Coherence in a cold atom photon transistor

Recent experiments have realized an all-optical photon transistor using a cold atomic gas. This approach relies on electromagnetically induced transparency (EIT) in conjunction with the strong interaction among atoms excited to high-lying Rydberg states. The transistor is gated via a so-called Rydberg spinwave, in which a single Rydberg excitation is coherently shared by the whole ensemble. In its absence the incoming photon passes through the atomic ensemble by virtue of EIT while in its presence the photon is scattered rendering the atomic gas opaque. An important current challenge is to preserve the coherence of the Rydberg spinwave during the operation of the transistor, which would enable for example its coherent optical read-out and its further processing in quantum circuits. With a combined field theoretical and quantum jump approach and by employing a simple model description we investigate systematically and comprehensively how the coherence of the Rydberg spinwave is affected by photon scattering. With large-scale numerical calculations we show how coherence becomes increasingly protected with growing interatomic interaction strength. For the strongly interacting limit we derive analytical expressions for the spinwave fidelity as a function of the optical depth and bandwidth of the incoming photon.

Single photon switchs might form a central building block of an all-optical quantum information processor [23][24][25]. The prime function of such switches is to control the transmission of an incoming photon through a single gate photon. One promising way to realize this is to store the gate photon in form of a gate (Rydberg) atom immersed in an atomic gas which is in a delocalized spinwave state [26][27][28]. The gate atom then prevents transmission of incident photons through the gas, while ideally the coherence of the Rydberg spinwave state is preserved [29][30][31][32]. The latter property would permit the subsequent coherent conversion of the Rydberg spinwave into a photon which would pave the way for gating the switch with superposition states that can also be subsequently retrieved. Currently, there is only a basic understanding of how the coherence of the Rydberg spinwave might be affected by the scattering of incoming photons and no systematic study of this important question exists.
In this work we address this outstanding issue within a simple model system. We study the propagation of a single photon under conditions of electromagnetically induced transparency (EIT) in a cold atomic gas in which a gate photon is stored as a Rydberg spinwave. An incident photon subsequently experiences a Rydberg mediated van der Waals (vdW) interaction with this stored gate atom which lifts the EIT condition and renders the atomic medium opaque. In this case the incident pho- 1. (a) EIT level scheme. The groundstate |1 , excited state |2 (decay rate γ) and Rydberg state |3 are resonantly coupled by a single photon field E(z, t) (with collective coupling strength g) and a classical field of Rabi frequency Ω. Initially a gate photon is stored as a spinwave in the Rydberg state |4 (indicated by the green circle). (b,c) Polarization profiles Pj(z, t) for a spinwave consisting of two possible gate atom positions Zj (j = 1, 2) and their dependence on the blockade radius R b and the system length L. (b) For L > R b and |Z2 − Z1| > 2R b the polarization profiles associated with the two gate atom positions are distinguishable. (c) When L < ∼ R b the polarization profile is independent of the gate atom position which leads to enhanced coherence of the stored spinwave.
ton is scattered incoherently off the Rydberg spinwave. We study the photon propagation and explore the dependence of Rydberg spinwave coherence on the interaction strength (parameterized by the blockade radius R b ), the system length L and bandwidth of the incident photon pulse. Our findings confirm that strong absorption, i.e. high gain, can be achieved already for large systems (L > R b ) while coherence of the spinwave is preserved only for sufficiently strong interactions, i.e. L < ∼ R b . Intuitively, this can be understood by regarding the scattering of the incoming photon as a measurement of the position of the gate atom. When L < ∼ R b this measurement is not able to resolve the position of the excitation and hence coherence of the Rydberg spinwave is maintained. Our study goes beyond this simple consideration arXiv:1505.02667v2 [quant-ph] 21 Oct 2015 by taking into account propagation effects, a realistic interaction potential and a finite photon band width. The results can therefore be considered as upper bounds for the fidelity with which a Rydberg spinwave can be preserved and re-converted into a photon in an experimental realization of a coherent cold atom photon switch.
The paper is organized as follows. In section II, we introduce a one-dimensional model system to study the propagation dynamics of single source photons in the atomic gas prepared in a Rydberg spinwave state. In Sec. III, the model system is solved numerically with realistic parameters. We identify the working regime for a single photon switch where the source photon is scattered completely. In Sec. IV, we numerically study the fidelity between the initial spinwave state and the final state after the source photon is scattered. Our calculation shows that the coherence of the spinwave is preserved when L ≥ R b while the final state becomes a mixed state when L < R b . In Sec. V, We provide analytical results for a coherent single photon switch (L ≥ R b ). We reveal that the transmission and switch fidelity depend nontrvially on the optical depth and bandwidth of the source photon field. We summarize in Sec. VI.

