Variance-Based Sensitivity Analysis of -type Quantum Memory Kai Shinbrough12and Virginia O. Lorenz12 1Department of Physics University of Illinois at Urbana-Champaign

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Variance-Based Sensitivity Analysis of Λ-type Quantum Memory

Kai Shinbrough1,2∗and Virginia O. Lorenz1,2

1Department of Physics, University of Illinois at Urbana-Champaign,

1110 West Green Street, Urbana, IL 61801, USA

2Illinois Quantum Information Science and Technology (IQUIST) Center,

University of Illinois at Urbana-Champaign, 1101 West Springﬁeld Avenue, Urbana, IL 61801, USA

(Dated: October 12, 2022)

The storage and retrieval of photonic quantum states, quantum memory, is a key resource for a

wide range of quantum applications. Here we investigate the sensitivity of Λ-type quantum memory

to experimental ﬂuctuations and drift. We use a variance-based approach, focusing on the eﬀects of

ﬂuctuations and drift on memory eﬃciency. We consider shot-to-shot ﬂuctuations of the memory

parameters, and separately we consider longer timescale drift of the control ﬁeld parameters. We

ﬁnd the parameters that a quantum memory is most sensitive to depend on the quantum memory

protocol being employed, where the observed sensitivity agrees with physical interpretation of the

protocols. We also present a general framework that is applicable to other ﬁgures of merit beyond

memory eﬃciency. These results have practical ramiﬁcations for quantum memory experiments.

I. INTRODUCTION

In the emerging ﬁeld of quantum technology, photons

play a critical role as carriers of quantum information

[1–3] and as the fundamental qubits for quantum com-

putation and information processing [4, 5]. Photons are,

however, diﬃcult to synchronize [6–8] and are subject to

losses in transmission [2, 9–11]. The ability to store and

retrieve photonic quantum states on demand—quantum

memory—provides a path forward to overcome these

challenges, and is therefore a critical enabling technology

for future quantum applications [12–14]. A considerable

body of work has been dedicated to quantum memories

based on atomic ensembles, where the three-level, reso-

nant, Λ-type atomic system is the most common [12–24].

In the ideal case, an optical quantum memory is capa-

ble of storing single-photon quantum states and retriev-

ing them on demand with high eﬃciency, high ﬁdelity,

long storage time, and broad bandwidth [12–14]. An-

other critical indicator of quantum memory performance,

however—which has largely been neglected until only re-

cently [25, 26]—is a memory’s sensitivity to experimen-

tal ﬂuctuations and drift. Fluctuations and drift in ex-

perimental parameters are invariably present in physical

quantum memory implementations, and a memory which

is more robust (less sensitive) to experimental noise is

more useful for real-world quantum applications. Here

we quantitatively address this aspect of Λ-type quantum

memory. (For an analysis of other types of memories, we

refer the reader to Refs. [25, 26].) We provide a variance-

based sensitivity analysis [27–34], which sheds light on

not only the sensitivity of an individual quantum mem-

ory implementation with device-speciﬁc ﬂuctuations and

drift, but also on the intrinsic sensitivity of diﬀerent phys-

ical Λ-type quantum memory protocols.

We consider quantum memory implementations with

memory parameters M= (d, τFWHMγ), where the opti-

∗kais@illinois.edu

cal depth dof the atomic ensemble and the intermediate

state coherence decay rate γ, scaled by the signal photon

duration τFWHM, are considered to be intrinsic and ﬁxed

properties of the memory. We then group the remaining

extrinsic, more readily tunable parameters as G, which

parameterize the optical control ﬁeld used in the memory

interaction, and which we assume have been optimized in

order to maximize memory eﬃciency. We further parti-

tion our analysis according to whether the parameters

of the control ﬁeld deﬁne a Gaussian temporal envelope

GG= (θ, ∆τctrl, τctrl

FWHM) or an arbitrary temporal shape

Gs= (ξ1, ..., ξN), as investigated in Ref. [35] (see Fig-

ure 1), where θ, ∆τctrl, and τctrl

FWHM correspond to the

Gaussian control ﬁeld pulse area, delay relative to the

signal ﬁeld, and duration, respectively, and the points ξi,

i= 1, ..., N correspond to interpolation points along the

temporal envelope of the control ﬁeld. Details on the nu-

merical calculation of memory eﬃciency given Mand G

can be found in Ref. [35].

