3721:. 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.
3753:
3688:
3676:
226:
5706:
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), which broke the previous record of 12.7 dB (2010).
5596:) 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.
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3766:
2616:
1646:
2325:
3535:
1895:
4788:
4320:
5767:
in the continuous variables (CV) regime has been increasing rapidly. 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. This is in contrast to quantum
3794:
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
5705:
ensembles. 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, fields, and the current state of the art for measurement enhancement is 20 dB. Generation
3614:
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.
4110:
2333:
1374:
4131:
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
1474:
5768:
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.
3782:
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
7202:
Eberle, Tobias; Steinlechner, Sebastian; Bauchrowitz, Jöran; Händchen, Vitus; Vahlbruch, Henning; Mehmet, Moritz; Müller-Ebhardt, Helge; Schnabel, Roman (June 22, 2010). "Quantum
Enhancement of the Zero-Area Sagnac Interferometer Topology for Gravitational Wave Detection".
2751:
1046:
3091:
2082:
2074:
4859:
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
2958:
4490:
3836:
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
5528:
3342:
5972:
Vahlbruch, Henning; Mehmet, Moritz; Chelkowski, Simon; Hage, Boris; Franzen, Alexander; Lastzka, Nico; Goßler, Stefan; Danzmann, Karsten; Schnabel, Roman (January 23, 2008). "Observation of
Squeezed Light with 10-dB Quantum-Noise Reduction".
412:
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
1763:
4574:
3277:
3702:
Different squeezed states of laser light in a vacuum depend on the phase of the light field. Images from the top: (1) Vacuum state, (2) Squeezed vacuum state, (3) Phase-squeezed state (4) Arbitrary squeezed state (5) Amplitude-squeezed
7661:
In literature different normalizations for the quadrature amplitudes are used. Here we use the normalization for which the sum of the ground state variances of the quadrature amplitudes directly provide the zero point quantum number
668:
4146:
2611:{\displaystyle \langle \alpha ,\zeta |{\hat {a}}^{\dagger }{\hat {a}}|\alpha ,\zeta \rangle =|\alpha |^{2}\cosh ^{2}(r)+(1+{|\alpha |}^{2})\sinh ^{2}(r)-({\alpha }^{2}e^{-i\theta }+{\alpha ^{*}}^{2}e^{i\theta })\cosh(r)\sinh(r)}
3952:
3728:. 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.
2826:
4136:. 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
5262:
3758:
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
1155:
1641:{\displaystyle {\hat {a}}^{\dagger }={\sqrt {\frac {m\omega }{2\hbar }}}\left(x-{\frac {ip}{m\omega }}\right)\qquad {\text{and}}\qquad {\hat {a}}={\sqrt {\frac {m\omega }{2\hbar }}}\left(x+{\frac {ip}{m\omega }}\right)}
497:
3605:
3334:
7036:
Bohnet, J. G.; Cox, K. C.; Norcia, M. A.; Weiner, J. M.; Chen, Z.; Thompson, J. K. (July 13, 2014). "Reduced spin measurement back-action for a phase sensitivity ten times beyond the standard quantum limit".
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176:
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5326:
486:
7734:
7375:
Louchet-Chauvet, Anne; Appel, Jürgen; Renema, Jelmer J; Oblak, Daniel; Kjaergaard, Niels; Polzik, Eugene S (June 28, 2010). "Entanglement-assisted atomic clock beyond the projection noise limit".
3791:
219:
1943:
4579:
2320:{\displaystyle \langle \alpha ,\zeta |{\hat {a}}^{2}|\alpha ,\zeta \rangle =\alpha ^{2}\cosh ^{2}(r)+{\alpha ^{*}}^{2}e^{2i\theta }\sinh ^{2}(r)-(1+2{|\alpha |}^{2})e^{i\theta }\cosh(r)\sinh(r)}
4855:
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. An
2631:
926:
4963:
2966:
6252:
1954:
841:
4845:
403:
2834:
6924:
Hosten, Onur; Engelsen, Nils J.; Krishnakumar, Rajiv; Kasevich, Mark A. (January 11, 2016). "Measurement noise 100 times lower than the quantum-projection limit using entangled atoms".
4371:
6659:
Grote, H.; Danzmann, K.; Dooley, K. L.; Schnabel, R.; Slutsky, J.; Vahlbruch, H. (2013). "First Long-Term
Application of Squeezed States of Light in a Gravitational-Wave Observatory".
343:
5415:
3784:
1442:
906:
3170:
1711:
730:
5594:
4363:
3944:
3132:
3530:{\displaystyle {\hat {D}}(\alpha ){\hat {S}}(\zeta )={\hat {S}}(\zeta ){\hat {S}}^{\dagger }(\zeta ){\hat {D}}(\alpha ){\hat {S}}(\zeta )={\hat {S}}(\zeta ){\hat {D}}(\gamma )}
7090:
Lücke, Bernd; Peise, Jan; Vitagliano, Giuseppe; Arlt, Jan; Santos, Luis; Tóth, Géza; Klempt, Carsten (April 17, 2014). "Detecting
Multiparticle Entanglement of Dicke States".
297:
1890:{\displaystyle (\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 }}
5680:
5647:
5044:
4119:
superpositions. Single mode squeezed states are typically generated by degenerate parametric oscillation in an optical parametric oscillator, or using four-wave mixing.
1406:
1110:
1077:
262:
5006:
4783:{\displaystyle {\begin{aligned}\rho _{1}&=\mathrm {Tr} _{2}\\&={\frac {1}{\cosh ^{2}(r)}}\sum _{n=0}^{\infty }\tanh ^{2n}(r)|n\rangle \langle n|,\end{aligned}}}
1143:
536:
6220:
5124:
3653:
1466:
3178:
1669:
6202:
Eqn (15). Note that in this reference, the definition of the squeeze operator (eqn. 12) differs by a minus sign inside the exponential, therefore the expression of
5559:
5407:
5380:
5353:
5178:
5151:
5091:
4140:
in its original formulation in terms of continuous position and momentum observables. A two-mode squeezed vacuum (TMSV) state can be mathematically represented as,
769:
6975:
Cox, Kevin C.; Greve, Graham P.; Weiner, Joshua M.; Thompson, James K. (March 4, 2016). "Deterministic
Squeezed States with Collective Measurements and Feedback".
4315:{\displaystyle |{\text{TMSV}}\rangle =S_{2}(\zeta )|0,0\rangle =\exp(\zeta ^{*}{\hat {a}}{\hat {b}}-\zeta {\hat {a}}^{\dagger }{\hat {b}}^{\dagger })|0,0\rangle }
5064:
1755:
1735:
544:
118:
98:
78:
58:
4105:{\displaystyle |{\text{SMSV}}\rangle ={\frac {1}{\sqrt {\cosh r}}}\sum _{n=0}^{\infty }(-e^{i\phi }\tanh r)^{n}{\frac {\sqrt {(2n)!}}{2^{n}n!}}|2n\rangle }
6555:
Villar, A. S.; Cruz, L. S.; Cassemiro, K. N.; Martinelli, M.; Nussenzveig, P. (2005). "Generation of Bright Two-Color
Continuous Variable Entanglement".
6616:
Schumaker, Bonny L.; Caves, Carlton M. (May 1, 1985). "New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation".
7314:
Leroux, Ian D.; Schleier-Smith, Monika H.; Vuletić, Vladan (June 25, 2010). "Orientation-Dependent
Entanglement Lifetime in a Squeezed Atomic Clock".
5714:
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
2762:
1369:{\displaystyle {\hat {D}}(\alpha )=\exp(\alpha {\hat {a}}^{\dagger }-\alpha ^{*}{\hat {a}})\qquad {\text{and}}\qquad {\hat {S}}(\zeta )=\exp {\bigg }}
5187:
3778:
For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in
427:
were first produced in the mid 1980s. At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e.
3828:
involved in the process. Recent studies have looked into multimode squeezed states showing quantum correlations among more than two modes as well.
3799:
For the squeezed vacuum state the photon number distribution displays odd-even-oscillations. This can be explained by the mathematical form of the
6407:
Heidmann, A.; Horowicz, R.; Reynaud, S.; Giacobino, E.; Fabre, C.; Camy, G. (1987). "Observation of
Quantum Noise Reduction on Twin Laser Beams".
5604:
There has been a whole variety of successful demonstrations of squeezed states. The first demonstrations were experiments with light fields using
7584:
Menicucci, Nicolas C.; Flammia, Steven T.; Pfister, Olivier (September 22, 2008). "One-Way
Quantum Computing in the Optical Frequency Comb".
6816:
Machida, S.; Yamamoto, Y.; Itaya, Y. (March 9, 1987). "Observation of amplitude squeezing in a constant-current driven semiconductor laser".
3623:: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the
3540:
3820:
Squeezed states of light are broadly classified into single-mode squeezed states and two-mode squeezed states, depending on the number of
3285:
5649:
crystal; similarly travelling wave phase-sensitive amplifiers generate spatially multimode quadrature-squeezed states of light when the
5741:
5561:
is reduced relative to the coherent (unentangled) state. The second factor is how much the coherence (the length of the Bloch vector,
6047:"Detection of 15 dB Squeezed States of Light and their Application for the Absolute Calibration of Photoelectric Quantum Efficiency"
6046:
6330:
kitagawa, Akira; Takeoka, Masahiro; Sasaki, Masahide; Chefles, Anthony (2006). "Entanglement evaluation with Fisher information".
6773:
Wineland, D. J.; Bollinger, J. J.; Heinzen, D. J. (July 1, 1994). "Squeezed atomic states and projection noise in spectroscopy".
4864:
to provide a phase reference, whereas squeezed vacuum is considered more suitable for quantum enhanced sensing applications. The
3847:
3669:
Measured quantum noise of the electric field (3π-interval shown for the first two states; 4π-interval for the last three states)
138:
5093:
will correspond to the population difference in the two level system, i.e. for an equal superposition of the up and down state
3804:
4886:
For squeezing of two-level neutral atom ensembles it is useful to consider the atoms as spin-1/2 particles with corresponding
4498:
5267:
3725:
6858:
O. V. Misochko, J. Hu, K. G. Nakamura, "Controlling phonon squeezing and correlation via one- and two-phonon interference,"
6720:
The LIGO Scientific Collaboration (2011). "A gravitational wave observatory operating beyond the quantum shot-noise limit".
430:
7665:
6166:
Nieto, Michael Martin; Truax, D. Rodney (1997). "Holstein‐Primakoff/Bogoliubov Transformations and the Multiboson System".
3693:
229:
181:
2746:{\displaystyle |\zeta ,\alpha \rangle ={\hat {D}}(\alpha )|\zeta \rangle ={\hat {D}}(\alpha ){\hat {S}}(\zeta )|0\rangle }
1906:
1041:{\displaystyle |\alpha ,\zeta \rangle ={\hat {S}}(\zeta )|\alpha \rangle ={\hat {S}}(\zeta ){\hat {D}}(\alpha )|0\rangle }
3086:{\displaystyle \langle \zeta ,\alpha |{\hat {a}}^{\dagger }{\hat {a}}|\zeta ,\alpha \rangle =|\alpha |^{2}+\sinh ^{2}(r)}
5409:
direction, we can define the Wineland criterion for squeezing, or the metrological enhancement of the squeezed state as
2069:{\displaystyle \langle \alpha ,\zeta |{\hat {a}}|\alpha ,\zeta \rangle =\alpha \cosh(r)-\alpha ^{*}e^{i\theta }\sinh(r)}
4896:
6225:
5825:
2953:{\displaystyle \langle \zeta ,\alpha |{\hat {a}}^{2}|\zeta ,\alpha \rangle =\alpha ^{2}-e^{i\theta }\cosh(r)\sinh(r)}
784:
4872:
gravitational wave detectors use squeezed vacuum to achieve enhanced sensitivity beyond the standard quantum limit.
