Squeezed coherent state

In physics, a squeezed coherent state is any state of the quantum mechanical Hilbert space such that the uncertainty principle is saturated. That is, the product of the corresponding two operators takes on its minimum value:

Husimi distribution of the squeezed coherent state

\Delta x\Delta p={\frac {\hbar }2}

The simplest such state is the ground state $|0\rangle$ of the quantum harmonic oscillator. The next simple class of states that satisfies this identity are the family of coherent states $|\alpha \rangle$ .

Often, the term squeezed state is used for any such state with $\Delta x\neq \Delta p$ in "natural oscillator units". The idea behind this is that the circle denoting a coherent state in a quadrature diagram (see below) has been "squeezed" to an ellipse of the same area. ^[1] ^[2] ^[3] ^[4] ^[5]

Mathematical definition

Animated position-wavefunction of an 2dB amplitude-squeezed coherent state of α=3.

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 {(x-x_{0})^{2}}{2w_{0}^{2}}}+ip_{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 of squeezed coherent states

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

|\alpha ,\zeta \rangle =D(\alpha )S(\zeta )|0\rangle

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

D(\alpha )=\exp(\alpha {\hat a}^{\dagger }-\alpha ^{*}{\hat a})\qquad {\text{and}}\qquad S(\zeta )=\exp({\frac {1}{2}}(\zeta ^{*}{\hat a}^{2}-\zeta {\hat a}^{{\dagger 2}}))

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 {{\frac {m\omega }{2\hbar }}}}\left(x-{\frac {{\mathrm {i}}p}{m\omega }}\right)\qquad {\text{and}}\qquad {\hat a}={\sqrt {{\frac {m\omega }{2\hbar }}}}\left(x+{\frac {{\mathrm {i}}p}{m\omega }}\right)

For a real $\zeta$ , the uncertainty in $x$ and $p$ are given by

(\Delta x)^{2}={\frac {\hbar }{2m\omega }}\mathrm {e} ^{-2\zeta }\qquad {\text{and}}\qquad (\Delta p)^{2}={\frac {m\hbar \omega }{2}}\mathrm {e} ^{2\zeta }

Therefore, a squeezed coherent state saturates the Heisenberg Uncertainty Principle $\Delta x\Delta p={\frac {\hbar }{2}}$ , with reduced uncertainty in one of its quadrature components and increased uncertainty in the other.

Examples of squeezed coherent states

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 at the phase shifted by $\pi /2$ .

Figure 1: Measured quantum noise of the electric field of different squeezed states depends on the phase of the light field. For the first two states a 3π-interval is shown; for the last three states, belonging to a different set of measurements, it is a 4π-interval.^[6]

Figure 2: Oscillating wave packets of the five states.

Figure 3: Wigner functions of the five states. The ripples are due to experimental inaccuracies.

From the top:

Vacuum state
Squeezed vacuum state
Phase-squeezed state
arbitrary squeezed state
Amplitude-squeezed state

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 of squeezed states

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.

Figure 4: Experimental photon number distributions for an amplitude-squeezed state, a coherent state, and a phase squeezed state reconstructed from measurements of the quantum statistics. Bars refer to theory, dots to experimental values.^[6] (source: link 1)

Figure 5: Pegg-Barnett phase distribution of the three states.

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.^[7]

Figure 6: Reconstructed and theoretical photon number distributions for a squeezed-vacuum state. A pure squeezed vacuum state would have no contribution from odd-photon-number states. The non-zero contribution in the above figure is because the detected state is not a pure state - losses in the setup convert the pure squeezed vacuum into a mixed state.^[6] (source: link 1)

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 of squeezed states

Based on the number of modes

Squeezed states of light are broadly classified into single-mode squeezed states and two-mode squeezed states,^[8] 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,

|SMSV\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, this can be expanded as,

|SMSV\rangle ={\frac {1}{{\sqrt {\cosh r}}}}\sum _{{n=0}}^{\infty }(-\tanh r)^{n}{\frac {{\sqrt {(2n)!}}}{2^{n}n!}}|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,^[9] or using four-wave mixing.^[5]

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..^[10] More recently, two-mode squeezing was generated on-chip using a four-wave mixing OPO above threshold.^[11] 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.^[12]^[13] A two-mode squeezed vacuum (TMSV)state can be mathematically represented as,

|TMSV\rangle =S_{2}(\zeta )|0\rangle =\exp(\zeta ^{*}{\hat {a}}{\hat {b}}-\zeta {\hat {a}}^{\dagger }{\hat {b}}^{\dagger })|0\rangle

and in the photon number basis as,

|TMSV\rangle ={\frac {1}{\cosh r}}\sum _{{n=0}}^{\infty }(\tanh r)^{n}|nn\rangle

If the individual modes of a TMSV are considered separately (i.e., $|nn\rangle =|n\rangle |n\rangle$ , say by tracing over or absorbing the other mode), then the remaining mode is left in a thermal state.

