Quantum Coherence and Control in One-And Two-Photon Optical Systems

arXiv:quant-ph/0010001v2 3 Oct 2000 aionaIsiueo ehooy aaea A915 US 91125, CA Pasadena, [email protected] Technology, address: Electronic of Institute California ∗ die yD.Wle .Lwec) nexperimental Dr. Kwiat. an of G. Lawrence), supervision Paul the E. under investigation Walter theoretical and Dr. 2000, by June College, advised Dartmouth (A.B. thesis dergraduate tt nsm pca ae,adt euedchrnein decoherence reduce to cases. and general cases, polarization more coherent special a some preserve in to state op- [5]. shown an be such operation since will ex- justified coupling eration is control” the rapidly “quantum term of by The eigenstates decoherence and the “bang- of introduce changing so-called control then of quantum We implementation bang” optical state. an unpolarized polarization examine definite an a to from state evolves non- photon effects, the decoherence Such to coupling whereby rise unitary give modes. errors” a “phase frequency of dissipative and action polarization the under between state photon gle the of evolution birefrin- decoherence the controllable state. of observing polarization a process and to the “environment” photons investigated examine gent single we be Here, subjecting can by system circuits 4]. accessible 3, quantum easily [2, simple polarization an which including provide in modes, modes, Photon spatial which and effects quantum [1]. proposed decoherence of algorithms implementation to practical leads the limit environmental freedom unobserved of and quan- states degrees quan- proposed (“qubit”) between of bit Coupling tum realization technologies. the information tum to obstacle nificant 1 rsn drs:Nra rdeLbrtr fPyis12-33 Physics of Laboratory Bridge Norman address: Present hsc iiin -3 o lmsNtoa aoaoy Lo Laboratory, [email protected]. Email: National USA. Alamos 87545, Los NM Alamos, P-23, Division, Physics [ nscinIIw ilitoueapriua two- particular a introduce will we III section In sin- a of evolution the investigate we II, section In sig- a is systems quantum two-state in Decoherence Note hsmnsrp stknfo h uhrsun- author’s the from taken is manuscript This : unu oeec n oto noe n w-htnoptica two-photon and one- in control and coherence Quantum ne h cino iia opig h est operator numerically. density obtained The are coupling. solutions similar and a Hilbe of the action of the decoher subspace under “decoherence-free” of a types of certain existence “bang- the from of information implementation an polarization that opera ton show We ( density observed freedom. the between of develop coupling degrees non-dissipative we a case, to first subjected the In perimentally. eivsiaechrnei n-adtopoo pia sys optical two-photon and one- in coherence investigate We 1 eateto hsc n srnm,DrmuhClee Ha College, Dartmouth Astronomy, and Physics of Department .INTRODUCTION I. ] hsc iiin -3 o lmsNtoa aoaoy Lo Laboratory, National Alamos Los P-23, Division, Physics nrwJ Berglund J. Andrew A; s , in(htnfeunymds.I ilb hw that shown be will It representa- information modes). free- for frequency of utilized (photon degrees not tion to are which modes) information-carrying dom polarization of coupling (photon will the states “environmental” describe In to term used states. the be basis argument, a qubit following to between leads the information freedom, phase of degrees of those fol- loss over states, trace environmental a by to lowed systems coupling quantum when in arises decoherence which of study the to relevant oee yasml n hsclyitiiemodification apparatus. intuitive the physically of re- and frequency DFS simple this the a the and by that covered suppressed In shown effectively be be can states. will correlation it two-photon case, these co- experimental the of affect conserva- properties which energy correlations herence particular, frequency In imposes de- tion particular . this mechanism complicate further coherence 9]), [8, in (see down-conversion entangled sources in such produced pairs freedom, a photons of hyper-entangled Photon is degrees as frequency state and 7]. polarization this of both [6, space is, Hilbert polarization That the photon of decoherence above. (DFS) subspace collective mentioned decoherence-free to type the immune of is sym- its properties, to due metry that, state polarization-entangled photon ini iernet“niomn”floe yatrace decoherence. a to by leads followed frequency “environment” polariza- over birefringent and a frequency in photon tion between coupling tary) 2 egho h nietlgt si nublne ihlo i Michelson coherence unbalanced an the [10], in it (see beyond as light, ferometer with separated incident interfere the are optics of to length modes classical ability polarization the uncorrelat in when loses of counterpart components composed well-known frequency light a quasi-monochromatic has whereby effect This nti eto,w ilso htnndsiaie(uni- non-dissipative that show will we section, this In I OEEC N “BANG-BANG” AND COHERENCE II. tsaeo w-htnplrzto states polarization two-photon of space rt oaiain n nbevd(frequency) unobserved and polarization) .Rve fsnl-htndecoherence single-photon of Review A. OTO:TEOEPOO CASE ONE-PHOTON THE CONTROL: ne ntescn ae einvestigate we case, second the In ence. ersnaini eeoe analytically developed is representation ∗ o ersnigasnl htnstate photon single a representing tor ag unu oto rtcspho- protects control quantum bang” es ohtertclyadex- and theoretically both tems, lms M855 USA 87545, NM Alamos, s oe,N 35,UAand USA 03755, NH nover, § 7.5.8). systems l 2 hs eut are results These nter- self ed 2 even non-dissipative coupling between these modes leads partial trace over frequency degrees of freedom: to a loss of information in the qubit states. A single photon characterized by its frequency spec- ρ(x) = dω ω U(x)ρ(0)U†(x) ω (5a) h | | i trum and polarization can be represented by the state Z ket 2 = c c∗ χ χ i j | iih j | i,j=1 2 X Ψ = cj χj dωA(ω) ω (1) 2 i[ϕi(ω,x) ϕj (ω,x)] | i | i⊗ | i dω A(ω) e − . (5b) j=1 Z × | | X Z Since this process realizes the entangling of qubit states where χ , j 1, 2 , are orthonormal polarization basis | j i ∈{ } with environmental states (frequency modes which are states with complex amplitudes cj , and A(ω) is the com- not involved in information representation or manipula- plex amplitude corresponding to the frequency ω, nor- tion), we expect to observe decoherence effects in the off- malized so that diagonal elements of the qubit state (polarization) density matrix [6]. In order to investigate the coherence dω A(ω) 2 =1. (2) properties of a photon under the action of such an oper- | | Z ator, we observe that in a completely mixed (or “deco- hered”) state, the density matrix is simply a multiple of For simplicity, we assume that the frequency spectrum is the identity matrix. The diagonal elements of ρ(x) are independent of polarization, so that we need not index indexed by i = j. For these values of i and j, the argu- A(ω) by polarization mode. In physical terms, this means ment of the exponential in Eq. 5b vanishes and by the that polarization and frequency are not entangled. normalization condition (Eq. 2), we have ρii(0) = ρii(x). Since Ψ represents a pure state, the density operator In other words, the diagonal elements of ρ(x) are unaf- can be written| i as ρ = Ψ Ψ (see [11], 3.4), so that we fected by U(x). have the initial state | ih | § The off-diagonal elements of the density matrix are indexed by the values i = 1, j = 2 and i = 2, j = 1. To 2 compute these explicitly, we must define the functions ρ (x =0) = c c∗ χ χ ϕj (ω, x). In the case of a single photon in a birefringent, ω i j | iih j | i,j=1 linearly dispersive crystal of thickness x (e.g., a quartz X crystal), dω1dω2A(ω1)A∗(ω2) ω1 ω2 . (3) ⊗ | ih | nj x ZZ ϕ (ω, x)= ω (6) j c The subscript indicates that ρω includes frequency de- where nj is the index of refraction corresponding to po- grees of freedom. The usual 2 2 density matrix repre- 3 (n2 n1)x × − senting the polarization state of the photon is given by larization j. Making the substitution τ = c and writing f(ω)= A(ω) 2, the off-diagonal elements of the ρij = cicj∗. density matrix are| given| by We will now seek the dependence of the density operator ρ(x) on some spatially extended “environment,” iωτ by introducing the operator U(x) which we require to be ρ12(x) = ρ12(0) dωf (ω)e− both linear and unitary. Furthermore, we demand that Z = ρ12(0) ∗(τ) (7a) U(x) have eigenkets χj with (frequency-dependent) F eigenvalues U (ω, x) where,| i as before, j 1, 2 indexes ∞ iωτ j ρ21(x) = ρ21(0) dωf (ω)e the polarization mode. Since U(x) exhibits∈{ frequency-} Z−∞ and polarization-dependent eigenvalues, it introduces a = ρ21(0) (τ) (7b) coupling between qubit states (polarization modes) and F non-qubit states (frequency modes). As shown in Ap- where (τ) is the Fourier transform of f(ω) (up to a F pendix A 1, U(x) introduces pure“phase errors” on po- constant, depending on convention). larization modes χj , so that Evidently, ρ†(0) = ρ(0) implies that ρ†(x) = ρ(x), so | i that Eqs. 7a and 7b preserve the required hermiticity of U(x) χ = eiϕj (ω,x) χ (4) | j i | j i where the phase factor ϕj (ω, x) is a real-valued function 3 Here, we neglect dispersion, the variation of n with ω. By as- of ω and x. suming |χj i to be an eigenstate of U (Eq. 4), we require the optic axis of the birefringent element to be aligned with one of Now we can calculate the spatial dependence of the the polarizations |χj i. In the case where the |χji represent cir- density operator under the influence of U(x) and follow- cular (elliptical) polarizations, this means that the environment ing a frequency-insensitive measurement, effected by a is optically active as opposed to (as well as) birefringent. 3

