X(0))}\sqrt{4{\sigma }^2(X(0)){\sigma }^2(P(0))-{\hslash }^2},$$

$${\sigma }^2(X(0))=\hslash /(2Re\lambda ),{\sigma }^2(P(0))=\hslash |\lambda {|}^2/(2Re\lambda ),$$

$$<\psi (0)|\{\Delta X(0),\Delta P(0)\}|\psi (0)>$$

$$=\mp \sqrt{4{\sigma }^2(X(0)){\sigma }^2(P(0))-{\hslash }^2},ifIm\lambda =\pm |Im\lambda |$$

$$t_M=\frac{m}{{\sigma }^2(P(0))}\sqrt{4{\sigma }^2(X(0)){\sigma }^2(P(0))-{\hslash }^2},$$

$$H=\hslash \omega a^{†}a=\frac{1}{2}\hslash \omega (p^2+x^2-1),$$

$$a=(x+ip)/\sqrt{2},a^{†}=(x-ip)/\sqrt{2}.$$

$${\sigma }^2(x(t))={\cos }^2(\omega t){\sigma }^2(x(0))+{\sin }^2(\omega t){\sigma }^2(p(0))$$

$$+\frac{1}{2}\sin (2\omega t)<\psi (0)|\{\Delta x(0),\Delta p(0)\}|\psi (0)>.$$

$$|{\sigma }^2(x(t))-{\cos }^2(\omega t){\sigma }^2(x(0))-{\sin }^2(\omega t){\sigma }^2(p(0))|$$

$$\le \frac{1}{2}|\sin (2\omega t)|\sqrt{4{\sigma }^2(x(0)){\sigma }^2(p(0))-1}$$

$$SCS:(b-\beta )|\psi (0)>=0,withb=\mu a+\nu a^{†},$$

$$\alpha \equiv <\psi (0)|a|\psi (0)>,\beta =\mu \alpha +\nu {\alpha }^{\ast },$$

$$\mu =\cosh r,\nu =e^{i\theta }\sinh r,$$

$$|\psi (0)>=|\alpha ,r\exp (i\theta )>\equiv D(\alpha ,a)S(\xi )|0>,$$

$$D(\beta ,b)=D(\alpha ,a)=\exp (\alpha a^{†}-{\alpha }^{\ast }a),$$

$$\displaystyle S(\xi)=\exp{\frac{1}{2}\big{(}\xi^{*}a^{2}-\xi a^{\dagger 2}\big{)}},\>\xi\equiv r\exp{(i\theta)}.$$

$$D^{†}(\alpha ,a)aD(\alpha ,a)=a+\alpha ,$$

$$D^{†}(\beta ,b)bD(\beta ,b)=b+\beta ,$$

$$S(\xi )a{S}^{†}(\xi )=a\cosh r+a^{†}{e}^{i\theta }\sinh r=b.$$

$$\displaystyle\sigma^{2}(x(0))=1/2\big{(}cosh(2r)-cos(\theta)sinh(2r)\big{)}$$

Full text

———————Dedicated to the 125th birth anniversary of S. N. Bose.

History. S. N. Bose’s 1924 paper “Planck’s Law and Hypothesis of Light Quanta ” founded quantum statistics even before quantum mechanics was born. Naturally, it is one of the pillars of quantum optics . Here we construct quantum states of optical quadratures which are non-spreading and hence useful for accurate monitoring of small disturbances.

Much before the actual discovery of gravitational waves Abbott it was realised that accurate monitoring of position of an oscillator and of a free mass, including intrinsic quantum uncertainties, are important for gravitational wave interferometers Thorne . Monitoring accuracy is significantly restricted due to the nearly ubiquitous “spreading of wave packets ” suggested by the heuristic standard quantum limit (SQL) (Braginsky ,Caves1980 ).Fortunately, Yuen Yuen discovered that there are contractive states of free masses for which the SQL is incorrect.Recently one of us (SMR) SMR2018 has obtained rigorous quantum limits (RQL) on monitoring free masses ,oscillators and photon quadratures ,and the corresponding maximally contractive states. Consistent with these RQL we present a large class of generic coherent states and generic contractive states likely to be useful for accurate quantum monitoring.

