UNCLASsiFiED//ren ornciA L uoe omly Defense Intelligence Reference Document D efense F utures 11 Jan uary 2011 ICOD: 10 August 2010 DIA-08-1102-007 Quantum Tomography of Negative Energy States in the Vacuum U N CLASSI FI E D/ /FUK UI I 1MA L UD E ONLY UNCLASSIFIED//SD R fiK K iA L UPC ONLY" Quantum Tomography of Negative Energy States in the Vacuum The Defense Intelligence Reference Document provides n on -substan tive but authoritative referen ce in form ation related to in telligen ce topics or m ethodologies. Prepared by: Technology Warning Division (DWO-4) Defense Warning Office Directorate for Analysis Defense Intelligence Agency Author: AAP Person 58 COPYRIGHT W ARN IN G: Further dissem in ation of the photographs in this publication is n ot authorized. This product is on e of a series of advan ced techn ology reports produced in FY 2010 un der the Defen se In telligen ce Agen cy, Defen se W arn in g Office's Advan ced Aerospace W eapon s System Application s (AAW SA) Program . Com m en ts or question s pertain in g to this docum en t should be addressed to AAP Person 1 AAW SA Program M an ager, Defen se In telligen ce Agen cy, ATTN : JUIAF - bl/W O-J, Bldg. 6000, W ashin gton D.C. 20340-5100. ii UNCLASSIFIED//RQR CFH MA h M6i QNL¥ UNCLASSIFIED//EQR OFFICIA L UG C OMh¥ Contents Introduction............................................... 1 REVIEW OF NEGATIVE (or SUB -VACUUM) ENERGY...................................................3 Overview.......................................................... 3 Examples of Negative (Sub-Vacuum) Energy Found in Nature................... 4 B asic Notions of the Quantum Field Theory of Light......................... 5 B asic 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 B asic Notions of Quantum Optical Homodyne Tomography.................... 16 Wigner Functions..................................................................... 17 B eam Splitters..................... 24 Photodiodes.................................................................................................27 B alanced Homodyne Detection...................... 27 Outline of Experimental Procedure..............................................................32 B ALANCED HOMODYNE SYSTEMS FOR MEASURING NEGATIVE (SUB -VACUUM) ENERGY................................................................... 33 Time-Domain B alanced Homodyne System....................................................33 B alanced Homodyne System for Casimir Cavities..................................... 36 CONCLUSION.............................................................................................................43 ACKNOWLEDGEMENTS.............................................................................................45 REFERENCES........................................................................ 46 iii U NCLASSI FI ED//FUK UI 11L1A L UJL UH L I U NCLASSI FIED/ / FOR OFFICIA L USE BMW 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 Schrddinger-Cat States....................................22 Figure 8. Schematic of an Ideal Lossless B eam Splitter........................ 25 Figure 9. Illustration of a Fictitious B eam Splitter..................................................26 Figure 10. Schematic of a B alanced Homodyne Detector........................... 29 Figure 11. B alanced Homodyne Detector Using Fictitious B eam Splitters.............31 Figure 12. B alanced Homodyne Detector Using A Single Effective Fictitious B eam Splitter....................................................................................................... 32 Figure 13. Time-Domain B alanced Homodyne Detector............................ 34 Figure 14. Experimentally Measured Squeezed State.............................................35 Figure 15. B alanced Homodyne Detector with a Local Oscillator..........................38 Figure 16. Diagram of Casimir Cavity with B HD Photodiodes ...............................40 Figure 17. Experimental Setup of B HD Photodiodes and LO Field.........................40 Figure 18. Detailed Schematic of Experimental B HD Apparatus...........................41 Figure 19. Predicted Casimir Spectral Density......................................................41 Figure 20. Predicted Suppression of Vacuum Fluctuations in dB ..........................42 iv UN CLASSI FIED//BQR QEMCJA h MSB OML¥ UNCLASSIFIED//TOW OFFICIA L USE ONLY Quantum Tomography of Negative Energy States in the Vacuum Introduction Future aerospace vehicles could have an advan ced propulsion system that uses n egative quan tum vacuum en ergy to m odify the spacetim e geom etry in the im m ediate vicin ity surroun din g the vehicle in order to in duce faster-than -light m otion via traversable worm holes or warp drives, or even levitation via an tigravity [1, 2]. These exotic propulsion con cepts are well-kn own in m ain stream gen eral relativity an d quan tum field theory research. The n otion of a physical state with n egative en ergy is n ot fam iliar in the realm of classical physics. However, it is n ot rare in quan tum field theory to have quan tum states with n egative en ergy den sity or a n egative en ergy flux. Even for a quan tum scalar field in the flat M in kowski spacetim e, it can be proved that the existen ce of quan tum states with n egative en ergy den sity is in evitable [3], Although all kn own form s of classical m atter have n on -n egative en ergy den sity, it is n ot so in quan tum field theory. A gen eral quan tum state can be a superposition of particle n um ber eigen states an d m ay have a n egative expectation value of en ergy den sity in certain spacetim e region s due to quan tum coheren ce effects [3], These con sideration s rem ain true even for quan tum fields in a curved spacetim e where the effects of gravitation al fields, or equivalen tly, acceleration s, can be observed due to the m ass of astron om ical bodies or the m otion s of astron om ical bodies. There are two key exam ples of specially prepared quan tum vacuum states that are kn own to produce sm all am oun ts of n egative en ergy den sity in the laboratory. These are the well-kn own Casim ir effect an d the squeezed vacuum states of the electrom agn etic field. The form er is a static quan tum vacuum effect while the latter is a tim e-dom ain quan tum vacuum effect. There are several other exam ples of special quan tum vacuum or particle states that produce n egative en ergy den sity, but they are beyon d the scope of this report because they rem ain m athem atical curiosities or are n ot practicable to im plem en t in the laboratory in the foreseeable future. W e already m ake sm all am oun ts of n egative en ergy in the laboratory via the Casim ir effect an d squeezed electrom agn etic vacuum states, but we do n ot yet kn ow if we can access larger am oun ts for exten ded periods of tim e over exten ded spatial distribution s for the purpose of m odifyin g spacetim e for aerospace propulsion application s. It will be n ecessary to first explore the quan tum n ature of the Casim ir effect an d squeezed electrom agn etic vacuum states to determ in e whether we can m easure an d spatially m ap their n egative en ergy den sity. This is a n ecessary first step to take before begin n in g an y study on producin g large quan tities of n egative en ergy because we will first n eed to kn ow how to m easure an d spatially m ap n egative en ergy in order to properly con trol it after producin g it. This is the m otivation for this report. W e n eed to firm up our un derstan din g of how lab detectors will respon d to n egative en ergy in situ. A first step in this direction was already taken by Han sen et al. [4] in 2001 for the tim e-dom ain n egative en ergy pulses in squeezed electrom agn etic vacuum states, an d m ore recen tly M arecki [5, 6] gen eralized the an alysis of the output of balan ced hom odyn e detectors (BHDs) for the case of static n egative en ergy states 1 UNCLASSIFIED//TOR. OFFICIA L U3E ONLY UNCLASSIFIED//FQR QmCIA L UD E ONL¥ in side Casim ir cavities. The m ost im portan t feature of these devices is their ability to quan tify the quan tum vacuum fluctuation s of the electric field because the output of BHDs provides in form ation on the on e- an d two-poin t fun ction s of arbitrary states of quan tum fields. M arecki com puted the two-poin t fun ction an d the associated spectral den sity for the groun d state of the quan tum electric field in Casim ir geom etries, an d predicts a position - an d frequen cy-depen den t pattern of BHD respon ses if a device of this type is placed in side a Casim ir cavity. The proposed device allows for the direct detection of quan tum vacuum fluctuation s an d provides a spatial m appin g of the n egative en ergy con tain ed in side the cavity, which will be sum m arized in this report. 2 UNCLASSIFIED//FUK UH UL1A L U3E ONLY UNCLASSIFIED//H QR OFFICIA L USE ONLV REVIEW OF NEGATIVE (or SUB -VACUUM) ENERGY Overview The im plem en tation of faster-than -light (FTL) in terstellar travel via traversable worm holes or warp drives or other an tigravity forces for propulsion , gen erally requires the en gin eerin g of spacetim e in to very specialized local geom etries surroun din g the im m ediate vicin ity of the aerospace vehicle un dergoin g this type of m otion . The an alysis of these via the gen eral relativistic field equation plus the resultan t source m atter equation s of state dem on strates that such geom etries require the use of "exotic" m atter in order to produce the requisite FTL or an tigravity spacetim e m odification . Exotic m atter is gen erally defin ed by gen eral relativity physics to be m atter that possesses (ren orm alized) n egative en ergy den sity (som etim es n egative stress-ten sion = outward pressure, a.k.a. gravitation al repulsion or an tigravity), an d this is a very m isun derstood an d m isapplied term by the n on -gen eral relativity com m un ity. W e clear up this m iscon ception by defin in g what n egative en ergy is, where it can be foun d in n ature, an d we also review the two prim ary experim en tal con cepts that are kn own to produce n egative en ergy in the laboratory. Also, it has been claim ed that FTL an d an tigravity spacetim es are n ot plausible because exotic m atter violates the gen eral relativistic en ergy con dition s. However, it has been shown that this is a spurious issue. The iden tification , m agn itude, an d production of exotic m atter is seen to be a key techn ical challen ge, however. FTL an d an tigravity spacetim es also possess features that challen ge the n otion s of causality an d there are alleged con strain ts placed upon them by quan tum effects. Referen ce [1] reviews an d sum m arizes these issues with an assessm en t on the presen t state of their resolution . W hat exactly is "exotic" m atter? In classical physics the en ergy den sity of all observed form s of m atter (fields) is n on -n egative. W hat is exotic about the type of m atter that m ust be used to produce traversable worm hole, warp drive, or an tigravity spacetim es is that it m ust have n egative en ergy den sity an d/or n egative flux [7]. The en ergy den sity is "n egative" in the sen se that the con figuration of m atter fields we m ust deploy to produce a traversable worm hole, warp drive, or an tigravity effect m ust have an en ergy den sity, pE (= pc2, where p is the rest-m ass den sity), that is less than or equal to its pressures/ten sion s, px [8, 9].