ρ. Interms of∆n = n2 n1 and (τ), the density matrix eigenstates of the environmental coupling. In our case, at some later position −x becomesF a simple modification this corresponds to an interchange of the polarization ba- of the density matrix at x = 0: sis states χ1 and χ2 , which can be accomplished by an appropriate| i reflection| i or rotation operation. These x∆n ρ11(0) ρ12(0) ∗( ) exchanges are rapid in the sense that the period of flip- ρ(x)= n c . (8) x∆ F ping must be short compared to the decoherence time ρ21(0) ( c ) ρ22(0) F τ 1 and the time scales of any other relevant de- c ∼ δω Eq. 8 allows us to calculate the density matrix repre- coherence mechanisms (e.g. scattering, dissipation). So senting the polarization of an optical qubit subjected to called “bang-bang” control reduces the degree of deco- a non-dissipative frequency-polarization coupling, given herence by averaging out the time-dependent coupling the initial state density matrix, ρ(0), (written in the basis between qubit states and environmental states. of eigenmodes of the coupling) and the functional form In this section, the previous results are expanded to of the frequency-amplitude function A(ω).4 In general, include an implementation of “bang-bang” quantum con- A(ω) 2 is peaked at some central value ω with a finite trol by periodic exchanging of the environmental eigen- | | o width δω. Under these conditions, the Fourier transform states χj . The mathematical formulation is consider- | i (τ) (and the off-diagonal elements of ρ) will fall signifi- ably simplified if we move to the discrete regime in which F 1 the “environment” acts in a stepwise fashion, and the ex- cantly on the time scale τc δω . In other words, there is no coherent phase relationship∼ between polarization ba- change of eigenstates also occurs in discrete steps.6 To R sis states χ1 and χ2 after the photon has traveled a this end, we define to be a reflection (or rotation) op- | ic | i U erator which exchanges the eigenstates of the frequency- distance x> δω through the environment (x). To summarize, we have developed a prescription for polarization coupling: calculating the polarization state density matrix repre- R χ = χ (9a) senting a photon in a birefringent environment. From | 1i | 2i R χ = χ . (9b) this expression, we see that a photon with definite polar- | 2i ±| 1i ization phase information before entering such an envi- Here, the plus (minus) sign refers to a reflection (rota- ronment will, in general, lose phase information between tion). The choice of sign will not affect the results of a polarization basis states after traveling a suitably long measurement in the experimental case, and for concrete- distance. The physical mechanism which destroys such ness, we take the sign to be positive. phase information is the entanglement between polariza- We define , a “step-wise” operator analogous to U(x) tion modes and frequency modes, which is an instance in Sect. IIAU and also define to be the thickness of of the entanglement between qubit states and environ- the birefringent element (in ourL experiments, usually a mental degrees of freedom which are not useful for infor- quartz crystal); as before, nj is the refractive index corre- mation manipulation. Because we trace over frequency sponding to polarization state j, and ∆n = n n . This (environmental) degrees of freedom in this case, the phase 2 − 1 gives a relative phase shift of ϕ(ω) ϕ2(ω) ϕ1(ω) = coherence between the qubit basis states is destroyed. ∆n 7 ≡ − L c ω, so that we have χ = χ (10a) U| 1i | 1i B. “Bang-bang” quantum control of decoherence χ = eiϕ(ω) χ . (10b) U| 2i | 2i In the previous section, we showed that a qubit con- Eqs. 9a-9b and 10a-10b have the important conse- sisting of photon polarization modes χ may lose its ca- quence that | j i pacity for information representation under the action of (R )2 = eiϕ(ω) (11a) a unitary operator, U(x), which entangles frequency and U 2 iϕ(ω) polarization. In other words, decoherence occurs even †R† = e− (11b) U for non-dissipative “phase errors” in which the qubit basis states are eigenstates of the environmental coupling, where it is understood that such operator identities are only meaningful when the operators are applied to kets or and there is no chance for a bit flip error [1].5 bras (see Appendix A 2 a for a proof of these identities). In [5], the authors describe this result in the general case of qubit states coupled to environmental degrees of freedom and introduce a scheme for reducing such decoherence effects via rapid, periodic exchanging of the 6 Developing the general continuous case in analogy with Sect. II A by including a rotation operator acting simultaneously with U(x) is considerably complicated, since this rotation operator will not, in general, commute with U(x) for different x values. 4 7 n1 Actually, we need only know the complex square of the Here, we omit the phase shift of ϕ1(ω) = Lc ω, which is 2 frequency-amplitude function, |A(ω)| . common to Eqs. 10a and 10b. Such a global phase shift is 5 Note however, that phase errors in the basis {|χj i} may appear never observable and cannot affect the coherence properties of ( χ1 χ2 ) as bit-flip errors in another basis, e.g. { | i±| i }. the polarization states. √2 4

In this discrete regime, we consider “rapid” exchang- The subscript QC (for “Quantum Control”) indicates ing of eigenstates to correspond to the alternating action the case in which R is included. Note that ρQC is depen- of the operators and R. Regarding and R as optical dent only on the parity of N, so that under the influence elements (for example,U a quartz crystalU and a half-wave of the quantum control procedure, any number of cycles plate), we consider a cavity scheme in which a single pho- through this particular birefringent environment causes ton passes through this two-element unit N times. at most a decoherence equal to that of the first pass. In Symbolically, after N passes through the system, we fact, Eq. 16a shows that an even number of cycles causes have (compare Eq. 5a), no decoherence whatsoever. By making the identification x = N (the distance N N L ρ(N)= dω ω (R ) ρ (0) †R† ω . (12) traveled through the crystal after N passes) in Eq. 8, we h | U ω U | i Z can directly compare the results of Sect. IIA with those Application of identities 11a and 11b leads to a simpli- of Sect. IIB: fication, depending on whether N is odd (N = 2m + 1) ρ11(0) ρ12(0) ∗ (Nτ0) or even (N =2m). ρ(N)= F . (17) ρ∗ (0) (Nτ0) ρ22(0) 12 F 2m 2m ρ(2m) = dω ω (R ) ρ (0) †R† ω Letting τc be the coherence time such that (τ) is h | U ω U | i F Z significantly reduced for τ > τc, then ρ(N) represents τ 2miϕ(ω) 2miϕ(ω) a partially mixed state for N > c .8 On the other = dω ω e ρ (0)e− ω τ0 h | ω | i Z hand, ρQC (N) never loses all phase information in the off-diagonal elements since N = 2m recovers the initial = dω ω ρ (0) ω h | ω | i state for all m. In this sense, R acts as a quantum control Z = ρ(N =0) (13a) element, suppressing decoherence and maintaining state purity indefinitely. 2m+1 2m+1 ρ(2m +1) = dω ω (R ) ρ (0) †R† ω We may ask what happens if the strength of the envi- h | U ω U | i Z ronmental coupling varies with x, so that varies on a U 2miϕ(ω) 2miϕ(ω) time scale comparable to the period of the exchange op- = dω ω e R ρω(0) †R†e− ω h | U U | i erator, R. For simplicity, we model this “slow-flipping” Z case by introducing two distinct operators which differ = dω ω R ρ (0) †R† ω h | U ω U | i only in their eigenvalues (not their eigenstates). In more Z concrete terms, we consider two coupling operators, = ρ(N = 1). (13b) U1 and 2, characterized by phase errors ϕ1(ω) and ϕ2(ω) Note that ρ(N = 1) is related to the continuous case of so that,U for j =1, 2: Sect. IIA by χ = χ (18a) Uj | 1i | 1i iϕ (ω) ρ(N =1)= dω ω Rρ(x = )R† ω . (14) χ = e j χ . (18b) h | L | i Uj | 2i | 2i Z R These relations lead to a simple analog of Eq. 5b (where The exchange operator acts in alternation between each of these two operators, so that the strength of the we use ρω(N = 0) ρω(x = 0), given by Eq. 3, as the input state in both↔ cases): coupling changes faster than the eigenstates of the coupling are exchanged. After N passes through the system, 2 we have ρ(N =0) = cicj∗ χi χj (15a) | ih | N N i,j=1 ρ(N) = dω ω ( ) ρ (0)( † †) ω (19a) X h | U2U1 ω U1 U2 | i 2 Z 2 ρ(N =1) = cic∗ dω A(ω) (R ) χi χj †R† . R R N R R N j | | U | ih | U ρQC (N) = dω ω ( 2 1) ρω(0)( 1† † 2† †) ω . i,j=1 Z h | U U U U | i X Z (15b) (19b) Finally, writing out ρ(N) in matrix form, substituting Intuitively, we expect to see a partial reduction in the for the Fourier transform of f(ω) = A(ω) 2, and de- degree of decoherence when the quantum control proce- F ∆n | | dure is implemented by introducing the exchange opera- noting by τ0 = L c the characteristic time parameter, we have tor.