4796:
5733:. This is an important problem in atomic clocks and other sensors that use small ensembles of cold atoms where the
4485:{\displaystyle |{\text{TMSV}}\rangle ={\frac {1}{\cosh r}}\sum _{n=0}^{\infty }(-e^{i\phi }\tanh r)^{n}|nn\rangle }
4127:
Two-mode squeezing involves two modes of the electromagnetic field which exhibit quantum noise reduction below the
1901:
348:
299:. These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with
7429:
5534:
This criterion has two factors, the first factor is the spin noise reduction, i.e. how much the quantum noise in
5523:{\displaystyle \chi ^{2}=\left({\frac {{\sqrt {N}}/2}{\Delta J_{z}}}{\frac {\left|J_{x}\right|}{N/2}}\right)^{2}}
4137:
5764:
302:
6372:
Wu, L.-A.; Xiao, M.; Kimble, H. J. (1987). "Squeezed states of light from an optical parametric oscillator".
5613:
4856:
1411:
853:
37:
7488:
Braunstein, Samuel L.; van Loock, Peter (June 29, 2005). "Quantum information with continuous variables".
3765:
3137:
1674:
1445:
676:
265:
5564:
4329:
3910:
3099:
3615:
The figures below give a nice visual demonstration of the close connection between squeezed states and
1945:, with reduced uncertainty in one of its quadrature components and increased uncertainty in the other.
274:
5757:
4887:
3752:
3624:
496:
6265:
6103:
Schnabel, Roman (2017). "Squeezed states of light and their applications in laser interferometers".
3903:
where the squeezing operator S is the same as introduced in the section on operator representations
3807:
and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.
424:
421:
of the same area. Note that a squeezed state does not need to saturate the uncertainty principle.
7755:
7750:
5652:
5619:
5011:
3736:
The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the
1382:
1086:
1054:
239:
4971:
3272:{\displaystyle {\hat {S}}(\zeta ){\hat {D}}(\alpha )\neq {\hat {D}}(\alpha ){\hat {S}}(\zeta )}
1119:
6871:
Ma, Jian; Wang, Xiaoguang; Sun, C.P.; Nori, Franco (December 2011). "Quantum spin squeezing".
5328:. Squeezing is here considered the redistribution of uncertainty from one variable (typically
515:
6205:
5096:
3825:
3630:
1451:
1113:
129:
6045:
Vahlbruch, Henning; Mehmet, Moritz; Danzmann, Karsten; Schnabel, Roman (September 6, 2016).
1654:
7603:
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7507:
7444:
7394:
7333:
7280:
7222:
7160:
7109:
7056:
6994:
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6890:
6825:
6782:
6739:
6678:
6625:
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6521:
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6416:
6381:
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5992:
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5862:
5745:
5537:
5385:
5358:
5331:
5156:
5129:
5069:
3620:
908:. In this sense, it is a generalization of the ground state as well as the coherent state.
747:
7265:
7180:
6506:
6081:
6012:
5921:
663:{\displaystyle \psi (x)=C\,\exp \left(-{\frac {(x-x_{0})^{2}}{2w_{0}^{2}}}+ip_{0}x\right)}
8:
5787:
5686:
current sources driving semiconductor laser diodes have led to amplitude squeezed light.
4881:
121:
7607:
7554:
7511:
7448:
7406:
7398:
7337:
7284:
7226:
7164:
7113:
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6998:
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6385:
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6284:
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plane represents the phase difference between the two states. This is also known as the
7627:
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5982:
5886:
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5719:
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3800:
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488:. As of 2017, squeeze factors larger than 10 (10 dB) have been directly observed.
125:
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83:
63:
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6000:
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5797:
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1146:
414:
6902:
6586:
6134:
4565:), then tracing over or absorbing one of the modes leaves the remaining mode in a
3687:
7562:
6451:
5908:
Observation of squeezed states generated by four wave mixing in an optical cavity
5777:
5698:
3790:
500:
Animated position-wavefunction of a 2dB amplitude-squeezed coherent state of α=3.
7292:
6837:
6533:
6428:
5949:
2821:{\displaystyle \langle \zeta ,\alpha |{\hat {a}}|\zeta ,\alpha \rangle =\alpha }
7648:
5792:
5715:
5683:
3779:
3710:
741:
269:
225:
7519:
5257:{\displaystyle \Delta J_{z}\cdot \Delta J_{y}\geq \left|\Delta J_{x}\right|/2}
3675:
7744:
7456:
7068:
6637:
6507:"Realization of the Einstein-Podolsky-Rosen paradox for continuous variables"
5882:
5734:
4566:
3741:
3714:
505:
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7353:
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7129:
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6393:
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which explicitly shows that the pure SMSV consists entirely of even-photon
1080:
7570:
7474:
6802:
6645:
7502:
6569:
6336:
6180:
4865:
3821:
7541:
Furusawa, A. (October 23, 1998). "Unconditional Quantum Teleportation".
6945:
3724:
In phase space, quantum mechanical uncertainties can be depicted by the
3663:
80:
of a particle, and the (dimension-less) electric field in the amplitude
7465:
4128:
4116:
3718:
847:
775:
733:
236:
Trivial examples, which are in fact not squeezed, are the ground state
33:
30:
6751:
5729:
Spin squeezed states of atoms can be used to improve the precision of
3731:
7201:
6292:
5874:
3696:
of the five states. The ripples are due to experimental inaccuracies.
3600:{\displaystyle \gamma =\alpha \cosh r+\alpha ^{*}e^{i\theta }\sinh r}
5737:
represents a fundamental limitation to the precision of the sensor.
6989:
6117:
5616:). This is achieved by a simple process of four-wave mixing with a
4495:
If the individual modes of a TMSV are considered separately (i.e.,
3329:{\displaystyle |\alpha ,\zeta \rangle \neq |\zeta ,\alpha \rangle }
737:
7598:
7389:
7328:
7217:
7104:
7051:
6923:
6885:
6859:
6734:
6673:
6472:
6357:
5987:
5740:
Various squeezed coherent states, generalized to the case of many
5689:
Squeezed states have also been realized via motional states of an
3717:
for a squeezed state is no longer independent of the phase of the
6719:
5722:). Amplitude-squeezed light can improve the readout of very weak
418:
232:
phase space distribution of a squeezed state of light with ζ=0.5.
22:
6406:
7374:
7151:
Rini, Matteo (September 6, 2016). "Synopsis: A Tight Squeeze".
6044:
5756:, and generally, particle production in curved backgrounds and
5694:
4869:
3737:
6505:
Ou, Z. Y.; Pereira, S. F.; Kimble, H. J.; Peng, K. C. (1992).
6554:
5971:
5922:"Generation of Squeezed States by Parametric Down Conversion"
5605:
3795:
the pure squeezed vacuum into a mixed state. (source: link 1)
1444:
are annihilation and creation operators, respectively. For a
6329:
7313:
6658:
5702:
732:
are constants (a normalization constant, the center of the
6264:
Breitenbach, G.; Schiller, S.; Mlynek, J. (May 29, 1997).
5853:
Walls, D. F. (November 1983). "Squeezed states of light".
5184:
picture. We can then define uncertainty relations such as
3893:{\displaystyle |{\text{SMSV}}\rangle =S(\zeta )|0\rangle }
771:, which is the reason why the state is called "squeezed".
171:{\displaystyle \Delta x\Delta p\geq {\frac {\hbar }{2}}\;}
6263:
5690:
2756:
Some expectation values for displaced squeezed state are
1948:
Some expectation values for squeezed coherent states are
7089:
6772:
6449:
4850:
4558:{\displaystyle |nn\rangle =|n\rangle _{1}|n\rangle _{2}}
7583:
5321:{\displaystyle \Delta J_{z}=\Delta J_{y}={\sqrt {N}}/2}
3627:
quadrature, that is, the field at the phase shifted by
124:). The product of the standard deviations of two such
6974:
3709:
As can be seen in the illustrations, in contrast to a
481:{\displaystyle \Delta ^{2}X\approx \Delta ^{2}X_{g}/2}
7729:{\displaystyle \Delta ^{2}X_{g}+\Delta ^{2}Y_{g}=1/2}
7668:
7035:
6266:"Measurement of the quantum states of squeezed light"
6228:
6208:
5655:
5622:
5567:
5540:
5418:
5388:
5361:
5334:
5270:
5190:
5159:
5132:
5099:
5072:
5052:
5014:
4974:
4899:
4799:
4577:
4501:
4374:
4332:
4149:
3955:
3913:
3850:
3633:
3543:
3345:
3288:
3181:
3140:
3102:
2969:
2837:
2765:
2634:
2336:
2085:
1957:
1909:
1766:
1743:
1723:
1677:
1657:
1477:
1454:
1414:
1385:
1158:
1122:
1089:
1057:
929:
856:
787:
750:
679:
547:
518:
433:
351:
305:
277:
242:
214:{\displaystyle \;\Delta X\Delta Y\geq {\frac {1}{4}}}
184:
141:
106:
86:
66:
46:
6815:
1938:{\displaystyle \Delta x\Delta p={\frac {\hbar }{2}}}
29:
is a quantum state that is usually described by two
7487:
7428:Kitagawa, Masahiro; Ueda, Masahito (June 1, 1993).
6504:
6318:
Measurement of the quantum states of squeezed light
3841:(SMSV) state can be mathematically represented as,
3771:
Pegg-Barnett phase distribution of the three states
3732:
Photon number distributions and phase distributions
1900:Therefore, a squeezed coherent state saturates the
7728:
6246:
6214:
5674:
5641:
5588:
5553:
5522:
5401:
5374:
5347:
5320:
5256:
5172:
5145:
5118:
5085:
5058:
5038:
5000:
4957:
4839:
4782:
4557:
4484:
4357:
4314:
4104:
3938:
3892:
3647:
3599:
3529:
3328:
3271:
3164:
3126:
3085:
2952:
2820:
2745:
2610:
2319:
2068:
1937:
1889:
1749:
1729:
1705:
1663:
1640:
1460:
1436:
1400:
1368:
1137:
1104:
1071:
1040:
900:
835:
763:
724:
662:
530:
480:
397:
337:
291:
256:
213:
170:
112:
92:
72:
52:
1361:
1281:
7742:
7649:Tutorial about quantum optics of the light field
4958:{\displaystyle J_{v}=\sum _{i=1}^{N}j_{v}^{(i)}}
6247:{\displaystyle \theta \rightarrow \theta +\pi }
6155:Walls, D.F. and G. J. Milburn, Quantum Optics.