* In m an y cases, these equation s of state are also kn own to possess an en ergy den sity that is algebraically n egative, i.e., the en ergy den sity an d flux are less than zero. It is on the basis of these con dition s that we call this m aterial property "exotic." The con dition for ordin ary, classical (n on -exotic) form s of m atter that we are all fam iliar with in n ature is that pE > pi an d/or pE > 0. These con dition s represen t two exam ples of what are variously called the "stan dard" en ergy con dition s which are com puted from the trace of the m atter stress-en ergy ten sor: W eak En ergy Con dition (W EC: pE > 0, pe+pi > 0), N ull En ergy Con dition (N EC: pE + pi S 0), Dom in an t En ergy Con dition (DEC), an d Stron g En ergy Con dition (SEC). These en ergy con dition s forbid n egative en ergy den sity between m aterial objects to occur in n ature, but they are m ere hypotheses. Hawkin g an d Ellis [10] form ulated the en ergy con dition s in order to establish a series of m athem atical hypotheses govern in g the behavior of * From this poin t forward, all Latin letters (e.g., i, j, k = 1...3) that appear as in dices on physical quan tities den ote the usual 3-dim en sion al space coordin ates, x*...x3, in dicatin g the spatial com pon en ts of vector or ten sor quan tities. 3 UNCLASSIFIED//rOR OFFICIA L USE ON MT 1 The stress-en ergy-m om en tum ten sor is a m atrix quan tity that en codes the den sity an d flux of en ergy an d m om en tum for an y type of m atter un der study. UNCLASSIFIED//FOR OFFICIA L USE ONLY collapsed-m atter sin gularities in their study of cosm ology an d black hole physics. M ore specifically, classical gen eral relativity allows on e to prove lots of gen eral theorem s about the behavior of m atter in gravitation al fields. However, real physical m atter is n ot "reason able" because the en ergy con dition s are in gen eral violated by sem iclassical quan tum effects (occurrin g at order q) [9].* M ore specifically, quan tum effects gen erically violate the average N EC (AN EC). Furtherm ore, it was discovered in 1965 that quan tum field theory has the rem arkable property of allowin g states of m atter con tain in g local region s of n egative en ergy den sity or n egative fluxes [3]. This violates the W EC, which postulates that the local en ergy den sity is n on ­ n egative for all observers. An d there are also gen eral theorem s of differen tial geom etry that guaran tee that there m ust be a violation of on e, som e, or all of the en ergy con dition s (m ean in g exotic m atter is presen t) for all FTL an d an tigravity spacetim es. However, all of the en ergy con dition hypotheses have been experim en tally tested in the laboratory an d experim en tally shown to be false - 25 years before their form ulation [11]. * Plan ck's reduced con stan t, n = 1.055 x 10 34 J s. In quan tum field theory, n egative en ergy is a m an ifestation of what is n ow called the "sub-vacuum " levels of the quan tum zero-poin t (or vacuum groun d state) fluctuation s that correspon d to an y particular quan tum field of m atter un der study. Hen ce, the en ergy correspon din g to sub-vacuum quan tum fluctuation s is n ow called "sub-vacuum en ergy": sub-vacuum en ergy = n egative en ergy. Further in vestigation in to this techn ical issue showed that violation s of the en ergy con dition s are widespread for all form s of both "reason able" classical an d quan tum m atter [12-16]. Furtherm ore, Visser [9] showed that all (gen eric) spacetim e geom etries violate all the en ergy con dition s. So the con dition that pE > pi an d/or pc > 0 m ust be obeyed by all form s of m atter in n ature is spurious. N egative en ergy has been produced in the laboratory an d this will be discussed in the followin g section s. Examples of Negative (Sub-Vacuum) Energy Found in Nature The exotic (en ergy con dition -violatin g) fields that are kn own to occur in n ature are: 1. Static, radially-depen den t electric or m agn etic fields. These are borderlin e exotic, if their ten sion were in fin itesim ally larger, for a given en ergy den sity [10, 17]. 2. Squeezed quan tum vacuum states: electrom agn etic an d other (n on -M axwellian ) quan tum fields [8, 18]. 3. Gravitation ally squeezed electrom agn etic vacuum fluctuation s [19]. 4. Casim ir effect, i.e., the Casim ir vacuum in flat, curved, an d topological spaces [20-28], 5. Other quan tum fields/states/effects. In gen eral, the local en ergy den sity in quan tum field theory can be n egative due to quan tum coheren ce effects [3]. Other exam ples that have been studied are Dirac field states: the superposition of two sin gle particle electron states an d the superposition of two m ulti-electron ­ positron states [29, 30], In the form er (latter), the en ergy den sities can be n egative when two sin gle (m ulti-) particle states have the sam e n um ber of 4 UNCLASSIFIED//TOR OTriCML UOE CNET UNCLASSIFIED//FQR QK K Ifth USE ONL¥ electron s (electron s an d positron s) or when on e state has on e m ore electron (electron -positron pair) than the other. Cosm ological in flation [9], cosm ological particle production [9], classical scalar fields [9], the con form al an om aly [9], an d gravitation al vacuum polarization [12-15] are am on g m an y other exam ples that also violate the en ergy con dition s. Sin ce the laws of quan tum field theory place n o stron g restriction s on n egative en ergies an d fluxes, then it m ight be possible to produce exotic phen om en a such as faster-than -light travel [31­ 33], traversable worm holes [8, 9, 34], violation s of the secon d law of therm odyn am ics [35, 36], an d tim e m achin es [9, 34, 37]. There are several other exotic phen om en a m ade possible by the effects of n egative en ergy, but they lie outside the scope of this report. In what follows, we con sider on ly item s 2 an d 4 in the previous list for the purpose of this report due to their ready applicability an d techn ical m aturity. W e will n ot exam in e the other item s in the list because they are theoretical curiosities that rem ain un der study by in vestigators. Basic Notions of the Quantum Field Theory of Light Before goin g further, it will be helpful to briefly outlin e the basic n otion s an d term in ology of the quan tum field theory of light (i.e., quan tum optics) because the con ten t of this report focuses on those aspects. Classically, light is electrom agn etic radiation that can be pictured as waves flowin g through space at the speed of light, c (= 3.0 x 103 m /s). The waves are n ot waves of an ythin g substan tive, but are in fact ripples in the state of a field. These waves carry en ergy, an d each wave has a specific direction , frequen cy an d polarization state. This is called a "propagatin g m ode of the electrom agn etic field." A sim ple m odel for this is the electrom agn etic oscillator. On e com plex-valued vector fun ction u(x,f) called a spatial-tem poral m ode com prises all classical wave aspects in cludin g polarization . The sim plest exam ple of a spatial-tem poral m ode is a plan e wave u(x,t) = u(}exp i^kx — Mt^ of polarization vector «o, an gular frequen cy co, an d wave vector k (defin ition : k2 = cxrlc2 ), where i is the un it com plex n um ber, an d x is the space coordin ate an d t is the tim e coordin ate. This m ode defin es a fram ework in space an d tim e that m ay be excited by the quan tum field "light." The m ode fun ction quan tifies the stren gth of on e excitation in space an d tim e. Also, the m ode fun ction obeys the laws of classical waves given by M axwell's equation s of electrodyn am ics. The choice of u(x,t) is m ade by the observer. The observer sin gles out on e m ode, on e quan tum object from the rest of the world to m ake a specific observation or m easurem en t. This object turn s out to be a harm on ic oscillator described by the an n ihilation operator a. A useful tool for m odelin g the propagatin g m ode of the electrom agn etic field in quan tum m echan ics is the ideal quan tum m echan ical harm on ic oscillator: a hypothetical charged m ass on a perfect sprin g oscillatin g back an d forth un der the action of the sprin g's restorin g force. The operator a stan ds for the quan tized am plitude with which u(x,t) can be excited. In classical optics it would be just a com plex n um ber a of m agn itude |a| an d phase arg(a). The quan tized am plitude a is n either predeterm in ed n or given by the observer 5 UNCLASSIFIED//FOK OFFICIA L USE ONLY UNCLASSIFIED//FOR OmCIA L UD E ONLY but depen ds on the state of u(x,t\ This state exists even if literally n othin g is in the m ode chosen by the observer. In this case, the light is just in the vacuum stated However, this "n othin g" can in deed cause sign ifican t physical effects as will be discussed in later section s. To m ake all this m ore precise, we postulate that the electric field stren gth E of the light field is given by E - u*(x,t)a + u(x,t)a an d that the am plitude operator a is a boson ic*’ an n ihilation operatorthat obeys the quan tum m echan ical com m utation relation i ri, fl ] = 1, where u(x,f) is the com plex con jugate of u(x,t) an d a? is the adjoin t (or con jugate) of a called the creation operator/1 The hat sym bol appearin g over quan tities den otes that they are quan tum operators (or observables). An other key elem en t of quan tum -oscillator physics is the photon n um ber operator h, which accoun ts for the n um ber of photon s (quan tized light particles) in the chosen u(x,t) an d is given by the quan tum m echan ical coun terpart of a classical m odulus-squared am plitude: h^a^a. Let us n ow in troduce a pair of operators, q an d p, called quadratures, They are defin ed as ^ = 2 l 2(rF + tz) an d p = i2 '^^-a^, which can be in verted to provide the addition al useful defin ition s a = 2~}n [q + ip} an d a -2~l ' [q-ip}. In optics q an d p correspon d to the in -phase an d the out-of-phase com pon en t of the electric field am plitude of u(x,t) (with respect to a referen ce phase). The boson ic com m utation relation dem on strates that q an d p are can on ically con jugate observables, [< 7,p] = zh. The quadratures q an d p can be regarded as the position an d the m om en tum of the quan tum electrom agn etic oscillator. They do n ot appear in real space but in the phase space span n ed by the com plex vibration al am plitude a of the quan tum electrom agn etic oscillator, an d they have n othin g to do with the position an d the m om en tum of a photon . However, the can on ical com m utation relation en titles us to treat q an d p as perfect exam ples of position - an d m om en tum -like quan tities in quan tum optics. Fin ally, we express the photon n um ber operator n in term s of the quadratures q an d p an d obtain , usin g the boson ic com m utation relation , the stan dard Ham ilton ian (or total en ergy) of the quan tum harm on ic (electrom agn etic) oscillator with un it m ass an d frequen cy: ^OSC “ ^ 2 2 2 ' (1) - Here we always m ean by "vacuum " sim ply "n o light" an d n ot an evacuated system . Boson or boson ic refers to quan tum particles that have in teger quan tum spin . +t In quan tum m echan ics, the vacuum is defin ed to be a state of n o (or zero) particles an d is den oted by the quan tum state eigen vector |0). By defin ition a "an n ihilates" the vacuum state: < 5|0) = 0. 