ρ11(0) ρ12(0) ρQC (2m) = (16a) ρ∗ (0) ρ22(0) 12 8 Here, “mixed” refers to the fact that the oﬀ-diagonal elements ρ11(0) ρ12∗ (0) (τ0) ρQC (2m +1) = F . of ρ are small so that Tr(ρ2) < 1. If the initial state, ρ(0), has ρ12(0) ∗ (τ0) ρ22(0) F diagonal elements of 1/2, then ρ represents a completely random (16b) ensemble for N ≫ τc . τ0 5

To see these results quantitatively, we make the follow- Recycling Mirror x ing substitution which reduces this problem (mathemat- PBS H+V

xx ically) to the previously solved one (see Appendix A 2 b xx xx for a derivation of these relations): xx H V R

xxxxxxxxxx 2 1 (20a) xxxxxxxxxx Compensator R URU ≡ U 2 1 QC (20b) U U ≡ U Polarizer so that, Detector

χ = χ (21a) U| 1i | 1i i(ϕ1(ω)+ϕ2(ω)) FIG. 1: Schematic diagram of experimental apparatus for pro- χ2 = e χ2 (21b) U| i | i ducing controllable decoherence in the H/V basis. PBS indicates a Polarizing Beam Splitter; R indicates the (removable) exchange element, a quarter-wave plate in this case. The in- H + V QC χ1 = χ1 (22a) put state is |45◦i ≡ | i | i . U | i | i √2 i(ϕ1(ω) ϕ2(ω)) QC χ = e − χ . (22b) U | 2i | 2i With these relations, each operator is mathematically quantum level by calculating probabilities from the den- analogous to the operator defined by Eq. 4, with ϕ(ω) sity operator, ρ(x)9: redefined as appropriate givenU relations 21a-22b and the details of the particular environment. In the case of bire- λ(x)= λ ρ(x) λ λ ρ(x) λ . (25) fringent crystals, the elements corresponding to and V |h | | i − h | | i| U1 2 differ only in their thickness (not in their orientation Note that we no longer include the denominator corre- withU respect to the polarization basis states χ ). Defin- sponding to the total number of photons in Eq. 24, since | j i ing j to be the thickness of a crystal represented by j , the denominator here is the trace of ρ in the λ , λ L U {| i | i} we have basis, which is unity. λ(x) quantifies the similarity between the state representedV by ρ(x) and the state λ j ∆n | i ϕ (ω)= L ω. (23) in the following sense: if ρ(x) represents an unpolar- j c ized state, it is a multiple of the identity matrix, and We see that before introducing the exchange operator, λ(x) = 0; at the opposite extreme, if the photon is inV a pure state with polarization λ, ρ(x) = λ λ , and the birefringent environment acts effectively as a single | ih | λ(x) = 1. If we choose to measure with respect to the birefringent crystal of thickness 1 + 2 (Eq. 21b). Af- L L stateV λ = 1 ( χ + χ ), we find that (See Appendix ter implementing “bang-bang” control by introducing the | i √2 | 1i | 2i exchange operator R, the environment looks effectively A3 for a proof) like a single crystal of thickness 1 2 (Eq. 22b). In L − L (x)=2Re( χ ρ(x) χ ) (26) the absence of quantum control, the photon is subjected V h 1| | 2i to the sum of the individual phase errors; in the presence of a quantum control element, the photon is subjected where we have dropped the subscript λ for this choice to the difference of the individual phase errors. In this of measurement, and Re indicates the real part. In this form, it is clear that visibility (x) depends on the case, where the strength of the environmental coupling V changes on roughly the same time scale as the exchang- off-diagonal elements of ρ(x), and so corresponds with ing of eigenstates, the quantum control procedure reduces our previous notion of coherence. If we write ρ in the H/V basis, measuring the visibility with respect to λ = but does not eliminate decoherence effects. 1 1 | i ( H + V )= 45◦ and λ = ( H V )= 45◦ √2 | i | i | i | i √2 | i−| i |− i gives us information about the off-diagonal elements of C. Experimental Demonstration of “Bang-Bang” ρ. Control To see the results of the previous sections experimentally, we seek a systematic and controllable implementation of the operators and R in the cavity scheme of It will be convenient in our experimental analysis to U measure the visibility, traditionally defined as Sect. II B. For the first case, in which there is a single operation which produces a phase error, we use a Michel- Nmax Nmin son polarization interferometer as a tunable decoherence = − (24) V Nmax + Nmin where Nmax (Nmin) denotes the maximum (minimum) number of photons passing through orthogonal settings 9 The notation {|λi, |λi}, is intended to suggest that these polar- of a polarizer. Denoting these polarization analyzer set- ization modes may, in general, represent any elliptical orthonor- tings by λ and λ , we define visibility at the single mal basis states. | i | i 6

mechanism in the H/V basis. A polarizing beam-splitter the exchange operator, R in Sect. II B. (PBS) separates the path of V - and H-polarized light. By recording photon counts in coincidence with the The unbalanced arms of the interferometer produce a diode pulse generator and using a polarizer at the detec- controllable, ω-dependent phase shift between H and tor, we can measure the visibility at 45 = 1 ( H + ◦ √2 V (see Fig. 1). | i | i | i | i V ). The frequency spectrum of laser light is restricted A photon in either path is reflected back to the PBS |usingi an interference filter with bandwidth δλ = 10 nm 13 and retraces its path to the input. The difference in path and central wavelength λo = 670 nm. For our calcu- lengths between arms of the interferometer will be de- lations, we use a Gaussian amplitude function with fre- noted by x. The unbalanced interferometer is the optical quency spread δω corresponding to δλ = 10 nm, central realization of the operator U of Sect. IIA and of Sect. frequency ω0 corresponding to λo = 670 nm, and nor- U 14 II B. malization factor Ao:

An R =0.97 reflector at the input recycles the light so 2 that a photon passes through the unbalanced arms of the ω0 ω A(ω)= Ao exp − . (27) 10 − δω/√2 interferometer more than once, in general. A partially " # reflective out-coupling mirror directs a photon out of the interferometer with a probability of R = 0.04 at each The corresponding coherence length of the photons is 2 pass.11 given by L λo = 45µm, so we expect substantial c ∼ δλ We send in light from a greatly attenuated diode laser decoherence for a path difference of 4.5 µm at cycle ∼ at λ = 670 nm, pulsed at 100 kHz. In our experiment, N = 10. we observe photons passing through the interferometer With this setup, we can measure vs. N as in Sect. V 10 times. The total distance traveled by a photon II B. Experimental and theoretical results are plotted in passing≤ through our interferometer 10 times is 5 m, so Fig. 2 for the case where the arms are unbalanced by 8 ∼ that any photon spends at most 10− s in the system. x = 0.00, 2.49, and 4.99µm. The experimental results EG&G SPCM-AQ silicon avalanche∼ photodiodes (dark show clearly that the introduction of the quantum control 1 count < 400 s− , efficiency 60%) are used as photon element results in a significant reduction of decoherence counters. Using electronic timing∼ techniques, we record in good agreement with the theory. However, practical photon counts only in a coincidence window of 50 ns be- difficulties with the interferometer limit the quantitative tween the pulse generator and detection event. These agreement between experimental and theoretical values timing techniques, together with spectral filters at the of QC , especially at large N. We see in Fig. 2(a) that V detectors, reduce background coincidence counts to < 1 the relative path difference is not 0 after N = 10 passes 1 s− . We graphically display photon counts vs. arrival through the interferometer due to imperfect balancing. time so that photons leaving the system after N cycles By determining the coherence length from the data in are recorded in the Nth peak. In a typical experiment, Fig. 2, we infer that the actual path length difference in our photon counters register no more than 5000 coinci- the interferometer was 0.7 µm (about one wavelength ∼± dence counts per second, and we conclude that 10, 000 of incident light) displaced from the nominal x-values. photons per second leave the system ( 0.1 per∼ pulse).12 This discrepancy then determines the bounding value of At the output, we count for a few seconds∼ (less than (N = 10) for all other plots. Theoretically, of course, V a minute in most cases). Although we cannot determine this value is exactly unity. the unknown polarization state of a single photon, we can In order to model the “slow-flipping” case, we intro- accurately find the statistical distribution of the ensemble duce an additional operator, 2, which will induce a sec- U over the observation time, from which useful information ond phase shift, also in the H/V basis. To accomplish about the density matrix can be inferred. this task, we include a birefringent quartz crystal whose optic axis is aligned with the axes of the PBS of the By placing a λ/4 waveplate with optic axis at 45◦ at original setup. We denote the optical path difference in the recycling reflector, we realize an optical “flipping” 15 operator. The λ/4-reflector-λ/4 combination takes H this crystal by xQ. We now use a λ/2-waveplate as to V and vice versa. This is the optical realization| ofi | i

13 Since the diode laser is weakly driven just above its lasing thresh- old, the intrinsic frequency spectrum of the photons is wide 10 We use R to denote the probability of reflection at a surface. 11 enough that these filters determine the overall frequency distri- For normal incidence, R = 0.04 at each air-glass interface. bution. Here, one surface of the out-coupling mirror is broad-band anti- 14 In practice, we find that the frequency spectrum is not as smooth reflection (BBAR) coated so that we have effectively only a single as a Gaussian. From the visibility curves, we infer that the fre- air-glass interface. quency spectrum has a substantial monochromatic (compared to 12 The probability that there are two photons in the system si- δω) component at ωo. multaneously is given by the Poisson distribution, P (2) = 15 − Instead of a single quartz crystal, we use a 1.046 mm crystal e 0.1(0.1)2 2! ∼ 0.005. These photon numbers are comparable to at 0◦ and a 0.850 mm crystal perpendicular to the first. The those used in current demonstrations of free-space quantum cryp- combination of these two crystals gives an effective path length tographic key distributions [12]. such that nearly complete decoherence occurs over 10 cycles. 7