5843:, "Quantum Noise", 3rd ed, Springer Berlin 2004
3815:
836:{\displaystyle {\hat {x}}+i{\hat {p}}w_{0}^{2}}
6615:
6312:
6310:
4840:{\displaystyle {\widetilde {n}}=\sinh ^{2}(r)}
3831:
3740:number distribution of the light wave and its
2625:for a quantum harmonic oscillator is given by
920:for a quantum harmonic oscillator is given by
736:, its width, and the expectation value of its
6870:
6316:G. Breitenbach, S. Schiller, and J. Mlynek, "
398:{\displaystyle \Delta X_{g}=\Delta Y_{g}=1/2}
7427:
6371:
5682:crystal is pumped in absence of any signal.
5599:
4793:with an effective average number of photons
4762:
4759:
4639:
4636:
4546:
4528:
4513:
4479:
4385:
4309:
4199:
4160:
4099:
3966:
3904:
3887:
3861:
3681:Oscillating wave packets of the five states.
3323:
3303:
3032:
2970:
2888:
2838:
2809:
2766:
2740:
2684:
2649:
2399:
2337:
2136:
2086:
2001:
1958:
1717:is squeezing parameter), the uncertainty in
1066:
1035:
979:
944:
417:space (see right) has been "squeezed" to an
286:
251:
6307:
6168:Fortschritte der Physik/Progress of Physics
4122:
6917:
6165:
5382:). If we consider a state pointing in the
911:
491:
185:
167:
7597:
7501:
7464:
7388:
7327:
7216:
7103:
7050:
6988:
6884:
6733:
6672:
6568:
6489:
6471:
6356:
6351:Lvovsky, A. I. (2014). "Squeezed light".
6335:
6179:
6116:
5986:
5948:
5902:
5900:
5763:Recently, the use of squeezed states for
566:
508:that satisfies the identity above is the
338:{\displaystyle \Delta x_{g}=\Delta p_{g}}
7540:
6102:
4875:
3789:
495:
224:
120:(phase 90°) of a light wave (the wave's
6350:
5910:, Phys. Rev. Lett. 55 (22), 2409 (1985)
7743:
7263:
6454:; Nussenzveig, P.; Lipson, M. (2015).
5897:
5744:, are used in various calculations in
5264:. For a coherent (unentangled) state,
3907:. In the photon number basis, writing
6450:Dutt, A.; Luke, K.; Manipatruni, S.;
5852:
4851:Based on the presence of a mean field
3726:Wigner quasi-probability distribution
1437:{\displaystyle {\hat {a}}^{\dagger }}
901:{\displaystyle x_{0}+ip_{0}w_{0}^{2}}
7150:
6040:
6038:
5046:is the single-spin operator in the
3165:{\displaystyle {\hat {D}}(\alpha )}
1706:{\displaystyle \zeta =re^{2i\phi }}
725:{\displaystyle C,x_{0},w_{0},p_{0}}
13:
7693:
7670:
7266:"Spectroscopy with squeezed light"
5919:
5589:{\displaystyle \left|J_{x}\right|}
5458:
5287:
5271:
5228:
5207:
5191:
4721:
4652:
4649:
4646:
4643:
4632:
4629:
4626:
4623:
4604:
4601:
4425:
4358:{\displaystyle \zeta =re^{i\phi }}
4006:
3939:{\displaystyle \zeta =re^{i\phi }}
3803:, that resembles the operator for
3127:{\displaystyle {\hat {S}}(\zeta )}
1916:
1910:
1874:
1835:
1809:
1770:
451:
435:
368:
352:
345:in "natural oscillator units" and
322:
306:
192:
186:
148:
142:
14:
7767:
7642:
7264:Polzik, E. S. (January 1, 1992).
6164:M. M. Nieto and D. Truax (1995),
6035:
5820:(Oxford University Press, 2000),
4365:, in the photon number basis as,
3810:
3172:do not commute with each other,
1860:
1595:
1515:
740:). The new feature relative to a
3787:due to experimental difficulty.
3785:photon number resolving detector
3764:
3751:
3686:
3674:
3662:
1902:Heisenberg uncertainty principle
292:{\displaystyle |\alpha \rangle }
7577:
7534:
7481:
7421:
7368:
7307:
7257:
7195:
7144:
7083:
7029:
6968:
6864:
6860:https://arxiv.org/abs/1011.2001
6852:
6809:
6766:
6713:
6652:
6609:
6548:
6498:
6491:10.1103/physrevapplied.3.044005
6443:
6400:
6365:
6344:
6323:
6257:
6158:
6149:
5709:
4138:Einstein-Podolsky-Rosen paradox
1831:
1825:
1564:
1558:
1468:, these operators are given by
1248:
1242:
774:The squeezed state above is an
744:is the free value of the width
7655:
7616:10.1103/PhysRevLett.101.130501
7346:10.1103/PhysRevLett.104.250801
7235:10.1103/PhysRevLett.104.251102
7181:11858/00-001M-0000-002B-87B5-3
7173:10.1103/PhysRevLett.117.110801
7122:10.1103/PhysRevLett.112.155304
7007:10.1103/PhysRevLett.116.093602
6691:10.1103/physrevlett.110.181101
6232:
6096:
6082:11858/00-001M-0000-002B-87B5-3
6074:10.1103/PhysRevLett.117.110801
6013:11858/00-001M-0000-0013-623A-0
6005:10.1103/PhysRevLett.100.033602
5965:
5913:
5846:
5830:
5810:
5765:quantum information processing
5667:
5661:
5634:
5628:
5031:
5025:
4950:
4944:
4834:
4828:
4769:
4752:
4748:
4742:
4699:
4693:
4661:
4657:
4618:
4614:
4538:
4520:
4503:
4469:
4459:
4430:
4376:
4296:
4292:
4280:
4261:
4242:
4230:
4211:
4186:
4182:
4176:
4151:
4089:
4061:
4052:
4040:
4011:
3957:
3880:
3876:
3870:
3852:
3524:
3518:
3512:
3503:
3497:
3491:
3479:
3473:
3467:
3458:
3452:
3446:
3437:
3431:
3419:
3409:
3403:
3397:
3385:
3379:
3373:
3364:
3358:
3352:
3310:
3290:
3266:
3260:
3254:
3245:
3239:
3233:
3221:
3215:
3209:
3200:
3194:
3188:
3159:
3153:
3147:
3121:
3115:
3109:
3080:
3074:
3048:
3039:
3019:
3012:
2994:
2983:
2947:
2941:
2932:
2926:
2875:
2862:
2851:
2796:
2789:
2779:
2733:
2729:
2723:
2717:
2708:
2702:
2696:
2677:
2673:
2667:
2661:
2636:
2605:
2599:
2590:
2584:
2575:
2509:
2503:
2497:
2481:
2470:
2462:
2450:
2444:
2438:
2415:
2406:
2386:
2379:
2361:
2350:
2314:
2308:
2299:
2293:
2271:
2260:
2252:
2237:
2231:
2225:
2171:
2165:
2123:
2110:
2099:
2063:
2057:
2022:
2016:
1988:
1981:
1971:
1842:
1832:
1777:
1767:
1571:
1485:
1422:
1392:
1356:
1341:
1316:
1296:
1267:
1261:
1255:
1239:
1233:
1202:
1189:
1177:
1171:
1165:
1132:
1126:
1099:
1093:
1059:
1028:
1024:
1018:
1012:
1003:
997:
991:
972:
968:
962:
956:
931:
812:
794:
604:
584:
557:
551:
279:
244:
1:
7407:10.1088/1367-2630/12/6/065032
6903:10.1016/j.physrep.2011.08.003
6587:10.1103/physrevlett.95.243603
6135:10.1016/j.physrep.2017.04.001
5803:
5701:, and spin states in neutral
5614:optical parametric oscillator
4857:optical parametric oscillator
36:having continuous spectra of
7563:10.1126/science.282.5389.706
5716:interferometric measurements
3816:Based on the number of modes
7:
7293:10.1103/PhysRevLett.68.3020
6838:10.1103/PhysRevLett.58.1000
6534:10.1103/physrevlett.68.3663
6456:"On-Chip Optical Squeezing"
6429:10.1103/physrevlett.59.2555
5950:10.1103/physrevlett.57.2520
5818:The Quantum Theory of Light
5771:
5675:{\displaystyle \chi ^{(2)}}
5642:{\displaystyle \chi ^{(3)}}
5039:{\displaystyle j_{v}^{(i)}}
3832:Single-mode squeezed states
3609:
1446:quantum harmonic oscillator
266:quantum harmonic oscillator
10:
7772:
6320:", Nature, 387, 471 (1997)
5758:Bogoliubov transformations
4888:angular momentum operators
4879:
1401:{\displaystyle {\hat {a}}}
1105:{\displaystyle D(\alpha )}
1072:{\displaystyle |0\rangle }
257:{\displaystyle |0\rangle }
100:(phase 0) and in the mode
7520:10.1103/RevModPhys.77.513
7490:Reviews of Modern Physics
6222:is modified accordingly (
5600:Experimental realizations
5001:{\displaystyle v={x,y,z}}
3946:this can be expanded as,
1138:{\displaystyle S(\zeta )}
7457:10.1103/PhysRevA.47.5138
7275:(Submitted manuscript).
7069:10.1038/nphoton.2014.151
6638:10.1103/PhysRevA.31.3093
6516:(Submitted manuscript).
5931:(Submitted manuscript).
5735:quantum projection noise
5355:) to another (typically
4123:Two-mode squeezed states
2623:displaced squeezed state
531:{\displaystyle \hbar =1}
425:Squeezed states of light
40:. Examples are position
7586:Physical Review Letters
7316:Physical Review Letters
7273:Physical Review Letters
7205:Physical Review Letters
7092:Physical Review Letters
6977:Physical Review Letters
6818:Physical Review Letters
6460:Physical Review Applied
6409:Physical Review Letters
6215:{\displaystyle \gamma }
6190:10.1002/prop.2190450204
6054:Physical Review Letters
5975:Physical Review Letters
5929:Physical Review Letters
5119:{\displaystyle J_{z}=0}
3648:{\displaystyle \tau /4}
1461:{\displaystyle \omega }
918:squeezed coherent state
912:Operator representation
512:(we work in units with
510:squeezed coherent state
492:Mathematical definition
27:squeezed coherent state
7730:
7430:"Squeezed spin states"
7377:New Journal of Physics
6795:10.1103/PhysRevA.50.67
6394:10.1364/JOSAB.4.001465
6248:
6216:
5906:R. E. Slusher et al.,
5676:
5643:
5590:
5555:
5524:
5403:
5376:
5349:
5322:
5258:
5174:
5147:
5120:
5087:
5060:
5040:
5002:
4959:
4933:
4841:
4784:
4725:
4559:
4486:
4429:
4359:
4316:
4106:
4010:
3940:
3894:
3796:
3744:distribution as well.
