Squeezed states of light

An oscillating physical quantity cannot have precisely defined values at all phases of the oscillation. This is true for the electric and magnetic fields of an electromagnetic wave, as well as for any other wave or oscillation (see figure right). This fact can be observed in experiments and is described by quantum theory. For electromagnetic waves usually just the electric field is considered, because it is the one that mainly interacts with matter.

Fig. 1. shows five different quantum states that a monochromatic wave could be in. The difference of the five quantum states is given by different electric field excitations and by different distributions of the quantum uncertainty along the phase ${\displaystyle \vartheta }$. For a displaced coherent state, the expectation (mean) value of the electric field shows an oscillation, with an uncertainty independent of the phase (a). Also the phase- (b) and amplitude-squeezed states (c) show an oscillation of the mean electric field, but here the uncertainty depends on phase and is squeezed for some phases. The vacuum state (d) is a special coherent state and is not squeezed. It has zero mean electric field for all phases and a phase-independent uncertainty. It has zero energy on average, i.e. zero photons, and is the ground state of the monochromatic wave we consider. Finally, a squeezed vacuum state has also a zero mean electric field but a phase-dependent uncertainty (e).

Generally, quantum uncertainty reveals itself through a large number of identical measurements on identical quantum objects (here: modes of light) that, however, give different results. Let us again consider a continuous-wave monochromatic light wave (as emitted by an ultra-stable laser). A single measurement of Ԑ${\displaystyle (\vartheta _{1})}$ is performed over many periods of the light wave and provides a single number. The next measurements of Ԑ${\displaystyle (\vartheta _{1})}$ will be done consecutively on the same laser beam. Having recorded a large number of such measurements we know the field uncertainty at ${\displaystyle \vartheta _{1}}$. In order to get the full picture, and for instance Fig.1(b), we need to record the statistics at many different phases ${\displaystyle 0<\vartheta _{i}<\pi }$.

The measured electric field strengths at the wave's phase ${\displaystyle \vartheta }$ are the eigenvalues of the normalized quadrature operator ${\displaystyle X_{\vartheta }}$, defined as[5]

$${\displaystyle {\hat {X}}_{\vartheta }={\frac {1}{\sqrt {2}}}\left[e^{-i\vartheta }{\hat {a}}+e^{i\vartheta }{\hat {a}}^{\dagger }\right]=\cos(\vartheta )\,{\hat {X}}+\sin(\vartheta )\,{\hat {Y}}}$$

where ${\displaystyle {\hat {a}}}$ and ${\displaystyle {\hat {a}}^{\dagger }}$ are the annihilation and creation operators, respectively, of the oscillator representing the photon. ${\displaystyle X_{\vartheta =0^{\circ }}\equiv X}$ is the wave's amplitude quadrature, equivalent to the position in optical phase space, and ${\displaystyle X_{\vartheta =90^{\circ }}\equiv Y}$ is the wave's phase quadrature, equivalent to momentum. ${\displaystyle X}$ and ${\displaystyle Y}$ are non-commuting observables. Although they represent electric fields, they are dimensionless and satisfy the following uncertainty relation:[6]

${\displaystyle \;\Delta ^{2}X\Delta ^{2}Y\geq {\frac {1}{16}}}$ ,

where ${\displaystyle \Delta ^{2}}$ stands for the variance. (The variance is the mean of the squares of the measuring values minus the square of the mean of the measuring values.) If a mode of light is in its ground state ${\displaystyle |0\rangle }$ (having an average photon number of zero), the uncertainty relation above is saturated and the variances of the quadrature are ${\displaystyle \Delta ^{2}X_{g}=\Delta ^{2}Y_{g}=1/4}$. (Other normalizations can also be found in literature. The normalization chosen here has the nice property that the sum of the ground state variances directly provide the zero point excitation of the quantized harmonic oscillator ${\displaystyle \Delta ^{2}X_{g}+\Delta ^{2}Y_{g}=1/2}$).