In 1926, referring to the general property of spreading of wave packets, H. A. Lorentz Lorentz said , in a letter to Schrödinger, ”because of this unavoidable blurring a wave packet does not seem to me to be very suitable for representing things to which we want to ascribe a rather permanent individual existence ”.In his reply Schrodinger constructed the now famous oscillator coherent state whose wave packet has a width (and shape) independent of time. At the 1927 Solvay conference Einstein used wave packet spreading to discuss the example of a particle passing through a narrow hole on to a hemispherical fluorescent screen which records the arrival of the particle (Fig. 1).

Suppose that a scintillation is seen at a point $P$ at time $t=T$, and suppose that the hole is so narrow that the wave packet corresponding to the particle is uniformly spread all over the screen at $t$ slightly less than $T$. Was the particle somewhere near $P$ at $t=T-\epsilon$ ($\epsilon$ small)? Ordinary quantum mechanics says that the probabilities at $t=T-\epsilon$ for the particle being found anywhere on the screen are uniform (and not particularly large in the vicinity of $P$). Thus the naive history corresponding to the reality of particle positions at each time is absent. Born’s rule says that the wave function gives probabilities of a particle being found somewhere if a measurement is made, and not of being somewhere.

The role of wave packet spreading in discussions of quantum foundations was further bolstered by heuristic arguments proposing that the accuracy of monitoring position of a free mass $m$ is limited by the standard quantum limit (SQL) (Braginsky ,Caves1980 ):

[TABLE]

where $\sigma^{2}(X(t))$ and $\sigma^{2}(P(t))$ denote variances of the Heisenberg representation position and momentum operators at time $t$.

For the free mass , the inequality (1) is actually an equality for Gaussian states,

[TABLE]

Equation ( 1) forms the basis of most discussions of spreading of wave packets. The non-causality of quantum mechanics is in sharp focus, because the same initial state is equally likely to result in clicks at widely separated points on the screen. Actually a causal “hidden variable” theory which yields the same probability densities of position and momentum as ordinary quantum mechanics exists Roy-Singh1995 . Nevertheless “non-classicality” of the trajectories is inevitable.

Both expanding and contracting wave packets represent departures from classicality or coherence represented by wave-packets of constant width. We shall see that the uncertainty principle limits both the possible rates of expansion and of contraction.

Rigorous quantum limits on contractive and expanding states for a free mass. Surprisingly, for free masses, Yuen discovered in 1983 Yuen a class of states called ’twisted coherent states’ which are ’contractive states’, i.e. states whose position uncertainty decreases with time for a certain duration. The SQL is incorrect for these states. One of us (SMR)SMR2018 obtained rigorous quantum limits (RQL) valid for all states including contractive states.

For any observable with Schrödinger operator $A$ (e.g. position $A=X$ or momentum $A=P$), and any Hamiltonian $H$, the Heisenberg operator $A(t)$ at time t and its variance $\sigma^{2}(A(t))$ are defined by,

[TABLE]

where $|{\psi(0)}>$ is the initial state.

For a free mass, $H=P^{2}/(2m)$ ; the Heisenberg equation yields,

[TABLE]

and hence,

[TABLE]

One obtains the SQL (Braginsky ,Caves1980 ) Eq. (1) if one assumes that the third term on the right-hand side, viz. the covariance $<\{\Delta X(0),\Delta P(0)\}>$ is non-negative. Yuen showed that the covariance is in fact negative for certain states. Nevertheless, rigorous quantum limits (RQL) can be obtained on the covariance, and hence on $\sigma^{2}(X(t))$.

Using $[\Delta X(0),\Delta P(0)]=i\hbar$ , we have,

[TABLE]

Cauchy-Schwarz inequality on the right-hand side yields,

[TABLE]

which is a rearrangement of the usual Schrödinger-Robertson uncertainty relation Kennard .

Substituting this into Eq. ( Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators.111Submitted to Physics News.) we have the rigorous quantum limits (RQL) SMR2018 on expansion and contraction of wave packets,

[TABLE]

valid for arbitrary states. The only states saturating the inequalities are those which obey

[TABLE]

with $Re\lambda>0$,

[TABLE]

and,

[TABLE]

The positive sign of $Im\lambda$ corresponds to maximally contractive (essentially Yuen states Yuen ),and the negative sign of $Im\lambda$ to **maximally expanding ** wave packets.