3649:
3601:
3531:
3330:
3273:
3166:
3128:
3087:
2954:
2822:
2747:
2621:The general form of a
2612:
2321:
2070:
1939:
1891:
1751:
1731:
1707:
1665:
1664:{\displaystyle \zeta }
1642:
1462:
1438:
1402:
1370:
1139:
1106:
1073:
1042:
916:The general form of a
902:
846:and the corresponding
837:
765:
726:
664:
532:
501:
482:
399:
339:
293:
258:
233:
215:
172:
114:
94:
74:
54:
7731:
6249:
6217:
5724:spectroscopic signals
5677:
5644:
5591:
5556:
5554:{\displaystyle J_{z}}
5525:
5404:
5402:{\displaystyle J_{x}}
5377:
5375:{\displaystyle J_{y}}
5350:
5348:{\displaystyle J_{z}}
5323:
5259:
5175:
5173:{\displaystyle J_{y}}
5148:
5146:{\displaystyle J_{x}}
5121:
5088:
5086:{\displaystyle J_{z}}
5061:
5041:
5003:
4960:
4913:
4876:Atomic spin squeezing
4842:
4785:
4705:
4560:
4487:
4409:
4360:
4317:
4107:
3990:
3941:
3895:
3826:electromagnetic field
3805:two-photon generation
3793:
3650:
3602:
3532:
3331:
3274:
3167:
3129:
3088:
2955:
2823:
2748:
2613:
2322:
2071:
1940:
1892:
1752:
1732:
1708:
1666:
1643:
1463:
1448:of angular frequency
1439:
1403:
1371:
1140:
1114:displacement operator
1107:
1074:
1043:
903:
838:
778:of a linear operator
766:
764:{\displaystyle w_{0}}
727:
665:
533:
499:
483:
400:
340:
294:
259:
228:
216:
173:
130:uncertainty principle
115:
95:
75:
55:
16:Type of quantum state
7666:
6226:
6206:
5920:Wu, Ling-An (1986).
5746:quantum field theory
5653:
5620:
5565:
5538:
5416:
5386:
5359:
5332:
5268:
5188:
5157:
5130:
5097:
5070:
5050:
5012:
4972:
4897:
4797:
4575:
4499:
4372:
4330:
4147:
3953:
3911:
3848:
3631:
3621:uncertainty relation
3541:
3343:
3286:
3179:
3138:
3100:
2967:
2835:
2763:
2632:
2334:
2083:
1955:
1907:
1764:
1741:
1721:
1675:
1655:
1475:
1452:
1412:
1383:
1156:
1120:
1087:
1055:
927:
854:
785:
748:
677:
545:
516:
431:
349:
303:
275:
240:
182:
139:
104:
84:
64:
44:
7608:2008PhRvL.101m0501M
7555:1998Sci...282..706F
7512:2005RvMP...77..513B
7449:1993PhRvA..47.5138K
7399:2010NJPh...12f5032L
7338:2010PhRvL.104y0801L
7285:1992PhRvL..68.3020P
7227:2010PhRvL.104y1102E
7165:2016PhRvL.117k0801V
7114:2014PhRvL.112o5304L
7061:2014NaPho...8..731B
6999:2016PhRvL.116i3602C
6946:10.1038/nature16176
6938:2016Natur.529..505H
6895:2011PhR...509...89M
6830:1987PhRvL..58.1000M
6787:1994PhRvA..50...67W
6744:2011NatPh...7..962L
6683:2013PhRvL.110r1101G
6630:1985PhRvA..31.3093S
6579:2005PhRvL..95x3603V
6526:1992PhRvL..68.3663O
6482:2015PhRvP...3d4005D
6421:1987PhRvL..59.2555H
6386:1987JOSAB...4.1465W
6285:1997Natur.387..471B
6127:2017PhR...684....1S
6066:2016PhRvL.117k0801V
5997:2008PhRvL.100c3602V
5941:1986PhRvL..57.2520W
5867:1983Natur.306..141W
5788:Optical phase space
5720:gravitational waves
5035:
4954:
4882:Spin squeezed state
897:
832:
632:
7726:
6374:J. Opt. Soc. Am. B
6244:
6212:
5783:Nonclassical light
5742:degrees of freedom
5672:
5639:
5586:
5551:
5520:
5399:
5372:
5345:
5318:
5254:
5170:
5143:
5116:
5083:
5056:
5036:
5015:
4998:
4955:
4934:
4837:
4780:
4778:
4555:
4482:
4355:
4326:and, writing down
4312:
4102:
3936:
3890:
3801:squeezing operator
3797:
3645:
3597:
3527:
3326:
3269:
3162:
3124:
3083:
2950:
2818:
2743:
2608:
2317:
2066:
1935:
1887:
1747:
1727:
1703:
1661:
1638:
1458:
1434:
1398:
1366:
1135:
1102:
1069:
1038:
898:
883:
833:
818:
761:
722:
660:
618:
528:
502:
478:
395:
335:
289:
268:and the family of
254:
234:
211:
168:
110:
90:
70:
50:
7549:(5389): 706–709.
7437:Physical Review A
7279:(20): 3020–3023.
6824:(10): 1000–1003.
6775:Physical Review A
6752:10.1038/nphys2083
6618:Physical Review A
6520:(25): 3663–3666.
6415:(22): 2555–2557.
6279:(6632): 471–475.
5935:(20): 2520–2523.
5861:(5939): 141–146.
5754:Hawking radiation
5718:(see for example
5610:non-linear optics
5507:
5472:
5446:
5308:
5066:-direction. Here
5059:{\displaystyle v}
4809:
4703:
4407:
4383:
4283:
4264:
4245:
4233:
4158:
4086:
4067:
3988:
3987:
3964:
3859:
3515:
3494:
3470:
3449:
3422:
3400:
3376:
3355:
3257:
3236:
3212:
3191:
3150:
3112:
3015:
2997:
2865:
2792:
2720:
2699:
2664:
2382:
2364:
2113:
1984:
1933:
1870:
1829:
1805:
1750:{\displaystyle p}
1730:{\displaystyle x}
1631:
1600:
1599:
1574:
1562:
1551:
1520:
1519:
1488:
1425:
1395:
1344:
1319:
1294:
1258:
1246:
1236:
1205:
1168:
1015:
994:
959:
815:
797:
634:
504:The most general
209:
165:
113:{\displaystyle Y}
93:{\displaystyle X}
73:{\displaystyle p}
53:{\displaystyle x}
7763:
7736:
7735:
7733:
7732:
7727:
7722:
7711:
7710:
7701:
7700:
7688:
7687:
7678:
7677:
7659:
7636:
7635:
7601:
7581:
7575:
7574:
7538:
7532:
7531:
7505:
7503:quant-ph/0410100
7485:
7479:
7478:
7468:
7443:(6): 5138–5143.
7434:
7425:
7419:
7418:
7392:
7372:
7366:
7365:
7331:
7311:
7305:
7304:
7270:
7261:
7255:
7254:
7220:
7199:
7193:
7192:
7148:
7142:
7141:
7107:
7087:
7081:
7080:
7054:
7039:Nature Photonics
7033:
7027:
7026:
6992:
6972:
6966:
6965:
6921:
6915:
6914:
6888:
6868:
6862:
6856:
6850:
6849:
6813:
6807:
6806:
6770:
6764:
6763:
6737:
6717:
6711:
6710:
6676:
6656:
6650:
6649:
6624:(5): 3093–3111.
6613:
6607:
6606:
6572:
6570:quant-ph/0506139
6552:
6546:
6545:
6514:Phys. Rev. Lett.
6511:
6502:
6496:
6495:
6493:
6475:
6447:
6441:
6440:
6404:
6398:
6397:
6369:
6363:
6362:
6360:
6348:
6342:
6341:
6339:
6337:quant-ph/0612099
6327:
6321:
6314:
6305:
6304:
6293:10.1038/387471a0
6270:
6261:
6255:
6253:
6251:
6250:
6245:
6221:
6219:
6218:
6213:
6201:
6183:
6181:quant-ph/9506025
6162:
6156:
6153:
6147:
6146:
6120:
6100:
6094:
6093:
6051:
6042:
6033:
6032:
5990:
5969:
5963:
5962:
5952:
5926:
5917:
5911:
5904:
5895:
5894:
5875:10.1038/306141a0
5850:
5844:
5834:
5828:
5816:Loudon, Rodney,
5814:
5798:Squeeze operator
5699:crystal lattices
5681:
5679:
5678:
5673:
5671:
5670:
5648:
5646:
5645:
5640:
5638:
5637:
5595:
5593:
5592:
5587:
5585:
5581:
5580:
5560:
5558:
5557:
5552:
5550:
5549:
5529:
5527:
5526:
5521:
5519:
5518:
5513:
5509:
5508:
5506:
5502:
5493:
5489:
5488:
5475:
5473:
5471:
5470:
5469:
5456:
5452:
5447:
5442:
5439:
5428:
5427:
5408:
5406:
5405:
5400:
5398:
5397:
5381:
5379:
5378:
5373:
5371:
5370:
5354:
5352:
5351:
5346:
5344:
5343:
5327:
5325:
5324:
5319:
5314:
5309:
5304:
5299:
5298:
5283:
5282:
5263:
5261:
5260:
5255:
5250:
5245:
5241:
5240:
5239:
5219:
5218:
5203:
5202:
5179:
5177:
5176:
5171:
5169:
5168:
5152:
5150:
5149:
5144:
5142:
5141:
5125:
5123:
5122:
5117:
5109:
5108:
5092:
5090:
5089:
5084:
5082:
5081:
5065:
5063:
5062:
5057:
5045:
5043:
5042:
5037:
5034:
5023:
5007:
5005:
5004:
4999:
4997:
4964:
4962:
4961:
4956:
4953:
4942:
4932:
4927:
4909:
4908:
4862:local oscillator
4846:
4844:
4843:
4838:
4824:
4823:
4811:
4810:
4802:
4789:
4787:
4786:
4781:
4779:
4772:
4755:
4738:
4737:
4724:
4719:
4704:
4702:
4689:
4688:
4675:
4667:
4660:
4655:
4635:
4621:
4613:
4612:
4607:
4591:
4590:
4564:
4562:
4561:
4556:
4554:
4553:
4541:
4536:
4535:
4523:
4506:
4491:
4489:
4488:
4483:
4472:
4467:
4466:
4448:
4447:
4428:
4423:
4408:
4406:
4392:
4384:
4381:
4379:
4364:
4362:
4361:
4356:
4354:
4353:
4321:
4319:
4318:
4313:
4299:
4291:
4290:
4285:
4284:
4276:
4272:
4271:
4266:
4265:
4257:
4247:
4246:
4238:
4235:
4234:
4226:
4223:
4222:
4189:
4175:
4174:
4159:
4156:
4154:
4129:shot noise level
4111:
4109:
4108:
4103:
4092:
4087:
4085:
4078:
4077:
4051:
4050:
4048:
4047:
4029:
4028:
4009:
4004:
3989:
3977:
3973:
3965:
3962:
3960:
3945:
3943:
3942:
3937:
3935:
3934:
3899:
3897:
3896:
3891:
3883:
3860:
3857:
3855:
3768:
3755:
3694:Wigner functions
3690:
3678:
3666:
3654:
3652:
3651:
3646:
3641:
3606:
3604:
3603:
3598:
3587:
3586:
3574:
3573:
3536:
3534:
3533:
3528:
3517:
3516:
3508:
3496:
3495:
3487:
3472:
3471:
3463:
3451:
3450:
3442:
3430:
3429:
3424:
3423:
3415:
3402:
3401:
3393:
3378:
3377:
3369:
3357:
3356:
3348:
3335:
3333:
3332:
3327:
3313:
3293:
3278:
3276:
3275:
3270:
3259:
3258:
3250:
3238:
3237:
3229:
3214:
3213:
3205:
3193:
3192:
3184:
3171:
3169:
3168:
3163:
3152:
3151:
3143:
3133:
3131:
3130:
3125:
3114:
3113:
3105:
3092:
3090:
3089:
3084:
3070:
3069:
3057:
3056:
3051:
3042:
3022:
3017:
3016:
3008:
3005:
3004:
2999:
2998:
2990:
2986:
2959:
2957:
2956:
2951:
2919:
2918:
2903:
2902:
2878:
2873:
2872:
2867:
2866:
2858:
2854:
2827:
2825:
2824:
2819:
2799:
2794:
2793:
2785:
2782:
2752:
2750:
2749:
2744:
2736:
2722:
2721:
2713:
2701:
2700:
2692:
2680:
2666:
2665:
2657:
2639:
2617:
2615:
2614:
2609:
2574:
2573:
2561:
2560:
2555:
2554:
2553:
2539:
2538:
2523:
2522:
2517:
2493:
2492:
2480:
2479:
2474:
2473:
2465:
2434:
2433:
2424:
2423:
2418:
2409:
2389:
2384:
2383:
2375:
2372:
2371:
2366:
2365:
2357:
2353:
2326:
2324:
2323:
2318:
2286:
2285:
2270:
2269:
2264:
2263:
2255:
2221:
2220:
2211:
2210:
2195:
2194:
2189:
2188:
2187:
2161:
2160:
2151:
2150:
2126:
2121:
2120:
2115:
2114:
2106:
2102:
2075:
2073:
2072:
2067:
2050:
2049:
2037:
2036:
1991:
1986:
1985:
1977:
1974:
1944:
1942:
1941:
1936:
1934:
1926:
1896:
1894:
1893:
1888:
1886:
1885:
1877:
1871:
1866:
1855:
1850:
1849:
1830:
1827:
1824:
1823:
1812:
1806:
1804:
1790:
1785:
1784:
1756:
1754:
1753:
1748:
1736:
1734:
1733:
1728:
1712:
1710:
1709:
1704:
1702:
1701:
1670:
1668:
1667:
1662:
1647:
1645:
1644:
1639:
1637:
1633:
1632:
1630:
1622:
1614:
1601:
1598:
1590:
1582:
1581:
1576:
1575:
1567:
1563:
1560:
1557:
1553:
1552:
1550:
1542:
1534:
1521:
1518:
1510:
1502:
1501:
1496:
1495:
1490:
1489:
1481:
1467:
1465:
1464:
1459:
1443:
1441:
1440:
1435:
1433:
1432:
1427:
1426:
1418:
1407:
1405:
1404:
1399:
1397:
1396:
1388:
1375:
1373:
1372:
1367:
1365:
1364:
1355:
1354:
1346:
1345:
1337:
1327:
1326:
1321:
1320:
1312:
1308:
1307:
1295:
1287:
1285:
1284:
1260:
1259:
1251:
1247:
1244:
1238:
1237:
1229:
1226:
1225:
1213:
1212:
1207:
1206:
1198:
1170:
1169:
1161:
1147:squeeze operator
1144:
1142:
1141:
1136:
1111:
1109:
1108:
1103:
1078:
1076:
1075:
1070:
1062:
1047:
1045:
1044:
1039:
1031:
1017:
1016:
1008:
996:
995:
987:
975:
961:
960:
952:
934:
907:
905:
904:
899:
896:
891:
882:
881:
866:
865:
842:
840:
839:
834:
831:
826:
817:
816:
808:
799:
798:
790:
770:
768:
767:
762:
760:
759:
731:
729:
728:
723:
721:
720:
708:
707:
695:
694:
669:
667:
666:
661:
659:
655:
651:
650:
635:
633:
631:
626:
613:
612:
611:
602:
601:
582:
537:
535:
534:
529:
487:
485:
484:
479:
474:
469:
468:
459:
458:
443:
442:
415:quadrature phase
404:
402:
401:
396:
391:
380:
379:
364:
363:
344:
342:
341:
336:
334:
333:
318:
317:
298:
296:
295:
290:
282:
263:
261:
260:
255:
247:
221:, respectively.
220:
218:
217:
212:
210:
202:
177:
175:
174:
169:
166:
158:
119:
117:
116:
111:
99:
97:
96:
91:
79:
77:
76:
71:
59:
57:
56:
51:
7771:
7770:
7766:
7765:
7764:
7762:
7761:
7760:
7741:
7740:
7739:
7718:
7706:
7702:
7696:
7692:
7683:
7679:
7673:
7669:
7667:
7664:
7663:
7660:
7656:
7645:
7640:
7639:
7582:
7578:
7539:
7535:
7486:
7482:
7432:
7426:
7422:
7373:
7369:
7312:
7308:
7268:
7262:
7258:
7200:
7196:
7149:
7145:
7088:
7084:
7034:
7030:
6973:
6969:
6932:(7587): 505–8.
6922:
6918:
6879:(2–3): 89–165.
6873:Physics Reports
6869:
6865:
6857:
6853:
6814:
6810:
6771:
6767:
6718:
6714:
6661:Phys. Rev. Lett
6657:
6653:
6614:
6610:
6557:Phys. Rev. Lett
6553:
6549:
6509:
6503:
6499:
6448:
6444:
6405:
6401:
6370:
6366:
6349:
6345:
6328:
6324:
6315:
6308:
6268:
6262:
6258:
6227:
6224:
6223:
6207:
6204:
6203:
6163:
6159:
6154:
6150:
6105:Physics Reports
6101:
6097:
6049:
6043:
6036:
5970:
5966:
5924:
5918:
5914:
5905:
5898:
5851:
5847:
5835:
5831:
5815:
5811:
5806:
5778:Negative energy
5774:
5712:
5660:
5656:
5654:
5651:
5650:
5627:
5623:
5621:
5618:
5617:
5602:
5576:
5572:
5568:
5566:
5563:
5562:
5545:
5541:
5539:
5536:
5535:
5514:
5498:
5494:
5484:
5480:
5476:
5474:
5465:
5461:
5457:
5448:
5441:
5440:
5438:
5437:
5433:
5432:
5423:
5419:
5417:
5414:
5413:
5393:
5389:
5387:
5384:
5383:
5366:
5362:
5360:
5357:
5356:
5339:
5335:
5333:
5330:
5329:
5310:
5303:
5294:
5290:
5278:
5274:
5269:
5266:
5265:
5246:
5235:
5231:
5227:
5223:
5214:
5210:
5198:
5194:
5189:
5186:
5185:
5164:
5160:
5158:
5155:
5154:
5137:
5133:
5131:
5128:
5127:
5104:
5100:
5098:
5095:
5094:
5077:
5073:
5071:
5068:
5067:
5051:
5048:
5047:
5024:
5019:
5013:
5010:
5009:
4981:
4973:
4970:
4969:
4943:
4938:
4928:
4917:
4904:
4900:
4898:
4895:
4894:
4884:
4878:
4853:
4819:
4815:
4801:
4800:
4798:
4795:
4794:
4777:
4776:
4768:
4751:
4730:
4726:
4720:
4709:
4684:
4680:
4679:
4674:
4665:
4664:
4656:
4642:
4622:
4617:
4608:
4600:
4599:
4592:
4586:
4582:
4578:
4576:
4573:
4572:
4549:
4545:
4537:
4531:
4527:
4519:
4502:
4500:
4497:
4496:
4468:
4462:
4458:
4440:
4436:
4424:
4413:
4396:
4391:
4380:
4375:
4373:
4370:
4369:
4346:
4342:
4331:
4328:
4327:
4295:
4286:
4275:
4274:
4273:
4267:
4256:
4255:
4254:
4237:
4236:
4225:
4224:
4218:
4214:
4185:
4170:
4166:
4155:
4150:
4148:
4145:
4144:
4125:
4088:
4073:
4069:
4068:
4049:
4043:
4039:
4021:
4017:
4005:
3994:
3972:
3961:
3956:
3954:
3951:
3950:
3927:
3923:
3912:
3909:
3908:
3879:
3856:
3851:
3849:
3846:
3845:
3834:
3818:
3813:
3776:
3775:
3774:
3773:
3772:
3769:
3761:
3760:
3756:
3734:
3707:
3706:
3705:
3704:
3699:
3698:
3697:
3691:
3683:
3682:
3679:
3671:
3670:
3667:
3637:
3632:
3629:
3628:
3612:
3579:
3575:
3569:
3565:
3542:
3539:
3538:
3507:
3506:
3486:
3485:
3462:
3461:
3441:
3440:
3425:
3414:
3413:
3412:
3392:
3391:
3368:
3367:
3347:
3346:
3344:
3341:
3340:
3309:
3289:
3287:
3284:
3283:
3249:
3248:
3228:
3227:
3204:
3203:
3183:
3182:
3180:
3177:
3176:
3142:
3141:
3139:
3136:
3135:
3104:
3103:
3101:
3098:
3097:
3065:
3061:
3052:
3047:
3046:
3038:
3018:
3007:
3006:
3000:
2989:
2988:
2987:
2982:
2968:
2965:
2964:
2911:
2907:
2898:
2894:
2874:
2868:
2857:
2856:
2855:
2850:
2836:
2833:
2832:
2795:
2784:
2783:
2778:
2764:
2761:
2760:
2732:
2712:
2711:
2691:
2690:
2676:
2656:
2655:
2635:
2633:
2630:
2629:
2566:
2562:
2556:
2549:
2545:
2544:
2543:
2528:
2524:
2518:
2513:
2512:
2488:
2484:
2475:
2469:
2461:
2460:
2459:
2429:
2425:
2419:
2414:
2413:
2405:
2385:
2374:
2373:
2367:
2356:
2355:
2354:
2349:
2335:
2332:
2331:
2278:
2274:
2265:
2259:
2251:
2250:
2249:
2216:
2212:
2200:
2196:
2190:
2183:
2179:
2178:
2177:
2156:
2152:
2146:
2142:
2122:
2116:
2105:
2104:
2103:
2098:
2084:
2081:
2080:
2042:
2038:
2032:
2028:
1987:
1976:
1975:
1970:
1956:
1953:
1952:
1925:
1908:
1905:
1904:
1878:
1873:
1872:
1856:
1854:
1845:
1841:
1826:
1813:
1808:
1807:
1794:
1789:
1780:
1776:
1765:
1762:
1761:
1742:
1739:
1738:
1722:
1719:
1718:
1691:
1687:
1676:
1673:
1672:
1656:
1653:
1652:
1623:
1615:
1613:
1606:
1602:
1591:
1583:
1580:
1566:
1565:
1559:
1543:
1535:
1533:
1526:
1522:
1511:
1503:
1500:
1491:
1480:
1479:
1478:
1476:
1473:
1472:
1453:
1450:
1449:
1428:
1417:
1416:
1415:
1413:
1410:
1409:
1387:
1386:
1384:
1381:
1380:
1360:
1359:
1347:
1336:
1335:
1334:
1322:
1311:
1310:
1309:
1303:
1299:
1286:
1280:
1279:
1250:
1249:
1243:
1228:
1227:
1221:
1217:
1208:
1197:
1196:
1195:
1160:
1159:
1157:
1154:
1153:
1121:
1118:
1117:
1088:
1085:
1084:
1058:
1056:
1053:
1052:
1027:
1007:
1006:
986:
985:
971:
951:
950:
930:
928:
925:
924:
914:
892:
887:
877:
873:
861:
857:
855:
852:
851:
827:
822:
807:
806:
789:
788:
786:
783:
782:
755:
751:
749:
746:
745:
716:
712:
703:
699:
690:
686:
678:
675:
674:
646:
642:
627:
622:
614:
607:
603:
597:
593:
583:
581:
577:
573:
546:
543:
542:
517:
514:
513:
494:
470:
464:
460:
454:
450:
438:
434:
432:
429:
428:
387:
375:
371:
359:
355:
350:
347:
346:
329:
325:
313:
309:
304:
301:
300:
278:
276:
273:
272:
270:coherent states
243:
241:
238:
237:
201:
183:
180:
179:
157:
140:
137:
136:
105:
102:
101:
85:
82:
81:
65:
62:
61:
45:
42:
41:
17:
12:
11:
5:
7769:
7759:
7758:
7756:Quantum states
7753:
7751:Quantum optics
7738:
7737:
7725:
7721:
7717:
7714:
7709:
7705:
7699:
7695:
7691:
7686:
7682:
7676:
7672:
7653:
7652:
7651:
7644:
7643:External links
7641:
7638:
7637:
7576:
7533:
7496:(2): 513–577.