6 UNCLASSIFIED//rOR OFFICIA L use ONLY UNCLASSIFIED//rOK OFFICIA L USE ONLY where the first an d secon d term s in the secon d lin e are the kin etic an d poten tial en ergies of the oscillator, respectively. The addition al 1/2 appearin g in the first lin e of Eq. (1) is called the vacuum zero-point energy for the reason to be explain ed in the n ext section . The first lin e of Eq. (1) is m ore com m on ly expressed in un its of en ergy (Joules) in quan tum m echan ics, which is obtain ed sim ply by m ultiplyin g the right-han d side by the photon en ergy Aco so that H^ = hw^ + ^. It is beyon d the scope of this report to elaborate further on the en tire subject of the quan tum optics. The reader should con sult Referen ce [38] for m ore in form ation . Basic Notions on the Origin of the Quantum Vacuum Zero-Point Fluctuations Here we discuss the basic n otion s of the quan tum vacuum zero-poin t fluctuation s (ZPF), which is an im portan t feature in quan tum optics. The origin of the ZPF is attributed to the Heisen berg Un certain ty Prin ciple. Accordin g to this prin ciple, q an d pare an y two con jugate observables that we are in terested in m easurin g, an d they obey the com m utation relation already shown in the previous section . Their correspon din g un certain ty relation is A^Ap>h/2, where Aq is the varian ce (a.k.a. un certain ty) of observable q an d Ap is that of the con jugate observable p. This relation states that if on e m easures observable q with very high precision (i.e., its un certain ty A^ is very sm all), then a sim ultan eous m easurem en t of observable p will be less precise (i.e., its un certain ty Ap is very large), an d vice versa. In other words, it is n ot possible to sim ultan eously m easure two con jugate observable quan tities with in fin ite precision . This m in im um un certain ty is n ot due to an y correctable flaws in m easurem en t, but rather reflects the in trin sic fuzzin ess in the quan tum n ature of en ergy an d m atter. Substan tial theoretical an d experim en tal work has shown that in m an y quan tum system s the lim its to m easurem en t precision is im posed by the quan tum vacuum ZPF em bodied within the un certain ty prin ciple. N owadays we rather see the Heisen berg Un certain ty Prin ciple as a n ecessary con sequen ce, an d therefore, a derived result of the wave n ature of quan tum phen om en a. The un certain ties are just a con sequen ce of the Fourier n ature of con jugate pairs of quan tities (observables). For exam ple, the two Fourier-wave-con jugates tim e an d frequen cy becom e the pair of quan tum -particle con jugates tim e an d en ergy an d the two Fourier-wave-con jugates displacem en t an d wave n um ber becom e the pair of quan tum -particle con jugates position an d m om en tum . The Heisen berg Un certain ty Prin ciple dictates that a quan tized electrom agn etic oscillator (a.k.a. a photon state) can n ever com e en tirely to rest, sin ce that would be a state of exactly zero en ergy, which is forbidden by the com m utation relation given in the previous section . In stead, every m ode of the field has Aco/2 as its average m in im um en ergy in the vacuum , an d this is called the zero-poin t en ergy (ZPE).** This ZPE term is added to the classical blackbody spectral radiation en ergy den sity p(w)r/w [i.e., the en ergy per un it volum e of radiation in the frequen cy in terval (co, co + Jco)] [25]: ** hio is the en ergy of a sin gle m ode (or photon ). 7 UNCLASSIFIED//EQR QmCIA L UOC ONgr UNCLASSI FIED/ / FD R OFFICIA L USE ONLY p(w) J® = —r^ 7 C C h®3 h® exp(h®/^Br)-l 2 h® , 4-----a® 2k2 c3 coth hw dw, (2) where fo is Boltzm an n 's con stan t (1.3807 x 10-23 J/K) an d Tis the absolute tem perature. The factor outside the square brackets in the first lin e of Eq. (2) is the den sity of m ode (or photon ) states (i.e., the n um ber of states per un it frequen cy in terval per un it volum e); the first term in side the square brackets is the stan dard Plan ck blackbody radiation en ergy per m ode; an d the secon d term in side the square brackets is the quan tum zero-poin t en ergy per m ode. Equation (2) is called the Zero­ Poin t Plan ck (ZPP) spectral radiation en ergy den sity. Plan ck first added the ZPE term to the classical blackbody spectral radiation en ergy den sity in 1912, although it was Ein stein , Hopf, an d Stern who actually recogn ized the physical sign ifican ce of this term in 1913 [25]. Direct spectroscopic eviden ce for the reality of ZPE was provided by M ulliken 's boron m on oxide spectral ban d experim en ts in 1924, several m on ths before Heisen berg first derived the ZPE for a harm on ic oscillator from his n ew quan tum m atrix m echan ics theory [39]. Followin g this lin e of reason in g, quan tum physics predicts that all of space m ust be filled with quan tum electrom agn etic ZPF creatin g a un iversal sea of zero-poin t en ergy. The other quan tum forces of n ature also have their own vacuum ZPF which con tributes to the un iversal sea of zero-poin t en ergy. But that is beyon d the scope of this report. Negative (Sub-Vacuum) Energy in Squeezed Light Substan tial theoretical an d experim en tal work has shown that in m an y quan tum system s the lim its to m easurem en t precision im posed by the quan tum vacuum ZPF can be breached by decreasin g the n oise in on e observable (or m easurable quan tity) at the expen se of in creasin g the n oise in the con jugate observable; at the sam e tim e the variation s in the first observable, say the en ergy, are reduced below the ZPF such that the en ergy becom es "n egative." "Squeezin g" is thus the con trol of quan tum fluctuation s an d correspon din g un certain ties, whereby on e can squeeze/reduce the varian ce of on e (physically im portan t) observable quan tity provided the varian ce in the (physically un im portan t) con jugate variable is stretched/in creased. The squeezed quan tity possesses an un usually low varian ce, m ean in g less varian ce than would be expected on the basis of the equipartition theorem . On e can in prin ciple exploit quan tum squeezin g to extract en ergy from on e place in the ordin ary vacuum at the expen se of accum ulatin g excess en ergy elsewhere [8], The squeezed state of the electrom agn etic field is a prim ary exam ple of a quan tum field that has n egative en ergy den sity an d n egative en ergy flux. Such a state becam e a physical reality in the laboratory as a result of the n on lin ear-optics techn ique of "squeezin g," i.e., of m ovin g som e of the quan tum -fluctuation s of laser light out of the 8 UNCLASSIFIED/ /FOR OFFICIA L UPC ONLY- UNCLASSIFIED//rOR OmCIA L UOE ONL¥ co5[< o(/-z/c)J part of the beam an d in to the sin^(a(t - z/c^ part [18, 40-44].§§ The observable that gets squeezed will have its fluctuation s reduced below the vacuum ZPF. The act of squeezin g tran sform s the phase space circular n oise profile characteristic of the vacuum in to an ellipse, whose sem im ajor an d sem im in or axes are given by un equal quadrature un certain ties (of the quan tized electrom agn etic oscillator operators). This applies to coheren t states in gen eral, an d the usual vacuum is also a coheren t state with eigen value zero. As this ellipse rotates about the origin with an gular frequen cy co, these un equal quadrature un certain ties m an ifest them selves in the electrom agn etic field oscillator en ergy by periodic occurren ces, which are separated by on e quarter cycle, of both sm aller an d larger fluctuation s com pared to the un squeezed vacuum . W e digress m om en tarily by n otin g that coheren t states, also called Glauber states, are the eigen states of the an n ihilation operator a: a|a) = a|a), (3) which have well-defin ed am plitudes |a| an d phases arg(a) (recall the discussion in Sect. IIB-1). They are called coheren t states because light fields in these states are perfectly coheren t, an d high-quality lasers gen erate such fields. This is an im portan t reason why high-quality laser light is an excellen t tool for experim en tal quan tum optics. Coheren t states com e as close as quan tum m echan ics allows to wave-like states of the electrom agn etic oscillator. Because the wave aspects of light are com m on ly regarded as classical, coheren t states are often called classical states. Furtherm ore, fields in statistical m ixtures of coheren t states (such as therm al fields) are classical as well, whereas an y state that can n ot be un derstood as an en sem ble of coheren t states is called nonclassical. The experim en tal gen eration an d application of n on classical light fields is the m ain subject of this report. Despite m uch recen t progress, producin g n on classical states of light is still extrem ely challen gin g because they are easily destroyed (reduced to classical) by an y kin d of losses. Furtherm ore, it turn s out that the vacuum is a coheren t state as well because it satisfies Eq. (3) for a = 0. In other words, the vacuum is a zero-am plitude coheren t state. W ith a little algebra we see directly from Eq. (3) that the m ean (i.e., quan tum expectation value of the) en ergy of a coheren t state with un it frequen cy is Equation (4) is the sum of the classical wave in ten sity |a|2 an d the vacuum zero-poin t en ergy 1/2. On e sim ply m ultiplies the right-han d side of Eq. (4) by Aco to put ^aj in to un its of en ergy. 55 z den otes the z-axis direction of beam propagation . UNCLASSIFIED//POR ePFIMA L U3E ONL¥ UNCLASSI FI ED//mn QrriG TA I UOE ONLY M orris an d Thorn e [8] an d Caves [45] poin t out that if on e squeezes the vacuum , i.e., if on e puts vacuum rather than laser light in to the in put port of a squeezin g device, then on e gets at the output an electrom agn etic field with weaker fluctuation s an d thus less en ergy den sity than the vacuum at location s where cos2I co(/ — z/c)J = 1 an d s/n 2[(o{/-z/c)J < < 1; but with greater fluctuation s an d thus greater en ergy den sity than the vacuum at location s where cw^(o(f-z/c)] < < 1 an d Jin 2^o)(f-z/c)]sl. Sin ce the vacuum is defin ed to have van ishin g en ergy den sity, an y region with less en ergy den sity than the vacuum actually has a n egative (ren orm alized) expectation value for the en ergy den sity. Therefore, a squeezed vacuum state con sists of a travelin g electrom agn etic wave that oscillates back an d forth between n egative en ergy den sity an d positive en ergy den sity, but has positive tim e-averaged en ergy den sity. In quan tum optics the squeezed state is gen erated by the un itary squeezin g operator: (5) where £ is a real n um ber that param eterizes the deviation of the varian ces & q an d Sp from their vacuum values an d is called the squeezin g param eter. From Eq. (5) we obtain the squeezed vacuum state |< p) = S(£)|O). The squeezin g operator S(^) is sim ply an evolution operator that describes the result of the n on lin ear squeezin g in teraction Ham ilton ian Him = ^^"d2-hd'^. The squeezin g param eter ^ con tain s the product of the am plitude b, the couplin g con stan t /, an d the in teraction tim e. But this is n ot the en tire story. Sin ce we will be dealin g with high-quality lasers in what follows, we also n eed to kn ow about an other im portan t quan tum optics operator that acts on coheren t states. W e in troduce the un itary displacem en t operator D(a) = eJ7j(aaf-a'a). /5(a) displaces the am plitude d by the com plex n um ber a accordin g to D\a) d D(a) = d + a ■ To show why /5(a) has an ythin g to do with coheren t states, we apply a n egative displacem en t to |a). From the basic property of /5(a), we see that «D (-a)|a) = d(-a)A!