Fig. 2 shows that for the initial state (13 ) with positive $Im\lambda$ , the state at time $t$ remains contractive upto $t=t_{M}/2$, where,

[TABLE]

and, for a given uncertainty product, by choosing $(t/m)\sigma^{2}(P(0))=\sigma(X(0))\sigma(P(0))$, $\sigma^{2}(X(t))$ can be made $\approx t\hbar^{2}/(4m\sigma(X(0))\sigma(P(0)))$ for a large uncertainty product, and can be much smaller than the heuristic standard quantum limit $\hbar t/m$ .

**Rigorous Quantum Limits on Monitoring Photon Quadratures . **

The single mode photon Hamiltonian is,

[TABLE]

where the quadrature operators $x,p$ are given by

[TABLE]

The Heisenberg equations of motion yield,

[TABLE]

As before, $[\Delta x(0),\Delta p(0)]=i$ and the Schrödinger-Robertson uncertainty relations yield the RQL SMR2018 ,

[TABLE]

which corresponds to Eqn.( Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators.111Submitted to Physics News. ) for a free mass. The extremal states saturating these RQL are complex Gaussians corresponding to to Equations (13), (14) ; both the maximally contractive (MCON) and maximally expanding (MEXP) states can be designated as ’twisted coherent states’ Yuen or ’squeezed coherent states’ (SCS) ,

[TABLE]

where $r>0$ is the squeezing parameter , $\theta$ is real and $|0>$ denotes the vacuum state ;here the unitary displacement operator $D(\alpha,a)$ and squeeze operator $S(\xi)$ are,

[TABLE]

They obey ,

[TABLE]

Explicit values for the standard deviations and covariance in the SCS (24) are then easily derived,

[TABLE]

For $r>0$, the state is squeezed ,i.e. $\sigma^{2}(x(0))<\sigma^{2}(p(0))$, if $cos(\theta)>0$, and the state is contractive for small positive $t$ if $sin(\theta)>0$ . The squeezed coherent states of negative covariance, being contractive , have been utilised in precision measurements with gravitational interferometers squeezed measurements .

Generic Coherent States (GCS) of arbitrarily large uncertainty product The Schrödinger coherent states $|\alpha>=D(\alpha,a)|0>$ have minimum uncertainty product $\sigma(x(0))\sigma(p(0))=1/2$, and time-independent $\sigma(x(t)),\sigma(p(t))$. Roy and Singh Roy-Singh1982 noted that the property of time-independent width of the wave packets also holds for the generalised coherent states ,

[TABLE]

We show here that the property of time-independent width of the wave packet holds for a class of states much larger than these displaced oscillator eigen states. We call this new class, “Generic coherent states” (GCS); they have arbitrarily large continuous values of the uncertainty product.

From the time development equation ( Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators.111Submitted to Physics News.), denoting expectation value of an operator $A$ in the initial state $|\psi>$ by $<A>$ , we see that $\sigma(x(t))$ is time-independent if and only if,

[TABLE]

Using,

[TABLE]

the GCS conditions are equivalent to,

[TABLE]

The GCS include the usual coherent states $\Delta a|\psi>=0$ as a special case.

We now have the theorem:

If $|\phi>$ is a normalized state obeying

[TABLE]

and

[TABLE]

where $\alpha$ is an arbitrary complex parameter, then $|\psi(\alpha,\phi)>$ is a generic coherent state (GCS).