1 Birefringent Crystal x PBS 0.9 H+V

xxx

0.8 xxx

xxx

xxx 0.7 H V 0.6 R Recycling Mirror Data xxxxxxxxxx xxxxxxxxxx

xxxxxxxxxx 0.5 Theory

Visibility 0.4 Data (QC) Theory (QC) 0.3 Polarizer

0.2 Detector

0.1 0 FIG. 3: Schematic diagram of experimental apparatus for the (a) 0 1 2 3 4 5 6 7 8 9 10 “slow-ﬂipping” case. As before, R indicates the exchange element, a half-wave plate in this case. The birefringent crystal, 1 a quartz plate, introduces a second phase error in the same 0.9 basis as the unbalanced interferometer arms. 0.8

0.7

0.6 Data Theory 0.5 Data (QC) 1 Visibility 0.4 Theory (QC) 0.9

0.3 0.8 0.7 0.2 0.6 Data 0.1 Theory 0.5 Data (QC)

0 Visibility 0.4 Theory (QC) 0 1 2 3 4 5 6 7 8 9 10 (b) 0.3 0.2 1 0.1 0.9 0 0.8 0 1 2 3 4 5 6 7 8 9 10 (a) 0.7 Data 0.6 Theory 0.5 Data (QC) 1 Visibility 0.4 Theory (QC) 0.9 0.3 0.8 0.2 0.7 Data 0.1 Theory 0.6 Data (QC) 0 0.5 Theory (QC) 0 1 2 3 4 5 6 7 8 9 10 Visibility 0.4 N (c) 0.3 0.2 FIG. 2: Experimental and theoretical curves showing V vs. N 0.1 for the case where (a) x = 0.0 µm, (b) x = 2.49 µm, and (c) 0 0 1 2 3 4 5 6 7 8 9 10 x = 4.99 µm, for the setup in Fig. 1. “QC” indicates the case (b) in which the quantum control procedure was implemented. In (a) (x = 0), the theoretical curves for both cases are constant at unity, since we expect no decoherence in this case. In 1 (c), the observed visibility curve (without quantum control) 0.9 falls to V ∼ 0.15 on the length scale of our observations. 0.8 We suspect that V falls to 0 on the length scale determined 0.7 0.6 Data by the monochromatic component of the frequency spectrum Theory 0.5 mentioned in footnote 14. The experimentally observed values Data (QC) Visibility 0.4 in the quantum control case alternate in the opposite sense to Theory (QC) 0.3 the theoretical values. This discrepancy is not yet understood. 0.2 0.1 0 0 1 2 3 4 5 6 7 8 9 10 the quantum control element which exchanges the eigen- (c) N states, H and V , at each pass both to the left and to the right (see Fig. 3). For the theoretical values, we used the FIG. 4: Experimental and theoretical curves showing V vs. same prescription as above, but with effective operators N for the “slow-flipping” case where xQ = 1.77 µm and (a) x . µm x . µm x . µm defined by Eqs. 21a-22b. Experimental and theoretical 1 = 2 1 , (b) 2 = 3 16 , and (c) 3 = 4 2 for the setup in Fig. 3. results are displayed in Fig. 4. 8

0.8 mm Quartz Recycling Mirror

"Slow-Flipping" Quantum Control

0.35

0.3 x3+xQ Analyzer R (rotator) 1.0 mm Quartz x2+xQ 0.25 Detector

x1+xQ m)

µ 0.2

0.15 FIG. 6: Schematic diagram of experimental apparatus for the Rate (1/ x3-xQ case in which the eigenstates of the polarization-frequency 0.1 coupling can be adjusted. As before, R indicates the exchange Measured Decoherence x2-xQ 0.05 element, an optically active 90◦ quartz rotator in this case. x1-xQ 0 The two birefringent crystals introduce diﬀerent phases errors 0 1 2 3 4 5 6 whose eigenstates can be adjusted independently by changing Predicted Decoherence Rate (µm) their orientation.

and FIG. 5: Measured (reciprocal) optical path difference inferred s = 2 H ρ(x) H 1 (29a) from the coherence lengths in Fig. 4 versus expected values h 1i h | | i− according to Sect. II B. Error bars in both directions are s = 2 45◦ ρ(x) 45◦ 1 h 2i h | | i− within the points. See the text for an explanation. = H ρ(x) V + V ρ(x) H (29b) h | | i h | | i s = 2 R ρ(x) R 1 h 3i h | | i− In Fig. 5, we have inferred the effective rate of decoher- = i( H ρ(x) V V ρ(x) H ). (29c) h | | i − h | | i ence (a reciprocal length, in this case) in the system by As usual, H and V denote horizontal and vertical po- fitting curves to each of Fig. 4 and (reciprocally) plotted | i | 1i larization, 45◦ = ( H + V ) denotes 45◦ polariza- these versus the rate of decoherence for a single pass pre- | i √2 | i | i dicted by Eqs. 21b and 22b. We expect to see an effective tion, and R = 1 ( H + i V ) denotes right-circular | i √2 | i | i decoherence rate given by the sum of xQ, the decoher- polarization. ence rate due to the quartz crystal, and xi, i 1, 2, 3 , Note that photon counting in the appropriate polar- ∈{ } the decoherence rate due to the interferometer, when the ization basis allows experimental determination of the quantum control procedure is not implemented; further- Stokes parameters, si . By photon counting and us- more, we expect to see a decoherence rate determined ing the relations 29a-29c,h i the density matrix of an arbi- by the difference of xQ and x when the quantum contrary polarization state can be reconstructed from these trol procedure is implemented. Here, the decoherence data.17 In contrast to measurements of with respect to “rate” of an optical element is directly proportional to some particular orthonormal basis λ ,V λ , this tomo- the reciprocal of the optical path difference between e- graphic technique allows complete{| determinationi | i} of the and o-polarizations. The linearity of this plot indicates polarization state. In fact, the visibility with respect that the quantum control element acts as predicted in to any basis λ can be found from the densityV matrix if | i Sect. II B. the transformation from χj to λ is known. We are limited in the type of decoherence we can ob- In the previous cases,| wei used| i an appropriately ori- serve by measuring . For example, a photon in the pure ented waveplate as a realization of the exchange operator, V state H has a visibility of 0 with respect to the 45◦ ba- R. However, the choice of orientation in both cases re- | i sis. We would like a more general method of quantifying quired prior knowledge of the eigenstates of the coupling decoherence effects. To this end, we can extend the var- between frequency and polarization. Note, however, that ious methods of polarization analysis in classical optics if we know only that the eigenstates of the coupling are to our more fundamental quantum mechanical setting. orthogonal, linear polarization states, then a rotation by With this idea in mind, we define the degree of polariza- 90◦ (as opposed to reflection about a specific axis) meets tion , and the Stokes parameters si in terms of ρ(x) the requirements of an exchange operator (Eqs. 9a-9b) in 16 P h i as all cases. We expect that an optically active 90◦ quartz rotator acts as a quantum control element, independent 2 2 2 = s1 + s2 + s3 (28) of the orientation of the quartz crystal that causes deco- P h i h i h i herence (since the eigenstates of our birefringent quartz p

16 These are a generalization of the Stokes parameters of classical optics to the quantum mechanical case. In classical optics, these 17 An extension of this technique of quantum tomography has been data can be used to reconstruct the coherency matrix, J, which demonstrated by reconstruction of two-photon density matrices completely characterizes the polarization state of the light (See in [13]. This technique will be useful in the two-photon state [10], §10.9). discussions of Sect. III. 9

1 0.9 1 0.9 0.8 0.8 0.7 Data 0.7 Data 0.6 Data (QC) P 0.6 Data (QC) 0.5 Theory P 0.5 Theory Theory (QC) 0.4 0.4 Theory (QC) 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 N N

FIG. 7: Theoretical and Experimental plots of P vs. N for FIG. 8: P vs. N for the case in which the strength and the environmental eigenstates of −10◦ in a linear cavity (see Fig. eigenstates of the polarization-frequency coupling change. In 6). The input polarization is 35◦. Clearly, introduction of the the case where we include the quantum control element, a 90◦ quartz rotator results in a preservation of photon polarization quartz rotator, the degree of polarization is preserved. information for this choice of environmental eigenstates.