Definition: Light is in a squeezed state, if (and only if) a phase ${\displaystyle \vartheta }$ exists for which ${\displaystyle \;\Delta ^{2}X_{\vartheta }<\Delta ^{2}X_{g}={\frac {1}{4}}}$.[6][7]

While coherent states belong to the semi-classical states, since they can be fully described by a semi-classical model,[8] squeezed states of light belong to the so-called nonclassical states, which also include number states (Fock states) and Schrödinger cat states.

Squeezed states (of light) were first produced in the mid 1980s.[9][10] At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e. ${\displaystyle \Delta ^{2}X_{\vartheta }\approx \Delta ^{2}X_{g}/2}$. Today, squeeze factors larger than 10 (10 dB) have been directly observed.[11][12][13] A limitation is set by decoherence, mainly in terms of optical loss.[8]

The squeeze factor in Decibel (dB) can be computed in the following way:

${\displaystyle -10\cdot \log {\frac {\Delta _{\mathrm {min} }^{2}X_{\vartheta }}{\Delta ^{2}X_{G}}}}$, where ${\displaystyle \Delta _{\mathrm {min} }^{2}X_{\vartheta }}$ is the smallest variance when varying the phase ${\displaystyle \vartheta }$ from 0 to ${\displaystyle \pi }$. This particular phase ${\displaystyle \vartheta }$ is called the squeeze angle.

Fig. 1(f): Left: Wigner function of a squeezed vacuum state. Right: Connection to Fig. 1 (e).

Quantum states such as those in Fig. 1 (a) to (e) are often displayed as Wigner functions, which are quasi-probability density distributions. Two orthogonal quadratures, usually ${\displaystyle X}$and ${\displaystyle Y}$, span a phase space diagram, and the third axes provides the quasi probability of yielding a certain combination of ${\displaystyle [X;Y]}$. Since ${\displaystyle X}$and ${\displaystyle Y}$are not precisely defined simultaneously, we cannot talk about a 'probability' as we do in classical physics but call it a 'quasi probability'. A Wigner function is reconstructed from time series of ${\displaystyle X(t)}$and ${\displaystyle Y(t)}$. The reconstruction is also called 'quantum tomographic reconstruction'. For squeezed states, the Wigner function has a Gaussian shape, with an elliptical contour line, see Fig.: 1(f).

Quantum uncertainty becomes visible when identical measurements of the same quantity (observable) on identical objects (here: modes of light) give different results (eigen values). In case of a single freely propagating monochromatic laser beam, the individual measurements are performed on consecutive time intervals of identical length. One interval must last much longer than the light's period; otherwise the monochromatic property would be significantly disturbed. Such consecutive measurements correspond to a time series of fluctuating eigen values. Consider an example in which the amplitude quadrature ${\displaystyle X}$ was repeatedly measured. The time series can be used for a quantum statistical characterization of the modes of light. Obviously, the amplitude of the light wave might be different before and after our measurement, i.e. the time series does not provide any information about very slow changes of the amplitude, which corresponds to very low frequencies. This is a trivial but also fundamental issue, since any data taking lasts for a finite time. Our time series, however, does provide meaningful information about fast changes of the light's amplitude, i.e. changes at frequencies higher than the inverse of the full measuring time. Changes that are faster than the duration of a single measurement, however, are invisible again. A quantum statistical characterization through consecutive measurements on some sort of a carrier is thus always related to a specific frequency interval, for instance described by ${\displaystyle f\pm \Delta f/2}$ with ${\displaystyle f>\Delta f/2>0.}$ Based on this insight, we can describe the physical meaning of the observable ${\displaystyle X_{\vartheta }}$ more clearly:[8]

Fig. 2: Normalized variances ${\displaystyle \Delta ^{2}X_{f,\Delta f}/\Delta ^{2}X_{f,\Delta f,g}}$ of modulation states of the same carrier light beam versus modulation frequency ${\displaystyle f}$. Here, the measurement band width ${\displaystyle \Delta f}$ is about 10 kHz. Each trace therefore describes about 200 mutually independent modulation modes.

The quantum statistical characterization using identical consecutive modes carried by a laser beam confers to the laser beam's electric field modulation within a frequency interval. The actual observable needs to be labeled accordingly, for instance as ${\displaystyle X_{\vartheta ,f,\Delta f}}$. ${\displaystyle X_{f,\Delta f}}$ is the amplitude (or depth) of the amplitude modulation and ${\displaystyle Y_{f,\Delta f}}$ the amplitude (or depth) of the phase modulation in the respective frequency interval. This leads to the doggerel expressions 'amplitude quadrature amplitude' and 'phase quadrature amplitude'.

