AAWSAP DIRD, Quantum Tomography of Negative Energy States in the Vacuum, January 2011

UNCLASSIFIED/ /FOA QFFI@IAL l:l!H! 9HL I Defense Intelligence Reference Document Defense Futures 11 January 2011 !COD: 10 August 2010 DIA-08-1102-007 Quantum Tomography of Negative Energy States in the Vacuum UNCLASSIFIED// FOR OPPICIJ!ct l:ISE 8P..,¥ UNCLASSIFIED//FOA OlifilGIAk WSE 8HLV Quantum Tomography of Negative Energy States in the Vacuum The Defense Intelligence Reference Document provides non-substantive but authoritative reference information related to intelligence topics or methodologies. Prepared by: Technology Warning Division (DW0-4) Defense Warning Office Directorate for Analysis Defense Intelligence Agency Author: AAP Person 58 COPYRIGHT WARNING: Further dissemination of the photographs in this publication is not authorized. This product is one of a series of advanced technology reports produced in FY 2010 under the Defense Intelligence Agency, Defense Warning Office's Advanced Aerospace Weapons System Applications AAWSA Pro ram. Comments or questions pertaining to this document should be addressed to MP Person 1 AAWSA Program Manager, Defense Intelligence Agency, ATTN: JUIAF ­ ashington D.C. 20340-5100. UNCLASSIFIED/}FQA QFFIGl,t.L: W&ii 9tlb¥ ii UNCLASSIFIED//EOR OFFIEIAk W&liii 8PtLY Contents Introduction ........................................................................................................... 1 REVIEW OF NEGATIVE (or SUB-VACUUM) ENERGY ................................................. 3 Overview ........................................................................................................ 3 Examples of Negative (Sub-Vacuum) Energy Found in Nature ....................... 4 Basic Notions of the Quantum Field Theory of Light ................................... 5 Basic Notions on the Origin of the Quantum Vacuum Zero-Point Fluctuations ............................................................................................... 7 Negative (Sub-Vacuum) Energy in Squeezed Light .................................... 8 Negative (Sub-Vacuum) Energy in the Casimir Effect................................13 QUANTUM OPTICAL HOMODYNE TOMOGRAPHY .................................................... 15 Observing Negative Energy in the Lab .......................................................... 15 Basic Notions of Quantum Optical Homodyne Tomography .......................... 16 Wigner Functions ..................................................................................... 17 Beam Splitters.......................................................................................... 24 Photodiodes ............................................................................................. 27 Balanced Homodyne Detection ................................................................. 27 Outline of Experimental Procedure........................................................... 32 BALANCED HOMODYNE SYSTEMS FOR MEASURING NEGATIVE (SUB-VACUUM) ENERGY ....................................................................................... 33 Time-Domain Balanced Homodyne System .................................................. 33 Balanced Homodyne System for Casimir Cavities ......................................... 36 CONCLUSION........................................................................................................ 43 ACKNOWLEDGEMENTS ......................................................................................... 45 REFERENCES ........................................................................................................ 46 UNCLASSIFIED// FOR OFFICIAL U:!I! fJHLY iii UNCLASSIFIED//FOA QFFI@IAL l:l!H! 9HLY Figures Figure 1. Illustration of a Squeezed State of Light ............................................... 13 Figure 2. Schematic of the Casimir Effect ............................................................. 14 Figure 3. Illustration of Quantum Optical Homodyne Tomography ....................... 16 Figure 4. Wigner Function for a Vacuum and for a Coherent State ....................... 19 Figure 5. Wigner Function of a Squeezed Vacuum ................................................ 20 Figure 6. Wigner Function of a Single Photon....................................................... 21 Figure 7. Quantum Tomography of Schrodinger-Cat States .................................. 22 Figure 8. Schematic of an Ideal Lossless Beam Splitter........................................ 25 Figure 9. Illustration of a Fictitious Beam Splitter ................................................ 26 Figure 10. Schematic of a Balanced Homodyne Detector...................................... 29 Figure 11. Balanced Homodyne Detector Using Fictitious Beam Splitters ............. 31 Figure 12. Balanced Homodyne Detector Using A Single Effective Fictitious Beam Splitter ................................................................................................................. 32 Figure 13. Time-Domain Balanced Homodyne Detector........................................ 34 Figure 14. Experimentally Measured Squeezed State ........................................... 35 Figure 15. Balanced Homodyne Detector with a Local Oscillator .......................... 38 Figure 16. Diagram of Casimir Cavity with BHD Photodiodes ............................... 40 Figure 17. Experimental Setup of BHD Photodiodes and LO Field ......................... 40 Figure 18. Detailed Schematic of Experimental BHD Apparatus ........................... 41 Figure 19. Predicted Casimir Spectral Density...................................................... 41 Figure 20. Predicted Suppression of Vacuum Fluctuations in dB.......................... 42 UNCLASSIFIED//EOR OEEICl.1.1. Uiliii ONI.¥ iv UNCLASSIFIED/ /FOR. OFFICIAL tt.!I!! OHL¥ Quantum Tomography of Negative Energy States in the Vacuum Introduction Future aerospace vehicles could have an advanced propulsion system that uses negative quantum vacuum energy to modify the spacetime geometry in the immediate vicinity surrounding the vehicle in order to induce faster-than-light motion via traversable wormholes or warp drives, or even levitation via antigravity [1, 2]. These exotic propulsion concepts are well-known in mainstream general relativity and quantum field theory research. The notion of a physical state with negative energy is not familiar in the realm of classical physics. However, it is not rare in quantum field theory to have quantum states with negative energy density or a negative energy flux. Even for a quantum scalar field in the flat Minkowski spacetime, it can be proved that the existence of quantum states with negative energy density is inevitable [3]. Although all known forms of classical matter have non-negative energy density, it is not so in quantum field theory. A general quantum state can be a superposition of particle number eigenstates and may have a negative expectation value of energy density in certain spacetime regions due to quantum coherence effects [3]. These considerations remain true even for quantum fields in a curved spacetime where the effects of gravitational fields, or equivalently, accelerations, can be observed due to the mass of astronomical bodies or the motions of astronomical bodies. There are two key examples of specially prepared quantum vacuum states that are known to produce small amounts of negative energy density in the laboratory. These