Squeezed Coherent States

In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position $x$ and momentum $p$ of a particle, and the (dimension-less) electric field in the amplitude $X$ (phase 0) and in the mode $Y$ (phase 90°) of a light wave (the wave's quadratures). The product of the standard deviations of two such operators obeys the uncertainty principle:

:$\Delta x \Delta p \geq \frac2\;$ and $\;\Delta X \Delta Y \geq \frac4$ , respectively.

Trivial examples, which are in fact not squeezed, are the ground state $, 0\rangle$ of the quantum harmonic oscillator and the family of coherent states $, \alpha\rangle$. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with $\Delta x_g = \Delta p_g$ in "natural oscillator units" and $\Delta X_g = \Delta Y_g = 1/2$.

The term squeezed state is actually used for states with a standard deviation below that of the ground state for one of the operators or for a linear combination of the two. The idea behind this is that the circle denoting the uncertainty of a coherent state in the quadrature phase space (see right) has been "squeezed" to an ellipse of the same area. Note that a squeezed state does not need to saturate the uncertainty principle.

Squeezed states of light were first produced in the mid 1980s.R. E. Slusher et al., ''Observation of squeezed states generated by four wave mixing in an optical cavity'', Phys. Rev. Lett. 55 (22), 2409 (1985)
At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e. $\Delta^2 X \approx \Delta^2 X_g/2$. As of 2017, a squeeze factor of 31 (15 dB) has been directly observed.

Mathematical definition

The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with $\hbar=1$)

:$\psi(x) = C\,\exp\left(-\frac + i p_0 x\right)$

where $C,x_0,w_0,p_0$ are constants (a normalization constant, the center of the wavepacket, its width, and the expectation value of its momentum). The new feature relative to a coherent state is the free value of the width $w_0$, which is the reason why the state is called "squeezed".

The squeezed state above is an eigenstate of a linear operator

:$\hat x + i\hat p w_0^2$

and the corresponding eigenvalue equals $x_0+ip_0 w_0^2$. In this sense, it is a generalization of the ground state as well as the coherent state.

Operator representation

The general form of a squeezed coherent state for a quantum harmonic oscillator is given by

:$, \alpha,\zeta\rangle = \hat(\zeta), \alpha\rangle = \hat(\zeta) \hat(\alpha), 0\rangle$

where $, 0\rangle$ is the vacuum state, $D(\alpha)$ is the displacement operator and $S(\zeta)$ is the squeeze operator, given by

:\hat(\alpha)=\exp (\alpha \hat a^\dagger - \alpha^* \hat a)\qquad \text\qquad \hat(\zeta)=\exp\bigg frac (\zeta^* \hat a^2-\zeta \hat a^)\bigg/math>

where $\hat a$ and $\hat a^\dagger$ are annihilation and creation operators, respectively. For a quantum harmonic oscillator of angular frequency $\omega$, these operators are given by

:$\hat a^\dagger = \sqrt\left(x-\frac\right)\qquad \text \qquad \hat a = \sqrt\left(x+\frac\right)$

For a real $\zeta$, (note that $\zeta = r e^$, where ''r'' is squeezing parameter), the uncertainty in $x$ and $p$ are given by

:$(\Delta x)^2=\frac\mathrm^ \qquad\text\qquad (\Delta p)^2=\frac\mathrm^$

Therefore, a squeezed coherent state saturates the Heisenberg uncertainty principle $\Delta x\Delta p=\frac$, with reduced uncertainty in one of its quadrature components and increased uncertainty in the other.

Some expectation values for squeezed coherent states are

:$\langle\alpha,\zeta , \hat a , \alpha,\zeta\rangle = \alpha \cosh(r) - \alpha^e^ \sinh(r)$

:$\langle\alpha,\zeta , ^2 , \alpha,\zeta\rangle = \alpha ^ \cosh^(r) +^e^ \sinh^(r) - (1+2^)e^ \cosh (r) \sinh (r)$

:$\langle\alpha,\zeta , ^\hat , \alpha,\zeta\rangle = , \alpha, ^2 \cosh^(r) + (1+^)\sinh^2 (r) - (^2 e^ + ^2 e^)\cosh (r) \sinh (r)$

The general form of a displaced squeezed state for a quantum harmonic oscillator is given by

:$, \zeta,\alpha\rangle = \hat(\alpha), \zeta\rangle = \hat(\alpha) \hat(\zeta), 0\rangle$

Some expectation values for displaced squeezed state are

:$\langle\zeta,\alpha , \hat a , \zeta,\alpha\rangle = \alpha$

:$\langle\zeta,\alpha , ^2 , \zeta,\alpha\rangle = \alpha ^ - e^ \cosh (r) \sinh (r)$

:$\langle\zeta,\alpha , ^\hat , \zeta,\alpha\rangle = , \alpha, ^2 + \sinh^2 (r)$

