Fundamental limitations in spin-ensemble quantum memories for cavity fields

Inhomogeneously broadened spin ensembles play an important role in present-day implementation of hybrid quantum processing architectures. When coupled to a cavity, such an ensemble may serve as a multimode quantum memory for the cavity field, and by employing spin-refocusing techniques the quantum-memory time can be extended to the coherence time of individual spins in the ensemble. In this paper, we investigate such a memory protocol capable of storing an unknown cavity-field state, and we examine separately the various constituents of the protocol: the storage and readout part, the memory hold time with the spin ensemble and cavity field decoupled, and the parts employing spin-refocusing techniques. Using both analytical and numerical methods, we derive how the obtainable memory performance scales with various physical parameters.

Figure 1

(a) The physical system under consideration: A spin ensemble is placed within a one-sided cavity, which may be driven externally by the field $\beta$ through a mirror with field-decay rate $\kappa$. (b) The quantum-memory protocol consists of storage and retrieval parts with a strongly and resonantly coupled spin-cavity system (shaded area), $\pi$ pulses for inverting the spin ensemble in a resonant but weakly coupled regime (hatched area), and parts (white) with effective spin-cavity decoupling. Due to the two $\pi$ pulses, a silenced spin echo occurs in the middle of the sequence. For later reference, $T_i$ denotes the duration of the $i$th part of the sequence.

Figure 2

(a) The solid curves show the evolution of $\langle {\widehat{a}}_{\mathrm{c}}\rangle$ (blue, upper curve at $t=0$) and $\langle \widehat{b}\rangle$ (red, lower curve at $t=0$) versus time for a coupled spin-cavity system with $g_{\mathrm{ens}}=2.5\Gamma$. The dashed lines show the free evolution if $g_{\mathrm{ens}}$ is changed to zero at $t=T_{\mathrm{swap}}$, and the dotted lines show the same but with a perfect inversion process included at $t=T_{\mathrm{swap}}$ (the blue dashed and dotted lines coincide at $\langle {\widehat{a}}_{\mathrm{c}}\rangle =0$). (b) The solid curves show the phase ${\varphi }_j$ of individual spin components for four different frequency classes with ${\Delta }_j$ taking the values (counting from the horizontal axis and away): $w$ (red), $2w$ (green), $3w$ (blue), and $4w$ (cyan). The inversion process implies ${\varphi }_j\to -{\varphi }_j$ at $t=T_{\mathrm{swap}}$ with the dashed lines representing the free evolution in the absence of the inversion process and the dotted lines (for the two lowest values of ${\Delta }_j$) the hypothetical free evolution of duration $T_{\mathrm{focus}}$ back in time to the effective point of origin. In panel (c), the thick solid line (red) shows the phase ${\varphi }_j$ versus ${\Delta }_j$ at $t=T_{\mathrm{swap}}$; the four dots correspond to the four cases examined in (b). The dashed line represents a fit of $T_{\mathrm{focus}}$ to the function ${\varphi }_j=\frac{\pi }{2}+{\Delta }_j{T}_{\mathrm{focus}}$ in the linear regime near ${\Delta }_j=0$, and the dotted lines follow ${\varphi }_j={\Delta }_j{T}_{\mathrm{swap}}$ and ${\varphi }_j={\Delta }_j{T}_{\mathrm{swap}}+\pi$. The thin gray curve represents (on an arbitrary vertical scale) the inhomogeneous distribution of Eq. (1) with the FWHM $w$ used in this example.

Figure 3

(a) The mean-value gain with $\kappa ={\gamma }_{\perp }=0$ and varying $g_{\mathrm{ens}}$. The solid red line, for guiding the eye, has a slope of $\approx -1.7$ in the double-logarithmic plot. (b) The gain for varying values of ${\gamma }_{\perp }$ and $T_{\mathrm{mem}}$ with $g_{\mathrm{ens}}=2.5\Gamma$, the different symbols represent different values of $T_{\mathrm{mem}}$. The solid curve is $y=0.997\exp (-x)$. (c) The gain versus $\kappa$ with ${\gamma }_{\perp }=0$ and $g_{\mathrm{ens}}=2.5\Gamma$. The solid line is $y=0.997\exp (-x)$. (d) For the rightmost data point in panel (c), the red solid curve shows the phase ${\varphi }_j$ versus ${\Delta }_j$ after the initial swap [in analogy to Fig. 2c]. The green dashed curve is a time-reversed swap, i.e., the desired target phase distribution before the retrieval part of the memory. The black dotted line is the actual phase distribution obtained by the refocusing process matching the desired slope for small $|{\Delta }_j|$.

Figure 4

All panels: ${\gamma }_{\perp }=0$ and $g_{\mathrm{ens}}=2.5\Gamma$. (a) The transverse-spin-component mean value versus time for the spin-refocusing protocol of Sec. 5 with $\kappa /w=0.075$ and ${\Delta }_{\mathrm{cs}}/w=20$. Red circles: Numerical simulation. Black solid line: Adiabatic model of Eqs. (E3) and (E5)–(E7). (b) For the same parameters as panel (a), the solid line shows the relative excess spin noise (RESN) versus time. The red dotted line denotes the steady-state value of Eq. (12). (c) $1-\mathcal{G}$ versus spin-cavity detuning ${\Delta }_{\mathrm{cs}}$ for $\kappa /w$ taking the values of 0.075 (red circles), 0.75 (green crosses), and 7.5 (blue diamonds). Open symbols correspond to ${\Delta }_{\mathrm{cs}}^{\prime }={\Delta }_{\mathrm{cs}}$ while the closed symbols (circles only) are obtained with ${\Delta }_{\mathrm{cs}}^{\prime }=-{\Delta }_{\mathrm{cs}}$. Solid lines show the prediction of Eq. (10), while the red dotted line corresponds to $1-\mathcal{G}=g_{\mathrm{ens}}^2/{\Delta }_{\mathrm{cs}}^2$. (d) With the same symbols as panel (c), the memory phase shift versus ${\Delta }_{\mathrm{cs}}$. Solid lines correspond to Eq. (11). (e) The RESN at the midpoint $t=2T$ versus ${\Delta }_{\mathrm{cs}}$ with symbols of panel (c). The open and closed circles are superimposed. Solid lines are theoretical according to Eq. (12). (f) The relation between the RESN at the midpoints and end points of the protocol. All data points were obtained for $\tilde{C}<0.06$ and shown with the symbols of panel (c).