are the well-known Casimir effect and the squeezed vacuum states of the electromagnetic field. The former is a static quantum vacuum effect wh ile the latter is a time-domain quantum vacuum effect. There are several other examples of special quantum vacuum or particle states that produce negative energy density, but they are beyond the scope of this report because they remain mathematical curiosities or are not practicable to im plement in the laboratory in the foreseeable future. We already make small amounts of negative energy in the laboratory via the Casimir effect and squeezed electromagnetic vacuum states, but we do not yet know if we can access larger amounts for extended periods of time over extended spatial distributions for the purpose of modifying spacetime for aerospace propulsion applications. It will be necessary to first explore the quantum nature of the Casimir effect and squeezed electromagnetic vacuum states to determine whether we can measure and spatially map their negative energy density. This is a necessary first step to take before beginning any study on producing large quantities of negative energy because we will first need to know how to measure and spatially map negative energy in order to properly control it after producing it. This is the motivation for this report. We need to firm up our understanding of how lab detectors will respond to negative energy in situ. A first step in this direction was already taken by Hansen et al. [4] in 2001 for the time-domain negative energy pulses in squeezed electromagnetic vacuum states, and more recently Marecki [5, 6] generalized the analysis of the output of balanced homodyne detectors (BHDs) for the case of static negative energy states UNCLASSIFIED// FOR OFFICIAL U.!I!! or~L' 1 UNCLASSIFIED/}FOA OFFl&IAl WSE er•tv inside Casimir cavities. The most important feature of these devices is their ability to quantify the quantum vacuum fluctuations of the electric field because the output of BHDs provides information on the one- and two-point functions of arbitrary states of quantum fields. Marecki computed the two-point function and the associated spectral density for the ground state of the quantum electric field in Casimir geometries, and predicts a position- and frequency-dependent pattern of BHD responses if a device of this type is placed inside a Casimir cavity. The proposed device allows for the direct detection of quantum vacuum fluctuations and provides a spatial mapping of the negative energy contained inside the cavity, which will be summarized in this report. UNCLASSIFIED//FOR OFFICIAL U.!! 8HLY 2 UNCLASSIFIED//iOR QFFl&IAL l:ISI!!! Brit I REVIEW OF NEGATIVE (or SUB-VACUUM) ENERGY Overview The implementation of faster-than-light (FTL) interstellar travel via traversable wormholes or warp drives or other antigravity forces for propulsion, generally requires the engineering of spacetime into very specialized local geometries surrounding the immediate vicinity of the aerospace vehicle undergoing this type of motion. The analysis of these via the general relativistic field equation plus the resultant source matter equations of state demonstrates that such geometries require the use of "exotic" matter in order to produce the requisite FTL or antigravity spacetime modification. Exotic matter is generally defined by general relativity physics to be matter that possesses (renormalized) negative energy density (sometimes negative stress-tension = outward pressure, a.k.a. gravitational repulsion or antigravity), and this is a very misunderstood and misapplied term by the non-general relativity community. We clear up this misconception by defining what negative energy is, where it can be found in nature, and we also review the two primary experimental concepts that are known to produce negative energy in the laboratory. Also, it has been claimed that FTL and antigravity spacetimes are not plausible because exotic matter violates the general relativistic energy conditions. However, it has been shown that this is a spurious issue. The identification, magnitude, and production of exotic matter is seen to be a key technical challenge, however. FTL and antigravity spacetimes also possess features that challenge the notions of causality and there are alleged constraints placed upon them by quantum effects. Reference [1] reviews and summarizes these issues with an assessment on the present state of their resolution. What exactly is "exotic" matter? In classical physics the energy density of all observed forms of matter (fields) is non-negative. What is exotic about the type of matter that must be used to produce traversable wormhole, warp drive, or antigravity spacetimes is that it must have negative energy density and/or negative flux [7]. The energy density is "negative" in the sense that the configuration of matter fields we must deploy to produce a traversable wormhole, warp drive, or antigravity effect must have an energy density, PE (= pc2, where pis the rest-mass density), that is less than or equal to its pressures/tensions, pi [8, 9].* In many cases, these equations of state are also known to possess an energy density that is algebraically negative, i.e., the energy density and flux are less than zero. It is on the basis of these conditions that we call this material property "exotic." The condition for ordinary, classical (non-exotic) forms of matter that we are all familiar with in nature is that PE> pi and/or PE;:::,: 0. These conditions represent two examples of what are variously called the "standard" energy conditions which are computed from the trace of the matter stress-energy tensort : Weak Energy Condition (WEC: PE;:::,: 0, PE+ Pi;:::,: 0), Null Energy Condition (NEC: PE+ Pi;:::,: 0), Dominant Energy Condition (DEC), and Strong Energy Condition (SEC). These energy conditions forbid negative energy density between material objects to occur in nature, but they are mere hypotheses. Hawking and Ellis [10] formulated the energy conditions in order to establish a series of mathematical hypotheses governing the behavior of • From this point forward, all Latin letters (e.g., i, j, k = 1...3) that appear as indices on physical quantities denote the usual 3-dimensional space coordinates, x• ...x3, indicating the spatial components of vector or tensor quantities. t The stress-energy-momentum tensor is a matrix quantity that encodes the density and flux of energy and momentum for any type of matter under study. UNCLASSIFIED/;<fQA OFFI@IAL l:ISf! 9Htl' 3 UNCLASSIFIED//POI': orr1c11tt U.!I! Oflt I collapsed-matter singularities in their study of cosmology and black hole physics. More specifically, classical general relativity allows one to prove lots of general theorems about the behavior of matter in gravitational fields. However, real physical matter is not "reasonable" because the energy conditions are in general violated by semiclassical quantum effects ( occurring at order Tl) [9].