Elsevier

Physics Reports

Volume 684, 24 April 2017, Pages 1-51
Physics Reports

Squeezed states of light and their applications in laser interferometers

https://doi.org/10.1016/j.physrep.2017.04.001Get rights and content

Abstract

According to quantum theory the energy exchange between physical systems is quantized. As a direct consequence, measurement sensitivities are fundamentally limited by quantization noise, or just ‘quantum noise’ in short. Furthermore, Heisenberg’s Uncertainty Principle demands measurement back-action for some observables of a system if they are measured repeatedly. In both respects, squeezed states are of high interest since they show a ‘squeezed’ uncertainty, which can be used to improve the sensitivity of measurement devices beyond the usual quantum noise limits including those impacted by quantum back-action noise. Squeezed states of light can be produced with nonlinear optics, and a large variety of proof-of-principle experiments were performed in past decades. As an actual application, squeezed light has now been used for several years to improve the measurement sensitivity of GEO  600 – a laser interferometer built for the detection of gravitational waves. Given this success, squeezed light is likely to significantly contribute to the new field of gravitational-wave astronomy. This Review revisits the concept of squeezed states and two-mode squeezed states of light, with a focus on experimental observations. The distinct properties of squeezed states displayed in quadrature phase-space as well as in the photon number representation are described. The role of the light’s quantum noise in laser interferometers is summarized and the actual application of squeezed states in these measurement devices is reviewed.

Introduction

Laser interferometers are used to monitor small changes in refractive indices, rotations, or surface displacements such as mechanical vibrations. They transfer a differential phase change between two light beams into a changing power of the output light, which is photo-electrically detected, for example by a photo diode. The light is produced in a lasing process that usually aims for a coherent (Glauber) state. In practice, laser light is often in a mixture of coherent states producing excess noise in the interferometric measurement. But even if the laser light is in a (pure) coherent state its detection is associated with noise, usually called ‘shot-noise’. This arises from the quantization of the electro-magnetic field, which, for a coherent state, results in Poissonian counting statistics of mutually independent photons.

If the coherent state is highly excited and thus the average number of photons n¯ per detection interval is large, the Poissonian distribution can be approximated by a Gaussian distribution with a standard deviation of ±n¯. During the past decades squeezed states of light have attracted a lot of attention because they can exhibit less quantum noise than a coherent state of the same coherent excitation, i.e. they can show sub-Poissonian counting statistic, see Fig. 1.

Squeezed states belong to the class of ‘non-classical’ states, which are considered to be at the heart of quantum mechanics. These states are defined as those that cannot be described as a mixture of coherent states. In this case, their Glauber–Sudarshan P-functions Sudarshan (1963), Glauber (1963) do not correspond to (classical) probability density functions, i.e. they are not positive-valued functions. As a ‘classical’ example, the P-function of a coherent state corresponds to a δ-function.

But the question remains what property of coherent states justifies the name ‘classical’, even though coherent states are quantum states and show quantum uncertainties. My answer to this question is the following. All experiments which only involve coherent states and mixtures of them allow for a description that uses a combination of classical pictures. As we will see below, this description swaps between two different classical pictures and is thus not truly classical but semi-classical. (A more precise description of the nature of coherent states uses the term ‘semi-classical’.)

Let us consider a laser interferometer that uses light in a coherent state. Firstly, the light beam is split in two halves by a beam splitter. The two beams travel along different paths and are subsequently overlapped on a beam splitter where they interfere exactly as classical waves would do. The electric fields superimpose, thereby producing the phenomenon of interference. Up to this point there is no reason to argue light might be composed of particles.

Secondly, the new (still coherent) beams that result from the interference are absorbed, for instance by a photo-electric detector. In the case of coherent states the detection process can be perfectly described in the classical particle picture in which the particles appear independently from each other in a truly random fashion, yielding the aforementioned Poisson statistic. During the detection process, no wave feature of the light is present. Let us have a closer look: A truly random (‘spontaneous’) event is an event that has not been triggered by anything in the past. This allows us to make a clear cut between the first part of the experiment, described by the classical wave picture, and the second part of the experiment, described by the classical particle picture. Both ‘worlds’ are disconnected. The subsequent application of two classical pictures is not truly classical, but ‘semi-classical’. It is indeed the observation that the photons occur individually with truly random statistics that allows this semi-classical description. In the case of a mixture of coherent states the photon statistics are super-Poissonian, which can be understood as a mixture of different Poissonian distributions. In the case of a slowly changing coherent state the mean value n¯ depends on time. In all these cases, the semi-classical description is appropriate. Let me point out that in this very reasonable description photons do not exist before they are detected, e.g. absorbed. Further note, that the famous double-slit experiment with coherent states also allows for the same semi-classical description.