For proof it suffices to note that

[TABLE]

and

[TABLE]

When $|\phi>=|n>$, we get the Roy-Singh Roy-Singh1982 generalised coherent states; but the possible states $|\phi>$ form a much larger set allowing arbitrarily large continuous values of the uncertainty product :

[TABLE]

It remains only to show that states $|\phi>$ giving arbitrary non-negative values of $\bar{n}$ exist. Let

[TABLE]

We assume $N\geq n+3$ and solve Equations (31) to get

[TABLE]

where the summations on the right-hand side are to be replaced by zero when the upper limit on $m$ is less than the lower limit. This pair of linear eqns. can be solved explicitly for $c_{n}$ and $c_{N}$, in terms of all the other non-zero $c_{m}$’s . We omit the explicit general solution because it is elementary. We just quote the solution for the special case $N=n+3$,

[TABLE]

It is elementary to check that the vast class of states ( 34) obeying (31 ) can yield any value of $\bar{n}$. E.g. if

[TABLE]

then, equations (31 ) are obeyed , and

[TABLE]

which can equal any value in the continuous interval $[0,3s]$.Thus the GCS with continuously varying uncertainties ( Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators.111Submitted to Physics News.) are obtained.

Squeezed generic coherent states (SGCS). Using the states $|\phi>$ to replace the vacuum state $|0>$ leads to the class of generic coherent states (GCS) with arbitrarily large continuous uncertainty products. We may similarly generalize the squeezed coherent states (SCS) of maximum possible magnitude of the covariance $|<\{\Delta x(0),\Delta p(0)\}>|$ (i.e. maximally contractive or maximally expanding state) to squeezed generic coherent states (SGCS) which can have any value of the covariance allowed by the uncertainty principle.

Consider, the states

[TABLE]

which are obtained by replacing $|0>$ in the SCS by the state $|\phi>$ . These states obey the SGCS conditions,

[TABLE]

which are obvious generalizations of the SCS conditions ( Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators.111Submitted to Physics News.).

Unlike the SCS wave functions, the SGCS wave functions are not complex Gaussians. E.g., when $|\phi>=|n>$ , using $(b^{\dagger}\>b-n)S(\xi)|n>=0$ we get the displaced and scaled oscillator eigen functions,

[TABLE]

For general $|\phi>$ (calculated in the above section), we obtain a generalization of the $SCS$ expressions,

[TABLE]

Time development of these generic contractive or expanding wave packets follows from Eqn. ( Generic Contractive States and Quantum Monitoring of Free Masses and Oscillators.111Submitted to Physics News. ) using $|\psi(\alpha,\xi,\phi)>$ as the initial state.

Overcompleteness of the SGCS. Let, $S(\xi)|\phi>\equiv|\psi(\xi,\phi)>$. Then,$|\psi(\alpha,\xi,\phi)>=D(\alpha,a)|\psi(\xi,\phi)>$. Hence,

[TABLE]

The integration over $\alpha$ and the fact that $|\psi(\xi,\phi)>$ is a normalized state yields the overcompleteness relation,

[TABLE]

Free Mass. The SGCS for the dimensionless photon variables $x,p$ can also be used as initial states for a free mass $m$ ,using $X=x\sqrt{\hbar/(m\omega)},P=p\sqrt{m\hbar\omega}$. We then find the time development equation for a free mass ,

[TABLE]

The third term on the right-hand side ,where the square root involves the dimensionless $x(0),p(0)$ of the last section ,exhibits all possible rates of contraction and expansion of wave packets allowed by the uncertainty principle.

Position Measurements On Free Masses and Harmonic Oscillators Using Contractive States. In order to exploit the new possibilities allowed by the contractive states which violate the SQL (but obey the RQL), the Ozawa measurement model for system-meter interaction Ozawa1988 which improves on the von Neumann model von Neumann has been used . The basic idea is to make successive measurements of appropriate duration with meters prepared in identical contractive states such that after each measurement the system is left in the contractive state in which the meter was prepared, and between measurements there is contractive evolution with the system Hamiltonian. Details of this can be found in ( Ozawa1988 , SMR2018 ) , of continuous measurement methods in Continuous , and of actual experimental realizations in squeezed measurements . The new generic coherent states (GCS) and the generic contractive states among the squeezed generic coherent states (SGCS) are expected to be useful for measurements necessary for accurate ‘quantum monitoring’.

Acknowledgements. One of us (SMR) thanks the Indian National Science Academy for the INSA honorary scientist position at HBCSE, TIFR, and Dipan Ghosh for the invitation to write this article and several editorial suggestions . Priyanshi Bhasin and Ujan Chakraborty thank the NIUS program of HBCSE for making this collaboration possible.