7480:
7420:
7367:
7322:(25): 250801.
7306:
7256:
7211:(25): 251102.
7194:
7159:(11): 110801.
7143:
7098:(15): 155304.
7082:
7045:(9): 731–736.
7028:
6967:
6916:
6863:
6851:
6808:
6765:
6722:Nature Physics
6712:
6667:(18): 181101.
6651:
6608:
6563:(24): 243603.
6547:
6497:
6442:
6399:
6364:
6343:
6322:
6306:
6256:
6243:
6240:
6237:
6234:
6231:
6211:
6174:(2): 145–156.
6157:
6148:
6095:
6060:(11): 110801.
6034:
5964:
5912:
5896:
5845:
5829:
5808:
5807:
5805:
5802:
5801:
5800:
5795:
5793:Quantum optics
5790:
5785:
5780:
5773:
5770:
5748:, for example
5711:
5708:
5684:Sub-Poissonian
5669:
5666:
5663:
5659:
5636:
5633:
5630:
5626:
5601:
5598:
5584:
5579:
5575:
5571:
5548:
5544:
5532:
5531:
5517:
5512:
5505:
5501:
5497:
5492:
5487:
5483:
5479:
5468:
5464:
5460:
5455:
5451:
5445:
5436:
5431:
5426:
5422:
5396:
5392:
5369:
5365:
5342:
5338:
5317:
5313:
5307:
5302:
5297:
5293:
5289:
5286:
5281:
5277:
5273:
5253:
5249:
5244:
5238:
5234:
5230:
5226:
5222:
5217:
5213:
5209:
5206:
5201:
5197:
5193:
5167:
5163:
5140:
5136:
5115:
5112:
5107:
5103:
5080:
5076:
5055:
5033:
5030:
5027:
5022:
5018:
4996:
4993:
4990:
4987:
4984:
4980:
4977:
4966:
4965:
4952:
4949:
4946:
4941:
4937:
4931:
4926:
4923:
4920:
4916:
4912:
4907:
4903:
4877:
4874:
4852:
4849:
4836:
4833:
4830:
4827:
4822:
4818:
4814:
4808:
4805:
4791:
4790:
4775:
4771:
4767:
4764:
4761:
4758:
4754:
4750:
4747:
4744:
4741:
4736:
4733:
4729:
4723:
4718:
4715:
4712:
4708:
4701:
4698:
4695:
4692:
4687:
4683:
4678:
4673:
4670:
4668:
4666:
4663:
4659:
4654:
4651:
4648:
4645:
4641:
4638:
4634:
4631:
4628:
4625:
4620:
4616:
4611:
4606:
4603:
4598:
4595:
4593:
4589:
4585:
4581:
4580:
4552:
4548:
4544:
4540:
4534:
4530:
4526:
4522:
4518:
4515:
4512:
4509:
4505:
4493:
4492:
4481:
4478:
4475:
4471:
4465:
4461:
4457:
4454:
4451:
4446:
4443:
4439:
4435:
4432:
4427:
4422:
4419:
4416:
4412:
4405:
4402:
4399:
4395:
4390:
4387:
4378:
4352:
4349:
4345:
4341:
4338:
4335:
4324:
4323:
4311:
4308:
4305:
4302:
4298:
4294:
4289:
4282:
4279:
4270:
4263:
4260:
4253:
4250:
4244:
4241:
4232:
4229:
4221:
4217:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4188:
4184:
4181:
4178:
4173:
4169:
4165:
4162:
4153:
4124:
4121:
4113:
4112:
4101:
4098:
4095:
4091:
4084:
4081:
4076:
4072:
4066:
4063:
4060:
4057:
4054:
4046:
4042:
4038:
4035:
4032:
4027:
4024:
4020:
4016:
4013:
4008:
4003:
4000:
3997:
3993:
3986:
3983:
3980:
3976:
3971:
3968:
3959:
3933:
3930:
3926:
3922:
3919:
3916:
3901:
3900:
3889:
3886:
3882:
3878:
3875:
3872:
3869:
3866:
3863:
3854:
3833:
3830:
3817:
3814:
3812:
3811:Classification
3809:
3780:sub-Poissonian
3770:
3763:
3762:
3757:
3750:
3749:
3748:
3747:
3746:
3733:
3730:
3711:coherent state
3701:
3700:
3692:
3685:
3684:
3680:
3673:
3672:
3668:
3661:
3660:
3659:
3658:
3657:
3644:
3640:
3636:
3611:
3608:
3596:
3593:
3590:
3585:
3582:
3578:
3572:
3568:
3564:
3561:
3558:
3555:
3552:
3549:
3546:
3526:
3523:
3520:
3514:
3511:
3505:
3502:
3499:
3493:
3490:
3484:
3481:
3478:
3475:
3469:
3466:
3460:
3457:
3454:
3448:
3445:
3439:
3436:
3433:
3428:
3421:
3418:
3411:
3408:
3405:
3399:
3396:
3390:
3387:
3384:
3381:
3375:
3372:
3366:
3363:
3360:
3354:
3351:
3337:
3336:
3325:
3322:
3319:
3316:
3312:
3308:
3305:
3302:
3299:
3296:
3292:
3280:
3279:
3268:
3265:
3262:
3256:
3253:
3247:
3244:
3241:
3235:
3232:
3226:
3223:
3220:
3217:
3211:
3208:
3202:
3199:
3196:
3190:
3187:
3161:
3158:
3155:
3149:
3146:
3123:
3120:
3117:
3111:
3108:
3094:
3093:
3082:
3079:
3076:
3073:
3068:
3064:
3060:
3055:
3050:
3045:
3041:
3037:
3034:
3031:
3028:
3025:
3021:
3014:
3011:
3003:
2996:
2993:
2985:
2981:
2978:
2975:
2972:
2961:
2960:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2917:
2914:
2910:
2906:
2901:
2897:
2893:
2890:
2887:
2884:
2881:
2877:
2871:
2864:
2861:
2853:
2849:
2846:
2843:
2840:
2829:
2828:
2817:
2814:
2811:
2808:
2805:
2802:
2798:
2791:
2788:
2781:
2777:
2774:
2771:
2768:
2754:
2753:
2742:
2739:
2735:
2731:
2728:
2725:
2719:
2716:
2710:
2707:
2704:
2698:
2695:
2689:
2686:
2683:
2679:
2675:
2672:
2669:
2663:
2660:
2654:
2651:
2648:
2645:
2642:
2638:
2619:
2618:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2572:
2569:
2565:
2559:
2552:
2548:
2542:
2537:
2534:
2531:
2527:
2521:
2516:
2511:
2508:
2505:
2502:
2499:
2496:
2491:
2487:
2483:
2478:
2472:
2468:
2464:
2458:
2455:
2452:
2449:
2446:
2443:
2440:
2437:
2432:
2428:
2422:
2417:
2412:
2408:
2404:
2401:
2398:
2395:
2392:
2388:
2381:
2378:
2370:
2363:
2360:
2352:
2348:
2345:
2342:
2339:
2328:
2327:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2284:
2281:
2277:
2273:
2268:
2262:
2258:
2254:
2248:
2245:
2242:
2239:
2236:
2233:
2230:
2227:
2224:
2219:
2215:
2209:
2206:
2203:
2199:
2193:
2186:
2182:
2176:
2173:
2170:
2167:
2164:
2159:
2155:
2149:
2145:
2141:
2138:
2135:
2132:
2129:
2125:
2119:
2112:
2109:
2101:
2097:
2094:
2091:
2088:
2077:
2076:
2065:
2062:
2059:
2056:
2053:
2048:
2045:
2041:
2035:
2031:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1990:
1983:
1980:
1973:
1969:
1966:
1963:
1960:
1932:
1929:
1924:
1921:
1918:
1915:
1912:
1898:
1897:
1884:
1881:
1876:
1869:
1865:
1862:
1859:
1853:
1848:
1844:
1840:
1837:
1834:
1822:
1819:
1816:
1811:
1803:
1800:
1797:
1793:
1788:
1783:
1779:
1775:
1772:
1769:
1746:
1726:
1700:
1697:
1694:
1690:
1686:
1683:
1680:
1660:
1649:
1648:
1636:
1629:
1626:
1621:
1618:
1612:
1609:
1605:
1597:
1594:
1589:
1586:
1579:
1573:
1570:
1556:
1549:
1546:
1541:
1538:
1532:
1529:
1525:
1517:
1514:
1509:
1506:
1499:
1494:
1487:
1484:
1457:
1431:
1424:
1421:
1394:
1391:
1377:
1376:
1363:
1358:
1353:
1350:
1343:
1340:
1333:
1330:
1325:
1318:
1315:
1306:
1302:
1298:
1293:
1290:
1283:
1278:
1275:
1272:
1269:
1266:
1263:
1257:
1254:
1241:
1235:
1232:
1224:
1220:
1216:
1211:
1204:
1201:
1194:
1191:
1188:
1185:
1182:
1179:
1176:
1173:
1167:
1164:
1134:
1131:
1128:
1125:
1101:
1098:
1095:
1092:
1068:
1065:
1061:
1049:
1048:
1037:
1034:
1030:
1026:
1023:
1020:
1014:
1011:
1005:
1002:
999:
993:
990:
984:
981:
978:
974:
970:
967:
964:
958:
955:
949:
946:
943:
940:
937:
933:
913:
910:
895:
890:
886:
880:
876:
872:
869:
864:
860:
844:
843:
830:
825:
821:
814:
811:
805:
802:
796:
793:
758:
754:
742:coherent state
719:
715:
711:
706:
702:
698:
693:
689:
685:
682:
671:
670:
658:
654:
649:
645:
641:
638:
630:
625:
621:
617:
610:
606:
600:
596:
592:
589:
586:
580:
576:
572:
569:
565:
562:
559:
556:
553:
550:
527:
524:
521:
493:
490:
477:
473:
467:
463:
457:
453:
449:
446:
441:
437:
410:squeezed state
394:
390:
386:
383:
378:
374:
370:
367:
362:
358:
354:
332:
328:
324:
321:
316:
312:
308:
288:
285:
281:
253:
250:
246:
223:
222:
208:
205:
200:
197:
194:
191:
188:
164:
161:
156:
153:
150:
147:
144:
109:
89:
69:
49:
15:
9:
6:
4:
3:
2:
7768:
7757:
7754:
7752:
7749:
7748:
7746:
7723:
7719:
7715:
7712:
7707:
7703:
7697:
7689:
7684:
7680:
7674:
7658:
7654:
7650:
7647:
7646:
7633:
7629:
7625:
7621:
7617:
7613:
7609:
7605:
7600:
7595:
7592:(13): 13501.