(-a)aD(-a)|a) = D (-a)(«-a)|a) = 0. (6) Equation (6) equals zero because of the defin ition Eq. (3) of coheren t states. This result im plies that D(-a)|a) = |0), which is the vacuum state. Therefore, coheren t states |a) are displaced vacua |a) = /5(a)|0). This does n ot m ean that coheren t states 10 UNCLASSIFIED//TOW OFFICIA L USE ONLY UNCLASSIFIED//ron OFFICIA L UD E eMt* are physically sim ilar to vacuum states, but in stead they have on ly som e quan tum n oise properties in com m on . It is a well kn own result in the quan tum field theory of light that the vacuum wave fun ction is a sim ple Gaussian fun ction of the quadratures (in either q or p represen tation ), an d thus coheren t states are also Gaussian [38]. Furtherm ore, a proof of Heisen berg's Un certain ty Prin ciple in con jun ction with the application of SIQ an d D(a) on the quadrature varian ces an d wave fun ction s showed that all m in im um un certain ty states are displaced Gaussian states such that they have displaced rescaled vacuum wave fun ction s. Con sequen tly, all m in im um un certain ty states are displaced squeezed vacua [18, 38]: |v) = O(a)S©|0). (7 ) The squeezin g in teraction HM is realized by the degen erate param etric am plification of the spatial-tem poral m ode. A crystal such as potassium titan yl phosphate (KTP) or lithium n iobate (LiN bOa) is pum ped by an other laser beam with am plitude b an d twice the frequen cy of the spatial-tem poral m ode (with am plitude a) of in terest. Accordin g to Hym, the "B" photon s (correspon din g to b) of the pum p beam are con verted in to pairs of "A" sign al photon s (correspon din g to a2 an d a'2) with a probability that depen ds on the couplin g con stan t % • The KTP or LiN bOa crystal acts like an electrom agn etic swin g, an d the pum p m odulates the oscillation of the "A" m ode at twice its frequen cy. The pum p am plifies the sign al param etrically m uch as a swin g is am plified by chan gin g the effective len gth at twice the frequen cy of the swin g. A classical swin g relies on tin y in itial fluctuation s (or "wobbles") that are in -phase with respect to the param etric pum p. In this way, the tin y fluctuation s are am plified; the swin g starts to oscillate. A quan tum swin g like the degen erate param etric am plifier experien ces at least the vacuum fluctuation s from the very begin n in g. Vacuum fluctuation s that are in -phase with respect to the pum p are am plified, whereas out-of-phase fluctuation s get de­ am plified or, in other words, squeezed. A squeezed vacuum requires a pum p for gen eration , an d, hen ce, when produced it carries en ergy. The n on lin ear crystal KTP or LiN bOs is a reson ator that is shaped like a cylin der with roun ded silvered en ds to reflect light. This reson ator acts to produce a secon dary lower frequen cy light beam in which the pattern of photon s is rearran ged in to pairs. The squeezed light em ergin g from the reson ator will con tain pulses of n egative en ergy in terspersed with pulses of positive en ergy. To quan tify the am oun t of squeezin g en ergy we 1) apply S(% ) to the quadratures an d fin d that it scales their eigen fun ction s;*" 2) we then substitute for a its quadrature decom position (given in Sect. IIB-1) an d substitute that result in to the scaled quadratures; an d then 3) do further algebra to derive how S(Q chan ges a: S\tyaS(ty=acosht)-arsinh^. W e substitute this last result in to Eq. (1) an d use Eq. (7) to calculate the quan tum expectation value in order to express the m ean en ergy of a squeezed state, an d obtain '" i.e., q gets squeezed an d p gets stretched. 11 UNCLASSIFIED/ /FUR UH H LIA L USE ONLY UNCLASSI FI ED//FOR OmCIA L USE ONLY (VI ^sqvac | v) = H 2 + | + ^^^ . (8) Equation (8) really describes the m ean photon n um ber of a sin gle m ode in a squeezed state, but on e sim ply m ultiplies the right-han d side by ho to get the m ean en ergy UAqvac) = hf.^|«|' +^ + .v/7z/r^. W e see in Eq. (8) that there are three term s con tributin g to the en ergy: the first term accoun ts for the coheren t en ergy given by |a|2, the secon d term is the vacuum zero-poin t en ergy 1/2, an d the third term quan tifies the fluctuation en ergy of squeezed states. The con tribution to this squeezin g en ergy origin ally com es from the pum p used to gen erate the squeezed light. It is stored in the en han ced fluctuation s of the an ti-squeezed com pon en t. Because both the squeezed an d the an ti-squeezed quadratures con tribute to the secon d lin e in Eq. (1), even a squeezed vacuum carries en ergy. However, Eq. (8) is n ot the fin al result because it on ly gives the m ean en ergy of a sin gle m ode in a squeezed state, while lasers an d n on lin ear crystal reson ators produce a very large n um ber of m odes. Equation (8) n eeds to be sum m ed (in tegrated) over the in fin ite n um ber of possible m odes; it m ust then be "ren orm alized" by sophisticated m athem atical techn iques in order to get rid of the divergen t (in fin ite) con tribution from the vacuum zero-poin t en ergy (a byproduct of takin g an in fin ite sum of m odes); an d then the result m ust be con verted in to un its of en ergy den sity by dividin g it by an appropriate volum e elem en t, because Ein stein 's gen eral theory of relativity requires an en ergy den sity (or pressure, both are in the sam e un its) to in duce spacetim e ben din g. The fin al result we seek is the en ergy den sity, pE-sqvac, given by Pfen n in g [46]: sinh S, ^sinh ^ + cosh ^ cos (2o(f - z/c)+3) J (J / m3), (9) where Ly is the volum e of a large box with sides of len gth L (i.e., we put the quan tum field in a box with periodic boun dary con dition s) an d 8 is the phase of squeezin g. Equation (9) shows that pE-sqvac falls below zero on ce every cycle when the con dition cosh^ > sinh^ is m et. It turn s out that this is always true for every n on zero value of ^, so pE-sqvac becom es n egative at som e poin t in the cycle for a gen eral squeezed vacuum state. See Figure 1 for an illustration . N ote in the figure that the blue troughs or valleys are the n egative en ergy pulses. On an other n ote, when a quan tum state is close to a squeezed vacuum state, there will alm ost always be som e n egative en ergy den sities presen t. An other way to gen erate n egative en ergy via squeezed light would be to m an ufacture extrem ely reliable light pulses con tain in g precisely on e, two, three, etc., photon s apiece an d com bin e them together to create squeezed states to order. Superim posin g m an y such states could theoretically produce bursts of in ten se n egative en ergy. Photon ic crystal research has already dem on strated the feasibility of usin g photon ic crystal waveguides (m ixin g together the classical an d quan tum properties of optical m aterials) to en gin eer light sources that produce beam s con tain in g precisely on e, two, three, etc., photon s. See Referen ce [1] for m ore details an d for the referen ces cited therein . 12 UNCLASSIFIED//rOR OFFICIA L UOC ONL¥ UNCLASSIFIED//EQn QITTCTA I UD E OH LT Figure 1. Illustration of a Squeezed State of Light, (courtesy of Lisa Burn ett) Negative (Sub-Vacuum) Energy in the Casimir Effect The Casim ir effect origin ates from the quan tum electrom agn etic vacuum ZPF. It is by far the easiest an d m ost well kn own way to gen erate (static) n egative en ergy in the lab. The Casim ir effect that is fam iliar to m ost people is the force that is associated with the quan tum vacuum electrom agn etic ZPF [47]. This is an attractive force that m ust exist between an y two n eutral (un charged), parallel, flat, con ductin g surfaces (e.g., m etallic plates) in a vacuum , This force has been well m easured an d it can be attributed to a m in ute im balan ce in the vacuum electrom agn etic ZPE den sity in side the cavity between the con ductin g surfaces versus the vacuum electrom agn etic ZPE den sity in the free-space region outside of the cavity [48-50], See Figure 2 for a schem atic of the Casim ir effect. 13 UNCLASSIFIED//EQR QmCIA L UD E CM Lit UNCLASSIFIED/ /FOR UPP1LIA L USE ONLY It turn s out that there are m an y differen t types of Casim ir effects foun d in quan tum field theory [20-22, 26-28, 51]. For exam ple, if on e in troduces a sin gle in fin ite plan e con ductor in to the M in kowski (flat spacetim e) vacuum by brin gin g it adiabatically from in fin ity so that whatever quan tum fields are presen t suffer n o excitation but rem ain in their groun d states, then the vacuum (electrom agn etic) stresses in duced by the presen ce of the in fin ite plan e con ductor produces a Casim ir effect. This result holds equally well when two parallel plan e con ductors (with separation distan ce d) are presen t, which gives rise to the fam iliar Casim ir effect in side a cavity. N ote that in both cases, the spacetim e m an ifold is m ade in com plete by the in troduction of the plan e con ductor boun dary con dition (s). The vacuum region put un der stress by the presen ce of the plan e con ductor(s) is called the Casim ir vacuum . The gen eric expression for the en ergy den sity of the Casim ir effect is pCE = -Ahcd^, where A = ^(D)/8n 2 in spacetim es of arbitrary dim en sion D [20-22]. The appearan ce of the zeta-fun ction C fD) is characteristic of expression s for vacuum stress-en ergy ten sors, 7^ -tt+ 1° our fam iliar 4-dim en sion al spacetim e (D = 4) we have that A = n 2/720 . To calculate 7^ for a given quan tum field is to calculate its associated Casim ir effect. W e should also poin t out that the m ethods used to obtain the quan tum vacuum electrom agn etic 7^ between parallel plan e con ductors can also be used when the con ductors are n ot parallel but are join ed together alon g a lin e of in tersection . If the con ductors have curved surfaces in stead, then on e obtain s results that are sim ilar to the case of in tersectin g con ductors. These geom etries have also been evaluated for the tt+ The Greek ten sor in dices (p,v = 0...3) den ote spacetim e coordin ates, x°...x3, such that xT-x3 = space coordin ates an d x° s tim e coordin ate. N ote in gen eral that 7’°" - pE (field energy density). 