As a further demonstration of “bang-bang” quan- crystal are orthogonal and linear, i.e., along the optic tum control of decoherence, we seek a scheme in which axes of the crystal).18 the strength and the eigenstates of the polarization- To investigate decoherence in an “adjustable” linear frequency coupling change between exchange operations. basis, we use a cavity with a photon passing through To reproduce these conditions, we simply place the 1.046 mm and 0.850 mm birefringent quartz crystals at quartz crystals of Fig. 6 at slightly different angles so each cycle. Since this thickness of quartz results in sub- that the eigenstates of the coupling are not constant at stantially faster decoherence than the unbalanced inter- each pass. The first (1.0 mm) crystal is oriented at 10◦ to ferometer of the previous cases, we use a narrow band- the horizontal while the second (0.8 mm) crystal is placed width filter of δλ = 1.5 nm to increase the coherence at 0◦ degrees. To solve for starting from the density P length of the photons. The exchange element, a 90◦ operator representation, we rewrite the first crystal oper- quartz rotator, is placed between the birefringent crys- ator as a matrix in the H/V basis using the appropriate tals (see Fig. 6). At the output, we measure the Stokes rotation matrices. The resulting output density matri- parameters by photon counting in the appropriate bases ces are calculated numerically and is extracted. The P and calculate to quantify the degree of coherence. By theoretical and experimental results are displayed in Fig. orienting bothP birefringent crystals so that the optic axis 8. makes an angle of 10◦ with the horizontal, and sending As a final demonstration of bang-bang control, we as- − linearly polarized light at 35◦ into the system (for max- sert that the same technique of exchanging orthogonal imum decoherence), we see that the rotator does indeed polarization states will also compensate for decoherence act as a quantum control element (Fig. 7). Note however, via dissipative (i.e., non-unitary) errors. Experimentally, that we need not rotate the optically active element by a we accomplish this task by inserting a neutral-density (ω- corresponding 10 , because its functionality is indepen- independent) filter into the transmission ( H ) arm of the ◦ | i dent of such a rotation. Put differently, the rotator acts balanced Michelson interferometer of Fig. 1. This filter as a quantum control operator for (slow) decoherence in results in 35% loss (after two passes) per cycle. Denoting all linear bases. this operator as T, letting T =1 0.35=0.65, and using 1 − For the theoretical curves in Fig. 7, we need not modify the input state ( H + V ), we have, after N cycles, √2 | i | i the previous calculations except to take the lower (negative) sign in the definition of R, Eq. 9b and include the α H + β V TN αT N H + β V . (30) narrower frequency spectrum. The invariance of physical | i | i −−−−→ | i | i systems under rotation allows us to rotate the entire ex- In the case when the quantum control procedure is im- T R perimental apparatus (the input polarization and quartz plemented, and alternate so that the state evolves in the following steps crystals) by 10◦ without adjusting the calculations. α H + β V T αT H + β V | i | i −−−−→ | i | i R αT V + β H 18 −−−−→ | i | i If the eigenstates of the environmental coupling are left- and T αT V + βT H right-circular polarizations, a λ/2 waveplate at any orientation | i | i −−−−→R acts as the exchange operator R. αT H + βT V (31) −−−−→ | i | i 10

coherence when the strength and/or the eigenstates of 1 the coupling vary between exchanges. 0.9 0.8 0.7 0.6 III. DECOHERENCE-FREE SUBSPACES OF TWO PHOTONS V 0.5 0.4 0.3 Data A. Polarization-frequency coupling in correlated Data (QC) 0.2 photon pairs Theory 0.1 Theory (QC) 0 It has been shown that the antisymmetric 2-qubit state 0 1 2 3 4 5 6 7 8 9 10 given by (in our optical notation) ψ = 1 ( χ χ − √2 1 2 N | i | i| i− χ2 χ1 ) is immune to collective decoherence, i.e. decoherence| i| i arising from an operation that is invariant under FIG. 9: Experimental and theoretical results displaying visi- qubit permutations [6]. That is, due to its symmetry bility vs. cycle for T = 65% transmission in the |Hi arm of properties, ψ− is a decoherence-free subspace (DFS) of the interferometer (see Fig. 1). The experimental data have 2 qubits. Therefore,| i it is reasonable that we expect the large error bars since photon numbers are quite low after 10 two-photon polarization state 1 ( H V V H ) to be passes, and counting statistics (which scale as the square root √2 | i| i−| i| i of the photon count) become significant. As before, we expect immune to decoherence of the type discussed in Sect. II, that the imperfectly balanced interferometer arms will reduce which arises from coupling to frequency degrees of free- the visibility below the theoretical values. Nevertheless, the dom that are unused for information representation. principle of the quantum control for reducing the effect of In this section, in order to investigate the possibility dissipation is clearly demonstrated. of a DFS, we will consider polarization phase information in a two-photon system. In non-linear optical ma- terials, spontaneous parametric down-conversion events so that can be used to produce spatially separated, polarization- entangled photon pairs with high fringe visibility [8, 9]. (TR)2(α H + β V ) = T (α H + β V ). (32) | i | i | i | i Here, we extend the results of Sect. II by presenting a method for calculating the density matrix representing TR 2 In this way, we prove the operator identity ( ) = T so the polarization of such a two-photon state in the pres- that, after an even number of cycles, the system evolves ence of a birefringent environment. Experimentally, we back to its original state but with a reduced probability of use the source detailed in [9] which produces correlated detection. Experimental and theoretical results for this photon pairs in the two paths labeled L and R (See Fig. case are displayed in Fig. 9. 10, p. 13). In summary, we have shown that coupling information- In order to keep track of frequency, spatial mode (L or carrying (polarization) degrees of freedom to unobserved R), and polarization, we define a shorthand notation for (frequency) degrees of freedom results in a loss of polar- indexing the 4 4 two-photon density matrix ρ(x). The ization phase information. We have shown in this sec- × element χij χkl is defined to be χi L χj R χk L χl R tion that an optical realization of “bang-bang” control where the| subscriptsih | L and R refer| toi the| lefti h and| h right| of decoherence does indeed preserve photon polarization photon paths respectively and χ1 and χ2 refer to the in the presence of such an “environment.” Our imple- polarization basis states. As a| concretei | example,i in the mentation of bang-bang control requires some knowledge H/V basis, the state V L H R is written as χ21 and of the eigenstates of the polarization-frequency coupling. V H H H is| writteni | i as χ χ . In| thisi way, We have demonstrated such control when the eigenstates | iL| iRh |Rh |L | 21ih 11| 19 each of the 16 entries of ρ is indexed by four numbers, of the coupling are linear and orthogonal. In addi- (ij,kl), where i,j,k,l 1, 2 . tion to preserving polarization phase information in the ∈{ } presence of a frequency-polarization coupling, we have In order to satisfy energy conservation, the frequen- shown that such bang-bang control similarly reduces de- cies of photon pairs produced in a down-conversion event must sum to the frequency of the pump beam, and their resultant energies become entangled (see [14] 22.4). De- § noting the pump frequency by ωo and the frequency of daughter photon j by ω (j 1, 2 ), we therefore require 19 In fact, in the PoincaréSphere representation of polarization (See j ∈{ } [10] §1.4), it is sufficient to know only the plane in which the eigenstates of the coupling lie: a 90◦ rotation about the normal to that plane satisfies the conditions of an exchange operator R. ωo ω1 = + ǫ (33a) The technique fails, however, for a decoherence mechanism whose 2 eigenstates coincide with those of R (i.e., which are parallel to ω ω = o ǫ (33b) this normal direction). 2 2 − 11

20 so that we have ω1 + ω2 = ωo in all cases. This fre- where we must be careful to treat ω± andω ˜± as functions quency “anti-correlation” will be important in an exper- of the variables of integration ǫ andǫ ˜. imental demonstration of a DFS. Taking cij to be the (in general, complex) amplitude In order to represent a birefringent crystal in each path, of the state χij and denoting the frequency spectrum we deﬁne the operator by A(ω), we| writei the initial (pure) state at x =0 as21

2 Ψ(0) = c χ | i ij | ij i i,j=1 U(x , x )= U (x ) U (x ) (37) X L R L L ⊗ R R ωo ωo ωo ωo dǫ A( + ǫ)A( ǫ) + ǫ ǫ . which depends on the distance, xL (xR), traveled through ⊗ 2 2 − | 2 i| 2 − i Z such a crystal in the L (R) photon path. It should be (34) noted that the in Eq. 36 represents a tensor product between the Hilbert⊗ spaces of photon polarization and For completeness, and to make this indexing scheme photon frequency, whereas the in Eq. 37 represents explicit, we write the reduced 4 4 density matrix repre- the tensor product of photon spatial⊗ modes L and R. In × 2 senting the pure polarization state Ψ = cij χij : analogy with Sect. II A, U (x ) associates a frequency- | i i,j=1 | i L L dependent phase, ϕj (ω, xL), to each polarization state 2 P c11 c11c21∗ c11c12∗ c11c22∗ j 1, 2 in the path labeled L: | | 2 ∈{ } c21c11∗ c21 c21c12∗ c21c22∗ ρ = Ψ Ψ = | | 2 . (35) | ih | c12c11∗ c12c21∗ c12 c12c22∗  | | 2 c22c11∗ c22c21∗ c22c12∗ c22   | |  iϕj (ω,xL) Note that there are six independent parameters cor- UL(xL) χj L =e χj L. (38) responding to the four complex amplitudes of the | i | i states χij constrained by the normalization condition 2| i cij = 1 and the freedom to choose the global (overall)| phase.| 22 P A similar relation holds for the path labeled R. The func- Introducing the shorthand notation ω± = ω±(ǫ) = ωo ωo tions ϕj (ω, xL/R) characterize the phase shift induced in 2 ǫ andω ˜± =ω ˜±(ǫ) = 2 ǫ˜, the density operator for± the initial state, including± frequency modes, can be either path (L or R). In this way, we require that the cou- written as pling between frequency and polarization share the same dependence on the parameters x and ω in either photon path. This requirement is reasonable since we wish to ex- ρω(0) = Ψ(0) Ψ(0) | ih | plore the robustness of the antisymmetric state against = c c∗ χ χ ij kl| ij ih kl| collective decoherence. i,j,k,lX + + dǫ d˜ǫ A(ω )A(ω−)A∗(˜ω )A∗(˜ω−) Making a frequency-insensitive measurement of polar- ⊗ ization after the photon in path L (R) has traveled a ZZ+ + ω ω− ω˜ ω˜− (36) distance xL (xR), we have 20 We have used× the | notationi| ih of|h [15] where| such frequency entan- so that initially hΨ|Ψi = 1 and ρ2 = ρ. glement is shown to cause nonlocal cancellation of dispersion to 22 For a partially mixed state, there are 15 independent parame- second order in optical systems. 21 ters corresponding to 4 real diagonal elements reduced to 3 by We impose the normalization condition the normalization constraint, and 12 complex amplitudes in the ωo ωo upper right entries. dǫ|A( + ǫ)|2|A( − ǫ)|2 = 1 2 2 Z