7591:
7587:
7580:
7572:
7568:
7564:
7560:
7556:
7552:
7548:
7544:
7537:
7529:
7525:
7521:
7517:
7513:
7509:
7504:
7499:
7495:
7491:
7484:
7476:
7472:
7467:
7462:
7458:
7454:
7450:
7446:
7442:
7438:
7431:
7424:
7416:
7412:
7408:
7404:
7400:
7396:
7391:
7386:
7383:(6): 065032.
7382:
7378:
7371:
7363:
7359:
7355:
7351:
7347:
7343:
7339:
7335:
7330:
7325:
7321:
7317:
7310:
7302:
7298:
7294:
7290:
7286:
7282:
7278:
7274:
7267:
7260:
7252:
7248:
7244:
7240:
7236:
7232:
7228:
7224:
7219:
7214:
7210:
7206:
7198:
7190:
7186:
7182:
7178:
7174:
7170:
7166:
7162:
7158:
7154:
7147:
7139:
7135:
7131:
7127:
7123:
7119:
7115:
7111:
7106:
7101:
7097:
7093:
7086:
7078:
7074:
7070:
7066:
7062:
7058:
7053:
7048:
7044:
7040:
7032:
7024:
7020:
7016:
7012:
7008:
7004:
7000:
6996:
6991:
6986:
6983:(9): 093602.
6982:
6978:
6971:
6963:
6959:
6955:
6951:
6947:
6943:
6939:
6935:
6931:
6927:
6920:
6912:
6908:
6904:
6900:
6896:
6892:
6887:
6882:
6878:
6874:
6867:
6861:
6855:
6847:
6843:
6839:
6835:
6831:
6827:
6823:
6819:
6812:
6804:
6800:
6796:
6792:
6788:
6784:
6780:
6776:
6769:
6761:
6757:
6753:
6749:
6745:
6741:
6736:
6731:
6727:
6723:
6716:
6708:
6704:
6700:
6696:
6692:
6688:
6684:
6680:
6675:
6670:
6666:
6662:
6655:
6647:
6643:
6639:
6635:
6631:
6627:
6623:
6619:
6612:
6604:
6600:
6596:
6592:
6588:
6584:
6580:
6576:
6571:
6566:
6562:
6558:
6551:
6543:
6539:
6535:
6531:
6527:
6523:
6519:
6515:
6508:
6501:
6492:
6487:
6483:
6479:
6474:
6469:
6466:(4): 044005.
6465:
6461:
6457:
6453:
6446:
6438:
6434:
6430:
6426:
6422:
6418:
6414:
6410:
6403:
6395:
6391:
6387:
6383:
6379:
6375:
6368:
6359:
6354:
6347:
6338:
6333:
6326:
6319:
6313:
6311:
6302:
6298:
6294:
6290:
6286:
6282:
6278:
6274:
6267:
6260:
6241:
6238:
6235:
6229:
6209:
6199:
6195:
6191:
6187:
6182:
6177:
6173:
6169:
6161:
6152:
6144:
6140:
6136:
6132:
6128:
6124:
6119:
6114:
6110:
6106:
6099:
6091:
6087:
6083:
6079:
6075:
6071:
6067:
6063:
6059:
6055:
6048:
6041:
6039:
6030:
6026:
6022:
6018:
6014:
6010:
6006:
6002:
5998:
5994:
5989:
5984:
5981:(3): 033602.
5980:
5976:
5968:
5960:
5956:
5951:
5946:
5942:
5938:
5934:
5930:
5923:
5916:
5909:
5903:
5901:
5892:
5888:
5884:
5880:
5876:
5872:
5868:
5864:
5860:
5856:
5849:
5842:
5838:
5833:
5827:
5826:0-19-850177-3
5823:
5819:
5813:
5809:
5799:
5796:
5794:
5791:
5789:
5786:
5784:
5781:
5779:
5776:
5775:
5769:
5766:
5761:
5759:
5755:
5751:
5747:
5743:
5738:
5736:
5732:
5731:atomic clocks
5727:
5725:
5721:
5717:
5707:
5704:
5700:
5696:
5692:
5687:
5685:
5664:
5657:
5631:
5624:
5615:
5611:
5607:
5597:
5582:
5577:
5573:
5569:
5546:
5542:
5515:
5510:
5503:
5499:
5495:
5490:
5485:
5481:
5477:
5466:
5462:
5453:
5449:
5443:
5434:
5429:
5424:
5420:
5412:
5411:
5410:
5394:
5390:
5367:
5363:
5340:
5336:
5315:
5311:
5305:
5300:
5295:
5291:
5284:
5279:
5275:
5251:
5247:
5242:
5236:
5232:
5224:
5220:
5215:
5211:
5204:
5199:
5195:
5183:
5165:
5161:
5138:
5134:
5113:
5110:
5105:
5101:
5078:
5074:
5053:
5028:
5020:
5016:
4994:
4991:
4988:
4985:
4982:
4978:
4975:
4947:
4939:
4935:
4929:
4924:
4921:
4918:
4914:
4910:
4905:
4901:
4893:
4892:
4891:
4889:
4883:
4873:
4871:
4867:
4863:
4858:
4848:
4831:
4825:
4820:
4816:
4812:
4806:
4803:
4773:
4765:
4756:
4745:
4739:
4734:
4731:
4727:
4716:
4713:
4710:
4706:
4696:
4690:
4685:
4681:
4676:
4671:
4669:
4609:
4596:
4594:
4587:
4583:
4571:
4570:
4569:
4568:
4567:thermal state
4550:
4542:
4532:
4524:
4516:
4510:
4507:
4476:
4473:
4463:
4455:
4452:
4449:
4444:
4441:
4437:
4433:
4420:
4417:
4414:
4410:
4403:
4400:
4397:
4393:
4388:
4368:
4367:
4366:
4350:
4347:
4343:
4339:
4336:
4333:
4306:
4303:
4300:
4287:
4277:
4268:
4258:
4251:
4248:
4239:
4227:
4219:
4215:
4208:
4205:
4202:
4196:
4193:
4190:
4179:
4171:
4167:
4163:
4143:
4142:
4141:
4139:
4135:
4130:
4120:
4118:
4096:
4093:
4082:
4079:
4074:
4070:
4064:
4058:
4055:
4044:
4036:
4033:
4030:
4025:
4022:
4018:
4014:
4001:
3998:
3995:
3991:
3984:
3981:
3978:
3974:
3969:
3949:
3948:
3947:
3931:
3928:
3924:
3920:
3917:
3914:
3906:
3884:
3873:
3867:
3864:
3844:
3843:
3842:
3840:
3829:
3827:
3823:
3808:
3806:
3802:
3792:
3788:
3786:
3781:
3767:
3754:
3745:
3743:
3739:
3729:
3727:
3722:
3720:
3716:
3715:quantum noise
3712:
3695:
3689:
3677:
3665:
3656:
3642:
3638:
3634:
3626:
3625:complementary
3622:
3618:
3607:
3594:
3591:
3588:
3583:
3580:
3576:
3570:
3566:
3562:
3559:
3556:
3553:
3550:
3547:
3544:
3521:
3509:
3500:
3488:
3482:
3476:
3464:
3455:
3443:
3434:
3426:
3416:
3406:
3394:
3388:
3382:
3370:
3361:
3349:
3320:
3317:
3314:
3306:
3300:
3297:
3294:
3282:
3281:
3263:
3251:
3242:
3230:
3224:
3218:
3206:
3197:
3185:
3175:
3174:
3173:
3156:
3144:
3118:
3106:
3077:
3071:
3066:
3062:
3058:
3053:
3043:
3035:
3029:
3026:
3023:
3009:
3001:
2991:
2979:
2976:
2973:
2963:
2962:
2944:
2938:
2935:
2929:
2923:
2920:
2915:
2912:
2908:
2904:
2899:
2895:
2891:
2885:
2882:
2879:
2869:
2859:
2847:
2844:
2841:
2831:
2830:
2815:
2812:
2806:
2803:
2800:
2786:
2775:
2772:
2769:
2759:
2758:
2757:
2737:
2726:
2714:
2705:
2693:
2687:
2681:
2670:
2658:
2652:
2646:
2643:
2640:
2628:
2627:
2626:
2624:
2602:
2596:
2593:
2587:
2581:
2578:
2570:
2567:
2563:
2557:
2550:
2546:
2540:
2535:
2532:
2529:
2525:
2519:
2514:
2506:
2500:
2494:
2489:
2485:
2476:
2466:
2456:
2453:
2447:
2441:
2435:
2430:
2426:
2420:
2410:
2402:
2396:
2393:
2390:
2376:
2368:
2358:
2346:
2343:
2340:
2330:
2329:
2311:
2305:
2302:
2296:
2290:
2287:
2282:
2279:
2275:
2266:
2256:
2246:
2243:
2240:
2234:
2228:
2222:
2217:
2213:
2207:
2204:
2201:
2197:
2191:
2184:
2180:
2174:
2168:
2162:
2157:
2153:
2147:
2143:
2139:
2133:
2130:
2127:
2117:
2107:
2095:
2092:
2089:
2079:
2078:
2060:
2054:
2051:
2046:
2043:
2039:
2033:
2029:
2025:
2019:
2013:
2010:
2007:
2004:
1998:
1995:
1992:
1978:
1967:
1964:
1961:
1951:
1950:
1949:
1946:
1930:
1927:
1922:
1919:
1913:
1903:
1882:
1879:
1867:
1863:
1857:
1851:
1846:
1838:
1820:
1817:
1814:
1801:
1798:
1795:
1791:
1786:
1781:
1773:
1760:
1759:
1758:
1757:are given by
1744:
1724:
1716:
1698:
1695:
1692:
1688:
1684:
1681:
1678:
1671:, (note that
1658:
1634:
1627:
1624:
1619:
1616:
1610:
1607:
1603:
1592:
1587:
1584:
1577:
1568:
1554:
1547:
1544:
1539:
1536:
1530:
1527:
1523:
1512:
1507:
1504:
1497:
1492:
1482:
1471:
1470:
1469:
1455:
1447:
1429:
1419:
1389:
1351:
1348:
1338:
1331:
1328:
1323:
1313:
1304:
1300:
1291:
1288:
1276:
1273:
1270:
1264:
1252:
1230:
1222:
1218:
1214:
1209:
1199:
1192:
1186:
1183:
1180:
1174:
1162:
1152:
1151:
1150:
1148:
1129:
1123:
1115:
1096:
1090:
1082:
1063:
1032:
1021:
1009:
1000:
988:
982:
976:
965:
953:
947:
941:
938:
935:
923:
922:
921:
919:
909:
893:
888:
884:
878:
874:
870:
867:
862:
858:
849:
828:
823:
819:
809:
803:
800:
791:
781:
780:
779:
777:
772:
756:
752:
743:
739:
735:
717:
713:
709:
704:
700:
696:
691:
687:
683:
680:
656:
652:
647:
643:
639:
636:
628:
623:
619:
615:
608:
598:
594:
590:
587:
578:
574:
570:
567:
563:
560:
554:
548:
541:
540:
539:
525:
522:
519:
511:
507:
506:wave function
498:
489:
475:
471:
465:
461:
455:
447:
444:
439:
426:
422:
420:
416:
411:
406:
392:
388:
384:
381:
376:
372:
365:
360:
356:
330:
326:
319:
314:
310:
283:
271:
267:
248:
231:
227:
206:
203:
198:
195:
189:
162:
159:
154:
151:
145:
135:
134:
133:
131:
127:
123:
107:
87:
67:
60:and momentum
47:
39:
35:
32:
31:non-commuting
28:
24:
19:
7657:
7589:
7585:
7579:
7546:
7542:
7536:
7493:
7489:
7483:
7440:
7436:
7423:
7380:
7376:
7370:
7319:
7315:
7309:
7276:
7272:
7259:
7208:
7204:
7197:
7156:
7152:
7146:
7095:
7091:
7085:
7042:
7038:
7031:
6980:
6976:
6970:
6929:
6925:
6919:
6876:
6872:
6866:
6854:
6821:
6817:
6811:
6781:(2): 67–88.