* More specifically, quantum effects generically violate the average NEC (ANEC). Furthermore, it was discovered in 1965 that quantum field theory has the remarkable property of allowing states of matter containing local regions of negative energy density or negative fluxes [3]. This violates the WEC, which postulates that the local energy density is non­ negative for all observers. And there are also general theorems of differential geometry that guarantee that there must be a violation of one, some, or all of the energy conditions (meaning exotic matter is present) for all FTL and antigravity spacetimes. However, all of the energy condition hypotheses have been experimentally tested in the laboratory and experimentally shown to be false - 25 years before their formulation [11]. In quantum field theory, negative energy is a manifestation of what is now called the "sub-vacuum" levels of the quantum zero-point (or vacuum ground state) fluctuations that correspond to any particular quantum field of matter under study. Hence, the energy corresponding to sub-vacuum quantum fluctuations is now called "sub-vacuum energy": sub-vacuum energy= negative energy. Further investigation into this technical issue showed that violations of the energy conditions are widespread for all forms of both "reasonable" classical and quantum matter [12-16]. Furthermore, Visser [9) showed that all (generic) spacetime geometries violate all the energy conditions. So the condition that PE> Pi and/or PE~ 0 must be obeyed by all forms of matter in nature is spurious. Negative energy has been produced in the laboratory and this will be discussed in the following sections. Examples of Negative (Sub-Vacuum) Energy Found in Nature The exotic (energy condition-violating) fields that are known to occur in nature are: 1. Static, radially-dependent electric or magnetic fields. These are borderline exotic, if their tension were infinitesimally larger, for a given energy density [10, 17]. 2. Squeezed quantum vacuum states: electromagnetic and other (non-Maxwellian) quantum fields [8, 18]. 3. Gravitationally squeezed electromagnetic vacuum fluctuations [19). 4. Casimir effect, i.e., the Casimir vacuum in flat, curved, and topological spaces [20-28). 5. Other quantum fields/states/effects. In general, the local energy density in quantum field theory can be negative due to quantum coherence effects [3]. Other examples that have been studied are Dirac field states: the superposition of two single particle electron states and the superposition of two multi-electron­ positron states [29, 30]. In the former (latter), the energy densities can be negative when two single (multi-) particle states have the same number of * Planck's reduced constant, TJ "' 1.055 x 10-34 J.s. UNCLASSIFIED//FOA QFFI&IAL 1:1!11! Oflt I 4 UNCLASSIFIED//EAR AfifilCIAk Wlili 9PtLY electrons (electrons and positrons) or when one state has one more electron (electron-positron pair) than the other. Cosmological inflation [9], cosmological particle production [9], classical scalar fields [9], the conformal anomaly [9], and gravitational vacuum polarization [12-15] are among many other examples that also violate the energy conditions. Since the laws of quantum field theory place no strong restrictions on negative energies and fluxes, then it might be possible to produce exotic phenomena such as faster-than-light travel [31­ 33], traversable wormholes [8, 9, 34], violations of the second law of thermodynamics [35, 36], and time machines [9, 34, 37]. There are several other exotic phenomena made possible by the effects of negative energy, but they lie outside the scope of this report. In what follows, we consider only items 2 and 4 in the previous list for the purpose of this report due to their ready applicability and technical maturity. We will not examine the other items in the list because they are theoretical curiosities that remain under study by investigators. Basic Notions of the Quantum Field Theory of Light Before going further, it will be helpful to briefly outline the basic notions and terminology of the quantum field theory of light (i.e., quantum optics) because the content of this report focuses on those aspects. Classically, light is electromagnetic radiation that can be pictured as waves flowing through space at the speed of light, c (= 3.0 x 108 m/s). The waves are not waves of anything substantive, but are in fact ripples in the state of a field. These waves carry energy, and each wave has a specific direction, frequency and polarization state. This is called a "propagating mode of the electromagnetic field." A simple model for this is the electromagnetic oscillator. One complex-valued vector function u(x,t) called a spatial-temporal mode comprises all classical wave aspects including polarization. The simplest example of a spatial-temporal mode is a plane wave u(x,t) = u0 exp[i(kx-wt)] of polarization vector uo, angular frequency co, and wave vector k (definition: k2 =u}/c2), where i is the unit complex number, and x is the space coordinate and tis the time coordinate. This mode defines a framework in space and time that may be excited by the quantum field "light." The mode function quantifies the strength of one excitation in space and time. Also, the mode function obeys the laws of classical waves given by Maxwell's equations of electrodynamics. The choice of u(x,t) is made by the observer. The observer singles out one mode, one quantum object from the rest of the world to make a specific observation or measurement. This object turns out to be a harmonic oscillator described by the annihilation operator a. A useful tool for modeling the propagating mode of the electromagnetic field in quantum mechanics is the ideal quantum mechanical harmonic oscillator: a hypothetical charged mass on a perfect spring oscillating back and forth under the action of the spring's restoring force. The operator astands for the quantized amplitude with which u(x,t) can be excited. In classical optics it would be just a complex number a of magnitude lal and phase arg(a) . The quantized amplitude a is neither predetermined nor given by the observer UNCLASSIFIED//FOR OFFICIICL tl.!l!! 9HLY 5 UNCLASSIFIED//FOR 8FFl&IAk le:ISE 8Hk¥ but depends on the state of u(x,t). This state exists even if literally nothing is in the mode chosen by the observer. In this case, the light is just in the vacuum state.§ However, this "nothing" can indeed cause significant physical effects as will be discussed in later sections. To make all this more precise, we postulate that the electric field strength Eof the light field is given by E=u*(x, t)a +u(x, t)a1 and that the amplitude operator a is a bosonic** annihilation operator that obeys the quantum mechanical commutation relation [a, at]= 1, where u*(x,t) is the complex conjugate of u(x,t) and at is the adjoint (or conjugate) of acalled the creation operator.tt The hat symbol appearing over quantities denotes that they are quantum operators (or observables). Another key element of quantum-oscillator physics is the photon number operator fi, which accounts for the number of photons (quantized light particles) in the chosen u(x,t) and is given by the quantum mechanical counterpart of a classical modulus-squared amplitude: n= a t cz. Let us now introduce a pair of operators, q and p, called quadratures. They are defined as q= 2- 112 (at+ a) and jJ = i2- I12 ( at - a) , which can be inverted to provide the additional useful definitions a= T 112 ( q+ip) and at =T I12 ( q- ip). In optics q and jJ correspond to the in-phase and the out-of-phase component of the electric field amplitude of u(x,t) (with respect to a reference phase). The bosonic commutation relation demonstrates that q and /J are canonically conjugate observables, [q, p] =ih. The quadratures q and p can be regarded as the position and the momentum of the quantum electromagnetic oscillator. They do not appear in real space but in the