Figure 5

(a) Illustration of the inversion schemes investigated, the varied parameters, and the symbols used for plotting. The first three cases employ an externally driven intracavity field with secant hyperbolic shape. In the last three cases, the entire spin ensemble is merely rotated by hand. (b) The average gain $\mathcal{G}$ versus the obtained excitation probability $p_{\mathrm{exc}}^{\mathrm{end}}$ at the end of the protocol. (c) The relative excess noise variance $\mathrm{REN}=2{\sigma }^2-1$ versus $p_{\mathrm{exc}}^{\mathrm{end}}$. In both panels (b) and (c), the solid lines are guides to the eye and the dashed horizontal lines correspond to ideal $\pi$ rotation for both inversion pulses.

Figure 6

(a) Gain $\mathcal{G}$ versus ${\gamma }_{\perp }$ for $\mu {\beta }_{\mathrm{sech}}=3w$ and various values of $w{T}_{\mathrm{mem}}=126$ (black asterisk), 146 (cyan pentagram), and 163 (magenta triangles). Solid lines represent fits to the function $\mathcal{G}={\mathcal{G}}_0{e}^{-{\gamma }_{\perp }(T_{\mathrm{mem}}-T_0)}$ with free parameters ${\mathcal{G}}_0$ and $T_0$. (b) The fitted $T_0$ is independent of $T_{\mathrm{mem}}$ for various values of $\mu {\beta }_{\mathrm{sech}}/w=3$ (red circles), 6 (green diamonds), 9 (blue squares), and $\infty$ (black triangles). (c) The relative excess noise $\mathrm{REN}=2{\sigma }^2-1$ versus ${\gamma }_{\perp }$ for various values of $\mu {\beta }_{\mathrm{sech}}$ using the same symbols as in panel (b). (d) The effect of ${\gamma }_{\perp }$ on the REN (symbols) in comparison to the guides to the eye (solid lines) from Fig. 5c. (e) Black circles: The fitted values of $T_0$ versus ${\beta }_{\mathrm{sech}}^{-1}$ following the linear fit: $T_0=4.0/{\beta }_{\mathrm{sech}}+0.92T_{\mathrm{swap}}$. Red squares: The slopes of the lines from panel (c) showing the susceptibility of the REN to ${\gamma }_{\perp }$. (f) Behavior of the collective spin variable $S_z$ versus time during the inversion process for the values of ${\beta }_{\mathrm{sech}}$ used in panels (b)–(e); the larger the ${\beta }_{\mathrm{sech}}$, the steeper the curve and the faster the inversion process.

Figure 7

(a) Illustration of various input states (gray circles) and output states (black circles) for a secant hyperbolic excitation with ${\chi }^{\max }=\mu {\beta }_{\mathrm{sech}}=9w$. The center of circles represents mean values of the cavity-field quadratures ${\widehat{X}}_{\mathrm{c}}$ and ${\widehat{P}}_{\mathrm{c}}$, and the radius of the circles represents the square root of the variances $\langle \delta {\widehat{X}}_{\mathrm{c}}^2\rangle$ and $\langle \delta {\widehat{P}}_{\mathrm{c}}^2\rangle$. (b) The same as panel (a) but with ${\chi }^{\max }$ reduced to $0.6\mu {\beta }_{\mathrm{sech}}$. Both gain and variances become asymmetric. (c) With the symbols defined in Fig. 5 the gain along the minor axis (open symbols) and major axis (closed symbols) is shown versus the final excitation probability $p_{\mathrm{exc}}^{\mathrm{end}}$. (d) Variances along the minor axis (open symbols) and major axis (closed symbols) versus $p_{\mathrm{exc}}^{\mathrm{end}}$. (e) The relationship between gain and variances; closed symbols: ${\sigma }_1^2$ versus ${\mathcal{G}}_1$; open symbols: ${\sigma }_2^2$ versus ${\mathcal{G}}_2$. (f) Qubit infidelity versus $p_{\mathrm{exc}}^{\mathrm{end}}$. The blue solid lines in panels (c), (d), and (f) correspond to the guide to the eye already shown in Fig. 5.

Figure 8

Comparison of numerical simulations with analytical results. Panel (a) shows $S_{\perp }=\sqrt{S_x^2+S_y^2}$ for three different values of $C$ (indicated on the figure). Within each family of curves, from above, the Lorentzian-distribution cutoff is varied between values ${\Delta }_{\mathrm{cut}}/\Gamma =20$ (red), 30 (green), 50 (blue), and 100 (cyan), while the lower (black) curve is the analytical result calculated from Eq. (D2). Panel (b) is arranged similar to panel (a), but the normalized spin noise is plotted instead. The analytical curves are from Eq. (D5). Panel (c) extends the $C=0.63$ curves from panel (a) for ${\Delta }_{\mathrm{cut}}/\Gamma =20$ (upper red) and 100 (lower cyan) in order to show spurious effects from the subensemble discretization. The artificial spin revival occurs at $t=\frac{2\pi }{d\Delta }\approx 71{\Gamma }^{-1}$.