6778:
6774:
6768:
6725:
6721:
6715:
6664:
6660:
6654:
6621:
6617:
6611:
6560:
6556:
6550:
6517:
6513:
6500:
6463:
6459:
6452:Gaeta, A. L.
6445:
6412:
6408:
6402:
6380:(10): 1465.
6377:
6373:
6367:
6346:
6325:
6276:
6272:
6259:
6171:
6167:
6160:
6151:
6108:
6104:
6098:
6057:
6053:
5978:
5974:
5967:
5932:
5928:
5915:
5907:
5858:
5854:
5848:
5841:Peter Zoller
5837:C W Gardiner
5832:
5817:
5812:
5762:
5750:Unruh effect
5739:
5728:
5713:
5710:Applications
5688:
5603:
5533:
5182:Bloch sphere
4967:
4885:
4854:
4792:
4494:
4325:
4133:
4126:
4114:
3902:
3838:
3835:
3819:
3798:
3777:
3735:
3723:
3708:
3613:
3338:
3095:
2755:
2622:
2620:
1947:
1899:
1714:
1650:
1378:
1081:vacuum state
1050:
917:
915:
845:
773:
672:
509:
503:
423:
409:
407:
235:
26:
20:
18:
7466:11094/77656
6728:(12): 962.
5693:in a trap,
4890:defined as
1651:For a real
1149:, given by
122:quadratures
38:eigenvalues
34:observables
7745:Categories
6990:1512.02150
6118:1611.03986
5804:References
5697:states in
4880:See also:
4117:Fock state
3719:light wave
3617:Heisenberg
848:eigenvalue
776:eigenstate
734:wavepacket
128:obeys the
7694:Δ
7671:Δ
7599:0804.4468
7528:118990906
7415:119112907
7390:0912.3895
7329:1004.1725
7218:1007.0574
7105:1403.4542
7052:1310.3177
6911:119239234
6886:1011.2978
6760:209832912
6735:1109.2295
6674:1302.2188
6473:1309.6371
6358:1401.4118
6242:π
6236:θ
6233:→
6230:θ
6210:γ
6143:119098759
5988:0706.1431
5883:1476-4687
5658:χ
5625:χ
5459:Δ
5421:χ
5288:Δ
5272:Δ
5229:Δ
5221:≥
5208:Δ
5205:⋅
5192:Δ
4915:∑
4826:
4807:~
4763:⟨
4760:⟩
4740:
4722:∞
4707:∑
4691:
4640:⟨
4637:⟩
4584:ρ
4547:⟩
4529:⟩
4514:⟩
4480:⟩
4453:
4445:ϕ
4434:−
4426:∞
4411:∑
4401:
4386:⟩
4351:ϕ
4334:ζ
4310:⟩
4288:†
4281:^
4269:†
4262:^
4252:ζ
4249:−
4243:^
4231:^
4220:∗
4216:ζ
4209:
4200:⟩
4180:ζ
4161:⟩
4100:⟩
4034:
4026:ϕ
4015:−
4007:∞
3992:∑
3982:
3967:⟩
3932:ϕ
3915:ζ
3888:⟩
3874:ζ
3862:⟩
3635:τ
3592:
3584:θ
3571:∗
3567:α
3557:
3551:α
3545:γ
3522:γ
3513:^
3501:ζ
3492:^
3477:ζ
3468:^
3456:α
3447:^
3435:ζ
3427:†
3420:^
3407:ζ
3398:^
3383:ζ
3374:^
3362:α
3353:^
3324:⟩
3321:α
3315:ζ
3307:≠
3304:⟩
3301:ζ
3295:α
3264:ζ
3255:^
3243:α
3234:^
3225:≠
3219:α
3210:^
3198:ζ
3189:^
3157:α
3148:^
3119:ζ
3110:^
3072:
3044:α
3033:⟩
3030:α
3024:ζ
3013:^
3002:†
2995:^
2980:α
2974:ζ
2971:⟨
2939:
2924:
2916:θ
2905:−
2896:α
2889:⟩
2886:α
2880:ζ
2863:^
2848:α
2842:ζ
2839:⟨
2816:α
2810:⟩
2807:α
2801:ζ
2790:^
2776:α
2770:ζ
2767:⟨
2741:⟩
2727:ζ
2718:^
2706:α
2697:^
2685:⟩
2682:ζ
2671:α
2662:^
2650:⟩
2647:α
2641:ζ
2597:
2582:
2571:θ
2551:∗
2547:α
2536:θ
2530:−
2515:α
2507:−
2495:
2467:α
2436:
2411:α
2400:⟩
2397:ζ
2391:α
2380:^
2369:†
2362:^
2347:ζ
2341:α
2338:⟨
2306:
2291:
2283:θ
2257:α
2235:−
2223:
2208:θ
2185:∗
2181:α
2163:
2144:α
2137:⟩
2134:ζ
2128:α
2111:^
2096:ζ
2090:α
2087:⟨
2055:
2047:θ
2034:∗
2030:α
2026:−
2014:
2008:α
2002:⟩
1999:ζ
1993:α
1982:^
1968:ζ
1962:α
1959:⟨
1928:ℏ
1917:Δ
1911:Δ
1883:ζ
1864:ω
1861:ℏ
1836:Δ
1821:ζ
1815:−
1802:ω
1792:ℏ
1771:Δ
1699:ϕ
1679:ζ
1659:ζ
1628:ω
1596:ℏ
1588:ω
1572:^
1548:ω
1531:−
1516:ℏ
1508:ω
1493:†
1486:^
1456:ω
1430:†
1423:^
1393:^
1349:†
1342:^
1332:ζ
1329:−
1317:^
1305:∗
1301:ζ
1277:
1265:ζ
1256:^
1234:^
1223:∗
1219:α
1215:−
1210:†
1203:^
1193:α
1187:
1175:α
1166:^
1130:ζ
1097:α
1067:⟩
1036:⟩
1022:α
1013:^
1001:ζ
992:^
980:⟩
977:α
966:ζ
957:^
945:⟩
942:ζ
936:α
813:^
795:^
591:−
579:−
571:
549:ψ
520:ℏ
452:Δ
448:≈
436:Δ
408:The term
369:Δ
353:Δ
323:Δ
307:Δ
287:⟩
284:α
252:⟩
199:≥
193:Δ
187:Δ
160:ℏ
155:≥
149:Δ
143:Δ
126:operators
7624:18851426
7354:20867356
7301:10045587
7243:20867358
7189:27661673
7138:38230188
7130:24785048
7077:67780562
7023:29467218
7015:26991175
6954:26751056
6846:10034306
6699:23683187
6603:13815567
6595:16384378
6542:10045765
6437:10035582
6198:14213781
6111:: 1–51.
6090:27661673
6021:18232978
5959:10033788
5772:See also
3610:Examples
1713:, where
738:momentum
7632:1307950
7604:Bibcode
7571:9784123
7551:Bibcode
7543:Science
7508:Bibcode
7475:9909547
7445:Bibcode
7395:Bibcode
7362:4514687
7334:Bibcode
7281:Bibcode
7251:9929939
7223:Bibcode
7161:Bibcode
7153:Physics
7110:Bibcode
7057:Bibcode
6995:Bibcode
6962:2139293
6934:Bibcode
6891:Bibcode
6826:Bibcode
6803:9910869
6783:Bibcode
6740:Bibcode
6707:3566080
6679:Bibcode
6646:9895863
6626:Bibcode
6575:Bibcode
6522:Bibcode
6478:Bibcode
6417:Bibcode
6382:Bibcode
6301:4259166
6281:Bibcode
6123:Bibcode
6062:Bibcode
6029:3938634
5993:Bibcode
5937:Bibcode
5891:4325386
5863:Bibcode
3824:of the
3759:values.
3537:, with
1145:is the
1112:is the
1079:is the
850:equals
419:ellipse
264:of the
178:and
23:physics
7630:
7622:
7569:
7526:
7473:
7413:
7360:
7352:
7299:
7249:
7241:
7187:
7136:
7128:
7075:
7021:
7013:
6960:
6952:
6926:Nature
6909:
6844:
6801:
6758:
6705:
6697:
6644:
6601:
6593:
6540:
6435:
6299:
6273:Nature
6196:
6141:
6088:
6027:
6019:
5957:
5889:
5881:
5855:Nature
5824:
5695:phonon
5606:lasers
5126:. The
4968:where
4870:GEO600
4866:AdLIGO
4134:et al.
3839:vacuum
3738:photon
3713:, the
3339:where
3096:Since
1379:where
1051:where
673:where
230:Wigner
7628:S2CID
7594:arXiv
7524:S2CID
7498:arXiv
7433:(PDF)
7411:S2CID
7385:arXiv
7358:S2CID
7324:arXiv
7269:(PDF)
7247:S2CID
7213:arXiv
7134:S2CID
7100:arXiv
7073:S2CID
7047:arXiv
7019:S2CID
6985:arXiv
6958:S2CID
6907:S2CID
6881:arXiv
6756:S2CID
6730:arXiv
6703:S2CID
6669:arXiv
6599:S2CID
6565:arXiv
6510:(PDF)
6468:arXiv
6353:arXiv
6332:arXiv
6297:S2CID
6269:(PDF)
6194:S2CID
6176:arXiv
6139:S2CID
6113:arXiv
6050:(PDF)
6025:S2CID
5983:arXiv
5925:(PDF)
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