Based on the presence of a mean field

Squeezed states of light can be divided into squeezed vacuum and bright squeezed light, depending on the absence or presence of a non-zero mean field (also called a carrier), respectively. Interestingly, an Optical Parametric Oscillator operated below threshold produces squeezed vacuum, whereas the same OPO operated above threshold produces bright squeezed light. Bright squeezed light can be advantageous for certain quantum information processing applications as it obviates the need of sending local oscillator to provide a phase reference, whereas squeezed vacuum is considered more suitable for quantum enhanced sensing applications. The AdLIGO and GEO600 gravitational wave detectors use squeezed vacuum to achieve enhanced sensitivity beyond the standard quantum limit.^[14]^[15]

Atomic spin squeezing

For squeezing of two-level neutral atom ensembles it is useful to consider the atoms as spin-1/2 particles with corresponding angular momentum operators defined as $J_{v}=\sum _{i=1}^{N}j_{v}^{(i)}$ where $v={x,y,z}$ and $j_{v}^{(i)}$ is the single-spin operator in the $v$ -direction. Here $J_{z}$ will correspond to the population difference in the two level system, i.e. for an equal superposition of the up and down state $J_z=0$ . The $J_{x}$ — $J_{y}$ plane represents the phase difference between the two states. This is also known as the Bloch sphere picture. We can then define uncertainty relations such as $\Delta J_{z}\cdot \Delta J_{y}\geq \left|\Delta J_{x}\right|/2$ . For a coherent (unentangled) state, $\Delta J_{z}=\Delta J_{y}={\sqrt {N}}/2$ . Squeezing is here considered the redistribution of uncertainty from one variable (typically $J_{z}$ ) to another (typically $J_{y}$ ). If we consider a state pointing in the $J_{x}$ direction, we can define the Wineland criterion ^[16] for squeezing, or the metrological enhancement of the squeezed state as

$\chi ^{2}=\left({\frac {{\sqrt {N}}/2}{\Delta J_{z}}}{\frac {\left|J_{x}\right|}{N/2}}\right)^{2}$ .

This criterion has two factors, the first factor is the spin noise reduction, i.e. how much the quantum noise in $J_{z}$ is reduced relative to the coherent (unentangled) state. The second factor is how much the coherence (the length of the Bloch vector, $\left|J_{x}\right|$ ) is reduced due to the squeezing procedure. Together these quantities tell you how much metrological enhancement the squeezing procedure gives. Here, metrological enhancement is the reduction in averaging time or atom number needed to make a measurement of a specific uncertainty. 20 dB of metrological enhancement means the same precision measurement can be made with 100 times fewer atoms or 100 times shorter averaging time.

Experimental realizations of squeezed coherent states

There has been a whole variety of successful demonstrations of squeezed states. The first demonstrations were experiments with light fields using lasers and non-linear optics (see optical parametric oscillator). This is achieved by a simple process of four-wave mixing with a $\chi ^{(3)}$ crystal; similarly traveling wave phase-sensitive amplifiers generate spatially multimode quadrature-squeezed states of light when the $\chi ^{(2)}$ crystal is pumped in absence of any signal. Sub-Poissonian current sources driving semiconductor laser diodes have led to amplitude squeezed light.^[17]

Squeezed states have also been realized via motional states of an ion in a trap, phonon states in crystal lattices, and spin states in neutral atom ensembles.^[18]^[19] Much progress has been made on the creation and observation of spin squeezed states in ensembles of neutral atoms and ions, which can be used to enhancement measurements of time, accelerations, and fields, and the current state of the art for measurement enhancement is 20 dB.^[20]^[20]^[21]^[22]^[23] Generation of spin squeezed states have been demonstrated using both coherent evolution of a coherent spin state and projective, coherence-preserving measurements. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Current state of the art in noise suppression, for laser radiation using squeezed light, amounts to 15 dB (as of 2016),^[24]^[25] which broke the previous record of 12.7 dB (2010).^[26]

Applications

Squeezed states of the light field can be used to enhance precision measurements. For example, phase-squeezed light can improve the phase read out of interferometric measurements (see for example gravitational waves). Amplitude-squeezed light can improve the readout of very weak spectroscopic signals.^[27]

Spin squeezed states of atoms can be used to improve the precision of atom clocks.^[28]^[29] This is an important problem in atomic clocks and other sensors that use small ensembles of cold atoms where the quantum projection noise represents a fundamental limitation to the precision of the sensor.^[30]

Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory, for example Unruh effect and Hawking radiation, and generally, particle production in curved backgrounds and Bogoliubov transformations.

Recently, the use of squeezed states for quantum information processing in the continuous variables (CV) regime has been increasing rapidly.^[31] Continuous variable quantum optics uses squeezing of light as an essential resource to realize CV protocols for quantum communication, unconditional quantum teleportation and one-way quantum computing.^[32]^[33] This is in contrast to quantum information processing with single photons or photon pairs as qubits. CV quantum information processing relies heavily on the fact that squeezing is intimately related to quantum entanglement, as the quadratures of a squeezed state exhibit sub-shot-noise quantum correlations.

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