Within some limitations, for instance set by the speed of the electronics, ${\displaystyle f}$ and ${\displaystyle \Delta f}$ can be freely chosen in course of data acquisition and, in particular, data processing. This choice also defines the measurement object, i.e. the mode that is characterized by the statistics of the eigen values of ${\displaystyle X_{f,\Delta f}}$ and ${\displaystyle Y_{f,\Delta f}}$. The measurement object thus is a modulation mode that is carried by the light beam. – In many experiments, one is interested in a continuous spectrum of many modulation modes carried by the same light beam.[14] Fig. 2 shows the squeeze factors of many neighboring modulation modes versus ${\displaystyle f}$. The upper trace refers to the uncertainties of the same modes being in their vacuum states, which serves as the 0 dB reference.

The observables in squeezed light experiments correspond exactly to those being used in optical communication. Amplitude modulation (AM) and frequency modulation (FM) are the classical means to imprint information on a carrier field. (Frequency modulation is mathematically closely related to phase modulation). The observables ${\displaystyle X_{f,\Delta f}}$ and ${\displaystyle Y_{f,\Delta f}}$ also correspond to the measurement quantities in laser interferometers, such as in Sagnac interferometers measuring rotation changes and in Michelson interferometers observing gravitational waves. Squeezed states of light thus have ample applications in optical communication and optical measurements. The most prominent and important application is in gravitational-wave observatories.[15][16][8] Arguably, it is the first end-user driven application of quantum correlations.[17] Squeezed light originally was not planned to be implemented in either Advanced LIGO nor in Advanced Virgo, but now it contributes a significant factor towards the observatories design sensitivities and increases the rate of observed gravitational-wave events.[2][3]

Fig. 6: Schematic of a squeezing resonator. The pumped nonlinear crystal inside the resonator attenuates the electric field at optical frequency ${\displaystyle \nu }$. This leads to perfect destructive interference for one quadrature angle that is carried by the optical frequency ${\displaystyle \nu }$ and propagates towards the left (left side of resonator). The pump light enters from the right and is simply retro-reflected. If the pump light intensity is kept below the resonator's oscillation threshold, its input and output powers are basically identical.

Timeline of experimentally achieved light squeezing values in the laboratory. Since the first demonstration in 1985 values have steadily improved.

Squeezed light is produced by means of nonlinear optics. The most successful method uses degenerate type I optical-parametric down-conversion (also called optical-parametric amplification) inside an optical resonator. To squeeze modulation states with respect to a carrier field at optical frequency ${\displaystyle \nu }$, a bright pump field at twice the optical frequency is focussed into a nonlinear crystal that is placed between two or more mirrors forming an optical resonator. It is not necessary to inject light at frequency ${\displaystyle \nu }$. (Such light, however, is required for detecting the (squeezed) modulation states). The crystal material needs to have a nonlinear susceptibility and needs to be highly transparent for both optical frequencies used. Typical materials are lithium niobate (LiNbO₃) and (periodically poled) potassium titanyl phosphate (KTP). Due to the nonlinear susceptibility of the pumped crystal material, the electric field at frequency ${\displaystyle \nu }$ is amplified and deamplified, depending on the relative phase to the pump light. At the pump's electric field maxima, the electric field at frequency ${\displaystyle \nu }$ is amplified. At the pump's electric field minima, the electric field at frequency ${\displaystyle \nu }$ is squeezed. This way, the vacuum state (Fig. 1e) is transferred to a squeezed vacuum state (Fig. 1d). A displaced coherent state (Fig. 1a) is transferred to a phase squeezed state (Fig. 1b) or to an amplitude squeezed state (Fig. 1c), depending on the relative phase between coherent input field and pump field. A graphical description of these processes can be found in.[8]

The existence of a resonator for the field at ${\displaystyle \nu }$ is essential. The task of the resonator is shown in Fig. 6. The left resonator mirror has a typical reflectivity of about ${\displaystyle r_{1}^{2}=90\%}$. Correspondingly ${\displaystyle {\sqrt {0.9}}}$ of the electric field that (continuously) enters from the left gets reflected. The remaining part is transmitted and resonates between the two mirrors. Due to the resonance, the electric field inside the resonator gets enhanced (even without any medium inside). ${\displaystyle 10\%}$ of the steady-state light power inside the resonator gets transmitted towards the left and interferes with the beam that was retro-reflected directly. For an empty loss-less resonator, 100% of the light power would eventually propagate towards the left, obeying energy conservation.