14 UNCLASSIFIED//FOR OFFICIA L UOC ONLY UNCLASSIFIED//EQR QmCIA L USE ON LT case of dielectric m edia. These particular cases will n ot be con sidered further sin ce there are techn ical subtleties in volved that com plicate the calculation s an d application of the differen t approaches. As a fin al n ote, n egative en ergy can be created by a sin gle m ovin g reflectin g (con ductin g) surface (a.k.a. a m ovin g m irror) via the dyn am ical Casim ir effect. A m irror m ovin g with in creasin g acceleration gen erates a flux of n egative en ergy that em an ates from its surface an d flows out in to the space ahead of the m irror [23, 52]. This is essen tially the sim ple case of an in fin ite plan e con ductor un dergoin g acceleration perpen dicular to its surface. If the acceleration varies with tim e, the con ductor will gen erally em it or absorb photon s (i.e., exchan ge en ergy with the vacuum ), even though it is n eutral. This is an exam ple of the well-kn own quan tum phen om en on of param etric excitation . The param eters of the quan tum electrom agn etic oscillators (e.g., their frequen cy distribution fun ction ) chan ge with tim e owin g to the acceleration of the m irror [53]. However, this effect is kn own to be exceedin gly sm all, an d it is n ot the m ost effective way to produce n egative en ergy for our purposes. W e will n ot con sider this schem e an y further. QUANTUM OPTICAL HOMODYNE TOMOGRAPHY Observing Negative Energy in the Lab N egative en ergy should be observable in lab experim en ts. A gen eric, n on -optical schem e for detectin g n egative en ergy in experim en ts was recen tly reported by Davies an d Ottewill [54] who studied the respon se of switched particle detectors to static n egative en ergy den sities an d n egative en ergy fluxes. Their m odel is based on a free (m assless) scalar field in flat 4-dim en sion al M in kowski spacetim e an d utilized a sim ple gen eralization of the stan dard m on opole detector, which is switched on an d off to con cen trate the m easurem en ts on periods of isolated n egative en ergy den sity (or n egative en ergy flux). The detector m odel in cludes an explicit switchin g factor whereby five differen t switchin g fun ction s (based on data win dowin g theory) are defin ed an d evaluated. In order to isolate the effects of n egative en ergy, a com parison is m ade for the respon se of a detector switched on an d off durin g a period of n egative en ergy den sity (or n egative en ergy flux) an d that switched on an d off in the vacuum . The results shed light on the respon se of m atter (detectors) to pulses of n egative en ergy of fin ite duration , an d they showed that n egative en ergy should have the effect of en han cin g de-excitation (i.e., in duce coolin g) of the detector. This is the opposite of our experien ce with detectors that un dergo excitation when en coun terin g "n orm al" m atter or en ergy, an d isolated detectors placed in a vacuum n aturally cool due to the usual therm odyn am ic reason s. But Davies an d Ottewill poin t out that the en han ced coolin g effect they discovered can n ot be used to draw a therm odyn am ic con clusion because their m odelin g was restricted to first order in perturbation theory. It is n ot possible at first order to determ in e whether the en han ced coolin g effects are due to the sm all violation of en ergy con servation expected in an y process in which a gen eral quan tum state collapses to an en ergy eigen state, or whether they predict a system atic reduction in the en ergy of the detector which has serious therm odyn am ic im plication s. However, Davies an d Ottewill poin t out that their results are m odel depen den t an d they foun d for their stan dard m on opole detector m odel that there is n ot always a sim ple relation ship 15 UN CLASSI FI ED//FOR OFFICIA L WOE ONLY UNCLASSI FIED/ / EQB nmCIA L USE CNET between the stren gth of the n egative en ergy den sity/flux an d the behavior of the detector. It is curious that Davies an d Ottewill did n ot con sider usin g quan tum optical hom odyn e tom ography as a tool to test their hypothesis, because this is already a m ature experim en tal disciplin e. In what follows we outlin e the basics of quan tum optical hom odyn e tom ography an d its application to detectin g an d m easurin g n egative en ergy den sity/flux states in squeezed light an d in the Casim ir effect. B asic Notions of Quantum Optical Homodyne Tomography Tom ography, from the Greek word for slice, is a m ethod to in fer the shape of a hidden object from its shadows (or projection s) un der various an gles. Quan tum tom ography is the application of this idea to quan tum m echan ics. In optical hom odyn e tom ography, the W ign er fun ction or, m ore gen erally, the quan tum state plays the role of the hidden object. The observable "quan tum shadows" are the quadrature distribution s an d are m easured usin g hom odyn e detection . From these distribution s the W ign er fun ction is recon structed. See Figure 3 for an illustration of quan tum optical hom odyn e tom ography. The vertical 2-dim en sion al plan e seen in the figure is fictitious an d is shown for illustrative purposes on ly. Figure 3. Illustration of Quantum Optical Homodyne Tomography (courtesy of Ulf Leon hardt). The W ign er fun ction (3-dim en sion al hill on the right) is recon structed in quan tum phase space (gridded plan e form ed by quadratures q an d p) from its experim en tally m easured projection s (curve in vertical 2-dim en sion al plan e), which represen ts the scan n in g process of tom ography. The vertical axis is the m agn itude of the W ign er (quasiprobability) fun ction . Quan tum tom ography was developed for the sim ple reason that a fun dam en tal feature of quan tum m echan ics preven ts us from seein g physical objects in their full quan tum com plexity. This is due to the in trin sic fuzzin ess in the quan tum n ature of en ergy an d m atter accordin g to the Heisen berg Un certain ty Prin ciple, which preven ts us from sim ultan eously an d precisely m easurin g the com plem en tary features (e.g., position an d m om en tum or en ergy an d tim e) com prisin g quan tum states. For this reason we can n ot 16 UNCLASSIFIED//FW OmCIA L UOC ONLY UNCLASSIFIED//TOW OFFICIA L USE ONLY directly observe quan tum states, an d so the true n ature of an in dividual quan tum system is hidden . However, n o prin cipal obstacle exists to observin g all com plem en tary aspects in a series of distin ct experim en ts on iden tically prepared quan tum objects. In the section s that follow, we briefly review the several parts that com prise the tom ography m achin ery, an d then put the whole picture together to un derstan d what the en tire process is. N o effort will be m ade for com pleten ess because the subject of quan tum tom ography takes up volum es of books. The reader will be referred to the key literature of im portan ce. Wigner Functions In classical optics the state of an electrom agn etic oscillator is perfectly described by the statistics of the classical am plitude a. The am plitude m ay be com pletely fixed (then the field is coheren t), or a m ay fluctuate (then the field is partially coheren t or in coheren t). In classical optics as well as in classical m echan ics, we can characterize the statistics of the com plex am plitude a or, equivalen tly, the statistics of the com pon en t position q an d m om en tum p by in troducin g a phase space distribution called the W ign er fun ction , W(q,p~).^ W(q,p) quan tifies the probability of fin din g a particular pair of q an d p values in their sim ultan eous m easurem en t. Kn owin g W(q,p) for a particular quan tum state that is un der study, all statistical quan tities of the electrom agn etic oscillator can be predicted by calculation . In this sen se W(q,p) describes the state in classical physics. The m otivation for in troducin g the W ign er fun ction was the desire to fin d a quan tum m echan ical description sim ilar to that in classical statistical physics. However, in quan tum m echan ics Heisen berg's Un certain ty Prin ciple preven ts on e from observin g position an d m om en tum sim ultan eously an d precisely. In addition to this, we also can n ot directly observe quan tum states either. N evertheless, we are perfectly en titled to use the con cept of quan tum states as if they were existin g en tities. W e use their properties to predict the statistics of observation s. It is well kn own that the quan tum m echan ical wave fun ction depen ds exclusively on either the position or the m om en tum an d con tain s n evertheless all the in form ation about the quan tum system un der study. However, E. W ign er showed that it is possible to defin e a form al quan tum m echan ical an alog to the classical distribution fun ction . He showed that we could use W(q,p) as a quan tum phase space distribution exclusively to calculate observables in a classical-like fashion . W ign er discovered that W(q,p) is a real-valued fun ction , but it is usually n ot just positive; it can also becom e n egative. This is a very n on classical behavior for a probability distribution . It is for this reason that W(q,p) cam e to be called a quasiprobability distribution . W(q,p) has several properties an d m athem atical postulates, but it turn s out that just on e postulate is sufficien t for the purposes of quan tum tom ography [38]. Usin g this postulate, it is assum ed that W(q,p) behaves like a join t probability distribution for q an d p without ever m en tion in g an y sim ultan eous observation of position an d f+X p+oO W(q,p)dp or W(q,p)dq *** Recall in Sect. IIB-1 that the real an d the im agin ary parts of the com plex am plitude a can be regarded as the position an d the m om en tum of the electrom agn etic oscillator. 17 UNCLASSIFIED//FOR OTriOIA L USE ONLY" UNCLASSI FI ED//FOA OFFICIA L USE ONLY m ust give the position or the m om en tum distribution , respectively. Furtherm ore, if on e perform s a phase shift 0 all com plex am plitudes a are shifted in phase,§§§ m ean in g that the com pon en ts q an d p rotate in the 2-dim en sion al phase space (q,p). A classical probability distribution for position an d m om en tum values would rotate accordin gly. This fact leads to the postulate that the position probability distribution pr(q,Q } after an arbitrary phase shift 0 should be [38] rt.e)s(«|W)p where a^ is the com plex con jugate of aw. The phase of the LO is 0, an d so we n ote from the defin ition of q^ that the m easured quan tity hi is in deed proportion al to ?6 because «21 = 2' ’ |aLO|g0, which is a result that has been verified by m ore sophisticated theories of hom odyn e detection [38]. A balan ced hom odyn e detector m easures q^. The referen ce phase 0 is provided by the LO an d can be varied by adjustin g the LO usin g a piezo-electrically m ovable m irror, for exam ple. An experim en tal m ethod for fin din g the scalin g of q0 in the differen ce curren t hi is to keep a record of the sum curren t because the sum of /i an d h is proportion al to |aLOf to leadin g order [38]. This can be experim en tally im portan t because the in ten sity of the LO is usually an un kn own quan tity. 28 UNCLASSIFIED/ /H UH UPF1LIA L USE ON LT UNCLASSI FI ED//FOR OmCIA L USE ONLY Figure 10. Schematic of a B alanced Homodyne Detector, (courtesy of Ulf Leon hardt) Furtherm ore, the balan ced hom odyn e detector is also an am plifier. The LO am plifies the sign al by the m utual optical m ixin g of the two. In other words, the hom odyn e detector is an in terferom eter that can be m easurably im balan ced by a sin gle photon in the sign al m ode because the referen ce field is very in ten se. A very im portan t techn ical advan tage of this is that the am plified sign al is well above the electron ic n oise floor of the photodiodes. The sign al am plitude is en han ced so that even the n oisy lin ear- respon se photodiodes can detect the quan tum features of the sign al with sin gle photon resolution . Because the LO serves as a coheren t am plifier, it also chooses the sign al m ode. The LO sin gles out on e spatial-tem poral (boson ic) m ode from the rest of the con tin uous quan tum field "light" (that m atches the LO field). In this way the observer separates the quan tum object (a sin gle optical m ode) from the rest of the world. The m ode fun ction is given by the spatial-tem poral shape of the LO beam at the detector surface an d durin g the m easurem en t tim e in terval [0,7]. The overall phase an d in ten sity of the LO is com prised in the com plex am plitude otto. Shiftin g the phase 0 = arg(ai.o) rotates the m easured ?n . The observer