ρ(x , x )= dω dω ω ω U(x , x )ρ (0)U†(x , x ) ω ω (39) L R 3 4h 3|h 4| L R ω L R | 3i| 4i ZZ which gives

+ − + − + 2 2 i(ϕi(ω ,xL)+ϕj (ω ,xR) ϕk(ω ,xL) ϕl(ω ,xR)) ρ(x , x ) = c c∗ χ χ dǫ A(ω ) A(ω−) e − − . (40) L R ij kl| ij ih kl| | | | | i,j,k,lX Z

Eq. 40 gives a prescription for calculating the density matrix representing polarization degrees of freedom 12

in the presence of non-dissipative “phase errors.” As in eral, with respect to another (possibly elliptical) basis Sect. IIB, we expect to observe decoherence effects in λ λ , the off-diagonal elements of ρ. Note that for the diag- {| i| i} 1 1 onal elements of ρ, indexed by i = k, j = l (see Eq. ( HH V V = ( λλ λλ . (42) 35), the argument of the exponential is identically zero √2 | i ± | i 6 √2 | i ± | i so that the diagonal elements do not change. Of course, this result holds only for the case of a unitary frequency- In particular, using the basis of left- and right-circular polarization, it can be shown that 1 ( HH + V V ) = polarization coupling in this particular orientation (i.e., √2 | i | i with eigenstates H and V ). 1 ( LR + RL ). So, just as the state 1 ( HV + VH ) | i | i √2 | i | i √2 | i | i Let us now choose the functions ϕj (ω, x) to be identical loses phase information in a birefringent crystal whose to those of Eq. 6. In other words, we choose to realize eigenmodes (for photons in either path) are H and V , a unitary polarization-frequency coupling by placing a the state 1 ( LR + RL ) loses phase information| i | ini √2 | i | i birefringent crystal across both photon paths with e- and a crystal whose eigenmodes are L and R . On these o-axes aligned with χ and χ .23 Letting denote | i | i | 1i | 2i L grounds, the only candidate for a DFS is the singlet state the thickness of the crystal, the phase difference between 1 ψ− = √ ( HV VH ), since, in analogy with the these functions evaluated at xL = xR = is given by | i 2 | i − | i L spin- 1 singlet state 1 ( ), this state is rota- 2 √2 | ↑↓i−| ↓↑i ∆n tionally invariant and looks the same when written in any ϕ2(ω, ) ϕ1(ω, )= ω L . (41) L − L c basis (see [11], 3.9). However, we have already shown § that the state ψ− loses phase information in this sys- In Appendix A 4, the 4 4 density matrix represen- | i tation is written for the× evolution of the Bell states tem. 1 1 φ± = ( HH V V ) and ψ± = ( HV VH ). | i √2 | i±| i | i √2 | i±| i From these, we see that the states ψ± become “inco- B. Recovering a decoherence-free subspace herent” in the presence of such a frequency-polarization| i coupling. Here, we do not make a quantitative measure of coherence (or mixture) but simply observe that The apparent contradiction between the prediction of the off-diagonal elements of the density matrix approach a DFS and the contrary result in the previous section zero for larger than the coherence length of the down- arises as a result of the frequency entanglement (anti- convertedL photons, given by c where δω is the width correlation) in down-converted photon pairs. We ex- ∼ δω pected to observe a DFS of two-photon states subjected of the frequency spectrum A(ω) 2. Therefore, these to identical phase errors. Frequency anti-correlation states lose phase coherence information| | in the presence breaks this qubit permutation symmetry since the phase of polarization-frequency coupling of this type: identi- errors are frequency-dependent. There is, however, a sim- cal birefringent crystals in the same orientation in both ple experimental alteration which compensates for this photon paths. frequency anti-correlation and allows us to observe a DFS On the other hand, the states φ do not undergo ± as was originally expected. decoherence for this particular coupling:| i they retain a Global phase shifts are never observable: only the definite phase relationship in the off-diagonal matrix el- phase difference induced between eigenstates of the cou- ements. We see from the density matrix representation pling operator enters into the calculations. As a result that the state 1 ( HH V V ) at the input looks like √2 | i ± | i of frequency anti-correlation, the phase shift experienced 1 i ωo L∆n 24 ( HH e 2 c V V ) at the analyzer. For this by a photon in path L with frequency ωo + ǫ corresponds √2 | i± | i 2 particular arrangement, a birefringent crystal with axes to that of an (identically polarized) photon of frequency ωo 25 along H and V , the states HH V V do not deco- 2 ǫ in path R. We have already seen that cancella- here: there is no decline in magnitude| i ± | of thei off-diagonal tion− of phase terms in the exponential of Eq. 40 accounts elements of the density matrix. for the coherence properties of a particular polarization Note, however, that neither of the states φ± can be state. In particular, an off-diagonal matrix element goes considered a decoherence-free subspace (DFS).| i In gen- to zero if not all ǫ terms cancel out, and the Fourier integral does not reduce to unity. Since frequency anti- correlation directly affects the sign of these phase terms, it also affects their cancellation or non-cancellation and 23 We can realize this situation experimentally either by placing consequently alters the predicted decoherence-free nature the same crystal across both paths, or equivalently by placing of the singlet state. “identical” crystals in each path and at the same orientation. For future convenience, we choose the second option. 24 The phase factor arises from the pump frequency ωo which we have assumed to be effectively monochromatic so that only a definite phase shift, and not decoherence, occurs. Including a 25 This equivalence holds only in the limit that we neglect dispersion frequency spectrum of finite width for the pump would result effects in the crystals so that ∆n of Eq. 41 is independent of in additional decoherence effects on a different length scale, the ω. This is true to a very good approximation over the 5 nm coherence length of the pump beam. bandwidth of the photons in our experiment. 13

By inducing a second anti-correlation, which we shall Birefringent Crystals term “path anti-correlation,” we recover the proposed Down-Conversion Crystal (BBO) DFS. This operation consists of forcing the phase dif- L ference in path L to be the negative of the correspond- Pump (UV) ing phase difference in path R (for identical frequencies). By twice reversing the sign of corresponding phase shifts R (once with frequency anti-correlation and a second time Phase Tuner with path anti-correlation) we effectively produce identi- Correlated Pairs cal phase errors in each path, and restore the permutation Analyzers symmetry of the polarization-frequency coupling. Detectors We are now faced with the experimental task of forcing phase shifts for corresponding polarizations to be of FIG. 10: Schematic diagram of an experimental apparatus for opposite sign in opposite photon paths. Experimentally, producing and analyzing correlated photon pairs. The phase we may simply rotate the crystal in one arm by 90◦. This tuner is used to produce the desired input state. The bire- operation exchanges ne and no and reverses the sign of fringent crystals have identical thickness, but can be oriented ∆n in Eq. 41.26 independently. For the experimental results, the crystal in To this end, we introduce a new (but familiar looking) path L was rotated by θL = −17◦, and the crystal in path R was rotated by θR = −107◦. The birefringent crystals give an operator defined by effective optical path difference between H and V of ∼ 140λo, where λo is the central wavelength of the down-converted pho- U˜ (x , x )= U⊥(x ) U (x ). (43) L R L L ⊗ R R tons (702 nm, in our case). U L⊥(xL) represents the crystal in path L which has been rotated and is now characterized by the phase function the states ψ± are possibly transformed at the pump ϕ˜j so that frequency,| butido not decohere. In particular, for the