For squeezed states Yuen (1976), Walls (1983) the situation is different. As before, the interference can be fully described by the classical wave picture. The result of the detection process, however, is different from that of mutually independent random events. It is also different from any super-Poissonian statistics that could be produced by mixing an arbitrary number of different and/or time-dependent Poissonian distributions. Instead, the squeezed probability distribution in Fig. 1 suggests that the probability of detecting a photon decreases with the more photons that are already detected in the same time interval over which a single measurement is integrated. From this observation, one must conclude that the photons do not individually appear in a random fashion upon detection. There must be ‘quantum’ correlations between the photons. These correlations must existed before detection, since there is no interaction between the photons during their detection. Pre-existing correlations between detected photons seem to imply that the photons themselves existed before detection, i.e. at times when interference occurred. In a semi-classical description, however, photons are classical particles and cannot interfere, for instance on a beam splitter. At this point, the semi-classical picture breaks down. Squeezed states are therefor ‘nonclassical’.

The failure of the semi-classical model described above generally certifies nonclassicality.

Squeezed states are usually not characterized by counting their photons, but by measuring canonical continuous-variable phase-space observables. Measurements are performed, as usual, on an ensemble of identical states, and quasi-probability density functions are calculated from the data. The Glauber–Sudarshan P-function is the quasi-probability density distribution over coherent states. If the P-function of a state is entirely positive, the state is a coherent state or a (classical) mixture of coherent states. The state is considered as semi-classical. If the P-function is not a positive-valued function, the state cannot be expressed as a (classical) mixture of coherent states and is thus nonclassical Gerry and Knight (2005), Vogel and Welsch (2006). A non-positive-valued P-function is the sufficient and necessary condition for the failure of the semi-classical model. The Wigner function is the quasi-probability phase-space representation over the canonical continuous-variable phase-space observables themselves (Gerry and Knight, 2005). The Wigner functions of squeezed states are entirely positive. Although subject to discussion, this fact does not mean that squeezed states are less nonclassical than Fock states or cat states, which not only have a nonclassical P-function but also a partially negative Wigner function. (A cat state is a quantum superposition of two macroscopically distinct states (Monroe, 2002), referring to Schrödinger’s-cat gedanken experiment (Schrödinger, 1935)). In practice, squeezed states can even be regarded as superior nonclassical states because they represent the only nonclassical state that has been produced in a steady state fashion.

In almost all experiments so far, the generation of Fock states and cat states involves a probabilistic event, such as the detection of a photon in another beam path, to herald these states. In fact, squeezed states provide the nonclassical resource for the probabilistic preparation of Fock states as well as cat states. But only the squeezed states themselves show a nonclassical effect in a stationary way: Limited only by the time duration and the frequency span of the mode that is in a squeezed state, the squeezing effect can be continuously observed independently of the time when the measurement is performed, and also independently of the measurement integration time. This fact is of great importance for applications of squeezed states in measurement devices since a squeezed-light-enhanced measurement remains unconditional and the effective measurement time is not reduced.

In past decades, squeezed states of light were used in many proof-of-principle experiments to research their potential for improving the sensitivity of laser interferometers Grangier et al. (1987), Xiao et al. (1987), McKenzie et al. (2002), Vahlbruch et al. (2005), Goda et al. (2008), Taylor et al. (2013) or the performance of imaging beyond the shot-noise limit Lugiato et al. (2002), Treps et al. (2003), both accompanied by a huge number of theoretical works. Potential applications in secure optical communication (quantum key distribution) were also proposed, and proof-of-principle experiments demonstrated Ralph (1999), Furrer et al. (2012), Gehring et al. (2015). This review restricts itself to the improvement of laser interferometers, since only here has the application of squeezed light gone beyond proof-of-principle. The gravitational-wave detector (GWD) GEO  600 has operated with squeezed light now for more than seven years, starting in 2010 Abadie (2011), Grote et al. (2013). GEO  600 is a 600  m long Michelson laser interferometer built for the detection of gravitational waves. These waves are audio-band and sub-audio-band changes of space–time curvature originating from cosmic events such as the merger of neutron stars or black holes, as detected recently (Abbott, 2016). In GWDs such as GEO  600 (Dooley et al., 2016), Advanced LIGO (Aasi, 2015), Advanced Virgo (Acernese, 2015), and KAGRA (Aso et al., 2013), conventional laser technology has been pushed to extremes over the past decades. Noise spectral densities normalized to space–time strain of less than 10−23 Hz12 have been measured (Abbott, 2016). Progress will continue and, based on the successful application in GEO  600, squeezed light is now widely accepted to provide a new additional technology to contribute to the new field of gravitational-wave astronomy. It was also successfully tested in one of the LIGO detectors in 2013 (Aasi, 2013) and is an integral part of the European design study for the 10 km Einstein-Telescope (Punturo et al., 2010).