defin es via the LO the fram e in space an d tim e that is subject to the field-quadrature m easurem en t. By tailorin g the shape of the LO beam high spatial-tem poral resolution can be achieved. Photodetection is usually n ot com pletely efficien t in practice so it is im portan t to describe the in fluen ce of in efficien cies on hom odyn e detection . This is easily don e by usin g the sim ple m odel for losses in direct photodetection that was given in Section IIIB-2. W e im agin e fictitious beam splitters to be placed in fron t of the two (assum ed ideal) detectors in the m easurem en t setup (see Figure 11). W e use 29 UNCLASSIFIED//TOR UU1L1A L U3E ONLY UNCLASSI FIED//FOP OFFICIA L UOE ONET 0' = ^' a + O-^)1 :a2 from Section IIIB-2 to defin e the an n ihilation operators of the detected light m odes «7= h' ~^^ + (1—H) ^! an d «’ -T]l_a'+(l-i])l_&2( where b} an d b^ are the an n ihilation operators of the vacua en terin g the secon d un used ports of the fictitious beam splitters. The an n ihilation operators a\ an d a\ describe the light m odes (or fields) em ergin g from the 50:50 beam splitter where the sign al is optically m ixed with the LO. Again , the LO is an in ten se field com pared with the sign al so it can be treated classically. Therefore, we do som e algebra to com pute the differen ce photon n um ber n 2l =h”2-n” = a? a" - a"^a", but retain on ly the leadin g term s with respect to ato, an d obtain the fin al result [38]: * 1/2 * ^21 — H ^LOnl/2a+(i-n),'2^+//c. (12) The sym bol HC in Eq. (12) den otes the Herm itian con jugate of the other part of an expression an d & = 2“'2 ^2-6, j. The fluctuation m ode operator b correspon ds to the optical m ixin g of the fictitious vacuum -n oise m odes bx an d b2, an d it obeys the boson ic com m utation relation ^,//J = l (e.g., see Sect. IIIB-2). Because the in terferen ce of vacuum with vacuum yields vacuum , the fluctuation m ode b can be regarded as a boson ic m ode, bein g in the vacuum state as well. 30 UNCLASSIFIED/ /FOR OFFICIA L UOC ONLY U N CLASSI FI E D/ / FOR OFFICIA L USE ONLY Figure 11. B alanced Homodyne Detector Using Fictitious B eam Splitters to Account for Detection Losses, (courtesy of Ulf Leon hardt) Equation (12) provides an addition al m odel for detection losses. Sim ilar to direct photon coun tin g, a fictitious vacuum field has to be added to the atten uated sign al in hom odyn e detection . This m ean s that we can replace the arran gem en t of two fictitious beam splitters in fron t of the photodetectors with just on e effective beam splitter in fron t of an ideal hom odyn e detector (see Figure 12). This effective beam splitter accoun ts for other kin ds of losses in cludin g m ode m ism atch, whereby the quan tum effects of both detection losses an d m ode m ism atch are com prised in an effective q. 31 UNCLASSIFIED//W» OmCIA L USE ONLY UNCLASSIFIED//POR OFFICIA L USE ONLY d etector local oscillator («LO) Figure 12. B alanced Homodyne Detector Using A Single Effective Fictitious B eam Splitter to Account for Detection Losses and Mode Mismatch, (courtesy of Ulf Leon hardt) The con sequen ce of an effective q an d Eq. (12) is that the m argin al distribution s pr(q,Q ) m ust becom e a fun ction of the effective q [38]: where the pr(x,B) in side the in tegral is defin ed by Eq. (10) an d x is a dum m y in tegration variable. Equation (13) defin es the m easured quadrature histogram s that are used to build the tran sm ission profiles in the tom ographic process, which is discussed in the followin g section . Outline of Experimental Procedure The key process of quan tum tom ography is to picture the "shape" of a quan tum object in phase space usin g the W ign er represen tation . The m argin al distribution s [Eq. (10) or (13)] correspon d to the tom ographic tran sm ission profiles of the W ign er fun ction W(q,p), i.e., to shadows projected on to a lin e in quan tum phase space. Because of the Heisen berg Un certain ty Prin ciple, we can n ot m easure sim ultan eously an d precisely the position q an d the m om en tum p, an d we can n ot observe the W ign er fun ction directly as a probability distribution . However, we can m easure the quadrature histogram s [i.e., the first lin e in Eq. (10)], an d by varyin g the phase 0 we observe the quan tum object un der differen t an gles. Given the pr{q$), the m athem atics of com puterized tom ography can be applied to deduce the W ign er fun ction . 32 UNCLASSI FIED/ / FOR OFFICIA L UOC ON MF UNCLASSIFIED//rOR OmOIA L UD E ONLY As discussed in the previous subsection , we can use balan ced hom odyn e detection to precisely m easure the quadratures ^0 of a spatial-tem poral m ode. As was also discussed in the previous subsection , the an gle 0 is defin ed by the phase of the local oscillator with respect to the sign al. The phase 0 can be varied usin g a piezo-electric tran slator. To m easure the quadrature distribution s, on e m ay fix 0 an d perform a series of hom odyn e m easurem en ts at this particular phase to build up a quadrature histogram . Then the LO phase should be chan ged in order to repeat the procedure at a n ew phase, an d so on . An other possibility is to m on itor the phase while it drifts or to sweep it in a kn own way. In an y case, the hom odyn e m easurem en t m ust be repeated m an y tim es on iden tically prepared light m odes (or on a con tin uous wave field) to gain sufficien t statistical in form ation about the quadrature values at a certain n um ber of referen ce phases. Fin ally, the W ign er fun ction is tom ographically recon structed from the experim en tal data. It is beyon d the scope of this report to sum m arize the en tire subject of experim en tal quan tum tom ography, its m athem atical basis an d procedures of quan tum state sam plin g, an d the correspon din g algorithm s an d n um erical recipes. The reader should see Referen ce [58] for the excruciatin g details. Balan ced hom odyn e detectors with local oscillators are am plifiers capable of detectin g an d quan tifyin g vacuum an d sub-vacuum fluctuation s. This is the subject of the two experim en tal approaches that will be discussed in the n ext section . B ALANCED HOMODYNE SYSTEMS FOR MEASURING NEGATIVE (SUB -VACUUM) ENERGY Time-Domain B alanced Homodyne System Squeezed states of light, which are "darker than vacuum ," have region s with sub­ vacuum fluctuation s. Slusher an d collaborators [40, 41] an d Robin son [42, 43] were the first to experim en tally observe these sub-vacuum region s. N um erous other experim en ts followed, which em ployed variation s on the experim en tal devices an d techn iques used to gen erate squeezed light an d m easure its sub-vacuum fluctuation pulses. Those early experim en tal devices later gave way to the developm en t an d use of balan ced hom odyn e detectors. For exam ple, Schn eider et al. [59] describe their com pact an d efficien t source of am plitude-squeezed light. Their experim en t used a sem i-m on olithic degen erate M gO:LiN bO3 optical param etric am plifier pum ped by a frequen cy-doubled N d:YAG laser at 532 n m . They em ployed in jection -seedin g of the am plifier by a 1064 n m wave to provide active stabilization of the cavity len gth an d stable operation . At a pum p power of 380 m W , their device detected a m axim um n oise reduction of 6.5 dB in the am plitude fluctuation s of the 0.2 m W 1064 n m wave, while the average detected n oise reduction in con tin uous operation over 14 m in utes was 6.2 dB. They reported a squeezin g of 7.2 dB in the em itted wave. However, m ost of these early an d m ore recen t series of balan ced hom odyn e detector (BHD) m easurem en ts have been perform ed in the frequen cy dom ain . A sign ifican t 33 UNCLASSIFIED//POR. OPPICIA L USE ONLY UNCLASSIFIEDZ/FOR QFFTOA L USE OMHF drawback of this approach is that it reveals in form ation about the quan tum state on ly within the sideban d chosen for the m easurem en t. Therefore, the m ethod is in com patible with other techn iques for characterizin g a quan tum state for which such precise selection of spectral m odes is im possible. Tim e-dom ain BHD resolves this lim itation . Han sen et al. [4] describe their experim en tal tim e-dom ain BHD device. They developed a pulsed BHD for precise m easurem en t of the electric field quadratures of pulsed optical quan tum states. A high level of com m on m ode suppression (> 85 dB) an d low electron ic n oise (730 electron s per pulse) in their device provides a sign al-to- n oise ratio of 14 dB for m easurem en t of the quan tum n oise of in dividual pulses. Their device achieved a sign al-to-n oise ratio of 14 dB at a pulse repetition rate of up to 1 M Hz, en ablin g high-accuracy quan tum m easurem en ts to be carried out in a short tim e. They perform ed a quan tum tom ography of the coheren t state as a test for their device, an d the W ign er fun ction an d den sity m atrix were recon structed with 99.5% fidelity while their detector exhibited 91% quan tum efficien cy. Their detection system can also be used for ultrasen sitive balan ced detection in con tin uous wave m ode. Figure 13 shows a schem atic of their tim e-dom ain BHD. The figure shows two polarizin g beam splitter (PBS) cubes, a 50:50 beam splitter (BS), two half-wave plates (1/2), two photodiodes (left-side in dotted box), an d the sign al processin g electron ics in side the dotted box. Figure 13. Time-Domain B alanced Homodyne Detector, (courtesy of P. Lodahl) As we discussed previously in Section s IIIB-4 an d IIIB-5, to perform BHD on e overlaps on a beam splitter the electrom agn etic wave whose quan tum state is to be m easured an d a relatively stron g LO wave in the m atchin g optical m ode. The two fields em ergin g from the beam splitter are in ciden t upon two high efficien cy photodiodes whose output photocurren ts are subtracted. The photocurren t differen ce is proportion al to the value of the electric field operator Ee in the sign al m ode, where 0 is the relative optical phase of the sign al an d the LO. In tradition al frequen cy-dom ain BHD, on e uses a certain frequen cy com pon en t of the differen ce sign al to determ in e the quadrature quan tum n oise of the optical state. The m easurem en t frequen cy is n orm ally chosen to be approxim ately 5 to 10 M Hz where the techn ical n oise is m in im ized. Figure 14 shows an exam ple of experim en tally m easured data for a typical (un disturbed) vacuum state an d a squeezed vacuum state usin g a tim e-dom ain BHD system . 34 UNCLASSIFIED//H UK UFP111A L UBE QNLV UNCLASSI FIED/ / FOR OFFICIA L USE ONLY Figure 14. Experimentally Measured Squeezed State, (courtesy of P. M arecki) This graph of vacuum dB n oise vs. relative optical phase an gle shows an experim en tally m easured squeezed state (plot (I)) an d a n orm al (un disturbed) vacuum state (plot (II)). The deep valleys with n egative dB values in plot (I) are sub-vacuum region s with sub-vacuum (n egative) en ergy den sity (see also, Figure 1 for a com parison ). W hen applied to pulsed sources, the frequen cy-dom ain BHD techn ique im plies that averagin g over m an y in dividual laser pulses takes place. However, in tim e-dom ain BHD, each laser pulse gen erates a sign al that is observed in real tim e an d yields a sin gle value of a field quadrature. Repeated m easurem en ts of a large n um ber of laser pulses produce a quan tum probability distribution associated with this quadrature. W hen tran sform -lim ited LO pulses are used, tim e-dom ain BHD gives the com plete in form ation about the quan tum state in the spatial-tem poral m ode that m atches that of the LO. Han sen et al. [4] poin t out that tim e-dom ain BHD is techn ically challen gin g, because 1) the electron ics m ust en sure tim e resolution of in dividual laser pulses an d 2) the m easured quadrature values m ust n ot be in fluen ced by low-frequen cy n oise. The detector m ust provide ultralow n oise, high subtraction , an d a flat am plification profile in the en tire frequen cy ran ge from DC to at least the LO pulse repetition rate. See Referen ce [4] for a com plete description of their device as shown in Figure 13. 