phase space spanned by the complex vibrational amplitude aof the quantum electromagnetic oscillator, and they have nothing to do with the position and the momentum of a photon. However, the canonical commutation relation entitles us to treat q and p as perfect examples of position- and momentum-like quantities in quantum optics. Finally, we express the photon number operator n in terms of the quadratures q and p and obtain, using the bosonic commutation relation, the standard Hamiltonian (or total energy) of the quantum harmonic ( electromagnetic) oscillator with unit mass and frequency: " - " 1 H ose= n+2 (1) § Here we always mean by "vacuum" simply "no light" and not an evacuated system. •· Boson or bosonic refers to quantum particles that have integer quantum spin . ., In quantum mechanics, the vacuum is defined to be a state of no (or zero) particles and is denoted by the quantum state eigenvector I0) . By definition a "annihilates" the vacuum state: aI0) =0 . UNCLASSIFIED//FOA OFFIGIAk lel&E 8Ptk\S 6 UNCLASSIFIED//POI\ OPPICIAL YSIE 8Htlf where the first and second terms in the second line are the kinetic and potential energies of the oscillator, respectively. The additional 1/2 appearing in the first line of Eq. (1) is called the vacuum zero-point energy for the reason to be explained in the next section . The first line of Eq. (1) is more commonly expressed in units of energy (Joules) in quantum mechanics, which is obtained simply by multiplying the right-hand side by the photon energy nw so that H ose =hw (ii +½) . It is beyond the scope of this report to elaborate further on the entire subject of the quantum optics. The reader should consult Reference [38] for more information. Basic Notions on the Origin of the Quantum Vacuum Zero-Point Fluctuations Here we discuss the basic notions of the quantum vacuum zero-point fluctuations (ZPF), which is an important feature in quantum optics. The origin of the ZPF is attributed to the Heisenberg Uncertainty Principle. According to this principle, q and pare any two conjugate observables that we are interested in measuring, and they obey the commutation relation already shown in the previous section. Their corresponding uncertainty relation is !1q!1p?:. h/2, where !1q is the variance (a.k.a. uncertainty) of observable q and !1p is that of the conjugate observable p. This relation states that if one measures observable q with very high precision (i.e., its uncertainty !1q is very small), then a simultaneous measurement of observable p will be less precise (i.e., its uncertainty !1p is very large), and vice versa . In other words, it is not possible to simultaneously measure two conjugate observable quantities with infinite precision. This minimum uncertainty is not due to any correctable flaws in measurement, but rather reflects the intrinsic fuzziness in the quantum nature of energy and matter. Substantial theoretical and experimental work has shown that in many quantum systems the limits to measurement precision is imposed by the quantum vacuum ZPF embodied within the uncertainty principle. Nowadays we rather see the Heisenberg Uncertainty Principle as a necessary consequence, and therefore, a derived result of the wave nature of quantum phenomena. The uncertainties are just a consequence of the Fourier nature of conjugate pairs of quantities (observables). For example, the two Fourier-wave-conjugates time and frequency become the pair of quantum-particle conjugates time and energy and the two Fourier-wave-conjugates displacement and wave number become the pair of quantum-particle conjugates position and momentum. The Heisenberg Uncertainty Principle dictates that a quantized electromagnetic oscillator (a.k.a. a photon state) can never come entirely to rest, since that would be a state of exactly zero energy, which is forbidden by the commutation relation given in the previous section. Instead, every mode of the field has liw/2 as its average minimum energy in the vacuum, and this is called the zero-point energy (ZPE) .** This ZPE term is added to the classical blackbody spectral radiation energy density p(w)dro [i.e., the energy per unit volume of radiation in the frequency interval (ro,w + dro)] [25]: *" hw is the energy of a single mode (or photon). UNCLASSIFIED//EOA QFFI&IAL YSE!! 9HL I 7 UNCLASSIFIED//FOR OFFICl"L tJ.!l!!! 8HL¥ oi [ hro hco] p(ro)dro = - ----- + - dco rc2c3 exp(hco/ k8T) - 1 2 (2) 3 = - hro-2 3 coth ( -- hro Jdro, 2n c 2k8 T where kp, is Boltzmann's constant (1. 3807 x 10-23 J/K) and Tis the absolute temperature. The factor outside the square brackets in the first line of Eq. (2) is the density of mode (or photon) states (i.e., the number of states per unit frequency interval per unit volume); the first term inside the square brackets is the standard Planck blackbody radiation energy per mode; and the second term inside the square brackets is the quantum zero-point energy per mode. Equation (2) is called the Zero­ Point Planck (ZPP) spectral radiation energy density. Planck first added the ZPE term to the classical blackbody spectral radiation energy density in 1912, although it was Einstein, Hopf, and Stern who actually recognized the physical significance of this term in 1913 [25]. Direct spectroscopic evidence for the reality of ZPE was provided by Mulliken's boron monoxide spectral band experiments in 1924, several months before Heisenberg first derived the ZPE for a harmonic oscillator from his new quantum matrix mechanics theory [39]. Following this line of reasoning, quantum physics predicts that all of space must be filled with quantum electromagnetic ZPF creating a universal sea of zero-point energy. The other quantum forces of nature also have their own vacuum ZPF which contributes to the universal sea of zero-point energy. But that is beyond the scope of this report. Negative (Sub-Vacuum) Energy in Squeezed Light Substantial theoretical and experimental work has shown that in many quantum systems the limits to measurement precision imposed by the quantum vacuum ZPF can be breached by decreasing the noise in one observable (or measurable quantity) at the expense of increasing the noise in the conjugate observable; at the same time the variations in the first observable, say the energy, are reduced below the ZPF such that the energy becomes "negative." "Squeezing" is thus the control of quantum fluctuations and corresponding uncertainties, whereby one can squeeze/reduce the variance of one (physically important) observable quantity provided the variance in the (physically unimportant) conjugate variable is stretched/increased. The squeezed quantity possesses an unusually low variance, meaning less variance than would be expected on the basis of the equipartition theorem . One can in principle exploit quantum squeezing to extract energy from one place in the ordinary vacuum at the expense of accumulating excess energy elsewhere [8]. The squeezed state of the electromagnetic field is a primary example of a quantum field that has negative energy density and negative energy flux. Such a state became a physical reality in the laboratory as a result of the nonlinear-optics technique of "squeezing," i.e., of moving some of the quantum-fluctuations of laser light out of the UNCLASSIFIED//FOA OFFI&iIAL YSIE 8HLY 8 UNCLASSIFIED//FOR 8FFI@IAL 1::191!! 8HLY cos[w(t- z!c)] part of the beam and into the sin[w(t- z!c)] part [18, 40-44].