iϕ˜j (ω,xL) initial state ρ(0) = ψ− ψ− , the output state is U⊥ χj L = e χj L (44a) | ih | L | i | i U χ = eiϕj (ω,xR) χ . (44b) 00 00 R j R j R L∆n | i | i iωo 1 0 1 e− c 0 ρ(0)  iω L∆n −  . (47) The phase functionsϕ ˜1 andϕ ˜2 in path L are related to → 2 0 e o c 1 0 the phase shifts ϕ1 and ϕ2 in path R by 00− 00     ϕ˜2(ω, ) ϕ˜1(ω, )= ϕ1(ω, ) ϕ2(ω, ). (45) 1 The state ψ− = ( HV VH ) at the input looks L − L L − L | i √2 | i − | i 1 i ωo L∆n 27 (compare Eq. 41). Keeping in mind that photon states like ( HV e 2 c VH ) at the output. √2 | i− | i with spatial mode L are indexed by i and k, we compute Recall now that the singlet state, ψ− can be written the analog of Eq. 40. as 1 ( λλ λλ ) where λ and λ | arei the (in general, √2 | i − | i | i | i elliptical) eigenmodes of any birefringent environment. ρ(xL, xR)= We conclude, therefore, that the state ψ− is generally + 2 2 | i cij ckl∗ χij χkl dǫ A(ω ) A(ω−) immune to decoherence of this type (i.e., collective deco- | ih | | | | | herence, produced here with perpendicular birefringent i,j,k,lX Z + − + − elements in each arm). By imposing an experimental re- i(ϕ˜i(ω ,xL)+ϕj (ω ,xR) ϕ˜k(ω ,xL) ϕl(ω ,xR)) e − − . × quirement which restores the symmetry of the coupling o(46) with respect to photon path permutation, we have recovered the singlet state as a DFS with respect to collective The 4 4 matrices representing the evolution of the decoherence. four Bell× states are displayed in Appendix A 4. From The technique of quantum tomography allows exper- these, we see that the states φ± at the input become imental reconstruction of two-photon polarization state mixed, or incoherent, in that| thei magnitude of the oﬀ- density matrices by an appropriate choice of polarization diagonal elements approaches zero. On the other hand, measurements (on members of an ensemble) [13]. With the experimental setup shown in Fig. 10 and using this technique, we compare the theoretical predictions with experimentally observed density matrices. Details of the 26 In general, we can produce a similar frequency-polarization cou- experiment and results will be reported elsewhere [16]. pling with any elliptical eigenmodes |λi, |λi by rotating these Here, we simply present the theroetical predictions. eigenstates into |Hi and |V i by passive optical elements (a unitary transformation), introducing a birefringent crystal aligned along H and V , and subsequently inverting the polarization transformation with more optical elements. Also, in this more general case, phase anti-correlation is realized by rotating the 27 In our experiment, we adjusted L (by slightly tilting one of the birefringent crystal in one arm by 90◦. quartz elements) such that this phase factor is 0 (modulo 2π). 14

Input Output polarization coupling. Since polarization degrees of freedom are analyzed and frequency degrees of freedom are unobserved, such a result demonstrates the robustness of 0.5 a decoherence-free subspace of two qubits against non- 0.25 1 dissipative errors arising from coupling with unutilized √ (|HHi + |VV i) 0 2 1 -0.25 (environmental) degrees of freedom. 2 -0.5

1 3 2 3 4 4 IV. SUMMARY AND CONCLUSIONS

0.5 In summary, we have developed a prescription for

0.25 1 calculating density matrices representing one- and two- √ (|HHi − |VV i) 0 2 1 photon states in the presence of unitary frequency- -0.25 2 -0.5 polarization coupling. With these methods, we have in- 1 3 2 vestigated decoherence in quantum systems arising from 3 4 4 coupling to unobserved (environmental) degrees of freedom. By inducing frequency-dependent phase errors in photon polarization states and tracing over frequency de- 0.5 grees of freedom, we observe information loss as a par- 0.25 1 ticular form of decoherence. √ (|HV i + |VHi) 0 2 1 -0.25 In the one photon case, we applied the results of [5] 2 -0.5 and established the feasibility and utility of “bang-bang” 1 3 2 3 4 quantum control of decoherence in an optical system. By 4 periodically exchanging the eigenstates of the “environmental” coupling, photon coherence (polarization information) is preserved. We observed this eﬀect in the case 0.5 in which the strength of the coupling is varied (the “slow- 0.25 1 ﬂipping” case); the case in which the basis of the cou- √ (|HV i − |VHi) 0 2 1 -0.25 pling is varied; and the case in which both the basis and 2 -0.5

1 3 the strength of the coupling are varied simultaneously. 2 3 4 Additionally, we demonstrated the utility of bang-bang 4 control in the presence of a non-unitary (dissipative) coupling operation. In all cases, we see a qualitative reduc- FIG. 11: Real part of polarization state density matrices for tion in decoherence for the case in which the quantum Bell states at the input of the experimental setup of Fig. 10 control procedure is implemented. as predicted by Eq. 46. Imaginary parts are zero for all In the two-photon case, we applied general results con- elements. cerning the existence of decoherence-free subspaces in [6, 7] to the case of polarization entangled photon pairs produced in a down-conversion scheme. We found that We seek to quantify decoherence properties in a single energy conservation in the down-conversion process gives parameter. To this end, we use the Fidelity of the trans- rise to frequency entanglement which alters the assump- mission process, defined here as F = T r(ρoutρin) where tions underlying the prediction of a DFS. In particular, ρin (ρout) represents the polarization density matrix at frequency anti-correlation breaks the permutation sym- the input (output) of the system [17]. For a mixed input metry of the environmental coupling between opposite 2 state, we must use F = T r( √ρinρout√ρin [18]. A photon paths by anti-correlating their respective phase Fidelity of 1 indicates a decoherence-free process. Nu- errors. We found that modifying the experimental appa- merical evaluation of Eq. 46p allows us to calculate the ratus to restore the symmetry of the phase errors allows Fidelity of the transmission process for coupling eigen- us to produce a truly collective decoherence process, and states rotated by an arbitrary angle. Such an analysis consequently we recovered the singlet state as a DFS. reveals a Fidelity of 1 for ψ− in a birefringent environ- In conclusion, we have investigated two procedures for ment with arbitray eigenstates,| i confirming the prediction reducing the decoherence of an optical qubit subjected that the singlet state is a decoherence-free subspace. to non-dissipative phase errors and non-unitary dissipa- In Fig. 11, we present a graphical representation of tive errors. The first procedure, “bang-bang” quantum the density matrices for each of the four Bell States pass- control, preserves polarization information in a single- ing through birefringent crystals in each photon path as photon by rapid, periodic control operations which ef- calculated from Eq. 46. fectively average out decoherence effects. The second We see that the singlet state does indeed maintain procedure requires entanglement between two photons full coherence in the presence of a unitary frequency- so that the resulting state exhibits symmetry properties 15 which are immune to collective decoherence. These cases b. Derivation of Equations 21a-22b together demonstrate the feasibility of both active and passive protection of coherence properties in an optical All four relations follow easily from the definitions of qubit. R and j (Eqs. 9a-9b and 18a-18b). Equation 21a is trivial, sinceU by definition, both operators and act U1 U2 as the identity on χ1 . For the other three, we apply the APPENDIX A: MATHEMATICAL ARGUMENTS definitions in succession.| i χ = χ (A5) 1. Eigenvalues of U(x) U2U1| 1i | 1i χ = eiϕ1(ω) χ U2U1| 2i U2 | 2i Unitarity restricts the possible eigenvalues, U (ω, x), i(ϕ1(ω)+ϕ2(ω)) j = e χ2 (A6) of U(x): | i R R χ = R R χ U2 U1| 1i U2 | 1i 1 = χj χj = R χ h | i U2| 2i iϕ2(ω) = χj U†(x)U(x) χj = Re χ h | | i | 2i iϕ2(ω) = χj Uj∗(ω, x)Uj (ω, x) χj = e ( χ ) h | | i ± | 1i = U (ω, x) 2. (A1) χ (A7) | j | → | 1i iϕ1(ω) R 2R 1 χ2 = R 2Re χ2 Since Uj(ω, x) is complex with modulus 1, we may write U U | i U | i iϕ1(ω) = R 2e χ1 U (ω, x)= eiϕj (ω,x) (A2) ± U | i j = Reiϕ1(ω) χ ± | 1i iϕ1(ω) where ϕj is a real-valued function of ω and x, and the = e χ2 subscript refers to polarization mode j. ± | i iϕ2(ω) i(ϕ1(ω) ϕ2(ω)) = e e − χ ± | 2i i(ϕ1(ω) ϕ2(ω)) e − χ (A8) 2. Operator Relations with R and U → | 2i Again, if R represents a reflection (rotation) we use the a. Proof of Equations 11a, 11b upper (lower) sign. From these results, we see that 2 1 can be treated as a single operator with an associatedU U U Since is a linear operator, we need only show that phase error of ϕ1(ω)+ ϕ2(ω). Factoring out the global U phase shift of ϕ (ω) in the last two relations (indicated by Eqs. 11a and 11b hold for the basis states, χ1,ω and 2 R | i the arrow in eqs. A7 and A8), we see that R R can χ2,ω . From the definitions of and (Eqs. 9a-10b), U2 U1 | i U be treated as a single operator QC with an associated we have U phase error of ϕ1(ω) ϕ2(ω). R R χ ,ω = R R χ ,ω − U U| 1 i U | 1 i = R χ ,ω U| 2 i 3. Experimental Measures of Coherence = eiϕ(ω)R χ ,ω | 2 i = eiϕ(ω) χ ,ω . (A3a) a. Proof of Equation 26 ± | 1 i Similarly, We begin by defining λ in Eq. 25 to be an equal su- R R χ ,ω = eiϕ(ω)R R χ ,ω perposition of the basis states χj : U U| 2 i U | 2 i | i iϕ(ω) = e R χ1,ω 1 ± U| i λ = ( χ1 + χ2 ) (A9a) = eiϕ(ω)R χ ,ω | i √2 | i | i ± | 1 i iϕ(ω) 1 = e χ ,ω . (A3b) λ = ( χ1 χ2 ) . (A9b) ± | 2 i | i √2 | i − | i 2 Thus we have, (R ) = eiϕ(ω) where the upper (lower) Then we have sign occurs when UR represents± a polarization reflection (x) = λ ρ(x) λ λ ρ(x) λ (rotation). V h | | i − h | | i 1 It follows immediately that = ( χ + χ )ρ(x)( χ + χ ) 2 { h 1| h 2| | 1i | 2i 2 iϕ(ω) †R† = e− . (A4) ( χ1 χ2 )ρ(x)( χ1 χ2 ) U ± − h | − h | | i − | i } = ( χ ρ(x) χ + χ ρ(x) χ ) Since equations 11a and 11b hold for the basis states they h 1| | 2i h 2| | 1i = ( χ ρ(x) χ + χ ρ(x) χ ∗) hold for all linear combinations of basis states, which is h 1| | 2i h 1| | 2i to say, for all states. = 2Re( χ ρ(x) χ ) (A10) h 1| | 2i 16 where Re indicates the real part.