Since $\hat(\zeta)$ and $\hat(\alpha)$ do not commute with each other,

: $\hat(\zeta) \hat(\alpha) \neq \hat(\alpha) \hat(\zeta)$

:$, \alpha, \zeta \rangle \neq , \zeta, \alpha \rangle$

where $\hat(\alpha)\hat(\zeta) =\hat(\zeta)\hat^(\zeta)\hat(\alpha)\hat(\zeta)= \hat(\zeta)\hat(\gamma)$, with $\gamma=\alpha\cosh r + \alpha^* e^ \sinh r$

Examples

Depending on the phase angle at which the state's width is reduced, one can distinguish amplitude-squeezed, phase-squeezed, and general quadrature-squeezed states. If the squeezing operator is applied directly to the vacuum, rather than to a coherent state, the result is called the squeezed vacuum. The figures below give a nice visual demonstration of the close connection between squeezed states and Heisenberg's uncertainty relation: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the complementary quadrature, that is, the field phase shifted by $\pi/2$.

As can be seen in the illustrations, in contrast to a coherent state, the quantum noise for a squeezed state is no longer independent of the phase of the light wave. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The probability distribution of a squeezed state is defined as the norm squared of the wave function mentioned in the last paragraph. It corresponds to the square of the electric (and magnetic) field strength of a classical light wave. The moving wave packets display an oscillatory motion combined with the widening and narrowing of their distribution: the "breathing" of the wave packet. For an amplitude-squeezed state, the most narrow distribution of the wave packet is reached at the field maximum, resulting in an amplitude that is defined more precisely than the one of a coherent state. For a phase-squeezed state, the most narrow distribution is reached at field zero, resulting in an average phase value that is better defined than the one of a coherent state.

In phase space, quantum mechanical uncertainties can be depicted by the Wigner quasi-probability distribution. The intensity of the light wave, its coherent excitation, is given by the displacement of the Wigner distribution from the origin. A change in the phase of the squeezed quadrature results in a rotation of the distribution.

Photon number distributions and phase distributions

The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the photon number distribution of the light wave and its phase distribution as well.

For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in sub-Poissonian light, whereas its phase distribution is wider. The opposite is true for the phase-squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution. Nevertheless, the statistics of amplitude squeezed light was not observed directly with photon number resolving detector due to experimental difficulty.

For the squeezed vacuum state the photon number distribution displays odd-even-oscillations. This can be explained by the mathematical form of the squeezing operator, that resembles the operator for two-photon generation and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.

Classification

Based on the number of modes

Squeezed states of light are broadly classified into single-mode squeezed states and two-mode squeezed states, depending on the number of modes of the electromagnetic field involved in the process. Recent studies have looked into multimode squeezed states showing quantum correlations among more than two modes as well.

Single-mode squeezed states

Single-mode squeezed states, as the name suggests, consists of a single mode of the electromagnetic field whose one quadrature has fluctuations below the shot noise level and the orthogonal quadrature has excess noise. Specifically, a single-mode squeezed ''vacuum'' (SMSV) state can be mathematically represented as,
:$, \text\rangle = S(\zeta), 0\rangle$
where the squeezing operator S is the same as introduced in the section on operator representations above. In the photon number basis, writing $\zeta = r e^$ this can be expanded as,
:$, \text\rangle = \frac \sum_^\infty (- e^ \tanh r)^n \frac , 2n\rangle$
which explicitly shows that the pure SMSV consists entirely of even-photon Fock state superpositions.
Single mode squeezed states are typically generated by degenerate parametric oscillation in an optical parametric oscillator, or using four-wave mixing.

Two-mode squeezed states

Two-mode squeezing involves two modes of the electromagnetic field which exhibit quantum noise reduction below the shot noise level in a linear combination of the quadratures of the two fields. For example, the field produced by a nondegenerate parametric oscillator above threshold shows squeezing in the amplitude difference quadrature. The first experimental demonstration of two-mode squeezing in optics was by Heidmann ''et al.''. More recently, two-mode squeezing was generated on-chip using a four-wave mixing OPO above threshold. Two-mode squeezing is often seen as a precursor to continuous-variable entanglement, and hence a demonstration of the Einstein-Podolsky-Rosen paradox in its original formulation in terms of continuous position and momentum observables. A two-mode squeezed vacuum (TMSV) state can be mathematically represented as,
:$, \text\rangle = S_2(\zeta), 0,0\rangle = \exp(\zeta^* \hat a \hat b - \zeta \hat a^\dagger \hat b^\dagger) , 0,0\rangle$,

and, writing down $\zeta = r e^$, in the photon number basis as,

:$, \text\rangle = \frac \sum_^\infty (-e^\tanh r)^n , nn\rangle$
If the individual modes of a TMSV are considered separately (i.e., $, nn\rangle=, n\rangle_1 , n\rangle_2$), then tracing over or absorbing one of the modes leaves the remaining mode in a thermal state

:\begin\rho_1 &= \mathrm_2 optical parametric oscillator