§§ The observable that gets squeezed will have its fluctuations reduced below the vacuum ZPF. The act of squeezing transforms the phase space circular noise profile characteristic of the vacuum into an ellipse, whose semimajor and semiminor axes are given by unequal quadrature uncertainties (of the quantized electromagnetic oscillator operators). This applies to coherent states in general, and the usual vacuum is also a coherent state with eigenvalue zero. As this ellipse rotates about the origin with angular frequency w, these unequal quadrature uncertainties manifest themselves in the electromagnetic field oscillator energy by periodic occurrences, which are separated by one quarter cycle, of both smaller and larger fluctuations compared to the unsqueezed vacuum. We digress momentarily by noting that coherent states, also called Glauber states, are the eigenstates of the annihilation operator a: ala) =ala), (3) which have well-defined amplitudes lal and phases arg(a) (recall the discussion in Sect. IIB-1). They are called coherent states because light fields in these states are perfectly coherent, and high-quality lasers generate such fields. This is an important reason why high-quality laser light is an excellent tool for experimental quantum optics. Coherent states come as close as quantum mechanics allows to wave-like states of the electromagnetic oscillator. Because the wave aspects of light are commonly regarded as classical, coherent states are often called classical states. Furthermore, fields in statistical mixtures of coherent states (such as thermal fields) are classical as well, whereas any state that cannot be understood as an ensemble of coherent states is called nonclassica/. The experimental generation and application of nonclassical light fields is the main subject of this report. Despite much recent progress, producing nonclassica l states of light is still extremely challenging because they are easily destroyed (reduced to classical) by any kind of losses. Furthermore, it turns out that the vacuum is a coherent state as well because it satisfies Eq. (3) for a= 0. In other words, the vacuum is a zero-amplitude coherent state. With a little algebra we see directly from Eq. (3) that the mean (i.e., quantum expectation value of the) energy of a coherent state with unit frequency is \Ha)= (alata + ½la) (4) =lal2 +½- Equation ( 4) is the sum of the classical wave intensity lal2 and the vacuum zero-point energy 1/2. One simply multiplies the right-hand side of Eq. ( 4) by liw to put (Ha) into units of energy. §§ z denotes the z-axis direction of beam propagation. UNCLASSIFIED/; PO" OPPICIICL U.!! OHL I' 9 UNCLASSIFIED//EAR AfifilEIAk WSE er•tv Morris and Thorne [8] and Caves [45] point out that if one squeezes the vacuum, i.e., if one puts vacuum rather than laser light into the input port of a squeezing device, then one gets at the output an electromagnetic field with weaker fluctuations and thus less energy density than the vacuum at locations where cos2[co(t- z/c)] ~ l and sin2 [ co(t- z/c)] < < 1; but with greater fluctuations and thus greater energy density than the vacuum at locations where cos2[co(t-z/c)] << 1 and sin2 [ co(t - z/c)]~1. Since the vacuum is defined to have vanishing energy density, any region with less energy density than the vacuum actually has a negative (renormalized) expectation value for the energy density. Therefore, a squeezed vacuum state consists of a traveling electromagnetic wave that oscillates back and forth between negative energy density and positive energy density, but has positive time-averaged energy density. In quantum optics the squeezed state is generated by the unitary squeezing operator: (5) where sis a real number that parameterizes the deviation of the variances !),,q and !),,p from their vacuum values and is called the squeezing parameter. From Eq. (5) we obtain the squeezed vacuum state lcp) = S(~)IO). The squeezing operator S(~) is simply an evolution operator that describes the result of the nonlinear squeezing interaction Hamiltonian H im =x( b*a2 - bat2 ). The squeezing parameter~ contains the product of the amplitude b, the coupling constant X, and the interaction time. But this is not the entire story. Since we will be dealing with high-quality lasers in what follows, we also need to know about another important quantum optics operator that acts on coherent states. We introduce the unitary displacement operator D(a)=exp(aa: -a·a). D(a) displaces the amplitude aby the complex number a according to b\a)aD(a)=a + a. To show why D(a) has anything to do with coherent states, we apply a negative displacement to la). From the basic property of D(a), we see that abc-a) la) =D(-a)Dt(-a)GD(-a) la) = be- a) (a- a)Ia) (6) = 0. Equation (6) equals zero because of the definition Eq. (3) of coherent states. This result implies that D(-a)la) = I0), which is the vacuum state. Therefore, coherent states Ja) are displaced vacua la) = D(a)J0). This does not mean that coherent states UNCLASSIFIED/i PO" OPPICIICL U.!! OHL I' 10 UNCLASSIFIED//FOA QFFIEJIAL YSI!! f>Ht'I" are physically similar to vacuum states, but instead they have only some quantum noise properties in common. It is a well known result in the quantum field theory of light that the vacuum wave function is a simple Gaussian function of the quadratures (in either q or jJ representation), and thus coherent states are also Gaussian [38]. Furthermore, a proof of Heisenberg's Uncertainty Principle in conjunction with the application of S(~) and D(a) on the quadrature variances and wave functions showed that all minimum uncertainty states are displaced Gaussian states such that they have displaced rescaled vacuum wave functions. Consequently, all minimum uncertainty states are displaced squeezed vacua [18, 38]: I\jf) = fJ ca)scs) Io) . (7) The squeezing interaction Hi"' is realized by the degenerate parametric amplification of the spatial-temporal mode. A crystal such as potassium titanyl phosphate (KTP) or lithium niobate (LiNbQ3) is pumped by another laser beam with amplitude band twice the frequency of the spatial-temporal mode (with amplitude a) of interest. According to H ;m , the "B" photons (corresponding to b) of the pump beam are converted into pairs of "A" signal photons (corresponding to a.2 and a,t2) with a probability that depends on the coupling constant X· The KTP or LiNb03 crystal acts like an electromagnetic swing, and the pump modulates the oscillation of the "A" mode at twice its frequency. The pump amplifies the signal parametrically much as a swing is amplified by changing the effective length at twice the frequency of the swing. A classical swing relies on tiny initial fluctuations (or "wobbles") that are in-phase with respect to the parametric pump. In this way, the tiny fluctuations are amplified; the swing starts to oscillate. A quantum swing like the degenerate parametric amplifier experiences at least the vacuum fluctuations from the very beginning. Vacuum fluctuations that are in-phase with respect to the pump are amplified, whereas out-of-phase fluctuations get de­ amplified or, in other words, squeezed. A squeezed vacuum requires a pump for generation, and, hence, when produced it carries energy. The nonlinear crystal KTP or LiNb03 is a resonator that is shaped like a cylinder with rounded silvered ends to reflect light. This resonator acts to produce a secondary lower frequency light beam in which the pattern of photons is rearranged into pairs. The