35 UNCLASSIFIED//EQR QFFIG IA L USE ONLY UNCLASsiFiED//rng ngri«A i uoe omw B alanced Homodyne System for Casimir Cavities W hat has n ot been experim en tally m easured yet are the sub-vacuum fluctuation s an d their (n egative) en ergy den sity in side a Casim ir cavity. M arecki [5, 6] theoretically evaluated the use of BHDs for this purpose. He proposed that a BHD can be used to detect an d spatially m ap the sub-vacuum fluctuation region in side a Casim ir cavity as well as m easure its n egative en ergy den sity spectrum . M arecki discovered that by exploitin g a trick with the subtraction of the output of balan ced photodiodes, it is possible to quan tify the fluctuation s of the quan tum field (even in the vacuum !), which un iquely addresses Davies an d Ottewill's [54] n egative en ergy detector hypothesis. The quan tity of in terest (to be m easured) is the fluctuation s of the quan tum electric field (.E^x^E^xj)} (for fields restricted to the frequen cy co of the local oscillator) for squeezed an d vacuum states, where E.{x,t) is the quan tum electric field operator (in groun d-state represen tation an d restricted in frequen cies) at the poin t x, t * r represen ts the tim e-depen den ce of E.(x,t), an d (,..)s stan ds for the expectation value with respect to an arbitrary in itial state S (vacuum , squeezed, groun d state, coheren t, therm al, etc.) of the quan tum radiation field un der study. (E^x^Ej^xj)^ IS also called a two-poin t fun ction . In quan tum field theory, the expectation value (or m atrix elem en t) com puted by in sertin g a product of two quan tum operators between two states, usually the vacuum states, is called a two-poin t fun ction . This quan tity suggests a "relation " between two states in the sam e dyn am ics, an d it expresses the fluctuation s of a quan tum field. The product of n -operators is called the n -poin t fun ction which expresses the higher m om en ts of the quan tum field fluctuation s. The goal of the experim en t is that a state S of the quan tum radiation field un der study n eeds to be characterized by its n -poin t fun ction s. The typical solution in quan tum optics is to use well-characterized quan tum system s in teractin g in a sim ple way with the quan tum radiation field. The detection schem e uses the sim ple m odel of a PIN jun ction photodiode in which a sin gle electron in teracts with the quan tum radiation field un der study. This sim ple in teraction m ean s that the state space of the electron can be severely restricted, the in teraction is assum ed to be lin ear in the quan tum field, an d so the Born approxim ation can be used [5, 6], The PIN jun ction m odel of the photodetection process is an electron in an in itial state |0)®5, with its boun d-state |o) well-localized aroun d a certain poin t Jc0, that gets excited to the con tin uum of scatterin g states |^) by the quan tum field state of in terest S such that the fin al states of the system are |^)®S .**** The excitation is caused by the lin ear (dipole approxim ation ) in teraction with the quan tum electric field which is **** The sym bol ® den otes the ten sor product of two quan tum eigen states such that |n ,, tt) = |aj®|«2} for factorized eigen states which correspon d to in depen den t m easurem en ts. 36 UNCLASSIFIED//FOR OFFICIA L UOE ONL¥ UNCLASSIFIED//YOU OFFICIA L USE ONLY ^dip-int ~ex' ®Ei(X’t) • &(O where e is the electron charge an d g(t) is a sm ooth test (or sm earin g) fun ction that is equal to 1 durin g the m easurem en t an d sm oothly van ishin g elsewhere. Usin g first-order tim e-depen den t perturbation theory, M arecki [5, 6] derived the probability of excitation : r r+°° ■■ / * r ^ r \ ^excU^)= f drdsg(r)g(s)G ,J(r-s)/E i(x,T )E :(x,s)) , (14) J—CO \ J / c where G'j(T- s) = jdq(0|x'(r)|^($|x’^^^I^) is the electron ic two-poin t fun ction , t an d s are dum m y tim e an d in tegration variables, an d ^dids^{r')g{s) is the tem poral sen sitivity in the m easurem en t process. The balan ced hom odyn e detector con sists of an arran gem en t of two photodiodes, whose outputs are subtracted, an d illum in ated with an auxiliary coheren t state of the radiation field (i.e., the local oscillator, LO; see Figure 15). Per the discussion in Section IIIB-4, the LO is used as a tool to in vestigate the properties of a certain state S of the quan tum radiation field un der study, an d so on a BHD the state S is optically m ixed with the coheren t LO state (see Referen ces [5] or [6] for further details). The quan tum field S de-balan ces the detector (stochastic process of m easurem en t). The expectation value of the observable correspon din g to the electron ic charge collected at the poin t P in Figure 15 (i.e., the BHD curren t) is the differen ce of excitation probabilities of the two photodiodes [5, 6]: (J)s = ^.(^aO-Z^X?’T), where position s x an d y correspon d to the position s x an d y in Figure 15. Further calculation s an d other theoretical con sideration s lead to the followin g fin al result for (J}s [5, 6]: {J^s = cct, • E l o • \^i(xdu) + E^yd^)^ where ae / depen ds on the electron ic structure of the PIN sem icon ductor in the photodiode, E[o is the electric field of the LO (correspon din g to F in Figure 15), to is the LO phase that can easily be varied in * r experim en ts, an d all field operators E^xd) are restricted to the frequen cy co of the LO. 37 UNCLASSI FI ED/ / FOR OmCIA L UOC ONLY UNCLASSIFIED//£OB ^EEICXAUU6MNfc¥ Figure 15. B alanced Homodyne Detector with a Local Oscillator, (courtesy of P. M arecki) The setup is arran ged so that the electric field F of the LO at position X has a reversed direction with respect to that at position j. If (J^ van ishes, then the varian ce of the BHD-output is ^J2^ which provides a characterization of the two-poin t fun ction of the state S. The varian ce is [5, 6]: , “ a'e/*^LO ^LO*^ ^i (^» ^o) ! ^i^y^^ ^j (^’ ^) + E j (?’ ^o) (15) where E^E^ is the power of the LO field. This expression shows that ^J‘^ scales quadratically with the am plitude of the electric field of the quan tum state S un der study an d thus lin early with the power of the LO field. The two-poin t fun ction s can be quan titatively estim ated by perform in g m easurem en ts with differen t powers of the LO. Therefore, BHDs with local oscillators are am plifiers that are capable of m easurin g the on e- an d two-poin t fun ction s of arbitrary states of quan tum fields (even for the vacuum ). For an experim en tal study of the vacuum state in side a Casim ir cavity, the station ary state is specified to be the groun d state {Grd) an d thus the on e-poin t fun ction {J}^^ van ishes. For station ary states the ^J2^(/ is related to the spectral den sity < j(To),x,y) which is defin ed as the Fourier tran sform of the two-poin t fun ction \E^xd(^E^y,tn}^ with respect to tim e; therefore, we have for groun d states [5, 6]: . . 2 it i* 2^Grd ^/(^M p(^/(^X,y)|£((5y^ , (16) 38 UNCLASSIFIED//POR OFFICIA L USE ONL¥ UNCLASSIFIED//ron OFFICIA L UOC ONLY where j(w) is the Fourier tran sform of # (r) an d is sharply peaked aroun d w = 0. In gen eral, alm ost all results of quan tum field theory in a vacuum state or un der the in fluen ce of extern al con dition s (i.e., in vacuum states "deform ed" by boun dary con dition s or extern al fields) are derivable from the spectral den sity. This quan tity is usually kn own an alytically, an d it is of great in terest to m easure it for in terestin g quan tum field states. Because the groun d state is station ary, the quan tum n oise in the Casim ir cavity is tim e­ in depen den t, i.e., it is in depen den t of the phase of the LO, an d thus the spectral den sity is also tim e-in depen den t. M arecki [5, 6] derived the diagon al part of the spectral den sity for the /-com pon en ts of the quan tum electric field between two parallel, perfectly con ductin g plates (position ed atx = 0 an d x = a) in a Casim ir cavity (see Figure 2): r r S r -iaV).(w,i,x) = -T J [2(conL)-2(co|2x-nL|)] (17) for y = 0, where A = 2a is twice the distan ce between the plates an d the fun ction Q(x) is defin ed as sinx cosx _ sinx ' 5 ~ ■X X" X N ote that the diagon al term s of the spectral den sity are the im portan t quan tities to be m easured because they will be dom in an t if the photodiodes are separated by a sufficien tly large distan ce [5, 6]. Spectral den sities reveal m uch fin er details of the quan tum groun d state than already-m easured Casim ir forces do. By explorin g the freedom of choosin g the location s of the photodiodes in side the Casim ir cavity as well as the polarization s, phases an d frequen cies of the LO, on e can obtain a detailed characterization of on e- an d two-poin t fun ction s of an y state S of the quan tum electric field. Therefore, an application of this particular type of BHD m easurem en t, via Equation s (16) an d (17), am oun ts to a tom ography of the groun d state of the Casim ir cavity. For the experim en tal detection of the Casim ir spectral den sity with a BHD-type device, the Casim ir cavity plates are separated by a = 1 m icrom eters while the photodiodes in side it are of subm icrom eter width in the x-direction an d subm illim eter len gth in the /-direction (see Figure 16). Photodiodes of several n an om eters in size have already been con structed an d their high quan tum efficien cy version s are un der developm en t, see Referen ce [60] an d the referen ces cited therein for m ore techn ical in form ation . As shown in Figure 16, a coheren t state in the TEI m ode of the Casim ir cavity with a very sm all waven um ber in the /-direction provides an appropriate LO. A BHD with such a LO an d the photodiodes located as shown in Figure 17 would be sen sitive on ly to the /- com pon en t of the quan tum electric field. Figure 18 shows a schem atic of the BHD apparatus with a LO. In the figure, the lin early polarized sign al field S (if presen t) is optically m ixed with a coheren t state (LO), which is polarized orthogon ally to S, on the polarizin g beam splitter (PBS1). The half wave plate (HW P) reflects the plan es of polarization with respect to its optical axis, thereby in ducin g a k/4 shift of the plan e of 39 UNCLASSI FIED/ / FOR OITICIA L UD E ONLY UNCLAssiFiED//ran ornciA L uoc ontY polarization of the sign al field S. The subsequen t PBS2 separates the two orthogon ally polarized sign als, which are detected at the photodiodes PDx an d PDy. The charge collected at poin t V (correspon din g to poin t P in Figure 15) provides a m easure of (J)^ (an d its higher m om en ts). N ote that the setup is arran ged in such a way, that if S happen ed to be a m on ochrom atic coheren t state, then it would be phase-m atched to the LO at the poin t x, but shifted in phase by it at the poin t y. Figure 19 displays M arecki's com puter m odel plot of the predicted Casim ir spectral den sity as a fun ction of the distan ce from the plates x an d the frequen cy co. For a com parison with quan tum optics literature, he plotted the n orm alized differen ce between the vacuum an d groun d state spectral den sity in the figure (see Referen ces [5] an d [6] for m ore detail). N ote in the figure that for co < nc/a, the Casim ir spectral den sity van ishes: g^/cd.A'M '^O, while discon tin uities in it appear at (3) = micla. Figure 20 displays the correspon din g com puter m odel plot by M arecki of the predicted "suppressed" vacuum fluctuation s in the groun d state relative to "un disturbed" vacuum fluctuation s (in absen ce of the plates) in dB, 10L(?