II. THE MODEL SYSTEM
Our model system is a one-dimensional, homogeneous gas consisting of N atoms, whose electronic levels are given in Fig. 1a. The photon fieldÊ(z, t) and the EIT control laser (Rabi frequency Ω) resonantly couple the groundstate |1 with the excited state |2 and |2 with the Rydberg state |3 . Following Ref. [34], we use polarization operatorsP (z, t) andŜ(z, t) to describe the slowly varying and continuum coherence of the atomic medium |1 2| and |1 3|, respectively. All the operatorsÔ(z, t) = {Ê(z, t) ,P (z, t) ,Ŝ(z, t)} are bosons and satisfy the equal time commutation relation, [Ô(z, t),Ô † (z , t)] = δ(z −z ). Initially, the atoms are prepared in a delocalized spinwave state with a single gate atom in the Rydberg state |4 , where k is the wavenumber of the spinwave and |Z i = |1 1 . . . 4 i . . . 1 N abbreviates many-body basis with the gated atom located at position Z i and the rest in the groundstate. The Rydberg spinwave state is created routinely in experiments [20-22, 35, 36]. When interacting with the incoming single photon, the general many-body state of this one-dimensional system is expanded as [2] |Ψ where ξ is probability amplitude of the initial spinwave state. In the weak field approximation, we will assume ξ = 1 at any moment. We have defined O(z, t) = Ô (z, t) , i.e. the expectation value of the operator O(z, t). Specifically one finds that E(z, t) is the probability amplitude in the one photon state, P (z, t) and S(z, t) are the amplitude of one atom in the |2 and |3 state, respectively. In order to develop a first intuition for the physics at work we first consider a spinwave that is delocalized merely over two atoms embedded in the atomic cloud (see Fig. 1b,c). We assume furthermore that the interaction between atoms in state |3 and the gate atom is infinite for distances smaller than the so-called blockade radius R b and zero otherwise. Outside the blockade region, the photon propagates (along the +z direction) as a dark-state polariton by virtue of EIT [34]. Inside the blockade region the medium behaves like an ensemble of two-level system. Here the incoming photon is building up a non-zero polarization P (z, t), whose modulus square is the probability density distribution for finding an atom in the decaying state |2 according to Eq. (1) [37]. Eventually, this leads to the loss of the incoming photon and makes the medium opaque. In order to understand how such photon scattering affects the coherence of the properties of the spinwave one needs to analyze the shape of the polarization profile. As shown in Fig. 1b this in general depends on the position of the gate atom when the system length is larger than the blockade radius R b < L.
Here, since L > 4R b and |Z 2 −Z 1 | > 2R b , it is possible to distinguish the profiles P j (z, t) which are associated with the two possible positions of the gate atom. Conversely, the polarization P j (z, t) becomes independent of the gate atom position when L < ∼ R b (see Fig. 1c). In this caseas discussed in detail later -the coherence of the spinwave will be preserved as one can not distinguish gate atoms from the scattered photon.
Let us now consider the actual photon propagation together with a realistic interaction potential. The dynamics of the system follows the master equation [34,38] where the first term on the right-hand side (RHS) is the evolution ofρ(t) under the effective HamiltonianĤ e = H p +Ĥ ap +Ĥ a , and the spontaneous decay (with rate γ) from the state |2 is governed by the second term. In the effective Hamiltonian, the photon propagation in the medium is governed by the Hamiltonian with the vacuum light speed c. The atom-photon coupling is described bŷ where g = √ N g s with g s being the single atom-photon coupling strength. The vdW interaction between an atom in the state |3 and the gate atom at position Z i iŝ The interaction potential depends on the gate atom position,V 6 gives the vdW interaction with C 6 being the dispersion coefficient.
For the case of a single incoming photon which we consider here the solution of the master equation (2) is [38] The first term on the RHS describes the unhindered photon propagation through the medium, while the second term accounts for the photon scattering, i.e. photon-loss from the medium.