GEO  600 has already taken several years of ‘squeezed’ observational data, which has increased its sensitivity at signal frequencies above 500 Hz. With the implementation of a squeezed light source in GEO  600, the application of nonclassical states in metrology has been pushed beyond merely proof-of-principle.

‘Two-mode squeezed states’ show a squeezed uncertainty in at least one joint continuous variable of two subsystems ‘A’ and ‘B’. Examples of joint variables are differences and sums of phase-space observables of A and B. Two-mode squeezed states not only belong to the class of nonclassical states but, due to their bi-partite character, also to the class of ‘inseparable’ or ‘entangled’ states. They are the ideal states to demonstrate the Einstein–Podolsky–Rosen paradox (Einstein et al., 1935), as first achieved in Ou et al. (1992). Apart from fundamental research on quantum mechanics, recent proof-of-principle experiments demonstrated their usefulness in interferometric measurements that go beyond the application of simple squeezed states Steinlechner et al. (2013), Ast et al. (2016). This experiment is the final topic of this review.

Section snippets

Observations on light fields in squeezed states

Generally there are two different kinds of observables that can be subject of a measurement performed on a quantum system. The first kind is associated with the system’s wave property. In optics, it corresponds to the electric field strength at a given phase angle ϑ. The according (dimensionless) operators are called the quadrature amplitudes Xˆϑ and have a continuous spectrum of eigenvalues. Quadrature amplitudes are measured in very good approximation with a balanced homodyne detector using

The quadrature amplitude operators

Consider a single mode of light at optical frequency ω. Its Hamilton operator reads Hˆω=ħωnˆ+12=ħωaˆωaˆω+12=ħωXˆω2+Yˆω2,where nˆ is the photon number operator, and aˆω and aˆω are the annihilation and creation operators, which obey the commutation rule aˆω,aˆω=1. The operator aˆω has a complex-valued dimensionless eigenvalue spectrum and corresponds to the complex amplitude αω in classical optics. Xˆω and Yˆω are the Hermitian amplitude and phase quadrature operators. The eigenvalues of the

Overview

Squeezed light was first produced in 1985 by Slusher et al. using four-wave-mixing in sodium atoms in an optical cavity (Slusher et al., 1985). Shortly after, squeezed light also was generated by four-wave-mixing in an optical fiber (Shelby et al., 1986) and by degenerate parametric down-conversion (PDC) in a 2nd-order nonlinear crystal placed in an optical cavity (Wu et al., 1986). The pumped cavity was operated below its oscillation threshold, i.e. the parametric gain did not fully compensate

Interferometric measurements

The purpose of a laser interferometer is the precise measurement of small changes of an optical path length with respect to a reference path. For this, the interferometer transfers the change of the phase difference between two light fields into an amplitude quadrature change of the interferometer’s output light. The latter can easily be detected by a single photo diode. Of general interest are differential changes of the optical path length that are much smaller than the laser wavelength,

The first application of squeezed light in an operating gravitational-wave detector

Squeezed states of light have been successfully used to improve the sensitivity of the gravitational-wave detector GEO  600 from 2010 up to the point when this Review was written Abadie (2011), Grote et al. (2013). After decades of proof-of-principle experiments Xiao et al. (1987), Grangier et al. (1987), McKenzie et al. (2002), McKenzie et al. (2004), Vahlbruch et al. (2005), Vahlbruch et al. (2006), Vahlbruch et al. (2007), Vahlbruch et al. (2008), Goda et al. (2008) the implementation of a

Quantum dense metrology

At first glance, the application of bi-partite (two-mode) squeezed states to a device whose goal is measuring a single observable seems meaningless. Squeezing the uncertainty of that observable should be the optimum one can do. This is indeed true when concerning just quantum noise, but recently it was discovered that in the presence of classical disturbances, bi-partite squeezing can improve such measuring devices (Steinlechner et al., 2013). The concept was named quantum dense metrology

Summary and outlook

In many cases, experiments that involve interference of quantum states can be described in a semi-classical way. This description uses the classical wave picture for the interference part of the experiment and subsequently the classical particle picture when the states transfer their energy to a detector, or more generally, to a thermal bath. This semi-classical description is not possible when using the specific class of ‘nonclassical’ states. Squeezed states of light are a prominent example

Acknowledgments

R.S. thanks  M. Ast, J. Bauchrowitz, C. Baune, S. Chelkowski, J. DiGuglielmo, A. Franzen, B. Hage, J. Harms, A. Khalaidovski, L. Kleybolte, N. Lastzka, M. Mehmet, S. Steinlechner, and H. Vahlbruch for their contributions, many fruitful discussions and their support with the figures, and J. Fiurášek for many valuable comments on the manuscript. Thanks are also due to Y. Chen, F. Khalili, and H. Miao for fruitful discussions within the quantum noise working group of the LIGO Scientific

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