^10[cyGn /(to,x,x)/ovae(co,x,x)] . Figure 16. Diagram of Casimir Cavity with B HD Photodiodes, (courtesy of P. M arecki) Side view: -1000 -500 0 y [pm] 500 1000 Figure 17. Experimental Setup of B HD Photodiodes and LO Field, (courtesy of P. M arecki) This setup is drawn on the plot of the /-com pon en t of the electric field of the TEI m ode of the Casim ir cavity. The m ode, servin g as the LO, propagates in the z- direction perpen dicular to the plot. 40 UNCLASSIFIED/ /FOR OFFICIA L UD E ONLY UNCLASSIFIED//TOft OFFICIA L USE OWL* Figure 18. Detailed Schematic of Experimental B HD Apparatus, (courtesy of P. M arecki) N ote that poin t V correspon ds to poin t P in Figure 15. PD x Figure 19. Predicted Casimir Spectral Density Gyy (courtesy of P. M arecki) gx/ is related to the expected output of a BHD with the LO polarized alon g the /-direction (parallel to the plates) for the groun d state in the Casim ir cavity. This is plotted as a fun ction of the position x e [0, a] between the plates (separation a = 1 gm is assum ed) an d the frequen cy w e [0, 4n c/a]. N egative values (suppression of fluctuation s) are shown in deep purple. 41 UNCI ASSIFIED//JQB OFFTCTAI ll« »*»^ U N CLASSI FI E D/ / FUR UFF1L1A L USE ONLY dB Figure 20. Predicted Suppression of Vacuum Fluctuations in dB . (courtesy of P. M arecki) Vacuum fluctuation s in the groun d state (for field operators restricted to the frequen cy m ) relative to vacuum fluctuation s (in the absen ce of the plates) for a BHD at x = 0.25 pm (solid lin e) an d x = 0.5 pm (dashed lin e) within the cavity. The frequen cy ran ge is m e [0, 4rc/a], The predicted spectral den sity pattern shown in Figure 19 is static, i.e., it is in depen den t of the LO phase an d in som e region s correspon ds to the suppression of vacuum fluctuation s by at least 3 dB. Such a behavior is allegedly forbidden by a theorem kn own as the Quan tum In equalities for quan tum fields without extern al con dition s (i.e., "un deform ed," or "un disturbed," vacuum states). The theorem states that region s with sub-vacuum fluctuation s m ust be followed by region s with greatly in creased vacuum fluctuation s n o m atter what the state of the quan tum field is. This has on ly been verified for sin gle-m ode squeezed light, see, e.g., Figures 1 an d 14. A m ajor con sequen ce of this theorem is that sub-vacuum fluctuation s, an d their correspon din g sub-vacuum (n egative) en ergy den sity, can n ot persist for lon g tim es. W hat is surprisin g here is that M arecki (private com m un ication , Leipzig Un iversity, Germ an y, 2010) claim s that the Quan tum In equalities should also apply to the case of static sub-vacuum fluctuation s, an d their correspon din g static sub-vacuum (n egative) en ergy den sity, in side Casim ir cavities. The efficacy of the Quan tum In equalities theorem in its application to curved spacetim e physics, an d m ore specifically faster- than -light spacetim e geom etries, has been argued in the literature in which serious theoretical shortcom in gs of the theorem have been iden tified by several in vestigators (see Referen ce [1] for the details). Therefore M arecki's proposed Casim ir cavity BHD experim en t provides a possible test of yet un explored gen eric quan tum field theoretic effects in Casim ir geom etries, com plem en tary to m easurem en ts of Casim ir forces. W e hope that experim en tal attem pts to verify his prediction s will follow. 42 UNCLASSI FI ED//EQR QITICIA L UG C OMh¥ UNCLASSI FIED/ / FOR. OFFICIA L USE ONLY CONCLUSION Future aerospace platform s m ay have propulsion system s that m odify their surroun din g spacetim e geom etry to im plem en t faster-than -light spaceflight (via traversable worm holes or warp drives) or produce levitation via an tigravity. To en gin eer such a m odification of local spacetim e requires the use of quan tum sub-vacuum fluctuation s an d their associated sub-vacuum (or n egative) en ergy den sity. There are two key exam ples of specially prepared quan tum vacuum states that are kn own to produce sm all am oun ts of sub-vacuum (n egative) en ergy den sity in the laboratory. These are the well-kn own Casim ir effect an d squeezed light. There are several other exam ples of special quan tum vacuum states or particle states that produce sub-vacuum (n egative) en ergy den sity, but they are still un der theoretical study. W e already m ake sm all am oun ts of sub-vacuum (n egative) en ergy in the laboratory via the Casim ir effect an d squeezed light, but we do n ot yet kn ow if we can access larger am oun ts for exten ded periods of tim e over exten ded spatial distribution s. The Quan tum In equalities theorem suggests that producin g large am oun ts of sub-vacuum (n egative) en ergy in "deform ed" vacuum states for exten ded periods of tim e in flat or curved spacetim es m ay n ot be possible. This claim rem ain s as yet un tested by experim en t while several in vestigators have stron g argum en ts showin g the theorem is in error in these particular cases. Quan tum optical hom odyn e tom ography can detect an d quan tify the fluctuation s in a variety of ("un disturbed") vacua as well as the sub-vacuum fluctuation s foun d in both squeezed light an d Casim ir cavities. Squeezed light has tim e-depen den t, altern atin g region s of sub-vacuum fluctuation s (a.k.a. two-poin t fun ction s) of the quan tum electric field. Casim ir geom etries provide en viron m en ts with n on -trivial position - an d frequen cy-depen den t, tim e-in depen den t, often sub-vacuum fluctuation s (two-poin t fun ction s) of the quan tum electric field. Balan ced hom odyn e detectors (BHD) with local oscillators are am plifiers that are capable of providin g detailed m easurem en ts of the sub-vacuum fluctuation s (the two- an d n -poin t fun ction s) of the states of the quan tum electrom agn etic field. N early a decade ago, Han sen et al. [4] reported on their experim en tal tim e-dom ain (or pulsed) BHD device that they developed to m ake precise m easurem en ts of the quan tum electric field quadratures of pulsed optical quan tum states (e.g., squeezed light). A m aster laser produced the local oscillator for this device. The device dem on strated a high level of com m on m ode suppression an d low electron ic n oise, which provided large en ough sign al-to-n oise ratio to m easure the quan tum n oise of in dividual pulses. The device exhibited over 90% quan tum efficien cy. However, their device was n ot design ed to directly m easure the en ergy den sity of the in dividual pulses. W e recom m en d that a research an d developm en t program be im plem en ted to m odify the design an d operation of the tim e-dom ain BHD device in order provide this im portan t data. It will be n ecessary to develop an d com m ercialize a portable tim e-dom ain BHD device for the purpose of detectin g, m easurin g, an d spatially m appin g the sub-vacuum (n egative) en ergy region s produced by a putative pulsed (or "AC") n egative en ergy gen erator that m ight be used for en gin eerin g the spacetim e surroun din g an aerospace platform for propulsion purposes. A n um ber of m odified tim e-dom ain BHD devices could also be assem bled in a sen sor array for surveillan ce an d detection of an y an om alous aerospace platform s that m ight use en gin eered spacetim e effects for propulsion . 43 UNCLASSI FIED//FOP nFFV™» ■■rrn...u UNCLASSI FI ED//rOR OFFICIA L USE ONLY W hat has n ot been experim en tally m easured yet are the sub-vacuum fluctuation s an d their correspon din g sub-vacuum (n egative) en ergy den sity in side a Casim ir cavity. Casim ir cavities produce static, or tim e-in depen den t, sub-vacuum fluctuation s an d (n egative) en ergy den sity. M arecki [5, 6] proposed a m odified BHD an d com puted the two-poin t fun ction an d the associated spectral den sity for the groun d state of the quan tum electric field in Casim ir geom etries, an d predicted a position - an d frequen cy­ depen den t pattern of BHD respon ses if a device of this type is placed in side a Casim ir cavity. He discovered that by exploitin g a trick with the subtraction of the output of two balan ced photodiodes, it is possible to quan tify an d m ap the sub-vacuum fluctuation s of the quan tum field an d its correspon din g en ergy den sity in side the cavity. His m odified BHD design uses the electric field of the TEI m ode of the Casim ir cavity as the local oscillator. M arecki also discovered that the sub-vacuum (n egative) en ergy den sity region s in side a Casim ir cavity violate the Quan tum In equalities theorem . W e recom m en d that an experim en tal program be im plem en ted to test M arecki's m odified BHD an d his prediction s for Casim ir geom etries. Usin g this device to also test the efficacy of the Quan tum In equalities theorem is a n ecessary part of the proposed experim en tal program . If such experim en ts are successful, then it will be n ecessary to follow up by im plem en tin g a program to develop an d com m ercialize a portable "m odified-M arecki BHD" device for the purpose of detectin g, m easurin g, an d spatially m appin g the sub-vacuum (n egative) en ergy region s produced by a putative static (or "DC") n egative en ergy gen erator that would be used for en gin eerin g the spacetim e surroun din g an aerospace platform for propulsion purposes. Because the Casim ir effect an d its associated n egative en ergy are in credibly feeble, such putative propulsion system s will n ot in volve the use of Casim ir cavities to produce a free-space distribution of n egative en ergy surroun din g the platform . Therefore, a m odified-M arecki BHD will require a high quality laser for the local oscillator an d the photodiodes are allowed to be m uch larger in size. A n um ber of m odified-M arecki BHD devices could also be assem bled in a sen sor array for surveillan ce an d detection of an y an om alous aerospace platform s that m ight use en gin eered spacetim e effects for propulsion . 44 UNCLASSI FIED/ /FUK UFF1«A L USE OH LY UNCLASSIFIED//FOK OFFICIA L USE ONLY ACKNOWLEDGEMENTS The author would like to than k Professors Ulf Leon hardt an d Piotr M arecki for con tributin g their lecture n otes, referen ces, an d experim en tal data to the con ten ts of this report. 45 UNCLASSI FIED//FOR OFFICIA L USE ONLY UNCLASSIFIED//FOft OFFICIA L USE ONLY REFERENCES [1] Davis, E. W . (2009), "Faster-Than -Light Approaches in Gen eral Relativity/' in Frontiers of Propulsion Science, eds. M . G. M illis an d E. W . Davis, Progress in Astron autics & Aeron autics Series, Vol, 227, Am erican In stitute of Aeron autics & Astron autics Press, Reston , VA, pp. 473-509. [2] Davis, E. W . 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