4. Explicit Calculation of Two-Photon Density Matrices

We begin by modeling two-photon density matrices including frequency anti-correlation eﬀects. Substituting the appropriate phase functions (Eq. 41) into Eq. 40, we have an explicit expression for ρ = ρ( , ): L L ω ω ωo ωo ωo ωo o 2 o 2 i(ϕi( +ǫ, )+ϕj ( ǫ, ) ϕk( +ǫ, ) ϕl( ǫ, )) ρ = c c∗ χ χ dǫ A( + ǫ) A( ǫ) e 2 L 2 − L − 2 L − 2 − L . (A11) ij kl| ij ih kl| | 2 | | 2 − | i,j,k,lX Z n o ⋆ We will now calculate ρ assuming the initial state matrix elements, cij ckl, are given by the density matrix representation of the four Bell States, 1 1 φ± = ( HH V V ), ψ± = ( HV VH ). (A12) | i √2 | i ± | i | i √2 | i ± | i

U ωo 2 ωo 2 In this way, we see the eﬀect of on each of these states. For brevity, let f(ǫ)= A( 2 + ǫ) A( 2 ǫ) . Then we have, | | | − |

L∆n iωo 1 00 1 1 00 e− c 1 0 00± 0 1 0 00± 0 φ± φ± =   (A13) | ih | 2  0 00 0  → 2 0 00 0 iω L∆n 100 1  e o c 00 1  ±  ±      00 00 00 00 2iǫ L∆n 1 0 1 1 0 1 0 1 dǫf(ǫ)e c 0 ψ± ψ± = ±  2iǫ L∆n ±  . (A14) | ih | 2 0 1 1 0 → 2 0 dǫf(ǫ)e c 1 0 ± − R 00 00 00± 00        R  Note that the dependence on ǫ (the variable of integration) drops out of the (11, 22) and (22, 11) terms of φ± φ± so that there is no decoherence due to the frequency distribution of the down-converted photons. However,| ǫihdoes| not drop out of the (12, 21) and (21, 12) terms of ψ± ψ± . Due to the finite width of the frequency term f(ǫ), for large , the integral in these terms approaches zero| inih the| same manner as the Fourier integrals of Sect. II A. Note L that the results developed above apply only to decoherence in the H/V basis. The states φ± look different in other bases, so they are subject to decoherence in general. | i In order to model decoherence in other bases, and also to model non-collective decoherence, we must include rotation operators in Eq. A11. We define an operator (θ ,θ ) which represents a rotation by θ in arm L and a rotation by R L R L θR in path R. It can be shown that in the present matrix notation, cos(θ ) cos(θ ) sin(θ ) cos(θ ) cos(θ ) sin(θ ) sin(θ ) sin(θ ) L R − L R − L R L R sin(θL) cos(θR) cos(θL) cos(θR) sin(θL) sin(θR) cos(θL) sin(θR) (θL,θR)= − − . (A15) R cos(θL) sin(θR) sin(θL) sin(θR) cos(θL) cos(θR) sin(θL) cos(θR) − − sin(θL) sin(θR) cos(θL) sin(θR) sin(θL)cos(θR) cos(θL)cos(θR)    1 It follows that − (θ ,θ ) = †(θ ,θ ) = ( θ , θ ). We can now generalize Eq. 39 by writing the reduced R L R R L R R − L − R density operator representing polarization degrees of freedom after an input state ρω(0) has passed through crystals of thicknesses xL and xR at angles θL and θR in the photon paths labeled L and R:

ρ(x , x ,θ ,θ )= dω dω ω ω (θ ,θ )U(x , x ) †(θ ,θ )ρ (0) (θ ,θ )U†(x , x ) †(θ ,θ ) ω ω . L R L R 3 4h 3|h 4|R L R L R R L R ω R L R L R R L R | 3i| 4i ZZ (A16)

Eq. A16 can be solved numerically given ϕj (ω, x) (j 1, 2 ), the eigenvalues which define U(xL, xR). With the same techniques, we can model density matrices∈{ including} both frequency and path anti-correlation effects. Substituting the modified phase functions (45) into Eq. 46, we have an expression which includes both frequency and path anti-correlation effects:

Again, using the four Bell States as input states, we see the eﬀect of U˜ on each of these states.

2iǫ L∆n 1 00 1 1 00 dǫf(ǫ)e c 1 0 00± 0 1 0 00± 0 φ± φ± =  R  (A18) | ih | 2  0 00 0  → 2 0 00 0 2iǫ L∆n 100 1  dǫf(ǫ)e− c 00 1  ±  ±      00 00 00R 00 L∆n iωo 1 0 1 1 0 1 0 1 e− c 0 ψ± ψ± = ±  iω L∆n ±  . (A19) | ih | 2 0 1 1 0 → 2 0 e o c 1 0 ± 00 00 00± 00        

Note that in this case, the dependence on ǫ (the variable ψ− , since it looks the same in every basis. of integration) drops out of the (21, 12) and (12, 21) terms | i of ψ± ψ± . On the other hand, ǫ does not drop out of | ih | the (11, 22) and (22, 11) terms of φ± φ± . Again, the results developed above apply only| to decoherenceih | in the ACKNOWLEDGEMENTS H/V basis. The analytical form of ρ is helpful in developing a physical picture of the decoherence process. In order to calculate the corresponding result for decoher- Experimental and theoretical work was performed in π ence in an arbitrary basis, we simply set θL = θR 2 the Quantum Information laboratory of P. G. Kwiat at in Eq. A16 and solve numerically. This corresponds− LANL. We wish to thank A. G. White for his willingness to the path anti-correlation procedure described earlier. to assist in all stages of this work. At Dartmouth, AJB These results will be presented elsewhere [16], however, gratefully acknowledges W. Lawrence and M. Mycek for the theoretical predictions are unchanged for the state oﬀering support, advice and feedback at various stages.

[1] W. G. Unruh, Phys. Rev. A 51, 992 (1995). [10] M. Born, and E. Wolf, Principles of Optics, (Cambridge [2] M. Reck, A. Zeilinger, H. Bernstein, and P. Bertani, Univ. Press, Cambridge, 7th Edition, 1999). Phys. Rev. Lett. 71, 58 (1994). [11] J. J. Sakurai, Modern Quantum Mechanics, Chapter 3, [3] N. J. Cerf, C. Adami, and P. G. Kwiat, Phys. Rev. A 57, (Addison-Wesley, Massachusetts, 2nd Edition, 1994). 1479 (1998) [12] W. T. Buttler et al., Phys. Rev. Lett. 81, 3283, (1998). [4] P. G. Kwiat, J. R. Mitchell, P. D. D. Schwindt, and A. [13] A. G. White, P. G. Kwiat, D. F. V. James, and P. Eber- G. White, J. of Mod. Opt. 47, 257 (2000). hard, Phys. Rev. Lett. 83, 3103 (1999). [5] L. Viola, and S. Lloyd, Phys. Rev. A 58, 2733 (1998). [14] L. Mandel, and E. Wolf, Optical Coherence and Quantum [6] P. Zanardi, Phys. Rev. A 56, 4445 (1997). Optics, (Cambridge Univ. Press, Cambridge, 1995). [7] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. [15] J. D. Franson, Phys. Rev. A 45, 3126 (1992). Lett. 81, 2594 (1998). [16] P. G. Kwiat, A. J. Berglund, J. B. Altepeter, and A. G. [8] P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. White, to appear in Science (2000). Sergienko, and Y. Shih, Phys. Rev. Lett. 75, 4337 (1995). [17] B. Schumacher, Phys. Rev. A 51, 2738 (1995). [9] P. G. Kwiat, E. Waks, A. G. White, I. Applebaum, and [18] R. Jozsa, J. Mod. Optics 41, 2315 (1994). P. Eberhard, Phys. Rev. A 60, R773 (1999).