squeezed light emerging from the resonator will contain pulses of negative energy interspersed with pulses of positive energy. To quantify the amount of squeezing energy we 1) apply S(~) to the quadratures and find that it scales their eigenfunctions;*** 2) we then substitute for aits quadrature decomposition (given in Sect. IIB-1) and substitute that result into the scaled quadratures; and then 3) do further algebra to derive how S(~) changes a: stcs)aS(s) =Gcoshs- atsinhs. We substitute this last result into Eq. (1) and use Eq. (7) to calculate the quantum expectation value in order to express the mean energy of a squeezed state, and obtain ,.. i.e., q gets squeezed and p gets stretched. UNCLASSIFIED// FOR OFFICIAL H.!l! 8PU:.lf 11 UNCLASSIFIED//FOA QFFI@IAL l:191! t>flt I (8) Equation (8) really describes the mean photon number of a single mode in a squeezed state, but one simply multiplies the right-hand side by /iro to get the mean energy (Asqvac ) = hw(la.12 + ½+sinh2~ )- We see in Eq. (8) that there are three terms contributing to the energy: the first term accounts for the coherent energy given by lal2, the second term is the vacuum zero-point energy 1/2, and the third term quantifies the fluctuation energy of squeezed states. The contribution to this squeezing energy originally comes from the pump used to generate the squeezed light. It is stored in the enhanced fluctuations of the anti-squeezed component. Because both the squeezed and the anti-squeezed quadratures contribute to the second line in Eq. (1), even a squeezed vacuum carries energy. However, Eq. (8) is not the final result because it only gives the mean energy of a single mode in a squeezed state, while lasers and nonlinear crystal resonators produce a very large number of modes. Equation (8) needs to be summed (integrated) over the infinite number of possible modes; it must then be "renormalized" by sophisticated mathematical techniques in order to get rid of the divergent (infinite) contribution from the vacuum zero-point energy (a byproduct of taking an infinite sum of modes); and then the result must be converted into units of energy density by dividing it by an appropriate volume element, because Einstein's general theory of relativity requires an energy density (or pressure, both are in the same units) to induce spacetime bending. The final result we seek is the energy density, PE-sqvac, given by Pfenning [ 46]: Pe.sq= = ( 21,co) sinh c; [sinh c; +cos hi:; cos ( 2ro(t - z/c) +8)] (J / m3) , (9) where L 3 is the volume of a large box with sides of length L (i.e., we put the quantum field in a box with periodic boundary conditions) and 8 is the phase of squeezing. Equation (9) shows that PE-sqvac falls below zero once every cycle when the condition cosh~ > sinh~ is met. It turns out that this is always true for every nonzero value of ; , so PE-sqvac becomes negative at some point in the cycle for a general squeezed vacuum state. See Figure 1 for an illustration. Note in the figure that the blue troughs or valleys are the negative energy pulses. On another note, when a quantum state is close to a squeezed vacuum state, there will almost always be some negative energy densities present. Another way to generate negative energy via squeezed light would be to manufacture extremely reliable light pulses containing precisely one, two, three, etc., photons apiece and combine them together to create squeezed states to order. Superimposing many such states could theoretically produce bursts of intense negative energy. Photonic crystal research has already demonstrated the feasibility of using photonic crystal waveguides (mixing together the classical and quantum properties of optical materials) to engineer light sources that produce beams containing precisely one, two, three, etc., photons. See Reference [1] for more details and for the references cited therein. UNCLASSIFIED//fOtt 8Ffl@IAL Y§E 8,.LY 12 UNCLASSIFIED//FOR QFFl&IAL l:ISI!!! 8flt I ENERGY SQUEEZED STATE DENSITY 0 POSITION Figure 1. Illustration of a Squeezed State of Light. (courtesy of Lisa Burnett) Negative (Sub-Vacuum) Energy in the Casimir Effect The Casimir effect originates from the quantum electromagnetic vacuum ZPF. It is by far the easiest and most well known way to generate (static) negative energy in the lab. The Casimir effect that is familiar to most people is the force that is associated with the quantum vacuum electromagnetic ZPF [47]. This is an attractive force that must exist between any two neutral (uncharged), parallel, flat, conducting surfaces (e.g., metallic plates) in a vacuum. This force has been well measured and it can be attributed to a minute imbalance in the vacuum electromagnetic ZPE density inside the cavity between the conducting surfaces versus the vacuum electromagnetic ZPE density in the free-space region outside of the cavity [ 48-50]. See Figure 2 for a schematic of the Casimir effect. UNCLASSIFIED//FOR QFFl&IAL l:ISI! 8flt I 13 UNCLASSIFIED//FOR OFFICUtt H.!l! er~tv Casimir Vacuum plates fluctuations Figure 2. Schematic of the Casimir Effect. It turns out that there are many different types of Casimir effects found in quantum field theory [20-22, 26-28, 51]. For example, if one introduces a single infinite plane conductor into the Minkowski (flat spacetime) vacuum by bringing it adiabatically from infinity so that whatever quantum fields are present suffer no excitation but remain in their ground states, then the vacuum (electromagnetic) stresses induced by the presence of the infinite plane conductor produces a Casimir effect. This result holds equally well when two parallel plane conductors (with separation distance d) are present, which gives rise to the familiar Casimir effect inside a cavity. Note that in both cases, the spacetime manifold is made incomplete by the introduction of the plane conductor boundary condition(s). The vacuum region put under stress by the presence of the plane conductor(s) is called the Casimir vacuum. The generic expression for the energy density of the Casimir effect is PCE =- Ahcd-4 , where A= C,(D)/8rr-2 in spacetimes of arbitrary dimension O [20-22]. The appearance of the zeta-function (,(D) is characteristic of expressions for vacuum stress-energy tensors, T.J:i~ .t t t In our familiar 4-dimensional spacetime (0 = 4) we have that A = rc2/720. To calculate T_J:i~ for a given quantum field is to calculate its associated Casimir effect. We should also point out that the methods used to obtain the quantum vacuum electromagnetic T;~~ between parallel plane conductors can also be used when the conductors are not parallel but are joined together along a line of intersection. If the conductors have curved surfaces instead, then one obtains results that are similar to the case of intersecting conductors. These geometries have also been evaluated for the ttt The Greek tensor indices (µ, v = 0...3) denote spacetime coordinates, >l'...x3, such that xt...x 3 = space coordinates and x0 "' time coordinate. Note in general that T00 = p E (field energy density). UNCLASSIFIED/,'FOR OFFI&I.t..k W&li OPtk\S 14 UNCLASSIFIED//FAA QFFl&IAL l:191! 