III. TRANSMISSION OF THE SOURCE PHOTON
To calculate (3) we first treat the dynamics under the effective Hamiltonian in the Heisenberg picture. To this end we obtain the equation of motion for the expectation values O(z, t) from the corresponding operator Heisenberg equation [39]. Note, that due to the linearity of the equations we can moreover calculate the expectation value for each component |Z j of the Rydberg spinwave, i.e. each of the possible positions of the gate atom, separately. This yields the set of equations where the index j labels the respective spinwave component. Alternatively, these equations can be obtained from a Heisenberg-Langevin approach [17]. We solve the coupled equations (4) through a Fourier transform yielding the formal solution for the polarization Here we have abbreviated T = t − z/c and introduced the electric susceptibility From χ j (z) one can actually extract the blockade radius as the critical distance at which the vdW interaction and the control laser are equally strong. This yields R b = |γC 6 /2Ω 2 | 1/6 [17].
The polarization (5) depends on the Fourier transform E 0 (ω) of the photon field at position z = 0. To be specific we take the photon pulse to be a Gaussian at t = 0 which is normalized in space, Here τ is the temporal duration of the pulse and z 0 is the initial central position (z 0 −cτ ). The band width of the pulse is then given by ∆ω = 1/τ . Note, that it is generally not possible to evaluate the formal solution (5) analytically. Moreover, numerical calculations are challenging since the involved time and length scales span several orders of magnitude [40].
Let us now calculate the photon transmission as a function of the pulse duration τ , which to our knowledge has not been examined previously.
We define the transmission of the photon pulse as T = ∞ 0 dt|E(L, t)| 2 / ∞ 0 dt|E(0, t)| 2 . In Fig. 2a, we show T as a function of the pulse width for two values of the atomphoton coupling strength g. For fixed pulse length τ , we find that stronger couplings generally are accompanied by a lower transmission. Furthermore, we observe that the transmission increases with decreasing pulse duration τ . This is due to the fact that the pulse contains increasingly more weight on frequency components, which are outside the absorption window of the medium. For the purpose of complete photon scattering, one thus has to utilize narrow frequency band pulses.
Next, we briefly discuss the dependence of the transmission T on the strength of the atom-photon coupling g. Fig. 2b shows data for two choices of the blockade radius, R b = L and R b = L/2. As expected, T decreases with increasing g. However, for the system parameters chosen here, there is virtually no dependence of T on the value of the blockade radius when g = 1000γ, where T ≈ 0. These findings indicate that one reaches the strong scattering regime when g γ and ∆ω γ. This is the working regime for the single photon switch where the medium becomes opaque for the incident photon.

IV. FIDELITY BETWEEN THE INITIAL AND FINAL STATE
Focusing on this regime, our next task is to investigate how the photon scattering influences the Rydberg spinwave. We quantify the difference between the initial Rydberg spinwaveρ i and the final stateρ f by the fidelity [41] As the initial spinwave is a pure state, this sim- τ ). This shows that a high fidelity can be obtained only if the polarization profiles P j (z, τ ) for each spinwave component are essentially equal: Only when A jk ∼ 1 and thus jk A jk ∼ N 2 the fidelity is close to one. This is the formal version of the intuitive statement that we made earlier in conjunction with the discussion of Fig. 1b,c. For completeness we provide a numerical example for which we choose R b = L/2 and select only two components of the spinwave, where the gate atom is located at either Z i = 0 or Z i = L. The resulting polarization profile |P (z, t)| 2 is shown in Fig. 3a,b. For Z i = 0, non-vanishing polarization is built up within the blockade region as long as the photon is inside the medium (Fig. 3a). Integrating over time we obtain the intensity I p (z) = ∞ 0 dτ |P (z, τ )| 2 which clearly shows a decay to zero within a blockade distance R b (see Fig. 3c). In contrast, for Z i = L, appreciable polarization is built up also outside R b and the profile is peaked at approximately z = L − R b (Fig. 3b,d). Clearly, both polarization profiles are strikingly different which in turn causes a loss of fidelity when the blockade radius is smaller than the system length. We verify this by numerically calculating the fidelity as a function of the blockade radius. The data is displayed in Fig. 4a, together with the corresponding transmission T . As anticipated, the fidelity decreases significantly below unity when R b is decreased with respect to L. Note, that the transmission is close to zero throughout.
A fidelity smaller than unity directly indicates the formation of a mixed state after the photon scattering. The  final state density matrix iŝ The final state can only be pure when |A jk | = 1 and hence F = 1. The formation of a mixed state is a consequence of the actual measurement of the gate atom position [42] which is performed by the photon scattering: When R b < L one in principle gains information on the position of the gate atom since the spatial uncertainty of its wave function is reduced from L to the blockade region. The final state is then a mixture of all states compatible with this additional information.