In this work we assume a typical scenario for ex-

perimental atomic-ensemble quantum memory, wherein

the memory parameters are ﬁxed with minimal long-

timescale drift at a given setpoint but may undergo non-

negligible shot-to-shot ﬂuctuations. This situation oc-

curs frequently in transient processes for generating dense

atomic ensembles, such as in light-induced atomic desorp-

tion (LIAD) [36, 37] or laser ablation, but applies to equi-

librium systems as well. We assume the optical parame-

ters of the control ﬁeld possess smaller shot-to-shot ﬂuc-

tuations (e.g., laser ﬁelds with locked frequency, power,

timing, etc.), but may either drift over time or may not

be set precisely for optimal memory performance. We

investigate the sensitivity of the memory performance to

the setting of these control parameters, including analysis

of correlations that exist between parameters, which may

allow, for example, for compensating a drop in eﬃciency

due to non-optimal setting of one parameter by modiﬁ-

cation of the remaining parameters. This latter analysis

may be important in situations where one parameter is

constrained experimentally, for example in the case of

arXiv:2210.05530v1 [quant-ph] 11 Oct 2022

@zA(z,⌧)=pdP (z, ⌧) (1)

@⌧P(z,⌧)=¯P(z,⌧)+pdA(z, ⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z, ⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z, ⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z,⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)=d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

@zA(z,⌧)=pdP (z,⌧) (1)

@⌧P(z,⌧)=¯P(z,⌧)+pdA(z,⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z,⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z,⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z,⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)=d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

@zA(z,⌧)=pdP (z, ⌧) (1)

@⌧P(z,⌧)=¯P(z,⌧)+pdA(z, ⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z, ⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z,⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z, ⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl ]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)= d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

@zA(z,⌧)=pdP (z,⌧) (1)

@⌧P(z,⌧)=¯P(z,⌧)+pdA(z,⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z,⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z,⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z,⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)=d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

@zA(z,⌧)=pdP (z, ⌧) (1)

@⌧P(z,⌧)=¯P(z,⌧)+pdA(z, ⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z, ⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z, ⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z, ⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl ]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)= d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

@zA(z,⌧)=pdP (z, ⌧) (1)

@⌧P(z,⌧)=¯P(z, ⌧)+pdA(z,⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z, ⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z, ⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z, ⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl ]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)= d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

⇠i

<latexit sha1_base64="17wB1RZFr6VNaz9cwmNgdjJUb/E=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0mkoseiF48VTFtoQ9lsN+3SzSbsTsRS+hu8eFDEqz/Im//GbZuDtj4YeLw3w8y8MJXCoOt+O4W19Y3NreJ2aWd3b/+gfHjUNEmmGfdZIhPdDqnhUijuo0DJ26nmNA4lb4Wj25nfeuTaiEQ94DjlQUwHSkSCUbSS330SPdErV9yqOwdZJV5OKpCj0St/dfsJy2KukElqTMdzUwwmVKNgkk9L3czwlLIRHfCOpYrG3AST+bFTcmaVPokSbUshmau/JyY0NmYch7Yzpjg0y95M/M/rZBhdBxOh0gy5YotFUSYJJmT2OekLzRnKsSWUaWFvJWxINWVo8ynZELzll1dJ86Lq1aqX97VK/SaPowgncArn4MEV1OEOGuADAwHP8ApvjnJenHfnY9FacPKZY/gD5/MH2leOuQ==</latexit>

⇠j

<latexit sha1_base64="0IHItgHpbniyecGrkGzDY3aZZ8Q=">AAAB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lE0WPRi8cKpi20oWy2k3btZhN2N2Ip/Q1ePCji1R/kzX/jts1BWx8MPN6bYWZemAqujet+O4WV1bX1jeJmaWt7Z3evvH/Q0EmmGPosEYlqhVSj4BJ9w43AVqqQxqHAZji8mfrNR1SaJ/LejFIMYtqXPOKMGiv5nSfefeiWK27VnYEsEy8nFchR75a/Or2EZTFKwwTVuu25qQnGVBnOBE5KnUxjStmQ9rFtqaQx6mA8O3ZCTqzSI1GibElDZurviTGNtR7Foe2MqRnoRW8q/ue1MxNdBWMu08ygZPNFUSaIScj0c9LjCpkRI0soU9zeStiAKsqMzadkQ/AWX14mjbOqd169uDuv1K7zOIpwBMdwCh5cQg1uoQ4+MODwDK/w5kjnxXl3PuatBSefOYQ/cD5/ANvbjro=</latexit>