614[1 case of dielectric media. These particular cases will not be considered further since there are technical subtleties involved that complicate the calculations and application of the different approaches. As a final note, negative energy can be created by a single moving reflecting (conducting) surface (a.k.a. a moving mirror) via the dynamical Casimir effect. A mirror moving with increasing acceleration generates a flux of negative energy that emanates from its surface and flows out into the space ahead of the mirror [23, 52]. This is essentially the simple case of an infinite plane conductor undergoing acceleration perpendicular to its surface. If the acceleration varies with time, the conductor will generally emit or absorb photons (i.e., exchange energy with the vacuum), even though it is neutral. This is an example of the well-known quantum phenomenon of parametric excitation. The parameters of the quantum electromagnetic oscillators (e.g., their frequency distribution function) change with time owing to the acceleration of the mirror [53]. However, this effect is known to be exceedingly small, and it is not the most effective way to produce negative energy for our purposes. We will not consider this scheme any further. QUANTUM OPTICAL HOMODVNE TOMOGRAPHY Observing Negative Energy in the Lab Negative energy should be observable in lab experiments. A generic, non-optical scheme for detecting negative energy in experiments was recently reported by Davies and Ottewill [54] who studied the response of switched particle detectors to static negative energy densities and negative energy fluxes. Their model is based on a free (massless) scalar field in flat 4-dimensional Minkowski spacetime and utilized a simple generalization of the standard monopole detector, which is switched on and off to concentrate the measurements on periods of isolated negative energy density (or negative energy flux). The detector model includes an explicit switching factor whereby five different switching functions (based on data windowing theory) are defined and evaluated. In order to isolate the effects of negative energy, a comparison is made for the response of a detector switched on and off during a period of negative energy density (or negative energy flux) and that switched on and off in the vacuum. The results shed light on the response of matter (detectors) to pulses of negative energy of finite duration, and they showed that negative energy should have the effect of enhancing de-excitation (i.e., induce cooling) of the detector. This is the opposite of our experience with detectors that undergo excitation when encountering "normal" matter or energy, and isolated detectors placed in a vacuum naturally cool due to the usual thermodynamic reasons. But Davies and Ottewill point out that the enhanced cooling effect they discovered cannot be used to draw a thermodynamic conclusion because their modeling was restricted to first order in perturbation theory. It is not possible at first order to determine whether the enhanced cooling effects are due to the small violation of energy conservation expected in any process in which a general quantum state collapses to an energy eigenstate, or whether they predict a systematic reduction in the energy of the detector which has serious thermodynamic implications. However, Davies and Ottewill point out that their results are model dependent and they found for their standard monopole detector model that there is not always a simple relationship UNCLASSIFIED//fl61t 6flfllelslct l:l!lfI 8,.LY 15 UNCLASSIFIED//FOR QFFI€ilAL l::l!H! 61\LI between the strength of the negative energy density/flux and the behavior of the detector. It is curious that Davies and Ottewill did not consider using quantum optical homodyne tomography as a tool to test their hypothesis, because this is already a mature experimental discipline. In what follows we outline the basics of quantum optical homodyne tomography and its application to detecting and measuring negative energy density/flux states in squeezed light and in the Casimir effect. Basic Notions of Quantum Optical Homodyne Tomography Tomography, from the Greek word for slice, is a method to infer the shape of a hidden object from its shadows (or projections) under various angles. Quantum tomography is the application of this idea to quantum mechanics. In optical homodyne tomography, the Wigner function or, more generally, the quantum state plays the role of the hidden object. The observable "quantum shadows" are the quadrature distributions and are measured using homodyne detection. From these distributions the Wigner function is reconstructed. See Figure 3 for an illustration of quantum optical homodyne tomography. The vertical 2-dimensional plane seen in the figure is fictitious and is shown for illustrative purposes only. Figure 3. Illustration of Quantum Optical Homodyne Tomography (courtesy of Ulf Leonhardt). The Wigner function (3-dimensional hill on the right) is reconstructed in quantum phase space (gridded plane formed by quadratures q and p) from its experimentally measured projections (curve in vertical 2-dimensional plane), which represents the scanning process of tomography. The vertical axis is the magnitude of the Wigner (quasiprobability) function. Quantum tomography was developed for the simple reason that a fundamental feature of quantum mechanics prevents us from seeing physical objects in their full quantum complexity. This is due to the intrinsic fuzziness in the quantum nature of energy and matter according to the Heisenberg Uncertainty Principle, which prevents us from simultaneously and precisely measuring the complementary features (e.g., position and momentum or energy and time) comprising quantum states. For this reason we cannot UNCLASSIFIED/;<fQA QFFl&l.t..k YSE 8,.L'&' 16 UNCLASSIFIED/ j FOR OFFICIAL tJ.!I!! 6HL'f directly observe quantum states, and so the true nature of an individual quantum system is hidden. However, no principal obstacle exists to observing all complementary aspects in a series of distinct experiments on identically prepared quantum objects. In the sections that follow, we briefly review the several parts that comprise the tomography machinery, and then put the whole picture together to understand what the entire process is. No effort will be made for completeness because the subject of quantum tomography takes up volumes of books. The reader will be referred to the key literature of importance. Wigner Functions In classical optics the state of an electromagnetic oscillator is perfectly described by the statistics of the classical amplitude a. The amplitude may be completely fixed (then the field is coherent), or a may fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude a or, equivalently, the statistics of the component position q and momentum p by introducing a phase space distribution called the Wigner function, W(q,p). "*" W(q,p) quantifies the probability of finding a particular pair of q and p values in their simultaneous measurement. Knowing W(q,p) for a particular quantum state that is under study, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense W(q,p) describes the state in classical physics. The motivation for introducing the Wigner function was the desire to find a quantum mechanical description similar to that in classical statistical physics. However, in quantum mechanics Heisenberg's Uncertainty Principle prevents one from observing position and momentum simultaneously and precisely. In addition to this, we also cannot directly observe quantum states either. Nevertheless, we are perfectly entitled to use the concept of quantum states as if they were existing entities. We use their properties to predict the statistics of observations. It