Quantum coherence and control in one- and two-photon optical systems Andrew J. Berglund∗ Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755, USA and Physics Division, P-23, Los Alamos National Laboratory, Los Alamos, NM 87545, USA We investigate coherence in one- and two-photon optical systems, both theoretically and ex- perimentally. In the first case, we develop the density operator representing a single photon state subjected to a non-dissipative coupling between observed (polarization) and unobserved (frequency) degrees of freedom. We show that an implementation of “bang-bang” quantum control protects pho- ton polarization information from certain types of decoherence. In the second case, we investigate the existence of a “decoherence-free” subspace of the Hilbert space of two-photon polarization states under the action of a similar coupling. The density operator representation is developed analytically and solutions are obtained numerically. [Note: This manuscript is taken from the author’s un- photon polarization-entangled state that, due to its sym- dergraduate thesis (A.B. Dartmouth College, June 2000, metry properties, is immune to collective decoherence advised by Dr. Walter E. Lawrence), an experimental of the type mentioned above. That is, this state is a and theoretical investigation under the supervision of Dr. decoherence-free subspace (DFS) of the Hilbert space of Paul G. Kwiat.1] photon polarization [6, 7]. Photon pairs entangled in both polarization and frequency degrees of freedom, such as hyper-entangled photons produced in down-conversion I. INTRODUCTION sources (see [8, 9]), further complicate this particular de- coherence mechanism . In particular, energy conserva- tion imposes frequency correlations which affect the co- Decoherence in two-state quantum systems is a sig- herence properties of these two-photon states. In the nificant obstacle to the realization of proposed quan- experimental case, it will be shown that this frequency tum information technologies. Coupling between quan- correlation can be effectively suppressed and the DFS re- tum bit (“qubit”) states and unobserved environmental covered by a simple and physically intuitive modification degrees of freedom leads to decoherence effects which of the apparatus. limit the practical implementation of proposed quantum algorithms [1]. Photon modes, including polarization and spatial modes, provide an easily accessible system in which simple quantum circuits can be investigated II. COHERENCE AND “BANG-BANG” [2, 3, 4]. Here, we examine the process of decoherence CONTROL: THE ONE-PHOTON CASE by subjecting single photons to a controllable birefrin- gent “environment” and observing the evolution of the A. Review of single-photon decoherence polarization state. In section II, we investigate the evolution of a sin- In this section, we will show that non-dissipative (uni- gle photon state under the action of a unitary coupling tary) coupling between photon frequency and polariza- between polarization and frequency modes. Such non- tion in a birefringent “environment” followed by a trace arXiv:quant-ph/0010001v2 3 Oct 2000 dissipative “phase errors” give rise to decoherence effects, over frequency leads to decoherence.2 These results are whereby the photon evolves from a definite polarization relevant to the study of decoherence in quantum systems state to an unpolarized state. We then introduce and which arises when coupling to environmental states, fol- examine an optical implementation of so-called “bang- lowed by a trace over those degrees of freedom, leads to a bang” quantum control of decoherence by rapidly ex- loss of phase information between qubit basis states. In changing the eigenstates of the coupling operation [5]. the following argument, the term “environmental” will The term “quantum control” is justified since such an op- be used to describe the coupling of information-carrying eration will be shown to preserve a coherent polarization states (photon polarization modes) to degrees of free- state in some special cases, and to reduce decoherence in dom which are not utilized for information representa- more general cases. tion (photon frequency modes). It will be shown that In section III we will introduce a particular two- 2 This effect has a well-known counterpart in classical optics ∗Present address: Norman Bridge Laboratory of Physics 12-33, whereby quasi-monochromatic light composed of uncorrelated California Institute of Technology, Pasadena, CA 91125, USA; frequency components loses the ability to interfere with itself Electronic address: [email protected] when polarization modes are separated beyond the coherence 1 Physics Division, P-23, Los Alamos National Laboratory, Los length of the incident light, as in an unbalanced Michelson inter- Alamos, NM 87545, USA. Email: [email protected]. ferometer (see [10], §7.5.8). 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) den- sity 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 op- erator 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.