@zA(z,⌧)=pdP (z, ⌧) (1)

@⌧P(z,⌧)=¯P(z,⌧)+pdA(z, ⌧)i⌦(⌧)

2B(z,⌧) (2)

@⌧B(z,⌧)=BB(z, ⌧)i⌦⇤(⌧)

2P(z,⌧),(3)

where dis the resonant optical depth of the memory,

¯=(i)/is the normalized complex detuning, and

P(z,⌧) and B(z, ⌧) are macroscopic ﬁeld operators rep-

resenting the atomic coherences |1i$|2iand |1i$|3i,

respectively, which are delocalized across the length of

the medium. In Eqs. (1)-(3), all frequency (time) scales

are normalized by (1/), and all length scales are nor-

malized by L. We assume that the coherence decay rate

corresponding to the |3i!|1itransition, B, is negligi-

ble during the storage and retrieval operations: B⌧1.

We iteratively solve these equations of motion us-

ing Heun’s method for evaluating the ⌧-derivatives and

Chebyshev spectral di↵erentiation for the z-derivatives.

After integration, we compare the population in Bto

the population in Ain in order to calculate the storage

eﬃciency for a particular choice of ⌦(⌧), as:

⌘=R1

0dz |B(z,⌧!1)|2

1 d⌧|Ain(⌧)|2,(4)

where in practice we truncate Ain(⌧) and B(z, ⌧) at

⌧end =4⌧FWHM,whereAin(⌧end) has dropped to

O(1010) of its maximum value. Thus Eqs. (1)-(3) in

combination with Eq. (4) deﬁne an objective function

that can be maximized with respect to the free param-

eters of ⌦(⌧). We parameterize the control ﬁeld Rabi

frequency—which we take to be real for simplicity—in

terms of its pulse area ✓=R1

1 d⌧⌦(⌧), temporal de-

lay ⌧ctrl relative to the arrival of the signal ﬁeld, and

duration ⌧ctrl

FWHM =2

p2ln2ctrl as:

⌦(⌧)=⌦0e[(⌧⌧ctrl)/2ctrl ]2,(5)

where ⌦0=✓/(2p⇡ctrl), and we optimize over the pa-

rameter space vector G⌘✓,⌧ctrl,⌧ctrl

FWHMusing a

Nelder-Mead simplex method, which rapidly identiﬁes

the eﬃciency maxima under these constraints, as veriﬁed

by deterministic searches of the same parameter space.

We deﬁne ⌧= 0 at the maximum of the signal ﬁeld.

Throughout this work we normalize the eﬃcien-

cies calculated via the method above by the protocol-

independent eﬃciency bound for a ﬁxed optical depth,

⌘opt, described in Refs. [10, 11, 14, 16] and elsewhere. In

brief, we calculate this eﬃciency bound by ﬁnding the

eigenvalues of the anti-normally ordered storage kernel

K(z,z0)= d

2ed(z+z0)/2I0(dpzz0),(6)

where I0(x) is the zeroth-order modiﬁed Bessel function

of the ﬁrst kind, and we discretize K(z,z0) on a 5000 ⇥

5000 point grid. For ﬁxed d, the largest eigenvalue 0

of this kernel represents the maximum achievable storage

eﬃciency at that optical depth, ⌘opt =0. By performing

this normalization, we aim to compare the eﬃciencies of

particular memory implementations independent of the

limitation imposed by ﬁnite optical depth.

III. RESULTS OF GAUSSIAN OPTIMIZATION

A. On Resonance (=0)

We ﬁrst consider the case of resonant interaction of

the optical ﬁelds with the atomic ⇤system (i.e., = 0).

At each optical depth and signal bandwidth, we optimize

over the control ﬁeld parameters G=✓,⌧ctrl,⌧ctrl

FWHM,

which fully deﬁne any Gaussian control ﬁeld through

Eq. (5). This allows us to show that the three known,

physically distinct quantum storage protocols for reso-

nant storage (see Appendix A for a brief overview of

the protocols) are smoothly connected via continuous

transformation of the control-ﬁeld parameters. This re-

sult is similar to that in Ref. [34], which demonstrated

ATS and EIT quantum memory behavior can be con-

nected through continuous transformation of the con-

trol ﬁeld Rabi frequency for ﬁxed memory parameters,

under the condition of either a constant control ﬁeld

or an interrupted control ﬁeld of varying linear slope.