is well known that the quantum mechanical wave function depends exclusively on either the position or the momentum and contains nevertheless a// the information about the quantum system under study. However, E. Wigner showed that it is possible to define a formal quantum mechanical analog to the classical distribution function. He showed that we could use W(q,p) as a quantum phase space distribution exclusively to calculate observables in a classical-like fashion. Wigner discovered that W(q,p) is a real-valued function, but it is usually not just positive; it can also become negative. This is a very nonclassical behavior for a probability distribution. It is for this reason that W(q,p) came to be called a quasiprobability distribution. W(q,p) has several properties and mathematical postulates, but it turns out that just one postulate is sufficient for the purposes of quantum tomography [38). Using this postulate, it is assumed that W(q,p) behaves like a joint probability distribution for q and p without ever mentioning any simultaneous observation of position and momentum. The reduced, or marginal, distributions J: W(q, p)dp or J: W(q, p)dq *" Recall in Sect. IIB-1 that the real and the imaginary parts of the complex amplitude o. can be regarded as the position and the momentum of the electromagnetic oscillator. UNCLASSIFIED/ifOR Offlf:!IAL Y.!I!! er•t I 17 UNCLASSIFIED//POI\ OPPICIJIIL YSIE 8Hllf must give the position or the momentum distribution, respectively. Furthermore, if one performs a phase shift 8 all complex amplitudes aare shifted in phase, §§§ meaning that the components q and p rotate in the 2-dimensional phase space (q,p). A classical probability distribution for position and momentum values would rotate accordingly. This fact leads to the postulate that the position probability distribution pr(q,8) after an arbitrary phase shift 8 should be [38] pr(q,0) =(qJOce)p 0\ e)J q) (10) =f: w(qcos0-psin0,qsin0+pcos0)dp, where p is the quantum density operator (or density matrix) which describes the statistical (or most general) state of a quantum system. The first line in Eq. (10) is the quantum expectation value of the phase-shifted p, which simply gives the probability distribution for the q-eigenstates to occur with probabilities p q (the elements of p). This single formula joins W(q,p) with quantum mechanics. It ties W(q,p) to observable quantities, and it links quantum states to observations. It is beyond the scope of this report to repeat the entire mathematical development of the explicit functional representations, identities, transformations and modifications of W(q,p). The reader should consult Reference [38] for more information. However, Figures 4 through 7 provide an example of what the experimentally reconstructed Wigner function visually looks like from the quantum optical homodyne tomography of the following cases of interest: a vacuum state, a coherent state, a squeezed vacuum state, a single photon, and Schrtidinger cat states. The Schrtidinger cat states are a very interesting case study of unusual nonclassical states of light that have been experimentally measured via quantum optical homodyne tomography. §§§ The unitary phase shifting operator is U(0) = exp(- i0ii), where ii is the photon number operator and e is the phase shift angle. Its action on the amplitude ais: u \ 8)11U(8) = 11e.>.p(- i0) . UNCLASSIFIED//FOR 0661CIAk W&IE 8HLY 18 UNCLASSIFIED//FOR 8FFl@lsllt U.!I! 014Li Figure 4. Wigner Function for a Vacuum (top) and for a Coherent State (bottom). This clearly shows that coherent states are just "displaced vacua" (Sect. IIB-3). Optical homodyne tomography was used to reconstruct the Wigner functions from experimental data (courtesy of Ulf Leonhardt). UNCLASSIFIED//FOR OFFICIAL U.!I! 9HLY 19 UNCLASSIFIED//FOR OFFl@IAL Y!H! 8HL'I 0 50 100 150 200 Time rmsl Figure S. Wigner Function of a Squeezed Vacuum. Wigner function (top) and quadrature fluctuations (bottom). This shows the experimentally reconstructed Wigner function of a significantly squeezed vacuum generated by parametric amplification (Sect. IIB-3). The noise trace (bottom) shows a part of the experimental data used to reconstruct the depicted Wigner function via optical homodyne tomography (courtesy of Ulf Leonhardt). UNCLASSIFIED/i POlt OPPlelJcL Y.!l!! 8HLY 20 -0.2 . . -.........."--... -2 ·,,-......__ -/ ,,"-.... o~ ......... ---- q 1 ~ -.... 2 .....___ p UNCLASSIFIED//FOR 8ffl@IJ!tt U.!I! 014Li 0.1 0 -0.l -0.3 0.1 0 _-:.·:l =-=----·~---- -0.3 -~------ "~ ........ ·-"-...____.__,.__ ........~........ -2 1 ~..,____ ----·­ • 0 -, -"? ------­ p Figure 6. Wigner Function of a Single Photon. The figure shows the experimentally reconstructed Wigner function as seen from above (top) and from below (bottom). Negative "probabilities" are clearly visible near the origin of the phase space, which demonstrates the nonclassical aspect of photons (courtesy of Ulf Leonhardt). UNCLASSIFIED//POlt OPPl@IJ!tt li!lf! 8HLY 21 ----------------------------- ------------------------ UNCLASSIFIED//PO" orr1e1J11t l:l!H! e ..tv 0.2 0 -0.2 q -4 '----­ -~---­ 0 ,.______ /J -0.2\ '--.. q -5 -------~ -2.5 0 p 2.s"'-...._____ ..... 5 Figure 7. Quantum Tomography of Schrodinger-Cat States. Top : q0 = 3. Two separated coherent amplitudes (peaks) are clearly visible. Bottom: qo = 4. The larger the separation of the amplitudes, the more rapid is the oscillation in t he quantum interference structure between the two peaks. Negative probabilities appear within the quantum interference structure. The experimental data used to reconstruct the depicted Wigner functions was provided by A. Furusawa and H. Yonezawa, University of Tokyo. ----- 4 We digress for the moment to explain what Schrodinger cat states are. Schrodinger's cat is a famous illustration of the principle of superposition in quantum theory that was proposed as a thought experiment by Erwin Schrodinger in 1935 . Schrodinger's cat UNCLASSIFIED//fOtt 8ffl@IAL Y§E 8,.LY 22 UNCLASSIFIED//FOR OFFICiltt tJ.!l! f>HLY serves to demonstrate the apparent conflict of what quantum theory tells us is true about the nature and behavior of matter on the quantum (atomic or subatomic) level compared with what we actually observe to be true about the nature and behavior of matter on the macroscopic level. Schrodinger's thought experiment is as follows: One places a living cat into a steel chamber along with a device containing a vial of hydrocyanic acid. There is also a very small amount of a radioactive substance inside the chamber. If even a single atom of the substance decays during the test period, then a relay mechanism will trip a hammer, which will in turn break the vial and kill the cat. The observer cannot know whether or not an atom of the radioactive substance has decayed, and consequently, cannot know whether the vial has been broken, the hydrocyanic acid released, and the cat killed. Since one cannot know, the cat is both dead and alive in a superposition of quantum states according to the quantum superposition principle. It is only when one breaks open the box and learns the condition of the cat that the superposition is lost, and the cat becomes either dead or alive. This situation is sometimes called quantum indeterminacy or the observer's paradox: the act of observation or measurement itself affects the outcome, so that the outcome as such does not exist unless, and until, the measurement is made. (That is, there is no single