The principle of the squeezing resonator is the following: The medium parametrically attenuates the electric field inside the resonator to such a value that perfect destructive interference is achieved outside the resonator for the attenuated field quadrature. The optimum field attenuation factor inside the resonator is slightly below 2, depending on the reflectivity of the resonator mirror.[8] This principle also works for electric field uncertainties. Inside the resonator, the squeeze factor is always less than 6 dB, but outside the resonator it can be arbitrarily high. If quadrature ${\displaystyle X_{f,\Delta f}}$ is squeezed, quadrature ${\displaystyle Y_{f,\Delta f}}$ is anti-squeezed – inside as well as outside the resonator. It can be shown that the highest squeeze factor for one quadrature is achieved if the resonator is at its threshold for the orthogonal quadrature. At threshold and above, the pump field is converted into a bright field at optical frequency ${\displaystyle \nu }$. Squeezing resonators are usually operated slightly below threshold, for instance, to avoid damage to the photo diodes due to the bright down-converted field.

A squeezing resonator works efficiently at modulation frequencies well inside its linewidth. Only for these frequencies highest squeeze factors can be achieved. At frequencies the optical-parametric gain is strongest, and the time delay between the interfering parts negligible. If decoherence was zero, infinite squeeze factors could be achieved outside the resonator, although the squeeze factor inside the resonator was less than 6 dB. Squeezing resonators have typical linewidths of a few tens of MHz up to GHz.[25]

Due to the interest in the interaction between squeezed light and atomic ensemble, narrowband atomic resonance squeezed light have been also generated through crystal[26] and the atomic medium.[27]

Fig. 7: Balanced homodyne detector. LO: local oscillator; PD: photo diode.

Squeezed states of light can be fully characterized by a photo-electric detector that is able to (subsequently) measure the electric field strengths at any phase ${\displaystyle \vartheta }$. (The restriction to a certain band of modulation frequencies happens after the detection by electronic filtering.) The required detector is a balanced homodyne detector (BHD). It has two input ports for two light beams. One for the (squeezed) signal field, and another for the BHDs local oscillator (LO) having the same wavelength as the signal field. The LO is part of the BHD. Its purpose is to beat with the signal field and to optically amplify it. Further components of the BHD are a balanced beam splitter and two photo diodes (of high quantum efficiency). Signal beam and LO need to be overlapped at the beam splitter. The two interference results in the beam splitter output ports are detected and the difference signal recorded (Fig. 7). The LO needs to be much more intense than the signal field. In this case the differential signal from the photo diodes in the interval ${\displaystyle f\pm \Delta f/2}$ is proportional to the quadrature amplitude ${\displaystyle X_{\vartheta ,f,\Delta f}}$. Changing the differential propagation length before the beam splitter sets the quadrature angle to an arbitrary value. (A change by a quarter of the optical wavelength changes the phase by  ${\displaystyle \pi /2}$.)

The following should be stated at this point: Any information about the electro-magnetic wave can only be gathered in a quantized way, i.e. by absorbing light quanta (photons). This is also true for the BHD. However, a BHD cannot resolve the discrete energy transfer from the light to the electric current, since in any small time interval a vast number of photons are detected. This is ensured by the intense LO. The observable therefore has a quasi-continuous eigenvalue spectrum, as it is expected for an electric field strength. (In principle, one can also characterize squeezed states, in particular squeezed vacuum states, by counting photons, however, in general the measurement of the photon number statistic is not sufficient for a full characterization of a squeezed state and the full density matrix in the basis of the number states has to be determined.)

• Spin squeezed state