Here we distinguish between the memory parameters

M⌘(d, ⌧FWHM), which represent the physical char-

acteristics of a particular quantum memory for the cho-

sen signal bandwidth, and the control ﬁeld parameters

G. In this formalism, Ref. [34] derived a connection be-

tween ATS and EIT storage for ﬁxed Mby varying G

[where, e.g., Gc=(⌦0) is a single-parameter vector in

the case of a constant control ﬁeld, ⌦(⌧)=⌦0]. Moti-

vated by this observation, we consider the distinct condi-

tion of Gaussian-shape control ﬁelds, and we show that

again ATS and EIT memory behavior can be connected

if we consider the transformation as a function of M,

where optimization of Gat each point in Mensures opti-

mal or near-optimal storage eﬃciency. Further, we show

the two protocols can be connected to the ‘absorb-then-

transfer’ protocol through the same continuous transfor-

mation. We show each protocol possesses a region of

optimality under the restriction of Gaussian pulses and

identify two regions where our optimization scheme is

most useful: one where the storage mechanism is given

by the ‘absorb-then-transfer’ protocol, but in the largely

unexplored non-adiabatic regime, and one between the

regions of eﬃcient ATS and EIT memory operation.

Figure 2 presents the main results of this section. In

Fig. 2(a) we show the normalized eﬃciencies achieved

through the optimization procedure described in Sec. II,

for memory parameters in the range d= 1 to 50 and

⌧FWHM=0to1.5, which we take to be representative

a) b)

GG=(✓,⌧ctrl,⌧ctrl

FWHM)

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Gs=(⇠1, ..., ⇠N)

<latexit sha1_base64="x2XXfxdAHlnYxGb5jxX1WMC8VZ8=">AAACCnicbZDLSsNAFIYn9VbrLerSzWgRKpSQSEU3QtGFrqSCvUATwmQ6aYdOLsxMxBK6duOruHGhiFufwJ1v46TNQlt/GPj4zznMOb8XMyqkaX5rhYXFpeWV4mppbX1jc0vf3mmJKOGYNHHEIt7xkCCMhqQpqWSkE3OCAo+Rtje8zOrte8IFjcI7OYqJE6B+SH2KkVSWq+/bAZIDjFh6NXYFPIcV+4G6VtUwjGpGN0euXjYNcyI4D1YOZZCr4epfdi/CSUBCiRkSomuZsXRSxCXFjIxLdiJIjPAQ9UlXYYgCIpx0csoYHiqnB/2IqxdKOHF/T6QoEGIUeKozW1zM1jLzv1o3kf6Zk9IwTiQJ8fQjP2FQRjDLBfYoJ1iykQKEOVW7QjxAHGGp0iupEKzZk+ehdWxYNePktlauX+RxFMEeOAAVYIFTUAfXoAGaAINH8AxewZv2pL1o79rHtLWg5TO74I+0zx83NZir</latexit>

FIG. 1. Control ﬁelds of (a) Gaussian shape, deﬁned by

the three parameters GG= (θ, ∆τctrl, τctrl

FWHM), where Ω0=

θ/(2√πσctrl ) and σctrl = 2√2 ln 2/τctrl

FWHM, and (b) arbitrary

shape, deﬁned by the Ninterpolation points Gs= (ξ1, ..., ξN).

limited laser power, which can often limit memory eﬃ-

ciency [38–45]. This type of memory sensitivity can be

interpreted as an indicator of the region of control ﬁeld

phase space where acceptable memory performance can

be achieved; low sensitivity implies a large acceptable re-

gion of control ﬁeld phase space, where the control ﬁeld

does not require careful ﬁne-tuning, and where restric-

tions on one parameter may be compensated for with

changes to the remaining parameters. Equivalently, this

type of memory sensitivity can be interpreted in terms

of the memory’s robustness to experimental drift, where

low sensitivity implies that, given optimal initial con-

trol ﬁeld settings, the memory will be robust to long-

timescale drift in the phase space surrounding the opti-

mal setpoint.