V. ANALYTICAL RESULTS FOR COHERENT PHOTON SWITCHS
In the remainder of the paper we will focus on the case of a coherent photon switch, i.e. R b ≥ L. Here the expression for the susceptibility of the medium simplifies to that of an ensemble of two-level atoms, χ j (z) ≈ −g/(ω + iγ/2) which permits the derivation of analytical results. For a narrow band width pulse we can derive explicit solutions to Eq. (4) that have no dependence on the position of the gate atom [43]. For example, the po- larization P (z, t) is given by where Ec(x) is the complementary error function. The corresponding time-integrated profile I p (z) agrees perfectly with the numerical result from Eq. (4) (see Fig. 4b).
The transmission T is given by where Er(x) is the error function and α = 4g 2 L/cγ is the optical depth of a resonant two-level medium. The excellent agreement between the analytical and numerical calculation is shown in Fig. 2b. Neglecting the finite band width of the photon pulse, i.e. when all the frequency components are in the absorption window, Eq. (7) reduces to the well-known form T ≈ e −α [17]. Finally, the fidelity can be expressed as a function of the optical depth and pulse band width This shows that indeed a small band width is a requirement for reaching a large fidelity. For example, the transmission is negligible (T ≈ 8 × 10 −4 ) when γτ = 5 and g = 1000γ according to the data in Fig. 2a. However, the fidelity is below unity (F = 0.94) due to non-negligible contributions from the terms accounting for the finite band width. In the limit of very long pulses one finds F ≈ 1 − T and thus the fidelity is solely determined by the transmission.

VI. SUMMARY
In summary, we have studied the coherence of a Rydberg spinwave in the operation of a signle photon switch. The current study is limited to a single gate atom and an incoming single-photon pulse, which permits the description of multi-photon scattering, however, only if the photons enter the switch sequentially. Addressing this limitation and extending the discussion to correlated and entangled photon pulses that fall in the operation regime of single photon transistors will be subject to future studies.

ACKNOWLEDGMENTS
The integration is in general difficult to carry out analytically due to the complicated form of the susceptibility. We overcome this difficulty by expanding the susceptibil-ity in powers of ω, First let us calculate the approximate solution for E(z, t). To carry out analytical calculations and at the same time take into account contributions due to the finite band width, we will keep terms up to the second order of ω in Eq. (A6). This yields the solution for E(z, t) .
This leads to the analytical form of the transmission (7) in the main text.
However it is difficult to calculate the fidelity from Eq. (A11) due to the presence of the error function. We thus calculate P (z, t) alternatively using the Fourier transform method. We note that the susceptibility χ appears at two places in Eq. (A5): one in front ofẼ(ω) and another one in the exponential function. In order to obtain an analytical result, we will expand the former susceptibility up to the second order of ω while the latter up to the linear order. After performing the inverse Fourier transform, we obtain the expression for P (z, t), P (z, t) = igE(z, t) 32c 2 τ 4 γ 9 c 2 γ 6 τ 4 + 8czγ 3 τ 2 g 2 − 4c 2 γ 3 τ 2 + 2cγ 5 τ 2 (ct − z + z 0 ) + 4 γ 2 (ct − z + z 0 ) + 4g 2 z 2 .(A12) Using Eq. (A12) the fidelity can be calculated analytically,