In the following sections, we restrict our discussion to

resonant Λ-type memory protocols, but the tools devel-

oped in this work are readily applicable to oﬀ-resonant

protocols, as well as other level systems and a wide range

of related techniques [46–48]. In Section II, we provide

deﬁnitions for several quantitative aspects of memory

sensitivity. In Section III we use these criteria to ana-

lyze the sensitivity of resonant Λ-type quantum memory

to ﬂuctuations in memory parameters, and in Section IV

we address sensitivity to improper setting of control ﬁeld

parameters or experimental drift.

II. VARIANCE-BASED SENSITIVITY

ANALYSIS

The sensitivity of classical systems is a much-discussed

subject with well-established theoretical and numerical

tools [27–34, 49]. In general, the task is to deter-

mine the sensitivity of a system with performance cri-

terion h(X,A) to changes in Ninput parameters X=

(x1, ..., xN) when internal system parameters Aare kept

ﬁxed. This performance criterion may correspond to any

desired single-valued metric of the system; in the case of

quantum memory, this may correspond to memory eﬃ-

ciency, ﬁdelity, storage time, etc. For the sake of brevity,

in Sec. III and Sec. IV we focus on memory eﬃciency as a

key performance criterion, but importantly other criteria

may be used and may be the subject of future work. In

this section, we provide an outline of the theoretical tools

used for a generic criterion h.

The most common method for determining the sensi-

tivity of h(X,A) to ﬂuctuations in the input parameters

proceeds as follows [27, 34]. We deﬁne center values for

the input parameters X, then draw many N-dimensional

ﬂuctuations ζstochastically from a known probability

distribution P(ζ), and average over these ﬂuctuations in

order to calculate the mean performance criterion

h(X) = Zdζ h(X+ζ, A)P(ζ) (1)

and the variance in the system performance

Vﬂuc

hX=Vζ[h(X+ζ, A)|A],(2)

where Vx[y(x, z)|z] = Rdx y2(x, z)P(x)−

[Rdx y(x, z)P(x)]2is the unconditional variance of

yobtained when xis allowed to vary and zis held

constant. In the absence of a tailored noise model, the

probability distribution for ﬂuctuations is commonly

approximated as an N-dimensional normal distribution

P(ζ)∼e−|ζ|2/(22)with standard deviation . The

resulting standard deviation in performance criterion h

can then be calculated, σﬂuc

h(X) = qVﬂuc

h(X).

The simple variance-based method above provides use-

ful information on the response of the system to short-

timescale, shot-to-shot ﬂuctuations in input parameters

around given central values X, which typically corre-

spond to the setpoints of the input parameters. Xcan

also correspond to control parameters, where the set-

point Xis assumed to be at or near the optimum val-

ues for system performance. The method above does

not provide detailed information on the local environ-

ment around the performance optimum, which may be

important for long-timescale drift or for determining

which parameter is most sensitive to experimental error.

The simplest method for determining a system’s sensitiv-

ity to these long-timescale changes in input parameters

X= (x1, ..., xN) is to vary each parameter one-at-a-time

(OAT), and to measure the resulting variance in the sys-

tem’s performance. This OAT analysis corresponds to

calculating the variances

VOAT

i=Vxi[h(X)|xj6=i] (3)

for each parameter xi, where xivaries over a ﬁnite range,

xi∈[xmin

i, xmax

i]. In Eq. (3) and in the following discus-

sion, we have suppressed the internal parameters that are

always held constant from the notation. Again, the stan-

dard deviation σOAT

i=pVOAT

imay be used to quantify

the change in system performance due to parameter xi.

The parameter xiwith the largest σOAT

ihas the largest

eﬀect on the performance criterion hand therefore the

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Variance-BasedSensitivityAnalysisof-typeQuantumMemoryKaiShinbrough1;2andVirginiaO.Lorenz1;21DepartmentofPhysics,UniversityofIllinoisatUrbana-Champaign,1110WestGreenStreet,Urbana,IL61801,USA2IllinoisQuantumInformationScienceandTechnology(IQUIST)Center,UniversityofIllinoisatUrbana-Champaign,1101West...

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Variance-Based Sensitivity Analysis of -type Quantum Memory Kai Shinbrough12and Virginia O. Lorenz12 1Department of Physics University of Illinois at Urbana-Champaign

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